Noncommutative Geometry and Physics 2005: Proceedings of the International Sendai-Beijing Joint Workshop

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noncommutative geometry and physics 2005 Proceedings of the International Sendai–Beijing Joint Workshop

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Ursula Carow-Watamura Satoshi Watamura Tohoku University, Japan Yoshiaki Maeda Hitoshi Moriyoshi Keio University, Japan Zhangju Liu Peking University, China Ke Wu Capital Normal University, China

noncommutative geometry and physics 2005 Proceedings of the International Sendai–Beijing Joint Workshop Sendai, Japan 1 – 4 November 2005 Beijing, China 7 – 10 November 2005

World Scientific NEW JERSEY

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NONCOMMUTATIVE GEOMETRY AND PHYSICS 2005 Proceedings of the International Sendai-Beijing Joint Workshop Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-270-469-6 ISBN-10 981-270-469-8

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ZhangJi - Noncommutative.pmd

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6/11/2007, 6:27 PM

Preface The workshop “Noncommutative Geometry and Physics 2005” was organized by mathematicians and physicists from Keio Universty, Tohoku University and Beijing University in cooperation. The first part of this workshop was held at Tohoku University in Sendai on November 1-4, and the second part was held at Beijing University in Beijing on November 7-11. This workshop was the fifth in a series of joint workshops for mathematicians and physicists working in noncommutative geometry, deformation quantization and related topics, with the aim to stimulate discussions and the exchange of new ideas between both disciplines. Since the subject of noncommutative geometry has undergone rapid developments in the past few years, one of the important functions of our meetings is to elucidate these recent advances and the current status of research projects from mathematical point of view as well as from the physics’ side. In physical applications of noncommutative geometry, many key results have emerged on solutions of field theory on noncommutative spaces. Therefore, this was naturally one of the main subjects of the workshop. This volume includes disscussions of solitons such as monopoles and instantons in noncommutative spaces as well as in nonanticommutative superspaces. All contributions in this volume were submitted by conference speakers and participants, and were duly refereed. The articles contain presentations of new results which have not appeared yet in professional journals, or comprehensive reviews including an original part of the present developments in those topics. Effort was to provide comprehensive introductions to each subject such that the volume becomes accessible to researchers and graduate students interested in mathematical areas related to noncommutative geometry and its impact on modern theoretical physics. The workshop was held in the framework and with the support of the 21st century Center of Excellence (COE) program at Keio University, “Integrative Mathematical Sciences: Progress in Mathematics v

vi

Preface

motivated by Natural and Social Phenomena”, and it was also supported by a Grant-in-Aid for Scientific Research (No.13135202) of the Japanese Ministry of Education, Culture, Sports, Science and Technology at Tohoku University. We are also grateful to the Tohoku University, Japan, and to the Beijing University and the Academia Sinica, China, for providing the lecture rooms and their facilities which made a smooth performance of the meeting possible. The World Scientific Publishing company has been very helpful in the production of this volume, and we would like to thank Ms. Zhang Ji for her editorial guidance. In this place we wish to express our special thanks to all authors for their continuous effort in preparing these articles and the referees for their valuable comments and suggestions. Mathematical section: Zhangju Liu (Beijing Univ.) Yoshiaki Maeda (Keio Univ.) Hitoshi Moriyoshi (Keio Univ.) Physical section: Ursula Carow-Watamura (Tohoku Univ.) Satoshi Watamura (Tohoku Univ.) Ke Wu (CNU, Beijing)

Contents

I

Preface

v

DEFORMATIONS AND NONCOMMUTATIVITY

1

1 Expressions of Algebra Elements and Transcendental Noncommutative Calculus Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka 3 2 Quasi-Hamiltonian Quotients as Disjoint Unions of Symplectic Manifolds Florent Schaffhauser 31 3 Representations of Gauge Transformation Groups of Higher Abelian Gerbes Kiyonori Gomi 55 4 Algebroids Associated with Pre-Poisson Structures Kentaro Mikami, Tadayoshi Mizutani

71

5 Examples of Groupoid Naoya Miyazaki

97

6 The Cohomology of Transitive Lie Algebroids Z. Chen, Z.-J. Liu

109

7 Differential Equations and Schwarzian Derivatives Hajime Sato, Tetsuya Ozawa, Hiroshi Suzuki

129

8 Deformation of Batalin-Vilkovsky Structure Noriaki Ikeda

151

vii

viii

II

Contents

DEFORMED FIELD THEORY AND SOLUTIONS

173

9 Noncommutative Solitons Olaf Lechtenfeld

175

10 Non-anti-commutative Deformation of Complex Geometry Sergei V. Ketov

201

11 Seiberg-Witten Monopole and Young Diagrams Akifumi Sako

219

12 Instanton Counting, Two Dimensional Yang-Mills Theory and Topological Strings Kazutoshi Ohta 239 13 Instantons in Non(anti)commutative Gauge Theory via Deformed ADHM Construction Takeo Araki, Tatsuhiko Takashima, Satoshi Watamur 253 14 Noncommuative Deformation and Drinfel’d Twisted Symmetry Yoshishige Kobayashi 261 c (2) k and Twisted Conformal Field 15 Affine Lie Superalgebra gl(2|2) Theory Xiang-Mao Ding, Gui-Dong Wang, Shi-Kun Wang 273 16 A Solution of Yang-Mills Equation on BdS Spacetime Xin’an Ren, Shikun Wang

289

17 Solitonic Information Transmission in General Relativity Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

297

18 Difference Discrete Geometry on Lattice Ke Wu, Wei-Zhong Zhao, Han-Ying Guo

301


Part I

DEFORMATIONS AND NONCOMMUTATIVITY

1

Expressions of algebra elements and transcendental noncommutative calculus Hideki Omori1 , Yoshiaki Maeda2 , Naoya Miyazaki3 , and Akira Yoshioka4 1

2

3

4

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama, 223-8825, Japan Department of Mathematics, Faculty of Economics, Keio University, Hiyoshi, Yokohama, 223-8521, Japan Department of Mathematics, Faculty of Science, Tokyo University of Science, Kagurazaka, Tokyo, 102-8601, Japan [email protected], [email protected], [email protected], [email protected] 2 Partially supported by Grant-in-Aid for Scientific Research (#18204006.), Ministry of Education, Science and Culture, Japan. 3 Partially supported by Grant-in-Aid for Scientific Research (#18540093.), Ministry of Education, Science and Culture, Japan. 4 Partially supported by Grant-in-Aid for Scientific Research (#17540096.), Ministry of Education, Science and Culture, Japan.

Abstract Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that 1 i~ uv in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set N+ 21 or −(N+ 21 ) . This may yield a more mathematical understanding of Dirac’s positron theory. A.M.S Classification (2000): Primary 53D55, 53D10; Secondary 46L65

1 Introduction Quantum theory is treated algebraically by Weyl algebras, derived from differential calculus via the correspondence principle. However, since the algebra is noncommutative, the so-called ordering problem appears. Orderings are treated in the physics literature of quantum mechanics (cf. [1]) as the rules of association from classical observables to quantum observables, which are supposed to be self-adjoint operators on a Hilbert space.

3

4

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Typical orderings are, the normal (standard) ordering, the anti-normal (antistandard) ordering, the Weyl ordering, and the Wick ordering in the case of complex variables. However, from the mathematical viewpoint, it is better to go back to the original understanding of Weyl, which says that orderings are procedures of realization of the Weyl algebra W~ . Since the Weyl algebra is the quotient algebra of the universal enveloping algebra of the Heisenberg Lie algebra, the Poincaré-Birkhoff-Witt theorem shows that this algebra can be viewed as an algebra defined on a space of polynomials. As we show in §1, this indeed gives product formulas on the space of polynomials which produce algebras isomorphic to W~ . This gives the unique way of expressions of elements, and as a result one can treat transcendental elements such as exponential functions, which are necessary to solve differential equations (cf. §2.2). However, we encounter several anomalous phenomena, such as elements with two different inverses (cf. §4) and elements which must be treated as double valued (cf. [16],[17]). 1 uv should be In this note, we treat the phenomenon which shows that i~ 1 1 viewed as an indeterminate living in the set N+ 2 or −(N+ 2 ). We reach this interpretation in two different ways, by analytic continuation of inverses of 1 z+ i~ uv, and by defining star gamma functions using various ordered expressions. We emphasise in this paper, that our approach to show the discrete picture 1 uv is not to use operator representation at all, but for the element z + i~ to express it in various orderings instead, under a leading principle that a physical object should be free from the choice of orderings(the ordering free principle), just as classical, geometric objects are expressed independent of the choice of local coordinates. Since similar discrete pictures of elements is familiar in quantum observables, treated as a self-adjoint operator, our observation gives for their justification for the operator theoretic formalism of quantum theory. However, in this note we restrict our ordering to a particular subset to avoid the multi-valued expressions. In some cases, we should be more careful about the convergence of integrals and the continuity of the product, so the detailed computations and the proof of continuity of the products will appear elsewhere.

2 K-ordered expressions for algebra elements We introcuce a method to realize the Weyl algebra via a family of expressions. This leads to a transcendental calculus in the Weyl algebra. 2.1 Fundamental product formulas and intertwiners Let SC (n) and AC (n) be the spaces of complex symmetric matrices and skewsymmetric matrices respectively, and MC (n)=SC (n)⊕AC (n). For an arbitrary

2 K-ordered expressions for algebra elements

5

fixed n×n-complex matrix Λ∈MC (n), we define a product ∗Λ on the space of u] by the formula polynomials C[u i~

f ∗Λ g = f e 2 (

P ←− ij − − → ∂ ui Λ ∂ uj )

g=

X (i~)k k!2k

k

Λi1 j1· · ·Λik jk ∂ui1· · ·∂uik f ∂uj1· · ·∂ujk g.

(1) u], ∗Λ ) is an associative algebra. It is known and not hard to prove that (C[u u], ∗Λ ) is determined by the skew-symmetric (a) The algebraic structure of (C[u part of Λ (in fact, by its conjugacy class A → t GAG). u], ∗Λ ) is isomorphic to the (b) In particular, if Λ is a symmetric matrix, (C[u usual polynomial algebra. Set Λ=K+J, K∈SC (n), J∈AC (n). Changing K for a fixed J will be called a deformation of expression of elements, as the algebra remains in the same isomorphism class. Example of computations: ui ∗Λ uj =ui uj +

i~ ij Λ , 2

ui ∗Λ uj ∗Λ uk =ui uj uk +

i~ ij (Λ uk +Λik uj +Λjk ui ). 2

By computing the ∗Λ -product using the product formula (1), every element of the algebra has a unique expression as a standard polynomial. We view these expressions of algebra elements as analogous to the “local coordinate expression” of a function on a manifold. Thus, changing K corresponds to a local coordinate transformation on a manifold. In this context, we call the product formula (1) the K-ordered expression by ignoring the fixed skew part J. Following a familiar notion in quantum we call the K-ordered mechanics, 0 −Im 0 Im , the Weyl ordering, , expression for the particular K=0, −Im 0 Im 0 normal ordering, anti-normal ordering, respectively. The intertwiner between a K-ordered expression and a K 0 -ordered expression, which we view as a local coordinate transformation, is given in a concrete form : Proposition 2.1 For symmetric matrices K, K 0 ∈ SC (n), the intertwiner is given by K0

IK (f ) = exp

i~ X 4

i,j

(K

0

ij

K K0 −K ij )∂ui ∂uj f (= I0 (I0 )−1 (f )),

(2)

K0

u]; ∗K+J ) → (C[u u]; ∗K 0 +J ) between algebras. giving an isomorphism IK : (C[u u] : Namely, for any f, g ∈ C[u K0

K0

K0

IK (f ∗K+J g) = IK (f ) ∗K 0 +J IK (g).

(3)

6

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka K

u], the set {I0 (f ); K∈SC (n)} forms a family of eleThus, for every f ∈C[u ments which are mutually intertwined. We denote this family by f∗ viewing K as an algebraic object, and we use often the notation :f∗ :K+J =I0 (f ). 0 −Im u], ∗Λ ) is called the Weyl algebra, , (C[u In the case n=2m and J= Im 0 with isomorphism class denoted by W2m . In fact, if J is non-singular, then u], ∗Λ ) is isomorphic to the Weyl algebra. (C[u 1 t(z+s i~ uk )

2.2 The star exponential function e∗

Using the various ordered expression of elements of algebra, we can treat elementary transcendental functions. For an object H∗ , the ∗-exponential func∗ ∗ tion etH is defined as the family :etH ∗ ∗ :Λ of real analytic solutions in t of the evolution equations d ft =:H∗ :Λ ∗Λ ft , f0 =1. (4) dt For instance, for every z∈C, we have 1 z+s i~ uk

:e∗

s

1

:Λ =ez :e∗ i~

uk

:Λ =ez es

2

kk 1 4i~ K

1

es i~ uk .

(5)

When we fix the skew part J of Λ, we often abbreviate the notation to : :K , ∗K for : :K+J , ∗K+J respectively. Since the exponential law 1 (z+w)+(s+t) i~ uk

:e∗

1 z+s i~ uk

:K =:e∗

1 w+t i~ uk

:K ∗K :e∗

:K

holds for every K, it is better to write 1 (z+w)+(s+t) i~ uk

e∗ 1 z+s i~ uk

by viewing :e∗

1 z+s i~ uk

=e∗

1 w+t i~ uk

∗e∗

:K as the K-ordered expression of the (ordering free) ex-

z+s

1

u

ponential element e∗ i~ k . Under this convention, one may write for instance ij :ui ∗uj :K =ui uj + i~ 2 (K+J) . s

1

u

We remark that even for the simplest exponential function e∗ i~ k , formula (5) gives the following (cf. [12]). 1 P 2n i~ uk Proposition 2.2 If Im K kk − , 2

(19)

1 . 2

(20)

1

1 1 e 2 tz i~ 2uv tanh 2 t dt, 1 e cosh 2 t

Re z
− 12 , which is also the holomorphic 1 domain for (z+ i~ uv)−1 −∗ . All of these results are easily proved for the Weyl ordering. However, if t 1 uv κ∈C−{κ≥1}∪{κ≤−1}, then :e∗i~ :κ is rapidly decreasing in t, and the same computation gives the following:

4 Inverses and their analytic continuation

15

1 uv)−1 Proposition 4.3 For every z with Re z>− 21 , the two inverses (z+ i~ +∗ 1 and (z− i~ uv)−1 −∗ are defined in the κ-ordered expression for κ∈C−{κ≥1} ∪ {κ≤−1}. −1 1 1 Note that (z+ i~ uv)−1 +∗ ∗(−z− i~ uv)−∗ diverges for any ordered expression. However, the standard resolvent formula gives the following:

Proposition 4.4 If z+w6=0, then 1 1 1 −1 (z+ uv)−1 uv) +(w− +∗ −∗ z+w i~ i~ 1 1 uv)∗(w− i~ uv). In particular, for every positive integer is an inverse of (z+ i~ n, and for every complex number z such that Re z> − (n+ 12 ), the κ-ordered −1 1 1 1 1 (1+ n1 (z+ i~ uv))−1 expression of 2n +∗ +(1− n (z+ i~ uv))−∗ gives an inverse of 1 1− n12 (z+ i~ uv)2∗ for κ∈C−{κ≥1}∪{κ≤−1}.

4.1 Analytic continuation of inverses 1 1 Recall that (z± i~ uv)−1 ±∗ are holomorphic on the domain Re z > − 2 . It is −1 −1 1 1 natural to expect that (z± i~ uv)±∗ =C(C(z± i~ uv))±∗ for any non-zero constant C. To confirm this, we set C=eiθ and consider the θ-derivative Z 0 1 uv) eiθ t(z± i~ dt. e∗ eiθ −∞

In the (κ, τ )-ordered expression, the phase part of the integrand is bounded in t and the amplitude is given by (1−κ)e

2eiθ tz , + (1+κ)e−eiθ t/2

eiθ t/2

κ6=1.

Hence, the integral converges whenever Re eiθ (z± 12 ) > 0, and by integration 1 by parts this convergence does not depend on θ. It follows that (z± i~ uv)−1 ±∗ 1 are holomorphic on the domain C−{t; −∞ − 21 t(z− i~ uv) dt= (z− uv)∗e∗ . 1−$00 z=− 21 i~ −∞

As suggested by these formulas, we extend the definition of the ∗-product as s 1 uv follows: For every polynomial p(u, v) or p(u, v)=e∗ i~ , Z 0 1 t(z± 1 uv) e∗ i~ dt. p(u, v)∗(z± uv)−1 = lim p(u, v) ∗ (25) +∗ N →∞ i~ −N Hence we have the formula 1 1 1 Re z > − 12 (z+ uv)∗(z+ uv)−1 = . +∗ 1−$00 z=− 21 i~ i~

(26)

Considering 1 1 1 1 ◦ n n n uv)∗(z+ uv)−1 uv)∗v n ∗(v ◦ )n ∗(z+ uv)−1 +∗ ∗v =(v ) ∗(z+ +∗ ∗v i~ i~ i~ i~ and using the formula (23), we have the following:

(v ◦ )n ∗(z+

Theorem 4.2 If we use definition (25) for the ∗-product, then 1 1 1 uv)∗(z+ uv)−1 = 1 1 +∗ 1− n! ( i~ u)n ∗$00 ∗v n i~ i~ 1 1 1 (z− uv)∗(z− uv)−1 = 1 1 −∗ 1− n! ( i~ v)n ∗$00 ∗un i~ i~ (z+

z6∈−(N+ 12 ) , z=−(n+ 21 )

(27)

z6∈−(N+ 12 ) . z=−(n+ 12 )

(28)

18


Although z=−(n+ 12 ) are all removable singularities for (27) and (28) as a function of z, it is better to retain these singular points. These formulas give in particular for every fixed positive integer m 1 1 1 1 1 z6∈−(N+m+ 12 ) −1 (1+ (z+ uv))∗(1+ (z+ uv))+∗ = 1 1 k k 1− k! ( i~ u) ∗$00 ∗v z=−(k+m+ 12 ) m i~ m i~ (29) for arbitrary k ∈ N. We state the following identity for later use: 1 1 1 $00 ∗v n ∗(−n− + uv)=$00 ∗( u∗v)∗v n =0. 2 i~ i~

(30)

5 An infinite product formula Recall the classical formula sin πx=πx ∞ Y

k=1

(1−

1 x2 )= k 2 2i

Z

x2 k=1 (1− k2 ).

Q∞

χ[−π,π] (t)eitx dt= lim

n→∞

Z Y n

Rewrite this as follows:

(1+

k=1

1 2 ∂ )δ(t)eitx dt, k2 t

where χ[−π,π] (t) is the characteristic function of the interval [−π, π]. It follows that χ[−π,π] (t)=2i lim

n→∞

n Y

k=1

(1+

1 2 ∂ )δ(t) k2 t

in the space of distributions. it 1 uv κ+1 |6=1 , so that :e∗ i~ :κ is not singular on t ∈ R, we For κ uch that | κ−1 compute as follows: Z Z 1 uv) ±it 1 uv) it(z± i~ :κ dt= χ[−π,π] (t)eitz :e∗ i~ :κ dt. χ[−π,π] (t):e∗ Fixing a cut-off function ψ(t) of compact support such that ψ=1 on [−π, π], we see that Z Z Y n 1 t(z± 1 uv) ±it 1 uv χ[−π,π] (t):e∗ i~ :κ dt=2i lim (1+ 2 ∂t2 )δ(t)ψ(t)etz :e∗ i~ :κ dt. n→∞ k k=1

Integration by parts gives Z n n Y Y 1 1 ±it 1 uv t(z± 1 uv) lim δ(t) (1+ 2 ∂t2 )ψ(t)etz :e∗ i~ :κ dt= lim :(1+ 2 ∂t2 )e∗ i~ :κ . n→∞ n→∞ k k k=1

k=1

Hence we have in the κ-ordered expression that Z n Y 1 1 1 uv) it(z± i~ dt=2i lim (1− 2 (z± uv)2 )∗ . χ[−π,π] (t)e∗ n→∞ k i~ k=1

5 An infinite product formula

19

Noting that sin∗ π(z±

1 1 uv)=π(z± uv)∗ i~ i~

Z

1 uv) it(z± i~

χ[−π,π] (t)e∗

dt ∈ Hol(C2 ),

we have n

sin∗ π(z±

Y 1 1 1 1 uv)=π(z± uv)∗ lim ∗ (1− 2 (z± uv)2 )∗ n→∞ i~ i~ k i~

(31)

k=1

in Hol(C2 ). In particular, we have κ+1 Proposition 5.1 In the κ-ordered expression with | κ−1 |6=1, we have n

sin∗ π(z+

Y 1 1 1 1 uv)=π(z+ uv)∗ lim ∗(1− 2 (z+ uv)2 ). n→∞ i~ i~ k i~ k=1

This is identically zero on the set z∈Z+ 12 . The formula in Proposition 5.1 may be rewritten as n

sin∗ π(z+

Y 1 1 1 1 1 1 (z+ i~ uv) uv)=π(z+ uv)∗ lim ∗(1− (z+ uv))∗e∗k n→∞ i~ i~ k i~ k=1 n Y

∗

k=1

1 1 1 − 1 (z+ i~ uv) ∗(1+ (z+ uv))∗e∗ k . k i~

In §6, we will define a star gamma function via the two different inverses mentioned previously and give an infinite product formula for the star gamma function. −1 1 1 1 uv)+∗ and with 1+ m (z+ i~ uv) 5.1 The product with (z+ i~

−1 +∗

1 1 uv)−1 First we consider the product (z+ i~ ±∗ ∗ sin∗ π(z+ i~ uv) in two different ways. One way is by defining:

(z+

1 1 uv)−1 uv) ±∗ ∗ sin∗ π(z+ i~ i~ n (32) Y 1 1 1 1 2 = lim (z+ uv)−1 ∗ (z+ uv)∗ ∗(1− (z+ uv) ) . ±∗ n→∞ i~ i~ k2 i~ k=1

Qn

1 uv)∗ Since (z+ i~ (27) and (30) give

k=1

1 ∗(1− k12 (z+ i~ uv)2 ) is a polynomial, Proposition 3.3,

∞

Y 1 1 1 1 uv)= ∗(1− 2 (z± uv)2 ). (z+ uv)−1 ±∗ ∗ sin∗ π(z± i~ i~ k i~ k=1

(33)

20


The second way is by defining 1 1 (z+ uv)−1 uv)= lim ±∗ ∗ sin∗ π(z+ N →∞ i~ i~

Z

0

1 t(z+ i~ uv)

e∗

∗ sin∗ π(z+

−N

1 uv). (34) i~

This may be written as the complex integral 1 2i

Z

0+πi

t(z+ 1 uv) e∗ i~ dt−

−∞+πi

1 2i

Z

0−πi

1 t(z+ i~ uv)

e∗

dt.

−∞−πi

Rπ 1 Adding − 21 −π eit(z+ i~ uv) dt to this expression gives the clockwise contour integral along the boundary of the domain D={z∈C; Re z 1 and the singular point z= log κ−1 +2πni has a positive real part.

Proposition 5.2 Suppose Re κ>0 and κ∈C−{κ≥1}∪{κ≤−1}. Then in the κ-ordered expression, we have Z 0 Z 1 1 1 π it(z+ i~ 1 t(z+ i~ uv) uv) lim e∗ ∗ sin∗ π(z+ uv)= dt. e∗ N →∞ −N i~ 2 −π According to (31) the right hand side gives the same result as (33), that is, Q∞ 1 1 2 ∗(1− 1 k2 (z+ i~ uv) ).

Proposition 5.3 Suppose Re κ>0 and κ∈C−{κ≥1}∪{κ≤−1}. Then in the 1 1 κ-ordered expression, the product sin∗ π(z+ i~ uv)∗(z+ i~ uv)−1 +∗ is an entire −1 1 function of z. Namely, all singularities of (z+ i~ uv)+∗ at −(N+ 12 ) are cancelled out in formulas (29) and (30). By a proof similar to that of Proposition 5.3, we obtain Proposition 5.4 Suppose Re κ>0, and κ∈C−{κ≥1}∪{κ≤−1}. Then in the κ-ordered expression,

6 Star gamma functions

sin∗ π(z−

21

1 1 uv)∗(z− uv)−1 −∗ i~ i~

is a well defined entire function of z. 1 1 In particular, sin∗ π(z+ i~ uv)∗(z 2 −( i~ uv)2 )−1 ±∗ is a holomorphic function of z in C. 1 1 1 (z+ i~ uv))−1 Consider next the product (1+ m +∗ ∗ sin∗ π(z+ i~ uv). Since

(1+

1 1 1 (z+ uv))−1 uv)−1 +∗ =m(m+z+ +∗ , m i~ i~

1 1 and sin∗ π(z+m+ i~ uv)=(−1)m sin∗ π(z+ i~ uv) by the exponential law, the product formula is essentially the same as above. Hence we see the following:

Proposition 5.5 Suppose Re κ>0, and κ∈C−{κ≥1}∪{κ≤−1}. Then in the 1 1 1 uv)∗(1+ m (z+ i~ uv))−1 κ-ordered expression, the product sin∗ π(z+ i~ +∗ is an entire function of z with no removable singularity. Remark 2 Suppose Re κ − 12 , the right hand side of (36) converges and is holomorphic with respect to z . However, Γ∗ (− 21 ± uv ~i ) is singular. Throughout this section, ordered expressions are always restricted to κ∈C−{κ≥1}∪{κ≤−1}. 6.1 Analytic continuation of Γ∗ (z ±

uv ) ~i

As with the usual gamma function, integration by parts gives the identity Γ∗ (z+1 ±

uv uv uv )=(z ± )∗Γ∗ (z ± ). ~i ~i ~i

(37)

Using

uv uv −1 uv )=(z ± ) ∗Γ∗ (z+1 ± ), ~i ~i ±∗ ~i and careful treating continuity inssues, we have Γ∗ (z ±

Proposition 6.2 Γ∗ (z ± C−{−(N+ 21 )}. τ (z± uv )

uv ~i )

extends to a holomorphic function on z ∈

~i ∗$00 =(z ± 21 )−1 $00 , we see the following remarkable feature Since e∗ of these star functions: Z N τ τ (z± uv ) uv ~i Γ∗ (z ± e−e e∗ )∗$00 ≡ lim dτ ∗$00 = Γ(z ± 21 )$00 , N →∞ −N ~i Z N uv τ (z± uv ~i ) (1−eτ )y−1 dτ ∗$00 = B(z ± 21 , y)$00 . , y)∗$00 ≡ lim e∗ B∗ (z ± N →∞ −N ~i (38)

6 Star gamma functions

23

6.2 An infinite product formula We see in the same notation as above Z 0 uv −1 uv τ (z± uv ~i ) e∗ dτ = z+ , B∗ (z± , 1) = ~i i~ ∗± −∞

1 Re z > − . 2

(39)

We now compute uv )Γ(y) = Γ∗ (z ± ~i

ZZ

τ σ τ (z± uv ~i ) σy −(e +e )

e∗

e e

dτ dσ.

R2

We change variables by setting eσ = et (1−es ),

τ = t+s,

where − ∞ < t < ∞, −∞ < s < 0.

Since eτ +eσ = et , this gives a diffeomorphism of R×R− onto R2 . The Jacobian 1 is given by dτ dσ = 1−e s dtds. Hence we have the fundamental relation between the gamma function and the beta function Z ∞Z 0 t uv t(y+z± uv s(z± uv ~i ) −e ~i ) )Γ(y) = e∗ e ∗e∗ (1−es )y−1 dtds Γ∗ (z ± ~i −∞ −∞ (40) uv uv =Γ∗ (y+z ± )∗B∗ (z ± , y). ~i ~i Integration by parts gives (z ±

uv uv uv )∗B∗ (z ± , y+1) = yB∗ (1+z ± , y+1). ~i ~i ~i

To prove this, note that uv d τ (z± uv τ (z± uv ~i ) ~i ) = (z ± , e∗ )∗e∗ dτ ~i lim e−e

τ →±∞

τ

uv +zτ ±τ ~i e∗

τ d −eτ e = −eτ e−e , dτ

1 = 0 for Re z > − . 2

uv uv Since B∗ (z ± uv ~i , y+1) = B∗ (z ± ~i , y)−B(1+z ± ~i , y), we have the functional equation y+z ± uv uv uv ~i , y) = ∗B∗ (z ± , y+1). (41) B∗ (z ± ~i y ~i

Iterate (41) to obtain B∗ (z ±

(y+z ± uv , y) = ~i

uv ~i )∗(y+1+z

y(y+1)

±

uv ~i )

∗B∗ (z ±

uv , y+2). ~i

Using the notation (a)n = a(a+1) · · · (a+n−1),

{A}∗n = A∗(A+1)∗ · · · ∗(A+n−1),

24


we have B∗ (z ±

{y+z ± uv uv uv ~i }∗n , y) = ∗B∗ (z ± , y+n). ~i (y)n ~i

(42)

Similarly, integration by parts gives the formula Γ∗ (1+z ±

uv uv uv ) = (z ± )∗Γ∗ (z ± ), ~i ~i ~i

1 for Re z > − . 2

(43)

Iterate (43) to obtain uv uv uv . ) = Γ∗ (z ± )∗ z ± ~i ~i ~i ∗n Qn uv −1 Lemma 6.1 B∗ (z ± uv k=0 ∗(k+z ± ~i )±∗ . ~i , n+1) = n!

(44)

Γ∗ (n+1+z ±

Proof. The right hand side of the above equality will be denoted by

n! . (±) {z± uv ~i }∗n+1

The case n = 0 is given by (39). Suppose the formula holds for n. For the case n+1, we see that uv B∗ (z ± , n+2) ~i Z 0 τ (z± uv ~i ) = (1−eτ )(1−eτ )n dτ = e∗ −∞

It follows that B∗ (z ±

n! {z ±

uv (±) ~i }∗n+1

−

n! {1+z ±

uv (±) ~i }∗n+1

.

uv (n+1)! , n+2) = . (±) ~i {z ± uv } ∗n+2 ~i

In this subsection, we give an infinite product formula for the ∗-gamma function. By Lemma 6.1, we see that Z 0 1 n! τ (z± uv ~i ) , Re z > − . e∗ (1 − eτ )n dτ = uv (±) 2 −∞ {z ± ~i }∗n+1 Replacing eτ by

1 τ0 ne ,

namely setting τ = τ 0 − log n in the left hand side, and (log n)(z± uv ~i )

multiplying both side by e∗ Z

log n

τ 0 (z± uv ~i )

e∗ −∞

, we have

n! 1 0 (log n)(z± uv ~i ) (1− eτ )n dτ 0 = . ∗e∗ (±) uv n {z ± ~i }∗n+1

(45)

Lemma 6.2 The Weyl ordering of the left hand side of (45) converges when R ∞ τ 0 (z± uv τ0 ~i ) −e n→∞ to −∞ e∗ e dτ 0 in Hol(C2 ).

6 Star gamma functions 0

25

τ0

Proof. Obviously, limn→∞ (1− n1 eτ )n =e−e uniformly on each compact subset as a function of τ 0 . In the Weyl ordering, it is easy to show that Z log n 0 Z ∞ 0 τ0 τ0 τ (z± uv τ (z± uv 0 ~i ) −e ~i ) −e e∗ e∗ lim e dτ = e dτ 0 n→∞

−∞

−∞

2

in Hol(C ). Thus it is enough to show that Z log n 0 τ0 1 0 τ (z± uv ~i ) e∗ lim (e−e −(1− eτ )n )dτ 0 =0 n→∞ −∞ n in Hol(C2 ). This is easy in the Weyl ordering. Applying the intertwiner gives the desired result. The right hand side of (45) equals for Re z > − 21 1 (log n−(1+ 12 +···+ n ))(z± uv ~i )

e∗

∗(z ±

n z± uv ~i z ± uv uv −1 Y ~i −1 k ∗e 1+ )∗± ∗ . ∗ ±∗ ~i k k=1

(log n−(1+ 1 +···+ 1 ))(z± uv )

2 n ~i The left hand side of (45) converges, and limn→∞ e∗ = −γ(z± uv ~i ) obviously, where γ is Euler’s constant. By the continuity of the ∗e∗ s uv multiplication e∗ ~i ∗, we have the convergence in Hol(C2 ) of

n Y uv −1 k1 (z± uv 1 ~i ) . ) ±∗ ∗e∗ lim ∗ 1+ (z ± n→∞ k ~i k=1

Hence we have the convergence in Hol(C2 ) of the infinite product formula Γ∗ (z+

∞ Y 1 −1 k1 (z+ i~ 1 uv 1 uv −γ(z+ uv uv) ~i ) 1+ ) = e∗ ∗ ∗ (z+ uv) ∗e ∗(z+ )−1 ∗ +∗ ~i ~i ∗+ k i~ k=1 (46) −

1

(z+ uv )

1 m ~i (z+ uv to both side of (46) and Fix m∈N. Multiplying (1+ m ~i )e∗ using the abbreviated notation Y Y 1 −1 k1 (a+ i~ 1 1 uv uv) (a+ uv) ∗e 1+ (z=a) = (z+ )−1 ∗ ∗ +∗ ~i +∗ k i~ k6=m

k6=m

we have uv 1 uv − 1 (z+ uv ~i ) (1+ (z+ ))∗e∗ m ∗Γ∗ (z+ ) m i~ ~i Q 1 (z=z) k6=m z6∈ − (N+m+ 12 ) (47) = Q 1 1 1 n n z= − (n+m+ 2 ) k6=m (z=−n−m− 2 )∗ 1− n! ( i~ u) ∗$00 ∗v

−1 1 1 where n∈N. As opposited to the case that (1+ m (z+ uv i~ ))+∗ ∗ sin∗ π(z+ i~ uv) is entire function (cf. Proposition 5.5), there are removable singularities with respect to z.

26


Multiplying (47), we have

− k1 (z+ uv ~i ) 1+ k1 (z+ uv to both sides of (46) and using ~i ) e∗

Q∞

k=1

N Y − 1 (z+ i~1 uv) 1 1 1 ∗Γ∗ (z+ uv) ∗ 1+ (z+ uv) ∗e∗ k N →∞ k i~ i~ k=1 z6∈ − (N+ 21 ) Pn 1 11 k = k 1− k=0 k! ( i~ u) ∗$00 ∗v , z= − (n+ 12 ),

lim

in Hol(C2 ), where n∈N.

1 uv) 7 Products with sin∗ π(z+ i~ 1 1 In this section we show that sin∗ π(z+ i~ uv)∗Γ∗ (z+ i~ uv) is well defined as an entire function of z. By recalling Euler’s reflection formula, this product may 1 be understood as Γ (1−(z+ . We define the product by the integral 1 uv)) ∗

2i sin∗ π(z+

i~

1 1 uv)∗Γ∗ (z+ uv) i~ i~ Z T0 1 1 τ τ (z+ uv ) uv) −πi(z+ i~ uv) πi(z+ i~ i~ −e∗ )∗e−e e∗ dτ = lim (e∗ 0 T,T →∞ −T Z ∞ τ (τ +πi)(z+ uv (τ −πi)(z+ uv i~ ) i~ ) = e−e (e∗ −e∗ )dτ. −∞

(48) The κ-ordered expression of (48) is given as follows: :(48):κ = By using e−e

Z

τ −πi

∞+πi

e−e

τ −πi

τ (z+ uv i~ )

e∗

dτ −

−∞+πi

=e−e

τ +πi

(

Z

Z

∞−πi

e−e

τ +πi

τ (z+ uv i~ )

e∗

dτ.

−∞−πi

, this is given by the integral

∞+πi

− −∞+πi

Z

∞−πi

τ

τ (z+ uv i~ )

)ee e∗

dτ.

−∞−πi

Note this is not a contour integral, but is defined for κ∈C−{κ≥1}∪{κ≤−1}. The following is our main result: 1 1 Theorem 7.1 sin∗ π(z+ i~ uv)∗Γ∗ (z+ i~ uv) is defined as an entire function 1 of z, vanishing at z∈N+ 2 in any κ-ordered expression such that Re κ 0 is not formulated as a σ-model. It may be better to think of our theory as a certain gauge theory in which higher-order differential forms provide “gauge fields”.

Representations of gauge transformation groups of higher abelian gerbes

61

3.2 Equation of motion To derive the equation of motion from the functional, we begin with some comments on Lie group structures on G(W ) and G(W )C : we can make G(W ) into an infinite dimensional Lie group whose Lie algebra g(W ) is the vector space g(W ) = A2k (W )/d(A2k−1 (W )) with the trivial Lie bracket. The exponential map exp : g(W ) → G(W ) is given by the following composition of homomorphisms: ι

A2k (W )/d(A2k−1 (W )) → A2k (W )/A2k (W )Z → G(W ). In a similar way, we can also make G(W )C into a complex Lie group. As the notation indicates, G(W )C gives rise to a complexification of G(W ). Since the tangent space of G(W )C is seen, we compute the equation of motion for the action functional EW to obtain: Lemma 3.2 A Deligne cohomology class f ∈ G(W )C such that δ(f )|∂W = 0 is a critical point of EW if and only if f satisfies: d∗ δ(f ) = 0, where d∗ = − ∗ d∗ is the formal adjoint of d : A2k+1 (W, C) → A2k+1 (W, C). d E (f + tα) for α ∈ g(W )C . Because δ(α) = dα, Proof. We compute dt t=0 W Stokes’ theorem leads to: Z Z d α ∧ d ∗ δ(f ), α ∧ ∗δ(f ) − 2`π EW (f + tα) = 2`π dt t=0

∂W

W

which implies the lemma. t u We note that f ∈ G(W )C always satisfies the equation dδ(f ) = 0. Thus, in the case where W has no boundary, f is a solution to the equation of motion if and only if δ(f ) is a harmonic form. So Lemma 2.2 allows us to identify the space of solutions with H 2k+1 (W, Z) × (H 2k (W, R)/H 2k (W, Z)). We also note that f ∈ G(W )C such that ∗δ(f ) = iδ(f ) or ∗δ(f ) = −iδ(f ) also gives rise to a solution to the equation of motion. This motivates us to introduce the following subgroups in G(W )C : Definition 3.3 We define the chiral subgroup G(W )+ C and the anti-chiral subgroup G(W )− to be the following subgroups in G(W ) C: C ∓ G(W )± C = Ker δ = {f ∈ G(W )C | δ(f ) ∓ i ∗ δ(f ) = 0}.

In the case of k = 0 and W is a Riemann surface, G(W )+ C is isomorphic to the group of holomorphic functions f : W → C/Z, and G(W )− C the group of anti-holomorphic functions f : W → C/Z. A 2k-form α ∈ A2k (W, C) satisfying the “self-dual” condition i ∗ δ(f ) = δ(f ) is called a chiral 2k-form (see [15, 20], for example). By means of the homomorphism ι in Lemma 2.2 (b), a chiral 2k-form induces an element in + G(W )+ C . This is the reason that G(W )C is named the chiral subgroup.

62

Kiyonori Gomi

3.3 Analogy of the Polyakov-Wiegmann formula The action functionals in the Wess-Zumino-Witten models obey the so-called Polyakov-Wiegmann formula. There is a similar formula for the functional EW . We define ΓW : G(W )C × G(W )C → C by Z ΓW (f, g) = 4`πi δ − (f ) ∧ δ + (g). W

Lemma 3.4 Suppose that ∂W = ∅. For f, g ∈ G(W )C we have: eEW (f ) · eEW (g) = (exp ΓW (f, g)) eEW (f +g) . Proof. A straight computation gives: Z EW (f + g) − EW (f ) − EW (g) + ΓW (f, g) = 2`πi

δ(f ) ∧ δ(g). W

Recall Rthat δ(f ), δ(g) ∈ A2k+1 (W, C)Z . Since ` is taken to be an integer, we have ` W δ(f ) ∧ δ(g) ≡ 0 in C/Z under the present assumption on W . t u The point R in the above proof is that W has no boundary: if W has a boundary, then W δ(f ) ∧ δ(g) is not necessarily an integer. To take into account contributions of the boundary, we introduce a complex line bundle. Definition 3.5 Let M be a compact oriented (4k + 1)-dimensional smooth manifold (without boundary). (a) We define the line bundle LM over G(M )C by LM = G(M )C × C. (b) We define the product structure LM × LM → LM by (f, z) · (g, w) = (f + g, zw exp 2`πiSM,C (f, g)), where SM,C : G(M )C × G(M )C → C/Z is defined to be Z SM,C (f, g) = f ∪g M

by using the cup product and the integration for smooth Deligne cohomology. Lemma 3.6 Suppose that ∂W 6= ∅. For f ∈ G(W )C we define an element eEW (f ) ∈ L∂W to be eEW (f ) = (f |W , exp EW (f )). Then we have: eEW (f ) · eEW (g) = (exp ΓW (f, g)) eEW (f +g) . Proof. By means of Lemma 2.4 (b), we have Z Z S∂W,C (f |∂W , g|∂W ) = (f ∪ g)|∂W = ∂W

Z δ(f ∪ g) = W

δ(f ) ∧ δ(g). W

Now this lemma follows from the formula in the proof of Lemma 3.4. t u


63

3.4 Central extension The product structure on LM induces a group multiplication on the complement LM \{0} of the image of the zero section. We denote the group by ˜ )C . Obviously, G(M ˜ )C is a central extension of G(M )C : G(M ˜ )C −→ G(M )C −→ 1. 1 −→ C∗ −→ G(M ˜ ) of G(M ): By restriction, we also obtain the central extension G(M ˜ ) −→ G(M ) −→ 1. 1 −→ U (1) −→ G(M ˜ ) = G(M ) × U (1) as We can express the group multiplication in G(M (f, u) · (g, v) = (f + g, uv exp 2`πiSM (f, g)), R where SM (f, g) ∈ R/Z is defied to be SM (f, g) = M f ∪ g by using again the cup product and the integration for smooth Deligne cohomology. Proposition 3.7 Let M be a compact oriented smooth (4k + 1)-dimensional ˜ ) is non-trivial as a central extension. manifold. If ` 6= 0, then G(M In [12], the proof in the case of ` = 1 is given. We can easily generalize the ˜ )C is also non-trivial, proof to the case of ` 6= 0. The central extension G(M ˜ since it is a complexification of G(M ). As an example, we consider the case of k = 0 and M = S 1 . In this ˜ 1) ∼ b (1)/Z2 , where case, G(S 1 ) ∼ = LU (1) as mentioned, and we have G(S = LU b LU (1)/Z2 is the universal central extension of LU (1), ([19]). 3.5 Toward the quantum theory To approach the quantum theory, we appeal to a method by using path integrals formally. In the case of ∂W = ∅, we may describe the partition function of our theory as: Z eEW (F ) DF,

ZW = F ∈G(M )C

where DF is a formal invariant measure on G(M )C . In the case of ∂W 6= ∅, the probability amplitude eEW (F ) is formulated as an element in L∂W . Hence the formal path integral gives the section ZW ∈ Γ (L∂W ) by Z ZW (f ) = eEW (F ) DF. F ∈G(W )C ,F |∂W =f

On a formal level, we can reconstruct the partition function for a (4k + 2)dimensional manifold W without boundary from the above sections by cutting W along a submanifold M of dimension 4k + 1. This suggests that the space

64

Kiyonori Gomi

of sections Γ (LM ) contains the quantum Hilbert space of our theory, as is in the Wess-Zumino-Witten models [9, 10]. By construction, the central extension G(M )C acts on the line bundle LM ˜ )C covering the action of G(M )C on itself. Though G(M )C is abelian, G(M ˜ )C is not. Hence the space of sections Γ (LM ) gives rise to a two sided G(M module. Compared to the Wess-Zumino-Witten models, one may expect that the quantum Hilbert space in Γ (LM ) is of the form “⊕λL ,λR HλL ⊗ HλR ”, ˜ )C , where HλL and HλR are certain left and right irreducible modules of G(M ˜ respectively. This motivates us to study representations of G(M )C , which is the subject of the next section.

4 Representations of smooth Deligne cohomology In this section, we deal with representations of smooth Deligne cohomology groups, culling out some results from [13]. After the statement of the classification, we explain a relationship between the representations and the quantum Hilbert space of a chiral 2-form due to Henningson [15]. We also consider an analogy of the space of conformal blocks in the Wess-Zumino-Witten model. 4.1 Representations of smooth Deligne cohomology First of all, we make a general remark: for a compact oriented (4k + 1)˜ ) is constructed by using dimensional manifold M , the central extension G(M 2`πiS(·,·) the group 2-cocycle e : G(M ) × G(M ) → U (1). Hence a representation ˜ ) such that the center U (1) acts as the scalar multiplication (˜ ρ, H) of G(M corresponds bijectively to a projective representation (ρ, H) of G(M ) with its ˜ )C . We cocycle e2`πiSM . There is a similar correspondence in the case of G(M use the correspondences freely in the following. For the smooth Deligne cohomology group G(M ), admissible representations of level ` are certain projective unitary representations on Hilbert spaces with their cocycle e2`πiSM . They are characterized by the representations of the subgroup A2k (M )/A2k (M )Z ⊂ G(M ) obtained by restriction. Generalizing straightly the proof of Theorem 1.1 given in [13], we can obtain the following classification of admissible representations: Theorem 4.1 Let M be a compact oriented (4k+1)-dimensional Riemannian manifold. For a positive integer `, admissible representations of G(M ) of level ` have the following properties: (a) An admissible representation is equivalent to a finite direct sum of irreducible admissible representations. (b) The number of the equivalence classes of irreducible admissible representations is (2`)b r, where b = b2k (M ) = b2k+1 (M ) is the Betti number, and r is the number of elements in the set {t ∈ H 2k+1 (M, Z)| 2` · t = 0}.


65

If H 2k+1 (M, Z) is torsion free, then the number of the equivalence classes of irreducible admissible representations of level ` is (2`)b . The outline of constructing these irreducible representations is as follows: 1. We let G 0 (M ) be the subgroup A2k (M )/A2k (M )Z in G(M ). Using the Riemannian metric on M , we decompose the subgroup G 0 (M ) as follows: G 0 (M ) = H2k (M )/H2k (M )Z × d∗ (A2k+1 (M )), where H2k (M ) is the group of harmonic 2k-forms on M , H2k (M )Z = H2k (M ) ∩ A2k (M )Z the subgroup of harmonic 2k-forms with integral periods, and d∗ : A2k+1 (M ) → A2k (M ) the formal adjoint of d given by d∗ = − ∗ d∗. Using the Riemannian metric again, we define an inner product ( , ) on d∗ (A2k+1 (M )) and a compatible complex structure J on the completion V of d∗ (A2k+1 (M )) such that: (ν, Jν 0 ) = `SM (ν, ν 0 ),

ν, ν 0 ∈ d∗ (A2k+1 (M )) ⊂ V.

2. We construct the projective representation (ρ, H) of d∗ (A2k+1 (M )). The representation is realized as the representation of the Heisenberg group associated to the symplectic form (·, J·) : V × V → R. The representation space H is a completion of the symmetric algebra S(W ), where W is the eigenspace in V ⊗ C of J with its eigenvalue i. 3. Let X (M ) denote the set of homomorphisms λ : H2k (M )/H2k (M )Z → R/Z. For λ ∈ X (M ), we construct the projective representation (ρλ , Hλ ) of G 0 (M ). The representation space is Hλ = H. The action of (η, ν) ∈ (H2k (M )/H2k (M )Z ) × d∗ (A2k+1 (M )) is ρλ (η, ν) = e2πiλ(η) ρ(ν). 4. We construct the projective representation (ρλ , Hλ ) as the representation induced from the representation (ρλ , Hλ ) of the subgroup G 0 (M ) ⊂ G(M ). The projective representations (ρλ , Hλ ) and (ρλ0 , Hλ0 ) are equivalent if and only if there is ξ ∈ H 2k+1 (M, Z) such that λ0 = λ + 2`s(ξ). Here the homomorphism s : H 2k+1 (M, Z) → X (M ) is defined by Z s(ξ)(η) = η ∧ ξR mod Z, M

where ξR is a de Rham representative of the real image of ξ. If H 2k+1 (M, Z) is torsion free, then s is an isomorphism by the Poincaré duality. Thus, among the representations (ρλ , Hλ ), we have (2`)b inequivalent representations. For example, we again consider the case of k = 0 and M = S 1 . Then we have the isomorphism G(S 1 ) ∼ = LU (1), and an admissible representation of G(S 1 ) of level ` gives rise to a positive energy representation of LU (1) of level 2`, and vice verse. This is because the construction of irreducible admissible representations outlined above coincides with that of irreducible positive energy representations given in [19]. The number of the equivalence classes of irreducible positive energy representations of LU (1) of level 2` is 2`, which is consistent with Theorem 4.1.

66

Kiyonori Gomi

4.2 Relationship to Henningson’s work The irreducible admissible representations in the case of k = 1 have certain relationship to the quantum Hilbert space of a chiral 2-form studied in a work of Henningson [15]. To explain the relationship, some notations in [15] will be used here without change. Let M be a compact oriented 5-dimensional Riemannian manifold such that H 2 (M, Z) and H 3 (M, Z) are torsion free. Then, by means of Theorem 4.1, the number of the equivalence classes of irreducible admissible representations of G(M ) of level ` = 1 is 2b . These irreducible representations are parameterized by the set Coker{2s : H 3 (M, Z) → X (M )} ∼ = (Z/2Z)b . In [15], Henningson studied the quantum Hilbert space V+ of a chiral 2form on R × M . The Hilbert space V+ can be expressed as a Hilbert space 0 0 tensor product: V+ = V 0 ⊗ V+ . The Hilbert space V+ admits a further decom0 0 position: V+ = ⊕a+ ∈H 3 (M,Z2 ) Va+ . Accordingly, we have the following decomposition into different version of the chiral theory: M V 0 ⊗ Va0+ . V+ = a+ ∈H 3 (M,Z2 )

Under the present assumption on M , the homomorphism s : H 3 (M, Z) → X (M ) induces the natural isomorphism: H 3 (M, Z2 ) ∼ = Coker{2s : H 3 (M, Z) → X (M )}. Hence we can naturally identify the parameterization space of the Hilbert spaces V 0 ⊗ Va0+ with that of the equivalence classes of irreducible admissible representations. If we take fixed lifting a+ ∈ H 3 (M, Z) of elements a+ ∈ H 3 (M, Z2 ), then (ρs(a+ ) , Hs(a+ ) ) represents the equivalence class of irreducible admissible representations corresponding to V 0 ⊗ Va0+ . In addition, we can find a natural isomorphism between V 0 ⊗ Va0+ and Hs(a+ ) . On the one hand, the construction in [15] implies that the Hilbert space V 0 is the completion of the symmetric algebra S(W ), so that V 0 = H. The Hilbert space Va0+ is spanned by |k+ , a+ i, (k+ ∈ H 3 (M, Z)). Thus, we have the following expression by using a Hilbert space direct sum: M d V 0 ⊗ Va0+ = H ⊗ C|k+ , a+ i. 3 k+ ∈H (M,Z)

On the other hand, it is shown in [13] that, for each λ ∈ X (M ), the representation Hλ |G 0 (M ) of G 0 (M ) obtained by the restriction of Hλ is expressed as the following Hilbert space direct sum: M d Hλ+2s(ξ) . Hλ |G 0 (M ) = 3 ξ∈H (M,Z)

Now the natural isomorphism V 0 ⊗ Va0+ → Hs(a+ ) follows from the obvious identification H ⊗ C|k+ , a+ i ∼ = Hs(a+ +2k+ ) .


67

4.3 An analogy of the space of conformal blocks Projective representations of G(M )C , rather than G(M ), concern the field theory in Section 3. It is known [19] that positive energy representations of LU (1) extend to that of LC∗ in a certain way. The following proposition generalizes the fact. Proposition 4.2 Let M be a compact oriented (4k + 1)-dimensional Riemannian manifold, and (ρ, H) an admissible representation of G(M ) of level `. Then there is an invariant dense subspace E ⊂ H, and (ρ, E) extends to a projective representation of G(M )C . We can prove the proposition above by a straight generalization of the proof in the case of ` = 1 described in [13]. By means of the representations of G(M )C , we consider below an analogy of the space of conformal blocks in higher dimensions. Before the consideration, we notice that Lemma 3.6 and Theorem 4.1 lead to: (i) For a compact oriented (4k+2)-dimensional Riemannian manifold W with boundary, the following map gives rise to a homomorphism: ˜ r+ : G(W )+ C → G(∂W )C ,

f 7→ (f |∂W , exp EW (f )).

(ii) For a compact oriented (4k + 1)-dimensional Riemannian manifold M , we can parameterize the equivalence classes of irreducible admissible representations of G(M ) of level ` by a finite set Λ` (M ). Now, for W and λ ∈ Λ` (M ), we define CB(W, λ) to be the vector space consisting of continuous linear maps ψ : Eλ → C such that ψ(˜ ρλ (r+ (f ))v) = + ψ(v) for all v ∈ Eλ and f ∈ G(M )C : +

CB(W, λ) = Hom(Eλ , C)G(W )C . As the simplest example, we let W = D 4k+2 be the standard (4k + 2)dimensional disk whose boundary is S 4k+1 . In a direct way, we can compute CB(W, λ) to obtain a finite dimensional vector space: in the case of k = 0, the parameterization set Λ` (S 1 ) is identified with Z/`Z, and we obtain: C, (λ = 0) CB(D2 , λ) ∼ = {0}. (λ 6= 0) In the case of k > 0, the parameterization set Λ` (S 4k+1 ) = {0} consists of a single element, and we obtain: CB(D4k+2 , 0) ∼ = C. At present, the only example available is the above one. Computations of other examples as well as a proof that CB(W, λ) is a finite dimensional vector space still remain as problems.

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Kiyonori Gomi

Acknowledgement. Thanks are due to organizers and audiences during International Workshop on Noncommutative Geometry and Physics 2005, November 1–4, 2005, at Tohoku University. The author’s research was supported by Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.

References 1. L. Breen, On the classification of 2-gerbes and 2-stacks. Asterisque No. 225 (1994), 160 pp. 2. J-L. Brylinski, Loop spaces, Characteristic Classes and Geometric Quantization. Birkh¨ auser Boston, Inc., Boston, MA, 1993. 3. J. Cheeger and J. Simons, Differential characters and geometric invariants. Lecture Notes in Math. 1167(1985), Springer Verlag, 50-80. 4. A. L. Carey, M. K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories. J. Geom. Phys. 21 (1997), no. 2, 183–197. 5. P. Deligne and D. S. Freed, Classical field theory. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 137–225, Amer. Math. Soc., Providence, RI, 1999. 6. J. L. Dupont and F. W. Kamber, Gerbes, simplicial forms and invariants for families of foliated bundles. Comm. Math. Phys. 253 (2005), no. 2, 253–282. 7. H. Esnault and E. Viehweg, Deligne-Be˘ılinson cohomology. Be˘ılinson’s conjectures on special values of L-functions, 43–91, Perspect. Math., 4, Academic Press, Boston, MA, 1988. 8. P. Gajer, Geometry of Deligne cohomology. Invent. Math. 127 (1997), no. 1, 155–207. 9. K. Gaw¸edzki, Conformal field theory: a case study. Conformal field theory (Istanbul, 1998), 55 pp., Front. Phys., 102, Adv. Book Program, Perseus Publ., Cambridge, MA, 2000. 10. K. Gaw¸edzki, Topological actions in two-dimensional quantum field theories. Nonperturbative quantum field theory (Cargèse, 1987), 101–141, NATO Adv. Sci. Inst. Ser. B Phys., 185, Plenum, New York, 1988. 11. J. Giraud, Cohomologie non-abélienne. Grundl. 179, Springer Verlag (1971). 12. K. Gomi, Central extensions of gauge transformation groups of higher abelian gerbes. J. Geom. Phys. to appear. hep-th/0504075. 13. K. Gomi, Projective unitary representations of smooth Deligne cohomology groups. math.RT/0510187. 14. K. Gomi and Y. Terashima, Higher-dimensional parallel transports. Math. Res. Lett. 8 (2001), no. 1-2, 25–33. 15. M. Henningson, The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions. J. High Energy Phys. 2002, no. 3, No. 21, 15 pp. 16. N. Hitchin, Lectures on special Lagrangian submanifolds. Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 151–182, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001. 17. M. K. Murray, Bundle gerbes. J. London Math. Soc. (2) 54 (1996), no.2, 403416. 18. R. Picken, A cohomological description of abelian bundles and gerbes. Twenty years of Bialowieza: a mathematical anthology, 217–228, World Sci. Monogr. Ser. Math., 8, World Sci. Publ., Hackensack, NJ, 2005.


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19. A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. 20. E. Witten, Five-brane effective action in M -theory. J. Geom. Phys. 22 (1997), no. 2, 103–133.


Algebroids associated with pre-Poisson structures Kentaro Mikami1 and Tadayoshi Mizutani2 1

2

Dep of Computer engineering, Akita University, Japan [email protected] Dep of Mathematics, Saitama University, Japan [email protected]

1 Introduction There are several ways to generalize Poisson structures. A Jacobi structure (or a local Lie algebra structure), in which we do not require the Leibniz identity for the bracket, and a Nambu-Poisson structure, where the brackets are not binary but n-ary operations satisfying a generalized Leibniz rule called fundamental identity, are well-known examples. Also, a Dirac structure is a natural generalization of a Poisson structure. As another direction of studying Poisson geometry, we would like to do some trial or attempt to generalize the concepts, ideas, or theories of Poisson geometry into some area where the Poisson condition is not fulfilled. In the first half of this note, we show briefly our trials in this context, namely in almost Poisson geometry. As we will see in short, a Poisson structure gives a Lie algebroid. It is natural to handle a Leibniz algebroid as generalization of a Lie algebroid. Thus, it is meaningful to study the fundamental properties of Leibniz algebra or super Leibniz algebra. In the second half of this note, after we recall some properties of Leibniz modules, we define super Leibniz algebras and super Leibniz modules keeping the exterior algebra bundle of the tangent bundle with Schouten bracket as a prototype of a super Lie algebra (and so a super Leibniz algebra). We will show that an abelian extension is controlled by the second super cohomology group. The notion of super Leibniz bundles is clear, but unfortunately we do not have the proper notion of anchor, so far. In near future, we hope we could find concrete examples of super Leibniz bundles tightly connected to the properties of Poisson geometry, and could understand what the anchor should be.

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Kentaro Mikami and Tadayoshi Mizutani

2 Review of elements of Poisson geometry Definition 1. By a Poisson structure or a Poisson bracket on a manifold M , we mean a binary operation on the function space C ∞ (M ) of M , C ∞ (M ) × C ∞ (M ) 3 (f, g) 7→ {f, g} ∈ C ∞ (M ) satisfying 1. R-bilinearity 2. skew-symmetry 3. Jacobi identity 4. Leibniz rule

{λf + µg, h} = λ{f, h} + µ{g, h} (λ, µ ∈ R) {g, f } = −{f, g} {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 {f, gh} = {f, g}h + g{f, h}.

We call a manifold with a specified Poisson structure as a Poisson manifold. Definition 2. On a Poisson manifold, the Hamiltonian vector field Xf of f ∈ C ∞ (M ) is given by hXf , dgi := {f, g} Proposition 1. X{f,g} = [Xf , Xg ] holds for f, g ∈ C ∞ (M ). This comes from Jacobi identity of the Poisson bracket. Example 1. Every manifold has the trivial Poisson structure {f, g} := 0. Example 2. For a given symplectic structure ω, the Poisson bracket is defined by {f, g} := hXf , dgi where ω [ (Xf ) := −df or Xf := −ω ] (df ) . It is well-known that the cotangent bundle T∗ (Q) of a manifold Q has a canonical symplectic structure. The Poisson bracket satisfies {τ ∗ f, τ ∗ g} = 0,

{X, τ ∗ g} = τ ∗ hX, dgi,

{X, Y } = [X, Y ]

where τ : T∗ (Q) → Q is the bundle projection, and f, g ∈ C ∞ (Q), X, Y ∈ Γ (T(M )) and considered as linear functions along the fibres of T∗ (M ), and [X, Y ] is the usual Lie bracket of X and Y . Example 3. Let g be a Lie algebra of finite dimension. Consider the dual space g∗ as the underlying manifold. Since an element of g is a linear function on g∗ , g is a subspace of C ∞ (g∗ ). forF, H ∈ C ∞ (g∗ ), andµ ∈ g∗ , the Poisson bracket is defined by δF δH , ], µi {F, H}(µ) := h[ δµ δµ δF d δF where (ν) := F (µ + tν)|t=0 (ν ∈ g∗ ) and ∈ g∗ ∗ ∼ = g. In fact, δµ X dt δµ {zj , zk } = cijk zi holds where (zi ) is a basis of g whose structure constants i

are (cijk ). This bracket is called Lie-Poisson bracket on the dual space of a Lie algebra.

Algebroids associated with pre-Poisson structures

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Definition 3. For a given Poisson bracket {·, ·}, there exists a unique π ∈ Γ (Λ2 T(M )) satisfying π(df, dg) = hπ, df ∧ dgi = {f, g} . π is called the Poisson 2-vector field (or the Poisson tensor field) of the Poisson bracket. Proposition 2. Conversely, for a given 2-vector field π, define a new bracket by {f, g} := π(df, dg) = hπ, df ∧ dgi . This bracket is Poisson, namely satisfies the Jacobi identity if and only if the Schouten bracket [π, π]S vanishes. We recall here the definition and the related properties of the Schouten bracket . Now on, we abbreviate Γ (Λ• T(M )) to Λ• T(M ), and so on. Definition 4. The Schouten bracket [·, ·]S is the homogeneous bi-derivadimM X tion on Λ• (T(M )) of degree −1 •=0

Λp T(M ) × Λq T(M ) 3 (P, Q) 7→ [P, Q]S ∈ Λp+q−1 (T(M )) uniquely defined by the following five conditions. Property (6) is called super Jacobi identity. 1. [f, g]S = 0 f, g ∈ Λ0 (T(M )) = C ∞ (M ) 2. [X, f ]S = hX, df i = Xf X ∈ Λ1 (T(M )), f ∈ Λ0 (T(M )) 3. [X, Y ]S = [X, Y ]Lie bracket X, Y ∈ Λ1 (T(M )) 4. [P, Q]S = −(−1)(p−1)(q−1) [Q, P ]S 5. [P, Q ∧ R]S = [P, Q]S ∧ R + (−1)(p−1)q Q ∧ [P, R]S 6. (−1)(p−1)(r−1) [[P, Q]S , R]S + (−1)(q−1)(p−1) [[Q, R]S , P ]S + (−1)(r−1)(q−1) [[R, P ]S , Q]S = 0, where the small letter p means the ordinary degree of the capital letter P , i.e., P ∈ Λp (T(M )). Remark 1. The Schouten bracket on the decomposable elements is given by [X1 ∧ · · · Xp ,Y1 ∧ · · · Yq ]S =

p X q X i

ci · · · Y1 ∧ · · · c (−1)i+j [Xi , Yj ] ∧ X1 ∧ · · · X Yj · · · ∧ Y q

j

for Xi , Yj ∈ Γ (T(M )) (p, q ≥ 1). We often abbreviate [·, ·]S to [·, ·]. The sign convention of the Schouten bracket here is different from that in Vaisman’s book [9].

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Given a 2-vector field π on M , we have a bundle map from T∗ (M ) to T(M ) defined by β 7→ π(β, ·) = ιβ π = β π . This map is denoted by π ] , π ˜ , or often π itself if there is no danger of confusion. The Hamilton vector field Xf is then written as π ] (df ).

3 Algebroids related with Poisson structures 3.1 Lie algebroids We start this section by a famous result by B. Fuchssteiner [2]. Theorem 1 ([2]). Given a Poisson 2-vector field π on M , define a bracket by {α, β}π := Lπ] (α) β − Lπ] (β) α − d(π(α, β)) for each α, β ∈ Γ (T∗ (M )). Then {·, ·}π yields a Lie algebra structure on Γ (T∗ (M )) and the following equality holds: {α, f β}π = hπ ] α, df iβ + f {α, β}π . Replacing T(M ) by a general vector bundle we obtain the notion of Lie algebroids. Definition 5. A vector bundle L over M is a Lie algebroid if and only if (a) Γ (L) is endowed with a Lie algebra bracket [·, ·] over R, i.e., [·, ·] is skewsymmetric R-bilinear and satisfies Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (b) there exists a bundle map (called anchor ) a : L → T(M ) which induces a : Γ (L) → Γ (T(M )) is a(f x) = f a(x) (c) and satisfying [x, f y] = ha(x), df iy + f [x, y] where x, y, z ∈ Γ (L), f ∈ C ∞ (M ). Example 4. (1) T∗ (M ) with the bracket {·, ·}π defined from the Poisson 2vector field π is a Lie algebroid whose anchor is π ] . (2) T(M ) is a Lie algebroid with the identity map as the anchor. Assume that a 2-vector field π is not necessarily Poisson. Then we look at the space {α ∈ Γ (T∗ (M )) | α

[π, π]S = 0} =: ker[π, π]S

and ask the questions. Is ker[π, π]S closed with respect to {·, ·}π ? Does {·, ·}π satisfy Jacobi identity?


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We would like to say this trial is one of pre-Poisson attempts. The answers for the both are yes and we get a theorem. Theorem 2 (Mikami-Mizutani [7]). If rank ker[π, π]S is constant, ker[π, π]S is a Lie algebroid having bracket defined by {α, β}π := Lπ] (α) β − Lπ] (β) α − d(π(α, β)) and the anchor α 7→ π ] (α). 3.2 Dirac structures As stated already, a 2-vector field π is Poisson if and only if [π, π]S = 0. On the other hand, T. Courant and A. Weinstein studied Poisson condition from more geometrical point of view. They handle the bundle homomorphism π ] : T∗ (M ) → T(M ). They claim that “Poisson condition is equivalent to some property of the graph of π ] , (Dirac structure)”, and generalize their discussion from a graph to a relation. On T(M ) ⊕ T∗ (M ), Courant([1]) defined h(Y1 , β1 ), (Y2 , β2 )i+ := iY1 β2 + iY2 β1 (fibre wise) 1 [[(Y1 , β1 ), (Y2 , β2 )]] := [Y1 , Y2 ] , LY1 β2 − LY2 β1 − d (iY1 β2 − iY2 β1 ) 2 T(e1 , e2 , e3 ) := h[[e1 , e2 ]], e3 i+ where (Yj , βj ) = ej ∈ Γ (T(M ) ⊕ T∗ (M )) (j = 1, 2, 3). Remark 2. In general, T is not tensor field, and only skew-symmetric in the first two arguments. Definition 6. A sub-bundle L ⊂ T(M ) ⊕ T∗ (M ) is an almost Dirac structure if L is maximally isotropic with respect to the pairing h·, ·i+ , i.e., L is a sub-bundle of rank dimM , and the restriction of h·, ·i+ to L × L is identically zero. Proposition 3. If L is an almost Dirac structure, T|L (e1 , e2 , e3 ) = S hβ1 , [Y2 , Y3 ]i + LY1 hβ2 , Y3 i

123

where ej = (Yj , βj ) ∈ Γ (L), (j = 1, 2, 3). Especially, T|L is tensorial, and skew-symmetric in 3 arguments. Example 5. Let π be an arbitrary 2-vector field on M . The graph of π ] , L = {(π ] (β), β) | β ∈ T∗ (M )} is an almost Dirac structure. T|L ((π ] (β1 ), β1 ), (π ] (β2 ), β2 ), (π ] (β3 ), β3 )) =

1 [π, π]S (β1 , β2 , β3 ) . 2

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Example 6. Let ω be a 2-form on M . The graph of ω [ , L = {(X, iX ω) | X ∈ T(M )} is almost Dirac. T|L ((X1 , iX1 ω), (X2 , iX2 ω), (X3 , iX3 ω)) = (dω)(X1 , X2 , X3 ) . The Courant bracket [[·, ·]] is skew-symmetric, but does not satisfy Jacobi identity. In fact, let (J1 , J2 ) be the components of Jacobiator, i.e., (J1 , J2 ) := S [[[[(Y1 , β1 ), (Y2 , β2 )]], (Y3 , β3 )]] . 123

Then, J1 = 0 holds, but J2 is complicated. Proposition 4. J2 is given explicitly, and the restriction of J2 to an almost Dirac structure L is given by J2 |L (· · · ) =

1 d T|L (· · · ) 2

Definition 7. An almost Dirac structure L is a Dirac structure if Γ (L) is closed by bracket [[·, ·]], i.e., it satisfies T|L ≡ 0. In the case of an almost Dirac structure defined by a 2-vector field, a Dirac structure gives a Poisson structure of the base manifold. In the case of a 2-form, a pre-symplectic structure. When L is almost Dirac, we consider the following “sub-bundle” ker(T|L ) : = {e ∈ L | T|L (e1 , e2 , e) = 0, e1 , e2 ∈ L} . Again we would like to say this is one of pre-Poisson trials. Theorem 3 (Mikami-Mizutani[8]). Let L be an almost Dirac structure and ker(T|L ) be of constant rank. Then ker(T|L ) is a Lie algebroid with the bracket [[·, ·]], and the anchor ρker(T|L ) , which is the restriction of the first projection ρ : T(M ) ⊕ T∗ (M ) → T(M ) to ker(T|L ). Application of Theorem 3 to Theorem 2 (Mikami-Mizutani[8]) We know that the graph of a general 2-vector field π is almost Dirac. Since 1 TL (·, (π ] (βj ), βj ), ·) = [π, π]S (·, βj , ·), we have ker TL := {(π ] (γ), γ) | γ ∈ 2 ker[π, π]S }. The bracket is computed as follows [[(π ] (α), α), (π ] (β), β)]] = ([π ] (α), π ] (β)], {α, β}π ) = (π ] ({α, β}π ), {α, β}π ) . If we pick up the first components of the elements of ker TL , this computation gives another proof of Theorem 2 of Mikami-Mizutani.


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4 Leibniz algebra and Leibniz algebroid Definition 8. A vector bundle A over M is a Leibniz algebroid if (a) Γ (A) is endowed with a Leibniz algebra structure over R, i.e., there is a binary operation [·, ·] : Γ (A) × Γ (A) → Γ (A) which is non skew in general and R-bilinear, and satisfying [[x, y], z] = [x, [y, z]] − [y, [x, z]] (Leibniz identity) where x, y, z ∈ Γ (A). (b) there exists a bundle map (called anchor ) a : A → T(M ) (c) the following compatibility condition holds: [x, f y] =ha(x), df iy + f [x, y] where x, y, z ∈ Γ (A), ha(x), df i = La(x) f for f ∈ C ∞ (M ). Remark 3. (1) The condition [[x, y], z] = [x, [y, z]] − [y, [x, z]] is equivalent to [x, [y, z]] = [[x, y], z] + [y, [x, z]] and this means [x, ·] satisfies Leibniz identity or left-derivation rule. (2) If it is skew-symmetric, Jacobi identity is equivalent to Leibniz identity as we see [[x, y], z] = −[[y, z], x] − [[z, x], y] = [x, [y, z]] − [y, [x, z]] (using skew-symmetry) (3) The difference between Lie algebroids and Leibniz algebroids is just relaxing the skew-symmetric property. But, we can see some hidden “almost” skew-symmetric property for Leibniz algebroid. The reason is: [[x, y], z] = [x, [y, z]] − [y, [x, z]] (Leibniz rule) this shows RHS is skew-symmetric in x and y, thus 0 = [[x, y] + [y, x], z]

(x, y, z ∈ Γ (A)).

(4) It follows a([x, y]) = [a(x), a(y)] (x, y ∈ Γ (A)) by the following computations. Since [y, f z] = (La(y) f )z + f [y, z], we have [x, [y, f z]] = [x, La(y) f z + f [y, z]] LHS =[[x, y], f z] + [y, [x, f z]] =(La([x,y])f )z + f [[x, y], z] + [y, (La(x) f )z + f [x, z]] =(La[x,y] f )z + f [[x, y], z] + (La(y) La(x) f )z+ + (La(x) f )[y, z] + (La(y) f )[x, z] + f [y, [x, z]] RHS =(La(x) La(y) f )z + (La(y) f )[x, z] + (La(x) f )[y, z] + f [x, [y, z]] Thus, we have La([x,y])f − La(x) La(y) f + La(y) La(x) f z = 0. This means we have the result.

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Example 7. Let M be a manifold. A := T(M ) ⊕ T∗ (M ). For a pair of sections of A, define a bracket by [(X, α), (Y, β)] := ([X, Y ], LX β − ιY dα) and the anchor a(X, α) := X. Then this is a Leibniz algebroid, but not a Lie algebroid. 4.1 Abelian extension of Leibniz algebra In this subsection, we shall study an algebraic property of a Leibniz algebra and related bi-module (a Leibniz bi-module), namely the second Leibniz cohomology group with coefficient in the Leibniz module. Definition 9 (Leibniz module). Let (g, [·, ·]) be a Leibniz algebra. A module A is called a g-bi module if A is a module where g acts from both, left and right, and it holds the following three conditions. (a · g) · h = a · [g, h] − g · (a · h) (g · a) · h = g · (a · h) − a · [g, h] [g, h] · a = g · (h · a) − h · (g · a) where g, h ∈ g and a ∈ A, and we denoted the left action of g on A by g · a, and the right action of g on A by a · g (g ∈ g, a ∈ A). When a Leibniz g-module A is given, we can construct cochain complex and cohomology groups which are called Leibniz cohomology groups (cf. J.-L. Lodays’s works, for example [4] or [5]). Definition 10. For each non-negative integer k, k-th cochain complex consists of k-multilinear maps from g × · · · × g to A and the coboundary operator | {z } k−times

is given by (δψ)(g1 , . . . , gk+1 ) =

k X

(−1)i−1 gi · ψ(. . . gbi . . .) + (−1)k+1 ψ(g1 , . . . , gk ) · gk+1

i=1

+

X

j

(−1)i ψ(. . . gbi . . . [gi , gj ] . . .) .

i<j

We denote by H k (g, A) the k-th cohomology group. We take a bi-linear map ψ : g × g → A and a linear exact sequence Π

0 −→ A −→ g0 −→ g −→ 0


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and a linear section s of Π, i.e., Π ◦ s = idg which gives a linear isomorphism g0 ∼ = s(g) ⊕ A. Define a multiplication on g0 by [[s(g1 ) + a1 , s(g2 ) + a2 ]] := s[g1 , g2 ] + ψ(g1 , g2 ) + g1 · a2 + a1 · g2 for g1 , g2 ∈ g and a1 , a2 ∈ A, where “·” is the Leibniz action of g to A. We let g 0 i := s(gi ) + ai ∈ g0 (i=1,2,3). We have [[[[g 0 1 , g 0 2 ]], g 0 3 ]] =[[s[g1 , g2 ] + ψ(g1 , g2 ) + g1 · a2 + a1 · g2 , s(g3 ) + a3 ]] =s[[g1 , g2 ], g3 ] + ψ([g1 , g2 ], g3 ) + [g1 , g2 ] · a3 + (ψ(g1 , g2 ) + g1 · a2 + a1 · g2 ) · g3 =s[[g1 , g2 ], g3 ] + ψ([g1 , g2 ], g3 ) + [g1 , g2 ] · a3 + ψ(g1 , g2 ) · g3 + (g1 · a2 ) · g3 + (a1 · g2 ) · g3 =s[[g1 , g2 ], g3 ] + ψ([g1 , g2 ], g3 ) + ψ(g1 , g2 ) · g3 + [g1 , g2 ] · a3 + (g1 · a2 ) · g3 + (a1 · g2 ) · g3 and [[g 0 1 , [[g 0 2 , g 0 3 ]]]] =[[s(g1 ) + a1 , s[g2 , g3 ] + ψ(g2 , g3 ) + g2 · a3 + a2 · g3 ]] =s[g1 , [g2 , g3 ]] + ψ(g1 , [g2 , g3 ]) + g1 · (ψ(g2 , g3 ) + g2 · a3 + a2 · g3 ) + a1 · [g2 , g3 ] =s[g1 , [g2 , g3 ]] + ψ(g1 , [g2 , g3 ]) + g1 · ψ(g2 , g3 ) + g1 · (g2 · a3 ) + g1 · (a2 · g3 ) + a1 · [g2 , g3 ] and so [[g 0 2 , [[g 0 1 , g 0 3 ]]]] =s[g2 , [g1 , g3 ]] + ψ(g2 , [g1 , g3 ]) + g2 · ψ(g1 , g3 ) + g2 · (g1 · a3 ) + g2 · (a1 · g3 ) + a2 · [g1 , g3 ] , [[·, ·]] satisfies Leibniz property, i.e., [[[[g 0 1 , g 0 2 ]], g 0 3 ]] = [[g 0 1 , [[g 0 2 , g 0 3 ]]]] − [[g 0 2 , [[g 0 1 , g 0 3 ]]]] if and only if [·, ·] satisfies Leibniz property and ψ([g1 , g2 ], g3 ) + ψ(g1 , g2 ) · g3 + [g1 , g2 ] · a3 + (g1 · a2 ) · g3 + (a1 · g2 ) · g3 =ψ(g1 , [g2 , g3 ]) + g1 · ψ(g2 , g3 ) + g1 · (g2 · a3 ) + g1 · (a2 · g3 ) + a1 · [g2 , g3 ] − ψ(g2 , [g1 , g3 ]) − g2 · ψ(g1 , g3 ) − g2 · (g1 · a3 ) − g2 · (a1 · g3 ) − a2 · [g1 , g3 ]

(1)

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for each g1 , g2 , g3 ∈ g and for each a1 , a2 , a3 ∈ A. The equation (1) is equivalent to the following 4 conditions: (a1 · g2 ) · g3 =a1 · [g2 , g3 ] − g2 · (a1 · g3 ) (g1 · a2 ) · g3 =g1 · (a2 · g3 ) − a2 · [g1 , g3 ] [g1 , g2 ] · a3 =g1 · (g2 · a3 ) − g2 · (g1 · a3 ) and ψ([g1 , g2 ], g3 ) + ψ(g1 , g2 ) · g3 =ψ(g1 , [g2 , g3 ]) + g1 · ψ(g2 , g3 ) − ψ(g2 , [g1 , g3 ]) − g2 · ψ(g1 , g3 ) .

(2)

The first three equations are just the property of the Leibniz action of the Leibniz algebra g on A, and the equation (2) is equivalent to ψ being a 2cocycle of Leibniz coboundary operator. Thus, we have Proposition 5. For a given Leibniz bracket [·, ·] on g and a Leibniz action of mikamiLieg on A, the new bracket [[·, ·]] on mikamiLieg 0 becomes a Leibniz bracket if and only if ψ is a 2-cocycle. From the definition, we see that [[A, A]] = {0} and Π is a Leibniz algebra homomorphism. We call such an exact sequence as an abelian Leibniz extension of g with the kernel A. Also, from the definition of the multiplication [[·, ·]], we see that [[s(g1 ), s(g2 )]] − s[g1 , g2 ] = ψ(g1 , g2 )

(∀g1 , g2 ∈ g).

Theorem 4 ([5]). For a given Leibniz algebra g and its bi-module A, the class of Leibniz abelian extensions up to Leibniz isomorphisms one-to-one correspond to the second Leibniz cohomology group H 2 (g, A). Example 8. Let g0 = Γ (T(M ) ⊕ T∗ (M )) be the Leibniz algebra with usual Courant bracket. The natural exact sequence of bundles T∗ (M ) → T(M ) ⊕ T∗ (M ) → T(M ) gives an exact sequence of Leibniz algebras 0 → A → g0 → g → 0

(3)

where g = Γ (T(M )) and A = Γ (T∗ (M )). The left action of g on A is given by X·α = LX α and the right action by α·X = −ιX dα. The cocycle corresponding to (3) is the zero cocycle. Indeed, taking ψ ≡ 0, we have [[(X, α), (Y, β)]] = ([X, Y ], ψ(X, Y ) + X · β + α · Y ) = ([X, Y ], LX β − ιY dα).


81

Example 9. We put g0 = Γ (Λq+1 (T(M )) and let F be a transversely oriented regular foliation of codimension q on M . Choose a transverse volume form ω of F which is a locally decomposable q-form that is, locally, ω = ω1 ∧ · · · ∧ ωq and F is defined by ω1 = · · · = ωq = 0. The bracket on g0 is defined by [P, Q]ω = LP (ω) Q + (−1)q+1 P (dω)Q. Then g0 is a Leibniz algebra and the ‘anchor’ ρ : P 7→ P (ω) is a Leibniz algebra homomorphism whose image is the space of vector fields tangent to the foliation. Take g = Imageρ. Thus, we have an exact sequence which is a central extension in the sense that [P, Q] = 0 for any Q ∈ g0 if P ∈ ker ρ, 0 → ker ρ → g0 → g → 0. An element in ker ρ is a q + 1-vector field contained in the ideal generated by Γ (Λ2 g). The corresponding 2-cocycle can be described as follows. Choose a q-vector field W which satisfies hω, W i ≡ 1 and take a 1-form γ satisfying dω = γ ∧ ω. The integrability assures that we have such a 1-form γ. The 2-cocycle on g corresponding to the above extension, which takes values in ker ρ is defined by ψ(X, Y ) = LX W + γ(X)W. Example 10. As in the previous example, we assume that ω1 and ω2 are transverse volume forms of (regular) foliations on M , of codimension q1 and q2 , respectively. Also,we assume ω1 and ω2 are transverse to each other and ω1 ∧ω2 defines a foliation of codimension q1 + q2 . We have the following bundle map which is the contraction by ω1 Iω

Λq1 +q2 +1 T(M ) →1 Λq2 +1 T(M ). This induces a Leibniz algebra homomorphism I˜ω

Γ (Λq1 +q2 +1 T(M )) →1 Γ (Λq2 +1 T(M )). It can be seen that ker I˜ω1 is consisting of q1 + q2 + 1-vector fields on M which are in the ideal (in the exterior algebra) generated by Γ (Λq2 +2 T(M )). Taking g0 = Γ (Λq1 +q2 +1 T(M )) and g = ImageI˜ω1 , we have an abelian extension of Leibniz algebras 0 → ker I˜ω1 → g0 → g → 0. The corresponding 2-cocycle is given by the following formula ψ(X, Y ) = LX(ω2 ) W1 ∧ Y + γ1 ((X(ω2 ))W1 ∧ Y

X, Y ∈ g,

where W1 is a q1 -vector field satisfying hω1 , W1 i ≡ 1 and γ1 is a 1-form satisfying dω1 = γ1 ∧ ω1 . Likewise as in the previous example, we can verify that ψ is a well-defined 2-cocycle having the values in ker I˜ω1 .

82


4.2 Super Leibniz algebra and its cochain complex We can rewrite super Jacobi identity of the Schouten bracket in the following form: [[P, Q]S , R]S = [P, [Q, R]S ]S − (−1)(p−1)(q−1) [Q, [P, R]S ]S for P ∈ Γ (Λp T(M )), Q ∈ Γ (Λq T(M )), and R ∈ Γ (Λr T(M )). Taking the above formula as a model, we generalize the notion of Leibniz algebras to that of graded (or super) Leibniz algebras. P Definition 11. A graded vector space g := j gj is a graded Leibniz algebra if there is a R-bilinear multiplication (we denote it by [·, ·] again) which satisfies the next two conditions. deg([x, y]) = deg(x) + deg(y) [[x, y], z] = [x, [y, z]] − (−1)deg(x) deg(y) [y, [x, z]] where deg(x) means the grade of x, i.e., x ∈ gdeg(x) . Example 11. If we consider the exterior algebra bundle of T(M ) and define deg(P ) := p − 1 (the reduced degree of P ) for P ∈ Γ (Λp T(M )), then we obtain a typical example of graded Leibniz algebras. We also construct super Leibniz algebras from Leibniz algebras just following the definition of Schouten bracket as follows. Example 12. Let g be a Leibniz algebra with the multiplication [·, ·]. Condim Xg 0 ∧j g, and define the degrees by deg(x) := sider the exterior algebra g := j=1

0 for each x ∈ g and deg(P ) := p − 1 for P ∈ ∧p g. In particular, deg(P ∧ Q) = deg(P ) + deg(Q) + 1 for homogeneous elements P and Q. It is convenient to consider the degree of ‘∧’ to be +1. Note that Q ∧ P = (−1)(deg(Q)+1)(deg(P )+1) P ∧ Q. A new multiplication on g0 is defined by [[x1 ∧ · · · xp , y1 ∧ · · · yq ]] :=

p X q X i

(−1)i+j [xi , yj ] ∧ x1 ∧ · · · xbi · · · y1 ∧ · · · ybj · · · ∧ yq

j

for xi , yj ∈ g (p, q ≥ 1). We extend it R-bi-linearly and we see that this satisfies Leibniz property. Indeed as we expected, we have the next three properties: [[x, y]] :=[x, y]

(x, y ∈ g)

[[P, Q ∧ R]] :=[P, Q] ∧ R + (−1)deg(P )(deg(Q)+1) Q ∧ [[P, R]] [[P ∧ Q, R]] :=P ∧ [Q, R] + (−1)(1+deg(Q) deg(R) [[P, R]] ∧ Q


83

As a complete analogue of Leibniz modules, we can define graded Leibniz modules. Definition 12. Let g be a P graded Leibniz algebra with the multiplication [·, ·]. A graded module A = j∈Z Aj is called (g-)graded Leibniz module if g operates on A on both sides, R-bilinearly (we use the same notations x · a or a · x as before) satisfying the following four conditions for the homogeneous elements. deg(x · a) = deg(a · x) = deg(x) + deg(a) [x, y] · a = x · (y · a) − (−1)deg(x) deg(y) y · (x · a) (x · a) · y = x · (a · y) − (−1)deg(x) deg(a) a · [x, y] (a · x) · y = a · [x, y] − (−1)deg(a) deg(x) x · (a · y) Definition 13. Let g be a graded Leibniz algebra and A a graded g-Leibniz bi-module. For each non-negative integer k, a k-multilinear map ψ from g ⊗ · · · ⊗ g to A is called homogeneous of degree p if it satisfies | {z } k−times

ψ(gd1 ⊗ · · · ⊗ gdk ) ⊂ Ap+d1 +···+dk

(∀dj ∈ Z,

j = 1 . . . k)

for a integer p. p is called the degree of ψ and denoted by deg(ψ). Now we define a R-multilinear operator by (δψ)(x1 , . . . , xk+1 ) :=(−1)k+1 ψ(x1 , . . . , xk ) · xk+1 +

k X

(−1)1+i+deg(xi )(deg ψ+

i=1

+

X

(−1)i+deg(xi )

i<j

P

ir ‘very off-diagonal’ spec(HimP ) = ∅         (k) spec(HkerP ) = R+  (k) spec(HGrP ) = {0 = λ1 < λ2 < · · · < λk }     1≤k≤r

k=0

(k)

   

    

spec(HimP ) = {0 = λk+1 < λk+2 < · · · < λr } (k) spec(HkerP )

= R+

 (0)  spec(HGrP )=∅

 





(0)

spec(HimP +kerP ) = R≥0 ∪ {λ− < 0}

   

‘slightly off-diagonal’

‘diagonal’

(96) These findings are visualized for r=4 in figure 4, with the following legend: double line solid segment dashed line

= b = b = b

negative eigenvalue zero eigenvalue admissible zero mode

−→ −→ −→

single instability (2r−1)C moduli phase modulus

Figures 5 and 6 show a numerical spectrum of the H (k) with cut-off size 30, also for the background Φ4 . Here, the legend is: boxes stars crosses circles

= b = b = b = b

Gr(P ) eigenvalues imP eigenvalues (k6=0) kerP modes (k6=0) diagonal modes (k=0)

−→ −→ −→ −→

# = min(r, k) # = max(r−k, 0) R+ continuum R≥0 ∪ {λ− }

192

Olaf Lechtenfeld

Fig. 4. Decomposition of perturbation around Φ4

15 12.5 10 7.5 5 2.5

Fig. 5. Discrete spectrum of H for Φ4

60

50

40

30

20

10

Fig. 6. Continuous spectrum of H for Φ4

Noncommutative Solitons

193

6.3 Single negative eigenvalue The numerical analysis for abelian diagonal backgrounds Φr revealed a single negative eigenvalue λ− among the diagonal fluctuations. It is found by diagonalizing the k=0 part of the Hessian,   1 −1 −1 3 −2      −2 5 −3     .. ..   . −3 .     . (0) .  , . 2r−3 −r+1 (97) (Hm` ) =     −r+1 −1 −r      −r +1 −r−1   .    −r−1 2r+3 . .   .. .. . . where I have emphasized in boldface the entries modified by the background. The result is indeed that spec(H (0) ) = {λ− } ∪ [0, ∞),

(98)

where λ− is computed as the unique negative zero of the determinant 1 Z ∞ −x Ir−1,r−1 (λ) − 2r Ir−1,r (λ) e dx Lk (x) Ll (x) with Ik,l (λ) := 1 x−λ 0 Ir,r−1 (λ) Ir,r (λ) − 2r

(99) being variants of the integral logarithm. The r complex zero eigenvalues of HGrP arise from turning on the location moduli αi of (35), while the r−1 complex zero eigenvalues of HimP point at non-Grassmannian classical solutions. Since H (0) is not non-negative, δ 2 E[Φ, φ(0) ] may vanish even if H (0) φ(0) 6= 0. 6.4 Instability in unitary sigma model The fluctuations φGrP are tangent to GrP ≡ Gr(r, H) and cannot lower the energy, as the BPS argument (31) had assured me from the beginning. Therefore, all solitons of Grassmannian sigma models are stable. On the other hand, an unstable mode of H occurred in imP ⊕kerP , indicating a possibility to continuously lower the energy E = 8πr of Φr along a path starting perpendicular to GrP . Indeed, there exists a general argument for any static soliton Φ = −2P inside the unitary sigma model, commutative or noncommutative. It goes as follows. Given a projector inclusion Pe ⊂ P (including Pe = 0), i.e. a ‘smaller’ projector Pe of rank re < r. Then, the path [Zak89] e

Φ(s) = ei s (P −P ) ( −2P ) =

− (1+ei s )P − (1−ei s )Pe

(100)

194

Olaf Lechtenfeld

connecting

Φ(0) = Φ = −2P

to

e = −2Pe Φ(π) = Φ

(101)

interpolates between static solitons in different Grassmannians inside U(H). Please note that the tangent vector (∂s Φ)(0) = −i(P −Pe ) is not an eigenmode of the Hessian. A quick calculation gives the energy along the path, 1 8π E[Φ(s)]

=

r+e r 2

+

r−e r 2 cos s

= r cos2

s 2

+ re sin2 2s .

(102)

For nonabelian noncommutative solitons the argument persists, with the topoe replacing r and re. Therefore, all solitons in unitary logical charges Q and Q sigma models eventually decay to the ‘vacua’ Q = 0, which belong to the constant (nonabelian) projectors.

7 d = 1+1 sine-Gordon solitons 7.1 Reduction to d = (1+1)θ : instantons In the remaining part of this lecture I look at the reduction from 1+2 to 1+1 dimensions, with the goal to generate new noncommutative solitons. However, naive reduction of the Ward solitons is not possible. Due to shape invariance, ∂s = 0, the one-soliton sector is already two-dimensional (in the rest-frame) but with Euclidean signature: ∂s = 0

↔

∂u + µ¯ µ ∂v − (µ+¯ µ) ∂x = 0

∂x = ν ∂w + ν¯ ∂w¯ , (103) hence I cannot simply put ∂x = 0 without killing the soliton entirely. Instead, the x dependence may be eliminated by taking the snapshot φ(x=0, y, t). Then, ∂s = 0 maps the remaining ty plane to the ww¯ plane as illustrated in figure 7. Because for vx 6=0 the soliton worldline pierces the xy plane as shown t

↔

s

x’ −> v

x

Fig. 7. Action of reduction ∂s = 0

in figure 8, the x=0 slice of the soliton is just an instanton!


195

y

l so

ito

n

instanton

− v>

x

Fig. 8. x=0 instanton snapshot of soliton

7.2 d = 1+1 sigma model metric Due to the x-derivatives in the Ward equation (41) the snapshot φ(x=0, y, t) will not satisfy this equation. Using (103) I find that instead it obeys the equation (1− µµ¯ ) ∂w (φ† ∂w¯ φ) − (1− µµ¯ ) ∂w¯ (φ† ∂w φ) = 0, (104) which is an extended sigma-model equation in 1+1 dimensions due to (w, w) ¯ ∼ (t, y). Comparison with (h(ij) + b[ij] ) ∂i (φ† ∂j φ) = 0 yields the metric  tt h  hyt

hty hyy

for i, j ∈ {t, y}

  1 | µ −µ|2 | µ1 |2 − |µ|2 µ¯ µ    = (µ+¯ µ)2 | 1 |2 − |µ|2 | 1 +µ|2 µ µ

(105)



(106)

and the magnetic field btt

bty

byt

byy

! =

0 1 −1 0

! .

(107)

The notation suggests a Minkowski signature, but a short computation says that µ 2 ≥ 0, (108) det(hij ) = Im Re µ hence the metric is Euclidean! Indeed, this very fact permits the Fock-space realization of the Moyal deformation, which follows.

196

Olaf Lechtenfeld

7.3 Moyal deformation in d = 1+1 In the present case I have no choice but to employ the time-space deformation [t, y]? = iθ

=⇒

[w, w] ¯ ? = 2θ

for µ ∈ / R or i R.

As before, I realize this algebra via the Moyal-Weyl correspondence √ w ↔ 2θ c such that [ c , c† ] =

(109)

(110)

on the standard Fock space H. In this way, the moving U? (1) soliton (59) becomes a gaussian instanton in the d = 1+1 U? (1) sigma model, after reexpressing w = w(y, t). The only exception occurs for vx = 0 (⇔ µ ∈ i R), i.e. motion in y direction only, because (51) then implies that [w, w] ¯ ? = 0. In fact, (47) shows (for x=0) that w ¯ ∼ w in this case, the rest frame degenerates •• to one dimension and there is no room left for a Heisenberg algebra. ∠ _ 7.4 Reduction to d = (1+1)θ : solitons So far, my attempts to construct noncommutative solitons in 1+1 dimensions by reducing such solitons in a d=1+2 model have failed. The lesson to learn is that the dimensional reduction must occur along a spatial symmetry direction of the d=1+2 configuration, i.e. along its worldvolume. In other words, the starting configuration should be spatially extended, or a d=1+2 noncommutative wave! Luckily, such wave solutions exist in the nonabelian Ward model [Lee89, Bie02]. Let me warm up with the commutative case and the sigma-model group of U(2). The Ward-model wave solutions Φ(u, v, x) dimensionally reduce to d=1+1 WZW solitons g(u, v) via Φ(u, v, x) = E eiα x σ1 g(u, v) e−iα x σ1 E †

for g(u, v) ∈ U(2)

(111)

and a constant 2×2 matrix E. The Ward equation for Φ descends to ∂v (g † ∂u g) + α2 (σ1 g † σ1 g − g † σ1 g σ1 ) = 0.

(112)

In a second step, I algebraically reduce g from U(2) to being U(1)-valued, allowing for an angle parametrization, i

g = e 2 σ3 φ .

(113)

The algebra of the Pauli matrices then simplifies (112) to ∂v ∂u φ + 4α2 sin φ = 0 which is nothing but the familiar sine-Gordon equation!

(114)


197

7.5 Integrable noncommutative sine-Gordon model Now I introduce the time-space Moyal deformation [t, y]? = iθ

[u, v]? = − 2i θ.

⇐⇒

(115)

The sine-Gordon kink must move in the y direction, which (we have learned) forbids a Heisenberg algebra (note the i above). Thus, no Fock-space formulation exists and I must content myself with the star product. Recalling the dimensional reduction (111) and (112) I must now solve ∂v (g † ? ∂u g) + α2 (σ1 g † ? σ1 g − g † σ1 ? g σ1 ) = 0.

(116)

The algebraic reduction U(2) → U(1) turns out to be too restrictive. In the i commutative case, the overall U(1) phase factor e 2 ρ of g decouples in (112), so I could have started directly with g ∈ SU(2) instead. In the noncommutative case, in contrast, this does not happen, and I am forced to begin with U? (2). Thus, I should not prematurely drop the overall phase and algebraically reduce g to U? (1) × U? (1), i

g(u, v) = e?2

ρ(u,v)

i

? e?2

σ3 ϕ(u,v)

(117)

.

With this, the 2×2 matrix equation (116) turns into the scalar pair − 2i ϕ

∂ v e?

i 2ϕ

∂ v e? ?

i

? ∂u e?2

ϕ

− i ϕ ∂ u e? 2

with the abbreviation

i −iϕ ϕ + 2iα2 sin? ϕ = −∂v e? 2 ? R ? e?2

2

i 2ϕ

− 2iα sin? ϕ = −∂v e? ? R ? − 2i ρ

R = e?

i

? ∂u e?2

− i ϕ e? 2

ρ

(118)

(119)

carrying the second angle ρ. For me, (118) are the noncommutative sineGordon (NCSG) equations. As a check, take the limit θ → 0, which indeed yields ∂v ∂u ρ = 0 and ∂v ∂u ϕ + 4α2 sin ϕ = 0. (120) 7.6 Noncommutative sine-Gordon kinks As an application I’d like to construct the deformed multi-kink solutions to the NCSG equations (118), e.g. via the associated linear system. First, consider the one-kink configuration, which obtains from the wave solution of the U? (2) Ward model by choosing x=0

as well as

µ = ip ∈ iR

Consequently, the co-moving coordinate becomes

=⇒

ν = 1.

(121)

198

Olaf Lechtenfeld

=: −i η. w = µ ¯u + µ1¯ v = −i (p u + 1p v) = −i √y−vt 1−v2

(122)

The BPS solution of the reduced Ward equation (116) is g = σ3 ( −2P )

with projector

P = T?

1 T † ?T

? T †,

(123)

where the 2×1 matrix function T (η) is subject to (∂η + α σ3 ) T (η) = 0.

(124)

Modulo adjusting the integration constant and (irrelevant) scaling factor, the general solution reads T =

e−αη

i eαη tanh 2αη

g =

i cosh 2αη

−2αη e −i 1 P = , 2 cosh 2αη i e+2αη

=⇒

i cosh 2αη

!

tanh 2αη

!

= E

i 2

i 2

ρ

e? ? e ? 0

ϕ

0 i 2

ρ

e? ?

−i ϕ e? 2

(125)

! E †.

7.7 One-kink configuration Since the expressions above depend on u and v only in the rest-frame combination η, it is clear that the deformation becomes irrelevant here, and the one- kink sector is commutative, effectively θ = 0 and ρ = 0. With E = √12 11 −11 the latest equation is solved by cos ϕ2 = tanh 2αη

tan ϕ4 = e−2αη (126) 2 which is precisely the standard sine-Gordon kink with velocity v = 1−p . 1+p2 With hindsight this was to be expected, since a one-soliton configuration in 1+1 dimensions depends on a single (real) co-moving coordinate. The deformation should reappear, however, in multi-soliton solutions. For instance, breather and two-soliton configurations seem to get deformed since pairs of rest-frame coordinates are subject to p [ηi , ηk ]? = −i θ (vi −vk ) (1−vi2 )(1−vk2 ). (127) and

sin ϕ2 =

1 cosh 2αη

=⇒

7.8 Tree-level scattering of elementary quanta Finally, it is of interest to investigate the quantum structure of noncommutative integrable theories, i.e. take into account the field excitations above the classical configurations. In my noncommutative sine-Gordon model (118) the elementary quanta are ϕ and ρ, and the Feynman rules for their scattering do get Moyal deformed. For illustrative purposes I concentrate on the ϕϕ → ϕϕ scattering amplitude in the vacuum sector. The kinematics of this process is


k1 = (E, p) ,

k2 = (E, −p) ,

k3 = (−E, p) ,

k4 = (−E, −p),

199

(128)

subject to the mass-shell condition E 2 − p2 = 4α2 . The action (which I did not present here) is non-polynomial; it contains hϕϕρi,

hρρρi,

hϕϕϕϕi,

hϕϕρρi,

hρρρρi

(129)

as elementary three- and four-point interaction vertices. Denoting ϕ propagators by solid lines and ρ propagators by dashed ones, there are the following four contributions to the ϕϕ → ϕϕ amplitude at tree level: 1

1 2

2

= − 2i p2 sin2 (θEp)

= 2iα2 cos2 (θEp)

4 3

1

4

2

3

1

2

= 2i E 2 sin2 (θEp) 3

4

= 0. 4

3

Taken together this means that Aϕϕ→ϕϕ = 2iα2

(130)

is causal. I can show that all other 2 → 2 tree amplitudes vanish. Hence, any θ dependence seems to cancel in the tree-level S-matrix! Furthermore, it can be established that there is no tree-level particle production in this model, just like in the commutative case. Although at tree-level I still probe only the classical structure of the theory, the absence of a deformation until this point is conspicuous: Could it be that the time-space noncommutativity in the sine-Gordon system is a fake, to be undone by a field redefinition? With this provoking question I close the lecture.

References [KS02]

[DN02] [Sza03] [Har01]

Konechny, A., Schwarz, A.S.: Introduction to matrix theory and noncommutative geometry, Phys. Rept., 360, 353 (2002) [hep-th/0012145, hep-th/0107251] Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory, Rev. Mod. Phys., 73, 977 (2002) [hep-th/0106048] Szabo, R.J.: Quantum field theory on noncommutative spaces, Phys. Rept., 378, 207 (2003) [hep-th/0109162] Harvey, J.A.: Komaba lectures on noncommutative solitons and D-branes, hep-th/0102076

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[Ham03] [Sza05] [War88]

Hamanaka, M.: Noncommutative solitons and D-branes, hep-th/0303256 Szabo, R.J.: D-branes in noncommutative field theory, hep-th/0512054 Ward, R.S.: Soliton solutions in an integrable chiral model in 2+1 dimensions, J. Math. Phys., 29, 386 (1988) [War90] Ward, R.S.: Classical solutions of the chiral model, unitons, and holomorphic vector bundles, Commun. Math. Phys., 128, 319 (1990) [IZ98a] Ioannidou, T.A., Zakrzewski, W.J.: Solutions of the modified chiral model in 2+1 dimensions, J. Math. Phys., 39, 2693 (1998) [hep-th/9802122] [LP01a] Lechtenfeld, O., Popov, A.D.: Noncommutative multi-solitons in 2+1 dimensions, JHEP, 0111, 040 (2001) [hep-th/0106213] [LP01b] Lechtenfeld, O., Popov, A.D.: Scattering of noncommutative solitons in 2+1 dimensions, Phys. Lett. B, 523 178 (2001) [hep-th/0108118] [LMPPT05] Lechtenfeld, O., Mazzanti, L., Penati, S., Popov, A.D., Tamassia, L.: Integrable noncommutative sine-Gordon model, Nucl. Phys. B, 705, 477 (2005) [hep-th/0406065] [DLP05] Domrin, A.V., Lechtenfeld, O., Petersen, S.: Sigma-model solitons in the noncommutative plane: Construction and stability analysis, JHEP, 0503, 045 (2005) [hep-th/0412001] [CS05] Chu, C.S., Lechtenfeld, O.: Time-space noncommutative abelian solitons, Phys. Lett. B, 625, 145 (2005) [hep-th/0507062] [KLP06] Klawunn, M., Lechtenfeld, O., Petersen, S.: Moduli-space dynamics of noncommutative abelian sigma-model solitons, hep-th/0604219 [Der64] Derrick, G.H.: Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., 5, 1252 (1964) [Wol02] Wolf, M.: Soliton antisoliton scattering configurations in a noncommutative sigma model in 2+1 dimensions, JHEP, 0206, 055 (2002) [hep-th/0204185] [Man82] Manton, N.S.: A remark on the scattering of BPS monopoles, Phys. Lett. B, 110, 54 (1982) [MS04] Manton, N.S., Sutcliffe, P.: Topological solitons. Cambridge University Press (2004) [IZ98b] Ioannidou, T.A., Zakrzewski, W.J.: Lagrangian formulation of the general modified chiral model, Phys. Lett. A, 249, 303 (1998) [hep-th/9802177] [DM05] Dunajski, M., Manton, N.S.: Reduced dynamics of Ward solitons, Nonlinearity, 18 1677 (2005) [hep-th/0411068] [LRU00] Lindstr¨ om, U., Roˇcek, M., von Unge, R.: Non-commutative soliton scattering, JHEP, 0012 004 (2000) [hep-th/0008108] [HLRU01] Hadasz, L., Lindstr¨ om, U., Roˇcek, M., von Unge, R.: Noncommutative multisolitons: Moduli spaces, quantization, finite theta effects and stability, JHEP, 0106 040 (2001) [hep-th/0104017] [GHS03] Gopakumar, R., Headrick, M., Spradlin, M.: On noncommutative multisolitons, Commun. Math. Phys., 233 355 (2003) [hep-th/0103256] [Zak89] Zakrzewski, W.J.: Low dimensional sigma models. Adam Hilger (1989) [Lee89] Leese, R.: Extended wave solutions in an integrable chiral model in 2+1 dimensions, J. Math. Phys., 30, 2072 (1989) [Bie02] Bieling, S.: Interaction of noncommutative plane waves in 2+1 dimensions, J. Phys. A, 35, 6281 (2002) [hep-th/0203269]

Non-anti-commutative Deformation of Complex Geometry Sergei V. Ketov Department of Physics, Tokyo Metropolitan University, Hachioji-shi, Tokyo 192–0397, Japan; [email protected]

Abstract In this talk I review the well known relation existing between extended supersymmetry and complex geometry in the non-linear sigma-models, and then briefly discuss some recent developments related to the introduction of the non-anti-commutativity in the context of the supersymmetric non-linear sigma-models formulated in extended superspace. This contribution is suitable for both physicists and mathematicians interesting in the interplay between geometry, supersymmetry and non(anti)commutativity.

1 Introduction Being a theoretical physicist, one gets used to mathematical tools, whose role in modern theoretical high-energy physics is indispensable and indisputable. Especially when the experimental base is limited or does not exist, the use of advanced mathematics to get new insights in physics is particularly popular. So it is no surprise that mathematical knowledge of theoretical physicists is quite high. However, the way of dealing with mathematics in theoretical physics is different from that commonly used by mathematicians, and even the motivation and goals are different, as is also well known. I would like to draw attention to another, less known fact that physical considerations may sometimes lead to new mathematics, or rediscovery of some famous mathematical facts. In my contribution to this workshop, aimed towards more cooperation and understanding between physicists and mathematicians, I would like to explain how investigation of supersymmetry in field theory of the non-linear sigma-models might have led to rediscovery of complex geometry and related mathematical tools. In addition, I would like to explain how introducing more fundamental structure (namely, non-anti-commutativity in superspace) leads 201

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to a new deformation of complex geometry, whose geometrical significance is yet to be understood. The paper is organized as follows. In sect.2 the basic notions of the nonlinear sigma-models are introduced. My presentation is ‘minimal’ on purpose, without going into details and/or many generalizations that might be easily added. I just summarize the basic ideas. In sect. 3, I introduce a simple superspace, and review the known relation between extended supersymmetry and complex geometry in the non-linear sigma-models, by getting all basic notions of complex geometry from a single and straightforward fieldtheoretical calculation. In sect. 4 some extended superspace techniques, making extended supersymmetry to be manifest, are briefly discussed. In sect. 5 some more superspace structure is added by introducing the notion of NonAnti-Commutativity (NAC) or ‘quantum superspace’, and its impact on the non-linear sigma-model target space is calculated. The simplest non-trivial explicit example of the NAC-deformed CP (1) metric is given in sect. 6. Our conclusion is sect. 7.

2 Non-linear sigma-models The Non-Linear Sigma-Model (NLSM) is a scalar field theory whose (multicomponent) scalar field φa (xµ ) is defined in a d-dimensional ‘spacetime’ or a ’worldvolume’ parametrized by local coordinates {xµ }, µ = 1, 2, . . . , d. The fields φa take their values in a D-dimensional Riemannian manifold M , called the NLSM target space, a = 1, 2, . . . , D. The NLSM field values φa can thus be considered as a set of (local) coordinates in M , whose metric is fielddependent. The NLSM format is the very general field-theoretical concept whose geometrical nature is the main reason for many useful applications of NLSM in field theory, string theory, condensed matter physics and mathematics (see e.g., the book [1] for much more). We assume the NLSM spacetime or worldvolume to be flat Euclidean space Rd , for simplicity, so that the NLSM action is supposed to be invariant under translations (with generators Pµ ) and rotations (with generators Mµν ) in Rd . Let ds2 = gab (φ)dφa dφb be a metric in M . Then a generic NLSM action is given by Z 1 Sbos.dbφ] = dd x L(∂µ φ, φ) , L = gab (φ)δ µν ∂µ φa ∂ν φb + m2 V (φ) , (1) 2 where summation over repeated indices is always implied. The function V (φ) is called a scalar potential in field theory with a mass parameter m. The higher derivatives of the field φ are not allowed in the Lagrangian L, with the notable exception of d = 2 where an extra (Wess-Zumino) term may be added to eq. (1): 1 LWZ = bab (φ)εµν ∂µ φa ∂ν φb . (2) 2

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The 2-form B = bab (φ)dφa ∧ dφb in eq. (2), is called a torsion potential in M , by the reason to be explained in the next sect. 1.3. In string theory, it is called a B-field (or a Kalb-Ramond field).

3 Supersymmetric NLSM There are two different ways to supersymmetrize the NLSM: either in the worldvolume, or in the target space. Here we only discuss the worldvolume supersymmetrization of NLSM, in the case of even d. 1 Then adding supersymmetry amounts to the extension of the Euclidean space motion group SO(d) × T d to a supergroup, with the key superalgebra relation ¯ • }+ = 2σ µ • Pµ δ i j , {Qiα , Q βj

(3)

αβ

¯ are chiral and anti-chiral where the additional fermionic supercharges Q and Q spinors of SO(d), repectively, in the fundamental representation of the internal U (N ) symmetry group, denoted by latin indices i, j = 1, 2, . . . , N . The N here is a number of supersymmetries, so that the N > 1 supersymmetry is called the extended one. The chiral σ-matrices in eq. (3) obey Clifford algebra in d dimensions. As regards the NLSM, it is not difficult to demonstrate by using only group-theoretical arguments that d ≤ 6 [1], and when d=2 then N ≤ 4 [2]. The model-independent technology for a construction of off-shell manifestly supersymmetric field theories is called superspace. To give an example, let us consider the simplest case of the N = 1 supersymmetric NLSM in d = 2. The two-dimensional complex coordinates z and z¯ can be extended by the anti-commuting (Grassmann) fermionic (spinor) coordinates θ and θ¯ to ¯ Tensor functions in superspace are called superform a superspace (z, z¯, θ, θ). fields. A superfield is always equivalent to a supermultiplet of the usual fields, e.g. ¯ = φ + θψ + θ¯ψ¯ + θθF ¯ Φ(z, z¯, θ, θ) , (4) in terms of the bosonic field components φ(z, z¯) and F (z, z¯), and the fermionic ¯ z¯). field components ψ(z, z¯) and ψ(z, The supercharges can be easily realized in superspace as the differential operators ∂ ¯ = ∂ − θ¯∂¯ , Q= − θ∂ , and Q (5) ∂θ ∂ θ¯ where we have introduced the notation ∂ = ∂z and ∂¯ = ∂z¯. It is not difficult to check that the covariant derivatives 1

In string theory, the world-sheet supersymmetrization is known as the NeveuSchwarz-Ramond (NSR) approach, whereas the target space supersymmetrization is called the Green-Schwarz (GS) approach.

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D=

∂ + θ∂ , ∂θ

¯ = ∂ + θ¯∂¯ , D ∂ θ¯

∂

and

∂¯ ,

(6)

all (anti)commute with the supercharges (5) indeed, which allows us to use them freely in the covariant superspace action. Then the unique N=1 supersymmetric extension of the bosonic NLSM action (1) is easily constructed in N=1 superspace as follows (we ignore a scalar potential here): Z ¯ b, S1 = d2 xd2 θ (gab + bab ) DΦa DΦ (7) where both gab (Φ) and bab (Φ) are now functions of the superfields Φa of eq. (4). The component fields F appear to be non-propagating (they are called to be auxiliary), since they satisfy merely algebraic equations of motions. They are supposed to be substituted by solutions to their ’equations of motion’. Having evaluated the Berezin integral in eq. (7), one gets eq. (1) as the only purely bosonic contribution that is modified by the fermionic terms, namely, by a sum of the covariant Dirac term and the quartic fermionic interaction whose field-dependent couplings are given by the curvature tensor with torsion. The B-field, in fact, enters the field action S1 only via its curl (= torsion in M ) (8) T a bc = − 32 g ad b[bc,d] , that, in its turn, enters the action only via the connections (in M ) a a Γ ±bc = ± T a bc . bc

(9)

By construction the two-dimensional action S1 is invariant under the N=1 supersymmetry transformations ¯ a, δsusy Φa = εQΦa + ε¯QΦ with QΦa | = ψ a

and

¯ a = ψ¯a , QΦ

(10) (11)

¯ independent) part of a superfield, where | denotes the leading (i.e. θ- and θwhile ε and ε¯ are the infinitesimal fermionic (Grassmann) N=1 supersymmetry transformation parameters. Eq. (11) can serve as the definition of the fermionic superpartners of the bosonic NLSM field φa . It is not difficult to generalize the NLSM (7) by adding a scalar superpotential in superspace, Z Spot. = m d2 xd2 θ W (Φ) , (12) with a arbitrary real function W (Φ) and a mass parameter m. It gives rise to the scalar potential


205

1 2 ab m g (φ)∂a W (φ)∂b W (φ) , 2

(13)

V (φ) =

while it does not modify the NLSM kinetic terms, as is already clear from dimensional reasons. A generic two-dimensional NLSM with an arbitrary Riemannian target space M (and no scalar potential) can always be N=1 supersymmetrized, as in eq. (7). When M is a Lie group manifold, there is a preferred (groupinvariant) choice for its metric and torsion, while such NLSM is called a WessZumino-Novikov-Witten (WZNW) model [1]. One may also introduce the socalled gauged WZNW models with a homogeneous target space G/H, where H is a subgroup of G. In differential geometry, it corresponds to the quotient construction [1]. The next relevant question is: which restrictions on the NLSM target space M , in fact, imply more supersymmetry, i.e. N > 1 ? To answer that question, all one needs is to write down the most general Ansatz for the second supersymmetry transformation law (it follows by dimensional reasons) in terms of the N=1 superfields as ¯ b, δ2 Φ = ηJ a b (Φ)DΦb + η¯J¯a b (Φ)DΦ

(14)

and then impose the invariance condition δ 2 S1 = 0 .

(15)

In equation (14), the η and η¯ are the infinitesimal parameters of the second supersymmetry, while J a b (Φ) and J¯a b (Φ) are some tensor functions to be fixed by eq. (15). It is straightforward (though tedious) to check that the condition (14) amounts to the following restrictions (see e.g., ref. [3]): a − ¯a ∇+ c J b = ∇c J b = 0 ,

and gbc J c a = −gac J c b ,

gbc J¯c a = −gac J¯c b .

(16) (17)

In addition, one gets the standard (on-shell) N=2 supersymmetry algebra (3) provided that (see e.g., ref. [3]) ¯ J] ¯ =0, J 2 = J¯2 = −1 and N a bc [J, J] = N a bc [J,

(18)

where we have introduced the Nijenhuis tensor N a bc [A, B] = Ad [b B a c],d + Aa d B d [b,c] + B d [b Aa c],d + B a d Ad [b,c] .

(19)

So we can already recognize (or re-discover) the basic notions of (almost) complex geometry, such as an (almost) complex structure, a hermitean metric, a covariantly constant (almost) complex structure, and an integrable complex structure (see e.g., ref. [4]). To be precise, we get the following theorem:

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a two-dimensional N=1 supersymmetric NLSM is actually (on-shell) N=2 supersymmetric, if and only if (1) it allows two (almost) complex structures, ¯ (2) the NLSM metric is hermitean with respect to each of them, and J and J, (3) each (almost) complex structure is covarianlty constant with respect to the asociated (±) connection, so that it is actually integrable. The integrability here means the existence of holomoprhic and antiholomorphic coordinates (i.e. the holomorphic transitions functions) after rewriting a complex structure to the diagonal form (with the eigenvalues i and −i). It should be noticed that the complex structures J and J¯ may not be ¯e 6= 0, because they are covariantly constant commuting with each other, dbJ, Jc with respect to the different connections in eq. (9), respectively. For instance, the mixed N=2 supersymmetry commutator ¯ c , ¯ea b DDΦ ¯ b + Γ b −cd DΦd DΦ dbδ(η), δ(¯ η )ceΦa = η η¯dbJ, Jc (20) is required to be vanishing by the N=2 supersymmetry algebra (3). It is ¯ b+ already true on-shell, i.e. when the NLSM equations of-motions, D DΦ b d ¯ c Γ −cd DΦ DΦ = 0, are satisfied, though it is also the case off-shell only if ¯e = 0. The complex structures J and J¯ may not therefore be simultanedbJ, Jc ously integrable, in general. If, however they do commute, then the existence of an off-shell N=2 extended superspace formulation of such N=2 NLSM with manifest N=2 supersymmetry is guaranteed.

4 NLSM in extended superspace To give the simplest example of the N=2 extended superspace in two dimensions (z, z¯), let’s introduce two (Grassmann) fermionic coordinates for each chirality, i.e. (z, θ + , θ− ) and (¯ z , θ¯+ , θ¯− ), and then the N=2 supercharges Q± ¯ ± , the N=2 superspace covariant derivatives D± and D ¯ ± , and N=2 and Q scalar superfields Φi (z, z¯, θ+ , θ− , θ¯+ , θ¯− ), like in the N=1 case (see the previous section), where now i = 1, 2, . . . , m. However, there is the immediate problem: a general (unconstrained) N=2 scalar superfield has a physical vector field component that is not suitable for the NLSM. The simplest way to remedy that problem is to use the (off-shell) N=2 chiral and anti-chiral superfields, subject to the constraints ¯ ±Φ = 0 D

and

D± Φ¯ = 0 ,

(21)

respectively. Their most general NLSM action is then given by Z Z Z ¯ + m d2 xd2 θ W (Φ) + m d2 xd2 θ¯ W ¯ (Φ) ¯ , (22) S2 = d2 xd2 θd2 θ¯ K(Φ, Φ) ¯ and a holomorphic in terms of a non-holomorphic kinetic potential K(Φ, Φ) superpotential W (Φ).


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A simple straightforward calculation of the NLSM metric out of eq. (22) reveals a K¨ ahler metric gi¯j = ∂i ∂¯j K with the K¨ ahler potential K, and no torsion. Therefore, K¨ ahler complex geometry could have been also re-discovered from the N=2 supersymmetric NLSM. When one adds the so-called twisted chiral N=2 superfields, subject to the following off-shell N=2 superspace constraints: ¯ + Φ˜ = D− Φ˜ = 0 and D ¯ − Φ˜ = D+ Φ˜ = 0 , D their most general N=2 superspace action, Z ˜ + obvious superpotential terms , ¯ Φ, ˜ Φ) S2,T = d2 xd2 θd2 θ¯ K(Φ, Φ,

(23)

(24)

would give rise to a non-trivial torsion too, though with the commuting com¯e = 0 [5]. Actually, the exchange Φ ↔ Φ ˜ corresponds to plex structures, dbJ, Jc the T-duality in string theory. ¯e 6= 0, one To get the most general N=2 supersymmetric NLSM with dbJ, Jc ˆ has to add the so-called semi-chiral (reducible) N=2 superfields, Φˆ and Φ, subject to the off-shell N=2 superspace constraints [6, 3] ¯ +D ¯ − Φˆ = 0 D

and

D+ D− Φˆ = 0 .

(25)

When asking for even more supersymmetry in a two-dimensional supersymmetric NLSM, one gets three linearly independent (almost) complex structures of each chirality, obeying a quaternionic algebra (see e.g., ref. [1]), ±(B)c

Ja±(A)b Jb

= −δ AB δac + εABC Ja±(C)c ,

where A, B, C = 1, 2, 3 , (26)

¯ They all must be covariantly constant, and similarly for J. ∇± J ± = 0 ,

(27)

respectively. In particular, N=3 supersymmetry implies N=4 supersymmetry. Unfortunately, a geometrical description of the two-dimensional N=4 supersymmetric NLSM with torsion is still incomplete. In the case of the vanishing torsion, an N=1 supersymmetric NLSM is, in fact, N=4 supersymmetric if and only if its target space is hyper-K¨ ahler (see e.g., ref. [1] for more details). When a supersymmetric NLSM in question is, in fact, a (gauged) WZNW model, then its N=4 supersymmetry implies that its target space must be a product of Wolf spaces [7, 3]. The Wolf space can be associated with any simple Lie group G. Let ψ be the highest weight root of G, and let (Eψ± , H) be the generators of the su(2)ψ subalgebra of Lie algebra of G (say, in Chevalley basis), associated with ψ. Then the Wolf space is given by the coset W olf space =

G , H⊥ ⊗ SU (2)ψ

(28)

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where we have introduced the SU (2)ψ Lie group of the Lie algebra su(2)ψ and the centralizer H⊥ of the SU (2)ψ in G. An efficient off-shell N=4 superspace description of all two-dimensional N=4 supersymmetric NLSMs does not exist, though the use of harmonic superspace [8] with the infinite number of auxiliary fields may be useful for describing a large class of the manifstly N=4 supersymmetric NLSM. It is also worth mentioning that the chiral generators of supersymmetry in two dimensions are independent, so that it is possible to have an unequal number of ’left’ and ‘right’ supersymmetries. It is called ’heterotic’ or (p, q) supersymmetry. 2 It is always possible to construct the minimal or (1/2, 0) supersymmetric extention of any NLSM. A generic (1/2, 1) supersymmetric NLSM can be formulated in (1/2, 1) superspace. Less is known about other (p, q) supersymmetric NLSM with n = 3, 4. Finally, there is a simple relation between extended supersymmetry and higher (d > 2) dimensions, which is just based on the representation theory of spinors and Clifford algebras in various dimensions. A supersymmetric NLSM can first be formulated in six or four dimensions, and then it can be rewritten to lower dimensions, by simply restricting all its fields to be dependent upon lower number of their worldvolume coordinates (this procedure is called dimensional reduction). The manifestly supersymmetric formulation of a higher-dimensional supersymmetric NLSM often requires the use of sophisticated (constrained) superfields [1]. In quantum theory, only two-dimensional NLSM are renormalizable, while their higher-dimensional counterparts are not. The same is true for the supersymmetric NLSM [1].

5 Non-anticommutative deformation of four-dimensional supersymmetric NLSM Non-Anti-Commutativity (NAC) or quantum superspace [9] is a natural extension of the ordinary superspace, when the fermionic superspace coordinates are assumed to obey a Clifford algebra instead of being Grassmann (i.e. anticommutative) variables [10]. The non-anticommutativity naturally arises in the D3-brane superworldvolume, in the type-IIB constant Ramond-Ramond type background, in superstring theory [11]. In four dimensions, the NAC deformation is given by {θα , θβ }∗ = C αβ , (29) where C αβ can be identified with a constant self-dual gravi-photon background [11]. The remaining N=1 superspace coordinates in the chiral basis (y µ = ¯ µ, ν = 1, 2, 3, 4 and α, β, . . . = 1, 2) can still (anti)commute, xµ + iθσ µ θ, •

•

•

•

dby µ , y ν ec = {θ¯α , θ¯β } = {θα , θ¯β } = dby µ , θα ec = dby µ , θ¯α ec = 0 . 2

It is conventional to set p + q = 2n.

(30)


209

provided we begin with a four-dimensional Euclidean 3 worldvolume having the coordinates xµ . A supersymmetric field theory in the NAC superspace was extensively studied in the recent past, soon after the pioneering paper [13]. The choice (30) preserves locality in a NAC-deformed field theory. The C αβ 6= 0 explicitly break the four-dimensional Euclidean invariance. The NAC nature of θ’s can be fully taken into account by using the (associative, but non-commutatvive) Moyal-Weyl-type (star) product of superfields,   ← αβ ∂ ∂ C  g(θ) , f (θ) ∗ g(θ) = f (θ) exp − (31) 2 ∂θα ∂θβ which respects the N=1 superspace chirality. nomial in the deformation parameter ,

4

The star product (31) is poly-

C αβ ∂f ∂g ∂2f ∂2g − det C 2 2 , α β 2 ∂θ ∂θ ∂θ ∂θ

f (θ) ∗ g(θ) = f g + (−1)degf

(32)

where we have used the identity det C = 12 εαγ εβδ C αβ C γδ ,

(33)

and the notation

∂ ∂ ∂2 = 14 εαβ α β . (34) ∂θ2 ∂θ ∂θ We also use the following book-keeping notation for 2-component spinors: θχ = θα χα ,

•

θ¯χ ¯ = θ¯• χ ¯α , α

θ 2 = θ α θα ,

•

θ¯2 = θ¯• θ¯α . α

(35)

The spinorial indices are raised and lowered by the use of two-dimensional Levi-Civita symbols. Grassmann integration amounts to Grassmann differentiation. The anti-chiral covariant derivative in the chiral superspace basis is • ¯ • = −∂/∂ θ¯α . The field component expansion of a chiral superfield Φ reads D α

Φ(y, θ) = φ(y) +

√

2θχ(y) + θ2 M (y) .

(36)

An anti-chiral superfield Φ in the chiral basis is given by √ ¯ θ) ¯ = φ(y) ¯ + 2θ¯χ(y) ¯ Φ(y µ − 2iθσ µ θ, ¯ + θ¯2 M(y) √ √ ¯ µ φ(y) ¯ ¯ ¯ θ¯2 − i 2σ µ θ∂ + θ2 θ¯2 4φ(y) , + 2θ iσ µ ∂µ χ(y) (37) where 4 = ∂µ ∂µ . The bars over fields serve to distinguish between the ‘left’ and ‘right’ components that are truly independent in Euclidean space. 3 4

The Atiyah-Ward space-time of signature (+, +, −, −) is also possible [12]. We use the left derivatives as a default, the right ones are explicitly indicated.

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The non-anticommutativity Cαβ 6= 0 also explicitly breaks half of the original N=1 supersymmetry [13]. Only the chiral subalgebra generated by the chiral supercharges (in the chiral basis) Qα = ∂/∂θα is preserved, with {Qα , Qβ }∗ = 0, thus defining what is now called N=1/2 supersymmetry. The use of the NAC-deformed superspace allows one to keep N=1/2 supersymmetry manifest. The N=1/2 supersymmetry transformation laws of the chiral and anti-chiral superfield components in eqs. (36) and (37) are as follows: √ √ δφ = 2εα χα , δχα = 2εα M , δM = 0 , (38) and δ φ¯ = 0 ,

• • √ δχ ¯α = −i 2(˜ σµ )αβ εβ ∂µ φ¯ ,

• √ ¯ = −i 2∂µ χ δM ¯ • (˜ σµ )αβ εβ ,

α

(39)

respectively, where we have introduced the N=1/2 supersymmetry (chiral) parameter εα . To the end of this section, we are going to demonstrate that, in the case of the supersymmetric NLSM, its NAC superworldvolume gives rise to the induced smearing or fuzzyness in the NLSM target space [14, 15]. Here we follow ref. [15] where the most general four-dimensinal supersymmetric NLSM with an arbitrary scalar potential in the NAC superspace was considered (without any gauge fields), with the action Z Z Z Z ¯ ¯ S[Φ, Φ] = d4 y d2 θd2 θ¯ K(Φi , Φj ) + d2 θ W (Φi ) + d2 θ¯ W (Φj ) . (40) This action is completely specified by the K¨ ahler superpotential K(Φ, Φ), the scalar superpotential W (Φ), and the anti-chiral superpotential W (Φ), in terms of some number n of chiral and anti-chiral superfields, i, ¯j = 1, 2, . . . , n. In Euclidean superspace the functions W (Φ) and W (Φ) are independent upon each other. The NAC-deromed action is formally obtained by replacing all superfield profucts in eq. (40) by their star products (31). The NAC-deformed extension of eq. (40) in four dimensions after a ‘Seiberg-Witten map’ (i.e. after an explicit computation of all star products) was found in a closed form (i.e. in terms of finite functions) in refs. [15, 16, 17]. Our four-dimensional results are in agreement with the results of ref. [14] in the case of the NAC-deformed N=2 supersymmetric two-dimensionl NLSM, after dimensional reduction to two dimensions. ¯ We use the following notation valid for any function F (φ, φ): ∂ s+t F , (41) ∂φi1 ∂φi2 · · · φis ∂ φ¯p¯1 ∂ φ¯p¯2 · · · ∂ φ¯p¯t R and the Grassmann integral normalisation d2 θ θ2 = 1. The actual deformation parameter, in the case of the NAC-deformed field theory (40), appears to be F,i1 i2 ···is p¯1 p¯2 ···¯pt =


√ c = − det C ,

211

(42)

where we have used the definition [13] det C = 21 εαγ εβδ C αβ C γδ .

(43)

As a result, unlike the case of the NAC-deformed supersymmetric gauge theories [13], the NAC-deformation of the NLSM field theory (40) appears to be invariant under Euclidean translations and rotations. A simple non-perturbative formula, describing an arbitrary NAC-deformed scalar superpotential V depending upon a single chiral superfield Φ, was found in ref. [16], Z 1 [V (φ + cM ) − V (φ − cM )] d2 θ V∗ (Φ) = 2c (44) χ2 − [V,φ (φ + cM ) − V,φ (φ − cM )] . 4cM The NAC-deformation in the single superfield case thus gives rise to the split of the scalar potential, which is controlled by the auxiliary field M . When using an elementary identity Z +1 ∂ f (x + a) − f (x − a) = a dξ f (x + ξa) , (45) ∂x −1 valid for any function f , we can rewrite eq. (44) to the equivalent form [14] Z +1 Z +1 Z 1 2 ∂2 ∂ 1 2 dξ V (φ + ξcM ) . dξ V (φ + ξcM ) − χ d θ V∗ (Φ) = 2 M ∂φ −1 4 ∂φ2 −1 (46) Similarly, in the case of several chiral superfields, one finds [14] Z ∂ ∂2 1 d2 θ V∗ (ΦI ) = 21 M I I Ve (φ, M ) − (χI χJ ) I J Ve (φ, M ) (47) ∂φ 4 ∂φ ∂φ in terms of the auxiliary pre-potential Z +1 Ve (φ, M ) = dξ V (φI + ξcM I ) .

(48)

−1

Hence the NAC-deformation of a generic scalar superpotential V results in its smearing or fuzziness controlled by the auxiliary fields M I . A calculation of the NAC deformed K¨ ahler potential Z Z Z ¯ (49) d4 y Lkin. ≡ d4 y d2 θd2 θ¯ K(Φi , Φj )∗ can be reduced to eqs. (44) or (47), when using a chiral reduction in superspace, with the following result [17]:

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Lkin. =

1 i 2 M Y,i

+ 21 ∂ µ φ¯p¯∂µ φ¯q¯Z,p¯ ¯q +

1 1 i j ¯p¯ 2 4φ Z,p¯ − 4 (χ χ )Y,ij

1 i µ − 12 i(χi σ µ χ ¯p¯)∂µ φ¯q¯Z,ip¯ ¯p¯)Z,ip¯ , ¯q − 2 i(χ σ ∂µ χ

(50)

where we have introduced the (component) smeared K¨ ahler pre-potential Z +1 ¯ ¯ M) = Z(φ, φ, dξ K ξ with K ξ ≡ K(φi + ξcM i , φ¯j ) , (51) −1

as well as the extra (auxiliary) pre-potential [14] Z +1 h i p¯ q¯ ¯p¯ ξ ¯ M, M) ¯ =M ¯ p¯Z,p¯− 1 (χ Y (φ, φ, ¯ χ ¯ )Z +c dξξ ∂ µ φ¯p¯∂µ φ¯q¯K,ξp¯ ,p¯ ¯q ¯q + 4φ K,p¯ 2 −1

(52) It is not difficult to check that eq. (50) does reduce to the standard (K¨ ahler) N=1 supersymmetric NLSM (cf. sect. 4) in the limit c → 0. Also, in the case ¯¯j , there is no deformation at of a free (bilinear) K¨ ahler potential K = δi¯j Φi Φ all. ¯ (Φ) ¯ ∗ imply, via The NAC-deformed scalar superpotentials W (Φ)∗ and W eqs. (47) and (48), that the following component terms are to be added to eq. (50): p¯ q¯ ¯ f ,i − 1 (χi χj )W f ,ij + M ¯ p¯W ¯ ,p¯ − 1 (χ Lpot. = 12 M i W ¯ )W,p¯q¯ , 4 2 ¯ χ

where we have introduced the smeared scalar pre-potential [14] Z +1 f (φ, M ) = W dξ W (φi + ξcM i ) .

(53)

(54)

−1

The anti-chiral superpotential terms are inert under the NAC-deformation. The ξ-integrations in the equations above represent the smearing effects. However, the smearing is merely apparent in the case of a single chiral superfield, which gives rise to the splitting (44) only. This can also be directly demonstrated from eq. (50) when using the identity (45) together with the related identity [17] Z +1 Z +1 ∂ f (x + a) + f (x − a) = dξ f (x + ξa) + a dξ ξf (x + ξa) . (55) ∂x −1 −1 The single superfield case thus appears to be special, so that a sum of eq. (50) and (53) can be rewritten to the bosonic contribution [17] ¯ µ φ¯ K ¯ ¯(φ + cM, φ) ¯ + K ¯ ¯(φ − cM, φ) ¯ Lbos. = + 1 ∂ µ φ∂ 2

+

1 ¯ 2 4φ

,φφ

,φφ

¯ + K ¯(φ − cM, φ) ¯ K,φ¯(φ + cM, φ) ,φ

¯ M ¯ − K ¯(φ − cM, φ) ¯ K,φ¯(φ + cM, φ) ,φ 2c ¯ 1 ¯ ∂W , + [W (φ + cM ) − W (φ − cM )] + M 2c ∂ φ¯

+

(56)


213

supplemented by the following fermionic terms [17]: Lferm. = − − − − − + − + + −

1 2 ¯ − K ¯ ¯(φ − cM, φ) ¯ χ ¯ K,φ¯φ¯ (φ + cM, φ) ,φφ 4c i ¯ − K ¯ ¯(φ − cM, φ) ¯ (χσ µ χ)∂ ¯ µ φ¯ K,φ¯φ¯(φ + cM, φ) ,φφ 2cM i ¯ − K ¯(φ − cM, φ) ¯ (χσ µ ∂µ χ) ¯ K,φ¯(φ + cM, φ) ,φ 2cM ¯ M ¯ − K ¯(φ − cM, φ) ¯ χ2 K,φφ¯(φ + cM, φ) ,φφ 4cM 1 2 µ ¯ ¯ ¯ + K ¯ ¯(φ − cM, φ) ¯ χ ∂ φ∂µ φ K,φφ¯φ¯(φ + cM, φ) ,φφφ 4M 1 ¯ µ φ¯ K ¯ ¯(φ + cM, φ) ¯ − K ¯ ¯ (φ − cM, φ) ¯ χ2 ∂ µ φ∂ ,φφ ,φ φ 2 4cM 1 2 ¯ ¯ + K ¯(φ − cM, φ) ¯ χ 4φ K,φφ¯(φ + cM, φ) ,φφ 4M 1 ¯ − K ¯(φ − cM, φ) ¯ χ2 4φ¯ K,φ¯(φ + cM, φ) ,φ 2 4cM 1 2 2 ¯ − K ¯ ¯ (φ − cM, φ) ¯ χ χ ¯ K,φφ¯φ¯(φ + cM, φ) ,φφφ 8cM 1 2 ¯ ,φ¯φ¯ . ¯2 W χ [W,φ (φ + cM ) − W,φ (φ − cM )] − 21 χ 4cM

(57)

¯ p¯ enter the action (50) linearly (as LaThe anti-chiral auxiliary fields M grange multipliers), while their algebraic equations of motion, 1 1 i j i ¯ 2 M Z,ip¯ − 4 (χ χ )Z,ij p¯ + W,p¯

=0,

(58)

¯ 5 are the non-linear set of equations on the auxiliary fields M i = M i (φ, φ). As a result, the bosonic scalar potential in components is given by [17] ¯ = 1 M iW f ,i Vscalar (φ, φ) . (59) 2 ¯ M =M (φ,φ)

Some comments are in order. The NAC-deformation just described is only possible in Euclidean superspace where the chiral and anti-chiral spinors are truly independent. The NAC-deformed NLSM is completely specified by a K¨ ahler function ¯ a chiral function W (Φ), an anti-chiral function W ¯ (Φ) ¯ and a constant K(Φ, Φ), deformation parameter c. As a matter of fact, we didn’t really use the constancy of c, so our results are valid even for any coordinate-dependent NAC deformation with c(y). 5

Equation (58) is not a linear system because the function Z is M -dependent.

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Sergei V. Ketov

Solving for the auxiliary fields in eq. (50) represents not only a technical but also a conceptual problem because of the smearing effects described by the ξ-integrations. To bring the kinetic terms in eqs. (50) or (56) to the standard NLSM form (i.e. without the second order derivatives), one has to integrate by parts that leads to the appearance of the derivatives of the auxiliary fields. This implies that one has to solve eq. (58) before integration by parts. Let ¯ be a solution to eq. (58), and let’s ignore fermions for simM i = M i (φ, φ) plicity (χiα = χ ¯p¯• = 0). Substituting the auxiliary field solution back to the α

Lagrangian (50) and integrating by parts yield ¯ = − 1 (∂µ φ¯p¯∂µ φq ) Lkin. (φ, φ) 2

Z

+1 −1

j +c2 ξ 2 M i K,ξpij ¯ M,q

−

1 ¯p¯ ¯q¯ 2 (∂µ φ ∂µ φ )

Z

h ξ i i ξ dξ K,ξpq ¯ + 2cξM,q K,pi ¯ + cξM K,piq ¯

i

+1 −1

h i j i 2 2 i ξ dξ 2cξK,ξpi M + c ξ M K M . ,¯ q ,¯ q ¯ ,pij ¯

(60) It is now apparent that the NAC-deformation does not preserve the original K¨ ahler geometry of eq. (40), though the absence of (∂µ φ)2 terms and the particular structure of various contributions to eq. (60) are quite remarkable. The action (60) takes the form of a generic NLSM, being merely dependent upon mixed derivatives of the K¨ ahler function, so that the original K¨ ahler gauge invariance of eq. (40), ¯ → K(φ, φ) ¯ + f (φ) + f¯(φ) ¯ , K(φ, φ)

(61)

¯ φ) ¯ is preserved. See ref. [16] for with arbitrary gauge functions f (φ) and f( more discussion about elimination of the auxiliary fields. As a result, the NAC deformation of the NLSM (40) amounts to the nonK¨ ahlerian and non-Hermitian deformation of the original K¨ ahlerian and Hermitian structures, which is controlled by the auxiliary field solution to eq. (58). In the case of a single chiral superfield, the deformed NLSM metric can be read off from the following kinetic terms [16]: ¯ µ φ) ∂ ∂ K(φ + cM (φ, φ), ¯ φ) ¯ + K(φ − cM (φ, φ), ¯ φ) ¯ ¯ = − 1 (∂µ φ∂ Lkin. (φ, φ) 2 ¯ ∂φ ∂ φ ¯ ¯ µ φ) ∂ cK,φ (φ + cM, φ) ¯ − cK,φ (φ − cM, φ) ¯ ∂M (φ, φ) + 12 (∂µ φ∂ ∂φ ∂ φ¯ ¯ µ φ) ¯ cK ¯(φ + cM (φ, φ), ¯ φ) ¯ − cK ¯(φ − cM (φ, φ), ¯ φ) ¯ − 21 (∂µ φ∂ ,φφ ,φφ ×

¯ ∂M (φ, φ) . ∂ φ¯ (62)


215

In the case of several superfields, the deformed NLSM can be read off from eq. (60), when assuming all the ξ-integrations to be performed with the auxiliary fields considered as spectators. We thus find a new (NAC) mechanism of deformation of complex geometry in the supersymmetric NLSM target space, by using a non-vanishing anti-holomorphic scalar potential W (Φ), because elimination of the auxiliary ¯ via their algebraic equations of motion in the NAC deformed fields M and M NLSM results in the deformed bosonic K¨ ahler potential depending upon C 0 and W (Φ). This feature is specific to the NAC deformation, because a scalar potential does not affect a K¨ ahler potential in the usual (undeformed) NLSM.

6 Example: NAC-deformed CP (1) model Let’s consider the simplest non-trivial example provided by a four-dimensional supersymmetric CP (1) NLSM with the (undeformed) K¨ ahler, Hermitian and symmetric target space characterized by the K¨ ahler potential ¯ = α ln(1 + κ−2 φφ) ¯ , K(φ, φ)

(63)

where two dimensional constants, α and κ, have been introduced, and with an arbitrary anti-holomorphic scalar superpotential W (Φ). An explicit solution to the auxiliary field equation (58) in this case reads [16] q ¯ + κ−2 φφ) ¯W ¯ ,φ¯ 2 α − α2 + 2cφ(1 M= , (64) ¯ ¯ 2c2 κ−2 φ¯2 W ,φ ¯ A straightforward calculation ¯ ,φ¯ = ∂ W ¯ /∂ φ. where we have used the notation W yields the following deformed NLSM kinetic terms [16]: ¯ µ φ¯ , Lkin. = −gφφ ∂µ φ∂µ φ − 2g ¯∂µ φ∂µ φ¯ − g ¯ ¯∂µ φ∂ φφ

φφ

(65)

where g

¯ φφ

=

q −α +

¯ ,φ¯ )2 −ακ−2 c2 φ¯2 (W q , 2 2 2 −2 2 −2 ¯ ¯ ¯ ¯ ¯ ¯ α + 2cφ(1 + κ φφ)W,φ¯ α + 2cφ(1 + κ φφ)W,φ¯

gφφ = 0 , g¯ ¯ = φφ

q α−

¯W ¯ ,φ¯ −2α−1 c2 (1 + κ−2 φφ) 2 q ¯ + κ−2 φφ) ¯W ¯ + κ−2 φφ) ¯W ¯ ,φ¯ ¯ ,φ¯ 2 α2 + 2cφ(1 α2 + 2cφ(1

¯ ¯ ,φ¯)3 (1 + κ−2 φφ) × 4c2 φ¯2 (W q ¯ + κ−2 φφ) ¯W ¯ ,φ¯ 2 (2W ¯ ,φ¯ + φ¯W ¯ ,φ¯φ¯ ) . + α α − α2 + 2cφ(1 (66)

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Sergei V. Ketov

It is worth noticing that det g = −(g ¯)2 . The most apparent feature φφ

gφφ = 0 is also valid in the case of a generic NAC-deformed NLSM (in the given parametrization).

7 Conclusion Our approach to the NAC-deformed NLSM is very general. The NAC deformation (i.e. smearing or fuzziness) of the NLSM K¨ ahler potential is controlled by the auxiliary fields M i entering the deformed K¨ ahler potential in the highly non-linear way. Both locality and Euclidean invariance are preserved, while no higher derivatives appear in the deformed NLSM action. One should distinguish between the NAC-deformation and N=1/2 supersymmetry. Though the NAC-deformation we considered is N=1/2 supersymmetric, the former is stronger than the latter. When requiring merely N=1/2 supersymmetry of a four-dimensional NLSM, it would give rise to much weaker restrictions on the NLSM target space. It is still the open question how to describe the NAC deformation of the NLSM metric in purely geometrical terms. This investigation was supported in part by the Japanese Society for Promotion of Science (JSPS). I am grateful to K. Ito, O. Lechtenfeld, P. Bowknegt and S. Watamura for useful discussions during the workshop.

References 1. Ketov S.V.: Quantum Non-Linear Sigma Models. Springer-Verlag, Berlin Heidelberg New-York (2000) 2. Spindel, P., Sevrin, A., Troost, W., van Proeyen, A.: Extended Supersymmetric Sigma Models on Group Manifolds. Nucl. Phys. B308, 662–698 (1988), and B311, 465–492 (1988) 3. Sevrin, A., Troost J.: Off-shell Formulation of N=2 Non-linear Sigma-models. Nucl. Phys. B492, 623–646 (1997) 4. Huybrechts D.: Complex Geometry. Springer-Verlag, Berlin Heidelberg NewYork (2004) 5. Gates, S.J. Jr., Hull, C., Roˇcek, M.: Twisted Multiplets and New Supersymmetric Non-linear Sigma Models. Nucl. Phys. B248, 157–186 (1984) 6. Buscher, T., Lindstr¨ om, U., Roˇcek, M.: New Supersymmetric Sigma-models with Wess-Zumino Terms. Phys. Lett. B202, 94–98 (1988) 7. Gates, S.J. Jr., Ketov, S.V.: No N=4 Strings on Wolf Spaces. Phys. Rev. D52, 2278–2293 (1995) 8. Galperin, A., Ivanov, E., Ogievetsky, V., Sokatchev, E.: Harmonic Superspace. Cambridge University Press (2001) 9. Brink, L., Schwarz J.: Quantum Superspace. Phys.Lett. B100, 310–312 (1981) 10. Klemm, D., Penati, S., Tamassia, L.: Non(anti)commutative Superspace. Classical and Quantum Grav. 20, 2905–2916 (2003)


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11. Ooguri, H., Vafa C.: Gravity induced C Deformation. Adv. Theor. Math. Phys. 7, 405–417 (2004) 12. Gates, S.J. Jr., Ketov, S.V., Nishino, H.: Self-dual Supersymmetry and Supergravity in Atiyah-Ward Space-time. Nucl. Phys. B716, 149–210 (1993) 13. Seiberg, N.: Noncommutative Superspace, N=1/2 Supersymmetry and String Theory. JHEP 0306, 010 (2003) 14. Alvarez-Gaume, L., Vazquez-Mozo, M.: On Non-anti-commutative N=2 Sigmamodels in Two Dimensions. JHEP 0504, 007 (2005) 15. Hatanaka, T., Ketov, S.V., Kobayashi, Y., Sasaki, S.: N=1/2 Supersymmetric Four-dimensional Non-linear σ-models from Non-anti-commutative Superspace. Nucl. Phys. B726, 481–493 (2005) 16. Hatanaka, T., Ketov, S.V., Kobayashi, Y., Sasaki, S.: Non-anti-commutative Deformation of Effective Potentials in Supersymmetric Gauge Theories. Nucl. Phys. B716, 88–104 (2005) 17. Hatanaka, T., Ketov, S.V., Sasaki, S.: Summing up Non-anti-commutative K¨ ahler Potential. Phys. Lett. B619, 352–358 (2005).


Seiberg-Witten Monople and Young Diagrams Akifumi Sako Department of Mathematics, Faculty of Science and Technology, Keio University 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan [email protected]

1 @Introduction The N = 2 supersymmetric gauge theory and related subjects have been studied by using nonperturbative approach [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. One big break through is given by Nekrasov [11, 12, 13] After them, many relating developments in N ≥ 2 Super Yang-Mills theories and string theories. One of the key ideas is the localization formula that is very strong tool. We can use the formula when fixed points of some groups action associated with symmetries are isolated. In this paper, we study the fixed points of the Seiberg-Witten monopole theory on noncommutative (N.C.) R4 . Its partition function is calculated by localization formula if its fixed points are isolated. The Seiberg-Witten duality conjecture implies that the preopotential on the patch of moduli space around the massless U (1) monopole theory is spread to the prepotential on the other patch that is low energy effective potential of SU (2) gauge theory. That is why, if we can get the result of the SeibergWitten monopole theory by the localization formula then we get the proof of the Seiberg-Witten duality. This is our motivation. In this article we report that N.C. cohomological field theories do not depend on N.C. parameter and this fact relates with dimensional reduction. For example, the partition function of N = 4 supersymmetric U (1) gauge theory is calculated by using D0-brane theory. Also, we show that the fixed points of the Seiberg-Witten theory is isolated and they are classified by Young diagrams. This article is based on the joint work with Toshiya Suzuki [14, 15].

2 N.C. Cohomological field theory We consider only N.C.R4 . Let N.C. parameter be iθ µν = [xµ , xν ]. At first, let us see the fact that the cohomological field theory is invariant under θ deformation [16, 17, 18]. Actions of cohomological field theories are given by BRS exact terms. The partition functions of the cohomological field theory 219

220

Akifumi Sako

is invariant under any infinitesimal operation, Z ˆ ˆ 0 = ±δ 0 δ, ˆ S = δV δδ Z 0 δ Zθ = DφDψDχDH Z = DφDψDχDH

variation that commute with the BRS

Z Dˆ δ − dx δV exp (−Sθ ) Z 0 ˆ δ − δ V exp (−Sθ ) = 0. 0

Here δˆ is a BRS operator and S is an action of some cohomological field theory defined by a gauge fermion V . Above equations show that the partition function of the cohomological field theory is invariant under infinitesimal deformation generated by δ 0 . Let introduce θ-deformation ; δθ θµν = θµν + δθµν .

(1)

If δθ commute with BRS operator, then the partition function is invariant under the θ-deformation. We can see that this statement is right for the N ≥ 2 supersymmetric N.C. Yang-Mills theory[14, 15]. Let us study what occur when the theory is invariant under the θ deformation. Here, we treat Moyal space that is defined by the Moyal product defined as

∗

θ

i← − − → := exp{ ∂µ θµν ∂ν } . 2

(2)

Let change variables as x µ xµ → x 0 = √ t dx =

√

∗

1 tdx0 , ∂µ = √ ∂µ0 , t

θ

−θ − → i← = exp{ ∂µ0 ∂ν0 } . 2 t

After this, do the θ-deformation θ → θ0 = tθ,

∗

θ0

− − → i← = exp{ ∂µ0 θ ∂ν0 } 2

From this procedure, we can interpret θ-deformation = Rescaling without ∗

(3)

That is to say Z Sθ 0 ∼

√ D 0D t dx L(

∗, √1t ∂x∂

0ν

),

(4)

Seiberg-Witten Monople and Young Diagrams

221

where ∗ is fixed under θ-deformation. Let’s consider “θ → ∞ ”= “t → ∞”. We find that kinetic terms (∂µ including terms) vanish. Using this phenomena, some calculations become easy. Indeed we can see such example. The Euler number of the moduli space of GMS soliton is calculated in [16, 17, 18] and it is equivalent to the partition function of balanced scalar cohomological field theory based on noncommutative plane. The moduli space of GMS soliton for real scalar field φ is Mm = {φ| bm φ ∗θ (φ − v1 ) ∗θ (φ − v2 ) · · · ∗θ (φ − vm ) = 0, bm > 0}. (5) Here, vi ∈ R. When the solutions of the algebraic equation x(x − v1 )(x − v2 ) · · · (x − vm ) = 0 are not degenerate, the Euler number of the space Mm is given by 1 : m is even number (6) χm = 0 : m is odd number. The partition function (Euler number of the moduli space) is invariant under the θ deformation. So, we can evaluate this partition function in the commutative limit, too, and this calculation is also easy. On the other hand, the case of large θ limit is a dimensional reduction to zero dimension. This is the theme of the next section.

3 Universality of Partition Functions Consider N.C.R2D . Let noncommutative parameters be (θ µν ) = ⊕θi 2i−1, 2i . As we saw in the previous section, in θ i → ∞, terms with derivative operators ∂x2i ∗ := −i(θi )−1 [x2i−1 , ∗] and −∂x2i−1 := −i(θi )−1 [x2i , ∗] become irrelevant in lagrangians. Therefore, if the theory is invariant under the θ-deformation, terms including ∂x2i or ∂x2i−1 can be removed. In the following, we consider only such cases. Operator expression of the N.C.R2D is constructed by using Fock basis. An arbitrary operator is expressed as X X n1 ···nD ˆ= O ··· Om |n1 , · · · , nD i hm1 , · · · , mD | . (7) 1 ···mD n1 ,m1

nD ,mD

This expression implies that N.C.field theories on the N.C.R2D are described by ∞ dimensional matrix models. Let us consider a matrix model given by n1 ···nD the Lagrangian L(O) where we take Om is a variable of path integration 1 ···mD and L(O) is a polynomial of O. Then we cannot distinguish the other model whose dynamical variables are n ···n

n

···n

i+1 D Om11 ···mi−1 i−1 mi+1 ···mD |n1 ,···,ni−1 ,ni+1 ,···,nD i hm1 ,···,mi−1 ,mi+1 ,···,mD |

222

Akifumi Sako

n1 ···nD |n1 , · · · , nD i hm1 , · · · , mD | , befrom the model whose variables are Om 1 ···mD cause both of them are ∞ dimensional matrices. That is why, in θi → ∞, there is no ∂zi or ∂z¯i and it is impossible to distinguish dynamical variables living in R2D from variables in R2D−2 . In other words, θi → ∞ is equivalent to the dimensional reduction corresponding to x2i−1 and x2i directions.

Claim Let Z2D and hOi2D be a partition function and VEV of O of a CohFT in N.C.R2D with D ≥ 1 s.t. δθ Z2D = 0 and δθ hOi2D = 0. Let Z2D−2 and hOi2D−2 be the partition function and VEV of O of a CohFT in N.C. R2D−2 , where they are given by dimensional reduction of Z2D and hOi2D . Then, Z2D = Z2D−2 , hOi2D = hOi2D−2 ,

(8)

i.e. the partition function of such theories do not change under dimensional reduction from 2D to 2D − 2. In this article, we do not manage topological terms to avoid difficulties to interpret them in dimensional reduced models. For example, as we will see in next section supersymmetric Yang-Mills theories are invariant under θ-deformation. From this fact, the following partition function of supersymmetric Yang-Mills theories on N.C. R2D are equivalent: 8dim 6dim 4dim 2dim 0dim ZN =2 = ZN =2 = ZN =4 = ZN =8 = Z∗∗∗ ,

(9)

Idim where ZN =J is a partition function of the N = J supersymmetric Yang-Mills theory in N.C. RI with arbitrary gauge group. Similarly, we get 4dim 2dim 0dim ZN =2 = ZN =4 = Z∗∗∗ .

(10)

4 Z of N =4 Super U (1) N.C.Theory As a well known fact, 0(+1)-dimension reduced Yang-Mills theory is D0-brane effective theory. From applying the above process (θ → ∞), we can expect that Z of N =4 supersymmetric U (1) theory is estimated by D0-brain theory and some correction from traceless part; Z[N = 4SU SY U (1)] = Z[∞D0 − brane] × Z[f inite] because D0-brane theory is given by 0-dim. reduction of N =1 10-dim YangMills theory. Detail estimations of this are in [14]. Final result is given by Z=

X 1 π2 = ζ(2) = . 2 d 6

d∈N

(11)


223

To extend this technique to the N = 2, d = 4 series like (10), it is natural to investigate the Seiberg-Witten model on N.C.R4 . We will show that the fixing points are isolated and we can apply the localization formula to get the partition functions in the following of this article.

5 @N = 2 SUSY Gauge Theory on N.C.R4 At first, we denote the set up the model of N = 2 supersymmetric gauge theory on N.C. R4 . SO(4) rotation of Euclidian space is locally isomorphic to SU (2)L × SU (2)R . N = 2 supersymmetric theory has SU (2)I R-symmetry. (R-symmetry is a symmetry ¯ αi of fermionic coordinate rotation.) The supersymmetric generators Qαi , Q ˙ have the indices i = 1, 2 for the R-symmetry. Totally N = 2 supersymmetric theory has following symmetry; H = SU (2)L × SU (2)R × SU (2)I .

(12)

The supersymmetric gauge multiplet is ψ1

Aµ

ψ2 .

(13)

φ Here ψ 1 , ψ 2 and ψ¯1 ,ψ¯1 are the Weyl spinors and their CPT conjugate. φ and φ¯ are scalar fields. Their quantum number of H are assigned as ψ 1 = (1/2, 0, 1/2), ψ 2 = (1/2, 0, 1/2), φ = (0, 0, 0), ψ¯1 = (0, 1/2, 1/2), ψ¯2 = (0, 1/2, 1/2), φ¯ = (0, 0, 0).

(14)

The action functional is given by L=

a a µαα − 41 Fµν Faµν − iψ¯αi ¯ ˙ Dµ ψαa i − Dµ φ¯a Dµ φa ˙ σ ¯ ψαi ]a − √i ψ¯α˙ ia [φ, ψ¯α˙ i ]a − 1 [φ, ¯ φ]2 , . − √i2 ψ αia [φ, 2 2

(15) (16)

The supersymmetric transformations with parameter ξ are given by δAµ = iξ αi σµαα˙ ψ¯α˙ i − iψ αi σµαα˙ ξ¯α˙ i , √ µ i β ˙ ¯αi ¯ αi, δψα i = σ µν + [φ, φ]ξ α ξβ Fµν + 2iσαα˙ Dµ φξ .. . Let us introduce topological twist. We use a diagonal subgroup SU (2)R0 in SU (2)R × SU (2)I .

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Akifumi Sako

K 0 := SU (2)L × SU (2)R0 .

(17)

Then a combination of spinors whose quantum number of H are (1/2, 0, 1/2)⊕ (0, 1/2, 1/2) has quantum number (1/2, 1/2) ⊕ (0, 1) ⊕ (0, 0) of K 0 . Note that ˙ ¯ (0, 0) is scalar and Q = αi Qαi ˙ is the BRS operator, that is the Fermionic scalar transformation generator. The BRS operator in this case is interpreted as an equivariant derivative operator as we will see soon. Similarly, spinor fields are twisted as 1 1 1 1 ψ i , ( , 0, ) → ψµ , ( , ) 2 2 2 2 1 1 i ¯ ψ , (0, , ) → χµν ⊕ η , (0, 1) ⊕ (0, 0) . 2 2

(18)

Their BRS transformations are given as ˆ µ δA ˆ µν δχ

= iψµ ,

= Hµν , ˆ δHµν = i[φ, χµν ],

ˆ µ δψ δˆφ¯

= −Dµ φ,

ˆ = 0, δφ

= iη, ˆ ¯ . δη = [φ, φ]

(19)

Next step, we consider a hypermultiplet. We introduce the Weyl fermions ψq and ψq˜† , and complex scalar fields q and q˜† ; ψq q˜† .

q ψq†˜

Their supersymmetric transformations are given by √ √ δq i = − 2ξ αi ψqα + 2ξ¯α˙ i ψ¯qα˜˙ , √ ¯ i ξαi , δψqα = − 2iσαµα˙ Dµ q i ξ¯α˙ i − 2φq √ ˙ σ µαα δ ψ¯α˙ = − 2i¯ Dµ q i ξαi + 2φq i ξ¯α˙ i , q˜

(20)

Let gauge representation of this Hypermultiplet be a fundamental representation. After topological twisting, their BRS transformations are given by ˆ α˙ = −ψ¯α˙ , δq ˆ † = −ψ¯qα˙ , δq q˜ α˙ α ˙ α˙ ˆ ¯ ˆ δ ψq˜ = −iφq , δ ψ¯qα˙ = iqα†˙ φ.

(21)

Using above fields, let us define the Seiberg-Witten monopole equations. The action with the hypermultiplet are defined by ˆ S = k − δΨ

(22)


where k is instanton number k=

1 8π 2

225

Z tr(FA ∧ FA ),

(23)

and Ψ is a gauge fermion; a a †α †α †α Ψ = −χµνa + {H+µν − s+µν } − χq {Hqα − sα } − {Hq − s }χqα ¯ a η a + Dµ φ¯a ψ µa − (−iq † φ)ψ ¯ α˙ − ψ † (iφq ¯ α˙ ) . +i[φ, φ] α ˙

q

q α˙

(24)

Here sµν (A, q, q † ) = Fa+µν + q † σ ¯ µν Ta q. sα (A, q) = σαµα˙ Dµ q α˙ = (6 Dq)α . After integration of H+µν and Hq , Bosonic action is Z √ 1 1 SB = d4x g |sµν |2 + |sα |2 + · · · . 4 2

(25)

(26)

The BPS eqs. are given by sµν (A, q, q † ) = 0 , sα (A, q) = 0 .

(27)

These equations are called Seiberg-Witten monopole equations. In the following, we investigate this theory whose gauge group is U (1) and defined space is N.C. R4 with [xµ , xν ] = iθµν . Here, θµν is an antisymmetric matrix  0  −θ1 µν (θ ) =   0 0

called N.C. parameter.  θ1 0 0 0 0 0  . 0 0 θ2  0 −θ2 0

(28)

(29)

We only use operator formalisms here, so all the fields R are operators act−1 ν ing on H. ∂µ is written by −iθµν [x , ∗] ≡ [∂ˆµ , ∗] and d2D x is written by det(θ)1/2 T rH . In the Fock space representation, fields are expressed as X Aµ = Aµ nm11nm2 2 |n1 , n2 ihm1 , m2 |, X ψµ = ψµ nm11nm2 2 |n1 , n2 ihm1 , m2 | , etc. Therefore, the BRS transformations are expressed as

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Akifumi Sako

ˆ µ n1 n2 = ψ µ n1 n2 , δA m1 m2 m1 m2

ˆ µ n1 n2 = (Dµ φ)n1 n2 , . . . , δψ m1 m2 m1 m2

(30)

−1 ν where Dµ ∗ := [∂ˆµ + iAµ , ∗ ] with ∂ˆµ := −iθµν x . The action functional without the topological term is defined as follows;

S = T rH L(Aµ , . . . ; ∂ˆzi , ∂ˆz¯i ) ˆ . = T rH trδΨ

(31)

Let prove the invariance under the θ-deformation, here. Let us change the dynamical variables as 1 1 1 ˜¯ 1 1 η → η˜, q → √ q˜, Aµ → √ A˜µ , ψµ → √ ψ˜µ , φ¯ → φ, θ θ θ θ θ 1 1 1 + ˜ + , φ → φ˜ ψq → √ ψ˜q , ˜+ , Hµν → H χ+ µν → χ θ µν θ µν θ 1 † 1˜ 1 † χq → χq , χq → χ˜q , Hq → Hq . θ θ θ Note that this changing does not cause nontrivial Jacobian from the path integral measure because of the BRS symmetry. Then, the action is 1 ˜ S θ2

1 L(Aµ , . . . ; ∂ˆzi , ∂ˆz¯i ) → 2 L(A˜µ , . . . ; −a†i , ai ) . (32) θ √ S depends on θ because ∂zi = − θ−1 [a†i , ]. In contrast, S˜ does not depend on θ, because all θ parameters are factorized out. S→

,

From the discussion in section 2, the BRS symmetry proves that the Z is invariant under the deformation of θ ; ˜ =0, δθ Z = −2(δθ)θ −3 hSi R where Z = DADψ . . . exp (−S[A, ψ, . . .]). This fact implies that Z can be determined by its dimensional reduced model. The dimensional reduced version of the Seiberg-Witten eqs. are P+µνρτ [Aρ , Aτ ]a + q¯ σ µν q † = 0 , σ µ Aµ q = 0 ,

(33) (34)

√ where P+µνρτ is a selfdual projection operator. Using q+ := (q1˙ + q2˙ )/ 2 and √ q− := (q1˙ − q2˙ )/ 2, the eqs. (33) are rewritten as ADHM eqs. : ∗t ∗t [Az1 , A†z1 ] + [Az2 , A†z2 ] + q− q− − q + q+ =0,

[Az1 , Az2 ] +

∗t q − q+

=0.

(35) (36)


227

6 D-brane interpretation The eqs. in the previous section are important even if we consider a finite size matrix model. Let’s consider the relations between dimensional reduction of Seiberg-Witten monopole eqs. and D-brane picture. The second order action of N brane N anti-brane system is given by Z 1 ¯ ) (N)µν ¯ (N (N ) (N )µν ¯2 . tr Fµν F + Fµν F + Dµ φDµ φ + (τ 2 − φφ) 4 (N )

¯) (N

¯

¯

Here Fµν ,Fµν are the curvature of A(N ) , A(N ) , and A(N ) , A(N ) are connec¯ tions corresponding to the open strings on D-brane and D-brane, respectively. Up to topological terms, we can rewrite this as Z n 1 ¯ ) (N ¯ (N ) (N ) (N tr Fµν F )µν + |Fz1 z¯1 + Fz2 z¯2 + (φφ¯ − τ )|2 2 o 2 +|Fz1 z2 | + |Dz¯¯1 φ|2 + |Dz¯¯2 φ|2 . ¯) (N

Case of Aµ

= 0, the E.O.M. is given as (N )

(N )

∗t Fz1 z¯1 + Fz2 z¯2 + q− q− =ζ , (N )

Fz 1 z 2 = 0 , Dz¯1 q− = 0 ,

(37) (38)

Dz¯2 q− = 0 ,

(39)

where we replace φ by q− . These are nothing but Seiberg-Witten eqs. under q+ = 0. This is the case that we will see later. Therefore we can understand ¯ configuration. that the solution of (34) is realized as the D3 − D3

7 Deformed BRS Operators To get the isolated fixed points, let us deform the BRS transformation to δ 2 Azi = δψzi = i[Azi , φ] − ii Azi ,

(40)

β˙

δ 2 qα˙ = δψqα˙ = −iφqα˙ + MR α˙ qβ˙ + iqα˙ b, 2 †α˙

δ q

= δψq

where β˙

MR α˙ = and

†α˙

†α˙

= q iφ − MR 0 i+ i+ 0

α˙

β˙ q

, + =

†β˙

†α˙

− ibq ,

1 + 2 , 2

(41) (42)

(43)

228

Akifumi Sako

   b= 



b1

  . 

b2 ..

.

(44)

bN 2

This deformation is made to make δ be the Lie derivative including group of global symmetries that generated by the above action with the parameter i , bi . Here we have to recall the fact that our BRS operator is interpreted as the equivariant derivative operator. In fact, ˜

SU (2),SU (2)

G G δ 2 = δ(−φ) + δ(b) + δ(1 ,2 )

,

(45)

that is to say δ 2 is given by the Lie derivatives. We do not change the gauge fermion Ψ , so the bosonic BPS equations do not change: ¯z2 z¯2 )q † − iζ1 = 0, σz1 z¯1 + σ µR ≡ i([Az1 , Az¯1 ] + [Az2 , Az¯2 ]) + q(¯ µC ≡ i[Az1 , Az2 ] + q¯ σz1 z2 q † = 0, (46) Dirac : (Az1 σ z1 + Az¯1 σ z¯1 + Az2 σ z2 + Az¯2 σ z¯2 )q = 0. For the later convenience, we introduce constant back ground ζ, here. The fixed point equations of the deformed BRS transformations for ψµ and ψq are i[Azi , φ] − ii Azi = 0, β˙

−iφqα˙ + MR α˙ qβ˙ + iqα˙ b = 0.

(47) (48)

The contributions to path integrals are given by neighborhood of the solutions of these equations.

8 Solutions In this section, we solve above eqs. (46)-(48). In the following of this article, we study only finite size matrix model given by the model in section 5 reduced to zero dimension and truncated into finite size matrices, for simplicity. At first, let us diagonalize φ by U (N ) gauge symmetry, φ = diag.(φ1 , φ2 , · · · , φN ).

(49)

Furthermore we assume φI does not degenerate. This is ensured from KempfNess Theorem, when we study above fixed points equations. µ−1 (0)/G * ) Closed GC orbit → non-degeneracy . (47) and (48) are fixed point equations for some torus actions. From (47) we see immediately that if and only if


φJ − φ I = i , A zi

IJ

229

(50)

could be non-zero. Also from (48) we see that if and only if, (±)

φI = b J ± + ≡ b J ,

(51)

q1˙ IJ and q2˙ IJ could be non-zero. From these facts and small calculations, we get the following lemma. Lemma 1. If µR = 0 and Fixed Point Eqs. have a solution, then φI takes (n1 ,n2 ) , given by any of ϕ (−) [bI

]

ϕ Each {ϕ

(−)

(n1 ,n2 ) (−)

[bI

(−)

(n1 ,n2 ) [bI

]

= bI

]

+ n 1 1 + n 2 2

n1 , n2 ∈ Z .

,

(52)

} assign some graphs G[b(−) ] . In Fig.1, the origin corresponds I

Fig. 1. G[ˆxI ]

to the eigenvalue ϕ to eigen values ϕ

(0,0) (−)

[bI ] (n1 ,n2 ) (−)

[bI

]

Fig. 2. G[ˆxI ] (−)

= bI , and other lattice points (n1 , n2 ) correspond

.

For given a set of G[b(−) ] , φ is written as I

   M  φ=  I  

ϕ



(n1 ,n2 ) (−) [bI ]

ϕ

    .   

(n01 ,n02 ) (−) [bI ]

ϕ

00 (n00 1 ,n2 ) (−)

[bI

]

..

.

We suppose that eigen values are arranged by order, ϕ

(n1 ,n2 ) (−) [bI ]

n2 >···>nN i<j

1

To obtain the net contribution to the prepotential, we need to exclude the contribution from the large number of the D1-branes. As we will see, the contribution from the infinite D1-branes corresponds to the perturbative part.

244

Kazutoshi Ohta

where the summation is done over integer sequences {ni } which satisfy a strongly decreasing condition n1 > n2 > · · · > nN . Observing the partition function (6), we find that it is a discretization of eigenvalues of the N ×N Hermite random matrix model with a quadratic potential ZMM =

Z Y N

dλi

Y i<j

i=1

µA

(λi − λj )2 e− 2gs

PN

i=1

λ2i

.

(7)

Comparing with the matrix model partition function, we find that the eigenvalues λi are quantized in units gs by gs ni in the 2 dimensional gauge theory partition function and the integrals over eigenvalues are replaced by summations. In this sense, we will call the partition function of the 2 dimensional Yang-Mills theory by a “discrete matrix model” in the following. Moreover, we find that the partition sum of the discrete matrix model is done over the sets of Young diagrams in the same way as Nekrasov’s partition function. The summation in (6) is over the integer sets which satisfy the strongly decreasing condition n1 > n 2 > · · · > n N ,

(8)

but if we set ni = ki − i + c, where c is an arbitrary constant, we can regard the summation as a summation of integer sequences which satisfy a weakly decreasing condition k1 ≥ k 2 ≥ · · · ≥ k N , (9) including equalities. So we can identify the integers {ki } with the number of rows in Young diagrams associated with the representation R of U (N ) group. Using this correspondence, if we rewrite the Vandermonde determinant and the quadratic potential in the discrete matrix model, they become the dimension of the representation R and second Casimir C2 (R), respectively. Therefore, by a suitable normalization, the partition function of the discrete matrix model (2 dimensional Yang-Mills theory) reduces to X µgs A (dim R)2 e− 2 C2 (R) . ZYM2 = (10) R

This is nothing but the original form of the 2 dimensional partition function given by Migdal.

4 Large N Limit Now, our purpose is to calculate the coefficients of the instanton expansion of the prepotential from counting the multiplicity of D1-branes. This can be obtained from the large N limit of the discrete matrix model or 2 dimensional Yang-Mills theory as we have seen. Next we would like to consider the

Instanton Counting, 2D YM Theory and Topological Strings

245

large N limit. Unfortunately, however, we obtain the prepotential only for 4 dimensional N =2 “U (1)” gauge theory, which is not so interesting from the point of view of gauge theory dynamics, if we use the quadratic potential like in (6). To obtain the prepotential for the general SU (r) theory, we need to extend the quadratic potential to (r+1)-th order potential W (Φ) which has r critical points. We will generally consider the (r+1)-th order potential in the following. Let us first consider dynamics of eigenvalues in the discrete matrix model with the generic potential. Here we introduce the density of the eigenvalues similar to the ordinary continuous matrix model ρ(x) =

N 1 X δ(x − gs ni ). N i=1

(11)

Using this eigenvalue density, the partition function (6) can be written as X e−S[ρ(x)] , (12) ZDMM = N ! {ρ(x)}

where Z Z N S[ρ(x)] = −N 2 − dxdy log |x − y|ρ(x)ρ(y) + dxW (x)ρ(x), gs

(13)

is an effective action adopting the effect from the Vandermonde determinant. If we look at the above effective action, we find that the Vandermonde determinant part works as a repulsive force between the eigenvalues and the potential provides an attractive force to the critical points2 . (See Fig. 2.) This fact means that the Vandermonde determinant represents the effect of the exclusion principle of free fermions if we identify the eigenvalues with the levels of states of the free fermions. In fact, the eigenvalues, which is a strongly decreasing integer sequence, can not take the same value with each other and so the densest configuration is a continuous integer sequence. The boundaries of the integer sequence, which are separated by fillings and vacancies of the eigenvalues, correspond to Fermi surfaces. For finite N , there exist the Fermi surfaces for both sides of each bunch of eigenvalues. We will call the densest configuration the “ground state”. Contribution from the ground state comes from the heat bath which is originally the contribution from the large N reduction of the D5-branes, and it will become the perturbative part of the 4 dimensional supersymmetric gauge theory. Therefore the net contribution 2

Strictly speaking, the force from the potential depends on a sign of the second derivative of the potential whether it is attractive or repulsive. However we think here that the eigenvalues are analytically continued and gather to any critical points in spite of the sign of the second derivative like in Dijkgraaf-Vafa theory [DV02].

246

Kazutoshi Ohta

nF

nF

Fig. 2. A distribution of the eigenvalues in the discrete matrix model with a quadratic potential. At the position of black dots, there exists the eigenvalues and white dots stand for vacant eigenvalues. nF represents the two Fermi surfaces. Using the so-called Maya/Young diagram correspondence, we can build the Young diagram above the eigenvalues. In the slant Young diagram, the right-down and right-up edges correspond to the black and white dots of the eigenvalues, respectively.

from the non-perturbative instanton correction is given by the residual part of the partition function divided by the ground state contribution. First of all, we would like to consider the contribution to the free energy (prepotential), but it is useful to introduce the following difference operator ∆gs f (x) ≡ f (x + gs ) − f (x),

(14)

since the eigenvalues take the discrete integer numbers. Using this difference operator, we can write the Vandermonde determinant part in the effective action (13) as Z Z 2 2 dxdyγ(x − y|gs )∆gs ρ(x)∆gs ρ(y), (15) N − dxdy log |x − y|ρ(x)ρ(y) = N where γ(x) satisfies ∆gs ∆−gs γ(x|gs ) = log x

(16)

The solution to this difference equation is very important later and a solution is given by γ(x|gs ) = − log Γ2 (x|gs ), (17) where Γ2 (x|gs ) is the so-called Barnes’ double gamma function. The double gamma function is an extension of the ordinary gamma function, but it also has an asymptotic expansion like the Stirling’s formula


γ(x|gs ) =

1 gs2

247

∞ g 2g−2 X B2g 1 1 2 3 s . (18) x log x − x2 − log x+ 2 4 12 2g(2g − 2) x g=2

As I mentioned before, this function relates to the perturbative part of the 4 dimensional supersymmetric gauge theory. At the same time, the asymptotic expansion (18) can be regarded as a genus expansion of closed string theory since powers of gs are proportional to the Euler number of closed Riemann surfaces. In fact, the function γ(x|gs ) is closely related to amplitudes of the topological B-model or c=1 non-critical string theory at self-dual radius. Originally, we have been considering the open string theory as gauge theory on the D-branes, but finally we have the closed string amplitude. So this model is a realization of the open/closed string duality. Now let us consider the large N limit after the above preparations. Here we assume that the eigenvalues gather around the r critical points of the potential. In particular, the ground state configuration is the situation where the eigenvalues in each bunch are closely P packed, and the distance between two Fermi surfaces is gs Ni , where Ni ( i Ni = N ) is the number of eigenvalues in each bunch and a sufficiently large number. So, if we consider the large N limit (the large Ni limit at the same time), the two Fermi surfaces in each bunch are separated far away and we can ignore effects between them. In addition, if we tune the coefficients of the potential expanded in terms of the order of Φ, the partition function of the discrete matrix model is decomposed into two parts coming from each Fermi surface (chiral decomposition) ZDMM ' |ZNekrasov (al ; gs )|2 ,

(19)

where ZNekrasov (al ; gs ) is Nekrasov’s partition function which appeared in (2), the Ω-background parameter is identified with the string coupling constant gs , and the moduli parameters of vacua al are determined by the positions of the critical points of the potential. The perturbative part in Nekrasov’s partition function Zpert (al ; gs ) is represented by the renormalized part from the ground state P Y Zpert (al ; gs ) = e l6=n γ(al −an ) = Γ2−1 (al − an |gs ), (20) l6=n

using the double gamma function. To see the contribution to the perturbative part of the prepotential, we should use the asymptotic expansion (18) and especially the term proportional to 1/gs2 is the contribution to the gauge theory without the graviphoton corrections. The residual terms in the gs expansion are regarded as the contribution from the graviphoton, the same as in Nekrasov’s formula. It is interesting that the perturbative calculation in gauge theory can be expressed in terms of the asymptotic expansion of the extended double gamma function.

248

Kazutoshi Ohta

5 Topological M-theory and Non-critical M-theory In this section, we would like to extend the previous discussions. It is basically an extension of the calculation of the prepotential in the 4 dimensional gauge theory to 5 dimensional theory compactified on S 1 . There will appear some interesting structures of topological or non-critical M-theory, which include all non-perturbative corrections of the topological or non-critical string theory, behind the extension. To extend the discrete matrix model partition function, we need ideas of “T-duality” in string theory and matrix model. Here the T-duality will apply to the Vandermonde determinant part in the (discrete) matrix model, but recalling that the Vandermonde determinant originally comes from the path integral measure of the adjoint scalar field (and its adjoint action), the T-duality means replacing the adjoint action with a covariant derivative. We are now considering that the T-dual direction is compactified on S 1 . So there exists the effect from infinitely many Kaluza-Klein modes. If we sum up all Kaluza-Klein modes in the Vandermonde determinant, we obtain the T-dualized measure ∞ Y Y

Y n (i + λi − λj )2 ' β i<j n=−∞ i<j

1 sinh β(λi − λj ) β

2 .

(21)

This measure is trigonometrically extended and equivalent with a measure of the unitary matrix model. In the discrete matrix model formulation, we have to discretize the above measure with a suitable potential. So we obtain the partition function. # " 2 Y1 X gs µA X 2 ni . Zq-DMM = sinh β(gs ni − gs nj ) exp − β 2 n >n >···>n i<j i 1

2

N

(22) We can understand this model as a q-deformation of the Vandermonde measure, which corresponds to the q-deformed dimension of the representation R. So it is called the q-deformed 2 dimensional Yang-Mills theory [AOSV05]. It is also known that the continuum limit of the q-deformed discrete matrix model is equivalent to the partition function of 3 dimensional (bosonic) Chern-Simons theory on S 3 [AKMV04]. We can follow the same analysis as the discrete matrix model, but the different point is the difference equation (16), which is now trigonometrically extended or q-deformed ∆gs ∆−gs γ˜(x|gs ; β) = log sinh βx.

(23)

Solution to this trigonometric difference equation is also expressed by using an extended gamma function. Here we need the triple sine function, which is made by gluing the triple gamma functions together. The gluing is an analog of the fact that the ordinary sine function can be expressed in terms of the product


249

of the gamma function and its reflection such as sin πx = πΓ −1 (x)Γ −1 (1 − x). If we use the triple sine function S3 (x|gs ; β), the above γ˜ (x|gs ; β) is written by its logarithm as γ˜ (x|gs ; β) = log S3 (x|gs ; β) Z ∞ iπ dt e−xt = B3,3 (x|gs ; β) − , − iπ 2 gs t 6 β t) −∞ t 4 sinh (1 − e 2

(24)

where B3,3 (x|gs ; β) is a 3rd order Bernoulli polynomial in x. Similar to the prepotential in the 4 dimensional gauge theory, which is expressed in terms of the double gamma function, we can express the perturbative part of the prepotential in the 5 dimensional supersymmetric gauge theory on S 1 using the triple sine function. Actually, the 3rd order polynomial in (24) is a characteristic of the 5 dimensional prepotential and the residues from the integral part give non-perturbative corrections proportional to e−2βnx (n ∈ Z). In addition, the function γ˜ (x|gs ; β) has a genus expansion in gs the same as the former case. This expansion now relates to the topological A-model amplitude. If we recall that γ(x|gs ) represents the topological B-model amplitude, the two models are related by the T-duality as expected. The discussion above is sufficient to see the relation between the ordinary topological string amplitudes, but we can find a more interesting feature if we examine the properties of the triple sine function. These are the residues in the integral of (24). If we evaluate more carefully the residues in the integral, we 2πix notice that they include terms proportional to e− gs n very non-trivially. The 2πix terms proportional to e− gs n vanish if we act with the difference operator. So these terms do not affect the difference equation (23) which we would like to 2πix consider first. However, the terms proportional to e− gs n naturally appear in the triple sine function and can be regarded as the non-perturbative effects in the topological string theory. So far, there is no proof that all non-perturbative corrections reduce to the triple sine function, but it strongly suggests the existence of the topological M-theory which gives all non-perturbative corrections from the point of view of the duality in string theory, since the radius β of S 1 and string coupling constant gs are symmetrically included in the triple sine function. On the other hand, the function γ(x|gs ) or double gamma function, which appears in the perturbative part of the 4 dimensional N =2 supersymmetric gauge theory as we pointed out in the previous section, are closely related to the c=1 non-critical string amplitude. The target space of the c=1 noncritical string is 2 dimensional, but the above uplift by the duality relation also suggests the existence of the non-critical M-theory [AK05, HK05] defined on higher 3 dimensional target space. There may appear q-deformed 2 dimensional Yang-Mills theory as a natural extension of the gamma function amplitudes. Finally, I would like to comment on connections to the non-perturbative formulation of M(atrix) theory by Banks, Fischler, Shenker and Susskind

250

Kazutoshi Ohta

(BFSS) [BFSS97] and type IIB string theory by Ishibashi, Kawai, Kitazawa and Tsuchiya (IKKT) [IKKT97]. The original BFSS and IKKT matrix model are considered to represent the 11 dimensional M-theory and 10 dimensional critical type IIB string theory, respectively. Now let us consider a non-perturbative formulation of the 7 dimensional topological M-theory or 6 dimensional topological B-model, where the number of matrices is reduced from the original matrix model formulation. Namely, we consider a partition function of a matrix quantum mechanics for five time-dependent k×k Hermite matrices Z R − 1 β dtTr( 12 Dt Y i Dt Yi + 14 [Y i ,Y j ]2 +··· ) (i, j = 1, . . . , 5), Z˜k = DY DΨ e g2 0

(25)

or a 0 dimensional matrix model with six k×k Hermite matrices Z ¯ Γ µ [Xµ ,Ψ ]) − 1 Tr 1 [X ,X ]2 + 1 Ψ Zk = DXDΨ e g2 ( 4 µ ν 2 (µ, ν = 1, . . . , 6).

(26)

Then the partition functions are exactly evaluated since the theories we are considering are topologically twisted, and we find that the partition functions of the discrete matrix model ZDMM or its q-deformation Zq-DMM are generating functions of the above size k partition function in the large N limit. Therefore, roughly speaking, the partition functions are related as X Z˜k q k |2 , (27) Zq-DMM ' | k

ZDMM ' |

X

Z k q k |2 ,

(28)

k

after the chiral decomposition in the large N limit3 . So the matrix model (25) or (26) actually describes the closed topological strings, since the partition functions give the amplitudes of the topological A/B-models on Calabi-Yau manifold as we have seen before. This is a non-trivial realization of the philosophy in [BFSS97] and [IKKT97]. Note also that the target space of the original matrix model is drastically changed to a different closed string target space. To summarize, if we truncate full string/M-theory to 7 or 6 dimensional topological string theory, or to 3 or 2 non-critical string theory, these are very nice models to show exactly the large N transition, open/closed string duality, counting BPS states, exact duality including all non-perturbative correction, and so on, since these models possess crucial integrable structures. So we can expect to understand essential properties of the whole string/M-theory if we investigate in detail these integrable models, which pick up good properties from the original string/M-theory. 3

To put it more precisely, we need to introduce extra matrices corresponding to the hypermuletiplets in order to control the positions of the eigenvalues of the matrices and obtain the moduli parameter dependence.


251

References [AK05]

Sergei Yu. Alexandrov and Ivan K. Kostov. Time-dependent backgrounds of 2d string theory: Non-perturbative effects. JHEP, 02:023, 2005. [AKMV04] Mina Aganagic, Albrecht Klemm, Marcos Marino, and Cumrun Vafa. Matrix model as a mirror of Chern-Simons theory. JHEP, 02:010, 2004. [AOSV05] Mina Aganagic, Hirosi Ooguri, Natalia Saulina, and Cumrun Vafa. Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys., B715:304–348, 2005. [BFSS97] Tom Banks, W. Fischler, S. H. Shenker, and Leonard Susskind. M theory as a matrix model: A conjecture. Phys. Rev., D55:5112–5128, 1997. [BT93] Matthias Blau and George Thompson. Lectures on 2-d gauge theories: Topological aspects and path integral techniques. Trieste HEP Cosmol., 0175:244, 1993. [BVS96] M. Bershadsky, C. Vafa, and V. Sadov. D-branes and topological field theories. Nucl. Phys., B463:420–434, 1996. [Con94] A. Connes. Noncommutative Geometry. Academic Press, 1994. [DV02] Robbert Dijkgraaf and Cumrun Vafa. Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys., B644:3–20, 2002. [EK82] Tohru Eguchi and Hikaru Kawai. Reduction of dynamical degrees of freedom in the large n gauge theory. Phys. Rev. Lett., 48:1063, 1982. [GV98] Rajesh Gopakumar and Cumrun Vafa. Topological gravity as large N topological gauge theory. Adv. Theor. Math. Phys., 2:413–442, 1998. [HK05] Petr Horava and Cynthia A. Keeler. Noncritical M-theory in 2+1 dimensions as a nonrelativistic fermi liquid. hep-th/0508024. [IKKT97] N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya. A large-N reduced model as superstring. Nucl. Phys., B498:467–491, 1997. [INOV03] Amer Iqbal, Nikita Nekrasov, Andrei Okounkov, and Cumrun Vafa. Quantum foam and topological strings. hep-th/0312022. [Mig75] A. A. Migdal. Recursion equations in gauge field theories. Sov. Phys. JETP, 42:413, 1975. [MMO05] Toshihiro Matsuo, So Matsuura, and Kazutoshi Ohta. Large N limit of 2d Yang-Mills theory and instanton counting. JHEP, 03:027, 2005. [MO06] So Matsuura and Kazutoshi Ohta. Localization on the D-brane, twodimensional gauge theory and matrix models. Phys. Rev., D73:046006, 2006. [Nek02] Nikita A. Nekrasov. Seiberg-Witten prepotential from instanton counting. hep-th/0206161. [NO03] Nikita Nekrasov and Andrei Okounkov. Seiberg-Witten theory and random partitions. hep-th/0306238. [ORV03] Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa. Quantum Calabi-Yau and classical crystals. hep-th/0309208 [Wit92] Edward Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9:303–368, 1992.


Instantons in Non(anti)commutative Gauge Theory via Deformed ADHM Construction Takeo Araki, Tatsuhiko Takashima and Satoshi Watamura Department of Physics Graduate School of Science Tohoku University Aoba-ku, Sendai 980-8578, Japan

Abstract We generalize the differential algebra on superspace to non(anti)commutative superspace by defining the deformed wedge product. Then, we formulate a non(anti)commutative version of the super ADHM construction which gives deformed instantons in N = 1/2 super Yang-Mills theory with U(n) gauge group.

1 Introduction It has been found that supersymmetric gauge theory defined on a kind of deformed superspace, called non(anti)commutative superspace, arises in superstring theory as a low energy effective theory on D-branes in the presence of constant graviphoton field strength [1]-[3]. In non(anti)commutative space, anticommutators of Grassmann coordinates become non-vanishing. Such a deformation of (Euclidean) four dimensional N = 1 super Yang-Mills (SYM) theory has been formulated by Seiberg [2] and it is sometimes called N = 1/2 SYM theory. It was argued by Imaanpur [4] that the anti-self-dual (ASD) instanton equations should be modified in the N = 1/2 SYM theory with self-dual (SD) non(anti)-commutativity. Solutions to those equations (deformed ASD instantons) have been studied by many authors [4]-[6]. It is well known that in the ordinary theory the instanton configurations of the gauge field can be obtained by the ADHM construction [7]. The authors of ref. [6] have studied string amplitudes in the presence of D(−1)-D3 branes with the background RR field strength and derived constraint equations for the string modes ending on D(−1)-branes, which are the ADHM constraints for the deformed ASD instantons. We show that we can obtain these constraints in the purely field 253

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and Satoshi Watamura

theoretic context, formulating a non(anti)commutative version of a superfield extension of the ADHM construction initiated by Semikhatov and Volovich [8]. We follow the notation and conventions in refs. [9, 10].

2 N = 1/2 SYM theory We briefly describe the non(anti)commutative deformation of N = 1 superspace and N = 1/2 SYM theory [2]. The non(anti)commutative deformation of N = 1 superspace is given by introducing non(anti)commutativity of the product of N = 1 superfields. This deformation is realized by the following Moyal type star product: f ∗ g = f exp(P )g,

1 ←− −→ P = − Qα C αβ Qβ , 2

(1)

where f and g are N = 1 superfields and Qα is the (chiral) supersymmetry generator. C αβ is the non-anticommutativity parameter and is symmetric: C αβ = C βα . The above star product gives the following relations among the chiral coordinates (y µ , θα , θ¯α˙ ): {θα , θβ }∗ = C αβ , [y µ , · ]∗ = 0, [θ¯α˙ , · }∗ = 0. Turning on such a deformation, the original action formulated in the N = 1 superfield formalism is deformed by the star product. The deformed N = 1 SYM theory has N = 1/2 supersymmetry, so that they are called N = 1/2 SYM theory. The action of N = 1/2 SYM theory is given by Z Z Z 1 4 2 α 2¯ ¯ α˙ ¯ S= (2) d x d θtrW ∗ W + d θtr W ∗ W α α˙ 16N g 2 where 1 ¯ ¯ α˙ −V Wα = − D e∗ ∗ Dα eV∗ , α˙ D 4

¯ α˙ = 1 Dα Dα eV∗ ∗ D ¯ α˙ e−V , W ∗ 4

n

(3)

z }| { ≡ and V ∗ · · · ∗ V . Here V = V a T a with V a the vector superfields and T the hermitian generators which are normalized as tr[T a T b ] = N δ ab . We may redefine the component fields of V in the WZ gauge such that the component gauge transformation becomes canonical (the same as the undeformed case). In [2], such a field redefinition is found and then the component action becomes Z h 1 1 4 ¯ σ µ Dµ λ + 1 D 2 S= tr d x − v µν vµν − iλ¯ 4N g 2 4 2 1 2 ¯ ¯ 2i i µν ¯ ¯ (4) − C vµν λλ + |C| (λλ) , 2 8 eV∗ a

P

1 n n!

where C µν ≡ C αβ (σ µν )α γ εβγ and |C|2 ≡ C µν Cµν .

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From the component action, we can see that the equations for SD instantons are unchanged compared to the undeformed case. Therefore, the SD instanton solutions are not affected by the deformation. On the other hand, the equations for ASD instantons should be modified. The action can be rewritten as [4] Z i h 1 2 1 i 1 2 1 µν SD µ 4 ¯ ¯ ¯ S= v + −i λ¯ σ D λ+ d x − λ λ tr C D + v v ˜ µ µν µν , (5) 4N g 2 2 µν 2 2 4 where v˜µν ≡ 12 εµνρσ vρσ . From this expression, we can see that configurations which satisfies the equations of motion and is connected to the ASD instantons when turning off the deformation are the solutions to the following deformed ASD instanton equations [4]: i SD ¯λ ¯ = 0, λ = 0, Dµ σ µ λ ¯ = 0, D = 0. vµν + Cµν λ 2

(6)

3 Differential algebra in the deformed superspace We take a geometrical approach to formulate the deformed super ADHM construction by generalizing the differential algebra: we extend the star product between superfields to the differential forms in superspace. The principle of our construction of the deformed differential algebra is that the operators Qα appearing in the star product are identified with the generators of supertranslation. As a result, the star product of differential forms is defined according to the representations of supersymmetry they belong to. Since the 1-form bases eA are supertranslation invariant, the action of Qα on eA is naturally defined as Qα (eA ) = 0. Then for a 1-form ω = eA ωA , it holds that Qα (ω) = (−)|A| eA Qα (ωA ). Using this action of Qα , we define the deformed wedge product of a p-form ωp and a q-form ωq as ∗ 1 ←− αβ −→ ωp ∧ ωq ≡ ωp ∧ exp − Qα C Qβ ωq , (7) 2 ← − → − where Q ( Q ) acts on ωp (ωq ) from the right (left) and the normal wedge product is taken for the resulting (transformed) differential forms. Note that ←− ω Qα = (−)|ω| Qα (ω). Hereafter we will suppress the wedge symbols. In the eA -basis, the product of the p- and q-form is simply given by the star product of the coefficients: ωp ∗ ωq = (−)(|A1 |+···+|Aq |)(|B1 |+···+|Bq |) eA1 · · · eAp eB1 · · · eBq ×(ωpAp ...A1 ∗ ωq Bq ...B1 ),

(8)

The exterior derivative d is defined as a map from a p-form to a p + 1-form by using the basis eA :

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dωp = eA1 · · · eAp eB DB ωpAp ...A1 p X + (−1)|Ar+1 |+···+|Ap | eA1 · · · deAr · · · eAp ωpAp ...A1

(9)

r−1

where deA is the same as the undeformed one. Let us mention about the relation of the above approach to the Hopf algebra approach taken in ref.[11, 12, 13]. As we described, the principle of the extention of the star product to the differential algebra is to identify the differential operator QA as the generator of the supersymmetry. This identification then defines the twist [14] of the supersymmetry algebra, which is a Hopf algebra including Poincare algebra. On the other hand the transformation of 1-forms is defined naturally by the action of QA as a Lie derivative and the action on the general forms follows by requireing the graded Leibniz rule, i.e., being the (graded) module algebra. This also leads to the twisted wedge product defined in eq.(7). We see that the deformed differential algebra defined above is consistent with the N = 1/2 SYM theory described in the previous section, in the sense that the curvature 2-from superfield will correctly reproduce the field strength ¯ α˙ in (3) (after imposing appropriate constraints as in the superfield Wα and W undeformed case [15]) based on the deformed differential algebra. Given a connection 1-form superfield φ, the curvature superfields FAB are obtained as the coefficient functions of the 2-form superfield F constructed in a standard way: F = dφ + φ ∗ φ. Therefore, we find the curvature superfields FAB as FAB = DA φB − (−)|A||B| DB φA − [φA , φB }∗ + TAB C φC ,

(10)

where TAB C is the torsion defined by deC = 21 eA eB TBA C with non-vanishing µ elements given by Tαβ˙ µ = Tβα = 2iσαµβ˙ . The proper constraints for the ˙ curvature superfields to give the N = 1/2 SYM theory turn out to be Fαβ = 0, Fα˙ β˙ = 0, Fαβ˙ = 0,

(11)

where the curvature superfields are given by (10) (see [15] for the undeformed case). We refer to these constraints as the Yang-Mills constraints. These constraints are solved in a parallel way to the undeformed case and the invariant action with respect to super- and gauge symmetry can be constructed which coincides with the action S given in (2). Therefore, imposing the Yang-Mills constraints (11), the N = 1/2 SYM theory can be correctly reproduced in a geometrical way based on the deformed differential algebra.

4 Review of the N = 1 super ADHM construction Before describing the deformed version, we briefly review the N = 1 super ADHM construction which was initiated by Semikhatov and Volovich [8]. Here we follow ref. [9].


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The U(n) (or SU(n)) k instanton configurations can be given by the ADHM construction [7]. Define ∆α (x) such as ∆α (x) = aα + xαα˙ bα˙

(12)

where aα and bα˙ are constant k × (n + 2k) matrices and xαα˙ ≡ ixµ σαµα˙ . We assume that ∆α has maximal rank everywhere except for a finite set of points. ˙ Its hermitian conjugate ∆†α ≡ (∆α )† is given by ∆†α (x) = a†α + b†β˙ xβα . Then the gauge field vµ is given by vµ = −2iv † ∂µ v, where v is the set of the normalized zero modes of ∆α : ∆α v = 0, v † v = 1n . For later use we define f as the inverse matrix of the quantity f −1 ≡ 21 ∆α ∆†α . In the superfield formalism, the ASD super instanton equations are equivalent to the following super ASD condition [8]: Fµα˙ = 0,

?Fµν = −Fµν .

(13)

Note that the latter equation follows from the former as long as the 2-form F satisfies the Bianchi identities and the (undeformed) Yang-Mills constraints. The super ADHM construction gives the solutions to (13) [9]. Define a superfield extension of ∆α (x): ˆα = ∆α (y) + θα M, ∆

(14)

where ∆α (y) is the zero dimensional Dirac operator in the ordinary ADHM construction with replacing xµ by the chiral coordinate y µ = xµ + iθσ µ θ¯ and M is a k × (n + 2k) fermionic matrix which includes the fermionic moduli. We suppose that ∆ˆα has a maximal rank almost everywhere as in the ordinary ADHM construction. Its ‡-conjugate [9] ∆ˆ‡α is found to be ∆ˆ‡α = ∆†α (y) + θα M† . As ∆ˆα has n zero modes we collect them in a matrix superfield vˆ[n+2k]×[n] and require that vˆ satisfies the normalization condition: ∆ˆα vˆ = 0, vˆ‡ vˆ = 1. (Its ‡-conjugate vˆ‡ satisfies vˆ‡ ∆ˆ‡α = 0.) Then the connection 1-form superfield φ is given by φ = −ˆ v ‡ dˆ v.

(15)

where d is exterior derivative of superspace. The connection φ defines the curvature ˆ α β d∆ˆβ vˆ, (16) F = dφ + φφ = vˆ‡ d∆ˆ‡α K ˆ βγ = ˆ α β is defined such that K ˆ −1 α β K ˆ −1 α β ≡ ∆ˆα ∆ˆ‡β and K where K ˆ −1 β γ = δ γ 1k . The curvature superfield Fµν becomes ASD if K ˆ satˆ αβ K K α ‡β β ˆ ˆ isfies ∆α ∆ ∝ δα and thus ˆ −1 α β = δαβ fˆ−1 K

(17)

where fˆ−1 ≡ 21 ∆ˆα ∆ˆ‡α is a k × k matrix superfield. There exists fˆ because ˆα has maximal rank. The above condition (17) leads we have assumed that ∆

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to both the bosonic and fermionic ADHM constraints. When eq. (17) holds, i.e., the parameters in ∆ˆα are satisfying both bosonic and fermionic ADHM constraints, we can check that the above F satisfies the Yang-Mills constraints and the super ASD condition. To ensure the WZ gauge of the superfields obtained by the super ADHM ¯ α˙ vˆ = 0 and construction, we impose on the zero mode vˆ of ∆ˆα the conditions D vˆ‡ ∂θ∂α vˆ = 0. Then vˆ is determined as vˆ = v + θ γ (∆†γ f Mv) + θθ( 21 M† f Mv), and the connection φµ in (15) correctly gives the super instanton configuration in the WZ gauge: Its lowest component is the instanton gauge field and the θ-component is the fermion zero mode.

5 Deformed super ADHM construction The deformed super ASD condition is found to be of the same form as the super ASD condition (13) but the product replaced with the star product (1): Fµα˙ = 0,

?Fµν = −Fµν ,

(18)

where the curvature superfields FAB are given by eq.(10). We can prove the equivalence of the condition (18) and the deformed equations (6). One would expect that solutions to eq. (18) can be constructed by the super ADHM construction, replacing each product with the star product (1). For the deformed super ASD instantons, φµ in the WZ gauge becomes ¯ (y). This leads us to adopt ∆ˆα in our super ADHM φµ = − 2i vµ + iθσµ λ construction with the same form as before: ˆα = ∆α (y) + θα M. ∆

(19)

Then, according to the ‡-conjugation rules, we have ∆ˆ‡α = ∆‡α (y) + θα M‡ . We collect the n zero modes of ∆ˆ into a matrix form u ˆ[n+2k]×[n] and require ˆα ∗ u it to be normalized with respect to the star product: ∆ ˆ = 0, u ˆ‡ ∗ u ˆ = 1n . ˆ ∗α β (α, β = 1, 2) as the “inverse” matrices of K ˆ −1 α β ≡ Define k×k matrices K ∗ ˆ‡β such that K ˆ −1 α β ∗ K ˆ ∗β γ = K ˆ ∗α β ∗ K ˆ −1 β γ = δ γ 1k . Then we have a ∆ˆα ∗ ∆ ∗ ∗ α ˆ‡α ∗ K ˆ ∗α β ∗ ∆ˆβ . relation u ˆ∗u ˆ‡ = 1n+2k − ∆ With the use of the zero modes u ˆ of ∆ˆα , the connection φ is given by ‡ φ = −ˆ u ∗ dˆ u, and the curvature 2-form becomes ˆ ∗α β ∗ d∆ˆβ ∗ u F = dφ + φ ∗ φ = u ˆ‡ ∗ d∆ˆ‡α ∗ K ˆ

(20)

ˆ ∗α β ∗ DB} ∆ˆβ ∗ u which reads FAB = −ˆ u‡ ∗ D[A ∆ˆ‡α ∗ K ˆ, especially Fµν = † αα ‡ ˙ ˆ β β˙ ˆ∗ u ˆ ∗ bα˙ σ ¯[µ K∗α σν] β β˙ b ∗ u ˆ. Thus Fµν becomes ASD (see eq. (18)) if K commutes with the sigma matrices σµ : ˆα ∗ ∆ ˆ‡β = K ˆ −1 α β ∝ δ β . ∆ ∗ α

(21)


259

ˆα is a Then we immediately find that Fα˙ β˙ = Fαβ˙ = 0 and Fµα˙ = 0, because ∆ chiral superfield. We can also check that Fαβ = 0 with the use of the constraint ˆα = εαβ (M + 4θ¯ ˙ bβ˙ ) and Dβ ∆ˆ‡α = δ α (M‡ + 4b† θ¯β˙ ), (21), the relations Dβ ∆ β β β˙ and the fact that Fαβ is symmetric with respect to α and β. Therefore, we have shown that the above described super ADHM construction gives curvature superfields that satisfy the Yang-Mills constraints (11) and the ASD conditions (18) if the condition (21) is imposed. ˆα ∗ ∆ ˆ‡β = ∆ˆα ∆ˆ‡β − 1 εαγ C γβ MM‡ , the requirement Since we can write ∆ 2 (21) leads to the following deformed bosonic ADHM constraint ∆α ∆‡β − 1 γβ MM‡ ∝ δαβ and the fermionic ADHM constraint ∆α M‡ + M∆‡ α = 2 εαγ C 0. These constraints agree with those in [6] obtained by considering string amplitudes. We can rewrite the deformed bosonic ADHM constraints in another form as follows. Let us denote ‡ ∆1 J[k]×[n] z¯2 1k + B2 ‡[k]×[k] z¯1 1k + B1 ‡[k]×[k] = , (22) ∆2 I[k]×[n] −z1 1k − B1[k]×[k] z2 1k + B2[k]×[k] where z1 ≡ y21˙ , z2 ≡ y22˙ and I ≡ ω2 , J ‡ ≡ ω1 , B1 ≡ a021˙ , B2 ≡ a022˙ . Then the bosonic ADHM constraints reads II ‡ − J ‡ J + [B1 , B1 ‡ ] + [B2 , B2 ‡ ] − C 12 MM‡ = 0, (23) 1 (24) IJ + [B2 , B1 ] − C 11 MM‡ = 0. 2 We can give an expression in terms of the ADHM data ∆α and M, of the general solution in the WZ gauge obtained by our construction, and have shown in [10] that it gives the known U(2) one instanton solution. In summary, we have correctly deformed the super ADHM construction to give solutions to the deformed ASD instantons in N = 1/2 SYM theory. We see that deformation terms emerge in the bosonic ADHM constraints (see also [6]), which are comparable with the U(1) terms due to space-space noncommutativity [16]. Our formulation reveals the geometrical meaning of those deformation terms as non(anti)commutativity of superspace. However, it needs a further study to clarify how those terms can be interpreted in the hyper-K¨ ahler quotient construction [17]. Acknowledgments: This research is partly supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 13640256, 13135202 and 14046201).

References 1. H. Ooguri and C. Vafa, Adv. Theor. Math. Phys. 7 (2003) 53, hep-th/0302109; Adv. Theor. Math. Phys. 7 (2004) 405, hep-th/0303063; J. de Boer, P. A. Grassi and P. van Nieuwenhuizen, hep-th/0302078.

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2. N. Seiberg, JHEP 0306 (2003) 010, hep-th/0305248. 3. N. Berkovits and N. Seiberg, JHEP 0307 (2003) 010, hep-th/0306226. 4. A. Imaanpur, JHEP 0309 (2003) 077, hep-th/0308171; JHEP 0312 (2003) 009, hep-th/0311137. 5. P. A. Grassi, R. Ricci and D. Robles-Llana, JHEP 0407 (2004) 065, hepth/0311155; R. Britto, B. Feng, O. Lunin and S. J. Rey, Phys. Rev. D 69, 126004 (2004), hep-th/0311275; S. Giombi, R. Ricci, D. Robles-Llana and D. Trancanelli, JHEP 0510 (2005) 021, hep-th/0505077. 6. M. Billo, M. Frau, I. Pesando and A. Lerda, JHEP 0405 (2004) 023, hepth/0402160. 7. M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld and Y. I. Manin, Phys. Lett. A 65 (1978) 185. 8. A. M. Semikhatov, JETP Lett. 35 (1982) 560 [Pisma Zh. Eksp. Teor. Fiz. 35 (1982) 452]; Phys. Lett. B 120 (1983) 171; I. V. Volovich, Phys. Lett. B 123 (1983) 329; Theor. Math. Phys. 54 (1983) 55 [Teor. Mat. Fiz. 54 (1983) 89]. 9. T. Araki, T. Takashima and S. Watamura, JHEP 0508 (2005) 065, hepth/0506112. 10. T. Araki, T. Takashima and S. Watamura, JHEP 0512 (2005) 044, hepth/0510088. 11. M. Chaichan, P. Kulish, K. Nishijima, and A. Tureanu, Phys. Lett. B604 (2004) 98. 12. P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp and J. Wess, Class. Quant. Grav. 22 (2005) 3511. 13. Y. Kobayashi, Article in this book and references therein. 14. V. G. Drinfeld, Leningrad Math. J. 1 (1990) 1419;Alg. Anal. 1 N 6 (1989)114. 15. R. Grimm, M. Sohnius and J. Wess, Nucl. Phys. B 133 (1978) 275. 16. N. Nekrasov and A. Schwarz, Commun. Math. Phys. 198 (1998) 689, hepth/980206. 17. N. J. Hitchin, A. Karlhede, U. Lindstr¨ om and M. Roˇcek, Commun. Math. Phys. 108 (1987) 535.

Noncommutative Deformation and Drinfel’d Twisted Symmetry Yoshishige Kobayashi Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan.

Abstract The Hopf algebraic construction of the non(anti)commutative superspace is discussed. In that context, Noncommutative (super)space arises as a representation of the modified Hopf algebra by the Drinfel’d twist. This construction gives the same result which is given by using the ordinary Moyal product, but has some advantages in considering the symmetry of the noncommutative theory. We construct the twisted super-Poincaré algebra, and obtain the several non(anti)commutative superspaces.

1 Introduction Noncommutative geometry has become an important subject in theoretical physics, ever since its realization was found in superstring theory[1]. Spacetime noncommutativity is deeply related to the structure of spacetime, and it may appear as an evidence of the quantized spacetime, or as a result of the quantum correction of a gravitational theory. We expect that the well-known theories, e.g. the standard model, to be modified on noncommutative spacetime. However, if we consider such theories, we often lose some symmetries, because the noncommutativity parameters are introduced into the theory by hand. In the majority of cases the noncommutativity parameters are dimensionful, and they often break the symmetry of the original (commutative) theory. Lorentz symmetry is rather important in particle physics. This symmetry is an experimental fact, and no definite evidence of its violation has been found from experimental tests or observations. The trouble is that certain classes of spacetime noncommutativity break Lorentz symmetry. Let us consider the following commutator relation between spacetime coordinates, 261

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[xµ , xν ] = iΘµν .

(1)

Here Θµν is a constant with antisymmetric indices. It is the most common noncommutativity which is extensively studied. This relation breaks Lorentz covariance of the theory, because no constant tensor is transformed in the Lorentz covariant way, except for the (Minkowski) metric and the Levi-Civita tensor εµνρσ . Usually it is explained by the argument that Θ µν is so small compared to our scale, that we cannot observe it. Therefore there is no inconsistency phenomenologically. But the explanation is not satisfactory theoretically, because an ordinary theory is built closely on the symmetry. One proposal for the problem is suggested[2]. In [2], the authors argued that Hopf algebra is a suitable framework for a description of noncommutative theory. In that context, we can think of a modified Lorentz symmetry, namely the twisted Lorentz symmetry, which is maintained on noncommutative space, even though the original Lorentz symmetry is broken indeed by the relation (1). On the other hand, another noncommutative geometry was obtained from superstring theory, that is the non-anticommutative superspace[3]. Superspace is a coordinate system which includes not only the ordinary spacetime coordinate but the anti-commutative, i.e. Grassmann number, coordinate. Supersymmetry, which is the symmetry between bosons and fermions, is naturally represented on superspace. It was revealed that in the graviphoton background, with taking a certain limit, the theory is described effectively by changing the anti-commutator relation, namely {θα , θβ } = C αβ 6= 0,

(2)

where θα is fermionic coordinates of superspace, α and β are spinor indices, and C αβ is a constant with symmetric indices. Like the spacetime noncommutativity, symmetry breaking occurs on noncommutative superspace. It is known that non-anticommutativity (2) breaks supersymmetry by half of the numbers of the degrees[3]. Our aim is to apply the Hopf algebraic method to a theory on superspace. Although superspace is different from ordinary spacetime owing to the anticommutative property, the Hopf algebra method can be applied with the applicable definitions. In this formulation, we can maintain the modified supersymmetry, namely the twisted supersymmetry. The next section is a very brief introduction to Hopf algebra and its twist. In section 3, following [2], we construct the twisted Poincaré algebra. The twist operation changes the product of the representation space, consequently the spacetime coordinate becomes noncommutative. In section 4, we consider the twisted super-Poincaré algebra on superspace. As the representations of the twisted super-Poincaré symmetry, several types of non(anti)commutative superspaces are obtained.

Noncommutative Deformation and Drinfel’d Twisted Symmetry

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2 Hopf algebra In this section, we review Hopf algebra and the Drinfel’d twist, to construct a modified symmetry algebra. For more details on Hopf algebra and the related subject, see references[4, 5, 6]. A Hopf algebra H(H, m, i, ∆, , γ; K) is a linear space H over K together with the five linear maps on the tensor product of H, namely m,i,∆, and γ. K is some field or ring. These maps, the product m : H ⊗ H → H, the unit i : K → H, the coproduct ∆ : H → H ⊗ H, the counit : H → K and the antipode γ : H → H, satisfy the following conditions. m ◦ (id ⊗ m) = m ◦ (m ⊗ id), m ◦ (i ⊗ id) = id = m ◦ (id ⊗ i),

(3) (4)

(id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆, (id ⊗ ) ◦ ∆ = id = ( ⊗ id) ◦ ∆,

(5) (6)

m ◦ (γ ⊗ id) ◦ ∆ = i ⊗ = m ◦ (id ⊗ γ) ◦ ∆.

(7)

Here id stands for the identity map of H. Next, we make a stage on which the symmetry algebra acts. A H-module, or a representation space, V is a vector space over K, on which the Hopf algebra H acts as an endomorphism of V , H : V → V . The action of h ∈ H on v ∈ V , h : a → a0 , denoted by h . a = a0 , satisfies g . (h . v) = (gh) . v,

(8)

for ∀v ∈ V ,∀g, h ∈ H. We consider the case that V is an algebra. To be compatible with the Hopf algebra H, we require the relations, h . (v · w) = h . m(v ⊗ w) = m ◦ ∆(h) . (v ⊗ w) = m(h(1) . v ⊗ h(2) . w).

(9)

In the last line we use Sweedler’s notation, X (i) (i) ∆(h) = h1 ⊗ h2 = h(1) ⊗ h(2) ,

(10)

i

The Drinfel’d twist is a systematic procedure to make another Hopf algebra from a given Hopf algebra by Drinfel’d[7]. Consider an invertible biproduct element F ∈ H ⊗H, here we call it a twist element. Using F, the new coproduct and antipode are defined as follows. ∆t (h) = F∆(h)F −1 , γt (h) = U γ(h)U

−1

.

The twist element should satisfy the following two relations,

(11) (12)

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F12 (∆ ⊗ id)F = F23 (id ⊗ ∆)F, ˆ = (id ⊗ )F. ( ⊗ id)F = 1

(13) (14)

The symbol F12 is used to represent the action of F on the first and second element, namely F12 = F ⊗ id = F[1] ⊗ F[2] ⊗ id, (15) and similar for F23 . Here we use Sweedler’s notation again, X (i) (i) F= F1 ⊗ F2 ≡ F[1] ⊗ F[2] .

(16)

i −1 −1 U and U −1 are given by U = F[1] γ(F[2] ) and U −1 = γ(F[1] )F[2] respectively, −1 −1 −1 where F ≡ F[1] ⊗ F[2] . Eq.(13) and (14) are crucial for the twist operation. Once the relations are satisfied, the algebra Ht (H, m, i, ∆t , , γt ; K) also satisfies Eq.(3)-(7), then Ht becomes a new Hopf algebra.

3 Twisted Poincar´ e symmetry Following [2], we consider the Poincaré algebra as the Lie algebra of the Lorentz symmetry and the translation symmetry. But this procedure also works for a general symmetry algebra. The Poincaré algebra P consists of the translational generators Pµ and Lorentz generators Mµν , which satisfy the following commutator relations, [Pµ , Pν ] = 0 [Mµν , Pρ ] = −iηρµ Pν + iηρν Pµ , [Mµν , Mρσ ] = iηνρ Mµσ − iηµρ Mνσ − iηνσ Mµρ + iηµσ Mνρ ,

(17)

where Greek characters are the indices of spacetime coordinate. The universal enveloping Poincaré algebra U(P) become a Hopf algebra with the following definitions. m(g ⊗ h) = gh, ˆ i(k) = k 1,

(18) (19)

ˆ+1 ˆ ⊗ g, ∆(g) = g ⊗ 1 (g) = 0,

(20) (21)

γ(g) = −g,

(22)

ˆ is used to represent the unit element for all g, h ∈ P. Here the hatted symbol 1 H, to distinguish it from the unit of the field K. In addition, ˆ =1 ˆ ⊗ 1, ˆ ∆(1) ˆ (1) = 1, ˆ = 1, ˆ γ(1)

(23) (24) (25)


265

ˆ For algebraic consistency, we require hold for 1. ∆(hg) = ∆(h)∆(g), (hg) = (h)(g),

(26) (27)

γ(hg) = γ(g)γ(h).

(28)

Using Eq. (26)-(28), the definitions (19)-(22) are extended to the whole U(P). Next we will twist the the Poincaré algebra by an appropriate twist element. Eq.(13) and (14) are nontrivial equations, though, the twist of a special form satisfies easily the equations. Let us consider the following F.   X F = exp  cij hi ⊗ hj  . (29) i,j

Here cij is some constant, and hi is an element in H. We assume that all hi commute with each other, namely [hi , hj ] = 0 ∀i, j. That means that all hi make some Abelian subsector in H. In fact, F satisfies Eq.(13). For more general twist, we will see it in the later section. The right-hand side of Eq.(29) is written in a formal expansion series, ˆ⊗1 ˆ + cij hi ⊗ hj + 1 (ckl hk ⊗ hl )(cmn hm ⊗ hn ) + · · · . F =1 2!

(30)

The counit condition (14) is satisfied clearly, because the second term and the subsequent terms are all annihilated by Eq.(21). Twisting the Hopf algebra does not change the algebra, but accompanies the modification of the multiplication rule in the representation space. The product is replaced by the star product, which is defined as v ? w ≡ m(F −1 . v ⊗ w).

(31)

Then the algebra of linear space V with the star product becomes consistent with the twisted algebra, i.e. V is the correct representation of the twisted Hopf algebra. Note that this product is associative, and obviously reduced to the ordinary product in the limit cij → 0. In the coordinate representation, Pµ and Mµν are written as derivative operators on the coordinate. Pµ = i∂µ , Mµν = i(xµ ∂ν − xν ∂µ ).

(32)

The idea is that the translation generator Pµ , which makes the Abelian subsector in the Poincaré algebra, is written by i∂µ . The twist, i µν F P P = exp Θ Pµ ⊗ P ν , (33) 2

266


is an appropriate twist element. In the coordinate representation, F P P is written as i (34) F P P = exp − Θµν ∂µ ⊗ ∂ν . 2 The above expression is very similar to the Moyal product, and in fact it works like that exactly. The product of spacetime coordinates is changed in such a way that xµ ? xν = m((F P P )−1 . xµ ⊗ xν ) i ρσ µ ν = m exp − Θ Pρ ⊗ Pσ . (x ⊗ x ) 2 i ρσ µ ν = m exp Θ ∂ρ ⊗ ∂σ (x ⊗ x ) 2 i ρσ µ ν µ ν = m x ⊗ x + Θ δρ ⊗ δ σ 2 i = xµ · xν + Θµν , 2

(35)

and therefore the spacetime coordinate becomes noncommutative, [xµ , xν ]? ≡ xµ ? xν − xν ? xµ = iΘµν 6= 0.

(36)

Now we turn to the modification of the symmetry algebra on such noncommutative representation space. While the coproduct of Pµ is same as before, the coproduct of Mµν is modified by the twist, P ˆ+1 ˆ ⊗ Mµν ∆P (Mµν ) = Mµν ⊗ 1 t 1 − Θρσ [(ηρµ Pν − ηρν Pµ ) ⊗ Pσ + Pρ ⊗ (ησµ Pν − ησν Pµ )] . 2 (37)

Consequently, the action of the Lorentz generator Mµν on the product in the representation space is changed. For example, the commutator is transformed such that −1 Mµν [xρ , xσ ]? = Mµν ◦ m (F PP ) xρ ⊗ xσ − xσ ⊗ xρ −1 = m (F PP ) ∆t (Mµν ) . xρ ⊗ xσ − xσ ⊗ xρ n h −1 ˆ+1 ˆ ⊗ (ixµ ∂ν − ixν ∂µ ) (ixµ ∂ν − ixν ∂µ ) ⊗ 1 = m (F PP ) 1 µ0 ν 0 + Θ (ηµ0 µ ∂ν − ηµ0 ν ∂µ ) ⊗ ∂ν 0 + ∂µ0 ⊗ (ην 0 µ ∂ν ) − ην 0 ν ∂µ ) 2 i xρ ⊗ x σ − x σ ⊗ x ρ = 0.

(38)


267

So the noncommutativity parameter Θ µν = [xρ , xσ ]? is annihilated by Mµν , which is a desirable behavior because Θ µν is constant. The product xµ xν in ordinary commutative spacetime is associated with x{µ ? xν} ≡ 12 (xµ ? xν + xν ? xµ ). Its transformation property is the same as xµ xν . A brief calculation shows that Mµν x{ρ ? xσ} = iηνρ x{µ ? xσ} − iηνσ x{ρ ? xµ} − iηµρ x{ν ? xσ} + iηµσ x{ν ? xρ} (39) Like above, despite the product changed by the star, the twisted symmetry generator acts on the noncommutative space like the original symmetry on commutative space. So once we find an appropriate twist, the twisted symmetry is guaranteed by construction for any theory on noncommutative space which this procedure can achieve.

4 Super-Poincar´ e algebra and Non(anti)commutative Superspace The Hopf algebraic method is useful for constructing a noncommutative theory. We apply it to a supersymmetric theory on non(anti)commutative superspace[8]. The extension to a supersymmetric case is almost straightforward, but some equipments are needed. • To obtain superspace noncommutativity, super-Poincaré algebra is used as a symmetry algebra, instead of the Poincaré algebra. • Since the super-Poincaré algebra contains fermionic generators, we should use Z2 graded Hopf algebra. The non(anti)commutative superspace is realized as the representation of the twisted super-Poincaré algebra. Because the fermionic generators anti-commute, the definition of the product of the Hopf algebra H should be modified with an appropriate sign flip. (a ⊗ b)(c ⊗ d) = (−1)|b||c| (ac ⊗ bd). a, b ∈ H Here |a| stands for the fermionic character of the element a, 0 if a is fermionic |a| = 1 if a is bosonic.

(40)

(41)

The Z2 grading is extended to all order of the product, so as to change a sign whenever fermionic elements jump over each other. For instance, (a ⊗ b ⊗ c)(d ⊗ e ⊗ f ) = (−1)|c|(|d|+|e|)+|b||d|(ad ⊗ be ⊗ cf ).

(42)

268


In addition, we allow not only complex numbers, but also Grassmann number ring to be the base K. Therefore, for consistency, we impose the anticommutative property of a fermionic number λ ∈ K with a fermionic element hi ∈ H, λh1 ⊗ h2 ⊗ h3 = (−1)|λ||h1 | h1 λ ⊗ h2 ⊗ h3 = (−1)|λ||h1 | h1 ⊗ λh2 ⊗ h3 = (−1)|λ|(|h1 |+|h2 |) h1 ⊗ h2 ⊗ λh3 .

(43)

With these definitions, the twist equation (13) of the extended version can be proved. Let us think the twist element which has the form, F = exp cij Gi ⊗ G0j . (44) Here cij is a constant, which can be a Grassmann number as well as a cnumber. G and G0 are generators, which can be either bosonic or fermionic. Assume that all G commute or anti-commute with each other, namely [Gi , Gj ]± ≡ Gi Gj − (−1)|Gi ||Gj | Gj Gi = 0

(45)

The left hand side(LHS) of the twist equation is expanded, F12 (∆0 ⊗ id)F

ˆ ⊗ Gi ⊗ G0j (46) ˆ exp cij Gi ⊗ 1 ˆ ⊗ G0j + 1 = exp cij Gi ⊗ G0j ⊗ 1

We consider the following commutator, ij ˆ ckl (Gk ⊗ 1 ˆ ⊗ G0l + 1 ˆ ⊗ Gk ⊗ G0l ) c Gi ⊗ G0j ⊗ 1, n kl 0 0 = cij ckl (−1)|c |(|Gi |+|Gj |)+|Gj ||Gk | Gi Gk ⊗ G0j ⊗ G0l o kl 0 +(−1)|c |(|Gi |+|Gj |) Gi ⊗ G0j Gk ⊗ G0l n ij 0 0 0 − ckl cij (−1)|c |(|Gl |+|Gk |)+(|Gi |+|Gj |)|Gl | Gk Gi ⊗ G0j ⊗ G0l o ij 0 0 0 +(−1)(|c |+|Gi |)(|Gk |+|Gl |)+|Gj ||Gl | Gi ⊗ Gk G0j ⊗ G0l = 0.

(47)

From the Baker-Campbell-Hausdorff formula, the right hand side(RHS) of Eq.(46) can be rewritten as ˆ + Gi ⊗ 1 ˆ ⊗ G0j + 1 ˆ ⊗ Gi ⊗ G0j . (48) (46) = exp cij Gi ⊗ G0j ⊗ 1 In a similar way, the RHS of Eq.(13) is F23 (id ⊗ ∆0 )F (49) ˆ ⊗ Gi ⊗ G0j exp cij (Gi ⊗ G0j ⊗ 1 ˆ + Gi ⊗ 1 ˆ ⊗ G0j ) = exp cij 1 ˆ ⊗ Gi ⊗ G0j + Gi ⊗ G0j ⊗ 1 ˆ + Gi ⊗ 1 ˆ ⊗ G0j = exp cij 1 (50)


269

Therefore, Eq.(13) is proved. The commutator relations of N = 1 super-Poincaré algebra are [Pµ , Pν ] = 0, [Mµν , Mρσ ] = iηνρ Mµσ − iηµρ Mνσ − iηνσ Mµρ + iηµσ Mνρ , [Mµν , Pρ ] = −iηρµ Pν + iηρν Pµ , ¯ α˙ ] = 0, [Pµ , Qα ] = 0, [Pµ , Q α˙ ¯ β˙ ¯ α˙ ] = i (¯ , [Mµν , Q σµν ) β˙ Q

β

[Mµν , Qα ] = i (σµν )α Qβ , ¯ ˙ } = 2σ µ Pµ . {Qα , Q

(51)

αβ˙

β

The generators are written as differential operators on superspace, Pµ = i∂µ , β

Mµν = i(xµ ∂ν − xν ∂µ ) − iθα (σµν )α

∂ ∂ α˙ − iθ¯α˙ (¯ σµν ) β˙ ¯ , ∂θβ ∂ θβ˙

∂ ˙ − σαµβ˙ θ¯β ∂µ , ∂θα ¯ α˙ = −i ∂ + θβ σ µ ∂µ . Q β α˙ ∂ θ¯α˙ Qα = i

(52)

¯ α˙ ) make an In super-Poincaré algebra, Pµ and Qα (otherwise Pµ and Q Abelian subalgebra. So any linear combination of the biproduct which includes only P and Q works as a twist, but not all combinations are meaningful and consistent with the noncommutativity. We take the following kinds of twists. • Q-Q Twist

F

QQ

1 = exp − C αβ Qα ⊗ Qβ 2

(53)

F QQ causes the desired non-anticommutative superspace, {θα , θβ }? = C αβ ,

(54)

[xµ , xν ]? =

˙ C αβ σαµγ˙ σβν δ˙ θ¯γ˙ θ¯δ ,

(55)

[xµ , θα ]? =

˙ −iC αβ σβµβ˙ θ¯β ,

(56)

the above results agree with [3]1 . • P -Q Twist i µα = exp λ (Pµ ⊗ Qα − Qα ⊗ Pµ ) . 2

F 1

PQ

Except for that they used the chiral coordinate base.

(57)

270

Yoshishige Kobayashi ˙

˙

[xµ , xν ]? = λµα σαν β˙ θ¯β − λνα σαµβ˙ θ¯β , [xµ , θα ]? = iλµα , α β θ , θ ? = 0.

(58)

In this twist, the noncommutativity parameter λµα is not an ordinary c-number, but a Grassmann number. This type of noncommutativity between spacetime coordinate and fermionic coordinate of the superspace has been discussed in the literature[9]. • Mixed Twist Of course, the linear combination of the previous three twists is a proper twist. i 1 i µν Θ Pµ ⊗ Pν + λµα (Pµ ⊗ Qα − Qα ⊗ Pµ ) − C αβ Qα ⊗ Qβ . 2 2 2 (59) The noncommutativity also result in the linear combination,

F mix = exp

˙

˙

˙

[xµ , xν ]? = iΘµν + C αβ σαµγ˙ σβν δ˙ θ¯γ˙ θ¯δ + λµα σαν β˙ θ¯β − λνα σαµβ˙ θ¯β , ˙

[xµ , θα ]? = iλµα − iC αβ σβµβ˙ θ¯β , α β θ , θ ? = C αβ .

(60)

¯ α˙ is Note that under the above twists, the coproduct of Mµν and Q changed, however we do not write down the explicit form here. They modify the action of the product of representation space. That gives the appropriate transformation relations of superspace coordinates on the corresponding non(anti)commutative superspace, in the twisted supersymmetric way.

5 Conclusion We constructed the Drinfel’d twisted (N = 1) super-Poincaré algebra by using Hopf algebra and the Drinfel’d twist operation. Several appropriate twist elements have been found, and the non(anti)commutative superspace is obtained as a representation space of the twisted algebra. It provides quite the same non(anti)commutative superspace which has been formulated by the Moyal product in a physical context previously. The crucial difference between the two procedures is the reinterpretation of the origin of noncommutativity. In the twist sense, noncommutativity is a consequence from the modification of the Hopf coalgebra. The structure of the algebra itself is not modified. As a result, the Hopf algebra construction always keeps the twisted symmetry. Here only N = 1 supersymmetry is treated, but the twist for the extended supersymmetry case had been discussed[8][10].


271

Hopf algebra and the twist method is systematic. Once we find the desired twist element, the procedure is almost automatic, and it gives the way to construct the representation space without ambiguity. It is hoped that this research contributes to understanding of symmetry on noncommutative space and provides new insights.

References 1. N. Seiberg and E. Witten, “String Theory and Noncommutative Geometry,” JHEP 9909, 032(1999)[hep-th/9908142]. 2. M. Chaichian, P. P. Kulish, K. Nishijima and A. Tureanu, “On a LorentzInvariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT,” Phys. Lett. B604, 98 (2004)[hep-th/0408069]. 3. N. Seiberg, “Noncommutative superspace, N = 1/2 supersymmetry, field theory and string theory,” JHEP 0306, 010 (2003) [hep-th/0305248]. 4. J. Fuchs, ”Affine Lie Algebras and Quantum Groups,” Cambridge University Press(1992), Cambridge. 5. V. Chari and A. Pressley, “A Guide to Quantum Groups,” Cambridge University Press(1994), Cambridge. 6. S. Majid, “Foundations of Quantum Group Theory,” Cambridge University Press(1995), Cambridge. 7. V. G. Drinfel’d, “Quasi-Hopf Algebras,” Leningrad Math. J. 1, 6, 1419(1990). 8. Y. Kobayashi and S. Sasaki, “Lorentz invariant and supersymmetric interpretation of noncommutative quantum field theory,” Int. J. Mod. Phys. A 2071757188(2005)[hep-th/0410164]. 9. D. Klemm, S. Penati and L. Tamassia “Non(anti)commutative superspace,” Class. Quant. Grav. 20, 2905-2916(2003)[hep-th/0104190]; J. de Boer, P. A. Grassi and P. van Nieuwenhuizen “Non-commutative superspace from string theory,” Phys. Lett. B574, 98-104(2003)[hep-th/0302078]; Y. Kobayashi and S. Sasaki, “Nonlocal Wess-Zumino model on nilpotent noncommutative superspace,” Phys. Rev. D 72065015(2005)[hep-th/0505011]. 10. B. M. Zupnik, “Twist-deformed supersymmetries in non-anticommutative superspaces,” Phys. Lett. B627, 208-216(2005)[hep-th/0506043]; M. Ihl and C. Saemann, “Drinfeld-twisted supersymmetry and non-anticommutative superspace,” JHEP 0601, 065 (2006)[arXiv:hep-th/0506057].


\(2) and Twisted Affine Lie superalgebra gl(2|2) k Conformal Field Theory Xiang-Mao Ding1 , Gui-Dong Wang1,2 , and Shi-Kun Wang1,3 1

2 3

Institute of Applied Mathematics, Academy of Mathematics and Systems Science Chinese Academy of Sciences, P.O.Box 2734, Beijing 100080, China Graduate School of the Chinese Academy of Sciences KLMM, AMSS, CAS, Beijing 100080, China [email protected]; [email protected]; [email protected].

Abstract In this paper, the differential operators realization of Lie superalgebra gl(2|2)(2) is given with coherent state methods. According to the differential operators realization of the Lie superalgebra, the free field realization of the affine (2) \ Kac-Moody superalgebra gl(2|2) k is given. The primary field of the superalgebra is also given. It is all know that the primary field corresponding to the highest weight representation of algebra. In our case, we obtain sixteen series[5], for the complexity of the expression and the limit of the space, we given a little more simple but more important primary fields. This corresponds to the highest weight representation of twisted Lie superalgebra psl(2|2)(2). Partially Supported by NKBRPC(2004CB31800,2006CB805905) and NNSFC(] 10231050; 10375087)

Keyword: superalgebra, differential operator realization, free field realization, primary field, conformal field theory

1 Introduction It is well known that the conformal group in two dimensions is the set of all analytic maps, where the group multiplication is the composition of maps. Obviously, this set is infinite dimensional, and the generators satisfy the communication relations of the Virasoro algebra. In fact, the algebraic theory of conformal field theory is the vertex algebra theory that was brought forward by Borcherds[1]. And the Chiral algebra that was put forward by V.G.Drinfeld is 273

274

Xiang-Mao Ding, Gui-Dong Wang, and Shi-Kun Wang

to use the algebraic geometry to study the conformal field theory. So the study of conformal field theory becomes an important problem in mathematics[6]. There are many conformal field theories, in which, the more important ones are minimal model, Ising model, WZW-model(Wess-Zumino-Witten model)etc. From the mathematical viewpoint, the most fruitful and the broadest researched was the WZW-model. The WZW-model is a Sigma model with Wess-Zumino (WZ)-term[3]. In this model, affine algebra and Virasoro algebra were connected by Sugawara construction and the correlation functions of the model satisfy the KZ-equation. In general, the WZW-model was defined on a ordinary manifold. Goddard, Kent and Olive([2]) found that any conformal field theory can be obtained by WZW-models and their coset models. So it has been an important and interesting problem in the recent twenty years. If we define the model on a super-manifold, then we will obtain the supersymmetric WZW-model. Super-symmetry was an attempt to unify the boson and fermions, that is, it was used to describe the transformation of boson and fermions and viceversa. In mathematics, it corresponds to the Z2 −graded algebra. The elementary reference about Z2 −graded algebra is the paper of V.G.Kac([5]). The work in the ref.[5] mainly describes the finite dimensional Lie superalgebra, and the finite dimensional Lie superalgebra corresponding to the symmetry of super-symmetric quantum mechanics. If one wants to describe the super-symmetric quantum field theory, then one should extend the finite dimensional superalgebra to the symmetric algebra which has infinite generators, called affine superalgebra.Virasoro algebra and current algebras are important algebraic structures in Conformal Field Theories (CFTs)[4, 6, 7] and string theory[8]. In this paper, we will investigate its structure. It is well known, Heisenberg algebra is the most important tool in the study of finite dimensional algebra. The free field realization, which is an infinite dimensional generalization of the Heisenberg algebra, is a powerful tool in many physical applications. This approach was first given by Wakimoto for the simplest case [ [9], and extended to sl(n) [ in [10], and other simple Lie algebras in sl(2) k k [11]. For twisted algebra, the free field approach was first realized in [12, 13], and extended to higher rank in [14]. In [15, 16], the method was generalized to the Kac-Moody superalgebra, and the study of some important Lie superalgebras in subsequent work [17, 18, 19, 20]. Using this approach, the correlation function can be easily evaluated. Beside the applications mentioned above, new features will arise in the representation theory of the Lie superalgebra. For example, for Lie superalgebra there are two kinds of representations, which are named the typical representation and the atypical representation, respectively. There is no counter part for the atypical representation in the ordinary Lie algebra. In this paper, we will consider the CFT based on the twisted current \(2) at general level k. The important ingredient of CFT superalgebra gl(2|2) k is the primary field, or the highest weight representation of the Kac-Moody

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k

275

(2)

\ , only the superalgebra. we think that for the twisted superalgebra gl(2|2) k atypical representation is surviving, and which are the chiral primary fields of Virasoro algebra with conformal dimension zero in the boundary theory. The paper is organized as follows. For conveniences, in section 2, we give some notations which will be used in the paper. In section 3, we will give the differential operators realization of the Lie superalgebra gl(2|2). In section 4, we derive the free field realization of the twisted Kac-Moody superalgebra \(2) , and construct a energy-momentum tensor of the current supergl(2|2) k algebra, which is CFT with zero central. Because of the complexity of the primary fields and the limit of space, in section 5, we only give the primary \(2) fields of Lie superalgebra of psl(2|2) k

2 Notations First of all, we will give the definition of Lie superalgebra gl(2|2). Take Eij , i, j = 1, 2, 3, 4 be the matrices with entry 1 at the i-th row and jth column, and zero elsewhere. Then gl(2|2) can be spanned by 2|2 matrices Eij , i, j = 1, 2, 3, 4, and its superalgebra structure is introduced by defining the Lie (anti-)bracket for any two matrices Eij and Ekl : [Eij , Ekl ] = δjk Eil − (−1)([i]+[j]+[k]+[l]) δil Ekj

(2.1)

where the Z2 -grading(superalgebra structure) is defined as [1] = [2] = 0, [3] = [4] = 1. It is well known that, unlike the ordinary Lie algebra, for a given Lie superalgebra, its Dynkin diagram is not unique, and not all of them have a finite order automorphism. For the given Lie superalgebra, if one can choose an appropriate Dynkin diagram, which has a finite order automorphism, the Lie superalgebra can be twisted by folding the Dynkin diagram. In the gl(2|2) case, there are three kinds of Dynkin diagrams, if we choose the following Dynkin diagram (for other choices, the process is similar):

⊗ ε1 − δ 1

g δ1 − δ 2

⊗ δ2 − ε 2

in which (εi |εj ) = δij , and (δi |δj ) = −δij . So that in the diagram ⊗ represent a null vector of fermionic type simple root. Now, we define an order-2 transformation on the Dynkin diagram τ : εi → −ε3−i ; δi → −δ3−i .

(2.2)

276


Obviously, the Dynkin diagram is invariant under the action of τ . Identically, we give the explicit action on the matrices as follows: τ (E11 ) = −E22 , τ (E22 ) = −E11 , τ (E33 ) = −E44 , τ (E44 ) = −E33 , τ (E13 ) = E42 , τ (E42 ) = E13 , τ (E34 ) = E34 , τ (E31 ) = −E24 , τ (E24 ) = −E31 , τ (E43 ) = E43 , τ (E14 ) = −E32 , τ (E32 ) = −E14 ,

(2.3)

τ (E12 ) = −E12 , τ (E41 ) = E23 , τ (E23 ) = E41 , τ (E21 ) = −E21 . It is easy to check that τ is an automorphism of the superalgebra gl(2|2) with order r = 2. So we obtain the twisted superalgebra gl(2|2), and we have gl(2|2) = gl(2|2)0 ⊕ gl(2|2)1 ,

(2.4)

in which gl(2|2)0 are spanned by 1 e1 = √ (E13 + E42 ), e2 = E34 , e3 = 2 H1 = E11 − E22 + E33 − E44 , 1 f1 = √ (E31 − E24 ), f2 = E43 , f3 = 2 H2 = E11 − E22 − E33 + E44 .

1 √ (E14 − E32 ), 2 1 √ (E41 + E23 ), 2

Here gl(2|2)0 is a fixed point sub-superalgebra under the automorphism. Obviously, this subalgebra is just the osp(2|2), or sl(2|1). If we twist the fixed subalgebra once more, we get its fixed subalgebra osp(2|1) [16]. So, twisting is an appropriate way to get a smaller algebra from a larger algebra. And the generators of gl(2|2)1 are 1 1 e1 = √ (E13 − E42 ), e2 = E12 , e3 = √ (E14 + E32 ), 2 2 H 1 = E11 + E22 + E33 + E44 , 1 1 f 1 = √ (E31 + E24 ), f 2 = E21 , f 3 = √ (E41 − E23 ), 2 2 H 2 = E11 + E22 − E33 − E44 . The (anti-)commutation relations of g(2|2) in this basis are as follows: [e1 , e2 ] = e3 , {e1 , f1 } =

1 H1 , {e1 , f3 } = f2 , [H2 , e1 ] = 2e1 , 2

1 (H1 − H2 ), [e2 , f3 ] = f1 , [H1 , e2 ] = 2e2 , 2 [H2 , e2 ] = −2e2 , [H1 , e3 ] = 2e3 , {e3 , f1 } = e2 , [e3 , f2 ] = e1 , (2.5) 1 {e3 , f3 } = H2 , [f1 , f2 ] = −f3 , [H2 , f1 ] = −2f1 , [H1 , f2 ] = −2f2 , 2 [H2 , f2 ] = 2f2 , [H1 , f3 ] = −2f3 , [e2 , f2 ] =


277

for ([gl(2|2)0 , gl(2|2)0 ]), and 1 H1 , [e1 , f 2 ] = −f3 , {e1 , f 3 } = f2 , 2 1 [H 2 , e1 ] = 2e1 , [e2 , f 1 ] = e3 , [e2 , f 2 ] = (H1 + H2 ), 2

{e1 , f 1 } =

[e2 , f 3 ] = −e1 , {e3 , f 1 } = e2 , [e3 , f 2 ] = f1 , {e3 , f 3 } =

1 H2 , 2

(2.6)

[H 2 , e3 ] = 2e3 , [H 2 , f 1 ] = −2f1 , [H 2 , f 3 ] = −2f3 , for ([gl(2|2)1 , gl(2|2)1 ]). At last 1 H 1 , [e1 , f 2 ] = f 3 , [H 2 , e1 ] = 2e1 , 2 [e2 , e1 ] = −e3 , [e2 , f 3 ] = f 1 , {e3 , e1 } = −e2 , [e3 , f 2 ] = −f 1 , 1 {e3 , f 3 } = H 1 , [H 2 , e3 ] = 2e3 , [H1 , e2 ] = 2e2 , [H1 , e3 ] = 2e3 , 2 [H1 , f 2 ] = −2f 2 , [H1 , f 3 ] = −2f 3 , [H2 , e1 ] = 2e1 , [H2 , e2 ] = 2e2 , 1 [H2 , f 1 ] = −2f 1 , [H2 , f 2 ] = −2f 2 , {f1 , e1 } = H 1 , [f1 , e2 ] = e3 , (2.7) 2 {f1 , f 3 } = −f 2 , [H 2 , f1 ] = −2f 1 , [f2 , e3 ] = −e1 , [f2 , f 1 ] = f 3 , 1 [f3 , e2 ] = −e1 , {f3 , e3 } = H 1 , {f3 , f 1 } = f 2 , [H 2 , f3 ] = −2f 3 . 2 {e1 , e3 } = e2 , {e1 , f 1 } =

for ([gl(2|2)0 , gl(2|2)1 ]). All of other relations are identically zero. So gl(2|2)0 and gl(2|2)1 satisfy [gl(2|2)i , gl(2|2)j ] ⊂ gl(2|2)(i+j)mod 2 .

(2.8)

For the twisted algebra gl(2|2) there is an endomorphism of any representation. For a given Killing form, we can construct the quadratic Casimir of gl(2|2) as C=

X

(−1)[j] Eij Eji

i,j

1 = H1 H2 − e 1 f 1 + f 1 e 1 − e 2 f 2 − f 2 e 2 − e 3 f 3 + f 3 e 3 2 1 + H 1 H 2 − e1 f 1 + f 1 e1 + e2 f 2 + f 2 e2 − e3 f 3 + f 3 e3 2

(2.9)

In fact, the quadratic Casimir is independent of the choice of the basis, it is useful to construct the energy-momentum operator.

278


3 Differential Operators realization of Lie superalgebra gl(2|2) To obtain a differential operators realization, we first construct a Fock space representation of gl(2|2). The Fock space is constructed by the actions of the lowering operators f1 , f2 , f3 f 1 , f 2 and f 3 on the highest weight state. Define the highest weight state |Λ > of the twisted algebra gl(2|2) by e1 |Λ >= e2 |Λ >= e3 |Λ >= e1 |Λ >= e2 |Λ >= e3 |Λ >= 0 Hi |Λ >= Λi |Λ >, H i |Λ >= Λi |Λ >, i = 1, 2.

(3.1)

Then the supergroup action of the operator eρ with vector ρ = θ 1 e1 + x 2 e2 + θ 3 e3 + θ 1 e1 + x 2 e2 + θ 3 e3 ,

(3.2)

on the highest state |Λ > generates a coherent state of the algebra, where x2 , x2 are bosonic coordinates satisfying [∂x2 , x2 ] = [∂x2 , x2 ] = 1, and θi , θj are fermionic coordinates obeying θi θj = −θj θi , θi θj = −θj θi and {∂θi , θj } = {∂θi , θ j } = δij , i, j = 1, 2, 3. This can be viewed as the super extension of the ordinary coherent state method. Please see [21] for more applications of coherent state method on representation theory and physics. Now we define T A eρ |Λ >= dT A eρ |Λ >,

(3.3)

where T A is any generator of gl(2|2) and dT A is the corresponding differential operator realization of the generator T A . By using the defining relations of twisted gl(2|2) and the Baker-Campbell-Hausdorff(BCH) formula, we obtain the differential operators representation of the twisted algebra. For positive simple roots, the expressions are very simple: 1 1 1 de1 = ∂θ1 − x2 ∂θ3 + θ3 ∂x2 + x2 θ1 ∂x2 2 2 12 1 1 1 de2 = ∂x2 + θ1 ∂θ3 + θ 1 ∂θ3 − θ1 θ1 ∂x2 2 2 6 1 de3 = ∂θ3 − θ1 ∂x2 2 1 1 1 de1 = ∂θ1 − x2 ∂θ3 − θ3 ∂x2 − x2 θ1 ∂x2 2 2 12 de2 = ∂x2 1 de3 = ∂θ3 + θ1 ∂x2 2 and the generators of the Cartan parts are DH1 = Λ1 − 2θ3 ∂θ3 − 2x2 ∂x2 − 2θ3 ∂θ3 − 2x2 ∂x2 DH2 = Λ2 − 2θ1 ∂θ1 + 2x2 ∂x2 − 2θ1 ∂θ1 − 2x2 ∂x2

(3.4)


DH 1 = Λ 1

279

(3.5)

DH 2 = Λ2 − 2θ1 ∂θ1 − 2θ3 ∂θ3 − 2θ1 ∂θ1 − 2θ3 ∂θ3 . But for the negative roots operators, the expressions of their realization are much more involved, 1 1 1 1 θ1 Λ1 + θ1 Λ1 + (θ3 − θ1 x2 )∂x2 − θ1 θ3 ∂θ3 2 2 2 2 1 1 1 1 − (θ1 x2 + θ1 θ3 θ1 )∂x2 − (θ1 θ3 + 2x2 − θ1 x2 θ1 )∂θ3 2 6 2 6 1 1 = x2 (Λ1 − Λ2 ) + θ3 ∂θ1 + θ 3 ∂θ1 − x2 (θ1 θ3 ∂x2 + θ3 θ1 ∂x2 ) 2 6 1 1 2 + x2 (θ1 ∂θ3 + θ1 ∂θ3 ) + x2 (θ1 ∂θ1 − 2x2 ∂x2 − θ3 ∂θ3 + θ1 ∂θ1 − θ3 ∂θ3 ) 4 2 1 1 1 1 = θ3 Λ2 + θ1 x2 (2Λ1 − Λ2 ) + ( θ3 + x2 θ1 )Λ1 + x2 ∂θ1 2 4 2 4 1 + [2θ1 θ3 ∂θ1 + 2x2 θ3 ∂x2 + (θ1 θ 3 − θ3 θ1 )∂θ1 − θ3 x2 ∂x2 − x2 x2 ∂θ3 ] 2 1 − [6θ1 x22 ∂x2 − 2θ1 x2 θ 1 ∂θ1 + (θ1 x2 x2 + 3θ1 θ3 θ3 )∂x2 + 3(θ1 x2 θ3 12 1 +x2 θ3 θ1 )∂θ3 ] + (4θ1 x22 θ1 ∂θ3 − 3θ1 x2 θ3 θ 1 ∂x2 ) 24 1 1 1 = θ1 Λ1 + θ1 Λ1 + x2 ∂θ3 + θ3 ∂x2 − θ1 θ 1 (x2 ∂θ3 − θ3 ∂x2 ) 2 2 12 1 − θ1 (x2 ∂x2 + θ3 ∂θ3 + x2 ∂x2 + θ3 ∂θ3 ) 2 1 1 1 = x2 (Λ1 + Λ2 ) − (θ1 θ 3 + θ3 θ1 )(Λ1 − Λ2 ) + (θ1 θ3 − θ 1 θ3 )Λ1 2 4 2 1 1 1 1 + θ1 x2 θ1 (2Λ1 − Λ2 ) − (θ3 θ3 − x2 θ3 θ1 − θ1 x2 θ3 + θ1 x22 θ1 )∂x2 6 2 2 4 1 1 1 −(θ1 x2 + θ1 θ3 θ1 )∂θ1 − (θ3 x2 + θ1 x2 θ3 θ 1 )∂θ3 − (θ 1 x2 − θ1 θ1 θ3 )∂θ1 2 12 2 1 1 −(x22 − θ1 θ3 θ1 θ3 )∂x2 − (x2 θ3 − θ1 x2 θ1 θ 3 )∂θ3 3 12 1 1 1 1 = θ3 Λ2 + x2 θ1 (2Λ1 − Λ2 ) + ( θ3 + θ1 x2 )Λ1 + θ 1 θ3 ∂θ1 2 4 2 4 1 1 1 1 +(−x2 + θ1 θ 3 − θ3 θ1 − θ1 x2 θ 1 )∂θ1 + (x2 θ3 − x22 θ1 )∂x2 2 2 6 2 1 1 1 1 +( x2 x2 + θ1 x2 θ3 + x2 θ3 θ1 − θ1 x22 θ 1 )∂θ3 2 4 4 6 1 1 1 1 +(− x2 θ3 + θ3 θ 1 θ3 − x2 θ 1 x2 + θ1 x2 θ 1 θ3 )∂x2 2 4 12 8

d f1 =

d f2

d f3

df

1

df 2

df

3

It is straightforward to check that the above differential operators satisfy the algebraic relations of the twisted algebra gl(2|2).

280


4 Free Field realization of Lie superalgebra gl(2|2) With the help of the differential operator representation of finite Lie algebra g, we can find the Wakimoto realization of the corresponding Kac-Moody algebra gˆ. (2)

\ For gl(2|2) k Kac-Moody superalgebra, the free field realization for arbitrary k will be given in terms of sixteen fields, namely two bosonic β − γ pairs, four fermionic b−c type pairs and four free scalar fields φ. The free fields obey the following OPEs: 1 , z−w 1 , = −γ 2 (z)β 2 (w) = z−w δij = ψj+ (z)ψi (w) = , i, j = 1, 3, z−w δij + = ψ j (z)ψ i (w) = , i, j = 1, 3, z−w = φ1 (z)φ2 (w) = −4ln(z − w)

β2 (z)γ2 (w) = −γ2 (z)β2 (w) = β 2 (z)γ 2 (w) ψi (z)ψj+ (w) +

ψ i (z)ψ j (w) φ1 (z)φ2 (w)

and all other Operator Product Expansions (OPEs) are trivial. The conformal + weights of βi (z), β i (z), ψi+ (z), ψ i (z), ∂φi (z) and ∂φi (z) are 1, and 0 for else. (2)

\ The free field realization of the gl(2|2) k current superalgebra is obtained by substitution the differential operators with certain kinds of fields. Concretely, the relations are, xi → γi (z), ∂xi → βi (z), xi → γ i (z), ∂xi → β i (z), +

θi → ψi (z), ∂θi → ψi+ (z), θi → ψ i (z), ∂θi → ψ i (z), √ √ Λi → −k∂φi (z), Λi → −k∂φi (z). For simplicity, we denote the generating functions of the Kac-Moody superalgebra as T A (z), if the finite Lie superalgebra generator is DT A . The expressions for the positive and Cartan currents are straightforward, 1 1 1 e1 (z) = ψ1+ (z) − γ2 (z)ψ3+ (z) + ψ 3 (z)β 2 (z) + γ2 (z)ψ 1 (z)β 2 (z) 2 2 12 1 1 1 + + e2 (z) = β2 (z) + ψ1 (z)ψ3 (z) + ψ 1 (z)ψ 3 (z) − ψ1 (z)ψ1 (z)β 2 (z) 2 2 6 1 + e3 (z) = ψ3 (z) − ψ 1 (z)β 2 (z) 2


281

1 1 1 + + e1 (z) = ψ 1 (z) − γ2 (z)ψ 3 (z) − ψ3 (z)β 2 (z) − γ2 (z)ψ1 (z)β 2 (z) 2 2 12 e2 (z) = β 2 (z) 1 + e3 (z) = ψ 3 (z) + ψ1 (z)β 2 (z) 2 √ H1 (z) = −k∂φ1 (z) − 2ψ3 (z)ψ3+ (z) − 2γ 2 (z)β 2 (z) +

−2ψ3 (z)ψ3 (z) − 2γ2 (z)β2 (z) √ + H2 (z) = −k∂φ2 (z) − 2ψ1 (z)ψ1+ (z) − 2ψ1 (z)ψ 1 (z) +2γ2 (z)β2 (z) − 2γ 2 (z)β 2 (z) √ H 1 (z) = −k∂φ1 (z) √ + + H 2 (z) = −k∂φ2 (z) − 2ψ1 (z)ψ 1 (z) − 2ψ3 (z)ψ 3 (z) −2ψ1 (z)ψ1+ (z) − 2ψ3 (z)ψ3+ (z). For the negative currents, the results are not so naive. Additional terms are needed to satisfy the Kac-Moody algebra. The negative currents are, √ √ 1 1 ψ1 (z) −k∂φ1 (z) + ψ 1 (z) −k∂φ1 (z) 2 2 1 1 +(ψ3 (z) − ψ1 (z)γ2 (z))β2 (z) − ψ1 (z)ψ3 (z)ψ3+ (z) 2 2 1 1 − (ψ1 (z)γ 2 (z) + ψ1 (z)ψ3 (z)ψ 1 (z))β 2 (z) 2 6 1 1 + − (2γ 2 (z) + ψ1 (z)ψ 3 (z) − ψ1 (z)γ2 (z)ψ 1 (z))ψ 3 (z) + k∂ψ1 (z) 2 6 1√ + f2 (z) = −k(∂φ1 (z) − ∂φ2 (z))γ2 (z) + ψ3 (z)ψ1+ (z) + ψ 3 (z)ψ 1 (z) 2 1 + + γ22 (z)(ψ1 (z)ψ3+ (z) + ψ 1 (z)ψ 3 (z)) 4 1 + + γ2 (z)(ψ1 (z)ψ1+ (z) − 2γ2 (z)β2 (z) − ψ3 (z)ψ3+ (z) + ψ 1 (z)ψ 1 (z) 2 1 + −ψ3 (z)ψ 3 (z)) − γ2 (z)(ψ1 (z)ψ 3 (z)β 2 (z) + ψ3 (z)ψ1 (z)β 2 (z)) 6 −(k + 1)∂γ2 (z) √ √ 1 1 f3 (z) = ψ3 (z) −k∂φ2 (z) + ψ1 (z)γ2 (z) −k(2∂φ1 (z) − ∂φ2 (z)) 2 4 √ 1 1 + +( ψ 3 (z) + γ2 (z)ψ 1 (z)) −k∂φ1 (z) + γ 2 (z)ψ 1 (z) 2 4 1 + [2ψ1 (z)ψ3 (z)ψ1+ (z) + 2γ2 (z)ψ3 (z)β2 (z) + (ψ1 (z)ψ 3 (z) 2 + + −ψ3 (z)ψ 1 (z))ψ 1 (z) − ψ3 (z)γ 2 (z)β 2 (z) − γ2 (z)γ 2 (z)ψ 3 (z)] 1 + − [6ψ1 (z)γ22 (z)β2 (z) − 2ψ1 (z)γ2 (z)ψ 1 (z)ψ 1 (z) 12

f1 (z) =

282


+(ψ1 (z)γ2 (z)γ 2 (z) + 3ψ1 (z)ψ3 (z)ψ 3 (z))β 2 (z) +

+3(ψ1 (z)γ2 (z)ψ3 (z) + γ2 (z)ψ3 (z)ψ 1 (z))ψ 3 (z)] 1 + + (4ψ1 (z)γ22 (z)ψ 1 (z)ψ 3 (z) − 3ψ1 (z)γ2 (z)ψ3 (z)ψ 1 (z)β 2 (z)) 24 1 1 1 1 5 +(k + )∂ψ3 (z) + (k − )γ2 (z)∂ψ1 (z) − (k + )ψ1 (z)∂γ2 (z) 2 2 6 2 3 √ √ 1 1 f 1 (z) = ψ 1 (z) −k∂φ1 (z) + ψ1 (z) −k∂φ1 (z) + γ 2 (z)ψ3+ (z) 2 2 1 +ψ3 (z)β2 (z) − ψ1 (z)ψ 1 (z)(γ2 (z)ψ3+ (z) − ψ 3 (z)β 2 (z)) 12 1 + − ψ 1 (z)(γ2 (z)β2 (z) + ψ3 (z)ψ3+ (z) + γ 2 (z)β 2 (z) + ψ 3 (z)ψ 3 (z)) 2 +k∂ψ1 (z) √ 1 f 2 (z) = γ 2 (z) −k(∂φ1 (z) + ∂φ2 (z)) 2 √ 1 − (ψ1 (z)ψ 3 (z) + ψ3 (z)ψ 1 (z)) −k(∂φ1 (z) − ∂φ2 (z)) 4 √ 1 + (ψ1 (z)ψ3 (z) − ψ 1 (z)ψ 3 (z)) −k∂φ1 (z) 2 √ 1 + ψ1 (z)γ2 (z)ψ 1 (z) −k(2∂φ1 (z) − ∂φ2 (z)) 6 1 −(ψ1 (z)γ 2 (z) + ψ1 (z)ψ3 (z)ψ 1 (z))ψ1+ (z) 2 1 −(ψ3 (z)ψ 3 (z) − γ2 (z)ψ3 (z)ψ 1 (z) 2 1 1 − ψ1 (z)γ2 (z)ψ 3 (z) + ψ1 (z)γ22 (z)ψ 1 (z))β2 (z) 2 4 1 −(ψ3 (z)γ 2 (z) + ψ1 (z)γ2 (z)ψ3 (z)ψ 1 (z))ψ3+ (z) 12 1 + −(ψ 1 (z)γ 2 (z) − ψ1 (z)ψ 1 (z)ψ 3 (z))ψ 1 (z) 2 1 −(γ 22 (z) − ψ1 (z)ψ3 (z)ψ 1 (z)ψ 3 (z))β 2 (z) 3 1 + −(γ 2 (z)ψ 3 (z) − ψ1 (z)γ2 (z)ψ 1 (z)ψ 3 (z))ψ 3 (z) + k∂γ 2 (z) 12 1 1 1 1 + kψ 3 (z)∂ψ1 (z) − (k − )γ2 (z)ψ 1 (z)∂ψ1 (z) − kψ3 (z)∂ψ 1 (z) 2 6 2 2 1 1 1 + (k − )γ2 (z)ψ1 (z)∂ψ1 (z) − (k + 2)ψ1 (z)ψ 1 (z)∂γ2 (z) 6 2 3 1 1 − (k + 1)ψ 1 (z)∂ψ3 (z) + (k + 1)ψ1 (z)∂ψ3 (z) 2 2 √ √ 1 1 f 3 (z) = ψ 3 (z) −k∂φ2 (z) + γ2 (z)ψ 1 (z) −k(2∂φ1 (z) − ∂φ2 (z)) 2 4


283

√ 1 1 + +( ψ3 (z) + ψ1 (z)γ2 (z)) −k∂φ1 (z) + ψ 1 (z)ψ 3 (z)ψ 1 (z) 2 4 1 1 1 +(−γ 2 (z) + ψ1 (z)ψ 3 (z) − ψ3 (z)ψ 1 (z) − ψ1 (z)γ2 (z)ψ1 (z))ψ1+ (z) 2 2 6 1 1 +(γ2 (z)ψ 3 (z) − γ22 (z)ψ1 (z))β2 (z) + ( γ2 (z)γ 2 (z) 2 2 1 1 1 + ψ1 (z)γ2 (z)ψ 3 (z) + γ2 (z)ψ3 (z)ψ 1 (z) − ψ1 (z)γ22 (z)ψ 1 (z))ψ3+ (z) 4 4 6 1 1 1 +(− γ 2 (z)ψ3 (z) + ψ3 (z)ψ 1 (z)ψ 3 (z) − γ2 (z)ψ 1 (z)γ 2 (z) 2 4 12 1 1 + ψ1 (z)γ2 (z)ψ 1 (z)ψ 3 (z))β 2 (z) + (k + )∂ψ 3 (z) 8 2 1 5 1 1 + (k − )γ2 (z)∂ψ 1 (z) − (k + )ψ 1 (z)∂γ2 (z). 2 6 2 3 The energy-momentum tensor corresponding to the quadratic Casimir C is given by 1 X (−1)[j] Eij (z)Eji (z) : T (z) = : 2k ij 1 = − [∂φ1 (z)∂φ2 (z) + ∂φ1 (z)∂φ2 (z)] + β2 (z)∂γ2 (z) 4 + +β 2 (z)∂γ 2 (z) − ψ1+ (z)∂ψ1 (z) − ψ 1 (z)∂ψ 1 (z) 1 + −ψ3+ (z)∂ψ3 (z) − ψ 3 (z)∂ψ 3 (z) − √ ∂ 2 φ1 (z). 2 −k

(4.1)

One can easily check that, for tensor T (z) and all currents (here we denote them as J(z)), we have the following relations: 2T (w) ∂T (w) + , 2 (z − w) z−w J(w) ∂J(w) T (z)J(w) = + . 2 (z − w) z−w

T (z)T (w) =

(4.2) (2)

\ We can see that T (z) is the energy-momentum tensor of the gl(2|2) k current superalgebra, with zero central charge. In this sense, the energymomentum tensor is a spin 2 primary filed with respect to itself, and respect to this tensor, all currents are spin 1 primary fields.

5 Primary Field of the Superalgebra The representation of the algebra is an important problem in Lie theory and theoretical physics. In this section, we will give the primary field, it corresponds to the representation of the algebra. As mentioned above, for the

284


complexity of the expression and the limit of the space, here we will just give the primary field that corresponds to the representation of the superalgebra psl(2|2)(2) . Let 1 V (z) = exp{ √ (β1 φ1 (z) + β2 φ2 (z) + β 1 φ1 (z) + β 2 φ2 (z))} k

(5.1)

in which β1 , β2 , β 1 and β 2 are four parameters. It is easy to see that the state V (z) is a highest state of the algebra. The conformal weight is 4=

2β2 (2β1 + 1) + 4β 1 β 2 k

(5.2)

2

If β22 = β 2 = 0, then the conformal weight 4 = 0 and the primary field corresponds to the atypical representation of the algebra. We will just consider this situation here. In this case, we obtained four linear independent series, 1. Bβn,m(z; 0) = β1 γ (z)β1 −n−2 γ2 (z)β1 −m−1 [4β1 γ 22 (z)γ2 (z) 3β1 − n 2 +(β1 − n)(β1 + m)γ 2 (z)γ2 (z)(ψ1 (z)ψ 3 (z) + ψ3 (z)ψ1 (z)) 1 − (β1 − n)(5β1 + 3m)γ 2 (z)γ22 (z)ψ1 (z)ψ1 (z) 6 +2(β1 − n)(β1 − m)γ 2 (z)ψ3 (z)ψ 3 (z) 1 + (β1 − n)(β1 − n − 1)(2β1 + m)γ2 (z)ψ3 (z)ψ 1 (z)ψ1 (z)ψ 3 (z)]V (z) 3 for this series we have β1 + m n,m−1 B (w; 0) z−w β 3β1 − n + 1 n−1,m Bβ (w; 0) f 2 (z)Bβn,m (w; 0) = − z−w f2 (z)Bβn,m (w; 0) =

2. f1 (z)Bβn,m (w; 0) =

1 F n,m (w; 0) z−w 1

in which F1n,m (w; 0) = β1 (β1 − m)γ 2 (w)β1 −n−1 γ2 (w)β1 −m−1 [−γ 2 (w)γ2 (w)ψ1 (w) + 2γ 2 (w)ψ3 (w) −(β1 − n)ψ1 (w)ψ3 (w)ψ 3 (w)

(5.3)


285

1 + (β1 − n)γ2 (w)ψ1 (w)ψ3 (w)ψ 1 (w)]V (w) 6 3. f 1 (z)Bβn,m (w; 0) =

1 n,m F 1 (w; 0) z−w

(5.4)

in which n,m

F1

(w; 0) = β1 (β1 − m)γ 2 (w)β1 −n−1 γ2 (w)β1 −m−1 [−γ 2 (w)γ2 (w)ψ 1 (w) + 2γ 2 (w)ψ 3 (w) 1 − (β1 − n)γ2 (w)ψ1 (w)ψ 1 (w)ψ 3 (w) 6 +(β1 − n)ψ 3 (w)ψ3 (w)ψ 1 (w)]V (w)

4. n,m

f1 (z)F 1

(w; 0) =

(β1 − m)(3β1 − n + 1) n−1,m+1 B1,1 (w; 0) z−w

(5.5)

in which n,m B1,1 (w; 0) =

β1 γ (w)β1 −n−2 γ2 (w)β1 −m−1 3β1 − n 2 [−2γ 22 (w)γ2 (w) + 2(β1 − m)γ 2 (w)ψ3 (w)ψ 3 (w) −(2β1 − m − n)γ 2 (w)γ2 (w)(ψ1 (w)ψ 3 (w) + ψ3 (w)ψ 1 (w)) 1 + (7β1 − 3m − 4n)γ 2 (w)γ22 (w)ψ1 (w)ψ 1 (w) 6 1 − (β1 − n − 1)(5β1 − 2m − 3n)γ2 (w)ψ1 (w)ψ 3 (w)ψ3 (w)ψ 1 (w)]V (w) 6 For this four series, the OPEs with rise operators and Cartan parts are as follows: for the serial Bβn,m (w; 0), we have 1 (β1 − n)(β1 + m) n+1,m−1 F1 (w; 0) z − w (3β1 − n)(β1 − m + 1) β1 − m n,m+1 e2 (z)Bβn,m (w; 0) = B (w; 0) z−w β

e1 (z)Bβn,m (w; 0) =

1 β1 − n n+1,m F (w; 0) z − w 3β1 − n 1 1 (β1 − n)(β1 + m) F n+1,m−1 (w; 0) e1 (z)Bβn,m (w; 0) = − z − w (3β1 − n)(β1 − m + 1) 1

e3 (z)Bβn,m (w; 0) =

286


1 (3β1 − n − 1)(β1 − n) n+1,m Bβ (w; 0) z−w 3β1 − n β1 − n n+1,m 1 F (w; 0) =− 3β1 − n z − w 1 2(2β1 − m − n) n,m =− Bβ (w; 0) z−w 2(2β1 + m − n) n,m Bβ (w; 0) =− z−w −4β 2 n,m B (w; 0) = z−w β −4β 1 n,m B (w; 0) = z−w β

e2 (z)Bβn,m (w; 0) = e3 (z)Bβn,m (w; 0) H1 (z)Bβn,m (w; 0) H2 (z)Bβn,m (w; 0) H 1 (z)Bβn,m (w; 0) H 2 (z)Bβn,m (w; 0)

and for the serial F1n,m (w; 0), we have

e1 (z)F1n,m (w; 0)

e2 (z)F1n,m (w; 0) e3 (z)F1n,m (w; 0) e2 (z)F1n,m (w; 0) f2 (z)F1n,m (w; 0) f 2 (z)F1n,m (w; 0)

−(2β1 − m − n) n,m Bβ (w; 0) z−w (β1 + m)(β1 − n) n,m + B13 (w; 0) z−w β1 − m n,m+1 F (w; 0) = z−w 1 (β1 − m)(β1 − n) n,m+1 β1 − m n,m+1 (w; 0) Bβ (w; 0) − B11 = z−w z−w β1 − n n+1,m = F (w; 0) z−w 1 1 (β1 + m)(β1 − m) n,m−1 = F1 (w; 0) z−w β1 − m + 1 3β1 − n + 1 n−1,m =− F1 (w; 0) z−w =

n,m

similarly for the serial F 1 n,m

e2 (z)F 1

n,m

e1 (z)F 1

n,m

e2 (z)F 1

n,m

e3 (z)F 1

n,m

f2 (z)F 1

β1 − m n,m+1 F (w; 0) z−w 1 (β1 − m) n,m (β1 − n)(β1 − m) n,m = − Bβ (w; 0) − B31 (w; 0) z−w z−w β1 − n n+1,m = F (w; 0) z−w 1 β1 − m n,m+1 (β1 − n)(β1 − m) n,m+1 = B (w; 0) − B11 (w; 0) z−w β z−w 1 (β1 + m)(β1 − m) n,m−1 = F1 (w; 0) z−w β1 − m + 1

(w; 0) = (w; 0) (w; 0) (w; 0) (w; 0)

(w; 0) we have

\(2) and twisted CFT Affine Lie superalgebra gl(2|2) k n,m

f 2 (z)F 1

(w; 0) = −

287

3β1 − n + 1 n−1,m F1 (w; 0) z−w

n,m finally for the bosonic series B11 (w; 0) we have n,m (β1 − m + 1)e1 (z)B11 (w; 0) = −

(2β1 − m − n) n+1,m−1 1 F1 (w; 0) 3β1 − n z−w

β1 − m − 1 n,m+1 B11 (w; 0) z−w 1 n+1,m n,m (w; 0) = F (w; 0) (3β1 − n)e3 (z)B11 z−w 1 (2β1 − m − n) n+1,m−1 1 n,m (β1 − m + 1)e1 (z)B11 (w; 0) = F1 (w; 0) 3β1 − n z−w 1 β1 − n − 1 n+1,m 1 n,m e2 (z)B11 (w; 0) = (w; 0) − B11 B n+1,m (w; 0) z−w z − w 3β1 − n β 1 n,m F n+1,m (w; 0) (w; 0) = − (3β1 − n)e3 (z)B11 z−w 1 1 n,m B n,m−1(w; 0) (w; 0) = − (β1 − m + 1)f2 (z)B11 z−w β (β1 + m − 1)(β1 − m) n,m−1 + B11 (w; 0) z−w 3β1 − n + 1 n−1,m n,m (w; 0) = − (w; 0) f 2 (z)B11 B11 z−w n,m e2 (z)B11 (w; 0) =

It is easy to see that when β1 6= 0, the representation is non trivial but infinite dimensional. The nontrivial finite dimensional representation will exist when β2 6= 0, but as we have mentioned before, in this case the expression will be very complex and we will not discuss it here.

Acknowledgments: X.-M. Ding thanks Prof. A. Bellen for his warm invitation and great help during his stay in Trieste, where the work was partially complected. Thanks also to Prof. G. Lindi for his kindness. This work is (partially) supported by Inistero degli Affari Esteri- Direzione Generale per la Promozione la Cooperazione Culturale, and by Istituto Nazionale di Alta Matematica, francesco severi(INdAM), Roma. The work is also financially supported partly by Natural Science Foundation of China through the grands No.10231050 and No.10375087.

References 1. R. Borcherds, Vertex algebras, Kac-Moody algebras and the Monster, Proc.Natl.Acad.Sci.USA, 83(1986)3068-3071.

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2. P. Goddard, A. Kent and D. Olive, Virasoro Algebras and Coset Space Models, Phys.Lett. 152B(1985)88. 3. E. Witten, Commun. Math. Phys. 92, (1984) 445. 4. V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd Edition, Cambridge Univ. Press, Cambridge, 1990. 5. V.G.Kac, Lie superalgebras, Adva. in Math. (1977),26: 8-96. 6. A.A.Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nucl. Phys. B241, (1984) 333. 7. Ph. Di Francesco, P. Mathieu, D. Senehal, Conformal Field Theory, Springer, Berlin, 1997. 8. J. Polchinski, String Theory, Cambridge Univ, Press,Cambridge, 1998. 9. M. Wakimoto, Commun. Math. Phys. 104, (1986) 605. 10. B.L. Feigin and E. Frenkel, Commun. Math. Phys. 128, (1990) 161. 11. J.L. Petersen, J. Rasmussen and M. Yu, Nucl. Phys. B502, (1997) 649. 12. M. Szczesny, Math. Res. Lett. 9, (2002) 433. 13. X.-M. Ding, M. D. Gould and Y.-Z. Zhang, Phys. Lett. B523, (2001) 367. 14. L. Feher and B.G. Pusztai, Nucl. Phys. B674, (2003) 509. 15. X.-M. Ding, M. D. Gould and Y.-Z. Zhang, Phys. Lett. (2003), A318: 354. 16. X.-M. Ding, M. D. Gould, C. J. Mewton and Y.-Z. Zhang, Jour. Phys. A 36,(2003) 7649. 17. Yao-Zhong Zhang, Coherent State Construction of Representations of osp(2|2) and Primary Fields of osp(2|2) Conformal Field Theory, Phys.Lett.A327(2004)442-451. 18. Yao-Zhong Zhang and Mark D. Gould, A Unified and Complete Construction of All Finite Dimensional Irreducible Representations of gl(2|2), J.Math.Phys.46(2005)013505. 19. Yao-Zhong Zhang, Super Coherent States, Boson-Fermion Realizations and Representations of Superalgebras, hep-th/0405066. 20. Yao-Zhong Zhang, Xin Liu and Wen-Li Yang, Primary Fields and Screening Current of gl(2|2) Non-unitary Conformal Field Theory, Nucl.Phys.B704(2005)510526. 21. A. Perelomov, ”Generalized Coherent States and Their Applications”, SpringerVerlag, 1986.

A Solution of Yang-Mills Equation on BdS Spacetime Xin’an Ren1,2 and Shikun Wang1,3 1

2 3

Institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China Graduate school of Chinese Academy of Science, Beijing 100080, China KLMM, Academy of Mathematics and Systems Sciences,Chinese Academy of Sciences, Beijing 100080, China [email protected], [email protected]

Abstract Inspired by Prof. Lu’s lecture note [1], we will give the global solutions of Yang-Mills equation on de-Sitter spacetime with BHL metric, instead of the conformal metric. The project partially supported by NKBRPC(]2004CB31800, ]2006CB805905) and NNSFC(]10375087)

1 Introduction Recently, there are many authors [2-5] who have discussed the AdS/CF T correspondence. However, most of these discussions on AdS/CF T correspondence are based on the so-called Euclidean version of the AdS5 , or a 5dimensional unit ball B 5 . As was pointed out by Prof. Lu [6] the AdS/CF T correspondence of Lorentz version is the duality between a field theory on AdS5 and a conformal field theory on its boundary, the conformal space. It is well known that there are only three constant spacetimes, Minkowski, de-Sitter and anti-de-Sitter spacetimes with invariant groups being Poincare group, de-Sitter group SO(1, 4) and SO(2, 3). Besides various usual dS/AdS spacetimes with different metrics, there is a special one with most important properties analogue to Minkowski spacetime that should be paid attention to. It is the dS/AdS spacetime with Beltrami-Hua-Lu (BHL) metric, called BdS/BAdS spacetime [7,8]. Recent observations in cosmology show that our universe is in accelerated expansion [9]. This implies that the universe is probably asymptotically deSitter with positive cosmological constant Λ. So it is important to discuss the dS/CF T correspondence. However, first we should discuss the field theory on 289

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Xin’an Ren and Shikun Wang

de-Sitter spacetime. Prof. Lu has discussed the Yang-Mills equation on deSitter space with conformal flat metric in [1]. In this paper, we consider the BdS spacetime with BHL metric instead of conformal flat metric and find a global solution of Yang-Mills equation. This paper is organized as follows. In section 2, we give the definition of BdS spacetime with BHL metric in terms of classical domains and discuss the de-Sitter group. Yang-Mills equation on BdS is discussed and a global solution is given in section 3.

2 The Beltrami de-Sitter spacetime Originally, according the works of Professors Look, Tsou and Kuo, the deSitter spacetime can be regarded as the classical domain Dλ (1, 4), but here we firstly define the classical domains Dλ (m, n) = {X ∈ Rm×n |I − λXJX 0 > 0},

(1)

where J = {1, −1, · · · , −1} is diagonal matrix of order n and λ is a real constant. Let A, B, C, D be m × m, n × m, m × n, n × n matrices respectively and satisfy 0 I 0 A C I 0 A C , (2) = 0 − λJ B D 0 − λJ B D which means that AA0 − λCJC 0 = I,

AB 0 = λCJD0 ,

BB 0 − λDJD0 = −λJ,

(3)

λC 0 C − D0 JD = −J.

(4)

or λA0 A − B 0 JB = λI,

λA0 C = B 0 JD,

Therefore the transformation Y = (A + XB)−1 (C + XD)

(5)

must maps Dλ (m, n) one to one to itself because I − λY JY 0 = (A + XB)−1 (I − λXJX 0 )[(A + XB)−1 ]0 .

(6)

Moreover, there is in Dλ (m, n) a metric ds2 = tr[(I − λXJX 0 )−1 dX(J − λX 0 X)dX 0 ],

(7)

which is invariant under the transformation (2.5). Set X0 = −CD−1 , and the transformation can be written as

A Solution of Yang-Mills Equation on BdS Spacetime

Y = A−1 (I − λXJX00 )−1 (X − X0 )D.

291

(8)

From the last equation in (2.4), we have that D satisfies (DJD0 ) = (J − λX00 X0 )−1 and from the first and second equations in (2.4), it is not difficult to see that A satisfies (AA0 ) = (I − λX0 JX00 )−1 . (9) That is to say the transformation (2.5) can map every point X0 to Y = 0, which means Dλ (m, n) is homogeneous under the action of the group composed by transformation (2.8). In the case of Dλ (1, 4), X = (x0 , x1 , x2 , x3 ) and the condition (2.1) reduces to σ = σ(x, x) = 1 − ληij xi xj > 0, (10) where ηij (i, j = 0, 1, 2, 3) are elements in the diagonal matrix J . The metric in (2.7) reduces to ηjk ληjr ηks xr xs dX(J − λX 0 X)−1 dX 0 j k = g dx dx = ( + )dxj dxk , jk 1 − λXJX 0 σ σ2 (11) which is called the Beltrami-Hua-Lu (BHL) metric. Then λ is the Riemannian curvature of Dλ (1, 4). Moreover, by setting X0 = (a0 , a1 , a2 , a3 ), the transformation (2.8) has the form of ds2 =

y i = (σ(a, a))1/2

(xj − aj )Dji , σ(x, a)

(12)

where σ(x, a) = 1 − ληij xi aj and Dji satisfy ηpq Dip Djq = ηij +

ληir ηjs ar as . σ(a)

(13)

From the discussion above, we see that the de-Sitter group SO(1, 4) composed of the transformation (2.12) acts transitively on de-Sitter spacetime and leaves the BHL metric (2.11) invariant.

3 The connections and Yang-Mills equation on BdS spacetime In this section we will find the solutions of the Yang-Mills equation on BdS spacetime. First we compute the Christoffel symbol associated to the BHL metric ∂gki ∂gjk ∂gji 1 l + − , (1) = g li j k 2 ∂xk ∂xj ∂xi

292


which is an gl(4, R) connection. By formula (2.11), we can see easily that g ij = σ(x, x)(η ij − λxi xj ),

(2)

and ηij ∂σ 2λxi xj ∂σ λ ∂ ∂gij =− 2 − + 2 k (xi xj ) ∂xk σ ∂xk σ 3 ∂xk σ ∂x 4λ2 xi xj xk 2ληij xk + ληik xj + ληjk xi + = 2 σ σ3 Moreover, by (3.1), we have 1 2ληij xk + ληik xj + ληjk xi 4λ2 xi xj xk l il i l = σ(η − λx x ) + j k 2 σ2 σ3 2ληik xj + ληij xk + ληjk xi 1 + σ(η il − λxi xl ) 2 σ2 2ληkj xi + ληik xj + ληji xk − (3) σ2 4λ2 xi xj xk 2ληij xk + 2ληik xj 1 + = σ(η il − λxi xl ) 2 σ2 σ3 2λ2 xl xj xk λ l 2λ2 xl xj xk 2λ2 xl xj xk (1 − σ) δj xk + δkl xj + = − − 2 σ σ σ σ2 λ l = δ xk + δkl xj , (4) σ j where δjl is the Kronecker symbol and xi = ηij xj . Let 3 X gjk dxj dxk = ηab ω a ω b ds2 =

(5)

j,k=0

be a Lorentz metric on M , where (ηab ) = {1, −1, −1, −1} is a diagonal matrix and (a) ω a = ej dxj , (a = 0, 1, 2, 3) (6) and

∂ , (a = 0, 1, 2, 3) (7) ∂xj are the Lorentz coframe and the dual frame respectively. It is not difficult to see that ληjk a k 1 (a) √ x x δja + ej = √ (8) σ σ+ σ Xa = ej(a)

and ej(a)

=

√

σ

δaj

ληab b j √ x x − 1+ σ

From the Christoffel symbol, we get a so(1, 3)- connection as following

(9)

A Solution of Yang-Mills Equation on BdS Spacetime (a)

∂ek(b)

(a)

l

293

ek(b) j k ∂xj 1 λxa xl √ = √ δka + × σ σ+ σ √ λδ k xj λ 2 xb xj xk λ σ √ ηbj xk + δjk xb + √ √ 2 − √b − σ 1+ σ σ(1 + σ) a k λ λx xl λxb xk l l a √ √ + δ j xk + δ k xj δ b − δl + σ σ+ σ 1+ σ √ λ λ √ xa xb xj = δba xj + σδja xb + σ 1+ σ √ λ a λ σ a λ λ 2 xa xb xj 1 a √ δ xb − √ ηbj x − √ √ − √ δ b xj − +√ σ σ 1+ σ j 1+ σ σ(1 + σ) λ √ δja xb − ηbj xa . = σ+ σ

a Γbj = ek

+ el

Moreover, according to Theorem. 2.4.2 in [10], associated to the so(1, 3)a connection Γbj , there is a sl(2, C)-connection 1 a Aj = η cb Γcj σa σb∗ , 4

σb∗

0

= ¯ σb ,

=

0 1 −1 0

(10)

,

where σ0 =

1 0 0 1

, σ1 =

0 1 1 0

, σ2 =

0 −i i 0

, σ3 =

1 0 0 −1

.

(11)

This connection is globally defined in BdS spacetime because there is spin structure on it, which is implied by Dirac’s paper [11]. We are now in a position to compute the connection 1 cb a η Γcj σa σb∗ 4 λ 1 cb a a √ δ xc − ηcj x σa σb∗ = η 4 σ+ σ j λ √ δja xb − δjb xa σa σb∗ . = 4(σ + σ)

Aj =

(12)

It remains to prove that Aj satisfies the Yang-Mills equation defined by ∂Fjk r r g kl Fjk;l ≡ g kl + [A , F ] − F − = 0, (13) F l jk rk jr j l k l ∂xl where Fjk =

∂Aj ∂Ak − + [Aj , Ak ] ∂xj ∂xk

(14)

294


and

r

is the Christoffel symbol of the BHL metric. j l According to (3.11), the elements of Aj are odd functions of xj , which means that [Aj (x)]|x=0 = 0. Hence all elements of Fjk are even functions of xj . It is not difficult to see that all elements of Fjk;l are odd functions of xj and consequently [Fjk;l (x)]x=0 = 0. Since BdS spacetime is transitive under the group SO(1, 4), for any point x0 of BdS there is at least a transformation (2.13) which transform x0 to y = 0. Since both gjk and Fjk;l are covariant under the transformation (2.13), ∂xp ∂xq ∂xr 0 = [Fjk;l ]y=0 = UT (x)Fpq;r UT (x)−1 j k , (15) ∂y ∂y ∂y l x=x0

which implies that [Fpq;r ]x=x0 = 0. Since x0 can be an arbitrary point of BdS spacetime, we have Fjk;l = 0 and obviously it satisfies the Yang-Mills equation. It should be noted that an su(2)-connection can also be obtained by using the Reduction Theorem of Connections. Moreover, since the BHL metric gij is a solution of Einstein equation with cosmological constant Λ = 3λ, we get in fact a solution (gij , Aj ) of the Einstein-Yang-Mills equation defined as follows Rjk − 12 Rgjk + Λgjk = 0 g kl Fjk;l = 0, where Rjk and R are Ricci tensor and scalar curvature of BHL metric, respectively.

Acknowledgements We would like to thank Professors H. Y. Guo and K. Wu for valuable discussions and comments. Special thanks are given to Prof. Q. K. Lu who gave us a long series of lectures on conformal space and de-Sitter space. The first author would also like to thank Professor H. W. Xu for his encouragements and assistances.

References 1. Q. K. Lu, Global solutions of Yang-Mills equation, preprint. 2. J. Maldacena, The large N superconformal field theory and supergravity, Adv. Theor. Math. Phys., 2, 1998, 231. 3. J. L. Petersen, Introduction to the Maldacena conjecture on AdS/CF T , NBIHE-99-05. 4. E. Witten, Anti-de-Sitter space and holography, Adv. Theor. Math. Phys., 2, 1998, 253. 5. E. Witten and S.-T. Yau, Connectedness of the boundary in the AdS/CF T correspondence, hep-th/9910245.

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6. Q. K. Lu, A note to a paper of Dirac in 1935, Lecture at Morningside Center, Beijing. 7. H. Y. Guo, C. G. Huang, Z. Xu and B. Zhou, On Beltrami model of de Sitter spacetime, hep-th/0311156. 8. K. H. Look, C. L. Tsou and H.Y. Kuo, The kinematic effects in the classical domains and the red-shift phenomena of extra-galactic objects”, Acta Physica Sinica. 23, 1974, 225. 9. M. Tegmark, et all, Arxiv:astro/0310723. 10. Q. K. Lu, Differential Geometry and its application to physics, Science Press, Beijing, 1982. 11. P. A. M., Dirac, The electron wave equation in de-Sitter space, Ann. Math., 36, 1935, 657.


Solitonic Information Transmission in General Relativity Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, 55, Zhong Guan Cun Donglu, Beijing, China, 100080

Abstract An exact solution of the vacuum Einstein’s field equations is presented in which there exists a congruence of null geodesics whose shear behaves like a travelling wave of the KdV equation. On the basis of this exact solution, the feasibility of solitonic information transmission by exploiting the nonlinearity intrinsic to the Einstein field equations is discussed. Partially Supported by NKBRPC(2004CB31800,2006CB805905) and NNSFC(] 10231050; 10375087)

1 Introduction In electromagnetism, though the source free Maxwell equations are linear, the propagation of electromagnetic waves in a continuous media can display nonlinear behaviour due to the nonlinear structure of the media at the macroscopic level. When there is a balance between compression effect caused by nonlinearity and dispersive wave behaviour, a solitary wave occurs as a result of the interplay of these two competing factors (see for instance [1]). This is the mechanism underlying the formation of optical soliton propagating in an optical fibre. Optical soliton is highly stable under external interference and allows rapid and reliable information transmission, superior to conventional means of information transmission by means of cable. The present work aims to point out, in terms of an exact solution of the vacuum Einstein field equations, that instead of a continuous media, the nonlinear structure inherent in general relativity may also be exploited to achieve solitonic information transmission. The gravitational wave background itself acts as an effective continuous media to generate solitonic behaviour for light 297

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Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

propagation. Though the example to be presented below is highly idealistic and quite remote from practical implementation, still at least in theory it is a feasibility allowed by Einstein’s theory of general relativity. Throughout the present work, notations for the Newman Penrose formalism will be adopted from that in [2]

2 Description of the exact solution Consider a spacetime manifold whose metric may be expressed as ds2 = 2dU dV − e2η cosh ω(dx2 + e2χ dy 2 ) − 2e2η+χ sinh ωdxdy

(1)

where ζ, η, ω, χ are functions of the null coordinates U, V in general , ∂/∂x, ∂/∂y are two commuting spacelike Killing vector fields. Commutativity of ∂/∂x, ∂/∂y implies the existence of a spacelike two-surface whose tangent space at each point is spanned by ∂/∂x, ∂/∂y. While ∂/∂U, ∂/∂V are the two null normals of the spacelike two-surface spanned by ∂/∂x, ∂/∂y. Given the metric in (1), a Newman-Penrose tetrad may be set up naturally as ∂ , ∂U ∂ na = , ∂V √ 1 ∂ ∂ ma = √ e−η cosh ω(eiφ − ie−χ ), ∂x ∂y 2 ∂ ∂ 1 −η √ a cosh ω(e−iφ + ie−χ ) m ¯ = √ e ∂x ∂y 2 la =

(2)

where sin φ = tanh ω. With the Newman-Penrose tetrad in (2) and the metric in (1), it may further be computed that the spin coefficients κ = κ0 = 0. This means that the two null normals ∂/∂U, ∂/∂V define in a very natural way two null geodesic congruences orthogonal to the two-surface generated by ∂/∂x, ∂/∂y, and U, V are affine parameters of the respective geodesic congruences. Suppose one of the two null geodesic congruences defined above (say the one defined by ∂/∂V ) is shearfree and subject further to the vanishing of the Ricci curvature of the metric in (1), then η, w, χ become functions of U only. Further, the Newman Penrose equations consisting of a system of coupled differential equations may be reduced drastically to a second-order linear ODE of Sturm-Liouville type (detailed derivation is in [3], see also [4]), given by d2 f + |σ|2 f = 0 dU 2 where

(3)

Solitonic Information Transmission in General Relativity

299

f = e2η+χ , σ is the shear of the null geodesic congruence defined by ∂/∂U and its modulus is denoted by |σ|. σ is further given by 1 1 ∂w ∂χ i σ = (1 + i sinh w) + + i tanh w 2 ∂u 2 cosh w ∂u with 1 |σ| = 4 2

(

2

cosh ω

∂χ ∂u

2

+

∂w ∂u

2 ) (4)

Exact solutions of the Einstein field equations may then be constructed from (f, η, χ) subject to (3) and (4). The metric may be written as ds2 = 2dU dV − f cosh ω(e−χ dx2 + eχ dy 2 ) − 2f sinh ωdxdy.

(5)

It describes a type N spacetime of a self interacting gravitational wave background. The Weyl curvature components of the metric are all zero except d ∂σ ∂χ 1 ∂ω Ψ0 = 2 σ+ ln f + i sinh ω − dU ∂U cosh ω ∂U ∂U Among the vacuum solutions of the Einstein field equations constructed this way, there is a class of exact solutions described by the ordered pair 1 1 1 , ; 1, tanh U , sech2 U , (6) (f, |σ|2 ) = F 2 2 2 where F is the hypergeometric function. We note that the Sturm Liouville equation defined by (f, |σ|2 ) also gives rise to a form of travelling wave of the KdV equation [5], with the wave profile of the KdV wave given by |σ|2 . As the KdV equation originates from the isospectral deformation of a Sturm-Liouville equation, it is perhaps not too surprising to find a solution of this kind, in view of the Sturm-Liouville equation in (3) derived from Newman-Penrose equations. Away from the possible singular points of the Weyl curvature caused by focusing, the shear of the congruence of null geodesics defined by ∂/∂U behaves like the a solitary wave of the KdV equation (see Fig. 1 below), this suggests the feasibility of solitonic information transmission in the spacetime background defined by the metric in (5), whose functional form is specified by (6) and (4). The information to be transmitted is encoded into the shear, say for instance in its maximum amplitude. Like in the case of optical solitons, the information is faithfully transmitted without any distortion due to the solitonic behaviour of the information carrier. The class of examples described above suggests that, in general relativity, if the gravitational field is manipulated in the right way, solitonic behaviour for

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Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

light propagation is feasible. As a result, instead of an optical fibre, we may use a relativistic gravitational field as an effective continuous media for solitonic information transmission. Note also that we have actually described not a single but a class of spacetimes. Different choices of (f, η, χ) subject to (3) and (4) correspond to gravitational waves of different wave profile and polarisation. Our experience in the KdV equation also indicates that the solitonic behaviour described here is also likely to be stable.

I

I

V =0

U =0

1

Shear of the null geodesic congruence propagates like a KdV travelling wave

Figure 1: Propagation of the shear of the null geodesic congruence defined by ∂/∂U

References 1. Akira Hasegawa, Optical Solitons in Fibers (1989) (Springer-Verlag Press). 2. R. Penrose and W. Rindler, Spinors and Spacetime, Vol. 1, Camb. Univ. Press (1986) Chapter 4. 3. Z.Q. Kuang, Y.K.Lau and X.N.Wu, Gen. Rel. Grav. 31(9) (1999) 1327. 4. C .B. Liang, Gen. Rel. Grav. 27 (1995) 669. 5. P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Camb. Univ. Press (1989) Chapter 4.

1

Difference Discrete Geometry on Lattice Ke Wu1 , Wei-Zhong Zhao1 , and Han-Ying Guo2 1 2

Department of Mathematics, Capital Normal University, Beijing 100037, China Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China

Abstract This is an extended version of our previous paper [1] on difference discrete connection and curvature on cubic lattice. Some new results about properties of discrete topological number on bundles with Abelian gauge group are added. Keyword: discrete connection, discrete curvature, noncommutative calculus, lattice gauge theory, discrete Lax pair

1 Introduction The discrete systems play a very important role in various fields, so they are widely studied in different branches. One of the successful ways to deal with quantum gauge field theory non-perturbatively is the lattice gauge theory in high energy physics, which has opened up new directions in both physics and mathematics in order to deal with the gauge potentials as connections in a discrete manner. Among the integrable systems, there are certain discrete integrable ones which can be obtained as the discrete counterparts of the continuous systems by means of the integrable discretization method. For structure preserving algorithms or geometric algorithms in computational mathematics, the continuous systems are discretized such that some important properties like symplectic or multisymplectic structure, or the symmetry of the systems, are preserved in a discrete manner. There are many important discrete systems, and some of them do not even have a proper or unique continuum limit. In this paper, we focus on the problem how to get the discrete counterparts in a systematic manner for such a kind of continuous systems maintaining their important properties like the gauge potentials as connections, the symplectic or multisymplectic structures, the Lax pairs, the symmetries and so on. A simple and direct method to get a discrete counterpart for a given continuous system such as an ODE or a PDE is to discretize the independent variable(s) and let the dependent variables become discrete correspondingly

301

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Ke Wu, Wei-Zhong Zhao, and Han-Ying Guo

without specifying the specially chosen range. However, in most cases all important properties of the continuous system may be lost and the behaviour of the discrete systems are even hard to be controlled. For constructing the most (quarried??)/meaningful ones in all possible discretizations of the corresponding continuous systems, there is a working guide or a structure-preserving criterion: Namely, it is important to look at those discrete systems that preserve as much of the intrinsically important properties of the continuous system as possible (see, e.g. [2, 3, 4]). In the course of discretization, only a few of the most important properties, or ”structure” can be maintained. Thus, we have to select these “structures”, find their discrete counterparts and we have to know how to preserve them discretely at the lowest price to pay. There are, for example, two classes of conservation laws in canonical conservative mechanics. The first class is of phase-area conservation laws characterized by the symplectic preserving property. The other class is related to energy and all first integrals of the canonical equations. Thus, it is needed to know if it is possible to establish such a kind of discrete systems that they not only discretely preserving the “structure”, such as the symplectic structure, but also the energy conservation. And the question is whether it is possible to get these discrete systems by a discrete variational principle. In fact, as far as the discrete variation for the discrete mechanics is concerned, there are different approaches. In the usual approach (see, e.g. [5, 6]), only the discrete dependent variables are taken as the independent variational variables, while their differences are not. However, in the discrete variational principle proposed by two of the present authors and their collaborators recently [3, 4], the differences of the dependent variables as the discrete counterparts of derivatives are taken as the independent variational variables together with the discrete dependent variables themselves. Actually, this is just the discrete analogue of the variational principle for the continuous mechanics, where the derivatives of the dependent variables are dealt together with the independent variational variables. Thus, the difference discrete Legendre transformation can be made and the method can be applied to either discrete Lagrangian mechanics or its Hamiltonian counterpart, via the difference discrete Legendre transformation. The approach has been applied to the symplectic and multisymplectic algorithms in the both Lagrangian and Hamiltonian formalism. It has been also generalized to the case of variable steps in order to preserve the discrete energy in addition to the symplectic or the multisymplectic structure[4, 5, 7]. For the lattice gauge theory as the discrete counterpart of the gauge theory in continuous spacetime, the discrete gauge potentials, field strength and the action have been introduced in a manner almost completely different from (as least apparently) the ones in ordinary gauge fields or in the connection theory [8] in fibre bundle. Although the discrete connection theory has its own right (see, e.g. [9, 10]) as an application of the non-commutative geometry [11], is an important and interesting problem to investigate how the discrete

Difference Discrete Geometry on Lattice

303

gauge potential in the lattice gauge theory is introduced as a kind of discrete connection. Two of us with their collaborator had considered this issue in [12] very briefly in a way different from other relevant proposals (see, e.g. [13, 14]). In this paper, we study further the discrete connection and curvature on a regular lattice in a way similar to the connection and curvature on vector bundle. We define them in different but equivalent manner, find their gauge transformation properties, the Chern class in the Abelian case and related issues. We show that the discrete connection and curvature introduced here are completely equivalent to the ones in the lattice gauge theory on a regular lattice. We also apply these issues to the discrete integrable systems and show that their discrete Lax pairs and the discrete-curvature free conditions are certainly similar to their continuous cases. One of the key points in our approach is that the difference operators acting on the functions space over the regular lattice are still regarded as a kind of independent geometric objects and their dual should be the one forms, such that we can introduce the discrete tangent bundle over the lattice and the cotangent bundle as its dual, with the basis given by the difference operators and one-formes, respectively. They are just the discrete counterparts of the continuous case, where derivatives and the one-formes as their dual play the roles of the bases for the tangent bundle and the cotangent bundle, respectively. In order to do so, the non-commutative differential calculus on the function space over the lattice [12] has to be employed. Similarly, the discrete vector bundle over the regular lattice can also be set up. In the connection theory a la Cartan, the exterior differential of the basis of a vector space at a certain point should be expanded in terms of the basis, where the expanding coefficients are just the coefficients of the connection on the vector space. Since all counterparts of the basis, the exterior differential etc. are provided in the discrete vector bundle over the lattice, the discrete connection can also be introduced in a way similar to the one a la Cartan. This is our simple way to introduce the discrete connection. In the continuous case, there are several equivalence definitions for connection and curvature. Similarly, we introduce here some of the equivalent definitions for the discrete connection and curvature on the lattice. It should be noted that the definition of the discrete connection in terms of the decomposition of the tangent space of the discrete vector bundle may be written in a form without differences. Thus, it can be generalized to the case over the random lattices. The paper is organized as follows. In order to show the background and necessary preparation, we briefly recall the discrete mechanics and the noncommutative differential calculus on hypercubic lattice of high dimension in section 2 and section 3, respectively. In section 4, we discuss the discrete connection, curvature, Chern class and their gauge transformation properties. In section 5, we generalize one of the definitions for the discrete connection to the case of random lattice. Some applications to the lattice gauge theory and discrete integrable systems are given in section 6. We end with some remarks and discussions.

304


2 Difference Discrete Mechanics In order to introduce the structure of a discrete bundle, it is useful to review the formulism of Lagrangian mechanics and the discrete mechanics as its discrete counterpart. Let t ∈ T be the time, M an n-dimensional configuration space, and we take here a vector space for simplicity. A particle moving on the configuration space is denoted in terms of its generalized coordinates as q i (t) ∈ M and its generalized velocities q˙i (t) = dq i (t)/dt as an element in tangent bundle T M of M . The space of all the possible paths of a particle moving in configuration space is an infinite dimensional space. The Lagrangian of a system is a functional defined on this space and denoted as L(q i (t), q˙i (t)), i = 1, · · · , n. For simplicity, the Lagrangian in our discussion is of first order and independent of t. The action functional is Z t2 i S([q (t)]; t1 , t2 ) = dtL(q i (t), q˙i (t)). (1) t1

Here q i (t) describes a curve Cab with end points a and b, ta = t1 , tb = t2 , along which the motion of the system may take place. For the difference discrete Lagrangian mechanics, let us consider the case that “time” t is difference discretized t ∈ T → tk ∈ TD = {(tk , tk+1 = tk + ∆tk = tk + h,

k ∈ Z)}

(2)

and the step-lengths ∆tk = h are equal to each other for simplicity, while the n-dimensional configuration space Mk at each moment tk , k ∈ Z, is still continuous and smooth enough. S Let N be the set of all nodes on TD with index set Ind(N ) = Z, M = k∈Z Mk the configuration space on TD that is at least pierce wisely smooth enough. At the moment tk , Nk be the set of nodes neighboring to tk . Let Ik the index set of nodes of Nk including tk . The coordinates of Mk are denoted by q i (tk ) = q i(k) , i = 1, · · · , n. T (Mk ) the tangent bundle of S Mk in the sense that difference at tk is its base, T ∗ (Mk ) its dual. Let Mk = S l∈Ik Ml be the union of configuration spaces Ml at tl , l ∈ Ik on Nk , T Mk = l∈Ik T Ml the union of tangent bundles on Mk , F (T Mk ) and F (T Mk ) the functional spaces on each of them respectively, etc.. Sometime, it is also necessary to include the links, plaquettes etc. as well as the dual lattice, like in the lattice gauge theory, the mid-point scheme in the symplectic algorithm and so on. In such cases, the notations and conceptions introduced here should be generalized correspondingly. The above considerations should also make sense for the vector bundle over either the 1-dimensional lattice TD or the higher dimensional lattice as a discrete base manifold. In the difference variational approach and the definition of the difference discrete connection, these notions may be used. Now the discrete Lagrangian LD (k) on F (T (Mk )) reads


LD (k) = LD (q i(k) , ∆k q i(k) ),

305

(3)

with the difference ∆k q i(k) of q i(k) at tk defined by ∆k q i(k) :=

q i(k+1) − q i(k) 1 = (q i(k+1) − q i(k) ). tk+1 − tk h

The discrete action of the system is given by X h · LD (k) (q i(k) , ∆k q i(k) ). SD =

(4)

(5)

k∈Z

The discrete variation for q i(k) = q i (tk ) should be defined as i

δq i(k) := q 0 (tk ) − q i (tk ).

(6)

And the discrete variations for ∆k q i(k) are given by δ∆k q i(k) = ∆k δq i(k) .

(7)

Thus, the variations of the discrete Lagrangian can be calculated δLD (k) = [Lqi(k) ]δq i(k) + ∆k (pi (k+1) δq i(k) ),

(8)

where [Lqi(k) ] is the discrete Euler-Lagrange operator [Lqi(k) ] :=

∂LD (k) ∂LD (k−1) , −∆ i(k) ∂q ∂∆q i(k−1)

(9)

and pi (k) the discrete canonical conjugate momenta pi (k) :=

∂LD (k−1) . ∂∆q i(k−1)

And the variation of the discrete action can be written as X δSD = h[Lqi(k) ]δv q i(k) + ∆(pi (k+1) δv q i(k) ).

(10)

(11)

k

The variational principle requires δSD = 0, so the discrete Euler-Lagrange equations for q i(k) ’s follow as ∂LD (k) ∂LD (k−1) − ∆( ) = 0. ∂q i(k) ∂∆q i(k−1)

(12)

In order to transfer to the discrete Hamiltonian formalism, it is necessary to introduce the discrete canonical conjugate momenta according to the equation (10) and express the discrete Lagrangian by the discrete Hamiltonian via Legendre transformation

306


HD (k) := pi (k+1) ∆t q i(k) − LD (k) . Thus, the discrete action can be expressed as X SD = h · (pi (k+1) ∆t q i(k) − HD (k) ).

(13)

(14)

k

Now, Hamilton’s principle requires δv SD = 0, and we obtain the discrete canonical equations for pi (k) ’s and q i(k) ’s ∆q i(k) =

∂HD (k) , ∂pi (k+1)

∆pi (k) = −

∂HD (k) . ∂q i(k)

(15)

As we mentioned, the advantage of the difference discrete variational principle is based on keeping the difference operator as a discrete derivative operator. It is also clear that this approach can be applied to the field theory as well and it can be generalized to the total discrete variation with variable step-lengths [4, 7]. Actually, this key point will also play a central role in our proposal for the discrete connection and curvature. In the usual discrete variation, however, the Q × Q is used to indicate the vector field on a discrete space and the difference has not been dealt with as an independent variable (see, e.g. [5], [15]). The corresponding discrete action is n−1 X S= h · L(qk , qk+1 ), (16) k=0

where the Lagrangian L(qk , qk+1 ) is a functional on Q × Q. This is also the central idea of the resent proposal to the disconnection in [10]. Namely, using the tensor product Q×Q to study discrete tangent spaces of Q. In other words, the tangent vector q(t) ˙ at tk is represented by a pair of nodes (qk , qk+1 ) without introducing the difference operator. Thus, the difference discrete Legendre transformation and the discrete Hamiltonian formalism via the transformation cannot be formulated. In this case it is expected that the groupoid formalism may be used and there is a possibility to understand some of the geometric meaning of discrete models [16].

3 Difference and Differential Form on Lattice In this section, we recall the application of the differential calculus on discrete group [13] to the hypercubic lattice [12]. Although the result is similar to the one in [14], the key point is different. In our approach, the shift operator is regarded as the generator of a discrete Abelian group in each direction of the high dimensional hypercubic lattice. For simplicity, we focus on the lattice with equal spacing h = 1. Thus, the dimension of vector fields or differential forms is equal to the number of the shift operators of the lattice.


307

Let N and A be a lattice and the algebra of complex valued functions on N , respectively, then define the right and left shift operators Eµ , Eµ−1 at a node x ∈ N in the µ-direction by Eµ x = x + µ ˆ,

Eµ−1 x = x − µ ˆ,

(17)

and introduce a homeomorphism on the function space A, Eµ (f (x) · g(x)) = Eµ f (x) · Eµ g(x),

Eµ f (x) = f (x + µ ˆ),

f, g ∈ A, (18)

where (x − µ ˆ), x and (x + µ ˆ) are points on Nx and they are the nearest neighbors in the µ-direction, the dot denotes the multiplication in A. The tangent space at the node x of Nx is defined as T Nx := span{∆µ |x , µ = 1, · · · , n}, where the operator ∆µ is defined on the link between x and x + µ ˆ and its action on A is the differences in µ-th direction as, ∆µ f (x) := (Eµ − id)f = f (x + µ ˆ) − f (x).

(19)

The above difference operator is a discrete analogue of the basis ∂ν := ∂x∂ ν for a vector field X = X ν ∂ν in the continuous case. The action of a difference operator ∆µ in (19) on the functional space A satisfies the deformed Leibnitz rule ∆µ (f (x) · g(x)) = ∆µ f (x) · g(x + µ ˆ) + f (x)∆µ g(x).

(20)

For a given node x ∈ Nx , all ∆µ form a set of bases of the tangent space T Nx . Its dual space denoted as T ∗ Nx is a space of 1-forms with a set of bases dxµ defined on the link, too. They satisfy dxµ (∆ν ) = δνµ , which is also denoted as Ω 1 and Ω 0 = A like the continuous case. Thus, the tangent bundle and its dual cotangent bundle over N can be defined as [ [ T N := T Nx and T ∗ N := T ∗ Nx , (21) x∈N

x∈N

respectively. L n Let us construct the whole differential algebra Ω ∗ = Ω on T ∗ N as in n∈Z

continuous case. The exterior derivative operator dD : Ω k → Ω k+1 is defined as X dD ω = ∆α f dxα ∧ dxµ1 ∧ · · · ∧ dxµk ∈ Ω k+1 , (22) α

where ω = f dxµ1 ∧ · · · ∧ dxµk ∈ Ω k . When k = 0, Ω 0 = A, then d : Ω 0 → Ω 1 is given by

(23)

308


dD f =

X

∆α f dxα .

(24)

α

It is straightforward to prove that (a) : (dD f )(v) = v(f ), v ∈ T (N ), f ∈ Ω 0 , (b) : dD (ω ⊗ ω 0 ) = dD ω ⊗ ω 0 + (−1)degω ω ⊗ dD ω 0 ,

ω, ω 0 ∈ Ω ∗ ,

(25)

(c) : d2D = 0, provided that the following conditions hold (1) dxµ ∧ dxν = −dxν ∧ dxµ , (2) dD (dxµ ) = 0, (3) dxµ f = (Eµ f )dxµ , (no summation).

(26)

Thus, we set up a well defined differential algebra. Note that in order to do so the multiplication of functions and one-forms must be noncommutative.

4 Difference Discrete Connection and Curvature The discrete analogue of connection has been given by two of the present authors [12] and others (see, e.g. [14], [9], [10]). In this section we define the (difference) discrete connection in a simple way similar to that in the continuous case based on the noncommutative differential calculus introduced in the last section. As was mentioned, the key point is to replace the difference discrete exterior derivative by the covariant difference discrete derivative for the sections on a bundle. 4.1 Connection, Gauge Transformations and Holonomy Discrete Bundle and Discrete Sections Let P = P (M, G) be a principal bundle over an n-dimensional base manifold M isomorphic to Rn with a Lie group G as the structure group. Now consider its discrete counterpart in the following manner as a discrete principle bundle. Let the manifold M be discretized as Zn , i.e. the hypercubic lattice with equal spacing h = 1, and take such an M ' Zn (a lattice N ) as the discrete base space. For a node x ∈ N , there is a fiber Gx = π −1 (x) isomorphic to the Lie group G as the structure group. The union of all these fibers is called discrete principal bundle and denoted as Q(N, G): [ π −1 (x). (27) Q= x∈N

If the structure group G is a linear matrix group, for example, GL(m, R) or its subgroup, we may also define an associated vector bundle V =


309

V (N, Rm , GL(m, R)) to the discrete principal bundle Q(N, GL(m, R)). For any x ∈ N there is a fiber Fx = π −1 (x) isomorphic to an m-dimensional linear space Rm with the right action of GL(m, R) on the Fx . The union of all these fibers is called a discrete vector bundle: [ V = π −1 (x). (28) x∈N

The discrete field on a discrete bundle is a section h(x) ∈ G on discrete principal bundle or a ψ j (x) ∈ Rm on the vector bundle for all x ∈ Nx . A section on discrete bundle is a map, [ G:N = Nx −→ Q = Q(N, G), G : x 7→ G(x). (29) Similarly, we can define the discrete section on a discrete vector bundle. The union of the all sections is denoted as Γ (Q) for the discrete principal bundle or as Γ (V ) for the discrete vector bundle, respectively. One example for the discrete vector bundle is the tangent bundle T N over an n-dimensional hypercubic lattice N in eq.(21). The base manifold of this bundle is the hypercubic lattice, the fiber π −1 (x) over x ∈ N is an ndimensional vector space with basis {∆µ , µ = 1, · · · , n}. Another example is its dual bundle T ∗ N , its fiber is also the n-dimensional vector space with basis {dxµ , µ = 1, · · · , n}. For the discrete vector bundle the section space Γ (V ) is also the vector space. The same for the Γ (T ∗ N ). Their tensor product is given by Γ (T ∗ N × V ) = Γ (T ∗ N ) × Γ (V ).

(30)

As was mentioned in the previous section, the one form basis dxµ in the discrete case is defined on a point x ∈ N but it links a point with another nearby point x + µ ˆ. Then the section of one forms in Γ (T ∗ N ) is generally defined on Nx , its structure is very different from the section in continuous case. So we call it a discrete section. The section in Γ (V ) may be the discrete section as in Γ (T N ) of tangent vector bundle T N . However there is another possibility. Namely, the section is defined on the node only as in the discrete principal G-bundle. The direct product here can be simply understood as the discrete counterpart of the direct product in the continuous case in the above manner. Definition of Connection and Gauge Transformations Definition : A difference discrete connection or covariant difference discrete derivative is the linear map D : Γ (V ) −→ Γ (T ∗ N × V ), which satisfies the following condition:

(31)

310


D(s1 + s2 ) = Ds1 + Ds2 , D(as) = dD a ⊗ s + aDs,

(32)

where s, s1 , s2 ∈ Γ (V ) and a ∈ A. This is almost the same as the definition of the connection or the covariant derivative in the continuous case, which is basically equivalent to the connection a la Cartan. Since our discussion on all geometric quantities are in the local sense, for the simplicity, we can choose the basis {sα , 1 ≤ α ≤ m} as the basis of the linear space Γ (V ) and {dxµ ⊗ sα , 1 ≤ α ≤ m, 1 ≤ µ ≤ n} as the basis of the sections space Γ (T ∗ N × V ). Then the covariant derivative Dsα should be the linear expansion on {dxµ ⊗ sα }. Hence, we can define it in the local sense as X X µ − (Bµ )α (33) − (B)α Dsβ = β dx ⊗ sα β ⊗ sα = α

α,µ

or simply Ds = −B · s,

(34)

µ

where B = Bµ dx is the local expression of a discrete connection 1-form. For the continuous case, the connection 1-form is valued on a Lie algebra. However the 1-form B = Bµ dxµ here is matrix valued. Since all 1-forms are defined on the links, the coefficients Bµ are also defined on the link (x, x + µ ˆ) and can be written as Bµ (x) = B(x, x + µ ˆ). (35) P α For any section S = α a sα , we have X DS = (dD aα · sα + aα · Dsα ) X = (4µ aα dxµ ⊗ sα − aα (Bµ )βα dxµ ⊗ sβ ) (36) X = (4µ aβ − aα (Bµ )βα )dxµ ⊗ sβ X = (DDµ aβ )dxµ ⊗ sβ , where DDµ aβ = 4µ aβ − aα (Bµ )βα is called the discrete covariant derivative of the vector aα and X X µ DD aα = DDµ aα dxµ = (4µ aα − aβ (Bµ )α β )dx

(37)

(38)

is the discrete exterior covariant derivative of the vector aα . On the space Γ (V ), we can choose another linear basis or perform a linear transformation of the basis, i.e., take the gauge transformation as follows sα 7−→ s˜α = g(x)βα · sβ ,

x ∈ N,

(39)


311

where g(x)βα is the GL(m, R) gauge group valued function defined on the node x ∈ N . In order to keep section S being invariant, the coefficients aα of P the a general section S = aα sα should transform as aα 7−→ a ˜α = aβ · (g −1 (x))α β.

(40)

From the gauge invariance of S and DS, we can derive the gauge transformation property of DDµ aα , X β DS 7−→ (DDµ aγ )dxµ · (g −1 (x))α γ ⊗ g(x)α · sβ X (41) µ β = (DDµ aγ ) · (g −1 (x + µ ˆ))α γ dx ⊗ g(x)α · sβ , where the noncommutative commutation relation between function and 1form basis is used. Then the covariance of the covariant derivative follows DDµ aα 7−→ DDµ aγ · (g −1 (x + µ ˆ))α γ.

(42)

Thus, we get the gauge transformation property of the difference discrete connection 1-form as

or

Bµ (x)dxµ 7−→ g(x) · Bµ (x)dxµ · g −1 (x) + g(x) · dD g −1 (x),

(43)

Bµ (x) 7−→ g(x) · Bµ (x) · g −1 (x + µ ˆ)+g(x) · 4µ g −1 (x).

(44)

Together with the gauge transformation property of the coefficients of a vector field in (40), the gauge covariance of the derivative DDµ aγ is confirmed. Discrete Connection via Horizontal Tangent Vector For the vector bundle, there is another definition of connection. It is based on the decomposition of the total tangent vector of the bundle into the horizontal and vertical parts. Then the horizontal tangent vector invariant under the right operation of the structure group also defines a connection. In fact, the horizontal tangent vector is nothing but the covariant derivative. This definition can also be given in an analogous manner for the discrete case here. Let us consider the discrete vector bundle over a discrete base manifold N as a regular lattice with the fiber being a sufficiently smooth m-dimensional vector space Vx at x ∈ N , like the Vk used in section 2. As we discussed before, the basis of tangent space T Vx is Xα =

∂ , ∂aα

(1 ≤ α ≤ m),

(45)

where aα , 1 ≤ α ≤ m, are the coordinates of the fiber Vx . The basis of tangent vector on a discrete regular lattice is

312


4µ ,

(1 ≤ µ ≤ n).

(46)

Then the basis of the total tangent space of the discrete vector bundle is the union of (45) and (46). Similar to the continuous case, the vector space tangent to the fiber, i.e. the linear combination of basis in (45), is a vertical subspace of the total tangent space of the discrete vector bundle. Its complementary vectors of the vertical subspace in the total tangent of vector bundle are horizontal and constitute the horizontal subspace. The basis of horizontal subspace is as follows, β Xµ = 4µ − (Bµ )α βa

∂ , ∂aα

(1 ≤ µ ≤ n).

(47)

In comparison with the definition of difference discrete connection, it is easy to see that the horizontal vector is nothing but the covariant derivative in (37). This means that when we form the decomposition of tangent vector on the total bundle space we get the coefficients Bµ (x) of the discrete connection. For a given difference discrete connection or its coefficients Bµ (x), we can also get a decomposition of the total tangent vector space of bundle into vertical and horizontal parts sufficiently and necessarily. This shows that the difference discrete connection on a discrete vector bundle is equivalent to a decomposition of the total tangent vector space into vertical and horizontal subspaces as above. Similarly, we can define the basis of the dual space for the decomposition, i.e., the basis of the vertical and horizontal 1-form space, respectively, as β µ ω α = daα + (Bµ )α β a dx ,

(1 ≤ α ≤ m),

(48)

and ω µ = dxµ ,

(1 ≤ µ ≤ n).

(49)

They satisfy the following dual relation: ω α (Xβ ) = δβα ,

ω α (Xµ ) = 0,

ω µ (Xβ ) = 0,

ω µ (Xν ) = δνµ .

Covariant Derivative and Parallel Transport From the above discussions, we can get the difference discrete connection on discrete vector bundle through the definition of the absolute derivative or the horizontal tangent vector. Both lead to the difference discrete covariant derivative for the vectors, DDµ aβ = 4µ aβ − aα (Bµ )βα .

(50)

In terms of the relation between 4µ and Eµ , we obtain another expression for the covariant derivative as,


DDµ aγ (x) = Eµ aγ (x) − aβ (x) · [(Bµ (x))γβ + δβγ ],

313

(51)

where δβα is the unit matrix. Then we obtain another expression for the coefficient of the discrete connection, Uµ (x) = [(Bµ (x))γβ + δβγ ],

(52)

which is an element of some group, for example the group GL(m, R) in our discussions, and connects the points x and x + µ ˆ. We can also call Uµ (x) the discrete GL(m, R)-connection and express it as Uµ (x) = U (x, x + µ ˆ).

(53)

From the second expression of the coefficient of connection and the definition of covariant derivative for the vectors, we obtain the parallel transport of the section of vector aβ , provided that its covariant derivative is zero

or Namely,

DDµ aβ = 4µ aβ − aα (Bµ )βα = 0

(54)

Eµ aγ (x) − aβ (x) · (Uµ (x))γβ = 0.

(55)

aγ (x + µ ˆ) = aβ (x) · (U (x, x + µ ˆ))γβ .

(56)

This means that the parallel transport of the section of vector aβ (x) along the path x 7→ x + µ ˆ is expressed as aβ (x) 7−→ aβ (x + µ ˆ) = aγ (x) · (U (x, x + µ ˆ))βγ .

(57)

It is shown that there is a parallel transport on the discrete bundle along the curve on discrete base manifold for a given discrete connection on discrete bundle. Due to the discrete connection coefficient U (x, x + µ ˆ) as a group element, there are the following group properties of U (x, x + µ ˆ) along the path decomposition of (x, x + µ ˆ + νˆ) into (x, x + µ ˆ) and (x + µ ˆ, x + µ ˆ + νˆ), U (x, x + µ ˆ) · U (x + µ ˆ, x + µ ˆ + νˆ) = U (x, x + µ ˆ + νˆ).

(58)

From the inverse of the parallel transport, it also follows that (U (x, x + µ ˆ))−1 = U (x + µ ˆ, x).

(59)

The coefficient of the discrete connection U (x, x + µ ˆ) determines the parallel transport not only on a vector bundle but also on a GL(m, R) principal bundle. Therefore, it is also called a discrete GL(m, R) connection.

314


Parallel Transport on Discrete Principal Bundle The matrix structure group of a discrete vector bundle can be generalized to any Lie group G and the coefficients of a connection are the group G-valued with the right operations. Thus, we can get a difference discrete G-valued connection on a discrete principal G-bundle over the lattice. In this case, the concept of parallel transport can be extended to the discrete principal G-bundle. For a section h(x), the parallel transport with respect to a G-valued connection U (x, x + µ ˆ) reads h(x) 7−→ h(x + µ ˆ) = h(x) · U (x, x + µ ˆ).

(60)

All elements here belong to the Lie group G and the multiplication should be the group multiplication. The above equation can be expressed as h(x0 ) 7−→ h(x1 ) = h(x0 ) · U (x0 , x1 ),

(61)

h(x1 ) 7−→ h(x0 ) = h(x1 ) · U (x1 , x0 ),

(62)

U (x1 , x0 ) = U (x0 , x1 )−1 .

(63)

or which implies that

Similarly, the covariant derivative for the section h(x) can be given as DDµ h(x) = Eµ h(x) − h(x) · Uµ (x), and the covariant exterior derivative as X DD h(x) = DDµ h(x)dxµ = (Eµ h(x) − h(x) · Uµ (x))dxµ .

(64)

(65)

µ

If link (x, x + µ) ˆ appears as common boundary in two different faces, each of them could be understood as a local coordinate neighborhood and denoted as Uα , Uβ . Then the discrete connection U (x, x + µ ˆ) has different expressions U (α) (x, x + µ ˆ) and U (β) (x, x + µ ˆ) in Uα , Uβ respectively. They satisfy the following relations: −1 U (α) (x, x + µ ˆ) = gα,β (x) · U (β) (x, x + µ ˆ) · gα,β (x + µ ˆ),

(66)

which is nothing but the gauge transformation defined in (44), where gα,β (x) is the transition function between Uα , Uβ . If the link (x, x + µ ˆ) is the common boundary of three different faces or three local coordinate neighborhoods Uα , Uβ and Uγ . Then there will be three relations for the different expressions of discrete connection U (α) (x, x + µ ˆ), U (β) (x, x + µ ˆ) and U (γ) (x, x + µ ˆ) as follows, −1 U (α) (x, x + µ ˆ) = gα,β (x) · U (β) (x, x + µ ˆ) · gα,β (x + µ ˆ), −1 U (β) (x, x + µ ˆ) = gβ,γ (x) · U (γ) (x, x + µ ˆ) · gβ,γ (x + µ ˆ), (67) (γ) (α) −1 U (x, x + µ ˆ) = gγ,α (x) · U (x, x + µ ˆ) · gγ,α (x + µ ˆ),


315

The consistent condition for those transformation relations require that the transition functions should satisfy the cocycle condition or consistent condition, gα,β (x) · gβ,γ (x) · gγ,α (x) = id. (68) 4.2 Difference Discrete Curvature, Bianchi Identity and Abelian Chern Class Difference Discrete Curvature Based on the definition of the difference discrete connection 1-forms on discrete vector bundle, we can define the curvatures 2-forms similar to the continuous case [8], F = dD B + B ∧ B. (69) If we assume F =

1 Fµν dxµ ∧ dxν , it follows that 2

Fµν (x) = 4µ Bν (x) − 4ν Bµ (x) + Bµ (x) · Bν (x + µ ˆ) − Bν (x) · Bµ (x + νˆ). (70) Under the continuous limit, it is easy to see that this curvature should be the same as the usual formula of curvature. Since the definition is in a similar formulation except for the non-commutative exterior differential calculus, it is also easy to check the covariance property of the curvature under the gauge transformation (43) or (44) as follows F (x) 7−→ Fe(x) = g(x) · F (x) · g −1 (x),

(71)

Fµν (x) 7−→ Feµν (x) = g(x) · Fµν (x) · g −1 (x + µ ˆ + νˆ).

(72)

or the covariance behavior of its components

It is important to see that from the non-commutative property of differential calculus on lattice the shifting operator appears in the covariance of discrete curvature. This is a main difference between the continuous case and the discrete one. And it may lead to more difficulties in the discussion of the gauge covariance and invariance property of tensors in the discrete case. Curvature via Holonomy In the continuous case, the curvature may naturally appear in the homolomy consideration. As the (difference) discrete counterpart, the (difference) discrete curvature may also be described based on the holonomy consideration. Let us consider the square of the exterior covariant derivatives

316


(DD )2 h(x) = DD DDµ h(x)dxµ X (Eµ h(x) − h(x) · Uµ (x))dxµ = DD X µ Eν (Eµ h(x) − h(x) · Uµ (x))dxµ ∧ dxν = µνX (Eµ h(x) − h(x) · Uµ (x))dxµ ∧ Uν (x))dxν − µνX Uµ (x)dxµ ∧ Uν (x)dxν = h(x) µν X = h(x) Uµ (x) · Uν (x + µ ˆ)dxµ ∧ dxν

(73)

µν

=

X 1 h(x) [Uµ (x) · Uν (x + µ ˆ) − Uν (x) · Uµ (x + νˆ)]dxµ ∧ dxν . 2 µν

This leads to another expression for the curvature 2-form with its coefficients X F = U2 = Uµ (x)dxµ ∧ Uν (x)dxν , µν (74) 1 ˆ) − Uν (x) · Uµ (x + νˆ)]. Fµν = [Uµ (x) · Uν (x + µ 2 The zero curvature condition F = 0 is

or

Uµ (x) · Uν (x + µ ˆ) = Uν (x) · Uµ (x + νˆ),

(75)

Uµ (x) · Uν (x + µ ˆ) · Uµ−1 (x + νˆ) · Uν−1 (x) = 1,

(76)

or U (x, x + µ ˆ) · U (x + µ ˆ, x + µ ˆ + νˆ) · U (x + µ ˆ + νˆ, x + νˆ) · U (x + νˆ, x) = 1. (77) The expressions in the last formula is nothing but the holonomy group in geometry or the plaquette variable in lattice gauge theory. We will discuss them in the section 6 and show also that zero curvature condition is just the integrability condition for a discrete integrable system. Bianchi Identity and Abelian Chern Class Similar to the Bianchi identity in differential geometry, we can also derive the Bianchi identity for the difference discrete curvature DD F = dD F − F ∧ B + B ∧ F = 0, or in its components

(78)


317

ˆ = 0. (79) ελµν [4λ Fµν (x) − Fλµ (x) ∧ Bν (x + µ ˆ + νˆ) + Bλ (x) ∧ Fµν (x + λ)] For the Abelian case, one can define the following topological term as discrete Chern class [17], ck = F ∧ F ∧ · · · ∧ F,

(80)

which was used to discuss the chiral anomaly in the lattice gauge theory. The coefficient of the Abelian Chern class is εµ1 µ2 ···µ2k−1 µ2k Fµ1 µ2 (x)·Fµ3 µ4 (x+µˆ1 +µˆ2 ) · · · Fµ2k−1 µ2k (x+µˆ1 +µˆ2 +· · ·+µ2k−2 ˆ ) This equation first appeared in lattice gauge theory for the Abelian anomaly of chiral fermion in a quantum field theory [17].

5 Discrete Connection on G-Bundle over Random Lattice 5.1 Discrete Connection over Randam Lattice The definition of the discrete connection via the horizontal vector space in sect. 3 can be generalized to the one on a G-bundle Q(N, G) over a random lattice N . Let us consider the parallel transport of a section on such a G-bundle: h(x0 ) 7−→ h(x1 ) = h(x0 ) · U (x0 , x1 ),

(81)

where h(xj ) is the G-valued section defined on xj , j = 0, 1 and x0 , x1 are nearest neighbors. We can reexpress it equivalently as (x0 , h0 ) 7−→ (x1 , h1 ) = (x0 , h0 ) · U (x0 , x1 ),

(82)

where h0 = h(x0 ), h1 = h(x1 ) and right multiplication of U on the bundle acts only on the G-valued section h0 . It is easy to see that in these expressions there is no difference operator involved so that they could be make sense for the G-bundle over random lattice, if the discrete connection is properly introduced. On the other hand, if h0 and h1 satisfy eq.(81), it can be proved that the element (q0 , q1 ) = ((x0 , h0 ), (x1 , h1 )) ∈ Q × Q is just a horizontal vector on T Q, i.e. hor((x0 , h0 ), (x1 , h1 )) = ((x0 , h0 ), (x1 , h1 )), (83) where hor(∗, ∗) denotes the horizontal part of the (∗, ∗). In fact, this is almost the same as the one introduced in [10]. Thus, our definition for the discrete connection can be easily compared with the local expression A(x0 , x1 ) of the coefficients of a connection 1-form defined in [10]. Namely,

318


U (x0 , x1 ) = A(x0 , x1 )−1 then

(84)

((x0 , h0 ), (x1 , h0 · A(x0 , x1 )−1 ))

is a horizontal vector. According to the formulation in [10], we get ((x0 , h0 ), (x1 , h0 · A(x0 , x1 )−1 )) = h0 · i(x0 ,e) (A(x0 , x1 )−1 ) · ((x0 , e), (x1 , e)) = h0 · hor((x0 , e), (x1 , e)) = h0 · hor((x0 , e), q1 )

(85)

= h0 · hor((x0 , e), h−1 0 q1 ) = hor((x0 , h0 ), q1 ), which means that the horizontal vector ((x0 , h0 ), (x1 , h0 · A(x0 , x1 )−1 )) is the horizontal part of any vector (q0 , q1 ) with q0 is fixed and q1 is any point on the fiber of π −1 (x1 ). According to the definition of A(x0 , x1 ), we have A(x0 , x1 ) = Ad (x0 , e, x1 , e) . From the gauge transformation property of U (x0 , x1 ), it follows that under the gauge transformation g(x) A(x0 , x1 ) 7−→ g(x1 ) · A(x0 , x1 ) · g −1 (x0 ).

(86)

Ad (x0 , g(x0 ); x1 , g(x1 )) = g(x1 )Ad (x0 , e, x1 , e)g −1 (x0 ).

(87)

This leads to

Thus, we recover the property of the connection 1-form defined in [10] Ad (gq0 , hq1 ) = hAd (q0 , q1 )g −1 .

(88)

As was shown above, our definition of discrete connection is equivalent to that in [10] in the case of the cubic lattice. However, our definition for the discrete curvature is only for the hypercubic lattice, since it is based on the noncommutative differential calculus. How to extend those results to the case of random lattice is under investigation. 5.2 Topological Number in Two Dimension After integration of the Chern class of a principal G-bundle over a base manifold one will get the Chern number, also called topological number. In the discrete case we have got some local geometric information such as connection and curvature and some global properties such as the Chern class of U (1)


319

gauge theory. Can we get the similar Chern numbers in terms of discrete connections? Now we try to show the topological property of a discrete connection in the Abelian case using a simple example, i.e., to calculate its topological number in a two dimensional manifold which is topologically equivalent to the sphere and is discretized as a tetrahedron in the following picture, with notes A, B, C, D, links AB, BC, CA, · · · , and faces 4ABC, 4ABD, · · · , as follows:

A

D

B C

In the boundary of 4ABC there are three links and the holomomy expression F4ABC as F4ABC = U (A, B) · U (B, C) · U (C, A), and its logarithm ln F4ABC = ln U (A, B) + ln U (B, C) + ln U (C, A).

(89)

In this case the topological number is coming from the summation of ln F4ABC over the boundaries of all triangles, c=

1 X ln F4ABC . 2πi

(90)

4

It is not difficulty to see that the topological number c should be an integer and equal to c=

1 X (ln g4ABC,4ACD (A)+ln g4ACD,4ADB (A)+ln g4ADB,4ABC (A)), 2πi point

(91) where (ln g4ABC,4ACD (A) + ln g4ACD,4ADB (A) + ln g4ADB,4ABC (A)) is an integer since the consistency condition of gauge transformations as

320


g4ABC,4ACD (A) · g4ACD,4ADB (A) · g4ADB,4ABC (A) = 1. The discussion above can be easily used to the case of hypercubic lattice and for general case it will be given in forthcoming publications.

6 Applications 6.1 Lattice Gauge Theory and Difference Discrete Connection In the lattice gauge theory [18], the space-time is discretized as hypercubic lattice with equal spacing a in any direction in most cases. Suppose that Aµ is the gauge field or the connection on the continuous case. At each link on the lattice we introduce a discrete version of the path ordered product µ ˆ U (x, x + µ ˆ) ≡ Uµ (x) = eiaAµ (x+ 2 ) , (92) where µ ˆ is the vector in coordinate direction with length a and x is the point coordinate on the node of the hypercubic lattice which takes integer value only. The average field, which is denoted by Aµ (x + µ2ˆ ), is defined at the midpoint of the link (x, x + µ ˆ). Similarly, µ ˆ

ˆ, x). U (x, x − µ ˆ) ≡ U−µ (x) = e−iaAµ (x− 2 ) = U † (x − µ

(93)

If the connection Aµ is valued on the Lie algebra of SU (N ) with a hermitian basis, we have U † (x − µ ˆ, x) = U −1 (x − µ ˆ, x). The variable of a simplest Wilson loop called plaquette variable is expressed as the left side of (77), which is defined on the two dimensional square Wµν = Uµ (x) · Uν (x + µ ˆ) · Uµ† (x + νˆ) · Uν† (x).

(94)

It can be shown that the continue limit of Wµν is related to the Yang-Mills action a4 Re(1 − Wµν ) = Fµν F µν + O(a6 ) + · · · , 2 Im(Wµν ) = a2 Fµν + · · · . Therefore, plaquette variable Wµν should play some role of curvature in the discrete case as we discussed in previous section. However its continuous limit is related not only to the usual curvature but also to the Yang-Mills action as in the above expressions, there should be more geometric meaning in the theory of discrete connection and curvature than usual one.


321

6.2 Geometric Meaning of Discrete Lax Pair In order to understand the discrete connection on discrete bundle, we first discuss some geometric meaning of the Lax pair and the discrete Lax pair. In fact, this gives one a solid motivation and some consideration for the study of discrete connections. Let us start with the concept of a Lax pair of an integral system in continuous two dimensional case with 1-dimension time and 1-dimension space as follows ∂x ψ = ψ · A x , (95) ∂t ψ = ψ · A t , where ψ is a vector and Ax , At are matrix valued. The consistent condition for this linear system is ∂x At − ∂t Ax + [Ax , At ] = 0.

(96)

Now let us discretize the 2-dimensional space-time as a square lattice, R2 → Z2 . The section field ψ(x) on the vector bundle becomes the field ψ(m, n) the functions depending on two discrete variable, i.e., two integer (m, n). Naively the discrete Lax pair may be written as 4x ψ(m, n) = ψ(m, n) · Ax (m, n), 4t ψ(m, n) = ψ(m, n) · At (m, n),

(97)

The derivatives ∂x and ∂t with respect to x and t are replaced by difference operators 4x and 4t , respectively. The consistent condition for the discrete Lax pair, i.e., 4x 4t ψ(m, n) = 4t 4x ψ(m, n) (98) leads to 4x At (m, n)−4t Ax (m, n)+Ax (m, n)At (m+1, n)−At(m, n)Ax (m, n+1) = 0. (99) Using the shift operator Ex and Et we can also rewrite the discrete Lax pair (97) as Ex ψ(m, n) = ψ(m, n) · [1 + Ax (m, n)] = ψ(m, n) · Ux (m, n) Et ψ(m, n) = ψ(m, n) · [1 + At (m, n)] = ψ(m, n) · Ut (m, n),

(100)

where Ut (m, n) = 1 + At (m, n),

Ux (m, n) = 1 + Ax (m, n).

(101)

On requiring the corresponding consistent condition Ex Et ψ(m, n) = Et Ex ψ(m, n), a straightforward calculation leads to

(102)

322

or


Ux (m, n) · Ut (m + 1, n) = Ut (m, n) · Ux (m, n + 1),

(103)

Ux (m, n) · Ut (m + 1, n) · Ux−1 (m, n + 1) · Ut−1 (m, n) = 1.

(104)

If we use the relation of U and A in (101), we can derive the zero curvature condition (99) from (104). When we regard the quantities Ax , At , Ux and Ut as the discrete connections on discrete bundle, the equations (99) and (104) should be the zero curvature condition for these connections, and the left sides of (99) and (104) should be the extension of the curvature in the discrete case.

7 Remarks and Discussions As was mentioned previously, the study of discrete models is very important in both its own right as well as for its applications, although we mainly focus here on the discrete models as the discrete counterparts of the continuous cases. In order to get the discrete models that can keep the properties of continuous ones as much as possible, we may first consider those kinds of discrete models from their continuous counterparts with differences as discrete derivatives. These models can be given by replacing both, the continuous independent variables and their derivatives by their discrete independent variables as a regular lattice and their differences on the lattice, respectively. In general, for the discrete models on the regular lattices including the models just mentioned, it is natural to study first the properties of the function spaces on the lattices and the discrete bundles over the lattices both analytically and geometrically, such as discrete differential calculus, discrete metric, discrete Hodge operator, discrete connection and curvature, and so on. In doing so, we may follow a way similar to that in the continuous cases, as long as the differences are regarded as the discrete derivatives. In this paper, we have briefly reviewed the non-commutative differential calculus on hypercubic lattice, which have discussed by many groups. We have mainly introduced the (difference) discrete connections on discrete vector bundle in several manners, the parallel transport, the decomposition of the vector space into vertical and horizontal space, the covariant derivative on the section of vector bundle as well as the discrete curvature of the discrete connections. We have also studied their relation to the lattice gauge theory and applied to the Lax pairs for the discrete integrable systems. There are, of course, also many properties of the discrete models, which are apparently very different from the continuous ones. These should be investigated further. Although one of the definitions for the discrete connection can be extended to the case over the random lattice, however, for the discrete curvature on the random lattice in the lattice gauge theory, it is still open


323

whether it can be defined, at least formally, in a way similar to the continuous case. This is also a very interesting question and from its investigation we expect to get more results on lattices with no-trivial topology. Another very important problem is how to get the topological classes with non-Abelian group. Needless to say, more attention should be payed to those questions. The work is partly supported by NKBRPC(2004CB318000), Beijing JiaoWei Key project (KZ200310028010) and NSF projects (10375087, 10375038, 90403018, 90503002). The authors would like to thank Morningside Center for Mathematics, CAS. Part of the work was done during the Workshop on Mathematical Physics there.

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Noncommutative Geometry and Physics 2005: Proceedings of the International Sendai-Beijing Joint Workshop

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Noncommutative Geometry