Differential Geometry and Physics: Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics, Tianjin, ... August 2005 (Nankai Tracts in Mathematics)

Nankai Tracts in Mathematics

Vol. 10

DIFFERENTIAL GEOMETRY AND

PHYSICS Editors

Mo-Lin Ge & W e i p i n g Z h a n g

Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Iheoreti Physics

World Scientific

DIFFERENTIAL GEOMETRY AND

PHYSICS Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics

NANKAI TRACTS IN MATHEMATICS Series Editors: Yiming Long and Weiping Zhang Nankai Institute of Mathematics

Published Vol. 1

Scissors Congruences, Group Homology and Characteristic Classes by J. L Dupont

Vol. 2

The Index Theorem and the Heat Equation Method byY.LYu

Vol. 3

Least Action Principle of Crystal Formation of Dense Packing Type and Kepler's Conjecture by W. Y. Hsiang

Vol. 4

Lectures on Chern-Weil Theory and Witten Deformations by W. P. Zhang

Vol. 5

Contemporary Trends in Algebraic Geometry and Algebraic Topology edited by Shiing-Shen Chern, Lei Fu & Richard Hain

Vol. 6

Riemann-Finsler Geometry by Shiing-Shen Chern & Zhongmin Shen

Vol. 7

Iterated Integrals and Cycles on Algebraic Manifolds by Bruno Harris

Vol. 8

Minimal Submanifolds and Related Topics by Yuanlong Xin

Nankai Tracts in Mathematics - Vol. 10

DIFFERENTIAL GEOMETRY AND

PHYSICS Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics Tianjin, China

20 - 26 August 2005

Editors

Mo-Lin Ge & Weiping Zhang Chern Institute of Mathematics, Tianjin, China

\Hp World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

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Nankai Tracts in Mathematics — Vol. 10 DIFFERENTIAL GEOMETRY AND PHYSICS Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics Copyright © 2006 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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Dedicate to the memory of Professor Shiing-Shen Chern

Vll

FOREWORD The Nankai Mathematical Institute, whose grand new premises were inaugurated on the occasion of the 23rd conference on Differential Geometric Methods in Theoretical Physics, is the creation of the great Chinese mathematician Shing-Shen Chern. Unfortunately he did not live long enough to attend the conference, but his spirit was present throughout. Chern recognized many years ago the need for China to have its own centre for advanced mathematical research, a centre modelled on the Institute for Advanced Study at Princeton where Chern first went and on the Berkeley Institute (MSRI) which he later helped to establish. By his personal example and tireless efforts the Nankai Institute came into being and is well placed to play a leading role in the new China of the 21st century. The 2005 conference will no doubt be just the first of many subsequent meetings at Nankai which will strengthen the international links between Chinese mathematicians and their colleagues in other countries. I first met Chern in 1956, when I was a fresh Ph.D. on my first visit to the United States. He was very friendly and helpful and our association continued over subsequent years. When I was President of the London Mathematical Society in 1976, he came to London as the AMS bicentennial lecturer and brought me a Chinese poem, in beautiful calligraphy, which he had composed on the flight. Later he encouraged me to visit Nankai and meet some of his younger Chinese colleagues. He remained active till the very end and his friends were all very pleased when he was awarded the first Shaw Prize in Mathematics, in recognition of his pioneering role in modern differential geometry.

Michael Atiyah

FOREWORD This year, 2005, is the hundredth anniversary of Einstein's Annus Mirabilis. We recall his repeated emphasis on the need to geometrize the foundation of physics. It is thus especially appropriate this year to hold an International Conference on Differential Geometry Methods in Theoretical Physics. As a person associated with Nankai for many years, and as an early student and admirer of Professor S.S. Chern, I am particularly happy that this year's Conference site is his Nankai Institute of Mathematics. Professor Chern had eagerly anticipated his participation at this Conference. He is no longer with us, but his work and his spirit will be with this, and indeed with all future International Conference on Differential Geometry Methods in Theoretical Physics.

Chen Ning Yang

PREFACE The XXIII International Conference on Differential Geometric Methods in Theoretical Physics (XXIII DGMTP) was organized by Nankai Institute of Mathematics from August 20th to 26th, 2005. It was Professor S.S. Chern and Professor W. Nahm who proposed the XXIII DGMTP on the occasion of the 60th anniversary of Professor S.S. Chern's paper "Characteristic classes of Hermitian manifolds". Unfortunately, Professor S.S. Chern passed away in December 2004. So this Conference is in memory of Professor Chern dedicated by more than one hundred mathematicians and physicists actively working in the field, in particular differential geometry, topology, gauge theories, statistical mechanics, mathematical physics, and so on. The XXIII DGMTP was held in the new building of Nankai Institute of Mathematics. It was completed one month before the Conference and named Shiing-Shen Building in memory of Professor S.S. Chern who founded the Institute in 1985. The members of the International Advisory Committee include Professors Michael Atiyah, Jean-Michel Bismut, Shiing-Shen Chern, Alain Connes, Simon Donaldson, Ludvig Faddeev, Chaohao Gu, Vaughan F.R. Jones, Yuri. I. Manin, Edward Witten and Chen Ning Yang. We are greatly grateful to them for the very kind suggestions. We thank all of plenary and parallel session's speakers, not only for their bringing the newest developments in the frontier of the field, but also for their kind cooperation in many ways. We highly obliged to all of the session organizers including Professors Victor Batyrev, Jean-Pierre Bourguignon, Louis H. Kauffman, Xiao-Song Lin, Werner Nahm, Antti Niemi, Andrew Strominger, Fa Yueh Wu, Yong-Shi Wu and Xin Zhou for their most excellent jobs. We are indebted to the Ministry of Education of China who mainly supported the Conference. We sincerely thank Sir Michael Atiyah and Professor C.N. Yang who are close friends of Professor S.S. Chern for their kind contributions of the special preface and memory article.

xii

Last but not the least we thank World Scientific Publishing Co. for their generous support for the publication.

Mo-Lin Ge Weiping Zhang

CONTENTS

Foreword by Michael Atiyah

vii

Foreword by Chen Ning Yang

ix

Preface

xi

Plenary Lectures

1

Yangian and Applications C.-M. Bai, M.-L. Ge, K. Xue and H.-B. Zhang

3

The hypoelhptic Laplacian and Chern-Gauss-Bonnet J.-M. Bismut

38

S. S. Chern and Chern-Simons Terms R. Jackiw

53

Localization and Conjectures from String Duality K.F. Liu

63

Topologization of Electron Liquids with Chern-Simons Theory and Quantum Computation Z.H. Wang

106

Invited Contributions

121

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions H. Au-Yang and J. H. H. Perk

123

Theoretical Analysis of the Double Spin Chain Compound KCUCI3 M. T. Batchelor, X.-W. Guan and N. Oelkers

133

Applications of Geometric Cluster Algorithms H. W. J. Blote, Y. Deng and J. R. Heringa

142

Equivariant Cohomology and Localization for Lie Algebroids and Applications U. Bruzzo Directed Percolation in Two Dimensions: An Exact Solution L. C. Chen and F. Y. Wu Generalized Drinfeld Polynomials for Highest Weight Vectors of the Borel Subalgebra of the sl% Loop Algebra T. Deguchi On the Physical Significance of g-deformation in Many-body Physics J. P. Draayer, K. D. Sviratcheva, C. Bahri and A. I. Georgieva A Matrix Product Ansatz Solution of an Exactly Solvable Interacting Vertex Model A. A. Ferreira and F. C. Alcaraz A 2h-dimensional Model with Virasoro Symmetry P. Furlan and V.B. Petkova 3-dimensional Integrable Lattice Models and the BazhanovStroganov Model G. von Gehlen, S. Pakuliak and S. Sergeev

152

160

169

179

190

200

210

Exact Solution of Two Planar Polygon Models A.J. Guttmann and I. Jensen

221

Quasi-exact Solvability of Dirac Equations C.-L. Ho

232

Exotic Galilean Symmetry, Non-commutativity & the Hall Effect P. A. Horvdthy

241

The Energy-momentum and Related Topics in Gravitational Radiation W.-l. Huang and X. Zhang Quantum Operators and Hermitian Vector Fields J. Janyska and M. Modugno Electric-magnetic Duality Beyond Four Dimensions and in General Relativity B.L. Julia

248

256

266

Topology and Quantum Information L.H. Kauffman

273

Generalized Cohomologies and Differential Forms of Higher Order R. Kerner

283

Periodic Cellular Automata and Bethe Ansatz A. Kuniba and A. Takenouchi

293

An L 2 -Alexander-Conway Invariant for Knots and the Volume Conjecture W.P. Li and W.P. Zhang

303

Faddeev Knots, Skyrme Solitons, and Concentration-Compactness F.H. Lin and Y.S. Yang

313

Dynamics of Bose-Einstein Condensates W.-M. Liu

323

Twisted Space-Time Symmetry, Non-Commutativity and Particle Dynamics J. Lukierski and M. Woronowicz

333

Toeplitz Quantization and Symplectic Reduction X.N. Ma and W.P. Zhang

343

On Mysteriously Missing T-duals, H-flux and the T-duality Group V. Mathai and J. Rosenberg

350

Murphy Operators in Knot Theory H. R. Morton

359

Bethe Ansatz for the Open XXZ Chain from Functional Relations at Roots of Unity R.I. Nepomechie

367

Separation Between Spin And Charge in SU(2) Yang-Mills Theory A. J. Niemi

377

On Solutions of the One-dimensional Holstein Model F. Pan and J. P. Draayer

384

Recent Developments on Ising and Chiral Potts Model J. H. H. Perk and H. Au- Yang

389

Bethe Ansatz and Symmetry in Superintegrable Chiral Potts Model and Root-of-unity Six-vertex Model S.-S. Roan The Cyclic Renormalization Group G. Sierra

399

410

Bohm-Aharonov Type Effects in Dissipative Atomic Systems A.I. Solomon and S.G. Schirmer

420

Noncommutative Procedures in Spontaneous Breaking and Quantum Differentiation M. Suzuki

429

Symmetry

Lowner Equations and Dispersionless Hierarchies K. Takasaki and T. Takebe Multiparameter Quantum Deformations of Jordanian Type for Lie Superalgebras V.N. Tolstoy

438

443

XV11

A Correlation-Function Bell Inequality with Improved Visibility for 3 Qubits C.F. Wu, J.-L. Chen, L.C. Kwek and C.H. Oh

453

Topological Aspects of the Spin Hall Effect Y.-S. Wu

462

Positive Mass Theorems and Calabi-Yau Compactification N.Q. Xie

473

Analytic Torsion and an Invariant of Calabi-Yau Threefold K.-I. Yoshikawa

480

Differential Galois Groups of High Order Puchsian ODE's N. Zenine, S. Boukraa, S. Hassani and J.-M. Maillard

490

Conformal Triality of de Sitter, Minkowski and Anti-de Sitter Spaces B. Zhou and H.-Y. Guo

503

Some Observations on Gopakumar-Vafa Invariants of Some Local Calabi-Yau Geometries J. Zhou

513

Plenary Lectures

3

Yangian and Applications Cheng-Ming Bai, Mo-Lin Ge Theoretical Physics Division Chern Institute of Mathematics Nankai University Tianjin 300071, P.R. China Kang Xue, Hong-Biao Zhang Department of Physics Northeast Normal University Changchun 130024, PR- China In this paper, the Yangian relations are tremendously simplified for Yangians associated to SU(2), 5(7(3), 5 0 ( 5 ) and 5 0 ( 6 ) based on RTT relations that much benefit the realization of Yangian in physics. The physical meaning and some applications of Yangian have been shown.

1. Introduction Yangian was presented by Drinfel'd ([1-3]) twenty years ago. It receives more attention for the following reasons. It is related to the rational solution of Yang-Baxter equation and the RTT relation. It is a simple extension of Lie algebras and the representation theory of Y(SU(2)) has been given. Some physical models, say, two component nonlinear Schrodinger equation, Haldane-Shastry model and 1-dimensional Hubbard chain do have Yangian symmetry. Yangian may be viewed as the consequence of a "bi-spin" system. How to understand the physical meaning of Yangian is an interesting topic. In this paper, there is nothing with mathematics. Rather, we try to use the language of quantum mechanics and Lie algebraic knowledge to show the effects of Yangian. 2. Yangian and RTT Relations Let Q be a complex simple Lie algebra. The Yangian algebra Y{Q) associated to Q was given as follows ([1-3]). For a given set of Lie algebraic

4

C.-M. Bai, M.-L. Ge, K. Xue and H.-B.

Zhang

generators JM of Q the new generators J„ were introduced to satisfy [I\,In] = C\tiVIv,

Cx^v are structural constants;

(2.0.1)

[h^nJ^Cx^Jv;

(2.0.2)

and, for Q ^ sl(2): [J\, [Jn, h)) - [h, [Jfi, Jv}} = a\,ii,al3-({Ia, Ip, If},

(2.0.3)

where a

\\iva&i

=

T;k,\a,jGM/3rG„7pGcr7-p,

{xi, 12,13} = 2~2 xixjxk)

(2.0.4)

(symmetric summation);

(2.0.5)

or for G = sl(2): [[JA,JM].[^.^r]] + [[J f f ,Jr],[/A,^]] + 0-aTvaj31C\ilv){IonIiJ,J1}.

(2.0.6)

When Cx^v = i£xnV{\n,v = 1,2,3), equation (2.0.3) is identically satisfied from the Jacobian identities. Besides the commutation relations there are co-products as follows. A(JA)=/A®1 + 1®/A;

A( J\) = Jx ® 1 + 1 J\ + -jCx^h

(2-0.7)

® Iv.

(2.0.8)

Further, the Yangian can be derived through RTT relations where R is a rational solution of Yang-Baxter equation (YBE) ([1-12]). After lengthy calculations, we found the independent relations for Y(SU(2)), Y(SU(3)), Y(SO(5)) and Y(SO(6)) by expanding the RTT relations and also checked through equations (2.0.1)-(2.0.3) and (2.0.6) by substituting the structural constants ([13-17]), where RTT relation (Faddeev, Reshetikhin, Takhtajan — RFT [18]) satisfies R(u - v)(T(u) ® 1)(1 T(v)) = (1 T{v))(T(u)

l)R{u - v).

(2.0.9)

Yangian and Applications

2.1.

5

Y(SU(2))

Let P\2 be the permutation. Setting Ri2(u) = PRi2(u) rp(n)

T(u) = /+£«-"T.(n) n=l oo

21

(2.1.1)

= uPn + I;

rp(n)

^(n) -^22

Kr 0 l n , +r 3 ( n ) ), .(«)

n=l

T(«) 1 1 (rp(n) 2\ 0

(2.1.2)

T3(n))

and substituting the T(u) into RTT relation it turns out that only

/ ± = r W / 3 = lr,3(1) i

(2.1.3)

j ± = 7 l 2 \ j 3 = ir 3 (2)

(2.1.4)

are independent ones. The quantum determinant oo

detT{u) = Tn{u)T22{u gives

- 1) -T 1 2 (u)T 2 i(u - 1) = C0 + n^=«l - n C C 0 = l, Cj = T0(1) = t r T ^ , ^(2)

C2

[A,-^] = ie-Xuvh [-'A) •'nJ —

(2.1.7) are:

(A,/j,z/= 1,2,3); 1€\nvJv\

(2.1.5)

(2.1.6)

i 2 + T « ( i + ir 0 (1) ),

The independent commutation relations of Y(SU(2))

n

(2.1.8) (2.1.9)

and (A± = A\ ± iA2) [J 3 ,[J + ,J_]] = ( J _ J + - / - J + ) 7 3

(2.1.10)

that can be checked to generate all of relations of equations (2.0.1), (2.0.2) and (2.0.6) with the help of Jacobi identities. The co-product is given through (RFT) as AT ah = ^2 Tac ® Tcb-

(2.1.11)

6

C.-M. Bai, M.-L. Ge, K. Xue and H.-B.

Zhang

The simplest realization of Y{SU{2)) is N

I = 5 ^ 1 * (i : lattice indices), N

N

J = ^/x«Ii + 5 3 Wyli x L,-, i=l

(2.1.12)

(2.1.13)

i<j

where 1 i< 3 Wij = { 0 i = j (for any representation of SU(2))

(2.1.14)

or Wjfc = icot

—— (only for spin - , Haldane — Shastry model [19 — 21]),

(2.1.15) and ^ arbitrary constants. Noting that //» plays important role for the representation theory of Y(SU(2)) given by Chari and Pressley ([22-24]). The big difference between representations of Lie algebra and Yangian is in that in Yangian there appear free parameters \ii depending on models. Another example for single particle is finite W-algebra ([25-26]). Denoting by L and B angular momentum and Lorentz boost, respectively, as well as D the dilatation operator, the set of L and J satisfies Y(SU(2)) where ([13],[25]) I = L

(2.1.16)

J=IxB-i(D-l)B

(2.1.17)

and [Ja,J/3] = i e Q / 3 7 ( 2 I 2 - c ' 2 - 4 ) I 7 , 4

casimirof 50(4,2).

(2.1.18)

There are the following models whose Hamiltonians do commute with Y(SU(2)). • Two component nonlinear Schrodinger equation (Murakami and Wadati [27]) ii>t = ~i>xx + 2c|V|V,

(2.1.19)

Yangian and Applications

I = Jdxil>+(x)(l)apMx); J = -i / dxip£(x)(^)apipp(x)-^

/

7

(2-1-20)

dxdye(y-x)(^)pxi>p{x)il)i(y)il)a(x)ipx{y)(2.1.21)

• One-dimensional H u b b a r d model (for N —> co, [28]) JV

"

t=l

i=l

1

1

(2.1.22) J± = Ji

±iJz,

J+ =

Yf9itiatbi-UYieitiltll

J- = J2e^btaJ + uJ2e^IrI!> Js = l^e^iataj

- btbj) + U'£ei,jI?l7,

i,j

(2.1.23)

i j .

(2.1.24)

Essler, Korepin and Schoutens found the complete solutions ([29-30]) and excitation spectrum ([31]) of 1-D Hubbard model chain. • Haldane-Shastry model ([19-21]) whose Hamiltonian is given by a family. The first member is

where and henceforth the ' stands for i =fi j in the summation and Py- = 2(Si • Sj- + | ) , Zk = expi7r&, Zij = Zi- Zj. The next reads g

ZiZjZk

3 = i,j,k £ , ( z l z3 ^J )i ^ f c - 1 ) ' Zi Z kZk

(2 L26)

-

and HA4 == E \7ZfZfy ff

)(^« - 1) + H'A,

(2.1.27)

8

C.-M. Bai, M.-L. Ge, K. Xue and H.-B. Zhang

H'4 = -\H2 - 2^(p^nPij

- 1),

(2.1.28)

where Pijk

=

PijPjk

i PjkP,ki + P.ki-^ij,

Pijki = PijPjkPki + (cyclic for i,j,k and I).

(2.1.29)

The eigenvalues of H2 and H3 have been solved in Ref. [21] and numerical calculations were made for H^. The H2 and H3 were shown to be obtained in terms of quantum determinant ([32]). • Hydrogen atom (with and without monopole, [33])

where \x is mass, q = zeg, K = ze2 and g being monopole charge. • Super Yang-Mills Theory (N = 4): Y(SO(6)) ([34]) ^=2EEMJ)C+Ia

/i(i) = Ep /l (0) = l.

(2.1.31)

k=l

j

where P J is projector for the weight j of SU{2) and a stands for "lattice" index.

2.2.

Y(SU(3))

For the Yangian associated to SU(3), there are the following independent relations [h,In]=ihn*Iv,

[h,J»]=ihn»Jv

(\,ft,v

= l,--- ,8).

(2.2.1)

Define /£> = h ± ih, U{±1] =I6±

il7, Vil) = h T ih, ^j-I(s1} = h (2)

(2}

(2.2.2) (2")

and Jfj, represents the corresponding operator for 2± , U± , V± ' and /§ \ J3 \ After lengthy calculation one finds that based on RTT relation there is only one independent relation for Y(SU(3)) additional to equation (2.2.1):

[/ia),/ia)] = ^ ( { 4 1 ) . ^ i 1 , . ^ 1 ) } - { ^ 1 ) . ^ 1 ) . ^ 1 ) } )

(2-2-3)

Yangian and Applications

9

where {• • • } stands for the symmetric summation. The conclusion can be verified through both the Drinfel'd formula (Cx^ = ifx^v) and RTT relations with replacing P\2 in SU(2) by

where A^ are the Gell-Mann matrices. Setting OO

T(«) = ^ u - T

( n )

,

(2.2.5)

n=0

y(")

=

' 1 rp{n) rp(n) 1 rp(n) rp(n) -rp{n) rp(n) -rp{n) 1 J _ U 3i0 +J3 + 73J8 l ~%l1 4 S T(") I ;T(") I T W TC") _L 1 T^™) T^") ,'T( n ) n T ( " ) _I_ »T( ) T W X ,TW 1T(") 2TW J J J 1 4 +* 5 6 ~rll7 \/3 8 3 0

(2.2.6) and substituting them into RTT relation we find equations (2.2.1)-(2.2.3) are independent relations together with the co-product, for example,

A / f = i f ® 1 + 1 ® / ± 2(41} ® /£> - /£> ® #>) + hv™

® CT^ - C/^ ® V^1')

(2.2.7)

and others. The quantum determinant of T(u) which is 3 by 3 matrix for the fundamental representation of gl(3) takes the form det 3 T(u) = Tn{u){T22(u -T12(u){T21(u

- l)T33(u - 2) - T23{u)T32{u - 2)} - l)T33(u - 2) - T23(u - l)T31(u - 2)}

+T 13 (u){T 21 (u - l)T 3 2 (u - 2) - T22{u - l)T31{u - 2)} = £ ( - l ) p T l p i ( w ) T 2 p 2 ( W - l)T 3 p 3 (n - 2)

(2.2.8)

p

where p stands for all the possible arrangements of (pi,P2,P3)- In comparison with the quantum determinant

det2T(U)= JT ( j Z ^ u - M + * ) ( T W r W - ^ > r W ) , (2.2.9)

10

C.-M. Bai, M.-L. Ge, K. Xue and H.-B.

Zhang

now we have

det 3 T( U )= {)

Y a + ^ - l ) ' ^ ( P + g -l)! M -( m + i + f c + P + g ) M,i,=o (^-D"' (P-W (rp{k)(rp(m)rp(p)

iJll

rp{m)rr(p)\

J

^ 2 2 -'33

I rp{k),rp(m)rp{p)

23

1

rp(k) (rp(m)rp(p)

3 2 J — -^ 12 V J 21

rp(m)rp(p)

J

' - ' 1 3 \-'21 -'32 oo

22

J

J

33 ~

x

rr(m)rr(p)\

23

J

3li

x->

31//

= ^u-"Cn,

(2.2.10)

71=0

i.e., Co = 1, Ci = T0(1), C2 = T0(2) + T0(1) + 2(T 0 (1) ) 2 - I 2 ,

(2.2.11)

OO

I2 = E J A '

(2-2-12)

A=l

When we constrain detT(u) = 1 it leads to Y(SU{2)) and Y(SU(3)) that are formed by the set {h,J\}, A = 1,2,3 and A = 1,2, ••• ,8 for SU(2) and SU(3), respectively. An example of realization of Y(SU(3)) is the generalization of HaldaneShastry model ([19-21]) for the fundamental representation of generators of 517(3):

^ = I>f>

(2-2.13)

J» = Y, ViF? + A/„A„ £ WijFTFf,

(2.2.14)

i

i^ij

where Wij satisfies the same relation as in Haldane-Shastry model given in section 2.1 and F^ are the Gell-Mann matrices. 2.3. Y(SO(5))

and

Y(SO(6))

For SO(N) it holds [Lij,Lkl]=iC$+(z), Mv))+

= S(x - y)5a0.

(2.3.11)

Then

Iab = ^2lab(x),

Jab=

(2.3.12)

J2 <x ~ V^acix^cbiv) x,y,c^a,b

(2.3.13)

satisfies the commuting relations for Y(SO(5)). The following Hamiltonian of ladder model not only commutes with Iab, i.e., it possesses SO(5) symmetry, but also commutes with Jab-

H = H1 + Y/H2(x) + Y^H3(x); X

Hx = 2h Y,

(2.3.14)

X

[

H2(x) = C/(ncT - l-){nci _ I ) + ( c _» d) + V(nc - l)(nd - 1) + JS C • Sd

= I E 1 ^ + ( | J + \v)WUa - 2);

(2.3.16)

a 'Ti ;34 J

•^j({h3,Il6,he} +{hi,he,he} -{hi,

rr(2) r(2h [ i i2 i-'se J

+ {-^23,-^15,-^45} + {-^14,^25,-^35} - {I13,he,he)

- {A3,-^25,-^45}

hs, hs} - {hi, he, he});

-^{.{hh,hi,he\

(2-3.20)

+ {^15,-^24,-^46} + {-^26,-^13,-^35}

+ {-^26,-^14,-^45} — {-^25,-^13,-^36} — {-^25,-^14,-^46} (2'3-21)

- U l 6 , / 2 3 , / 3 5 } - {/l6,/24,/4 B }); (2) r(2)i

[I: 34 ^56 J

24

(i) JWX^ITW r(!) TWX^STW ({Ir(i) 45 ' 7-"13 ' J i6 } + ihb ' hi ,he I + {-";36 (1) r(l)

7-(l)l

+{I:36 i-"24 i ^ e } (1) r(l) r ( l ) l -W.-T:13 '-"le /

TW

r(Di

> J 14 ' M e J

/7-(l) r(l) /-(l)l r(l) ijW r(l) 7-UJl T-Ml M6 J l i 3 5 ' i 2 4 '-"26 J {•'35 i J 1M4 ''-"16 (1) r ( l ) r(l) i i 4 6 i-1:23 i i 2 6 })•

(2.3.22)

3. Applications of Yangian The first example was given by Belavin ([38]) in deriving the spectrum of nonlinear a model. Here we only show briefly some interpretations of Yangian through the particular realizations of Yangian. 3.1. Reduction

of

Y(SU(2))

The simplest realization of Y(SU(2)) is made of two-spin system with Si and S2 (any dimensional representations of SU(2)): J' =

U+V

[L + V

(/xSi x l + i / S 2 x l + 2ASi x S 2 ),

(3.1.1)

that contains the (antisymmetric) tensor interaction between Si and S2. For example, for Hydrogen atom Si = L and S2 = K (Lung-Lenz vector). For Si = S2 = 1/2, when \iv= A2,

(3.1.2)

we prove that after the following similar transformation

Y = A3'A~l,

A

"1 0 0 0" 0v iXO OtA v 0 .0 0 0 1.

(3.1.3)

14

C.-M. Bai, M.-L. Ge, K. Xue and H.-B.

Zhang

the Yangian reduces to 50(4): (p = v + i\ = \fv2 + Yi =

Mi 0 0 Li

Y2 =

M20 , M2 = 0 L2

Y3 =

±a30 1 , M3 = -cr3. 0 \a3

0 p p-1 0

Mi

\2eie)

Lx

Op'1 P0 1 0 -ip-1 ipO

0 -ip ip-1 0

(3.1.4)

and (3.1.5)

* • = & " > - ! •

Namely, under \iv = A2, the Y reduces to SO(4) by M± = Mx ± iM2, M+ = pa+, M_ = p _1 cr_. The scaled M± and M 3 still satisfy the SU(2) relations: [M3, M±] = ±M±, [M+,M_] = 2M 3

(3.1.6)

and there are the similar relations for L. It should be emphasized that here the new "spin" M (and L) is the consequence of two spin(^) interaction. As usual for two 2-dimensional representations of SU{2) (Lie algebra) 2 ® 2 = 3 (spin triplet) © 1 (singlet).

(3.1.7)

However, here we meet a different decomposition: 2 ® 2 = 2 ( M ) 0 2(L).

(3.1.8)

The idea can be generalized to SU(3)'s fundamental representation JX = ul$ + vl$ +

[F^, Fi

(A, p, v = 1,2, • • • ,8).

(3-1-9)

(3.1.10)

Under the condition uv = A2,

v + iX = p,

(3.1.11)

Yangian and Applications

15

and the similar transformation

Yll=AJliA-1/(u

1 0 0 0 0 0 0 0 0 0 v OiAOO 0 0 0 0 0 i/ 0 0 0 tA 0 0 OiAO i / 0 0 0 0 0 + v), A = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 i / 0 iAO 0 0 iA 0 0 0 (/ 0 0 0 0 0 OOiAO j / 0 0 0 0 0 0 0 0 0 1

(3.1.12)

the Yangian then reduces to " - X J_

Y(I-)

™ =T

0 0 pi- 0 0 /_

pl+ 0 0 p~lI. 0 0

Y{I+)

"A3 0 0 0 A3 0 . 0 0 A3

Y(h) = ~2

Y(U+) =

U+ 0 0 0 pU+ 0 0 0 p-xXJ.+ J

Y(V+) =

p~lV0 0

0 0 V- 0 0 pV.

0 0 h

A3 0 0 " 0 A3 0 0 0 A3 ~U0 0 p^U0 0

Y(U.)

Y{V-) =

pV- 0 0 0 V0 0 U p

0 0 pU. '

(3.1.13)

The usual decomposition through the Clebsch-Gordan coefficients for the representations of Lie algebra SU(3) is 3 (8) 3 = 6 © 3. However, here we have 33 = 3®3_©3,

(3.1.14)

and

En

A=l

U + V

T.J.

(3.1.15)

A=l

It is easy to check that the rescaling factor p does not change the commutation relations for SU(3) formed by I±, U±, V±, h and 1$. In general, we guess for the fundamental representation of SU(n) we shall meet >n = n © n © n + ----|-n (n times).

(3.1.16)

16

C.-M. Bai, M.-L. Ge, K. Xue and H.-B.

Zhang

Next we consider Yang-Mills gauge field for reduced Y(SU(2)). tensor wave function (x = {XI,X2,X3,XQ}), * ( I ) = ||iMx)|| (»,j = 1,2,3,4).

For a (3.1.17)

An isospin transformation yields *'(a;) = U(x)*(x),

U{x) = 1 - i6aJa,

(3.1.18)

where Ja = uSa ® 1 + vl Sa + 2XeabcSb ® Sc,

(3.1.19)

or [Ja]°f = ui^crySps

+ v ^ " ) / * * ^ + iaeabc(Sb)a7(Sc)ps.

(3.1.20)

Define Dp = dll+gAll,

(3.1.21)

i.e., [ A ^ a / 3 = SMVa/3 + < ^ W ^ < M * ) ,

AM - A%Ja.

(3.1.22)

The gauge-covariant derivative should preserve 5(D^)=0,

(3.1.23)

i.e., (-id^ix)

+ g5Al)[Ya)f8 - ig6a{x)Al\Jb,

Ja]fs = 0.

(3.1.24)

When uv = A2 and by rescaling Ya = (u + v)Ja,

(3.1.25)

we have 5AI = eabceb(x)A^(x)

i a + -d^ (x), 9

(3.1.26)

and i^„ = -[D„DV\ *£„ = 3 ^ - d ^

= F^Ya, + z^afcc^^.

(3.1.27) (3.1.28)

Here the tensor isospace space has been separated to two irrelevant irrel spaces, i.e # =

"*i 0~ where \I>i and ^ 2 are 2 x 2 wavefunction. 0 *5

Yangian and Applications

3.2. Illustrative examples: and Yangian

NMR

of Breit-Rabi

17

Hamiltonian

The Breit-Rabi Hamiltonian is given by H = K-S

+ fiB-S,

(3.2.1)

where 5 = \ and B = B(£) is magnetic field. The Hamiltonian can easily be diagonalized for any background angular momentum (or spin) K. The S stands for spin of electron and for simplicity K = Si(5i = 1/2) is an average background spin contributed by other source, say, control spin. Denoting by H = H0 + Hi(t),

H0 = a S i • S 2 , ffi(i) = /*B(t) • S 2 .

(3.2.2)

Let us work in the interaction picture: Hj = A*B(t) • (e i Q S l - S 2 S 2 e- i Q S l - S a ) = fjB(t) • J, J = ^ i S i + M 2 S 2 + 2A(Si x S 2 ) ,

(3.2.3) (3.2.4)

where fi\ = | ( 1 — cosa), /x2 = | ( 1 + cosa), A = ^sina. Obviously, here we have fj,ifX2 = A2. It is not surprising that the y(517(2)) reduces to 50(4) here because the transformation is fully Lie-algebraic operation. This is an exercise in quantum mechanics. For generalization we regard fi\ and /Z2 as independent parameters, i.e., drop the relation fiifX2 = A2. Looking at J = / x i S 1 + M 2 S 2 - i ( / i i + / i 2 ) ( S i + S 2 ) + 7 ( S 1 + S 2 ) + 2 A S 1 x S 2 . (3.2.5) When 7 = | , /x2 — Mi = cosa and A = ^sina, it reduces to the form in the interacting picture. Putting Sx + S 2 = S, 2A = - - ( / i is not Plank constant).

(3.2.6)

In accordance with the convention we have 2

h

1

J = 7 S + J2MiSi + - S i x S 3 - -(/ii + M2)S = 7 S + Y.

(3.2.7)

i=\

Since J —• ^S + J still satisfies Yangian relations, it is natural to appear the term 7S. The interacting Hamiltonian then reads HT(t) = - 7 B ( t ) • S - B(i) • Y.

(3.2.8)

18

C.-M. Bai, M.-L. Ge, K. Xue and H.-B. Zhang

When Hi = 0, h = 0, it is the usual NMR for spin 1/2. To solve the equation, we use dt

= Hi{t)*(t), |tf(t))=

£

aa(t)\Xa),

(3.2.9)

a=±,3;0

where {x±, X3} is the spin triplet and xo singlet. Setting B±(t) = Btit) ± iB2(t) = B i e T i w o t , and B 3 = const.

(3.2.10)

and rescaling by o±(t) = e ± - o t 6 ± ( i ) ,

(3.2.11)

we get .db±(t) ,da3(£) dt .da0(t) dt

1 — ; —

—

7 { ^ B i a 3 ( i ) T ( ^ o 7 - 1 - B3)b±{t)} ± - ^ = p _ B i a o ( i ) , 2^' 7B1 {b+(t) + b-(t)}--n-B3ao(t), \n+{^Bx\b-{t)

(3.2.12)

- 6+(*)]} + SaaaW,

where /x± = (fii — M2 ± i f ) , i-e., u)0 - 7S3 - 7 B 1 v^2^

6X(0 !*(*)>

= 0 at t = £-

(total spin = 0),

(3.2.19)

LLC

< s2 >= 2 at t = - (total spin = 1). (3.2.20) u> Under adiabatic approximation it can be proved that it appears Berry's phase. Obviously, only spin vector can make the stereo angle. The role of spin singlet here is a witness that shares energy of spin=l state. Actually, if B±(t) = Bo sin 6eTi"ot,

B3 = B0cos6,

(3.2.21)

and

lxn) = ITT>, |xi-i) = IU), lxio) = ^ ( | U ) + IIT», IXoo> = ^ ( | U > - U T » ,

(3-2-22)

then let us consider the eigenvalues of H = a S i • S 2 - 7 £ o S 3 - 9B0J3,

(3.2.23)

under adiabatic approximation which are E± = \ { ~

± y W f ^ + M - ) ,

(3.2.24)

and A (±) = [2(a 2 +

2 5

S 0 V+M-)]- 1 / 2 [(« 2 + < ? 2 B 0 W - ) 1 / 2 ± ^ ' ^

f™ = [2(a2+g2B2^^)}-'/2[^±(a2+g2B^+^)l/2Ta]1/2.

(3-2-25) (3.2.26)

We obtain the eigenstates of H besides \xu) (i = 1,2) |x±)=/i(±)|Xio)+/r)|xoo),

(3.2.27)

20

C.-M. Bai, M.-L. Ge, K. Xue and H.-B. Zhang

where |Xu(t)> = cos2 ^ | x i i ) + - ^ s i n f l e - ^ l x i o ) + s i n 2

ie-^lxi-i),

|Xi-iW> = sin2 ^ - o ' l x i i ) - - ^ s i n ^ e - ^ l x i o ) +cos 2

°-\Xi-i),

\x±(t)) = ^ / P H - s i n ^ e ^ ^ l x i i ) + V^cos^Xio) + s i n 0 e - i - o t | x 1 _ 1 ) } +#)|Xoo>.

(3.2.28)

We then obtain ( X n ( t % l x i i ( * ) > = -«"o(l -

cos