From Characters to Quantum Super-Spin Chains by Fusion

1
From Characters to Quantum Super-Spin Chains by
Fusion
Workshop Integrability and the Gauge /String
Correspondence

• V. Kazakov (ENS, Paris)
• Newton Institute, Cambridge, 12/12/07

with P.Vieira,
arXiv0711.2470 with A.Sorin and A.Zabrodin,
hep-th/0703147.
2
Motivation and Plan

• Classical and quantum integrability are
  intimately related (not only through classical
  limit!). Quantization discretization.
• Quantum spin chain Discrete
  classical Hirota dynamics for fusion
• of quantum states (according to
  representation theory)
• Based on Bazhanov-Reshetikhin (BR) formula for
  fusion of representations.
• Direct proof of BR formula was absent. We fill
  this gap.
• Solution of Hirota eq. for (super)spin chain in
  terms of Baxter TQ-relation
• More general and more transparent with SUSY new
  QQ relations.
• An alternative to algebraic Bethe ansatz
• all the way from R-matrix to nested Bethe
  Ansatz Equations

Klumper,Pearce 92, Kuniba,Nakanishi,92


Krichever,Lupan,Wiegmann, Zabrodin97
Bazhanov,Reshetikhin90 Cherednik88
V.K.,Vieira07
Kulish,Sklianin80-85
Tsuboi98
V.K.,Sorin,Zabrodin07
3
sl(KM) super R-matrix and Yang-Baxter
u
u

v
v
0
0
4
Fused R-matrix in any irrep ? of sl(KM)
vector irrep v in physical
space any l irrep in quantum space
u

• Idea of construction
• (easy for symmetric irreps)

5
Twisted Monodromy Matrix
ß2
ß1
ßN
l, ai
u
L ßi

l
l
u1
u2
uN
a2
? quantum space ?
a1
aN
auxiliary space

• Multiply auxiliary space
• by twist matrix

6
Twisted Transfer Matrix
polynomial of degree N

• Defines all conserved charges of
  (inhomogeneous) super spin chain

7
Bazhanov-Reshetikhin fusion formula
Bazhanov,Reshetikhin90 Cherednik87
for general irrep ??1,?2,,?a

• Expresses

through
in symmetric irreps

• Compare to Jacobi-Trudi formula for GL(KM)
  characters

- symmetric (super)Schur polynomials with
generating function
8
Proof of BR formula

• Left co-derivative D

• Definition

, where

• Or, in components

• More general, more absract

• Nice representation for R-matrix

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T-matrix and BR formula in terms of left
co-derivative

• Monodromy matrix

• Trasfer-matrix of chain without spins

• Trasfer-matrix of one spin

• Trasfer-matrix of N spins

10
Proof for one spin
Jacobi-Trudi formula for character
should be equal to

• First, check for trivial zeroes every 2x2 minor
  of two rows

is zero due to curious identity for symmetric
characters
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Proof of identity

• In term of generating f-n
• it reads

easy to prove using

• The remining linear polynomial can be read from
  large u asymptotics

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Proof for N spins

• BR determinant

has trivial factor with fixed zeroes
in virtue of the similar identity

• Easy to show by induction, that it is enough to
  prove it for all ?n 0

• The key identity! Should be a version of Hirota
  eq. for discrete KdV.

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Fixing T-matrix at uk8

• The rest of T-matrix is degree N polynomial
  guessed from

• Repeating for all uks we restore the standard
  T-matrix

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Proof of the main identity

• Consider

• One derivative

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Proof of the main identity

• One derivative

• Action of the derivative

• Two derivatives in components

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Graphical representation

• One derivative

• Three derivatives

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Proof of the main identity.

• Consider

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Comparison

• Notice that the difference is only in color of
  vertical lines.
• Identical after cyclical shift of upper indices
  to the right in 2-nd line
• (up to one line where red should be changed to
  green)

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Proving the identity .

• This completes our proof of Bazhanov-Reshetikhin
  formula

20
Hirota eq. from Jacobi relation for rectangular
tableaux
T(a,s,u) ??

• From BR formula, by Jacobi relation for det

we get Hirota eq.
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SUSY Boundary Conditions Fat Hook
a
K
T(a,s,u)?0
s
M

• All super Young tableaux of gl(KM) live within
  this fat hook

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Solution Generalized Baxters T-Q Relations
V.K.,Sorin,Zabrodin07

• Diff. operator generating all Ts for symmetric
  irreps

• Introduce shift operators

Baxters Q-functions
Qk,m(u)?j (u-uj )
k1,,K m1,M
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Undressing along a zigzag path (Kac-Dynkin
diagram)
V.K.,Sorin,Zabrodin07
undressing (nesting) plane (k,m)
k
(K,0)
9
(K,M)
n1

• At each (k,m)-vertex
• there is a Qk,m(u)

8
x
n2
6
7
3
4

• Change of path
• particle-hole duality

5
Tsuboi98
2
m
0
(0,M)
1
V.K.,Sorin,Zabrodin07

• Solution of Hirota equation with fat hook b.c.

• Using this and BR formula, we generate all TQ
  Baxter relations!

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Hirota eq. for Baxters Q-functions
(Q-Q relations)
k1,m
k1,m1
Zero curvature cond. for shift operators
k,m
k,m1
V.K.,Sorin,Zabrodin07
gl(KM) gl(K-1M) gl(km)
0
n
General nesting
n
n

• By construction T(u,a,s) and Qk,m(u) are
  polynomials in u.

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Bethe Ansatz Equations along a zigzag path

• BAEs follow from zeroes of various terms in
  Hirota QQ relation

and Cartan matrix along the zigzag path
1, if
where
-1, if
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Conclusions and Prospects

• We proved Bazhanov-Reshetikhin formula for
  general fusion
• We solved the associated Hirota discrete
  classical dynamics by
• generalized Baxter T-Q relations, found new
  Q-Q bilinear relations,
• reproduced nested TBA eqs. Fusion in quantum
  space done.
• Possible generalizations noncompact irreps,
  mixed (covariantcontravariant) irreps,
  osp(n2m) algebras.
• Trigonometric and elliptic(?) case.
• Non-standard R-matrices, like Hubbard or
  su(22) S-matrix in AdS/CFT, should be also
  described by Hirota eq. with different B.C.
• A potentially powerful tool for studying
  supersymmetric spin chains and 2d integrable
  field theories, including classical limits.
• Relation to KP, KdV. Matrix model applications?
• An alternative to the algebraic Bethe
  ansatz.

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Example Baxter and Bethe equations for sl(21)
with Kac-Dynkin diagram
Generating functional for antisymmetric irreps
T-matrix eigenvalue in fundamental irrep
Bethe ansatz equations