1. From Characters to Quantum Super-Spin Chains by Fusion
Workshop: Integrability and the Gauge /String Correspondence
From Characters to Quantum Super-Spin Chains by Fusion
V. Kazakov (ENS, Paris)
Newton Institute, Cambridge, 12/12/07
with P.Vieira, arXiv:
with A.Sorin and A.Zabrodin, hep-th/

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, Zabrodin’97]
[Bazhanov,Reshetikhin’90] [Cherednik’88]
[V.K.,Vieira’07]
[Kulish,Sklianin’80-85]
[Tsuboi’98]
[V.K.,Sorin,Zabrodin’07]

3. sl(K|M) super R-matrix and Yang-Baxteru u = v v

4. Fused R-matrix in any irrep λ of sl(K|M)
“l” β β′
“v”
vector irrep “v” in physical space
any “l“ irrep in quantum space u
Idea of construction
(easy for symmetric irreps)

5. Twisted Monodromy Matrixβ1 β2 βN
l, {αi} u
L {βi} = l l u1 u2 uN α2 α1
← quantum space → αN
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
Cherednik’87
Expresses
for general irrep λ={λ1,λ2,…,λa}
through
in symmetric irreps s
Compare to Jacobi-Trudi formula for GL(K|M) 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:

9. 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 should be equal to
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

11. 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

12. 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.

13. Fixing T-matrix at uk=∞
The rest of T-matrix is degree N polynomial guessed from
Repeating for all uk‘s we restore the standard T-matrix

14. Proof of the main identity
Consider
One derivative

15. Proof of the main identity
One derivative
Action of the derivative:
Two derivatives in components

16. Graphical representation
One derivative
Three derivatives

17. Proof of the main identity…….
Consider

18. 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)

19. Proving the identity ….
This completes our proof of Bazhanov-Reshetikhin formula

20. Hirota eq. from Jacobi relation for rectangular tableauxs
T(a,s,u): λ→
From BR formula, by Jacobi relation for det:
we get Hirota eq.

21. SUSY Boundary Conditions: Fat Hook
T(a,s,u)≠0 s M
All super Young tableaux of gl(K|M) live within this fat hook

22. Solution: Generalized Baxter’s T-Q Relations
[V.K.,Sorin,Zabrodin’07]
Diff. operator generating all T’s for symmetric irreps:
Introduce shift operators:
Baxter’s Q-functions:
Qk,m(u)=Πj (u-uj )
k=1,…,K
m=1,…M

23. Undressing along a zigzag path (Kac-Dynkin diagram)
[V.K.,Sorin,Zabrodin’07] k
undressing (nesting)
plane (k,m)
(K,0) 9
(K,M) n1
At each (k,m)-vertex
there is a Qk,m(u) x 8 n2 6 7 3 4
Change of path:
particle-hole duality 5
[Tsuboi’98] 2 m 1
(0,M)
[V.K.,Sorin,Zabrodin’07]
Solution of Hirota equation with fat hook b.c.:
Using this and BR formula, we generate all TQ Baxter relations!

24. Hirota eq. for Baxter’s Q-functions
(Q-Q relations)
k+1,m
k+1,m+1
Zero curvature cond.
for shift operators
k,m
k,m+1
[V.K.,Sorin,Zabrodin’07]
General nesting:
gl(K|M) gl(K-1|M) …… gl(k|m) … ∩ ∩ ∩
By construction T(u,a,s) and Qk,m(u) are polynomials in u.

25. Bethe Ansatz Equations along a zigzag path
BAE’s follow from zeroes of various terms in Hirota QQ relation
and Cartan matrix along the zigzag path
1, if
where
-1, if

26. 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 (covariant+contravariant) irreps, osp(n|2m) algebras.
Trigonometric and elliptic(?) case.
Non-standard R-matrices, like Hubbard or su(2|2) 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.

27. Example: Baxter and Bethe equations
for sl(2|1) with Kac-Dynkin diagram
Generating functional for antisymmetric irreps:
T-matrix eigenvalue in fundamental irrep:
Bethe ansatz equations: