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:0711.2470 with A.Sorin and A.Zabrodin, hep-th/0703147.

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]

sl(K|M) super R-matrix and Yang-Baxter u u = v v

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)

Twisted Monodromy Matrix β1 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

Twisted Transfer Matrix polynomial of degree N Defines all conserved charges of (inhomogeneous) super spin chain:

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

Proof of BR formula Left co-derivative D: Definition , where Or, in components: More general, more absract: Nice representation for R-matrix:

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

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

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

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.

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

Proof of the main identity Consider One derivative

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

Graphical representation One derivative Three derivatives

Proof of the main identity……. Consider

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)

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

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

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

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

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!

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

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

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.

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: