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 [Klumper,Pearce 92’], [Kuniba,Nakanishi,’92] • 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 [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]

u 0 sl(K|M) super R-matrix and Yang-Baxter u u = v v 0 0

β′ “v” “l” 0 β Fused R-matrix in any irrep λ of sl(K|M) 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 ← quantum space → αN α1 auxiliary space • Multiply auxiliary space • by twist matrix

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

s Bazhanov-Reshetikhin fusion formula Bazhanov,Reshetikhin’90 Cherednik’87 for general irrep λ={λ1,λ2,…,λa} • Expresses through in symmetric irreps • 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: , where • Definition • 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 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

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

SUSY Boundary Conditions: Fat Hook a K 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] 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 [Tsuboi’98] 2 m 0 (0,M) 1 [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:

From Characters to Quantum Super-Spin Chains by Fusion