Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics V. Kazakov (ENS, Paris) NBI, Copenhagen 18/04/07 with A.Sorin and A.Zabrodin, hep-th/

Motivation Classical and quantum integrability are intimately related (not only through the classical limit!) Quantization = discretization: Quantum spin chain Discrete classical Hirota dynamics We study SUSY spin chain via Hirota equation for fusion rules, with specific integrable boundary conditions. More general and more transparent with SUSY! An alternative to algebraic Bethe ansatz [Klumper,Pierce 92’], [Kuniba,Nakanishi,’92] [Kulish,Sklianin’80-85] [Krichever,Lupan,Wiegmann, Zabrodin’97]

Plan R-matrix and Yang-Baxter eq. for SUSY spin chain Bazhanov-Reshetikhin rel. & Hirota eq. for fusion SUSY boundary cond., Bäcklund transf. & undressing Baxter TQ relations & Hirota eq for Q-functions (QQ relations) SUSY nested Bethe ansätze & examples, gl(1|1), gl(2|1)… Fusion in quantum space… [Bazhanov,Reshetikhin’90]

Rα′β′Rα′β′ α βα β (u-v) u v α β α′α′ β′β′ gl(K|M) super R-matrix Unity: graded permutation: for even (odd) components

= u v α β α′α′ β′β′ α ′′ β ′′ 0 u v α β α′α′ β′β′ α ′′ β ′′ 0 γ ′′ γ′γ′ γ γ′γ′ γ Yang-Baxter relation for R-matrix

Monodromy Matrix and Transfer Matrix Transfer matrix = supertrace of monodromy matrix Defines all conserved charges of (inhomogeneous) super spin chain. T γ N, { β i } γ 0, { α i } = α2α2 γNγN β2β2 α1α1 β1β1 αNαN βNβN γ N-1 γ1γ1 γ2γ2 γ0γ0 ← quantum space → ↑ auxiliary space How to calculate it? u1u1 u2u2 uNuN

Fusion: Higher Irreps in Auxiliary Space u+2u+4 u u u-2 Projector to irrep : cross all lines with these rapidities, in lexicographic order along (super) Young tableau. Based on degeneracy of R-matrix into projectors at special values: Defines the transfer matrix in irrep λ : u u-2 u+4 u u+2 auxiliary quantum

T-matrix Eigenvalues as Quantum Characters “Conservation laws”: Bazhanov-Reshetikhin determinant formula: Expresses for general irrep λ throughfor the row s

a s T(a,s,u) → Hirota relation for rectangular tableaux T (u+1)T(u-1) T(u) s a From BR formula, by Jakobi relation for det:

Hirota relation Direct consequence of Bazhanov-Reshetikhin quantum character formula. Hirota eq. – integrable, Master equation of the soliton theory. The classical inverse scattering method can be applied. We use Hirota eq. to find all possible Baxter’s TQ relations and nested Bethe ansatz equations for superalgebras.

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

↑ First Lax pair of linear problems for, equiv. to Hirota eq. ↓ s a F(u) T(u) F(u+1) F(u) T(u+1) T(u)T(u+1)F(u) T(u+1) F(u+1) F(u) Bäcklund Transformation – I (BT-I) On the horizontal boundary: one can put F(K,s,u)=0. F(u)

a s K M K-1 Undressing by BT-I: vertical move T(a,s,u) ≡ T K,M (a,s,u) → F(a,s,u) ≡ T K-1,M (a,s,u) gl(K|M) gl(K-1|M) ∩ T K-1,M (a,s,u) also satisfies Hirota eq., but with shifted B.C. Notation:

s a T(u)F*(u) T(u+1) F*(u+1) T(u) F*(u+1) F*(u)F*(u+1)T(u) F*(u+1) T(u+1) T(u) Bäcklund Transformation - II ↑ Second Lax pair of linear problems ↓

a s K M M-1 Undressing by BT-II: horizontal move T K,M (a,s,u) → F*(a,s,u) ≡ T K,M-1 (a,s,u) …….→T k,m (a,s,u) One can repeat this procedure until the full undressing K,M=0: T 0,0 (u)=1. gl(K|M) gl(K-1|M) …… gl(k|m) … 0 ∩ ∩∩ Example: first of eqs. BT-II at a stage (k,m) of undressing: Nesting:

Extracting trivial zeroes T-functions become polynomials of the same power N=Length of spin chain,

a s k m T k,m (a,s,u)≠ 0 Q k,0 (u+a+m)Q 0,m (u-a-m)(-1) m(a-k) Q k,0 (u+s+k) Q 0,m (u-s-k) Q k,m (u-s) Q k,m (u+a) Boundary conditions….. B.C. respect Hirota equation. B.C. defined through Baxter’s Q-functions: k=1,…,K m=1,…M Q k,m (u)=Π j (u-u j ) [Tsuboi’97]

k m (K,M) (0,M) 0 (K,0) Full undressing along a zigzag path By construction T(u,a,s) are polynomials in u. Q k,m (u) and T k,m (u,a,s) are also polynomials of u. Analyticity: At each (k,m)-vertex there is a Q k,m (u)

Strategy Express T-functions through Q-functions. Find Q-functions from analyticity (polynomiality). This gives Nested Bethe Ansatz

Generalized Baxter’s T-Q Relations Diff. operator encoding all T’s for symmetric irreps: are shift operators on (k,m) plane. where From Hirota eq.:

k m (K,M) (0,M) 0 (K,0) x n1n1 n2n coordinate on (k,m) plane - unit vector in the direction of shift Generalized Baxter’s T-Q Relations [V.K.,Sorin,Zabrodin’07]

m k Q (u+2) Q (u) Q (u+2) Q (u) Hirota eq. for Baxter’s Q-functions k+1,m k+1,m+1 k,mk,m+1 Zero curvature cond. for shift operators [V.K.,Sorin,Zabrodin’07]

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

M K M-m K-k l′ km l km μ km Higher irreps in quantum space auxiliary quantum μ KM arbitrary polynomial

a s Q 1,0 (u+s+1) Q 0,1 (u-s-1) Q 1,0 (u+a+1)Q 0,1 (u-a-1) (-1) Q 1,1 (u-s) Q 1,1 (u+a) a-1 gl(1|1) algebra We reproduce BAE’s and Baxter’s TQ relations, including the irreps with continuous labels, in accordance with [Fendley,Intriligator’92]

a s Q 2,0 (u+s+2) Q 0,1 (u-s-2) Q 2,0 (u+a+1)Q 0,1 (u-a-1) (-1) a-2 Q 2,1 (u-s) Q 2,1 (u+a) T 2,1 (1,s,u) gl(2|1) algebra [Frahm,Pfanmüller’96] We reproduce BAE’s and Baxter’s TQ relations, including the irreps with continuous labels, in accordance with Related to Beisert’s su(2|2) S-matrix.

Applications and Problems Generalizations: noncompact irreps, mixed (covariant+contravariant) irreps, so(M|K), sp(M|K) algebras. Non-standard R-matrices, like Hubbard or su(2|2) S-matrix in AdS/CFT, should be also described by Hirota equation with different B.C. Beisert’05 Arutyunov,Frolov,Zamaklar’06 A powerful tool for constructing and studying supersymmetric spin chains and 2d integrable field theories, including classical limits. An alternative to the algebraic Bethe ansatz.

M K (K,M) (0,0) m k

a s μ\μ(a,s) μ(a,s)