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/0703147.

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′

α β(u-v) u

v

α

β

α′

β′

gl(K|M) super R-matrix

Unity: graded permutation:

for even (odd) components

=u

v

α

βα′

β′

α′′

β′′

0

u

v

α

β

α′

β′

α′′

β′′

0

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 γN ,{βi}

γ0, {αi} =

α2

γN

β2

α1

β1

αN

βN

γN-1γ1 γ2γ0

← quantum space →

↑ auxiliary space

How to calculate it?

u1 u2 uN

Fusion: Higher Irreps in Auxiliary Space

u+2 u+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 :

uu-2u+4

uu+2

auxiliary

quantum

T-matrix Eigenvalues as Quantum Characters

• “Conservation laws”:

• Bazhanov-Reshetikhin determinant formula:

• Expresses for general irrep λ

through for the row

s

a

sT(a,s,u) →

Hirota relation for rectangular tableaux

T (u+1)T(u-1) T(u) T (u)

T(u)

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+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) ≡ TK,M (a,s,u) → F(a,s,u) ≡ TK-1,M(a,s,u)

gl(K|M) gl(K-1|M)∩

TK-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

MM-1

Undressing by BT-II: horizontal move

TK,M (a,s,u) → F*(a,s,u) ≡ TK,M-1 (a,s,u) …….→Tk,m (a,s,u)

• One can repeat this procedure until the full undressing K,M=0: T0,0(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

Tk,m (a,s,u)≠ 0

Qk,0(u+a+m)Q0,m(u-a-m)(-1)m(a-k)

Qk,0(u+s+k) Q0,m(u-s-k)

Qk,m(u-s)

Qk,m(u+a)

Boundary conditions…..

• B.C. respect Hirota equation.

• B.C. defined through Baxter’s Q-functions:

k=1,…,k=1,…,KKm=1,…Mm=1,…M

Qk,m(u)=Πj (u-uj )

[Tsuboi’97]

k

m

(K,M)

(0,M)0

(K,0)

1

2

3 4 5

67

8

9

Full undressing along a zigzag path

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

• Qk,m (u) and Tk,m(u,a,s) are also polynomials of u.

Analyticity:

At each (k,m)-vertex

there is a Qk,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

n1

n2

1

2

3 4 5

67

8

9

- 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)

Q (u+2)

Q (u+2)

Q (u)

Hirota eq. for Baxter’s Q-functionsk+1,m k+1,m+1

k,m k,m+1

Zero curvature cond.for shift operators

[V.K.,Sorin,Zabrodin’07]

Bethe Ansatz Equations along a zigzag path

1, if

-1, ifwhere

and Cartan matrix along the zigzag path

• BAE’s follow from zeroes of various terms in Hirota QQ relation

M

K

M-m

K-kl′km

lkm

μkm

Higher irreps in quantum spaceauxiliary

quantum

μKM

arbitrary polynomial

a

s

Q1,0(u+s+1) Q0,1(u-s-1)

Q1,0(u+a+1)Q0,1(u-a-1) (-1)

Q1,1(u-s)

Q1,1(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

Q2,0(u+s+2) Q0,1(u-s-2)

Q2,0(u+a+1)Q0,1(u-a-1) (-1)a-2

Q2,1(u-s)

Q2,1(u+a)

T2,1 (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)