Workshop: Integrability and the Gauge /String Correspondence
From Characters to Quantum SuperSpin 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.
[Klumper,Pearce 92’],
[Kuniba,Nakanishi,’92]
[Krichever,Lupan,Wiegmann, Zabrodin’97]
•
Quantum spin chain
Discrete classical Hirota dynamics for fusion
of quantum states (according to representation theory)
[Bazhanov,Reshetikhin’90]
[Cherednik’88]
• Based on Bazhanov-Reshetikhin (BR) formula for fusion of representations.
• Direct proof of BR formula was absent. We fill this gap.
[Kulish,Sklianin’80-85]
[V.K.,Vieira’07]
[Tsuboi’98]
• Solution of Hirota eq. for (super)spin chain in terms of Baxter TQ-relation
[V.K.,Sorin,Zabrodin’07]
• 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
sl(K|M) super R-matrix and Yang-Baxter
u
0
u
u
=
v
v
0
0
Fused R-matrix in any irrep λ of sl(K|M)
β′
vector irrep “v” in physical space
“v”
any “l“ irrep in quantum space
“l”
u
0 β
• Idea of construction
(easy for symmetric irreps)
Twisted Monodromy Matrix
β1
L
l, {αi}
{β i}
=
l
βN
u
l
u1
auxiliary space
β2
α1
u2
uN
α2 ← quantum space → αN
• 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
Bazhanov,Reshetikhin’90
Cherednik’87
• Expresses
through
for general irrep λ={λ1,λ2,…,λa}
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
• Or, in components:
• More general, more absract:
• Nice representation for R-matrix:
,
where
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
Hirota eq. from Jacobi relation for rectangular tableaux
a
T(a,s,u):
λ→
s
• From BR formula, by Jacobi relation for det:
we get Hirota eq.
SUSY Boundary Conditions: Fat Hook
a
K
T(a,s,u)≠0
M
s
• 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
9
(K,0)
n1
x
(K,M)
8
n2
6
• At each (k,m)-vertex
there is a Qk,m(u)
7
3
4
• Change of path:
particle-hole duality
5
[Tsuboi’98]
2
0
1
(0,M)
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
k+1,m
k+1,m+1
(Q-Q relations)
Zero curvature cond.
for shift operators
k,m
gl(k|m) …
∩
gl(K-1|M) ……
[V.K.,Sorin,Zabrodin’07]
∩
∩
General nesting: gl(K|M)
k,m+1
• By construction T(u,a,s) and Qk,m(u) are polynomials in u.
0
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: