Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics

Alexander Sorin; Anton Zabrodin; Vladimir Kazakov

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Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics

Vladimir Kazakov, Alexander Sorin, Anton Zabrodin

High Energy Physics - Theory · Submitted 15 Mar 2007

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The authors' main claim We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M)-invariant $R$-matrix obey the Hirota bilinear difference equation.

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arXiv IDhep-th/0703147 (version 2)

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CitationAlexander Sorin, Anton Zabrodin, Vladimir Kazakov. "Supersymmetric Bethe Ansatz and Baxter Equations from Discrete Hirota Dynamics." arXiv:hep-th/0703147 (2007). https://pith.science/paper/VOZCYXH4

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We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M)-invariant $R$-matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Backlund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K\|M)-invariant $R$-matrix obey the Hirota bilinear difference equation.	result 0.95	0.94	0.74	asserted 1.00
s1	The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation.	contribution 0.61	0.87	0.14	asserted 1.00
s2	Our basic tool is a chain of Backlund transformations for the Hirota equation connecting quantum transfer matrices.	method 1.00	0.84	0.08	asserted 1.00
s3	This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.	contribution 0.48	0.91	0.18	asserted 0.93

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We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the gl(K|M)-invariant $R$-matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Backlund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super spin chains.

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Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve
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A Grassmannian exterior-form formalism solves the Hirota T-system on T-hooks and yields a twisted Quantum Spectral Curve that reproduces the gamma-twisted BMN vacuum energy at single-wrapping order.
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How the literature uses it

48 arXiv papers cite this in 171 sentences

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Builds on it 5 papers

The Bäcklund solution of Hirota equation for the $gl(M)$ algebras [ref] and $gl(M|N)$ super-algebras, for the fat hook boundary conditions in the representational $(a,s)$ space [ref] , was generalized to the case of a general $𝕋$-hook in [ref] .

Wronskian Solution for AdS/CFT Y-system · Introduction · 0.89
The algebraic form of the nested Bethe ansatz technique for the twisted $Y(gl(n))$ inhomogeneous spin chain [ref] can be naturally extended to the $Y(gl(N|M))$-case [ref] .

Supersymmetric quantum spin chains and classical integrable systems · Diagonalization of the T-operator via Bethe ansatz · 0.99
For example, to leading order, T_(u) ≃e^ip_1+e^ip_2+e^ip_3 , T_(u) ≃e^i(p_1+p_2)+e^i(p_2+p_3)+e^i(p_3+p_1) , T_(u)≃e^i(p_1+p_2+p_3) , so that if we define a set of exact quasimomenta $q_i$ by[Exploiting the similarity between this definition equation and 4.1 in [ref] we can easily generalize this algebraic equation to a more general $su(K|M)$ super group.

Complete 1-loop test of AdS/CFT · Derivation using the transfer matrices · 0.86

Uses its method or tool 6 papers

In this section following the lines of [ref] we develop a general algorithm to solve the Hirota equations expressing the functions $T_k,m^k',m'(a,s,u)$ in terms of boundary functions $Q^k',m'(u)$ and $Q̃_k,m(u)$.

Discrete Hirota dynamics for AdS/CFT · Integration of the Hirota equations · 0.93
The formal proof[ The Bäcklund procedure of [ref] can be used for it. ] of the fact that [ref],[ref] represents the complete solution of Hirota equation within the $𝕋$-hook will be published elsewhere, but, again, it is easy to convince oneself in the correctness of this formula on Mathematica.

Wronskian Solution for AdS/CFT Y-system · Generating functional for the $𝕋$-hook · 0.85
For that we use the Plücker relations which follow from ([ref]) and are explicitly given in [ref] : [equation] Reconstruction of the rest of  functions goes as follows: * From ([ref]) with $ij=12$ one finds [equation] * Relations ([ref]) for $ijk=123$, $124$, $213$ and $214$ can be explicitly written in view of ([ref]) as [equation] We can unambiguously and explicitly define $_13$, $_14$, $_23$ and $_24$ through $_1,_2,_12$ using their regularity above $Z_-1$, by inverting the linear difference operators in these Plücker relations.

Solving the AdS/CFT Y-system · Upper band · 0.89

Compares with it 5 papers

The normalization of the spectral parameter of the functions in [ref] are closer to the following (functions with prime): $ℱ_λ⊂μ^'I_K(x)= ℱ_λ⊂μ^I_K(xq^-m-n/2)$, ]

Solutions of the T-system and Baxter equations for supersymmetric spin chains · Analytic Bethe ansatz for the Perk-Schultz-type models and their fusion hierarchies revisited · 0.98
We remark that some of our results in this section overlap with those obtained in [ref] for the rational case $q=1$.

Baxter's Q-operators for supersymmetric spin chains · Lattice $R$-matrices · 1.00
Unlike the solution given in [ref] using the Bäcklund techniques of [ref] , the one we mean here would not contain, similarly to [ref], any infinite sums, in analogy to the solution of [ref] for the super-spin chains.

PSU(2,2|4) Character of Quasiclassical AdS/CFT · Conclusions · 0.82

Cites it as background 22 papers

The progress is largely facilitated by considering alternative formulations of BAE, such as Baxter's $TQ$-relation [ref] and the rational $Q$-system [ref] , which was based on various developments in the analytic $TQ$- and $QQ$-systems in the past few decades [ref] .

Spin-$s$ Rational $Q$-system · Introduction · 0.93
The connection between the Hirota equations [ref] and the Bethe Ansatz equations is well understood [ref] .

Baxter Q-Operators and Representations of Yangians · Bethe Equations · 0.90
The supersymmetric generalisation of these ideas was introduced in [ref] and coined to the form particularly relevant for us in [ref] .

Fast analytic solver of rational Bethe equations · Supersymmetric Q-systems · 0.84

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