scholarly journals Inner products in integrable Richardson-Gaudin models

2017 ◽  
Vol 3 (4) ◽  
Author(s):  
Pieter W. Claeys ◽  
Dimitri Van Neck ◽  
Stijn De Baerdemacker
Keyword(s):  
Domain Wall ◽  
Integrable Models ◽  
Partition Functions ◽  
Dual Representation ◽  
Determinant Formula ◽  
Quadratic Equations ◽  
Conserved Charges ◽  
Inner Products ◽  
Gaudin Models ◽  
Bethe Equations

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula.Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

scholarly journals Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
Simons Theory ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Yang Mills ◽  
Stable Domain ◽  
Coxeter Number ◽  
Topological Quantum ◽  
Simply Connected

Abstract 4d$$ \mathcal{N} $$ N = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

scholarly journals SUPERSYMMETRIC TWO-BOSON EQUATION, ITS REDUCTIONS AND THE NONSTANDARD SUPERSYMMETRIC KP HIERARCHY

1995 ◽  
Vol 10 (32) ◽  
pp. 4563-4599 ◽  
Author(s):  
J.C. BRUNELLI ◽  
ASHOK DAS
Keyword(s):  
Integrable Models ◽  
Lax Representation ◽  
Kp Equation ◽  
Kp Hierarchy ◽  
Hamiltonian Structures ◽  
Conserved Charges ◽  
Nonlinear Schrödinger ◽  
Field Redefinition

In this paper, we review various properties of the supersymmetric Two-Boson (sTB) system. We discuss the equation and its nonstandard Lax representation. We construct the local conserved charges as well as the Hamiltonian structures of the system, and show how this system leads to various other known supersymmetric integrable models under appropriate field redefinition. We discuss the sTB and the supersymmetric nonlinear Schrödinger (sNLS) equations as constrained, nonstandard supersymmetric Kadomtsev-Petviashvili (sKP) systems and note that the nonstandard sKP systems naturally unify all the KP and mKP flows while leading to a new integrable supersymmetrization of the KP equation. We construct the nonlocal conserved charges associated with the sTB system and show that the algebra of charges corresponds to a graded, cubic algebra. Also, we note that the sTB system has a hidden supersymmetry making it an N=2 extended supersymmetric system.

scholarly journals Domain wall partition functions and KP

2009 ◽  
Vol 2009 (03) ◽  
pp. P03017 ◽  
Author(s):  
O Foda ◽  
M Wheeler ◽  
M Zuparic
Keyword(s):  
Domain Wall ◽  
Partition Functions

scholarly journals QUASI-EXACTLY SOLVABLE DEFORMATIONS OF GAUDIN MODELS AND "QUASI-GAUDIN ALGEBRAS"

1998 ◽  
Vol 13 (04) ◽  
pp. 281-292 ◽  
Author(s):  
A. G. USHVERIDZE
Keyword(s):  
Bethe Ansatz ◽  
Complete Integrability ◽  
Integrable Models ◽  
Exact Solvability ◽  
New Class ◽  
Algebraic Bethe Ansatz ◽  
Exactly Solvable ◽  
Algebraic Deformation ◽  
Bethe Ansatz Solution ◽  
Gaudin Models

A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra.

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