Supersymmetric vacua and Bethe ansatz

By Nikita A.Nekrasov, Samson L.Shatashvili. 2009

341 citations recorded by the source corpus.

This note is a short announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N = 4 gauge theories with matter, softly broken by the twisted masses down to N = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg SU(2) XXX spin chain is mapped to the two dimensional U(N) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model are related to the four dimensional theory compactified on T. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models as well as the dynamical spin chains such as the Hubbard model. Compactifications of four dimensional N = 2 theories on a two-sphere lead to the instanton-corrected Bethe equations. We propose a completely novel way for the Yangian, quantum affine, and elliptic algebras to act as a symmetry of a union of quantum field theories.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b77b7f4c9922570e57f541e8da3796a5342c7a38

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Citations in the corpus

Comments on twisted indices in 3d supersymmetric gauge theories (2016, methodology): “The ‘holomorphic’ properties of the low-energy theory on M are determined by the effective twisted superpotential, which can be obtained by integrating out all the massive fields at generic values of ua, including all the Kaluza-Klein (KK) modes on S 1 [10, 46].”
The omega deformation, branes, integrability and Liouville theory (2010, background): “This correspondence has recently been extended [4, 5] to a much wider class of examples.”
Quantum geometry of refined topological strings (2011, background): “Indeed, in [7] Nekrasov and Shatashvili observed that the gauge theory in the Ω-background in this limit is connected to various quantum integrable systems in a highly non-trivial way, a feature shared by various two-dimensional supersymmetric gauge systems [49, 50, 51, 52, 53, 26].”
Gauge Theories Labelled by Three-Manifolds (2011, background): “3) with W̃ a sum of dilogarithm functions, is highly reminiscent of recent work relating effective 2d field theories to quantum integrable systems [36, 37, 38].”
Quantum Geometry and Quiver Gauge Theories (2013, background): “3) which can be identified with the Yang-Yang function of some quantum integrable system [4], [5], [6], [7].”
BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters (2015, background): “It was proposed in [89, 97, 98] that there are symmetries acting between different quantum field theories, for example changing the gauge groups.”
Defects and quantum Seiberg-Witten geometry (2014, methodology): “At finite Q we will consider the Nekrasov-Shatashvili limit 2 → 0 of the normalized expectation value of the defect D(i)ρ (~a,Q, 1, 2,~t) = lim 2→0 Z(i)ρ (~a,Q, 1, 2,~t) Z(a,Q, 1, 2) , (4.5) 8Some work is this direction is now being done in [41].”
N$$ \mathcal{N} $$ = 1 supersymmetric indices and the four-dimensional A-model (2017, methodology): “The linear ambiguity in ua reflects the fact that Ua is a constrained twisted chiral fields, with the twisted F -term given by the twodimensional gauge flux [32, 43].”
Review of AdS/CFT Integrability: An Overview (2010, background): “Furthermore, supersymmetric vacua in D = 2, N = 4 gauge theories with matter can be described by Bethe ansätze [20,21].”
3-Manifolds and 3d Indices (2011, background): “There are three such quantities that are related in a surprising manner: the moduli space L[T ] of supersymmetric vacua for massive deformations of the SCFT compactified on a circle [7–9], the ellipsoid partition function Zb[T ] [10–12] and the refined index I[T ] [13–17].”
Darboux coordinates, Yang-Yang functional, and gauge theory (2011, methodology): “n studied there we have half the supersymmetry, so the three dimensional theory has only four supercharges, and upon compactiﬁcation on S1 ∞ generates a twisted superpotential, of the form studied in [14],[15] W∞(α;ν,γ) ∼ X ℓ Li2(eℓ(α,ν)) (19) where the sum is over the charged matter ﬁelds, and ℓis ”