Quantization of Integrable Systems and Four Dimensional Gauge Theories (original) (raw)
Related papers
Quantum Integrability and Supersymmetric Vacua
Progress of Theoretical Physics Supplement, 2009
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. Examples include: the Heisenberg SU (2) XXX spin chain which is mapped to the two dimensional U (N ) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model which are related to the four dimensional theory compactified on T 2 . A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as T * Gr(N, L) and the ring of quantum integrals of motion of various spin chains. 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 like Hubbard model. We give the gauge-theoretic interpretation of Drinfeld polynomials and Baxter operators. The two-sphere compactifications of the four dimensional N = 2 theories lead to the instanton corrected Bethe equations. We suggest the Yangian, quantum affine, and elliptic algebras are a completely novel kind of symmetry of the (collections of the) interacting quantum field theories. To Prof. T. Eguchi on the occasion of his 60th anniversary a On leave of absence from ITEP, Moscow, Russia 0
On the integrability of four dimensional $ \mathcal{N}=2 $ gauge theories in the omega background
Journal of High Energy Physics, 2013
We continue to investigate the relationship between the infrared physics of N = 2 supersymmetric gauge theories in four dimensions and various integrable models such as Gaudin, Calogero-Moser and quantum spin chains. We prove interesting dualities among some of these integrable systems by performing different, albeit equivalent, quantizations of the Seiberg-Witten curve of the four dimensional theory. We also discuss conformal field theories related to N = 2 4d gauge theories by the Alday-Gaiotto-Tachikawa (AGT) duality and the role of conformal blocks of those CFTs in the integrable systems. As a consequence, the equivalence of conformal blocks of rank two Toda and Novikov-Wess-Zumino-Witten (WZNW) theories on the torus with punctures is found.
Two-dimensional gauge theories and quantum integrable systems
Arxiv preprint arXiv:0711.1472, 2007
In this paper the relation between 2d topological gauge theories and Bethe Ansatz equations is reviewed. 1 In addition we present some new results and clarifications. We hope the relations discussed here are particular examples of more general relations between quantum topological fields theories in dimensions d ≤ 4 and quantum integrable systems.
On three dimensional quiver gauge theories and integrability
Journal of High Energy Physics, 2013
In this work we compare different descriptions of the space of vacua of certain three dimensional N = 4 superconformal field theories, compactified on a circle and mass-deformed to N = 2 in a canonical way. The original N = 4 theories are known to admit two distinct mirror descriptions as linear quiver gauge theories, and many more descriptions which involve the compactification on a segment of four-dimensional N = 4 super Yang-Mills theory. Each description gives a distinct presentation of the moduli space of vacua. Our main result is to establish the precise dictionary between these presentations. We also study the relationship between this gauge theory problem and integrable systems. The space of vacua in the linear quiver gauge theory description is related by Nekrasov-Shatashvili duality to the eigenvalues of quantum integrable spin chain Hamiltonians. The space of vacua in the four-dimensional gauge theory description is related to the solution of certain integrable classical many-body problems. Thus we obtain numerous dualities between these integrable models.
Quantum integrable systems from supergroup gauge theories
Journal of High Energy Physics, 2020
In this note, we establish several interesting connections between the super- group gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Calogero-Moser system by considering certain orbifolded instanton partition functions of their corresponding supergroup gauge theories. We also derive an exotic generalization of 𝔰𝔩(2) XXX spin chain arising from the instanton partition function of SQCD with super- gauge group, and study its Bethe ansatz equation.
Supersymmetric Vacua and Bethe Ansatz
Nuclear Physics B - Proceedings Supplements, 2009
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 2. 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.
Supersymmetric Yang-Mills theory and integrable systems
Nuclear Physics B, 1996
The Coulomb branch of N = 2 supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the N = 2 SU(n) gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model has an SL(2, Z) S-duality group (with the central element -1 of SL(2,Z) acting as charge conjugation); SL(2, Z) permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.
Journal of High Energy Physics, 2014
We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on C 2 × S 2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S 2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W N algebrae, thus providing a gauge theoretical proof of AGT correspondence. *
Quantization of the Wess-Zumino-Witten model on a circle
Physics Letters B, 1991
A systematic and direct derivation of the Poisson brackets for the chiral group elements in the Wess-Zumino-Witten model, based on a compact Lie group, is given. The quantization of these relations leads to a unitary version of the quantum Yang-Baxter equation, which is discussed in detail for SU (2). The matrix element of the chiral group elements are identified with projected chiral vertex operators.