Two-dimensional gauge theories and quantum integrable systems

Physical Review D, 1991

The path-integral generalization of the Duistermaat-Heckman integration formula is investigated for integrable models. It is shown that for models with periodic classical trajectories the path integral reduces to a form similar to the finite-dimensional Duistermaat-Heckman integration formula. This provides a relation between exactness of the stationary-phase approximation and Morse theory. It is also argued that certain integrable models can be related to topological quantum theories. Finally, it is found that in general the stationary-phase approximation presumes that the initial and final configurations are in difFerent polarizations. This is exemplified by the quantization of the SU(2) coadjoint orbit.

Physics Letters B, 1990

Two-dimensional euclidean (topological) quantum Yang-Mills theory on the compact manifold in the Lorentz gauge is analysed in the framework of the covariant path-integral approach. The Nicolai map for the partition function and for the Wilson loop observables is explicitly given. Topological quantum field theory (TQFT) is a fascinating and fashionable subject nowadays. Each "theory of nothing", i.e. possessing zero degrees of freedom from the "non-topological point of view" (particularly, a theory with a local symmetry), is a potential candidate for TQFT. Apparently, there are three categories of TQFTs: ( 1 ) TQFT with (very large) topological symmetry, e.g. topological Chern-Witten theory in four dimensions

XVIth International Congress on Mathematical Physics, 2010

We study four dimensional N = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional N = 2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N = 2 theory. The ε-parameter of the Ω-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L 2 -spectrum. We very briefly discuss the quantization of Hitchin system.

Arxiv preprint hep-th/9310144, 1993

1Presented at the 1993 Trieste Summer School in High Energy Physics and Cosmology, 14 June - 30 July 1993. 2e-mail: [email protected] 3e-mail: [email protected] ... 2 Yang-Mills and Topological Gauge Theories in Two Dimensions 4

Nuclear Physics B, 1991

We give a formulation of two-dimensional topological gravity without matter in terms of a supersymmetric conformally invariant field theory and derive a path integral expression for the physical amplitudes. A careful analysis of the contact terms of the physical operators reveals the presence of a non-commutative algebra, isomorphic to the Virasoro algebra. We show that this algebra completely determines all the amplitudes at arbitrary genus, which coincide with those of the one-matrix model at the k = 1 critical point .

1998

In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N = 2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N = 2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in terms of Seiberg-Witten invariants. Generalizations of Donaldson-Witten theory are reviewed, and the structure of the vacuum expectation values of their observables is analyzed in the context of duality for the simplest case.

Cambridge University Press eBooks, 1997

The aim of this work is to explain what a topological quantum field theory (TQFT) is and the relation between TQFTs in dimension 2 and Frobenius algebras. More precisely, it is shown that there exists an equivalence of categories between the category of 2-dimensional TQFTs on the one hand, and the category of commutative Frobenius algebras on the other. The work begins with a review of the basic concepts of category theory and, in particular, of the theory of symmetric monoidal categories, a categorical analogue of the abelian monoids. In the next two chapters the categories of cobordisms and of commutative Frobenius algebras are introduced. Both are examples of monoidal categories. Finally, in the last chapter TQFTs are defined and the above mentioned theorem proved. At the end of the work, some explicit examples of TQFTs in dimension 2 are described and the corresponding topological invariants of surfaces computed.

Physical Review D, 2008

Four dimensional BF theory admits a natural coupling to extended sources supported on two dimensional surfaces or string world-sheets. Solutions of the theory are in one to one correspondence with solutions of Einstein equations with distributional matter (cosmic strings). We study new (topological field) theories that can be constructed by adding extra degrees of freedom to the two dimensional world-sheet. We show how two dimensional Yang-Mills degrees of freedom can be added on the world-sheet, producing in this way, an interactive (topological) theory of Yang-Mills fields with BF fields in four dimensions. We also show how a world-sheet tetrad can be naturally added. As in the previous case the set of solutions of these theories are contained in the set of solutions of Einstein's equations if one allows distributional matter supported on two dimensional surfaces. These theories are argued to be exactly quantizable. In the context of quantum gravity, one important motivation to study these models is to explore the possibility of constructing a background independent quantum field theory where local degrees of freedom at low energies arise from global topological (world-sheet) degrees of freedom at the fundamental level.

International Journal of Modern Physics A, 2012

We consider a two-parameter family of ℤ2 gauge theories on a lattice discretization [Formula: see text] of a three-manifold [Formula: see text] and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space Γ. We show that there is a region Γ0 ⊂ Γ where the partition function and the expectation value 〈WR(γ)〉 of the Wilson loop can be exactly computed. Depending on the point of Γ0, the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of [Formula: see text]. The Wilson loop on the other hand, does not depend on the topology of γ. However, for a subset of Γ0, 〈WR(γ)〉 depends on the size of γ and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.

1999

Description of two three-dimensional topological quantum field theories of Witten type as twisted supersymmetric theories is presented. Low-energy effective action and a corresponding topological invariant of three-dimensional manifolds are considered.