Four-dimensional Gauge Theories Research Articles

Electric-magnetic duality of Abelian gauge theory on the four-torus, from the fivebrane on T 2 × T 4, via their partition functions

We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T 4 with flat metric using Dirac quantization. In addition to an $$ \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) $$ symmetry, it possesses $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ symmetry that is electromagnetic S-duality. We show explicitly how this $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2, 0) tensor theory, by computing the partition function of a single fivebrane compactified on T 2 times T 4, which has $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right)\times \mathrm{S}\mathrm{L}\left(4,\;\mathcal{Z}\right) $$ symmetry. If we identify the couplings of the abelian gauge theory $$ \tau =\frac{\theta }{2\pi }+i\frac{4\pi }{e^2} $$ with the complex modulus of the T 2 torus $$ \tau ={\beta}^2+i\frac{R_1}{R_2} $$ , then in the small T 2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4d gauge theory. In this way the $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ symmetry of the 6d tensor partition function is identified with the S-duality symmetry of the 4d gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the $$ \mathrm{S}\mathrm{L}\left(2,\;\mathcal{Z}\right) $$ acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.

Gauge theories and dessins d’enfants: beyond the torus

Dessin d'Enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant.

Alpha- and Omega-deformations from fluxes in M-theory

We discuss an SL(2,R) family of deformed N=2 four-dimensional gauge theories which we derive from a flux background in M-theory. In addition to the Omega-deformation this family includes a new deformation, which we call the Alpha-deformation, which can be viewed as an S-dual to the Omega-deformation. We study these gauge theories in two ways: by constructing a non-Abelian (but UV-complete) Lagrangian, and by their strong coupling lift to M-theory where their low-energy dynamics can be determined by examining the equation of motion of a single M5-brane wrapped on a Riemann surface.

Darboux coordinates, Yang-Yang functional, and gauge theory

The moduli space of flat SL 2 connections on a punctured Riemann surface Σ with fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates in which the generating function of the variety of SL 2 -opers is identified with the universal part of the effective twisted superpotential of the corresponding four-dimensional N=2 supersymmetric theory subject to the two-dimensional Ω-deformation. This allows defining the Yang-Yang functionals for the quantum Hitchin system in terms of the classical geometry of the moduli space of local systems for the dual gauge group and relating it to the instanton counting of the four-dimensional gauge theories in the rank-one case.

Ω-deformation and quantization

We formulate a deformation of Rozansky-Witten theory analogous to the $\Omega$-deformation. It is applicable when the target space $X$ is hyperk\"ahler and the spacetime is of the form $\mathbb{R} \times \Sigma$, with $\Sigma$ being a Riemann surface. In the case that $\Sigma$ is a disk, the $\Omega$-deformed Rozansky-Witten theory quantizes a symplectic submanifold of $X$, thereby providing a new perspective on quantization. As applications, we elucidate two phenomena in four-dimensional gauge theory from this point of view. One is a correspondence between the $\Omega$-deformation and quantization of integrable systems. The other concerns supersymmetric loop operators and quantization of the algebra of holomorphic functions on a hyperk\"ahler manifold.

Monopoles in superloop space

In this paper, we will analyse a four-dimensional gauge theory with supersymmetry in superloop space formalism. We will thus obtain an expression for the connection in the infinite-dimensional superloop space. We will then use this connection to obtain an expression for the curvature of the infinite-dimensional superloop space. We will also show that this curvature is proportional to the Bianchi identity in spacetime. Thus, in the absence of a monopole, this curvature will vanish. However, it will not vanish if the superloop intersects the world-line of a monopole because the Bianchi identity will not hold at that point.

Gauge-invariant functional measure for gauge fields onCP2

We introduce a general parametrization for non-Abelian gauge fields on the four-dimensional space ${\mathbb{C}\mathbb{P}}^{2}$. The volume element for the gauge-orbit space or the space of physical configurations is then investigated. The leading divergence in this volume element is obtained in terms of a higher dimensional Wess-Zumino-Witten action, which has previously been studied in the context of K\"ahler-Chern-Simons theories. This term, it is argued, implies that one needs to introduce a dimensional parameter to specify the integration measure, a step which is a nonperturbative version of the well-known dimensional transmutation in four-dimensional gauge theories.

Reading between the lines of four-dimensional gauge theories

Starting with a choice of a gauge group in four dimensions, there is often freedom in the choice of magnetic and dyonic line operators. Different consistent choices of these operators correspond to distinct physical theories, with the same correlation functions of local operators in R^4. In some cases these choices are permuted by shifting the theta-angle by 2pi. In other cases they are labeled by new discrete theta-like parameters. Using this understanding we gain new insight into the dynamics of four-dimensional gauge theories and their phases. The existence of these distinct theories clarifies a number of issues in electric/magnetic dualities of supersymmetric gauge theories, both for the conformal N=4 theories and for the low-energy dualities of N=1 theories.

BPS states in the duality web of the Omega deformation

Abstract In this note, we study different limits of an Ω-deformed (2, 0) six-dimensional gauge theory realized in an M-theory fluxtrap background. Via a chain of dualities, we connect the Ω-deformed sym to a new four-dimensional gauge theory which we refer to as the reciprocal gauge theory. This theory has several properties in common with Liouville field theory, such as its gauge coupling b 2 = ϵ 2/ϵ 1, and its behavior under S-duality. Finally, we realize the bps states on the sym side of the agt correspondence and follow them along the chain of dualities. In the fluxtrap frame, we are dealing with two distinct types of states localized in different radial positions, while in the reciprocal frame, we find single states carrying both charges localized in one place which appear to be perturbatively stable. Our microscopic picture of the small-b limit exhibits semiclassically bps bound states, which are not visible at the level of the partition function.

On three dimensional quiver gauge theories and integrability

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.

The toroidal block and the genus expansion

Abstract We study the correspondence between four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories in the case of $ \mathcal{N}={2^{*}} $ gauge theory. We emphasize the genus expansion on the gauge theory side, as obtained via geometric engineering from the topological string. This point of view uncovers modular properties of the one-point conformal block on a torus with complexified intermediate momenta: in the large intermediate weight limit, it is a power series whose coefficients are quasimodular forms. The all-genus viewpoint that the conformal field theory approach lends to the topological string yields insight into the analytic structure of the topological string partition function in the field theory limit.

$ {{\mathcal{W}}_3} $ irregular states and isolated $ \mathcal{N}=2 $ superconformal field theories

We explore the proposal that the six-dimensional (2, 0) theory on the Riemann surface with irregular punctures leads to a four-dimensional gauge theory coupled to the isolated $ \mathcal{N}=2 $ superconformal theories of Argyres-Douglas type, and to two-dimensional conformal field theory with irregular states. Following the approach of Gaiotto-Teschner for the Virasoro case, we construct $ {{\mathcal{W}}_3} $ irregular states by colliding a single SU(3) puncture with several regular punctures of simple type. If n simple punctures are colliding with the SU(3) puncture, the resulting irregular state is a simultaneous eigenvector of the positive modes L n , . . . , L 2n and W 2n , . . . , W 3n of the $ {{\mathcal{W}}_3} $ algebra. We find the corresponding isolated SCFT with an SU(3) flavor symmetry as a nontrivial IR fixed point on the Coulomb branch of the SU(3) linear quiver gauge theories, by confirming that its Seiberg-Witten curve correctly predicts the conditions for the $ {{\mathcal{W}}_3} $ irregular states. We also compare these SCFT’s with the ones obtained from the BPS quiver method.

Four-dimensional superconformal index reloaded

We consider the four-dimensional N≥1 superconformal index and its generalization to the lens space. We discuss reductions of the latter to the three-dimensional N≥2 sphere partition function, the threedimensional N≥2 superconformal index, and the two-dimensional N≥(2, 2) sphere partition function. We apply these reductions to a class of four-dimensional N=1 superconformal field theories dual to toric Calabi-Yau manifolds, and we find surprising connections with integrable spin chains and hyperbolic geometry. We comment on the problem of classifying infrared fixed points of four-dimensional and threedimensional supersymmetric gauge theories.

5D SYM and 2D q-deformed YM

We study the AGT-like conjectured relation of a four-dimensional gauge theory on S3×S1 to two-dimensional q-deformed Yang–Mills theory on a Riemann surface Σ by using a five-dimensional N=2 supersymmetric Yang–Mills theory on S3×Σ, following the conjectured relation of the six-dimensional N=(2,0) theory on S1 to the five-dimensional Yang–Mills theory. Our results are in perfect agreement with both of the conjectures.

Limit cycles in four dimensions

We present an example of a limit cycle, i.e., a recurrent flow-line of the beta-function vector field, in a unitary four-dimensional gauge theory. We thus prove that beta functions of four-dimensional gauge theories do not produce gradient flows. The limit cycle is established in perturbation theory with a three-loop calculation which we describe in detail.

An Anomaly Associated with Ward—Takahashi Identity for Pseudo-Tensor Current in QED

We study the quantum anomaly for the transverse Ward—Takahashi relations in the four-dimensional gauge theory QED by using Fujikawa's method in which the anomaly is fundamentally a variation of the functional integral measure under transformation. A regulator which leads to a finite result for the anomaly is proposed. It is shown that a novel anomaly associated with transverse Ward—Takahashi identity of pseudo-tensor current is derived due to a set of infinitesimal transverse transformation of field variables.

All one-loop amplitudes in $ \mathcal{N}=6 $ superconformal Chern-Simons theory

We exploit a recently found connection between special triple-cut diagrams and tree-level recursive diagrams to derive a general formula capturing the multi-particle factorisation of arbitrary one-loop amplitudes in the ABJM theory. This formula contains certain anomalous contributions which are reminiscent of the so-called non-factorising contributions appearing in the factorisation of one-loop amplitudes in four-dimensional gauge theory. In the second part of the paper we derive a recursion relation for the supercoefficients of one-loop amplitudes in ABJM theory. By applying this recursion relation, any one-loop supercoefficient can be reduced to special triple-cut diagrams involving at least one four-point tree amplitude. In turn, this implies that any one-loop supercoefficient can be derived from tree-level recursive diagrams.

An E 7 surprise

We explore some curious implications of Seiberg duality for an SU(2) four-dimensional gauge theory with eight chiral doublets. We argue that two copies of the theory can be deformed by an exactly marginal quartic superpotential so that they acquire an enhanced E7 flavor symmetry. We argue that a single copy of the theory can be used to define an E7-invariant superconformal boundary condition for a theory of 28 five-dimensional free hypermultiplets. Finally, we derive similar statements for three-dimensional gauge theories such as an SU(2) gauge theory with six chiral doublets or Nf=4 SQED.

Light scalars in strongly-coupled extra-dimensional theories

The low-energy dynamics of five-dimensional Yang-Mills theories compactified on S^1 can be described by a four-dimensional gauge theory coupled to a scalar field in the adjoint representation of the gauge group. Perturbative calculations suggest that the mass of this elementary scalar field is protected against power divergences, and is controlled by the size of the extra dimension R. As a first step in the study of this phenomenon beyond perturbation theory, we investigate the phase diagram of a SU(2) Yang-Mills theory in five dimensions regularized on anisotropic lattices and we determine the ratios of the relevant physical scales. The lattice system shows a dimensionally reduced phase where the four-dimensional correlation length is much larger than the size of the extra dimension, but still smaller than the four-dimensional volume. In this region of the bare parameter space, at energies below 1/R, the non-perturbative spectrum contains a \emph{light} scalar state. This state has a mass that is independent of the cut-off, and a small overlap with glueball operators. Our results suggest that light scalar fields can be introduced in a lattice theory using compactified extra dimensions, rather than fine tuning the bare mass parameter.

The omega deformation from string and M-theory

Abstract We present a string theory construction of Omega-deformed four-dimensional gauge theories with generic values of ϵ1 and ϵ2. Our solution gives an explicit description of the geometry in the core of Nekrasov and Witten’s realization of the instanton partition function, far from the asymptotic region of their background. This construction lifts naturally to M-theory and corresponds to an M5-brane wrapped on a Riemann surface with a selfdual flux. Via a 9-11 flip, we finally reinterpret the Omega deformation in terms of noncommutative geometry. Our solution generates all modified couplings of the Ω-deformed gauge theory, and also yields a geometric origin for the quantum spectral curve of the associated quantum integrable system.