Four-dimensional Gauge Theories Research Articles

Connecting Polyakov loops to the thermodynamics ofSU(Nc)gauge theories using the gauge-string duality

We show that in four-dimensional gauge theories dual to five-dimensional Einstein gravity coupled to a single scalar field in the bulk the derivative of the single heavy quark free energy in the deconfined phase is $dF_{Q}(T)/dT \sim -1/c_s^2(T)$, where $c_s(T)$ is the speed of sound. This general result provides a direct link between the softest point in the equation of state of strongly-coupled plasmas and the deconfinement phase transition described by the expectation value of the Polyakov loop. We give an explicit example of a gravity dual with black hole solutions that can reproduce the lattice results for the expectation value of the Polyakov loop and the thermodynamics of SU(3) Yang-Mills theory in the (non-perturbative) temperature range between $T_c$ and $3T_c$.

Penner type matrix model and Seiberg-Witten theory

We discuss the Penner type matrix model recently proposed by Dijkgraaf and Vafa for a possible explanation of the relation between four-dimensional gauge theory and Liouville theory by making use of the connection of the matrix model to two-dimensional CFT. We first consider the relation of gauge couplings defined in UV and IR regimes of N_f = 4, N = 2 supersymmetric gauge theory being related as $q_{{\rm UV}}={\vartheta_2(q_{{\rm IR}})^4/\vartheta_3(q_{{\rm IR}})^4}$. We then use this relation to discuss the action of modular transformation on the matrix model and determine its spectral curve. We also discuss the decoupling of massive flavors from the N_f = 4 matrix model and derive matrix models describing asymptotically free N = 2 gauge theories. We find that the Penner type matrix theory reproduces correctly the standard results of N = 2 supersymmetric gauge theories.

Liouville Correlation Functions from Four-Dimensional Gauge Theories

We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.

Entanglement entropy in Abelian gauge theories

The definition of geometric entanglement entropy associated with some region in space is discussed for the case of gauge theories. It is argued that since in gauge theories elementary excitations look like loops (closed electric strings) rather than points (particles), the boundaries of the regions should also carry some nonzero entropy. This entropy counts the number of strings which cross these boundaries. Explicit calculations of such entropy are carried out in the limits of infinitely strong and weak couplings of three- and four-dimensional ZN gauge theories. In three dimensions we find that the entropy is a constant which does not depend on the region, while in four dimensions the familiar area law for the entropy is recovered.

Nonperturbative scales in AdS/CFT

The cusp anomalous dimension is a ubiquitous quantityin four-dimensional gauge theories, ranging from QCD to maximally supersymmetric Yang–Mills theory, and it is one of the most thoroughly investigated observables in the AdS/CFT correspondence. In planar SYM theory, its perturbative expansion at weak coupling has a finite radius of convergence while at strong coupling it admits an expansion in inverse powers of the 't Hooft coupling which is given by a non-Borel summable asymptotic series. We study the cusp anomalous dimension in the transition regime from strong to weak coupling and argue that the transition is driven by nonperturbative, exponentially suppressed corrections. To compute these corrections, we revisit the calculation of the cusp anomalous dimension in planar SYM theory and extend the previous analysis by taking into account nonperturbative effects. We demonstrate that the scale parameterizing nonperturbative corrections coincides with the mass gap of the two-dimensional bosonic O(6) sigma model embedded into the AdS5 × S5 string theory. This result is in agreement with the prediction coming from the string theory consideration.

Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory

We use the real-time finite-temperature AdS/CFT correspondence to compute the effect of general R2 corrections to the gravitational action in AdS space on the shear viscosity of the dual gauge theory. The R2 terms in AdS5 are determined by the central charges of the CFT. We present an example of a four-dimensional gauge theory in which the conjectured lower bound of 1/4π on the viscosity-to-entropy ratio is violated for finite N.

Coset space dimensional reduction and Wilson flux breaking of ten-dimensional $\mathcal{N}=1$ , E 8 gauge theory

We consider a $\mathcal{N}=1$ supersymmetric E 8 gauge theory, defined in ten dimensions and we determine all four-dimensional gauge theories resulting from the generalized dimensional reduction à la Forgacs–Manton over coset spaces, followed by a subsequent application of the Wilson flux spontaneous symmetry-breaking mechanism. Our investigation is constrained only by the requirements that (i) the dimensional reduction leads to the potentially phenomenologically interesting, anomaly-free, four-dimensional E 6, SO10 and SU5 GUTs and (ii) the Wilson flux mechanism makes use only of the freely acting discrete symmetries of all possible six-dimensional coset spaces.

A counterexample to thea-`theorem'

The conclusion of the original paper was wrong, due to the incorrect assumption that the low-energy limit at the strongly-coupled point consists of a single, coupled SCFT. By taking into account the fact that the low-energy limit consists of multiple decoupled parts, it was later shown in arXiv:1011.4568 that there is no violation of the a-theorem in this system. Furthermore, the a-theorem itself was convincingly demonstrated in arXiv:1107.3987, and the argument presented there has been further refined. The rest of this paper is kept as it was, for some parts of the discussions might still be of interest. Original abstract: We exhibit a renormalization group flow for a four-dimensional gauge theory along which the conformal central charge 'a' increases. The flow connects the maximally superconformal point of an N=2 gauge theory with gauge group SU(N+1) and N_f=2N flavors in the ultraviolet, to a strongly-coupled superconformal point of the SU(N) gauge theory with N_f=2N massless flavors in the infrared. Our example does not contradict the proof of the a-theorem via a-maximization, due to the presence of accidental symmetries in the infrared limit. Nor does it contradict the holographic a-theorem, because these gauge theories do not possess weakly-curved holographic duals.

Second order hydrodynamics of a CFT plasma from boost invariant expansion

We compute finite coupling correction to a non-linear second order hydrodynamic coefficient in the boost invariant expansion of the N = 4 supersymmetric Yang–Mills plasma. The result is universal for a large class of strongly coupled four-dimensional conformal gauge theories.

Gauging spacetime symmetries on the worldsheet and the geometric Langlands program — II

We study the two-dimensional twisted (0, 2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence—at the level of the holomorphic chiral algebra—between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical -algebras and a level relation which underlie a geometric Langlands correspondence for G = SL(N, C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G = SL(N, C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G = SL(N, C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.

Full fermion-boson vertex function derived in terms of symmetry relations in Abelian gauge theory

Nonperturbative studies such as confinement and dynamical chiral symmetry breaking need the nonperturbative interacting vertex functions. In this paper, an approach to determining the full fermion-boson vertex function in four-dimensional Abelian gauge theory is presented: this full vertex function is derived in terms of a set of normal (longitudinal) and transverse Ward-Takahashi relations for the fermion-boson (vector) and axial-vector vertices in the momentum space in the case of massless fermion. Such a derived fermion-boson vertex function should be satisfied both perturbatively and nonperturbatively. The fact that such a derived full fermion-boson vertex function to one-loop order holds indeed is proven and the nonperturbative form of this vertex is also under discussion.

Gauging spacetime symmetries on the worldsheet and the geometric Langlands program

We study the two-dimensional twisted (0,2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence - at the level of the holomorphic chiral algebra - between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical W-algebras and a level relation which underlie a geometric Langlands correspondence for G=SL(N,C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G=SL(N,C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G=SL(N,C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.

Topological gauge theories on local spaces and black hole entropy countings

We study cohomological gauge theories on total spaces of holomorphic line bundles over complex manifolds and obtain their reduction to the base manifold by $U(1)$-equivariant localization of the path integral. We exemplify this general mechanism by proving via exact path integral localization a reduction for local curves conjectured in hep-th/0411280, relevant to the calculation of black hole entropy/Gromov–Witten invariants. Agreement with the four-dimensional gauge theory is recovered by taking into account in the latter non-trivial contributions coming from one-loop fluctuation determinants at the boundary of the total space. We also study a class of abelian gauge theories on Calabi–Yau local surfaces, describing the quantum foam for the $A$-model, relevant to the calculation of Donaldson–Thomas invariants.

Orientifolds in 𝒩 = 2 Liouville theory and its mirror

We consider unoriented strings in the supersymmetric SL(2, )/U(1) coset, which describes the two-dimensional Euclidean black hole, and its mirror dual = 2 Liouville theory. We analyze the orientifolds of these theories from several complementary points of view: the parity symmetries of the worldsheet actions, descent from known AdS3 parities, and the modular bootstrap method (in some cases we can also check our results against known constraints coming from the conformal bootstrap method). Our analysis extends previous work on orientifolds in Liouville theory, the AdS3 and SU(2) WZW models and minimal models. Compared to these cases, we find that the orientifolds of the two dimensional Euclidean black hole exhibit new intriguing features. Our results are relevant for the study of orientifolds in the neighborhood of NS5-branes and for the engineering of four-dimensional chiral gauge theories and gauge theories with SO and Sp gauge groups with suitable configurations of D-branes and orientifolds. As an illustration, we discuss an example related to a configuration of D4-branes and O4-planes in the presence of two parallel fivebranes.

A quantization of twistor Yang-Mills theory through the background field method

Four-dimensional Yang-Mills theory formulated through an action on twistor space has a larger gauge symmetry than the usual formulation, which in previous work was shown to allow a simple gauge transformation between textbook perturbation theory and the Cachazo-Svr\ifmmode \check{c}\else \v{c}\fi{}ek-Witten rules. In this paper we study nonsupersymmetric twistor Yang-Mills theory at loop level using the background field method. For an appropriate partial quantum field gauge choice it is shown that the calculation of the effective action is equivalent to (the twistor lift of) the calculation in ordinary Yang-Mills theory in the Chalmers and Siegel formulation to all orders in perturbation theory. A direct consequence is that the twistor version of Yang-Mills theory is just as renormalizable in this particular gauge. As applications an explicit calculation of the Yang-Mills beta function and some preliminary investigations into using the formalism to calculate S-matrix elements at loop level are presented. In principle the technique described in this paper generates consistent quantum completions of the Cachazo-Svr\ifmmode \check{c}\else \v{c}\fi{}ek-Witten rules. However, by inherent limitations of the partial gauge choice employed here, this offers in its current form mainly simplifications for tree-level forestry. The method is expected to be applicable to a wide class of four-dimensional gauge theories.

On the consistency of coset space dimensional reduction

In this Letter we consider higher-dimensional Yang–Mills theories and examine their consistent coset space dimensional reduction. Utilizing a suitable ansatz and imposing a simple set of constraints we determine the four-dimensional gauge theory obtained from the reduction of both the higher-dimensional Lagrangian and the corresponding equations of motion. The two reductions yield equivalent results and hence they constitute an example of a consistent truncation.

Real forms of complex higher spin field equations and new exact solutions

We formulate four-dimensional higher spin gauge theories in spacetimes with signature ( 4 − p , p ) and non-vanishing cosmological constant. Among them are chiral models in Euclidean ( 4 , 0 ) and Kleinian ( 2 , 2 ) signature involving half-flat gauge fields. Apart from the maximally symmetric solutions, including de Sitter spacetime, we find: (a) SO ( 4 − p , p ) invariant deformations, depending on one continuous and infinitely many discrete parameters, including a degenerate metric of rank one; (b) non-maximally symmetric solutions with vanishing Weyl tensors and higher spin gauge fields, that differ from the maximally symmetric solutions in the auxiliary field sector; and (c) solutions of the chiral models furnishing higher spin generalizations of type D gravitational instantons, with an infinite tower of Weyl tensors proportional to totally symmetric products of two principal spinors. These are apparently the first exact 4D solutions with non-vanishing massless higher spin fields.

TRANSVERSE WARD–TAKAHASHI RELATION FOR THE FERMION–BOSON VERTEX FUNCTION IN FOUR-DIMENSIONAL ABELIAN GAUGE THEORY

In this paper, we give a general formula of the transverse Ward–Takahashi relation for the fermion–boson vertex function in momentum space in four-dimensional Abelian gauge theory, with the complete expression of the integral-term involving the nonlocal axial-vector vertex function, from which the corresponding one-loop expression is derived straightforwardly. Then we deduce carefully this transverse Ward–Takahashi relation to one-loop order in d dimensions, with d = 4 + ∊. The result shows that this relation holds in d dimensions if the combination ελμνργργ5 in d dimensions is replaced with -½{γλ, σμν} by the identity of γ-matrices, and there is no additional piece proportional to (d-4) to include for this relation to hold in four dimensions. This result is further confirmed by an explicit computation of terms in this transverse WT relation to one-loop order. We also discuss the momentum space representation of the nonlocal term in the transverse WT relation.

Gauged supergravity from type IIB string theory on [formula omitted] manifolds

We first construct a consistent Kaluza–Klein reduction ansatz for type IIB theory compactified on Sasaki–Einstein manifolds Y p , q with Freund–Rubin 5-form flux giving rise to minimal N = 2 gauged supergravity in five dimensions. We then investigate the R-charged black hole solution in this gauged supergravity, and in particular study its thermodynamics. Based on the gauge theory/string theory correspondence, this non-extremal geometry is dual to finite temperature strongly coupled four-dimensional conformal gauge theory plasma with a U ( 1 ) R -symmetry charge chemical potential. We study transport properties of the gauge theory plasma and show that the ratio of shear viscosity to entropy density in this plasma is universal. We further conjecture that the universality of shear viscosity of strongly coupled gauge theory plasma extends to non-zero R-charge chemical potential.

Confining k-string tensions with D-branes in super Yang–Mills theories

Abstract We discuss confining k strings in four-dimensional gauge theories using D5 branes in AdS 5 × S 5 , and D3 branes in Klebanov–Strassler and Maldacena–Nunez backgrounds. We present two results: The first that confining k string tensions in N = 4 can be calculated using D5 branes in AdS 5 × S 5 with a cut-off in the bulk AdS . Using an embedding of R 2 times S 4 ⊂ S 5 , we show that the D5 brane replicates a string of rank k in the antisymmetric representation. The second result shows that the S-dual calculation to hep-th/0111078 reproduces the action in the Klebanov–Strassler and Maldacena–Nunez backgrounds exactly, while providing a more natural manifestation of the string charge k .