2D Yang-Mills Research Articles

Hidden local symmetry and infinite tower of vector mesons for baryons

In an effort to access dense baryonic matter relevant for compact stars in a unified framework that handles both single baryon and multibaryon systems on the same footing, we first address a holographic dual action for a single baryon focusing on the role of the infinite tower of vector mesons deconstructed from five dimensions. To leading order in 't Hooft coupling $\lambda=N_c g_{\rm YM}^2$, one has the Bogomol'nyi-Prasad-Sommerfield (BPS) Skyrmion that results when the warping of the bulk background and the Chern-Simons term in the Sakai-Sugimoto D4/D8-${\bar{\rm D8}}$ model are ignored. The infinite tower was found by Sutcliffe to induce flow to a conformal theory, i.e., the BPS. We compare this structure to that of the SS model consisting of a 5D Yang-Mills action in warped space and the Chern-Simons term in which higher vector mesons are integrated out while preserving hidden local symmetry and valid to $O(\lambda^0)$ and $O(p^4)$ in the chiral counting. We point out the surprisingly important role of the $\omega$ meson that figures in the Chern-Simons term that encodes chiral anomaly in the baryon structure and that may be closely tied to short-range repulsion in nuclear interactions.

Unitary Chern-Simons matrix model and the Villain lattice action

We use the Villain approximation to show that the Gross-Witten model, in the weak- and strong-coupling limits, is related to the unitary matrix model that describes $U(N)$ Chern-Simons theory on ${S}^{3}$. The weak-coupling limit corresponds to the $q\ensuremath{\rightarrow}1$ limit of the Chern-Simons theory while the strong-coupling regime is related to the $q\ensuremath{\rightarrow}0$ limit. In the latter case, there is a logarithmic relationship between the respective coupling constants. We also show how the Chern-Simons matrix model arises by considering two-dimensional Yang-Mills theory with the Villain action. This leads to a $U(1{)}^{N}$ theory which is the Abelianization of 2d Yang-Mills theory with the heat-kernel lattice action. In addition, we show that the character expansion of the Villain lattice action gives the $q$ deformation of the heat kernel as it appears in $q$-deformed 2d Yang-Mills theory. We also study the relationship between the unitary and Hermitian Chern-Simons matrix models and the rotation of the integration contour in the corresponding integrals.

The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere

Based on the construction by Hosomichi, Seong and Terashima we consider N = 1 supersymmetric 5D Yang-Mills theory with matter on a five-sphere with radius r. This theory can be thought of as a deformation of the theory in flat space with deformation parameter r and this deformation preserves 8 supercharges. We calculate the full perturbative partition function as a function of $ {{{r} \left/ {{g_Y^2}} \right.}_M} $ , where $ {g_Y}_M $ is the Yang-Mills coupling, and the answer is given in terms of a matrix model. We perform the calculation using localization techniques. We also argue that in the large N-limit of this deformed 5D Yang-Mills theory this matrix model provides the leading contribution to the partition function and the rest is exponentially suppressed.

Derivation of an Abelian effective model for instanton chains in 3D Yang-Mills theory

In this work, we derive a recently proposed Abelian model to describe the interaction of correlated monopoles, center vortices, and dual fields in three dimensional SU(2) Yang-Mills theory. Following recent polymer techniques, special care is taken to obtain the end-to-end probability for a single interacting center vortex, which constitutes a key ingredient to represent the ensemble integration.

Resummation scheme for 3d Yang-Mills and the two-loop magnetic mass for hot gauge theories

Perturbation theory for non-Abelian gauge theories at finite temperature is plagued by infrared divergences caused by magnetic soft modes $\sim g^2T$, which correspond to the fields of a 3d Yang-Mills theory. We revisit a gauge invariant resummation scheme to solve this problem by self-consistent mass generation using an auxiliary scalar field, improving over previous attempts in two respects. First, we generalise earlier SU(2) treatments to SU(N). Second, we obtain a gauge independent two-loop gap equation, correcting an error in the literature. The resulting two-loop approximation to the magnetic mass represents a $\sim 15%$ correction to the leading one-loop value, indicating a reasonable convergence of the resummation.

Non-abelian action for multiple five-branes with self-dual tensors

We construct an action for non-abelian 2-form in 6-dimensions. Our action consists of a non-abelian generalization of the abelian action of Perry and Schwarz for a single five-brane. It admits a self-duality equation on the field strength as the equation of motion. It has a modified 6d Lorentz symmetry. On dimensional reduction on a circle, our action gives the standard 5d Yang-Mills action plus higher order corrections. Based on these properties, we propose that our theory describes the gauge sector of multiple M5-branes in flat space.

Bubble Divergences from Twisted Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory and 3d Riemannian quantum gravity, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined because of a phenomenon known as ‘bubble divergences’. In this paper, we extend recent results of the authors to the cases where these divergences cannot be understood in terms of cellular cohomology. We introduce in its place the relevant twisted cohomology, and use it to compute the divergence degree of the partition function. We also relate its dominant part to the Reidemeister torsion of the complex, thereby generalizing previous results of Barrett and Naish-Guzman. The main limitation to our approach is the presence of singularities in the representation variety of the fundamental group of the complex; we illustrate this issue in the well-known case of two-dimensional manifolds.

Four Dimensional Superconformal Index fromq-Deformed Two Dimensional Yang-Mills Theory

We show that the superconformal index (the partition function on the three-sphere times a circle) of a certain class of 4D supersymmetric field theories is exactly equal to a partition function of q-deformed nonsupersymmetric 2D Yang-Mills theory.

Quantum ’t Hooft Loops of SYM $${{\mathcal N}=4}$$ as Instantons of YM2 in Dual Groups SU(N) and SU(N)/ZN

A relation between circular 1/2 BPS ’t Hooft operators in 4d \({{\mathcal N}=4}\) SYM and instantonic solutions in 2D Yang-Mills theory (YM2) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM2, belonging to representations of the dual group L G. This conjecture has been tested in the case G = U(N) = L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G = SU(N) and L G = SU(N)/Z N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out.

Correlators of local operators and 1/8 BPS Wilson loops on S 2 from 2d YM and matrix models

We propose that, in N=4 Super Yang-Mills theory, correlation functions of certain 1/8 BPS Wilson loops and local operators inserted on a S^2 in space-time may be computed in terms of analogous observables in the "zero-instanton" sector of 2d Yang-Mills theory. The Wilson loops are mapped to the standard Wilson loops of the 2d theory, as recently conjectured, while the local operators are mapped to powers of the 2d field strength. We give several perturbative checks of the correspondence, and derive from 2d Yang-Mills a two-matrix model for the correlator of a local operator and a Wilson loop of arbitrary shape. We show that the strong coupling planar limit of the two-matrix model precisely agrees with a string theory calculation in AdS_5 x S^5.

Bubble Divergences from Cellular Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory, the Ponzano-Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called `bubble divergences'. A common expectation is that the degree of these divergences is given by the number of `bubbles' of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian -- in both cases, the divergence degree is given by the second Betti number of the 2-complex.

Notes on supersymmetric Wilson loops on a two-sphere

We study a recently discovered family of 1/8-BPS supersymmetric Wilson loops in $ \mathcal{N} = 4 $ super Yang-Mills theory and their string theory duals. The operators are defined for arbitrary contours on a two-sphere in space-time, and they were conjectured to be captured perturbatively by 2d bosonic Yang-Mills theory. In the AdS dual, they are described by pseudo-holomorphic string surfaces living on a certain submanifold of AdS 5 × S 5. We show that the regularized area of these string surfaces is invariant under area preserving diffeomorphisms of the boundary loop, in agreement with the conjecture. Further, we find a connection between the pseudo-holomorphicity equations and an auxiliary σ-model on S 3, which may help to construct new 1/8-BPS string solutions. We also show that the conjectured relation to 2d Yang-Mills implies that a connected correlator of two Wilson loops is computed by a Hermitian Gaussian two-matrix model. On the AdS dual side, we argue that the connected correlator is described by two disconnected disks interacting through the exchange of supergravity modes, and we show that this agrees with the strong coupling planar limit of the two-matrix model.

Can fermions save large N dimensional reduction?

This paper explores whether Eguchi-Kawai reduction for gauge theories with adjoint fermions is valid. The Eguchi-Kawai reduction relates gauge theories in different numbers of dimensions in the large $N$ limit provided that certain conditions are met. In principle, this relation opens up the possibility of learning about the dynamics of 4D gauge theories through techniques only available in lower dimensions. Dimensional reduction can be understood as a special case of large $N$ equivalence between theories related by an orbifold projection. In this work, we focus on the simplest case of dimensional reduction, relating a 4D gauge theory to a 3D gauge theory via an orbifold projection. A necessary condition for the large N equivalence between the 4D and 3D theories to hold is that certain discrete symmetries in the two theories must not be broken spontaneously. In pure 4D Yang-Mills theory, these symmetries break spontaneously as the size of one of the spacetime dimensions shrinks. An analysis of the effect of adjoint fermions on the relevant symmetries of the 4D theory shows that the fermions help stabilize the symmetries. We consider the same problem from the point of view of the lower dimensional 3D theory and find that, surprisingly, adjoint fermions are not generally enough to stabilize the necessary symmetries of the 3D theory. In fact, a rich phase diagram arises, with a complicated pattern of symmetry breaking. We discuss the possible causes and consequences of this finding.

A gauge invariant infrared stabilization of 3D Yang–Mills gauge theories

We demonstrate that the inversion method can be a very useful tool in providing an infrared stabilization of 3D gauge theories, in combination with the mass operator A2 in the Landau gauge. The numerical results will be unambiguous, since the corresponding theory is ultraviolet finite in dimensional regularization, making a renormalization scale or scheme obsolete. The proposed framework is argued to be gauge invariant, by showing that the nonlocal gauge invariant operator A2min, which reduces to A2 in the Landau gauge, could be treated in 3D, in the sense that it is power counting renormalizable in any gauge. As a corollary of our analysis, we are able to identify a whole set of power counting renormalizable nonlocal operators of dimension two.

Pion in the holographic model with 5D Yang-Mills fields

We study the pion in the holographic model of Hirn and Sanz which contains two Yang-Mills fields defined in the background of the sliced AdS space. The infrared boundary conditions imposed on these fields generate the spontaneous breaking of the chiral symmetry down to its vector subgroup. Within the framework of this model, we get an analytic expression for the pion form factor and a compact result for its radius. We also extend the holographic model to include the Chern-Simons term which is required to reproduce the appropriate axial anomaly of QCD. As a result, we calculate the anomalous form factor of the pion and predict its ${Q}^{2}$ slope for the kinematics when one of the photons is almost on shell. We also observe that the anomalous form factor with one real and one virtual photon is given by the same analytic expression as the electromagnetic form factor of a charged pion. One of the advantages of the present model is that it does not require an infrared boundary counterterm to correctly reproduce the anomaly of QCD.

Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System

Given g and f = gg′, we consider solutions to the following non linear wave equation : $$\left\{ \begin{array}{l}\displaystyle u_{tt} - u_{rr} - \frac 1 r u_r = - \frac{f(u)}{r^2},\\(u, u_t)|_{t =0} = (u_0,u_1). \end{array}\right.$$ Under suitable assumptions on g, this equation admits non-constant stationary solutions : we denote Q one with least energy. We characterize completely the behavior as time goes to ±∞ of solutions (u, u t ) corresponding to data with energy less than or equal to the energy of Q : either it is (Q, 0) up to scaling, or it scatters in the energy space. Our results include the cases of the 2 dimensional corotational wave map system, with target $${{\mathbb S}^2}$$ , in the critical energy space, as well as the 4 dimensional, radially symmetric Yang-Mills fields on Minkowski space, in the critical energy space.

Bagger-Lambert theory for general Lie algebras

We construct the totally antisymmetric structure constants f ABCD of a 3-algebra with a Lorentzian bi-invariant metric starting from an arbitrary semi-simple Lie algebra. The structure constants f ABCD can be used to write down a maximally superconformal 3d theory that incorporates the expected degrees of freedom of multiple M2 branes, including the ``center-of-mass mode described by free scalar and fermion fields. The gauge field sector reduces to a three dimensional BF term, which underlies the gauge symmetry of the theory. We comment on the issue of unitarity of the quantum theory, which is problematic, despite the fact that the specific form of the interactions prevent the ghost fields from running in the internal lines of any Feynman diagram. Giving an expectation value to one of the scalar fields leads to the maximally supersymmetric 3d Yang-Mills Lagrangian with the addition of two U(1) multiplets, one of them ghost-like, which is decoupled at large g YM.

TOWARDS (DE)CONSTRUCTING 4D YANG-MILLS THEORY

We investigate 3D Yang-Mills theory coupled to an adjoint scalar, which can be regarded as a zeroth order approximation in (de)constructing 4D Yang-Mills theory. We develop a new algorithm to obtain the renormalized Hamiltonian with the Karabali-Nair variable, by carefully identifying finite local counterterms. We also discuss how this formalism can be applied to obtain glueball spectrum.

Free fermions and thermal AdS/CFT

The dynamics of finite temperature U(N) gauge theories on $S^3$ can be described, at weak coupling, by an effective unitary matrix model. Here we present an exact solution to these models, for any value of $N$, in terms of a sum over representations. Taking the large $N$ limit of this solution provides a new perspective on the deconfinement transition which is supposed to be dual to the Hawking-Page transition. The large $N$ phase transition manifests itself here in a manner similar to the Douglas-Kazakov phase transition in 2d Yang-Mills theory. We carry out a complete analysis of the saddle representation in the simplest case involving only the order parameter ${\rm Tr}U$. We find that the saddle points corresponding to thermal $AdS$, the small black hole and the large black hole can all be described in terms of free fermions. They all admit a simple phase space description {\it a la} the BPS geometries of Lin, Lunin and Maldacena.

Branes, anti-branes and Brauer algebras in gauge-gravity duality

We propose gauge theory operators built using a complex Matrix scalar which are dual to brane-anti-brane systems in $AdS_5 \times S^5 $, in the zero coupling limit of the dual Yang-Mills. The branes involved are half-BPS giant gravitons. The proposed operators dual to giant-anti-giant configurations satisfy the appropriate orthogonality properties. Projection operators in Brauer algebras are used to construct the relevant multi-trace Matrix operators. These are related to the ``coupled representations'' which appear in 2D Yang-Mills theory. We discuss the implications of these results for the quantum mechanics of a complex matrix model, the counting of non-supersymmetric operators and the physics of brane-anti-brane systems. The stringy exclusion principle known from the properties of half-BPS giant gravitons, has a new incarnation in this context. It involves a qualitative change in the map between brane-anti-brane states to gauge theory operators. In the case of a pair of sphere giant and anti-giant this change occurs when the sum of the magnitudes of their angular momenta reaches $N$.