Two-dimensional Yang-Mills Theory Research Articles

The phase diagram of Toverline{T} -deformed Yang-Mills theory on the sphere

We study the large-N dynamics of Toverline{T} -deformed two-dimensional Yang-Mills theory at genus zero. The 1/N-expansion of the free energy is obtained by exploiting the associated flow equation and the complete phase diagram of the theory is derived for both signs of the rescaled deformation parameter τ. We observe a third-order phase transition driven by instanton condensation, which is the deformed version of the familiar Douglas-Kazakov transition separating the weakly-coupled from the strongly-coupled phase. By studying these phases, we compute the deformation of both the perturbative sector and the Gross-Taylor string expansion. Nonperturbative corrections in τ drive the system into an unexplored disordered phase separated by a novel critical line meeting tangentially the Douglas-Kazakov one at a tricritical point. The associated phase transition is induced by the collision of large-N saddle points, determining its second-order character.

Exact Toverline{T} deformation of two-dimensional Yang-Mills theory on the sphere

We study the Toverline{T} deformation of two-dimensional Yang-Mills theory at genus zero by carrying out the analysis at the level of its instanton representation. We first focus on the perturbative sector by considering its power expansion in the deformation parameter μ. By studying the resulting asymptotic series through resurgence theory, we determine the nonperturbative contributions that enter the result for μ < 0. We then extend this analysis to any flux sector by solving the relevant flow equation. Specifically, we impose boundary conditions corresponding to two distinct regimes: the quantum undeformed theory and the semiclassical limit of the deformed theory. The full partition function is obtained as a sum over all magnetic fluxes. For any μ > 0, only a finite portion of the quantum spectrum survives and the partition function reduces to a sum over a finite set of representations. For μ < 0, nonperturbative contributions regularize the partition function through an intriguing mechanism that generates nontrivial subtractions.

Symmetry defects and orbifolds of two-dimensional Yang–Mills theory

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

Tensor network approach to two-dimensional Yang–Mills theories

Abstract We propose a novel tensor network representation for two-dimensional Yang–Mills theories with arbitrary compact gauge groups. In this method, tensor indices are given directly by group elements with no direct use of the character expansion. We apply the tensor renormalization group method to this tensor network for SU(2) and SU(3), and find that the free energy density and the energy density are accurately evaluated. We also show that the singular value decomposition of a tensor has a group-theoretic structure and can be associated with the character expansion.

From mathcal{N} = 4 Super-Yang-Mills on \u211d\u21194 to bosonic Yang-Mills on \u211d\u21192

We study the four-dimensional mathcal{N} = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge mathcal{Q} , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ2 submanifold. This paves the way for understanding mathcal{N} = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.

Toverline{T} -deformation of q-Yang-Mills theory

We derive the Toverline{T} -perturbed version of two-dimensional q-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the Toverline{T} -deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large N factorization into chiral and anti-chiral sectors. For the U(N) gauge theory on the sphere, we show that the large N phase transition persists, and that it is of third order and induced by instantons. The effect of the Toverline{T} -deformation is to decrease the critical value of the ’t Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for (q, t)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large N limit of Yang-Mills theory, showing that the Toverline{T} -deformation decreases the contribution of the Boltzmann entropy.

Five-dimensional cohomological localization and squashed q-deformations of two-dimensional Yang-Mills theory

We revisit the duality between five-dimensional supersymmetric gauge theories and deformations of two-dimensional Yang-Mills theory from a new perspective. We give a unified treatment of supersymmetric gauge theories in three and five dimensions using cohomological localization techniques and the Atiyah-Singer index theorem. We survey various known results in a unified framework and provide simplified derivations of localization formulas, as well as various extensions including the case of irregular Seifert fibrations. We describe the reductions to four-dimensional gauge theories, and give an extensive description of the dual two-dimensional Yang-Mills theory when the three-dimensional part of the geometry is a squashed three-sphere, including its extension to non-zero area, and a detailed analysis of the resulting matrix model. The squashing parameter b yields a further deformation of the usual q-deformation of two-dimensional Yang-Mills theory, which for rational values b2 = p/s yields a new correspondence with Chern-Simons theory on lens spaces L(p, s).

Entanglement entropy and the large N expansion of two-dimensional Yang-Mills theory

Two-dimensional Yang-Mills theory is a useful model of an exactly solvable gauge theory with a string theory dual at large N. We calculate entanglement entropy in the 1/N expansion by mapping the theory to a system of N fermions interacting via a repulsive entropic force. The entropy is a sum of two terms: the “Boltzmann entropy,” log dim(R) per point of the entangling surface, which counts the number of distinct microstates, and the “Shannon entropy,” − Σ pR log pR, which captures fluctuations of the macroscopic state. We find that the entropy scales as N2 in the large N limit, and that at this order only the Boltzmann entropy contributes. We further show that the Shannon entropy scales linearly with N, and confirm this behaviour with numerical simulations. While the term of order N is surprising from the point of view of the string dual — in which only even powers of N appear in the partition function — we trace it to a breakdown of large N counting caused by the replica trick. This mechanism could lead to corrections to holographic entanglement entropy larger than expected from semiclassical field theory.

Entanglement branes, modular flow, and extended topological quantum field theory

Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the “entanglement brane” we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular ow. As a byproduct we find that the negativity distinguishes between the “log dim R” edge term and the “Shannon” edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.

Higgs and Coulomb branch descriptions of the volume of the vortex moduli space

BPS vortex systems on closed Riemann surfaces with arbitrary genus are embedded into two-dimensional supersymmetric Yang-Mills theory with matters. We turn on a background R-gauge fields to keep half of rigid supersymmetry (topological A-twist) on the curved space. We consider two complementary descriptions; Higgs and Coulomb branches. The path integral reduces to the zero mode integral by the localization in the Higgs branch. The integral over the bosonic zero modes directly gives an integral over the volume form of the moduli space, whereas the fermionic zero modes are compensated by an appropriate operator insertion. In the Coulomb branch description with the same operator insertion, the path integral reduces to a finite-dimensional residue integral. The operator insertion automatically determines a choice of integral contours, leading to the Jeffrey-Kirwan residue formula. This result ensures the existence of the solution to the BPS vortex equation and explains the Bradlow bounds of the BPS vortex. We also discuss a generating function of the volume of the vortex moduli space and show a reduction of the moduli space from semi-local to local vortices.

Two-Dimensional Yang–Mills Theory on Surfaces with Corners in Batalin–Vilkovisky Formalism

In this paper we recover the non-perturbative partition function of 2D~Yang-Mills theory from the perturbative path integral. To achieve this goal, we study the perturbative path integral quantization for 2D~Yang-Mills theory on surfaces with boundaries and corners in the Batalin-Vilkovisky formalism (or, more precisely, in its adaptation to the setting with boundaries, compatible with gluing and cutting -- the BV-BFV formalism). We prove that cutting a surface (e.g. a closed one) into simple enough pieces -- building blocks -- and choosing a convenient gauge-fixing on the pieces, and assembling back the partition function on the surface, one recovers the known non-perturbative answers for 2D~Yang-Mills theory.

The Large-N Limit for Two-Dimensional Yang–Mills Theory

The analysis of the large-$N$ limit of $U(N)$ Yang-Mills theory on a surface proceeds in two stages: the analysis of the Wilson loop functional for a simple closed curve and the reduction of more general loops to a simple closed curve. In the case of the 2-sphere, the first stage has been treated rigorously in recent work of Dahlqvist and Norris, which shows that the large-$N$ limit of the Wilson loop functional for a simple closed curve in $S^{2}$ exists and that the associated variance goes to zero. We give a rigorous treatment of the second stage of analysis in the case of the 2-sphere. Dahlqvist and Norris independently performed such an analysis, using a similar but not identical method. Specifically, we establish the existence of the limit and the vanishing of the variance for arbitrary loops with (a finite number of) simple crossings. The proof is based on the Makeenko-Migdal equation for the Yang-Mills measure on surfaces, as established rigorously by Driver, Gabriel, Hall, and Kemp, together with an explicit procedure for reducing a general loop in $S^{2}$ to a simple closed curve. The methods used here also give a new proof of these results in the plane case, as a variant of the methods used by L\'{e}vy. We also consider loops on an arbitrary surface $\Sigma$. We put forth two natural conjectures about the behavior of Wilson loop functionals for topologically trivial simple closed curves in $\Sigma.$ Under the weaker of the conjectures, we establish the existence of the limit and the vanishing of the variance for topologically trivial loops with simple crossings that satisfy a "smallness" assumption. Under the stronger of the conjectures, we establish the same result without the smallness assumption.

A new stochastic computing paradigm for nonlinear Painlevé II systems in applications of random matrix theory

The aim of the present work is to investigate the stochastic numerical solutions of nonlinear Painleve II systems arising from studies of two-dimensional Yang-Mills theory, growth processes through fluctuation formulas in statistical physics, soft-edge random matrix distributions using the strength of bio-inspired heuristics through artificial neural networks (ANNs), genetic algorithm (GA)-based evolutionary computations and interior-point techniques (IPTs). A new mathematical modelling of the system is formulated through ANNs by defining an error function that exactly satisfies the initial conditions. The weights of ANN models optimized through a memetic computing approach that is based on a global search with GAs, and IPTs are used for an efficient local search. The designed scheme is substantiated through comparative analysis with a fully explicit Range-Kutta numerical procedure on nonlinear Painleve II systems by taking different magnitudes of forcing factors. The accuracy and convergence of the proposed scheme are validated through statistics performed on large numbers of simulations.

Testing the criterion for correct convergence in the complex Langevin method

Recently the complex Langevin method (CLM) has been attracting attention as a solution to the sign problem, which occurs in Monte Carlo calculations when the effective Boltzmann weight is not real positive. An undesirable feature of the method, however, was that it can happen in some parameter regions that the method yields wrong results even if the Langevin process reaches equilibrium without any problem. In our previous work, we proposed a practical criterion for correct convergence based on the probability distribution of the drift term that appears in the complex Langevin equation. Here we demonstrate the usefulness of this criterion in two solvable theories with many dynamical degrees of freedom, i.e., two-dimensional Yang-Mills theory with a complex coupling constant and the chiral Random Matrix Theory for finite density QCD, which were studied by the CLM before. Our criterion can indeed tell the parameter regions in which the CLM gives correct results.

Entanglement branes in a two-dimensional string theory

What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large N . The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the modular Hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an entanglement brane, or E-brane.

Partition function of N=2 supersymmetric gauge theory and two-dimensional Yang-Mills theory

We study four-dimensional $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theory with $2N$ fundamental hypermultiplets in the self-dual $\Omega$-background. The partition function simplifies at special points of the parameter space and is related to the partition function of two-dimensional Yang-Mills theory on $S^{2}$. We also consider the insertion of a Wilson loop operator in two-dimensional Yang-Mills theory and find the corresponding operator in the four-dimensional $\mathcal{N}=2$ gauge theory.

Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory

We derive the analog of the large $N$ Gross-Taylor holomorphic string expansion for the refinement of $q$-deformed $U(N)$ Yang-Mills theory on a compact oriented Riemann surface. The derivation combines Schur-Weyl duality for quantum groups with the Etingof-Kirillov theory of generalized quantum characters which are related to Macdonald polynomials. In the unrefined limit we reproduce the chiral expansion of $q$-deformed Yang-Mills theory derived by de Haro, Ramgoolam and Torrielli. In the classical limit $q=1$, the expansion defines a new $\beta$-deformation of Hurwitz theory wherein the refined partition function is a generating function for certain parameterized Euler characters, which reduce in the unrefined limit $\beta=1$ to the orbifold Euler characteristics of Hurwitz spaces of holomorphic maps. We discuss the geometrical meaning of our expansions in relation to quantum spectral curves and $\beta$-ensembles of matrix models arising in refined topological string theory.

New phase transitions in Chern–Simons matter theory

Applying the machinery of random matrix theory and Toeplitz determinants we study the level k, U(N) Chern–Simons theory coupled with fundamental matter on S2×S1 at finite temperature T. This theory admits a discrete matrix integral representation, i.e. a unitary discrete matrix model of two-dimensional Yang–Mills theory. In this study, the effective partition function and phase structure of the Chern–Simons matter theory, in a special case with an effective potential namely the Gross–Witten–Wadia potential, are investigated. We obtain an exact expression for the partition function of the Chern–Simons matter theory as a function of k, N, T, for finite values and in the asymptotic regime. In the Gross–Witten–Wadia case, we show that ratio of the Chern–Simons matter partition function and the continuous two-dimensional Yang–Mills partition function, in the asymptotic regime, is the Tracy–Widom distribution. Consequently, using the explicit results for free energy of the theory, new second-order and third-order phase transitions are observed. Depending on the phase, in the asymptotic regime, Chern–Simons matter theory is represented either by a continuous or discrete two-dimensional Yang–Mills theory, separated by a third-order domain wall.

Equation of state of two-dimensional Yang-Mills theory

We study the pressure, $P$, of $\mathrm{SU}(N)$ gauge theory on a two-dimensional torus as a function of area, $A=l/t$. We find a crossover scale that separates the system on a large circle from a system on a small circle at any finite temperature. The crossover scale approaches zero with increasing $N$ and the crossover becomes a first-order transition as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and $l\ensuremath{\rightarrow}0$ with the limiting value of $\frac{2Pl}{(N\ensuremath{-}1)t}$ depending on the fixed value of $Nl$.

Entanglement entropy and nonabelian gauge symmetry

Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang–Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.