Q-deformed Yang-Mills Theory Research Articles

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.

Phase diagram of q-deformed Yang-Mills theory on S2 at non-zero \u03b8-angle

We study the phase diagram of q-deformed Yang-Mills theory on S2 at non-zero θ-angle using the exact partition function at finite N . By evaluating the exact partition function numerically, we find evidence for the existence of a series of phase transitions at non-zero θ-angle as conjectured in [hep-th/0509004].

On irregular singularity wave functions and superconformal indices

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU(N). As an application, we give closed-form expressions for the Schur indices of all (AN − 1, AN (n − 1)−1) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the (AN, AM) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of “Type IV” AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple “monopole vev RG” formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

Wilson punctured network defects in 2D q-deformed Yang-Mills theory

In the context of class S theories and 4D/2D duality relations there, we discuss the skein relations of general topological defects on the 2D side which are expected to be counterparts of composite surface-line operators in 4D class S theory. Such defects are geometrically interpreted as networks in a three dimensional space. We also propose a conjectural computational procedure for such defects in two dimensional SU(N) topological q-deformed Yang-Mills theory by interpreting it as a statistical mechanical system associated with ideal triangulations.

On the superconformal index of Argyres–Douglas theories

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres–Douglas (AD) superconformal field theories (SCFTs): the and the theories. While these SCFTs can be realized at special points on the Coulomb branch of certain gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the and indices by using a relation to two-dimensional q-deformed Yang–Mills theory and data from the class construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the and SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S1 reduction of our indices in a separate paper.

Self-dual strings and 2D SYM

We study the system of M2-branes suspended between parallel M5-branes using ABJM model with a natural half-BPS boundary condition. For small separation between M5-branes, the worldvolume theory is shown to reduce to a 2D N=(4,4) super Yang-Mills theory with some similarity to q-deformed Yang-Mills theory. The gauge coupling is related to the position of the branes in an interesting manner. The theory is considerably different from the 2D theory proposed for multiple "M-strings". We make a detailed comparison of elliptic genus of the two descriptions and find only a partial agreement.

On the 6d origin of discrete additional data of 4d gauge theories

Starting with a choice of gauge algebras, specification of a 4d gauge theory involves additional data, namely the gauge groups and the discrete theta angles. Equivalently, one needs to specify the set of charges of allowed line operators. In this note, we study how these additional data are represented in 6d, when the 4d theory in question is an N=4 super Yang-Mills theory or an N=2 class S theory. We will see that the Z_N symmetry of the so-called T_N theory plays an important role. As a byproduct, we will find that the superconformal index of class S theories can be refined so that it can give 2d q-deformed Yang-Mills theory with different gauge groups associated to the same gauge algebra.

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.

3d partition function as overlap of wavefunctions

We compute the partition function on S^3 of 3d N=4 theories which arise as the low-energy limit of 4d N=4 super Yang-Mills theory on a segment or on a junction, and propose its 1d interpretation. We show that the partition function can be written as an overlap of wavefunctions determined by the boundary conditions. We also comment on the connection of our results with the 4d superconformal index and the 2d q-deformed Yang-Mills theory.

A Note on q-Deformed Two-Dimensional Yang–Mills and Open Topological Strings

In this note we make a test of the open topological string version of the OSV conjecture in the toric Calahi–Yau manifold X = O(–3) → P2 with background D4-hranes wrapped on Lagrangian suhmanifolds. The D-brane partition function reduces to an expectation value of some inserted operators of a q-deformed Yang–Mills theory living on a chain of P1 's in the base P2 of X. At large N this partition function can be written as a sum over squares of chiral blocks, which are related to the open topological string amplitudes in the local P2 geometry with branes at both the outer and inner edges of the toric diagram. This is in agreement with the conjecture.

Chern–Simons matrix models, two-dimensional Yang–Mills theory and the Sutherland model

We derive some new relationships between matrix models of Chern–Simons gauge theory and of two-dimensional Yang–Mills theory. We show that q-integration of the Stieltjes–Wigert matrix model is the discrete matrix model that describes q-deformed Yang–Mills theory on S2. We demonstrate that the semiclassical limit of the Chern–Simons matrix model is equivalent to the Gross–Witten model in the weak-coupling phase. We study the strong-coupling limit of the unitary Chern–Simons matrix model and show that it too induces the Gross–Witten model, but as a first-order deformation of Dyson's circular ensemble. We show that the Sutherland model is intimately related to Chern–Simons gauge theory on S3, and hence to q-deformed Yang–Mills theory on S2. In particular, the ground-state wavefunction of the Sutherland model in its classical equilibrium configuration describes the Chern–Simons free energy. The correspondence is extended to Wilson line observables and to arbitrary simply laced gauge groups.

Black holes, instanton counting on toric singularities and q-deformed two-dimensional Yang–Mills theory

We study the relationship between instanton counting in N = 4 Yang–Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang–Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang–Mills theory and Chern–Simons theory on generic Lens spaces, and use it to show that the known instanton counting is only reproduced when the Chern–Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.

Branes, black holes and topological strings on toric Calabi-Yau manifolds

We develop means of computing exact degerenacies of BPS black holes on toric Calabi-Yau manifolds. We show that the gauge theory on the D4 branes wrapping ample divisors reduces to 2D q-deformed Yang-Mills theory on necklaces of P^1's. As explicit examples we consider local P^2, P^1 x P^1 and A_k type ALE space times C. At large N the D-brane partition function factorizes as a sum over squares of chiral blocks, the leading one of which is the topological closed string amplitude on the Calabi-Yau. This is in complete agreement with the recent conjecture of Ooguri, Strominger and Vafa.

Chern-Simons theory onS1-bundles: abelianisation and q-deformed Yang-Mills theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.

Black-holes, topological strings and large N phase transitions

The counting of microstates of BPS black-holes on local Calabi-Yau of the form \U0001d4aa(p−2)⊕\U0001d4aa(−p) → S2 is explored by computing the partition function of q-deformed Yang-Mills theory on S2. We obtain, at finite N, the instanton expansion of the gauge theory. It can be written exactly as the partition function for U(N) Chern-Simons gauge theory on a Lens space, summed over all non-trivial vacua, plus a tower of non-perturbative instanton contributions. In the large N limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces to the trivial sector and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point, instantons are enhanced and the theory undergoes a phase transition into a strong coupling regime. The transition from the strong coupling phase to the weak coupling phase is of third order.

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

We continue our study of the large N phase transition in q-deformed Yang-Mills theory on the sphere and its role in connecting topological strings to black hole entropy. We study in detail the chiral theory defined in terms of uncoupled single U(N) representations at large N and write down the resulting partition function by means of the topological vertex. The emergent toric geometry has three Kaehler parameters, one of which corresponds to the expected fibration over the sphere. By taking a suitable double-scaling limit we recover the chiral Gross-Taylor string expansion. To analyse the phase transition we construct a matrix model which describes the chiral gauge theory. It has three distinct phases, one of which should be described by the closed topological string expansion. We verify this expectation by explicit comparison between the matrix model and the chiral topological string free energies. We also show that the critical point in the pertinent phase of the matrix model corresponds to a divergence of the topological string perturbation series.

Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory

We examine the problem of counting bound states of BPS black holes on local Calabi-Yau threefolds which are fibrations over a Riemann surface by computing the partition function of q-deformed Yang-Mills theory on the Riemann surface. We study in detail the genus zero case and obtain, at finite N, the instanton expansion of the gauge theory. It can be written exactly as the partition function for U(N) Chern-Simons gauge theory on a Lens space, summed over all non-trivial vacua, plus a tower of non-perturbative instanton contributions. The correspondence between two and three dimensional gauge theories is elucidated by an explicit mapping between two-dimensional Yang-Mills instantons and flat connections on the Lens space. In the large N limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces exactly to the trivial flat connection sector with instanton contributions exponentially suppressed, and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point all non-trivial vacua contribute, instantons are enhanced and the theory appears to undergo a phase transition into a strong coupling regime. We rederive these results by performing a saddle-point approximation to the exact partition function. We obtain a q-deformed version of the Douglas-Kazakov equation for two-dimensional Yang-Mills theory on the sphere, whose one-cut solution below the transition point reproduces the resolved conifold geometry. Above the critical point we propose a two-cut solution that should reproduce the chiral-antichiral dynamics found for black holes on the Calabi-Yau threefold and the Gross-Taylor string in the undeformed limit. The transition from the strong coupling phase to the weak coupling phase appears to be of third order.

Black holes, q-deformed 2d Yang–Mills, and non-perturbative topological strings

We count the number of bound states of BPS black holes on local Calabi–Yau three-folds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a q-deformed Yang–Mills theory on the Riemann surface. Following the recent connection between the black hole entropy and the topological string partition function, we find that for a large black hole charge N, up to corrections of O ( e − N ) , Z BH is given as a sum of a square of chiral blocks, each of which corresponds to a specific D-brane amplitude. The leading chiral block, the vacuum block, corresponds to the closed topological string amplitudes. The subleading chiral blocks involve topological string amplitudes with D-brane insertions at ( 2 g − 2 ) points on the Riemann surface analogous to the Ω points in the large N 2d Yang–Mills theory. The finite N amplitude provides a non-perturbative definition of topological strings in these backgrounds. This also leads to a novel non-perturbative formulation of c = 1 non-critical string at the self-dual radius.

Theory of q-deformed forms. III. q-deformed Hodge star, inner product, adjoint operator of exterior derivative, and self-dual yang-mills equation

In this paper we introduce the q-deformed Hodge star operator, q-deformed inner product, and q-deformed adjoint of the q-deformed exterior derivative and investigate their properties. Using this mathematical background, we construct the q-deformed self-dual Yang-Mills theory.

Q-trace for quantum groups and q-deformed Yang-Mills theory

The definitions of orbits and q-trace for the quantum groups are introduced. Then the q-trace is used to construct the invariants for the quantum group orbits and to formulate the q-deformed Yang-Mills theory. The amusing formal relation of the Weinberg type mixing angle with the quantum group deformation parameter is discussed.