Deformed Yang-Mills Theory Research Articles

Perturbative aspects of deformed Yang-Mills theory

Centre-stabilised $SU(N)$ Yang-Mills theories on $\mathbb{R}^3 \times S^1$ are QCD-like theories that can be engineered to remain weakly-coupled at all energy scales by taking the $S^1$ circle length $L$ to be sufficiently small. In this regime, these theories admit effective long-distance descriptions as Abelian $U(1)^{N-1}$ gauge theories on $\mathbb{R}^3$, and semiclassics can be reliably employed to study non-perturbative phenomena such as colour confinement and the generation of mass gaps in an analytical setting. At the perturbative tree level, the long-distance effective theory contains $(N-1)$ free photons with identical gauge couplings $g^2_3 \equiv g^2/L$. Vacuum polarisation effects, from integrating out heavy charged fields, lift this degeneracy to give $\floor{\frac{N}{2}}$ distinct values: $g^2(\frac{2}{L})\lesssim g_{3,\ell}^2 L \lesssim g^2(\frac{2\pi}{NL}) $. In this work, we calculate these corrections to one-loop order in theories where the centre-symmetric vacuum is stabilised by $2\leq n_f \leq 5$ massive adjoint Weyl fermions with masses of order $m_\lambda \sim \frac{2\pi}{NL}$, (also known as "deformed Yang-Mills,") and show that our results agree with those found in previous studies in the $m_\lambda \to 0$ limit. Then, we show that our result has an intuitive interpretation as the running of the coupling in a "lattice momentum" in the context of the non-perturbative "emergent latticised fourth dimension" in the $N\to \infty$, fixed-$NL$ limit.

Gauge theory geography: charting a path between semiclassical islands

We study two semiclassical limits of SU(2) Yang-Mills theory on a spatial torus with a ’t Hooft twist: the “femtouniverse,” where all \U0001d54b3 directions are small, and deformed Yang-Mills theory on \U0001d54b2 × \U0001d54a1, with small \U0001d54a1 and large or infinite \U0001d54b2. Carefully defining the symmetries, we show that the classical ground states, while different, have the same transformation properties under the 1-form center symmetry and parity. We argue that this is behind the identical multi-branch θ-dependent vacuum structure of these theories. We then calculate the one-loop potential for the \U0001d54a1-holonomy in the presence of twists on \U0001d54b2. We use it to study the quantum stability of the semiclassical ground states in gauge theories with massive or massless adjoint fermions on spatial \U0001d54b2 × \U0001d54a1, with a twist in the \U0001d54b2. The results point towards some interesting features worthy of further study.

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.

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.

String tensions in deformed Yang-Mills theory

We study k-strings in deformed Yang-Mills (dYM) with SU(N) gauge group in the semiclassically calculable regime on {mathbb{R}}^3times {mathbb{S}}^1 . Their tensions Tk are computed in two ways: numerically, for 2 ≤ N ≤ 10, and via an analytic approach using a re-summed perturbative expansion. The latter serves both as a consistency check on the numerical results and as a tool to analytically study the large-N limit. We find that dYM k-string ratios Tk/T1 do not obey the well-known sine- or Casimir-scaling laws. Instead, we show that the ratios Tk/T1 are bound above by a square root of Casimir scaling, previously found to hold for stringlike solutions of the MIT Bag Model. The reason behind this similarity is that dYM dynamically realizes, in a theoretically controlled setting, the main model assumptions of the Bag Model. We also compare confining strings in dYM and in other four-dimensional theories with abelian confinement, notably Seiberg-Witten theory, and show that the unbroken {mathbb{Z}}_N center symmetry in dYM leads to different properties of k-strings in the two theories; for example, a “baryon vertex” exists in dYM but not in softly-broken Seiberg-Witten theory. Our results also indicate that, at large values of N, k-strings in dYM do not become free.

Representation dependence of k -strings in pure Yang-Mills theory via supersymmetry

We exploit a conjectured continuity between super Yang-Mills on $\mathbb R^3\times \mathbb S^1$ and pure Yang-Mills to study $k$-strings in the latter theory. As expected, we find that Wilson-loop correlation functions depend on the N-ality of a representation ${\cal R}$ to the leading order. However, the next-to-leading order correction is not universal and is given by the group characters, in the representation ${\cal R}$, of the permutation group. We also study W-bosons in super Yang-Mills and show that they are deconfined on the string worldsheet, and therefore, can change neither the string N-ality nor its tension. This phenomenon mirrors the fact that soft gluons do not screen probe charges with non-zero N-ality in pure Yang-Mills. Finally, we comment on the scaling law of $k$-strings in super Yang-Mills and compare our findings with strings in Seiberg-Witten theory, deformed Yang-Mills theory, and holographic studies that were performed in the 't Hooft large-$N$ limit.

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.

On the global structure of deformed Yang-Mills theory and QCD(adj) on ℝ 3 × S 1 $$ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}^1 $$

Spatial compactification on $$ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}_L^1 $$ at small $$ {\mathbb{S}}^1 $$ -size L often leads to a calculable vacuum structure, where various “topological molecules” are responsible for confinement and the realization of the center and discrete chiral symmetries. Within this semiclassically calculable framework, we study how distinct theories with the same $$ \mathrm{S}\mathrm{U}\left({N}_c\right)/{\mathrm{\mathbb{Z}}}_k $$ gauge group (labeled by “discrete θ-angles”) arise upon gauging of appropriate $$ {\mathrm{\mathbb{Z}}}_k $$ subgroups of the one-form global center symmetry of an SU(N c ) gauge theory. We determine the possible $$ {\mathrm{\mathbb{Z}}}_k $$ actions on the local electric and magnetic effective degrees of freedom, find the ground states, and use domain walls and confining strings to give a physical picture of the vacuum structure of the different $$ \mathrm{S}\mathrm{U}\left({N}_c\right)/{\mathrm{\mathbb{Z}}}_k $$ theories. Some of our results reproduce ones from earlier supersymmetric studies, but most are new and do not invoke supersymmetry. We also study a further finite-temperature compactification to $$ {\mathrm{\mathbb{R}}}^2\times {\mathbb{S}}_{\beta}^1\times {\mathbb{S}}_L^1 $$ . We argue that, in deformed Yang-Mills theory, the effective theory near the deconfinement temperature β c ≫ L exhibits an emergent Kramers-Wannier duality and that it exchanges high- and low-temperature theories with different global structure, sharing features with both the Ising model and S-duality in $$ \mathcal{N}=4 $$ supersymmetric Yang-Mills theory.

Two-dimensional Yang-Mills theory on recursive infinite genus surfaces

The partition function of Euclidean Yang-Mills theory on two dimensional surfaces is given by the Migdal formula. It involves the area and topological characteristics of the surface. We consider this theory on a class of infinite genus surfaces that are constructed recursively. We make use of this recursive structure to compute the partition functions (with or without additional Wilson loops) on such surfaces. Our method also works for the quantum deformed Yang-Mills theory.

Refined Chern-Simons theory and (q, t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit

We study the relationship between refined Chern-Simons theory on lens spaces S^3/Z_p and (q,t)-deformed Yang-Mills theory on the sphere S^2. We derive the instanton partition function of (q,t)-deformed U(N) Yang-Mills theory and describe it explicitly as an analytical continuation of the semi-classical expansion of refined Chern-Simons theory. The derivations are based on a generalization of the Weyl character formula to Macdonald polynomials. The expansion is used to formulate q-generalizations of beta-deformed matrix models for refined Chern-Simons theory, as well as conjectural formulas for the chi_y-genus of the moduli space of U(N) instantons on the surface O(-p)--->P^1 for all p which enumerate black hole microstates in refined topological string theory. We study the large N phase structures of the refined gauge theories, and match them with refined topological string theory on the resolved conifold.

Center-stabilized Yang-Mills theory: Confinement and largeNvolume independence

We examine a double trace deformation of $SU(N)$ Yang-Mills theory which, for large $N$ and large volume, is equivalent to unmodified Yang-Mills theory up to $O(1/{N}^{2})$ corrections. In contrast to the unmodified theory, large $N$ volume independence is valid in the deformed theory down to arbitrarily small volumes. The double trace deformation prevents the spontaneous breaking of center symmetry which would otherwise disrupt large $N$ volume independence in small volumes. For small values of $N$, if the theory is formulated on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ with a sufficiently small compactification size $L$, then an analytic treatment of the nonperturbative dynamics of the deformed theory is possible. In this regime, we show that the deformed Yang-Mills theory has a mass gap and exhibits linear confinement. Increasing the circumference $L$ or number of colors $N$ decreases the separation of scales on which the analytic treatment relies. However, there are no order parameters which distinguish the small and large radius regimes. Consequently, for small $N$ the deformed theory provides a novel example of a locally four-dimensional pure-gauge theory in which one has analytic control over confinement, while for large $N$ it provides a simple fully reduced model for Yang-Mills theory. The construction is easily generalized to QCD and other QCD-like theories.

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.