Hooft Flux Research Articles

Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism.

Magnetic excitations play a crucial role in understanding the color confinement of 4D Yang-Mills theory, and we have the monopole and the center vortex as plausible candidates to explain its mechanism. Under suitable compactified setups of 4D Yang-Mills theory, we can achieve different weakly coupled descriptions of confinement phenomena: The monopole mechanism takes place on R^{3}×S^{1} with the double-trace deformation, and the center-vortex mechanism is effective on R^{2}×T^{2} with the 't Hooft flux. We unify these two semiclassical descriptions by showing the explicit relation between the monopole and center vortex.

Semiclassics for the QCD vacuum structure through T2-compactification with the baryon-’t Hooft flux

We study QCD vacuum structure with the topological θ angle using a recently proposed semiclassical approach on ℝ2 × T2 with the ’t Hooft and baryon magnetic fluxes. Under the assumption of adiabatic continuity in this setup, the confining vacuum can be described by the dilute gas of center vortices. With this semiclassical approach, we derive the 2d effective description at small T2 and successfully explain the reasonable theta dependence of the QCD vacuum: in the one-flavor QCD at θ = π, the CP symmetry is spontaneously broken for quark mass above a critical value and restored for a subcritical mass, while the CP symmetry is always spontaneously broken in the multi-flavor QCD at θ = π. From our semiclassical description, we discuss implications to the 4d chiral Lagrangian and propose how the η′ meson should be incorporated in consistent with known global structures: the periodicity of the η′ should be extended from the naive one 2π to 2πN. Additionally, we revisit the phase diagram of Nf = 1 + 1 and Nf = 1 + 1 + 1 QCD on the up and down quark mass plane, confirming and refining the existence of the CP-broken Dashen phase.

Non-supersymmetric duality cascade of QCD(BF) via semiclassics on ℝ2× T2 with the baryon-’t Hooft flux

We study the phase diagrams of the bifundamental QCD (QCD(BF)) of different ranks, which is the 4d SU(N1) × SU(N2) gauge theory coupled with a bifundamental Dirac fermion. After discussing the anomaly constraints on possible vacuum structures, we apply a novel semiclassical approach on ℝ2× T2 with the baryon-’t Hooft flux to obtain the concrete dynamics. The 2d effective theory is derived by the dilute gas approximation of center vortices, and it serves as the basis for determining the phase diagram of the model under the assumption of adiabatic continuity. As an application, we justify the non-supersymmetric duality cascade between different QCD(BF), which has been conjectured in the large-N argument. Combined with the semiclassics and the large-N1,2 limit, we construct the explicit duality map from the parent theory, SU(N1) × SU(N2) QCD(BF), to the daughter theory, SU(N1) × SU(N2− N1) QCD(BF), including the correspondence of the coupling constants. We numerically examine the validity of the duality also for finite N1,2 within our semiclassics, finding a remarkable agreement of the phase diagrams between the parent and daughter sides.

Fractional topological charge in lattice Abelian gauge theory

AbstractWe construct a non-trivial $U(1)/\mathbb {Z}_q$ principal bundle on T4 from the compact U(1) lattice gauge field by generalizing Lüscher’s constriction so that the cocycle condition contains $\mathbb {Z}_q$ elements (the ’t Hooft flux). The construction requires an admissibility condition on lattice gauge field configurations. From the transition function so constructed, we have the fractional topological charge that is $\mathbb {Z}_q$ one-form gauge invariant and odd under the lattice time reversal transformation. Assuming a rescaling of the vacuum angle θ → qθ suggested from the Witten effect, our construction provides a lattice implementation of the mixed ’t Hooft anomaly between the $\mathbb {Z}_q$ one-form symmetry and the time reversal symmetry in the U(1) gauge theory with matter fields of charge $q\in 2\mathbb {Z}$ when θ = π, which was studied by Honda and Tanizaki [J. High Energy Phys. 12, 154 (2020)] in the continuum framework.

Polyakov model in ’t Hooft flux background: a quantum mechanical reduction with memory

We construct a compactification of Polyakov model on T2 × ℝ down to quantum mechanics which remembers non-perturbative aspects of field theory even at an arbitrarily small area. Standard compactification on small T2 × ℝ possesses a unique perturbative vacuum (zero magnetic flux state), separated parametrically from higher flux states, and the instanton effects do not survive in the Born-Oppenheimer approximation. By turning on a background magnetic GNO flux in co-weight lattice corresponding to a non-zero ’t Hooft flux, we show that N-degenerate vacua appear at small torus, and there are N − 1 types of flux changing instantons between them. We construct QM instantons starting with QFT instantons using the method of replicas. For example, SU(2) gauge theory with flux reduces to the double-well potential where each well is a fractional flux state. Despite the absence of a mixed anomaly, the vacuum structure of QFT and the one of QM are continuously connected. We also compare the quantum mechanical reduction of the Polyakov model with the deformed Yang-Mills, by coupling both theories to TQFTs. In particular, we compare the mass spectrum for dual photons and energy spectrum in the QM limit. We give a detailed description of critical points at infinity in the semi-classical expansion, and their role in resurgence structure.

Semiclassics with ’t Hooft flux background for QCD with 2-index quarks

We study quantum chromodynamics including the two-index symmetric or anti-symmetric quark (QCD(Sym/ASym)) on small ℝ2× T2 with a suitable magnetic flux. We first discuss the ’t Hooft anomaly of these theories and claim that discrete chiral symmetry should be spontaneously broken completely to satisfy the anomaly matching condition. The T2 compactification with the magnetic flux preserves the ’t Hooft anomaly, and the 2d effective theory is constrained by the same anomaly of 4d QCD(Sym/ASym). We demonstrate the spontaneous breakdown of chiral symmetry using the dilute gas of center vortices, which confirms the prediction of the ’t Hooft anomaly. We also find that each vacuum maintains the charge conjugation symmetry, and this gives affirmative support for the nonperturbative large-N orientifold equivalence between QCD(Sym/ASym) and mathcal{N} = 1 supersymmetric SU(N) Yang-Mills theory.

Cluster Expansion and Resurgence in the Polyakov Model.

In the Polyakov model, a nonperturbative mass gap is formed at leading-order semiclassics by instanton effects. By using the notions of critical points at infinity, cluster expansion, and Lefschetz thimbles, we show that a third-order effect in semiclassics gives an imaginary ambiguous contribution to the mass gap, which is supposed to be real and unambiguous. This is troublesome for the original analysis, and it is difficult to resolve this issue directly in quantum field theory (QFT). However, we find a new compactification of the Polyakov model to quantum mechanics, by using a background 't Hooft flux. The compactification has the merit of remembering the monopole instantons of the full QFT within Born-Oppenheimer approximation, while the periodic compactification does not. In the quantum mechanical limit, we prove the resurgent cancellation of the ambiguity in three-instanton sector against ambiguity in the Borel resummation of the perturbation theory around one instanton. Assuming that this result holds in QFT, we provide a large-order asymptotics of perturbation theory around perturbative vacuum and instanton.

Center vortex and confinement in Yang–Mills theory and QCD with anomaly-preserving compactifications

Abstract We construct an anomaly-preserving compactification of 4D gauge theories, including SU(N) Yang–Mills theory, $\mathcal {N}=1$ supersymmetric Yang–Mills theory, and quantum chromodynamics (QCD), down to 2D by turning on the ’t Hooft flux through T2. This provides a new framework to analytically calculate nonperturbative properties such as confinement, chiral symmetry breaking, and the multi-branch structure of vacua. We give a semiclassical description of these phenomena based on the center vortex and show that it enjoys the same anomaly-matching condition as the original 4D gauge theory. We conjecture that the weak-coupling vacuum structure on small $T^2 \times \mathbb {R}^2$ is adiabatically connected to the strong-coupling regime on $\mathbb {R}^4$ without any phase transitions. In QCD with fundamental quarks as well, we can turn on the ’t Hooft flux either by activating the SU(Nf)V symmetry twist for Nf = N flavors or by introducing a magnetic flux of baryon number U(1)B for arbitrary Nf flavors. In both cases, the weak-coupling center-vortex theory gives a prediction consistent with the chiral Lagrangian of 4D QCD.

Strongly coupled QFT dynamics via TQFT coupling

We consider a class of quantum field theories and quantum mechanics, which we couple to ℤN topological QFTs, in order to classify non-perturbative effects in the original theory. The ℤN TQFT structure arises naturally from turning on a classical background field for a ℤN 0- or 1-form global symmetry. In SU(N) Yang-Mills theory coupled to ℤN TQFT, the non-perturbative expansion parameter is exp[−SI/N] = exp[−8π2/g2N] both in the semi-classical weak coupling domain and strong coupling domain, corresponding to a fractional topological charge configurations. To classify the non-perturbative effects in original SU(N) theory, we must use PSU(N) bundle and lift configurations (critical points at infinity) for which there is no obstruction back to SU(N). These provide a refinement of instanton sums: integer topological charge, but crucially fractional action configurations contribute, providing a TQFT protected generalization of resurgent semi-classical expansion to strong coupling. Monopole-instantons (or fractional instantons) on T3 × {S}_L^1 can be interpreted as tunneling events in the ’t Hooft flux background in the PSU(N) bundle. The construction provides a new perspective to the strong coupling regime of QFTs and resolves a number of old standing issues, especially, fixes the conflicts between the large-N and instanton analysis. We derive the mass gap at θ = 0 and gaplessness at θ = π in mathbbm{CP} 1 model, and mass gap for arbitrary θ in mathbbm{CP} N−1, N ≥ 3 on ℝ2.

Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD

We show that 22d adjoint QCD, an SU(N)SU(N) gauge theory with one massless adjoint Majorana fermion, has a variety of mixed ’t Hooft anomalies. The anomalies are derived using a recent mod 22 index theorem and its generalization that incorporates ’t Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of NN, are matched by spontaneous chiral symmetry breaking. We find that massless 22d adjoint QCD confines for N >2N>2, except for test charges of NN-ality N/2N/2, which are deconfined. In other words, \mathbb Z_NℤN center symmetry is unbroken for odd NN and spontaneously broken to \mathbb Z_{N/2}ℤN/2 for even NN. All of these results are confirmed by explicit calculations on small \mathbb{R}\times S^1ℝ×S1. We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when NN is even. Furthermore, for most values of NN, 22d massive adjoint QCD describes a non-trivial symmetry-protected topological (SPT) phase of matter, including certain cases where the number of interacting Majorana fermions is a multiple of 88. As a result, it fits into the classification of (1+1)(1+1)d SPT phases of interacting Majorana fermions in an interesting way.

Equivariant Verlinde Algebra from Superconformal Index and Argyres–Seiberg Duality

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory $T[\Sigma,G]$ on $L(k,1) \times S^1$, the other is the $^LG$ "equivariant Verlinde formula", or equivalently partition function of $^LG_{\mathbb{C}}$ complex Chern-Simons theory on $\Sigma\times S^1$. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally $G$ and its Langlands dual $^LG$. When $G$ is not simply-connected, we provide a recipe of computing the index of $T[\Sigma,G]$ as summation over indices of $T[\Sigma,\tilde{G}]$ with non-trivial background 't Hooft fluxes, where $\tilde{G}$ is the simply-connected group with the same Lie algebra. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for $G=SU(2)$ or $SO(3)$. In the end, as an application of this newly found relation, we consider the more general case where $G$ is $SU(N)$ or $PSU(N)$ and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the $SU(3)$ equivariant Verlinde coefficients.

Quantum vacua of 2d maximally supersymmetric Yang-Mills theory

We analyze the classical and quantum vacua of 2d mathcal{N}=left(8,8right) supersymmetric Yang-Mills theory with SU(N) and U(N) gauge group, describing the worldvolume interactions of N parallel D1-branes with flat transverse directions {mathrm{mathbb{R}}}^8 . We claim that the IR limit of the SU(N) theory in the superselection sector labeled M (mod N) — identified with the internal dynamics of (M, N)-string bound states of the Type IIB string theory — is described by the symmetric orbifold mathcal{N}=left(8,8right) sigma model into {left({mathrm{mathbb{R}}}^8right)}^{D-1}/{mathbb{S}}_D when D = gcd(M, N) > 1, and by a single massive vacuum when D = 1, generalizing the conjectures of E. Witten and others. The full worldvolume theory of the D1-branes is the U(N) theory with an additional U(1) 2-form gauge field B coming from the string theory Kalb-Ramond field. This U(N) + B theory has generalized field configurations, labeled by the mathrm{mathbb{Z}} -valued generalized electric flux and an independent {mathrm{mathbb{Z}}}_N -valued ’t Hooft flux. We argue that in the quantum mechanical theory, the (M, N)-string sector with M units of electric flux has a {mathrm{mathbb{Z}}}_N -valued discrete θ angle specified by M (mod N) dual to the ’t Hooft flux. Adding the brane center-of-mass degrees of freedom to the SU(N) theory, we claim that the IR limit of the U(N) + B theory in the sector with M bound F-strings is described by the mathcal{N}=left(8,8right) sigma model into {mathrm{Sym}}^Dleft({mathrm{mathbb{R}}}^8right) . We provide strong evidence for these claims by computing an mathcal{N}=left(8,8right) analog of the elliptic genus of the UV gauge theories and of their conjectured IR limit sigma models, and showing they agree. Agreement is established by noting that the elliptic genera are modular-invariant Abelian (multi-periodic and meromorphic) functions, which turns out to be very restrictive.

Ground states of supersymmetric Yang-Mills-Chern-Simons theory

We consider minimally supersymmetric Yang-Mills theory with a Chern-Simons term on a flat spatial two-torus. The Witten index may be computed in the weak coupling limit, where the ground state wave-functions localize on the moduli space of flat gauge connections. We perform such computations by considering this moduli space as an orbifold of a certain flat complex torus. Our results agree with those obtained previously by instead considering the moduli space as a complex projective space. An advantage of the present method is that it allows for a more straightforward determination of the discrete electric 't Hooft fluxes of the ground states in theories with non-simply connected gauge groups. A consistency check is provided by the invariance of the results under the mapping class group of a (Euclidean) three-torus.

BPS partition functions in N = 4 Yang-Mills theory on T 4

We consider N = 4 Yang-Mills theory on a flat four-torus with the R-symmetry current coupled to a flat background connection. The partition function depends on the coupling constant of the theory, but when it is expanded in a power series in the R-symmetry connection around the loci at which one of the supersymmetries is unbroken, the constant and linear terms are in fact independent of the coupling constant and can be computed at weak coupling for all non-trivial 't Hooft fluxes. The case of a trivial 't Hooft flux is difficult because of infrared problems, but the corresponding terms in the partition function are uniquely determined by S-duality.

One-loop effective potential inM4×T2with and without ’t Hooft flux

We review the basic notions of compactification in the presence of a background flux. In extra-dimentional models with more than five dimensions, Scherk and Schwarz boundary conditions have to satisfy 't Hooft consistency conditions. Different vacuum configurations can be obtained, depending whether trivial or non-trivial 't Hooft flux is considered. The presence of the "magnetic" background flux provide, in addition, a mechanism for producing four-dimensional chiral fermions. Particularizing to the six-dimensional case, we calculate the one-loop effective potential for a U(N) gauge theory on M4 x T2. We firstly review the well known results of the trivial 't Hooft flux case, where one-loop contributions produce the usual Hosotani dynamical symmetry breaking. Finally we applied our result for describing, for the first time, the one-loop contributions in the non-trivial 't Hooft flux case.

Automorphic properties of (2, 0) theory on T 6

We consider ADE-type (2, 0) theory on a family of flat six-tori endowed with flat Sp(4) connections coupled to the R-symmetry. Our main objects of interest are the components of the `partition vector' of the theory. These constitute an element of a certain finite dimensional vector space, carrying an irreducible representation of a discrete Heisenberg group related to the 't Hooft fluxes of the theory. Covariance under the SL_6(Z) mapping class group of a six-torus amounts to a certain automorphic transformation law for the partition vector, which we derive. Because of the absence of a Lagrangian formulation of (2, 0) theory, this transformation property is not manifest, and gives useful non-trivial constraints on the partition vector. As an application, we derive a shifted quantization law for the spatial momentum of (2, 0) theory on a space-time of the form R x T5. This quantization law is in agreement with an earlier result based on the relationship between (2, 0) theory and maximally supersymmetric Yang-Mills theory together with certain geometric facts about gauge bundles.

Comments on orientifolds without vector structure

We revisit type I compactifications with a Spin(32)/Z2 gauge bundle that admits no vector structure. We elucidate the relation of this Z2 obstruction to discrete B- field flux and to 't Hooft flux and clarify some subtleties in the T-duality transformation to type IIA intersecting D-brane models. We reexamine the earliest 3-generation GUT model on magnetized D-branes and show its consistency when a discrete B-flux is switched on. We further generalize partially known results for toroidal models to type I compactifications without vector structure and their mirror dual type IIA orientifolds on genuine Calabi-Yau manifolds. We illustrate this by working out the example of the quintic in some detail.

Zero-energy states of 𝒩 = 4 SYM onT3:S-duality and the mapping class group

We continue our studies of the low-energy spectrum of N=4 super-Yang-Mills theory on a spatial three-torus. In two previous papers, we computed the spectrum of normalizable zero-energy states for all choices of gauge group and all values of the electric and magnetic 't Hooft fluxes, and checked its invariance under the SL_2(Z) S-duality group. In this paper, we refine the analysis by also decomposing the space of bound states into irreducible unitary representations of the SL_3(Z) mapping class group of the three-torus. We perform a detailed study of the S-dual pairs of theories with gauge groups Spin(2n+1) and Sp(2n). The predictions of S-duality (which commutes with the mapping class group) are fulfilled as expected, but the proof requires some surprisingly intricate combinatorial infinite product identities.

A new class of rank breaking orbifolds

We describe field-theory T 2 / Z n orbifolds that offer new ways of breaking SU ( N ) to lower rank subgroups. We introduce a novel way of embedding the point group into the gauge group, beyond the usual mapping of torus and root lattices. For this mechanism to work the torus Wilson lines must carry nontrivial 't Hooft flux. The rank lowering mechanism proceeds by inner automorphisms but is not related to continuous Wilson lines and does not give rise to any associated moduli. We give a complete classification of all possible SU ( N ) breaking patterns. We also show that the case of general gauge group can already be understood entirely in terms of the SU ( N ) case and the knowledge of standard orbifold constructions with vanishing 't Hooft flux.

Bound states in Script N = 4 SYM onT3: Spin(2n) and the exceptional groups

The low energy spectrum of (3+1)-dimensional N = 4 supersymmetric Yang-Mills theory on a spatial three-torus contains a certain number of bound states, characterized by their discrete abelian magnetic and elecric 't Hooft fluxes. At weak coupling, the wave-functions of these states are supported near points in the moduli space of flat connections where the unbroken gauge group is semi-simple. The number of such states is related to the number of normalizable bound states at threshold in the super symmetric matrix quantum mechanics wit h 1 6 supercharges based on this unbroken group. Mathematic ally, the determination of the spectrum relies on the classification of almost commuting triples with semi-simple centralizers. We complete the work begun in a previous paper, by computing the spectrum of bound states in theories based on the even-dimensional spin groups and the exceptional groups. The results satisfy the constraints of S-duality in a rather non-trivial way.