Hooft Flux Research Articles

Low-energy spectrum of Script N = 4 super-Yang-Mills onT3: flat connections, bound states at threshold, andS-duality

We study (3+1)-dimensional N=4 supersymmetric Yang-Mills theory on a spatial three-torus. The low energy spectrum consists of a number of continua of states of arbitrarily low energies. Although the theory has no mass-gap, it appears that the dimensions and discrete abelian magnetic and electric 't Hooft fluxes of the continua are computable in a semi-classical approximation. The wave-functions of the low-energy states are supported on submanifolds of the moduli space of flat connections, at which various subgroups of the gauge group are left unbroken. The field theory degrees of freedom transverse to such a submanifold are approximated by supersymmetric matrix quantum mechanics with 16 supercharges, based on the semi-simple part of this unbroken group. Conjectures about the number of normalizable bound states at threshold in the latter theory play a crucial role in our analysis. In this way, we compute the low-energy spectra in the cases where the simply connected cover of the gauge group is given by SU(n), Spin(2n+1) or Sp(2n). We then show that the constraints of S-duality are obeyed for unique values of the number of bound states in the matrix quantum mechanics. In the cases based on Spin(2n+1) and Sp(2n), the proof involves surprisingly subtle combinatorial identities, which hint at a rich underlying structure.

Confinement and the center of the gauge group

The question of the role of the center of the gauge group in the phenomenon of confinement in Yang-Mills theory is addressed. The investigation is performed from the most general perspective of considering all possible choices for the gauge symmetry group. In this context, an interesting role is played by G(2) Yang-Mills theory: the simplest pure gauge theory with a trivial center and without 't Hooft flux vortices. Numerical simulations show the presence of a first order finite temperature deconfinement phase transition in G(2) Yang-Mills theory in (3+1) dimensions. Interestingly, the G(2) gauge symmetry can be broken to an SU(3) subgroup by the Higgs mechanism. We investigate the relation between the deconfinement phase transition in G(2) and SU(3) Yang-Mills theories by numerical simulations in the G(2) gauge-Higgs system.

Towards the solution of noncommutative YM2: Morita equivalence and large N-limit

In this paper we shall investigate the possibility of solving U(1) theories on the non-commutative (NC) plane for arbitrary values of $\theta$ by exploiting Morita equivalence. This duality maps the NC U(1) on the two-torus with a rational parameter $\theta$ to the standard U(N) theory in the presence of a 't Hooft flux, whose solution is completely known. Thus, assuming a smooth dependence on $\theta$, we are able to construct a series rational approximants of the original theory, which is finally reached by taking the large $N-$limit at fixed 't Hooft flux. As we shall see, this procedure hides some subletities since the approach of $N$ to infinity is linked to the shrinking of the commutative two-torus to zero-size. The volume of NC torus instead diverges and it provides a natural cut-off for some intermediate steps of our computation. In this limit, we shall compute both the partition function and the correlator of two Wilson lines. A remarkable fact is that the configurations, providing a finite action in this limit, are in correspondence with the non-commutative solitons (fluxons) found independently by Polychronakos and by Gross and Nekrasov, through a direct computation on the plane.

Lattice gauge fields and discrete noncommutative Yang-Mills theory

We present a lattice formulation of noncommutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely nonperturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and noncommutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic noncommutative gauge theories in the continuum can be regularized nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft flux. In the case of irrational noncommutativity parameters, the rank of the gauge group of the commutative lattice theory must be sent to infinity in the continuum limit. As a special case, the construction includes the recent description of noncommutative Yang-Mills theories using twisted large $N$ reduced models. We study the coupling of noncommutative gauge fields to matter fields in the fundamental representation of the gauge group using the lattice formalism. The large mass expansion is used to describe the physical meaning of Wilson loops in noncommutative gauge theories. We also demonstrate Morita equivalence in the presence of fundamental matter fields and use this property to comment on the calculation of the beta-function in noncommutative quantum electrodynamics.

On the hierarchy between non-commutative and ordinary supersymmetric Yang-Mills

Non-commutative supersymmetric Yang-Mills with rational Theta is dual to an ordinary supersymmetric Yang-Mills with a 't Hooft flux. It is believed that the simplest description is via the ordinary supersymmetric Yang-Mills. We claim, however, that the two descriptions form a hierarchy. The SYM description is the proper description in the ultra violet while the non-commutative description takes over in the infra red.

Gauge bundles and Born-Infeld on the non-commutative torus

In this paper, we describe non-abelian gauge bundles with magnetic and electric fluxes on higher-dimensional non-cummutative tori. We give an explicit construction of a large class of bundles with non-zero magnetic 't Hooft fluxes. We discuss Morita equivalence between these bundles. The action of the duality is worked out in detail for the four-torus. As an application, we discuss Born-Infeld on this torus, as a description of compactified string theory. We show that the resulting theory, including the fluctuations, is manifestly invariant under the T-duality group SO(4,4; Z ). The U-duality invariant BPS mass-formula is discussed shortly. We comment on a discrepancy of this result with that of a recent calculation.

The Vafa–Witten theory for gauge group $SU(N)$

We derive the partition function for the Vafa-Witten twist of the $\mathcal{N}=4$ supersymmetric gauge theory with gauge group SU(N) (for prime $N$) and arbitrary values of the 't Hooft fluxes $v\in H^{2}(X,\mathbb{Z}_{N})$ on Kahler four-manifolds with $b^{+}_2>1$.

Duality in twisted [formula omitted] = 4 supersymmetric gauge theories in four dimensions

We consider a twisted version of the four-dimensional N = 4 supersymmetric Yang-Mills theory with gauge groups SU(2) and SO(3) and bare masses for two of its chiral multiplets, thereby breaking N = 4 down to N = 2. Using the wall-crossing technique introduced by Moore and Witten within the u-plane approach to twisted topological field theories, we compute the partition function and all the topological correlation functions for the case of simply connected spin four-manifolds of simple type. By including 't Hooft fluxes, we analyse the properties of the resulting formulae under duality transformations. The partition function transforms in the same way as the one first presented by Vafa and Witten for another twist of the N = 4 supersymmetric theory in their strong coupling test of S-duality. Both partition functions coincide on K3. The topological correlation functions turn out to transform covariantly under duality, following a simple pattern which seems to be inherent in a general type of topological quantum field theories.

SU(N) gauge theories with C-periodic boundary conditions (I). Topological structure

C-periodic boundary conditions are introduced to SU( N) gauge theory on a torus. C-periodic fields are replaced by their charge conjugates when they are shifted over the boundary. As for periodic boundary conditions the most general C-periodic boundary condition includes twist. The topological structure with C-periodic boundary conditions is quite different from the periodic case. In the periodic case twist leads to the ℤ N 't Hooft flux sectors. In the C-periodic case with even N the symmetry of the flux sectors is reduced to ℤ 2. For odd N the flux sectors are eliminated completely. Furthermore, the topological charge is an integer when N is odd, whereas it can be a half-integer when N is even.

Kinematics of periodic structures in SU( N) gauge theories

We study periodicity by use of the holonomy group. For a general field configuration, which spans the whole SU( N) group, we show that periodicity of gauge-invariant quantities implies that either the flux is quantized or that there is no flux at all. Finally we show that in a special gauge, the quantized 't Hooft flux can be expressed as a line integral over one of the components of the vector potential. The other components then have “Higgs-like” zeros, with winding numbers related to the 't Hooft flux.

Topological field theory on a lattice, discrete theta-angles and confinement

We study a topological field theory describing confining phases of gauge theories in four dimensions. It can be formulated on a lattice using a discrete 2-form field talking values in a finite abelian group (the magnetic gauge group). We show that possible theta-angles in such a theory are quantized and labeled by quadratic functions on the magnetic gauge group. When the theta-angles vanish, the theory is dual to an ordinary topological gauge theory, but in general it is not isomorphic to it. We also explain how to couple a lattice Yang-Mills theory to a TQFT of this kind so that the ’t Hooft flux is well-defined, and quantized values of the theta-angles are allowed. The quantized theta-angles include the discrete theta-angles recently identified by Aharony, Seiberg and Tachikawa.