An L2-Index Theorem for Dirac Operators on S1×R3

Collier

et al. 2013

We study finite-temperature N = 1 SU (2) super Yang-Mills theory, compactified on a spatial circle of size L with supersymmetric boundary conditions. In the semiclassical small-L regime, a deconfinement transition occurs at T c 1/L. The transition is due to a competition between non-perturbative topological "molecules"-magnetic and neutral bioninstantons-and electrically charged W -bosons and superpartners. Compared to deconfinement in non-supersymmetric QCD(adj) [1], the novelty is the relevance of the light modulus scalar field. It mediates interactions between neutral bions (and W -bosons), serves as an order parameter for the Z (L) 2 center symmetry associated with the non-thermal circle, and explicitly breaks the electric-magnetic (Kramers-Wannier) duality enjoyed by non-supersymmetric QCD(adj) near T c . We show that deconfinement can be studied using an effective twodimensional gas of electric and magnetic charges with (dual) Coulomb and Aharonov-Bohm interactions, or, equivalently, via an XY-spin model with a symmetry-breaking perturbation, where each system couples to the scalar field. To study the realization of the discrete Rsymmetry and the Z remaining unbroken at the transition. Thus, the SYM transition appears similar to the one in SU (2) QCD(adj) [1] and is also likely to be characterized by continuously varying critical exponents.

Section: Nonperturbative Dynamics At Zero Temperaturementioning

confidence: 99%

Deconfinement in $ \mathcal{N}=1 $ super Yang-Mills theory on $ {{\mathbb{R}}^3}\times {{\mathbb{S}}^1} $ via dual-Coulomb gas and “affine” XY-model

Collier

et al. 2013

“…We stress again that the + and − charge monopole-instanton solutions are both self-dual and that the corresponding antiself-dual antimonopole solutions carry opposite magnetic charges. In a theory with massless adjoint fermions, the monopole-instantons have fermionic zero modes (see [42,43] for the relevant index theorem) and generate, instead of the usual monopole operator e −S 0 e ±iσ , operators of the form:…”

Section: Jhep04(2012)040mentioning

confidence: 99%

2d affine XY-spin model/4d gauge theory duality and deconfinement

Ünsal

2012

103

We introduce a duality between two-dimensional XY-spin models with symmetry-breaking perturbations and certain four-dimensional SU(2) and SU(2)/Z 2 gauge theories, compactified on a small spatial circle R 1,2 × S 1 , and considered at temperatures near the deconfinement transition. In a Euclidean set up, the theory is defined on R 2 × T 2 . Similarly, thermal gauge theories of higher rank are dual to new families of "affine" XY-spin models with perturbations. For rank two, these are related to models used to describe the melting of a 2d crystal with a triangular lattice. The connection is made through a multicomponent electric-magnetic Coulomb gas representation for both systems. Perturbations in the spin system map to topological defects in the gauge theory, such as monopoleinstantons or magnetic bions, and the vortices in the spin system map to the electrically charged W -bosons in field theory (or vice versa, depending on the duality frame). The duality permits one to use the two-dimensional technology of spin systems to study the thermal deconfinement and discrete chiral transitions in four-dimensional SU(N c ) gauge theories with n f ≥1 adjoint Weyl fermions.

“…This is because (see also the following Sections) confinement is associated with the generation of a mass gap of the dual photon field-to which only magnetically charged objects can contribute (recall that σ is sourced by magnetic charge). The other topological solutions, the fundamental BBS and KK monopoles, have fermionic zero modes (for the relevant index theorems, see [31,32]) and hence also do not generate mass for the dual photon. However, as was shown in [2], BPS-KK molecules (bions) will generate a mass term for the dual photon and lead to confinement of electric charges.…”

Section: The Vacuum Of the Small-s 1 Regime Of Qcd(adj) Theoriesmentioning

confidence: 99%

“…We further note that the "almost"-BPS nature of the BPS and KK monopole solutions relevant in our case does not affect the leading long-distance behavior of the fermion zero modes. First, the index theorem [31,32] does not depend on the existence of a mass gap for A4 (i.e., it does not require (anti-)self-duality of the background) and, second, the long-distance asymptotics of the fermion zero mode is the one given by (4.25), as can be seen from the expression for the general non-BPS monopole adjoint zero modes [40].…”

Section: Computation Of S Intmentioning

confidence: 99%

Microscopic structure of magnetic bions

2011

Abstract:Magnetic bions-stable bound states of monopoles and twisted ("Kaluza-Klein") monopoles, carrying two units of magnetic charge-have been shown to be the leading cause of confinement and mass gap in four-dimensional gauge theories with massless adjoint fermions compactified on R 1,2 × S 1 , at least at small S 1 . In this paper, we study in detail the bion mechanism and the scales involved for an SU (2) gauge group, using traditional QCD instanton methods. We represent the vacuum functional as the partition function of a bion-anti-bion plasma and obtain the next-to-leading dependence of the mass gap on the S 1 size L at fixed strong-coupling scale Λ. We find that, at small ΛL, the mass gap is an increasing function of L for theories with four massless adjoint Weyl fermions, a case left undetermined by the previous leading-order analysis, and comment on the approach to R