GEOMETRY AND QUANTIZATION OF TOPOLOGICAL GAUGE THEORIES

1990 ◽  
Vol 05 (24) ◽  
pp. 4721-4752 ◽  
Author(s):  
DANNY BIRMINGHAM ◽  
MATTHIAS BLAU ◽  
GEORGE THOMPSON

A general method for constructing interesting topological gauge theories in arbitrary dimensions is presented. The basic framework upon which these models are built is given by the geometrical data of the "universal bundle with connection" of Atiyah and Singer. The models considered include theories which represent the moduli spaces of flat connections and solutions to the Yang-Mills equations. The former theories correspond to supersymmetric versions of the recently introduced BF systems. In all cases we show explicitly that the quantization can be carried out through the construction of an off-shell nilpotent BRST operator, thus guaranteeing the metric independence of these models.

1993 ◽  
Vol 08 (03) ◽  
pp. 573-585 ◽  
Author(s):  
MATTHIAS BLAU ◽  
GEORGE THOMPSON

We rederive the recently introduced N=2 topological gauge theories, representing the Euler characteristic of moduli spaces ℳ of connections, from supersymmetric quantum mechanics on the infinite-dimensional spaces [Formula: see text] of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces, and introduce supersymmetric quantum mechanics actions modeling the Riemannian geometry of submersions and embeddings, relevant to the projections [Formula: see text] and inclusions [Formula: see text] respectively. We explain the relation between Donaldson theory and the gauge theory of flat connections in three dimensions and illustrate the general construction by other two- and four-dimensional examples.


2012 ◽  
Vol 10 (01) ◽  
pp. 1220021
Author(s):  
R. CARTAS-FUENTEVILLA

It is shown that if the contribution of flat connections on the dimension of the moduli spaces of Yang–Mills instantons and anti-instantons is appropriately taken into the account, then the inadmissible cases of negative dimensions may be reduced to zero-dimensional moduli spaces, corresponding to a collection of points, and whose counting will correspond to the Donaldson invariant of the base manifold. These results will lead to a possible description of that invariant in terms of flat connections with diverse applications, for example for testing the conjecture on its equivalence to the Seiberg–Witten invariant, and for the study of the qualitative and quantitative aspects of the gauge/gravity duality.


Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor

A geometrical derivation of Abelian and non- Abelian gauge theories. The Faddeev–Popov quantisation. BRST invariance and ghost fields. General discussion of BRST symmetry. Application to Yang–Mills theories and general relativity. A brief history of gauge theories.


2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Guido Festuccia ◽  
Anastasios Gorantis ◽  
Antonio Pittelli ◽  
Konstantina Polydorou ◽  
Lorenzo Ruggeri

Abstract We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $$ \mathcal{N} $$ N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.


2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Jean-Nicolas Lang ◽  
Stefano Pozzorini ◽  
Hantian Zhang ◽  
Max F. Zoller

Abstract Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to nf fermions with arbitrary masses.


2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Max Hübner

Abstract M-theory on local G2-manifolds engineers 4d minimally supersymmetric gauge theories. We consider ALE-fibered G2-manifolds and study the 4d physics from the view point of a partially twisted 7d supersymmetric Yang-Mills theory and its Higgs bundle. Euclidean M2-brane instantons descend to non-perturbative effects of the 7d supersymmetric Yang-Mills theory, which are found to be in one to one correspondence with the instantons of a colored supersymmetric quantum mechanics. We compute the contributions of M2-brane instantons to the 4d superpotential in the effective 7d description via localization in the colored quantum mechanics. Further we consider non-split Higgs bundles and analyze their 4d spectrum.


2014 ◽  
Vol 29 (30) ◽  
pp. 1450184 ◽  
Author(s):  
Alexander Reshetnyak

A consistent quantum treatment of general gauge theories with an arbitrary gauge-fixing in the presence of soft breaking of the BRST symmetry in the field–antifield formalism is developed. It is based on a gauged (involving a field-dependent parameter) version of finite BRST transformations. The prescription allows one to restore the gauge-independence of the effective action at its extremals and therefore also that of the conventional S-matrix for a theory with BRST-breaking terms being additively introduced into a BRST-invariant action in order to achieve a consistency of the functional integral. We demonstrate the applicability of this prescription within the approach of functional renormalization group to the Yang–Mills and gravity theories. The Gribov–Zwanziger action and the refined Gribov–Zwanziger action for a many-parameter family of gauges, including the Coulomb, axial and covariant gauges, are derived perturbatively on the basis of finite gauged BRST transformations starting from Landau gauge. It is proved that gauge theories with soft breaking of BRST symmetry can be made consistent if the transformed BRST-breaking terms satisfy the same soft BRST symmetry breaking condition in the resulting gauge as the untransformed ones in the initial gauge, and also without this requirement.


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