Equivariant dimensional reduction and quiver gauge theories

We review recent applications of equivariant dimensional reduction techniques to the construction of Yang-Mills–Higgs–Dirac theories with dynamical mass generation and exactly massless chiral fermions.     

New example of infinite family of quiver gauge theories

We construct a new infinite family of quiver gauge theories which blow down to the Xp,q$X^{p,q}$ quiver gauge theories found by Hanany, Kazakopoulos and Wecht. This family includes a quiver gauge theory for the third del Pezzo surface. We show, using Z-minimization, that these theories generically have irrational R-charges. The AdS/CFT correspondence implies that the dual geometries are irregular toric Sasaki–Einstein manifolds, although we do not know the explicit metrics.     

Nearly conformal gauge theories in finite volume

We report new results on nearly conformal gauge theories with fermions in the fundamental representation of the SU(3)$\mathit{SU}(3)$ color gauge group as the number of fermion flavors is varied in the Nf=4–16$N_f=4\text{–}16$ range. To unambiguously identify the chirally broken phase below the conformal window we apply a comprehensive lattice tool set in finite volumes which includes the test of Goldstone pion dynamics, the spectrum of the fermion Dirac operator, and eigenvalue distributions of random matrix theory. We also discuss the theory inside the conformal window and present our first results on the running of the renormalized gauge coupling and the renormalization group beta function. The importance of understanding finite volume zero momentum gauge field dynamics inside the conformal window is illustrated. Staggered lattice fermions are used throughout the calculations.     

Broken conformal ward identities in non-abelian gauge theories

Integrated broken conformal Ward identities for non-abelian gauge theories are derived. There exist anomalous terms linked to the ones in the Callan-Symanzik equations. Also there is a gauge dependent term that can be predicted classically and a new term involving Faddeev-Popov ghosts. These two terms cannot be neglected asymptotically relative to the other terms for gauge variant Greens functions.     

A new approach to the perturbative treatment of anomalous chiral gauge theories

The Ward identities following from the conservation of the source-current of anomalous chiral gauge theories on full quantum level are used to define a new kind of perturbation expansion for such theories. Zero modes are found to play a crucial role.     

Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

Complexification of gauge theories

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification G_C of G. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.     

Renormalization of gauge theories

Gauge theories are characterized by the Slavnov identities which express their invariance under a family of transformations of the supergauge type which involve the Faddeev Popov ghosts. These identities are proved to all orders of renormalized perturbation theory, within the BPHZ framework, when the underlying Lie algebra is semisimple and the gauge function is chosen to be linear in the fields in such a way that all fields are massive. An example, the SU2 Higgs Kibble model is analyzed in detail: the asymptotic theory is formulated in the perturbative sense, and shown to be reasonable, namely, the physical S operator is unitary and independent from the parameters which define the gauge function.     

Quantization of gauge theories

A path-integral procedure for quantizing gauge theories is proposed (on a heuristic level). The Hilbert space of physical states is constructed. Each physical state is represented by an infinite set of gauge equivalent configurations. All physical transition amplitudes are defined. In this approach, the “natural” value of parameter θ is zero.     

No‐go result for supersymmetric gauge theories in the causal approach

We consider the massless supersymmetric vector multiplet in a purely quantum framework and propose a power counting formula. Then we prove that the interaction Lagrangian for a massless supersymmetric non‐Abelian gauge theory (SUSY‐QCD) is uniquely determined by some natural assumptions, as in the case of Yang‐Mills models. The result can be easily generalized to the case when massive multiplets are present, but one finds out that the massive and the massless Bosons must be decoupled, in contradiction with the standard model. Going to the second order of perturbation theory produces an anomaly which cannot be eliminated. We make a thorough analysis of the model working only with the component fields.     

Residual gauge invariance of Hamiltonian lattice gauge theories

The time-independent residual gauge invariance of Hamiltonian lattice gauge theories is considered. Eigenvalues and eigenfunctions of the unperturbed Hamiltonian are found in terms of Gegenbauer's polynomials. Physical states which satisfy the subsidiary condition corresponding to Gauss' law are constructed systematically.     

Finiteness of the Coulomb gauge QCD perturbative effective action

At 2-loop order in the Coulomb gauge, individual Feynman graphs contributing to the effective action have energy divergences. It is proved that these cancel in suitable combinations of graphs. This has previously been shown only for transverse external fields. The calculation results in a generalization of the Christ–Lee term which was inserted into the Hamiltonian.     

Surface description of gauge theories

The partition function of non-abelian gauge theory is expressed, in the continuum limit, as a sum over surfaces which are swept out by the propagation of electric flux rings. Each flux surface is described by a two-dimensional continuum gauge theory, confined to that particular surface. The gauge field can then be integrated out; however, for closed and intersecting surfaces interesting complications arise, which reveal an algebraic structure typical of strong-coupling lattice gauge theory.     

On gauge theories of mass

The classical Einstein–Hilbert action in general relativity extends naturally to a blow-up (in the sense of algebraic geometry) of the usual space of pseudo-Riemannian metrics; this presents the metric tensor _gik$g_{ik}$ as a kind of Goldstone boson associated to the real scalar field defined by its determinant. This seems to be quite compatible with the Higgs mechanism in the standard model of particle physics.