Dynamical supersymmetry breaking and gauge anomalies

Some aspects of supersymmetric gauge theories and discussed. It is shown that dynamical supersymmetry breaking does not occur in supersymmetric QED in higher dimensions. The cancellation of both local (perturbative) and global (non-perturbative) gauge anomalies are also discussed in supersymmetric gauge theories. We argue that there is no dynamical supersymmetry breaking in higher dimensions in any supersymmetric gauge theories free of gauge anomalies. It is also shown that for supersymmetric gauge theories in higher dimensions with a compact connected simple gauge group, when the local anomaly-free condition is satisfied, there can be at most a possibleZ ₂ global gauge anomaly in extended supersymmetricSO(10) (or spin (10)) gauge theories inD=10 dimensions containing additional Weyl fermions in a spinor representation ofSO(10) (or spin (10)). In four dimensions with local anomaly-free condition satisfied, the only possible global gauge anomalies in supersymmetric gauge theories areZ ₂ global gauge anomalies for extended supersymmetricSP(2N) (N=rank) gauge theories containing additional Weyl fermions in a representation ofSP(2N) with an odd 2nd-order Dynkin index.     

The cancellation of world-sheet anomalies in the D = 10 Green-Schwarz heterotic string sigma model

We determine the two-dimensional Weyl, Lorentz and κ-anomalies in the D = 10 Green-Schwarz heterotic string sigma model, in an SO(1, 9) Lorentz-covariant background gauge, and prove their cancellation.     

Lorentz anomaly in arbitrary dimensions

It is shown that the Lorentz invariance is broken in gauge theories of chiral Weyl fermions in flat space-time via one-loop quantum corrections. Abelian gauge fields contribute to this anomaly in even dimensions larger than or equal to four and non-Abelian gauge fields do in even dimensions larger than or equal to six. The anomaly is proportional toD/2–1 power to the charge, whereD is a number of space-time dimensions.     

Topology of quantum gauge fields and duality (I): Yang-Mills-Higgs system in 2 + 1 dimensions

The basic role of the representation of the gauge group in characterizing the topological excitations of the vacuum is pointed out. For SU(N) gauge fields on a lattice, the topological excitations are monopoles in the adjoint representation of the dual group ¹SU(N). This leads to a dual representation of the Yang-Mills-Higgs system in 2 + 1 dimensions. For SU(3) the deal theory in a scalar theory with discrete Weyl symmetry S₃. In the presence of adjoint Higgs fields the Weyl symmetry is broken in the Higgs phase but restored by pseudo-particles in the confinement phase.     

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2n dimensions is related to the non-trivial topology of gauge orbit space. The form of the anomaly and its normalization are shown to follow from a familiar index theorem for a certain (2n + 2)-dimensional Dirac operator. We are thus able to recover and give topological meaning to a variety of results concerning anomalies in 4- and higher-dimensional theories.     

Conformal invariance in Weyl gravity

A conformal-invariant model of Weyl gravity, based on a nondecomposable representation of the conformal group, allows one to have a conformal-invariant propagator in an arbitrary gauge, as well as a conformal-invariant gauge-fixing term in the Lagrangian approach. It is shown that in the gauge-invariant sector this theory coincides with ordinary Weyl gravity (with conformal-noninvariant gauge fixing). The corresponding BRST transformations are found and are used for derivation of the Slavnov-Taylor identities.     

The algebra of the Gauss law in d = 4 massless axial QED with a WZ term

The consequences of adding an anomaly-reproducing Wess-Zumino action to d = 4 massless axial QED are examined in the Weyl gauge. It is found that to the relevant leading order in ħ the Jacobi identity for the hamiltonian and the Gauss law remains broken.     

General relativity and gauge theories

The theory of general relativity is presented in the form of a gauge field theory by use of the group SL(2,C). The following topics are discussed: (1)Spinor representation of the group SL(2,C); (2)Connection between spinors and tensors; (3)Maxwell, Weyl and Riemann Spinors; (4)Classification of Maxwell spinor; (5)Classification of Weyl spinor; (6)Isotopic spin and gauge fields; (7)Lorentz invariance and the gravitational field; (8)SL(2,C) invariance and the gravitational field; (9)Gravitational field equations.     

The extended conformal Einstein field equations with matter: The Einstein–Maxwell field

A discussion is given of the conformal Einstein field equations coupled with matter whose energy–momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know a priori the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain (i) a new proof of the stability of Einstein–Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein–Maxwell spacetimes.     

Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the (n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G:

• •there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric;
• •in that gauge, a simple constraint on the conformal scalar curvature holds;
• •a non-exact Einstein–Weyl structure may exist only if the (n−1)-dimensional homogeneous space G/H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H;
• •the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′.

The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint.We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the (n−2)-dimensional space G/(H×H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU(n/2) being, thanks to our results, invariant under U(1)×SU(n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.     

On Weyl invariance conditions in string theory coupled with two-dimensional gravity

The unified theory of string and two-dimensional quantum gravity is considered. We introduce nontrivial dynamics for the two-dimensional metric g_μν from the very beginning and calculate the path integral over the string coordinates and g_μν without taking into account the order of integrations. Throughout the paper we use two different kinds of gauges - the covariant one of the harmonic type and also the conformal gauge, where the original (D + 1)-dimensional sigma model with quantum gravity becomes the (D+2)-dimensional sigma model on the classical background of g_μν The general symmetries of the theory consist in the reparametrizations of the target space coordinates, in the conformal transformations of the metric and in the usual 2d diffeomorphisms. These symmetries do not disturb the structure of the background fields in the (D+2) -dimensional formulation. On the other hand the related arbitrariness of the renormalization does not affect the qualitative structure of the loop contributions to the Weyl anomaly. In the theory with quantum gravity the parameter a′ does not play as the parameter of the loop expansion. That is why the one-loop conditions of the Weyl invariance differs from the well known effective equations which arise in the standard approach when g_μν is not quantized simultaneously with the string coordinates. Therefore, despite the new conditions of the Weyl invariance for the background fields are different from the standard effective equations, our result does not contradict to the standard approach. The new one-loop conditions of the Weyl invariance are much more complicated and contain the higher derivatives in the dilaton sector.     

Conformally Flat Spacetimes and Weyl Frames

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordstr?m scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.     

Structure of supergravity

It is shown how to construct actions and kinetic multiplets out of scalar multiplets of supergravity. The multiplets containing the scalar and the conformal tensor are obtained and yield the gauge actions of Einstein and Weyl supergravity. The complete supersymmetric R² and Bel-Robinson invariants are also obtained as well as general n-loop counterterms.     

Electronic Properties of Strained Double‐Weyl Systems

The effects of strains on the low‐energy electronic properties of double‐Weyl phases are studied in solids and cold‐atom optical lattices. The principal finding is that deformations do not couple, in general, to the low‐energy effective Hamiltonian as a pseudoelectromagnetic gauge potential. The response of an optical lattice to strains is simpler, but still only one of the several strain‐induced terms in the corresponding low‐energy Hamiltonian can be interpreted as a gauge potential. Most interestingly, the strains can induce a nematic order parameter that splits a double‐Weyl node into a pair of Weyl nodes with the unit topological charges. The effects of deformations on the motion of wavepackets in the double‐Weyl optical lattice model are studied. It is found that, even in the undeformed lattices, the wavepackets with opposite topological charges can be spatially split. Strains, however, modify their velocities in a very different way and lead to a spin polarization of the wavepackets.     

Tractors,mass, and Weyl invariance

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus—a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. Breitenlohner–Freedman stability bounds for Anti-de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories—which rely on the interplay between mass and gauge invariance—are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s?2$s?2$ are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s?2$s?2$ we give tractor equations of motion unifying massive, massless, and partially massless theories.     

Gauge group of gravity,spinors, and anomalies

A discussion is given of the gravitational anomalies that arise from coupling Weyl spinors to gravity, treating the metric, the soldering form, and the connection as independent dynamical variables. This system is strictly analogous to Weyl spinors coupled to Yang-Mills fields and a nonlinear sigma model. The larger gauge group of this formulation is seen to lie at the root of the equivalence between Einstein and Lorentz anomalies.On leave of absence from SISSA, Trieste, Italy.     

Weyl geometry as a particle geometry

It is shown that a stress-tensor which is a curvature source for a Weyl geometry most contain an antisymmetric component. Otherwise the Weyl geometry is equivalent to a Riemannian geometry via a gauge transformation.     

Reformulation of general relativity as a gauge theory

The starting point is a spinor affine space-time. At each point, two-component spinors and a basis in spinor space, called “spin frame,” are introduced. Spinor affine connections are assumed to exist, but their values need not be known. A metric tensor is not introduced. Global and local gauge transformations of spin frames are defined with GL(2) as the gauge group. Gauge potentials B_μ are introduced and corresponding fields F_μν are defined in analogy with the Yang-Mills case. Gravitational field equations are derived from an action principle. Incases of physical interest SL(2, C) is taken as the gauge group, instead of GL(2). In the special case of metric space-times the theory is identical with general relativity in the Newman-Penrose formalism. Linear combinations of B_μ are generalized spin coefficients, and linear combinations of F_μν are generalized Weyl and Ricci tensors and Ricci scalar. The present approach is compared with other formulations of gravitation as a gauge field.     

Supersymmetric gauge theories in quantum mechanics

A general construction for supersymmetric U(1) gauge invariant Hamiltonians in quantum mechanics is given. For a given number of fermionic and bosonic degrees of freedom it is shown that for four supercharges the interactions are determined uniquely, and coincide with the dimensionally reduced N = 1, d = 3 + 1 supersymmetric electrodynamics. With two supercharges one gets models which cannot be obtained through dimensional reduction. For two special choices of a parameter one recovers the dimensionally reduced d = 1 + 1 Weyl supersymmetric and Majorana supersymmetric electrodynamics.     

Classical versus quantum trace for supersymmetric gauge fields

I calculate the effective action for a supersymmetric Schwinger model in a two-dimensional supergravity background and prove that the gauge superfield does not contribute to the trace anomaly in spite of its classical Weyl non-invariance.