Richard J. Szabo

Lattice gauge fields and discrete noncommutative Yang-Mills theory

We present a lattice formulation of non-commutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of non-commutative field theories is demonstrated at a completely non-perturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and non-commutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic non-commutative gauge theories in the continuum can be regularized non perturbatively by means of ordinary lattice gauge theory with t Hooft flux. In the case of irrational non-commutativity 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 non-commutative Yang-Mills theories using twisted large-N reduced models. We study the coupling of non-commutative 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 non-commutative 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 non-commutative quantum electrodynamics.

Finite N matrix models of noncommutative gauge theory

We describe a unitary matrix model which is constructed from discrete analogs of the usual projective modules over the noncommutative torus and use it to construct a lattice version of noncommutative gauge theory. The model is a discretization of the noncommutative gauge theories that arise from toroidal compactification of Matrix theory and it includes a recent proposal for a non-perturbative definition of noncommutative Yang-Mills theory in terms of twisted reduced models. The model is interpreted as a manifestly star-gauge invariant lattice formulation of noncommutative gauge theory, which reduces to ordinary Wilson lattice gauge theory for particular choices of parameters. It possesses a continuum limit which maintains both finite spacetime volume and finite noncommutativity scale. We show how the matrix model may be used for studying the properties of noncommutative gauge theory.

Nonperturbative dynamics of noncommutative gauge theory

Abstract We present a nonperturbative lattice formulation of noncommutative Yang–Mills theories in arbitrary even dimension. We show that lattice regularization of a noncommutative field theory requires finite lattice volume which automatically provides both an ultraviolet and an infrared cutoff. We demonstrate explicitly Morita equivalence of commutative U( p ) gauge theory with p · n f flavours of fundamental matter fields on a lattice of size L with twisted boundary conditions and noncommutative U(1) gauge theory with n f species of matter on a lattice of size p · L with single-valued fields. We discuss the relation with twisted large N reduced models and construct observables in noncommutative gauge theory with matter.

Symmetry, gravity and noncommutativity

We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative Yang–Mills theory can be related to a gravity theory. The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. We also analyse certain generic features of noncommutative gauge theories on D-branes in curved spaces, treating several explicit examples of superstring backgrounds.

Brane Descent Relations in K-theory

Abstract The various descent and duality relations among BPS and non-BPS D-branes are classified using topological K-theory. It is shown how the descent procedures for producing type-II D-branes from brane–antibrane bound states by tachyon condensation and (−1) F L projections arise as natural homomorphisms of K-groups generating the brane charges. The transformations are generalized to type-I theories and type-II orientifolds, from which the complete set of vacuum manifolds and field contents for tachyon condensation is deduced. A new set of internal descent relations is found which describes branes over orientifold planes as topological defects in the worldvolumes of brane–antibrane pairs on top of planes of higher dimension. The periodicity properties of these relations are shown to be a consequence of the fact that all fundamental bound state constructions and hence the complete spectrum of brane charges are associated with the topological solitons which classify the four Hopf fibrations.

Induced dilaton in topologically massive quantum field theory

We consider the conformally-invariant coupling of topologically massive gravity to a dynamical massless scalar field theory on a three-manifold with boundary. We show that, in the phase of spontaneously broken Lorentz and Weyl symmetries, this theory induces the target space zero mode of the vertex operator for the string dilaton field on the boundary of the three-dimensional manifold. By a further coupling to topologically massive gauge fields in the bulk, we demonstrate directly from the three-dimensional theory that this dilaton field transforms in the expected way under duality transformations so as to preserve the mass gaps in the spectra of the gauge and gravitational sectors of the quantum field theory. We show that this implies an intimate dynamical relationship between T-duality and S-duality transformations of the quantum string theory. The dilaton in this model couples bulk and worldsheet degrees of freedom to each other and generates a dynamical string coupling.

Isometric embeddings and noncommutative branes in homogeneous gravitational waves

We characterize the worldvolume theories on symmetric D-branes in a six-dimensional Cahen–Wallach pp-wave supported by a constant Neveu–Schwarz 3-form flux. We find a class of flat noncommutative Euclidean D3-branes analogous to branes in a constant magnetic field, as well as curved noncommutative Lorentzian D3-branes analogous to branes in an electric background. In the former case, the noncommutative field theory on the branes is constructed from first principles, related to dynamics of fuzzy spheres in the worldvolumes, and used to analyse the flat space limits of the string theory. The worldvolume theories on all other symmetric branes in the background are local field theories. The physical origins of all these theories are described through the interplay between isometric embeddings of branes in the spacetime and the Penrose–Guven limit of AdS3 × S3 with Neveu–Schwarz 3-form flux. The noncommutative field theory of a non-symmetric spacetime-filling D-brane is also constructed, giving a spatially varying but time-independent noncommutativity analogous to that of the Dolan–Nappi model.

Fermionic quantum gravity

Abstract We study the statistical mechanics of random surfaces generated by N×N one-matrix integrals over anti-commuting variables. These Grassmann-valued matrix models are shown to be equivalent to N×N unitary versions of generalized Penner matrix models. We explicitly solve for the combinatorics of txa0Hooft diagrams of the matrix integral and develop an orthogonal polynomial formulation of the statistical theory. An examination of the large N and double scaling limits of the theory shows that the genus expansion is a Borel summable alternating series which otherwise coincides with two-dimensional quantum gravity in the continuum limit. We demonstrate that the partition functions of these matrix models belong to the relativistic Toda chain integrable hierarchy. The corresponding string equations and Virasoro constraints are derived and used to analyse the generalized KdV flow structure of the continuum limit.

Bottleneck surfaces and worldsheet geometry of higher-curvature quantum gravity

We describe a simple lattice model of higher-curvature quantum gravity in two dimensions and study the phase structure of the theory as a function of the curvature coupling. It is shown that the ensemble of flat graphs is entropically unstable to the formation of baby universes. In these simplified models the growth in graphs exhibits a branched polymer behaviour in the phase directly before the flattening transition.

Spectral geometry of heterotic compactifications

The structure of heterotic string target-space compactifications is studied using the formalism of the non-commutative geometry associated with lattice vertex operator algebras. The spectral triples of the non-commutative spacetimes are constructed and used to show that the intrinsic gauge field degrees of freedom disappear in the low-energy sectors of these spacetimes. The quantum geometry is thereby determined in much the same way as for ordinary superstring target spaces. In this setting, non-Abelian gauge theories on the classical spacetimes arise from the K-theory of the effective target spaces.