arXiv subjects

Richard J. Szabo

Publications and source records attributed to Richard J. Szabo.

196 records · Page 11Linked to original sources

arXiv1998

Gradient Flow in Logarithmic Conformal Field Theory

We establish conditions under which the worldsheet beta-functions of logarithmic conformal field theories can be derived as the gradient of some scalar function on the moduli space of running coupling constants. We derive a renormalization group invariant version of this function and relate it to the usual Zamolodchikov C-function expressed in terms of correlation functions of the worldsheet energy-momentum tensor. The results are applied to the example of D-brane recoil in string theory.

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arXiv1997

Noncommutative Geometry and Spacetime Gauge Symmetries of String Theory

We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a universal gauge group which overlies all of the dynamical symmetries of the string spacetime. We also describe some aspects of toroidal compactifications with a light-like coordinate and show how certain generalized Kac-Moody symmetries, such as the Monster sporadic group, arise as gauge symmetries of the resulting spacetime and of superstring theories.

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arXiv1997

Electric-magnetic Duality in Noncommutative Geometry

The structure of S-duality in U(1) gauge theory on a 4-manifold M is examined using the formalism of noncommutative geometry. A noncommutative space is constructed from the algebra of Wilson-'t Hooft line operators which encodes both the ordinary geometry of M and its infinite-dimensional loop space geometry. S-duality is shown to act as an inner automorphism of the algebra and arises as a consequence of the existence of two independent Dirac operators associated with the spaces of self-dual and anti-selfdual 2-forms on M. The relations with the noncommutative geometry of string theory and T-duality are also discussed.

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arXiv1997

Target Space Duality in Noncommutative Geometry

The structure of spacetime duality and discrete worldsheet symmetries of compactified string theory is examined within the framework of noncommutative geometry. The full noncommutative string spacetime is constructed using the Frohlich-Gawedzki spectral triple which incorporates the vertex operator algebra of the string theory. The duality group appears naturally as a subgroup of the automorphism group of the vertex operator algebra and spacetime duality is shown to arise as the possibility of associating two independent Dirac operators, arising from the chiral structure of the worldsheet theory, to the noncommutative geometry.

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arXiv1997

Matrix Strings in Two-dimensional Yang-Mills Theory

We describe the structure of string vacuum states in the supersymmetric matrix model for M theory compactified on a circle in the large-N limit. We show that the theory admits topological instanton field configurations which at short-distance scales reduce to ordinary Yang-Mills instantons that interpolate between degenerate vacua of the theory. We show that there exists further classes of hadronic strings associated with the D-string super-fields. We discuss the relationships between these non-perturbative string states and rigid QCD strings, critical strings, and membrane states.

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arXiv1997

Liouville Dressed Weights and Renormalization of Spin in Topologically Massive Gravity

We examine the relations between observables in two- and three-dimensional quantum gravity by studying the coupling of topologically massive gravity to matter fields in non-trivial representations of the three-dimensional Lorentz group. We show that the gravitational renormalization of spin up to one-loop order in these theories reproduces the leading orders of the KPZ scaling relations for quantum Liouville theory. We demonstrate that the two-dimensional scaling dimensions can be computed from tree-level Aharonov-Bohm scattering amplitudes between the charged particles in the limit where the three-dimensional theory possesses local conformal invariance. We show how the three-dimensional description defines scale-dependent weights by computing the one-loop order anomalous magnetic moment of fermions in a background electromagnetic field due to the renormalization by topologically massive gravity. We also discuss some aspects concerning the different phases of three-dimensional quantum gravity and argue that the topological ones may be related to the branched polymer phase of two-dimensional quantum gravity.

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arXiv1997

Dynamical Description of Spectral Flow in N=2 Superconformal Field Theories

We show how the spectral flow between the Neveu-Schwarz and Ramond sectors of N=2 superconformal field theories can be described in three dimensions in terms of the propagation of charged particles coupled to a a Chern-Simons gauge theory. Quantum mechanical mixing between the degenerate Chern-Simons vacua interpolates between the different boundary conditions of the two sectors and so provides a dynamical picture for the GSO-projection.

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arXiv1996

Curvature Matrix Models for Dynamical Triangulations and the Itzykson-DiFrancesco Formula

We study the large-N limit of a class of matrix models for dually weighted triangulated random surfaces using character expansion techniques. We show that for various choices of the weights of vertices of the dynamical triangulation the model can be solved by resumming the Itzykson-Di Francesco formula over congruence classes of Young tableau weights modulo three. From this we show that the large-N limit implies a non-trivial correspondence with models of random surfaces weighted with only even coordination number vertices. We examine the critical behaviour and evaluation of observables and discuss their interrelationships in all models. We obtain explicit solutions of the model for simple choices of vertex weightings and use them to show how the matrix model reproduces features of the random surface sum. We also discuss some general properties of the large-N character expansion approach as well as potential physical applications of our results.

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arXiv1996

Fermionic Matrix Models

We review a class of matrix models whose degrees of freedom are matrices with anticommuting elements. We discuss the properties of the adjoint fermion one-, two- and gauge invariant D-dimensional matrix models at large-N and compare them with their bosonic counterparts which are the more familiar Hermitian matrix models. We derive and solve the complete sets of loop equations for the correlators of these models and use these equations to examine critical behaviour. The topological large-N expansions are also constructed and their relation to other aspects of modern string theory such as integrable hierarchies is discussed. We use these connections to discuss the applications of these matrix models to string theory and induced gauge theories. We argue that as such the fermionic matrix models may provide a novel generalization of the discretized random surface representation of quantum gravity in which the genus sum alternates and the sums over genera for correlators have better convergence properties than their Hermitian counterparts. We discuss the use of adjoint fermions instead of adjoint scalars to study induced gauge theories. We also discuss two classes of dimensionally reduced models, a fermionic vector model and a supersymmetric matrix model, and discuss their applications to the branched polymer phase of string theories in target space dimensions D>1 and also to the meander problem.

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arXiv1996

Equivariant Localization of Path Integrals

We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localization ideas are related to classical integrability and how they can be formally extended to derive explicit localization formulas for path integrals in special instances using BRST quantization techniques. Various loop space localizations are presented and related to notions in quantum integrability and topological field theory. We emphasize the common symmetries that such localizable models always possess and use these symmetries to discuss the range of applicability of the localization formulas. A number of physical and mathematical applications are presented in connection with elementary quantum mechanics, Morse theory, index theorems, character formulas for semi-simple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemann surfaces, supersymmetric quantum field theories, two-dimensional Yang-Mills theory, conformal field theory, cohomological field theories and the loop expansion in quantum field theory. Some modern techniques of path integral quantization, such as coherent state methods, are also discussed. The relations between equivariant localization and other ideas in topological field theory, such as the Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented.

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arXiv1996

Polymer Statistics and Fermionic Vector Models

We consider a variation of $O(N)$-symmetric vector models in which the vector components are Grassmann numbers. We show that these theories generate the same sort of random polymer models as the $O(N)$ vector models and that they lie in the same universality class in the large-$N$ limit. We explicitly construct the double-scaling limit of the theory and show that the genus expansion is an alternating Borel summable series that otherwise coincides with the topological expansion of the bosonic models. We also show how the fermionic nature of these models leads to an explicit solution even at finite-$N$ for the generating functions of the number of random polymer configurations.

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arXiv1994

Critical Behaviour of a Fermionic Random Matrix Model at Large-N

We study the large-$N$ limit of adjoint fermion one-matrix models. We find one-cut solutions of the loop equations for the correlators of these models and show that they exhibit third order phase transitions associated with $m$-th order multi-critical points with string susceptibility exponents $γ_{\rm str}=-1/m$. We also find critical points which can be interpreted as points of first order phase transitions, and we discuss the implications of this critical behaviour for the topological expansion of these matrix models.

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arXiv1994

Canonical BF-type Topological Field Theory and Fractional Statistics of Strings

We consider BF-type topological field theory coupled to non-dynamical particle and string sources on spacetime manifolds of the form $\IR^1\times\MT$, where $\MT$ is a 3-manifold without boundary. Canonical quantization of the theory is carried out in the Hamiltonian formalism and explicit solutions of the Schrödinger equation are obtained. We show that the Hilbert space is finite dimensional and the physical states carry a one-dimensional projective representation of the local gauge symmetries. When $\MT$ is homologically non-trivial the wavefunctions in addition carry a multi-dimensional projective representation, in terms of the linking matrix of the homology cycles of $\MT$, of the discrete group of large gauge transformations. The wavefunctions also carry a one-dimensional representation of the non-trivial linking of the particle trajectories and string surfaces in $\MT$. This topological field theory therefore provides a phenomenological generalization of anyons to (3 + 1) dimensions where the holonomies representing fractional statistics arise from the adiabatic transport of particles around strings. We also discuss a duality between large gauge transformations and these linking operations around the homology cycles of $\MT$, and show that this canonical quantum field theory provides novel quantum representations of the cohomology of $\MT$ and its associated motion group.

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arXiv1994

Equivariant Localization, Spin Systems and Topological Quantum Theory on Riemann Surfaces

We study equivariant localization formulas for phase space path integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the non-trivial homology classes of the phase space. We explicitly construct the coherent states in the canonical quantum theory and show that the Hilbert space is finite dimensional with the wavefunctions carrying a projective representation of the discrete homology group of the phase space. The corresponding coherent state path integral then describes the quantum dynamics of a novel spin system given by the quantization of a non-symmetric coadjoint Lie group orbit. We also briefly discuss the geometric structure of these quantum systems.

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arXiv1993

Phase Space Isometries and Equivariant Localization of Path Integrals in Two Dimensions

By considering the most general metric which can occur on a contractable two dimensional symplectic manifold, we find the most general Hamiltonians on a two dimensional phase space to which equivariant localization formulas for the associated path integrals can be applied. We show that in the case of a maximally symmetric phase space the only applicable Hamiltonians are essentially harmonic oscillators, while for non-homogeneous phase spaces the possibilities are more numerous but ambiguities in the path integrals occur. In the latter case we give general formulas for the Darboux Hamiltonians, as well as the Hamiltonians which result naturally from a generalized coherent state formulation of the quantum theory which shows that again the Hamiltonians so obtained are just generalized versions of harmonic oscillators. Our analysis and results describe the quantum geometry of some two dimensional systems.

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arXiv1993

The Spectrum of Topologically Massive Quantum Electrodynamics

We discuss the possibility for the spectrum of topologically massive quantum electrodynamics with spinor matter fields to contain unexpected and unusual stable particle excitations for certain values of the topological photon mass. The new field theoretical phenomena arising from this novel spectral structure are briefly discussed.

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