U(N|M) quantum mechanics on Kaehler manifolds

We study the extended supersymmetric quantum mechanics, with supercharges transforming in the fundamental representation of U(N|M), as realized in certain one-dimensional nonlinear sigma models with Kaehler manifolds as target space. We discuss the symmetry algebra characterizing these models and, using operatorial methods, compute the heat kernel in the limit of short propagation time. These models are relevant for studying the quantum properties of a certain class of higher spin field equations in first quantization.Comment: 21 pages, a reference adde

Quantum theories of (p,q)-forms

We describe quantum theories for massless (p,q)-forms living on Kaehler spaces. In particular we consider four different types of quantum theories: two types involve gauge symmetries and two types are simpler theories without gauge invariances. The latter can be seen as building blocks of the former. Their equations of motion can be obtained in a natural way by first-quantizing a spinning particle with a U(2)-extended supersymmetry on the worldline. The particle system contains four supersymmetric charges, represented quantum mechanically by the Dolbeault operators and their hermitian conjugates. After studying how the (p,q)-form field theories emerge from the particle system, we investigate their one loop effective actions, identify corresponding heat kernel coefficients, and derive exact duality relations. The dualities are seen to include mismatches related to topological indices and analytic torsions, which are computed as Tr(-1)^F and Tr[(-1)^F F] in the first quantized supersymmetric nonlinear sigma model for a suitable fermion number operator F.Comment: 44 pages, 2 figures, a reference adde

Quantum theory of massless (p,0)-forms

We describe the quantum theory of massless (p,0)-forms that satisfy a suitable holomorphic generalization of the free Maxwell equations on Kaehler spaces. These equations arise by first-quantizing a spinning particle with a U(1)-extended local supersymmetry on the worldline. Dirac quantization of the spinning particle produces a physical Hilbert space made up of (p,0)-forms that satisfy holomorphic Maxwell equations coupled to the background Kaehler geometry, containing in particular a charge that measures the amount of coupling to the U(1) part of the U(d) holonomy group of the d-dimensional Kaehler space. The relevant differential operators appearing in these equations are a twisted exterior holomorphic derivative and its hermitian conjugate (twisted Dolbeault operators with charge q). The particle model is used to obtain a worldline representation of the one-loop effective action of the (p,0)-forms. This representation allows to compute the first few heat kernel coefficients contained in the local expansion of the effective action and to derive duality relations between (p,0) and (d-p-2,0)-forms that include a topological mismatch appearing at one-loop.Comment: 32 pages, 3 figure

On a New Method for Computing Trace Anomalies

I describe a new method for computing trace anomalies in quantum field theories which makes use of path-integrals for particles moving in curved spaces. After presenting the main ideas of the method, I discuss how it is connected to the first quantized approach of particle theory and to heat kernel techniques. (Talk given at Journees Relativistes '93).Comment: 4 pages, LaTeX, USITP-93-1

Path integrals in curved space and the worldline formalism

We describe how to construct and compute unambiguously path integrals for particles moving in a curved space, and how these path integrals can be used to calculate Feynman graphs and effective actions for various quantum field theories with external gravity in the framework of the worldline formalism. In particular, we review a recent application of this worldline approach and discuss vector and antisymmetric tensor fields coupled to gravity. This requires the construction of a path integral for the N=2 spinning particle, which is used to compute the first three Seeley-DeWitt coefficients for all p-form gauge fields in all dimensions and to derive exact duality relations.Comment: 16 pages, to appear in the proceedings of "Path Integrals 2005: from Quantum Information to Cosmology" (Prague, June 6-10, 2005), references adde

A note on 2D chiral gravity and chiral bosons

Quantization of two dimensional chiral matter coupled to gravity induces an effective action for the zweibein field which is both Weyl and Lorentz anomalous. Recently, the quantization of this induced action has been analyzed in the light-cone gauge as well as in the conformal gauge. An apparent mismatch between the results obtained in the two gauges is analyzed and resolved by properly treating the Lorentz field as a chiral boson.Comment: 7 pages, plain TeX, USITP-92-09 (minor correction in the list of references

Dimensional regularization of nonlinear sigma models on a finite time interval

We extend dimensional regularization to the case of compact spaces. Contrary to previous regularization schemes employed for nonlinear sigma models on a finite time interval (``quantum mechanical path integrals in curved space'') dimensional regularization requires only a covariant finite two-loop counterterm. This counterterm is nonvanishing and given by R/8.Comment: 9 pages, 7 figures, LaTeX, minor changes in text and reference

A Locally Supersymmetric Action for the Bosonic String

Recently Berkovits and Vafa have shown that the bosonic string can be viewed as the fermionic string propagating in a particular background. Such a background is described by a somewhat unusual $N=1$ superconformal system. By coupling it to $N=1$ supergravity I construct a local supersymmetric action for the bosonic string.Comment: 7 pages, NBI-93-6