Symplectic Geometry of Supersymmetry and Nonlinear Sigma Model

Recently it has been argued, that Poincar\'{e} supersymmetric field theories admit an underlying loop space hamiltonian (symplectic) structure. Here shall establish this at the level of a general $N=1$ supermultiplet. In particular, we advocate the use of a superloop space and explain the necessity of using nonconventional auxiliary fields. As an example we consider the nonlinear $\sigma$-model. Due to the quartic fermionic term, we conclude that the use of superloop space variables is necessary for the action to have a hamiltonian loop space interpretation.Comment: 9 pages, UU-ITP 30/9

Competition in European Telecom Markets

In recent years, the European telecommunications market has witnessed major developments, with rapid expansion in access to telecommunications networks and a surge in the number of available services and applications. While many factors have contributed to the transformation of the telecommunications industry, competition has played a key role in driving telecom players to invest in new technologies, to innovate and to offer new services. Increased competitive pressure is being felt across all market segments, even though significant differences remain across services and countries. Broadband roll-out has allowed operators to offer multiple-play services, thereby transforming traditional segment boundaries and competitive market structures.competition; access; convergence; multiple-play; fixed telephony; mobile services; broadband; VoIP; MVNO

The B model as a twisted spinning particle

The B-twisted topological sigma model coupled to topological gravity is supposed to be described by an ordinary field theory: a type of holomorphic Chern-Simons theory for the open string, and the Kodaira-Spencer theory for the closed string. We show that the B model can be represented as a PARTICLE theory, obtained by reducing the sigma model to one dimension, and replacing the coupling to topological gravity by a coupling to a twisted one-dimensional supergravity. The particle can be defined on ANY Kahler manifold--it does not require the Calabi-Yau condition--so it may provide a more generalized setting for the B model than the topological sigma model. The one-loop partition function of the particle can be written in terms of the Ray-Singer torsion of the manifold, and agrees with that of the original B model. After showing how to deform the Kahler and complex structures in the particle, we prove the independence of this partition function on the Kahler structure, and investigate the origin of the holomorphic anomaly. To define other amplitudes, one needs to introduce interactions into the particle. The particle will then define a field theory, which may or may not be the Chern-Simons or Kodaira-Spencer theories.Comment: 25/17 Pages big/little (LaTeX), TAUP-2192-94, CERN-TH.7402/9

Determinant bundles, boundaries, and surgery

In this note we specialize and illustrate the ideas developed in the paper math.DG/0201112 of the first author ("Index theory, eta forms, and Deligne cohomology ") in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regularized determinant of the Dirac Laplacian obtained by the second author and K. Wojciechowski.Comment: 23 page

Cohomological Partition Functions for a Class of Bosonic Theories

We argue, that for a general class of nontrivial bosonic theories the path integral can be related to an equivariant generalization of conventional characteristic classes.Comment: 9 pages; standard LATEX fil

Equivariant Kaehler Geometry and Localization in the G/G Model

We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents on K\"ahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formula. We also show that the supersymmetry is related to a non-linear generalization (q-deformation) of the ordinary moment map of symplectic geometry in which a representation of the Lie algebra of a group $G$ is replaced by a representation of its group algebra with commutator $[g,h] = gh-hg$. In the large $k$ limit it reduces to the ordinary moment map of two-dimensional gauge theories.Comment: LaTex file, 40 A4 pages, IC/94/108 and ENSLAPP-L-469/9

A new short proof of the local index formula and some of its applications

We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler's rescaling with Greiner's approach of the heat kernel asymptotics. As application we can easily compute the Connes-Moscovici cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index formula (even case) and the Atiyah-Patodi-Singer spectral flow formula (odd case).Comment: v5: more typos fixed; 19 page

Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces

The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.Comment: LaTeX, 44 pages, to appear in Comm. Math. Phy