12 research outputs found
The functional of super Riemann surfaces -- a "semi-classical" survey
This article provides a brief discussion of the functional of super Riemann
surfaces from the point of view of classical (i.e. not "super-) differential
geometry. The discussion is based on symmetry considerations and aims to
clarify the "borderline" between classical and super differential geometry with
respect to the distinguished functional that generalizes the action of harmonic
maps and is expected to play a basic role in the discussion of "super
Teichm\"uller space". The discussion is also motivated by the fact that a
geometrical understanding of the functional of super Riemann surfaces from the
point of view of super geometry seems to provide serious issues to treat the
functional analytically
Symmetries and conservation laws of a nonlinear sigma model with gravitino
We show that the action functional of the nonlinear sigma model with
gravitino considered in a previous article [18] is invariant under rescaled
conformal transformations, super Weyl transformations and diffeomorphisms. We
give a careful geometric explanation how a variation of the metric leads to the
corresponding variation of the spinors. In particular cases and despite using
only commutative variables, the functional possesses a degenerate super
symmetry. The corresponding conservation laws lead to a geometric
interpretation of the energy-momentum tensor and supercurrent as holomorphic
sections of appropriate bundles.Comment: 27 page
Regularity of Solutions of the Nonlinear Sigma Model with Gravitino
We propose a geometric setup to study analytic aspects of a variant of the
super symmetric two-dimensional nonlinear sigma model. This functional extends
the functional of Dirac-harmonic maps by gravitino fields. The system of
Euler--Lagrange equations of the two-dimensional nonlinear sigma model with
gravitino is calculated explicitly. The gravitino terms pose additional
analytic difficulties to show smoothness of its weak solutions which are
overcome using Rivi\`ere's regularity theory and Riesz potential theory.Comment: 24 pages. This is a revised version, with some typos corrected. To
appear in Commun. Math. Phy
Super Gromov-Witten Invariants via torus localization
In this article we propose a definition of super Gromov-Witten invariants by
postulating a torus localization property for the odd directions of the moduli
spaces of super stable maps and super stable curves of genus zero. That is, we
define super Gromov-Witten invariants as the integral over the pullback of
homology classes along the evaluation maps divided by the equivariant Euler
class of the normal bundle of the embedding of the moduli space of stable spin
maps into the moduli space of super stable maps. This definition sidesteps the
difficulties of defining a supergeometric intersection theory and works with
classical intersection theory only. The properties of the normal bundles, known
from the differential geometric construction of the moduli space of super
stable maps, imply that super Gromov-Witten invariants satisfy a generalization
of Kontsevich-Manin axioms and allow for the construction of a super small
quantum cohomology ring. We describe a method to calculate super Gromov-Witten
invariants of of genus zero by a further geometric torus
localization and give explicit numbers in degree one when dimension and number
of marked points are small
Geometric analysis of the Yang-Mills-Higgs-Dirac model
The harmonic sections of the Kaluza-Klein model can be seen as a variant of
harmonic maps with additional gauge symmetry. Geometrically, they are realized
as sections of a fiber bundle associated to a principal bundle with a
connection. In this paper, we investigate geometric and analytic aspects of a
model that combines the Kaluza-Klein model with the Yang-Mills action and a
Dirac action for twisted spinors. In dimension two we show that weak solutions
of the Euler-Lagrange system are smooth. For a sequence of approximate
solutions on surfaces with uniformly bounded energies we obtain compactness
modulo bubbles, namely, energy identities and the no-neck property hold.Comment: 31 page
Supergeometry, super Riemann surfaces and the superconformal action functional
This book treats the two-dimensional non-linear supersymmetric sigma model or spinning string from the perspective of supergeometry. The objective is to understand its symmetries as geometric properties of super Riemann surfaces, which are particular complex super manifolds of dimension 1|1. The first part gives an introduction to the super differential geometry of families of super manifolds. Appropriate generalizations of principal bundles, smooth families of complex manifolds and integration theory are developed. The second part studies uniformization, U(1)-structures and connections on Super Riemann surfaces and shows how the latter can be viewed as extensions of Riemann surfaces by a gravitino field. A natural geometric action functional on super Riemann surfaces is shown to reproduce the action functional of the non-linear supersymmetric sigma model using a component field formalism. The conserved currents of this action can be identified as infinitesimal deformations of the super Riemann surface. This is in surprising analogy to the theory of Riemann surfaces and the harmonic action functional on them. This volume is aimed at both theoretical physicists interested in a careful treatment of the subject and mathematicians who want to become acquainted with the potential applications of this beautiful theory
Moduli spaces of SUSY curves and their operads
This article is dedicated to the generalization of the operad of moduli
spaces of curves to SUSY curves. SUSY curves are algebraic curves with
additional supersymmetric or supergeometric structure. Here, we focus on the
description of the relevant category of graphs and its combinatorics as well as
the construction of dual graphs of SUSY curves and the supermodular operad
taking values in a category of moduli spaces of SUSY curves with Neveu-Schwarz
and Ramond punctures