arXiv subjects

Enno Keßler

Publications and source records attributed to Enno Keßler.

12 recordsLinked to original sources

arXiv2023

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 $\mathbb{P}^n$ 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.

math.AG↗

arXiv2023

Torus Actions on Moduli Spaces of Super Stable Maps of Genus Zero

We construct smooth $\mathbb{C}^*$-actions on the moduli spaces of super $J$-holomorphic curves as well as super stable curves and super stable maps of genus zero and fixed tree type such that their reduced spaces are torus invariant. Furthermore, we give explicit descriptions of the normal bundles to the fixed loci in terms of spinor bundles and their sections. Main steps to the construction of the $\mathbb{C}^*$-action are the proof that the charts of the moduli space of super $J$-holomorphic curves obtained by the implicit function theorem yield a smooth split atlas and a detailed study of the superconformal automorphism group of $\mathbb{P}_{\mathbb{C}}^{1|1}$ and its action on component fields.

math.DG↗

arXiv2022

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.

math-ph↗

arXiv2022

Moduli spaces of SUSY curves and their operads

This article generalizes 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 SUSY operad taking values in a category of moduli spaces of SUSY curves with Neveu-Schwarz and Ramond punctures.

math.AG↗

arXiv2021

Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures

In this article we define stable supercurves and super stable maps of genus zero via labeled trees. We prove that the moduli space of stable supercurves and super stable maps of fixed tree type are quotient superorbifolds. To this end, we prove a slice theorem for the action of super Lie groups on Riemannian supermanifolds and discuss superorbifolds. Furthermore, we propose a Gromov topology on super stable maps such that the restriction to fixed tree type yields the quotient topology from the superorbifolds and the reduction is compact. This would, possibly, lead to the notions of super Gromov-Witten invariants and small super quantum cohomology to be discussed in sequels.

math.DG↗

arXiv2020

Super $J$-holomorphic Curves: Construction of the Moduli Space

Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $Φ\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ to $J$. Such a super $J$-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super $J$-holomorphic curves as a smooth subsupermanifold of the space of maps $M\to N$.

math.DG↗

arXiv2017

From harmonic maps to the nonlinear supersymmetric sigma model of quantum field theory. At the interface of theoretical physics, Riemannian geometry and nonlinear analysis

Harmonic maps from Riemann surfaces arise from a conformally invariant variational problem. Therefore, on one hand, they are intimately connected with moduli spaces of Riemann surfaces, and on the other hand, because the conformal group is noncompact, constitute a prototype for the formation of singularities, the so-called bubbles, in geometric analysis. In theoretical physics, they arise from the nonlinear $σ$-model of quantum field theory. That model possesses a supersymmetric extension, coupling a harmonic map like field with a nonlinear spinor field. In the physical model, that spinor field is anticommuting. In this contribution, we analyze both a mathematical version with a commuting spinor field and the original supersymmetric version. Moreover, this model gives rise to a further field, a gravitino, that can be seen as the supersymmetric partner of a Riemann surface metric. Altogether, this leads to a beautiful combination of concepts from quantum field theory, structures from Riemannian geometry and Riemann surface theory, and methods of nonlinear geometric analysis.

math.DG↗

arXiv2017

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 Rivière's regularity theory and Riesz potential theory.

math.DG↗

arXiv2017

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.

math.DG↗

arXiv2015

Super Riemann Surfaces and the Super Conformal Action Functional

Riemann surfaces are two-dimensional manifolds with a conformal class of metrics. It is well known that the harmonic action functional and harmonic maps are tools to study the moduli space of Riemann surfaces. Super Riemann surfaces are an analogue of Riemann surfaces in the world of super geometry. After a short introduction to super differential geometry we will compare Riemann surfaces and super Riemann surfaces. We will see that super Riemann surfaces can be viewed as Riemann surfaces with an additional field, the gravitino. An extension of the harmonic action functional to super Riemann surfaces is presented and applications to the moduli space of super Riemann surfaces are considered.

math.DG↗

arXiv2015

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 Teichmüller 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.

math.DG↗

arXiv2015

Super Riemann surfaces, metrics, and gravitinos

The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of non-linear super symmetric sigma models, the action functional underlying string theory, can be obtained from a geometric action functional on super Riemann surfaces. All invariances of the super symmetric action functional are explained in super geometric terms and the action functional is a functional on the moduli space of super Riemann surfaces.

math-ph↗