Supermanifolds, rigid manifolds and mirror symmetry

“…Sigma-models having supermanifolds as target spaces were considered in an interesting paper [5]. However, the approach of [5] leads to a conclusion that in the case when the target space of A-model is a supermanifold the contribution of rational curves to correlation functions vanishes (i.e.…”

Section: Introductionmentioning

confidence: 99%

Sigma-models having supermanifolds as target spaces

Schwarz¹

1996

Lett Math Phys

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We study a topological sigma-model (A-model) in the case when the target space is an (m0|m1)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m0 − m1)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.Our goal is to study a two-dimensional topological σ-model (A-model). Sigma-models having supermanifolds as target spaces were considered in an interesting paper [5]. However, the approach of [5] leads to a conclusion that in the case when the target space of A-model is a supermanifold the contribution of rational curves to correlation functions vanishes (i.e. these functions are essentially trivial). In our approach A-model having a (m 0 |m 1 )-dimensional supermanifold as a target space is not trivial, but it is equivalent to an A-model with (m 0 − m 1 )-dimensional target space. We hope, that this equivalence can be used to understand better the mirror symmetry, because it permits us to replace most interesting target spaces with supermanifolds having non-trivial Killing vectors and to use T -duality.We start with a definition of A-model given in [1]. This definition can be applied to the case when the target space is a complex Kähler supermanifold M . Repeating the consideration of [1] we see that the correlation functions can be expressed in terms of rational curves in M , i.e. holomorphic maps of CP 1 into M . (We restrict ourselves to the genus 0 case and assume that the situation is generic; these restrictions will be lifted in a forthcoming paper [8]).Let us consider for simplicity the case when (m 0 |m 1 )-dimensional complex supermanifold M corresponds to an m 1 -dimensional holomorphic vector bundle α over an m 0 -dimensional complex manifold M 0 (i.e. M can be obtained

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“…According to the usual rules, the mirror image of such spaces has no Kähler moduli and hence it cannot be a usual CY manifold. In [6] Sethi argued that the dual of a rigid CY is instead given by a CY super-manifold.…”

Section: Introductionmentioning

confidence: 99%

The WZW-model: From supergeometry to logarithmic CFT

2006

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We present a complete solution of the WZW model on the supergroup GL(1|1). Our analysis begins with a careful study of its minisuperspace limit ("harmonic analysis on the supergroup"). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the representation theory of the gl(1|1) current algebra and propose an Ansatz for the state space of the WZW model. The latter is established through an explicit computation of the correlation function. We show in particular, that the 4-point functions of the theory factorize on the proposed set of states and that the model possesses an interesting spectral flow symmetry. The note concludes with some remarks on generalizations to other supergroups.

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Supermanifolds, rigid manifolds and mirror symmetry

Cited by 71 publications

References 31 publications

Generalized Homological Mirror Symmetry and cubics

Generalized Homological Mirror Symmetry and cubics

Sigma-models having supermanifolds as target spaces

The WZW-model: From supergeometry to logarithmic CFT

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