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85 D -branes,strings,vertexoperatorsandgaugefieldsonthesehypersurfaces. X de-terminedbyareflexivepolytope,thatareCalabi-Yau.Inthisnote,wewillprovesomeparticularpropertiesof Itiswell-knownthatacompacttoricmanifoldisnotCalabi-Yau.However,Batyrev[3]showedtheex

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JGSP47(2018) 85–104

SPANNING CLASS IN THE CATEGORY OF BRANES

ANDRÉS VIÑA

Communicated by Jean-Louis Clerc

Abstract. Given a generic anticanonical hypersurfaceY of a toric variety de- termined by a reflexive polytope, we define a line bundleLonY that generates a spanning class in the bounded derivative categoryDb(Y). From this fact, we deduce properties of some spaces of strings related with the braneL. We prove a vanishing theorem for the vertex operators associated to strings stretching from branes of the formL⊗ito nonzero objects inDb(Y). We also define a gauge field onLwhich minimizes the corresponding Yang-Mills functional.

MSC: 81T30, 14F05, 14M05

Keywords:B-branes, derived categories of sheaves, toric varieties

Contents

1 Introduction 85

2 Calabi-Yau Hypersurfaces 87

3 B-Branes on the Hypersurface 90

4 Vertex Operators 92

5 Yang-Mills Fields onY 93

6 Equivariant Chern Class 99

References 103

1. Introduction

It is well-known that a compact toric manifold is not Calabi-Yau. However, Batyrev [3] showed the existence of anticanonical hypersurfaces, in the toric variety X de- termined by a reflexive polytope, that are Calabi-Yau. In this note, we will prove some particular properties of D-branes, strings, vertex operators and gauge fields on these hypersurfaces.

doi: 10.7546/jgsp-47-2018-85-104

85

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