Abstract

We find an interpretation of the recent connection found between topological strings on Calabi-Yau threefolds and crystal melting: Summing over statistical mechanical configuration of melting crystal is equivalent to a quantum gravitational path integral involving fluctuations of Kähler geometry and topology. We show how the limit shape of the melting crystal emerges as the average geometry and topology of the quantum foam at the string scale. The geometry is classical at large length scales, modified to a smooth limit shape dictated by mirror geometry at string scale and is a quantum foam at area scales ∼ gsα'.

Highlights

  • The idea that quantum gravity should lead to wild fluctuations of topology and geometry at the Planck scale is an old idea [1,2]

  • Even with the advent of superstring theory as a prime candidate for a theory of quantum gravity, we still have a long way to go to understand the geometry of space at short distances

  • It is natural to try to interpret the configurations of melting crystal as target space field theory of the A-model, which should be a gravitational theory involving fluctuations of topology and geometry

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Summary

Introduction

The idea that quantum gravity should lead to wild fluctuations of topology and geometry at the Planck scale is an old idea [1,2]. It is natural to try to interpret the configurations of melting crystal as target space field theory of the A-model, which should be a gravitational theory involving fluctuations of topology and geometry. There is no one to one correspondence between the sheaves on X and blown up geometries X, and in some sense there are more sheaves than geometries We interpret this mismatch as a lesson of quantum string theory for quantum gravity: The path-integral space for quantum gravity should include classical topologies and geometries. That for topological strings a gauge theory is the fundamental description of gravity at all scales including the Planck scale, where it leads to a quantum gravitational foam. It is necessary for understanding the nature of the quantum foam.

The Basic Idea
There is exact sequence of sheaves:
Toric Geometry Preliminaries
Examples
Twisted masses
Application to C3
Physical Interpretation of the Result
More General Toric Geometries and the Topological Vertex
Target space theory viewpoint
Supersymmetric Yang-Mills theory on six-manifold
Toric localization
Gauge vertex
Torsion free sheaves and general Calabi-Yau’s
Full Text
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