2 TOM BRIDGELAND

and tilting; this is used in Section 6 to describe other examples of spaces of stabil-

ity conditions relating to local Calabi-Yau varieties and certain non-commutative

algebras. The ﬁnal, rather speculative section is concerned with extra structures

that may exist on spaces of stability conditions.

2. Background from string theory

The main players in the mirror symmetry story are N=2 superconformal ﬁeld

theories, henceforth simply referred to as SCFTs. We mathematicians have little

intuition for what these are, and until we do mirror symmetry will remain a mystery.

This section contains a collection of statements about SCFTs that I have learnt more

or less by rote by reading physics papers and talking to physicists. I do not claim to

have any idea of what an SCFT really is and all statements in this section should be

taken with a hefty pinch of salt. Nonetheless, I believe that in the not-too-distant

future mathematicians will be able to deﬁne SCFTs and prove (or disprove) such

statements, and I hope that in the meantime the general picture will be of some

value in orienting the reader for what follows. Bourbakistes should skip to Section

3.

2.1. Abstract SCFTs. An SCFT has associated to it a pair of topological

conformal ﬁeld theories (TCFTs), the so-called A and B models. It is now fairly

well-understood mathematically that a TCFT corresponds to a Calabi-Yau A∞

category. The branes of the theory are the objects of this category. For more

precise statements the reader can consult Costello’s paper [21]. However it is

possible that the TCFTs that arise as topological twistings of SCFTs have some

additional properties. Recall that any A∞ category has a minimal model obtained

by taking cohomology with respect to the diﬀerential d = m1. One thing we

shall always require is that this underlying cohomology category is triangulated; as

explained in [9] this is a kind of completeness condition.

The moduli space M of SCFTs up to isomorphism is expected to be a reason-

ably well-behaved complex space. Studying this space M is a good way to try to

understand what sort of information an SCFT contains. There is an even better

behaved space N lying over M which physicists often refer to as Teichm¨ uller space,

consisting of SCFTs with some sort of framing. This is the place for example where

ratios of central charges of branes are well-deﬁned (see Section 2.4 below). One

might perhaps hope that N is a complex manifold with a discrete group action

whose quotient is M.

There is an involution on the space M, called the mirror map, induced by

an involution of the N = 2 superconformal algebra. This involution has the ef-

fect of exchanging the A and B models. The mirror map permutes the connected

components of M in a non-trivial way. At general points of M there are two folia-

tions corresponding to theories with ﬁxed A or ﬁxed B model respectively. Clearly

these must be exchanged by the mirror map. In many examples it seems that the

Teichm¨ uller space N has a local product structure inducing the two foliations on

M. Note that this implies that the two topological twists together determine the

underlying SCFT. The suggestion that this last statement might hold in general

seems to be extremely unpalatable to most physicists; it would be interesting to see

some examples where it breaks down.

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