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
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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