Proceedings of Symposia in Pure Mathematics

Spaces of stability conditions

Tom Bridgeland

Abstract. Stability conditions are a mathematical way to understand Π-

stability for D-branes in string theory. Spaces of stability conditions seem to

be related to moduli spaces of conformal field theories. This is a survey article

describing what is currently known about spaces of stability conditions, and

giving some pointers for future research.

1. Introduction

Stability conditions on triangulated categories were introduced in [12]. The

motivation was to understand Douglas’ work on Π-stability for D-branes in string

theory [23, 24]. Since then a fair number of examples have been computed and

the definition has been further scrutinized. The aim of this paper is twofold: firstly

to survey the known examples of spaces of stability conditions, and secondly to

float the idea that there is some yet-to-be-discovered construction that will allow

one to define interesting geometric structures on these spaces. Understanding this

construction seems to me to be the logical next step, but there are certainly other

interesting questions that can be asked, and throughout the paper I have tried

to point out some of the many parts of the story that have yet to be properly

understood.

It is perhaps fair to say that the whole subject of stability conditions has a

slightly temporary feel to it because the definition itself looks a bit unnatural. On

the other hand many of the examples that have been computed are extremely neat,

and the whole idea of extracting geometry from homological algebra is, to me at

least, a very attractive one. Certainly, the agreement between spaces of stability

conditions and moduli spaces of conformal field theories is impressive enough to

suggest that stability conditions do indeed capture some part of the mathematics

of string theory. My own feeling is that at some point in the near future the notion

of a stability condition will be subsumed into some more satisfactory framework.

The detailed contents of the paper are as follows. Section 2 is about moduli

spaces of superconformal field theories. This section does not constitute rigorous

mathematics but rather provides background to what follows. Section 3 contains

the basic definitions concerning stability conditions; it is eﬀectively a summary of

the contents of [12]. Section 4 lists the known examples of spaces of stability con-

ditions on smooth projective varieties. Section 5 contains material on t-structures

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Proceedings of Symposia in Pure Mathematics

Volume 80.1, 2009

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http://dx.doi.org/10.1090/pspum/080.1/2483930