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
1
Proceedings of Symposia in Pure Mathematics
Volume 80.1, 2009
1
http://dx.doi.org/10.1090/pspum/080.1/2483930
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