18 mai 2021

Mardi 18 mai 14:00-15:15 Boris Pioline (LPTHE (Paris 6))
Attractor invariants for local Calabi-Yau threefolds

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Résumé : The Donaldson-Thomas invariants $\Omega(\gamma,z)$ associated to
the bounded derived category of coherent sheaves $D(X)$
on a Calabi-Yau threefold $X$ are the mathematical incarnation of the
indices counting supersymmetric (or BPS) states in string theory
compactified on $X$. They depend on the Chern vector $\gamma$ and exhibit
discontinuities as the Kahler moduli $z$ are varied. The attractor
invariants $\Omega_\star(\gamma)$ are special instances of $\Omega(\gamma,z)$ where
$z$ corresponds to the self-stability (or attractor) condition.
When $X$ is a crepant resolution of a toric singularity,
$\Omega(\gamma,z)$ can be computed by using the equivalence between $D(X)$
and the category $D(Q)$ of representations of a quiver with potential. For
such $X$ ,I will present evidence for the « Attractor conjecture », which
states that the attractor invariants are essentially (but not completely) trivial. Using
the attractor flow tree formulae, this conjecture determines
the full set of DT invariants for any $\gamma$ and $z$. When $X$ is the
canonical bundle over a Fano surface $S$, I will argue that suitable
generating series of DT invariants in the anti-attractor chamber must
exhibit mock modular properties.

Attractor invariants for local Calabi-Yau threefolds  Version PDF