BPS black holes, topological strings and Donaldson-Thomas invariants – TopStringDT

Calabi-Yau manifolds, the technical term for 6-dimensional spaces used as inner dimensions for compact- ifications of string theories, have a very rich typology, and the Donaldson-Thomas invariants relevant to the entropy of 4-dimensional black holes are very difficult to compute in general. Gromov-Witten invariants, which play an analogous role for 5-dimensional black holes, are more easily computable, thanks in part to the mirror symmetry between pairs of Calabi-Yau spaces, and are controlled by topological string theory, a simplified version of the original superstrings. The two types of invariants are related by so-called ’wall-crossing’ formulas, interpreted physically as the formation or dissolution of black hole bound states. Moreover, the duality symmetries of string theories predict subtle number-theoretical relations between these invariants, called modularity relations. In the case of non-compact, toric Calabi-Yau manifolds, representation theory provides an alternative approach to the calculation of Donaldson-Thomas invariants, which are no longer directly linked to black hole microstates (because gravity is decoupled) but instead count supersymmetric magnetic (or in general dyonic) monopoles.

With the help of our PhD and postdoctoral students and in collaboration with physicist and mathematician colleagues from other countries, we were able to exploit these links between physics and mathematics and, for the first time, calculate infinite families of Donaldson-Thomas invariants for compact Calabi-Yau manifolds with a small number of parameters, including the paradigmatic example of the quintic manifold, and verify the modular properties predicted by physics. We pushed the calculation of Gromov-Witten invariants beyond existing limits and opened a new window onto the non-perturbative regime of topological string theory.

These results open new perspectives on the mathematical aspects of string theories (much cheaper to explore than their phenomenological consequences) and suggest new connections between different fields of physics and mathematics. In particular, the direct relationship we have highlighted, between Donaldson-Thomas invariants counting BPS states on one side, and the Stokes coefficients controlling the singularities of topological amplitudes in the Borel plane, is a major step in understanding non-perturbative effects in string and field theory.

Elucidating the microscopic origin of the entropy of black holes is a key objective of any putative theory of quantum gravity. String theory approached this milestone 25 years ago, with the first quantitative description of the micro-states of supersymmetric black holes (also known as BPS black holes) in terms of bound states of extended solitons (known as D-branes) wrapped around minimal submanifolds in the internal compactification space. The spectacular quantitative agreement between the (logarithm of the) number of micro-states and the Bekenstein-Hawking entropy has sparked an extremely fruitful dialogue between high energy physicists and mathematicians. However, a full accounting of the micro-states of BPS black holes has only be obtained to date in the case of string vacua with maximal or half-maximal supersymmetry. The main objective of this project is to perform a similar exact accounting in the far more challenging case of four-dimensional superstring vacua with N=2 supersymmetry, which is the minimal amount of supersymmetry which allows the very existence of BPS states, and to build new bridges between physics and mathematics in the process.

String vacua with N=2 supersymmetry in four dimensions arise primarily by compactifying type IIA strings on a Calabi-Yau threefold (CY3) X. BPS black holes then arise from bound states of D0-branes, D2-branes wrapped on curves, D4-branes wrapped on divisors, and D6-branes wrapped on X. In mathematical terms, they correspond to stable objects in the derived category of coherent sheaves D(X), and are counted by the generalized Donaldson-Thomas (DT) invariants associated to the category D(X). The DT invariants depend sensitively on the Kähler moduli of X,and exhibit a complicated pattern of jumps across walls in Kähler moduli space, reflecting the appearance or disappearance of multi-centered black hole bound states. Fortunately, there exist distinguished choices of the moduli where the problem simplifies. At the attractor point, most bound states are absent and the computation of the resulting attractor index becomes more tractable. Once these attractor invariants are determined, the DT invariants for any other moduli can be deduced using the attractor flow tree formula. At the large radius point, DT invariants are related to the topological string partition function, for which many computational methods have been developped.

The main goal of this project will be to derive explicit results and structural properties of generalized Donaldson-Thomas invariants and topological string partition functions of Calabi-Yau threefolds, both in the compact and non-compact cases. In the first case, we shall rely on the equivalence between the category coherent sheaves D(X) and the category D(Q,W) of representations of quiver with potential, which allows to convert the computation of DT invariants into a representation-theoretic problem, related to the combinatorics of molten crystals. In this context, we shall aim to prove the Attractor Conjecture recently put forward by the scientific coordinator and his collaborators, which predicts a very simple structure for the attractor indices associated to toric CY threefolds. We shall also derive the modularity properties of generating series of such invariants, by constructing a suitable Vertex Operator Algebra acting on the cohomology of the moduli space of quiver representations. In the compact case, we shall investigate the relation between the topological string partition function and the spectrum of BPS states at other special points in moduli space such as K-points and rank-two attractor points, and determine generating series of attractor invariants for one-parameter families of CY threefolds by exploiting mock modular properties and string dualities.