ANR Project TopStringDT, Contract number: ANR-21-CE31-0021, Nov. 1st, 2021 to January 30, 2026
The scientific goal of this ANR project was to combine and develop methods from string theory and algebraic geometry to compute microscopic degeneracies of supersymmetric black holes in string theory compactifications on Calabi-Yau manifolds. In less technical terms: examine black holes under the mathematical microscope.
Since the work of Bardeen, Carter, Bekenstein, and Hawking in the 70s, it is well known that black holes satisfy the fundamental principles of thermodynamics, and in particular, possess an entropy proportional to the area of their horizon, measured in Planck scale units. By analogy with conventional thermodynamic systems, it is natural to think that this entropy corresponds to the existence of an exponential number of quantum microstates, all with the same macroscopic properties. Superstring theory defines a coherent quantum theory of gravity, constrained by supersymmetry, and thus provides a coherent framework for addressing this question. In the late 1990s, Strominger, Vafa, and others realized that in this setting, the microstates of supersymmetric black holes were described by extended solitonic objects, called Dirichlet branes, wrapped in the compact directions of spacetime required by string theory. The number of these microstates is an algebraic invariant of the space of internal dimensions, called the Donaldson-Thomas (DT) invariant, of great interest for both physics and mathematics.
Calabi-Yau manifolds, the technical term for 6-dimensional spaces used as inner dimensions for compactifications 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.
All publications are available on [arXiv] and [HAL]
[arXiv:2110.06652] Annales Henri Poincare 23 (2022) 3633
On the existence of scaling multi-centered black holes
by P. Descombes, B. Pioline
[arXiv:2204.02207] Adv.Theor.Math.Phys. 27 (2023) 683
Modular bootstrap for D4-D2-D0 indices on compact Calabi-Yau threefolds
by S. Alexandrov, N. Gaddam, J. Manschot, B. Pioline
[arXiv:2204.06506] JHEP 09 (2022) 019
Time reversal and CP invariance in Calabi-Yau compactifications
by K. Bonisch, M. Elmi, A. Kashani-Poor, A. Klemm
[arXiv:2210.10712] Commun.Math.Phys. 405 (2024) 108
BPS Dendroscopy on Local P2
P. Bousseau, P. Descombes, B. le Floch, B. Pioline
[arXiv:2211.14601], JHEP 07 (2023) 208
Affine characters at negative level and elliptic genera of non-critical strings
by D. J. Duque, A. Kashani-Poor
[arXiv:2212.08655] Commun.Math.Phys. 405 (2024) 62
Topological Strings on Non-Commutative Resolutions
by S. Katz, A. Klemm, T. Schimannek, E. Sharpe
[arXiv:2212.04503] JHEP 03 (2023) 090
The discrete Green-Schwarz mechanism in 6D F-Theory and Elliptic Genera of Non-Critical String
by M. Dierigl, P. Oehlmann, T. Schimannek
[arXiv:2301.08066] Commun.Num.Theor.Phys. 18 (2024) 49
Quantum geometry, stability and modularity
by S. Alexandrov, S. Feyzbakhsh, A. Klemm, B. Pioline, T. Schimannek
[arXiv:2305.19916] SciPost Phys. 16 (2024) 079
Non-perturbative topological string theory on compact Calabi-Yau 3-folds
by J. Gu, A. Kashani-Poor, A. Klemm, M. Marino
New non-commutative resolutions of determinantal Calabi-Yau threefolds from hybrid GLSM
by S. Katz, T. Schimannek
[arXiv:2311.17638] SIGMA 20 (2024) 073
Resurgence of Refined Topological Strings and Dual Partition Functions
by S. Alexandrov, M. Marino, B. Pioline
[arXiv:2312.12629], to appear in CNTP
Quantum geometry and mock modularity
by S. Alexandrov, S. Feyzbakhsh, A. Klemm, B. Pioline
[arXiv:2408.02994], to appear in ATMP
Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
by C. Doran, B. Pioline, T. Schimannek
[arXiv:2412.16140] JHEP 06 (2025) 253
Borel singularities and Stokes constants of the topological string free energy on one-paramete
by S. Douaud, A. Kashani-Poor
[arXiv:2502.04606] JHEP 07 (2025) 123
An Airy Tale at Large N
by N. Bobev, P. de Smet, J. Hong, V. Reys, X. Zhang
Hyperbolic localization of the Donaldson-Thomas sheaf
by P. Descombes
[arXiv:2507.08511], to appear in JHEP
Black Hole Quantum Mechanics and Generalized Error Functions
by B. Pioline, R. Raj
The twisted geometry of 6d F-theory vacua with discrete gauge symmetries
by D. J. Duque, A. Kashani-Poor, T. Schimannek
Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds
by B. Pioline, T. Schimannek
Supergravity anomaly equations from modularity of Calabi--Yau threefolds
by C. F. Cota