Geometry and Automorphicity of Supersymmetric Partitions

13 Feb 2023, 09:00 → 18 Feb 2023, 11:00 Asia/Tokyo

Main Lecture Hall (Kavli Institute for Physics and Mathematics of the Universe, University of Tokyo (Kashiwa Campus))

This hybrid workshop will be a focused event aiming to bring together physicists and mathematicians working on the interface of string theory, quantum field theory, geometry, representation theory and number theory - with particular emphasis on the study of partition functions - for a fruitful exchange of ideas. The event will consist of one or two talks per day leaving the rest of the workshop time free for discussions and interactions with other participants.

Confirmed speakers (so far): 

1. Noah Arbesfeld (Imperial College, London & Kavli IPMU)
2. Gabriel Cardoso (CAMSGD, Lisbon)
3. Miranda Cheng (Academia Sinica & U.Amsterdam) (Remote) 
4. Suresh Govindarajan (IIT Madras)
5. Suresh Nampuri (IST Lisbon)
6. Caner Nazaroglu (University of Cologne)
7. Davide Passaro (U.Amsterdam) (Remote)
8. Boris Pioline (LPTHE)
9. Kaiwen Sun (KIAS)
10. Masahito Yamazaki (Kavli IPMU)

Please note: On-site participation is moderated and subject to the Covid-19 policy of the University of Tokyo. We expect to be able to host a maximum of 60 participants on-site.

Registration deadline for onsite participation: 31st Dec 2022, or until max. capacity is reached. 

Preference for on-site participation will be given to faculty members, postdocs, as well as graduate/PhD students working on topics related to the theme of the workshop. 

For online participation: Registration deadline: 1st February 2023.

 

Organizers: Ioana Coman, Abhiram Kidambi, Emily Nardoni

Poster_GASP.pdf

Monday 13 February

1 Registration

2 BPS Algebras for Toric Calabi-Yau Manifolds (Masahito Yamazaki)

I will discuss old and new developments for BPS state counting problems for toric Calabi-Yau manifolds. The main ingredients will include quivers, crystals and quiver Yangians.

3 Informal Discussion Session

12:00 Lunch

4 Discussion Session

15:00 Tea & Coffee Break

5 Informal Discussion Session

6 State Counting and Topology with Quantum Modular Forms (Miranda Cheng)

Quantum modular forms are functions with delicate modular properties that generalize mock modular forms. The q-series 3-manifold invariants provide new insights and computational tools in 3-manifold topology, 3d SQFT, and M-theory compactifications. In this talk I will survey the relation between these q-series invariants and quantum modular form.

Tuesday 14 February

7 Modularity of BPS indices on Calabi-Yau threefolds (Boris Pioline)

Unlike in cases with maximal or half-maximal supersymmetry, the spectrum of BPS states in type II string theory compactified on a Calabi-Yau threefold with generic SU(3) holonomy remains partially understood. Mathematically, the BPS indices coincide with the generalized Donaldson-Thomas invariants associated to the derived category of coherent sheaves, but they are rarely known explicitly. String dualities indicate that suitable generating series of rank 0 Donaldson-Thomas invariants counting D4-D2-D0 bound states should transform as vector-valued mock modular forms, in a precise sense. I will spell out and test these predictions in the case of one-modulus compact Calabi-Yau threefolds such as the quintic hypersurface in $P^4$, where rank 0 DT invariants can (at least in principle) be computed from Gopakumar-Vafa invariants, using recent mathematical results by S. Feyzbakhsh and R. Thomas.

8 Informal Discussion Session

12:00 Lunch

9 Discussion Session

15:00 Tea & Coffee Break

10 Searching for a new type of Lie algebra (Suresh Govindarajan)

The generating functions (and refinements thereof) of the
degeneracies of quarter BPS states in four-dimensional N=4
supersymmetric theories that arise from type II compactifications on K3XT2 and its asymmetric CHL orbifolds are genus-two Siegel modular forms. In some of the cases, the walls of marginal stability across which two-centred BPS states decay into single centered ones lead to rank-three Lorentzian root lattices with Weyl vector. In all but three examples, the square-roots of generating functions, are the Weyl-Kac-Borcherds denominator formula for some Borcherds-Kac-Moody (BKM) Lie superalgebra. Rank-three Lorentzian lattices with Weyl vectors have been classified long ago by Nikulin. The three examples that do not have an Lie algebraic interpretation have Weyl vectors of hyperbolic type. Gritsenko and Nikulin have a no-go theorem that states that such Lorentzian lattices are not related to any BKM Lie superalgebra.

The dyon generating functions lead to potential denominator formulae for a new kind of Lie superalgebra. We study these denominator formulae in terms of an affine sl(2) subalgebra and a Borcherds extension of the affine sl(2) subalgebra. We discuss our studies on the decompositon of the potential denominator formula in terms of the characters of both sub-algebras. An important result is the appearance of fermionic roots with unusual behaviour. We are able to characterise the multiplicity of various imaginary simple roots in terms of vector valued modular forms for which we can give closed formulae in some cases.

Wednesday 15 February

11 Computations in the algebro-geometric approach to Vafa-Witten theory (Noah Arbesfeld)

I'll present Tanaka-Thomas's algebro-geometric approach to Vafa-Witten invariants of projective surfaces. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components; S-duality implies conjectural symmetries between these contributions.

I'll then explain work in progress with M. Kool and T. Laarakker on the "vertical" or "monopole" component, which can be regarded as a nested Hilbert scheme on a surface. Namely, we apply a recent blow-up identity of Kuhn-Leigh-Tanaka to obtain constraints on Vafa-Witten invariants of the vertical component predicted by Göttsche-Kool-Laarakker. One consequence is a complete formula for refined invariants of this component in rank 2.

10:30 Coffee and Tea Break

12 Informal Discussion Session

12:30 Lunch

13 Optional Discussion Session

13:30 Optional IPMU Colloquium (Math meets computer graphics)

15:00 Tea and Coffee

14 Logarithmic vertex operator algebras and 3 manifold invariants (Davide Passaro)

Logarithmic vertex operator algebras (Log-VOAs) are vertex operator algebras that admit reducible but indecomposable modules. They formalize the underlying mathematical structure of logarithmic conformal field theories and have been used to study various phenomena including the quantum Hall effect, percolation and limits of the Q-Potts and O(n) lattice models. Recently a connection between characters of certain Log-VOAs and the q-series 3-manifold invariant was discovered. In this talk I will describe a class of Log-VOAs called logarithmic extensions of minimal models and I will demonstrate the relation that the characters of these Log-VOAs have with the q-series 3-manifold invariant.

15 Discussion Session

Thursday 16 February

16 "The gravitational path integral for N=4 BPS black holes from black hole microstate counting: part 1 (Gabriel Cardoso)

The degeneracies of 1/4 BPS black holes in four-dimensional D=4 heterotic string theory are given in terms of the Fourier coefficients of the meromorphic Siegel modular form $1/\Phi_{10}$.
In the first part of this talk, we show how to obtain an exact expression for these degeneracies by using the symplectic symmetries of $1/\Phi_{10}$ to construct a fine-grained Rademacher type expansion
which expresses these BPS degeneracies as a regularized sum over residues of the poles of $1/\Phi_{10}$. The construction uses two distinct
$SL(2, \mathbb{Z})$ subgroups of $Sp(4,\mathbb{Z})$ which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein.

17 Informal Discussion Session

12:00 Lunch

18 The gravitational path integral for N=4 BPS black holes from black hole microstate counting: part 2 (Suresh Nampuri)

TBA

15:00 Tea and Coffee

19 Discussion Session (Rossello)

20 Informal Discussion

Friday 17 February

21 Integrals of Meromorphic Jacobi Forms and Mock/False Modular Forms at Higher Depth (Caner Nazaroglu)

Integrals involving meromorphic Jacobi forms appear in physical applications such as black hole counting, N=2 Schur indices, and elliptic genera of 2D CFT’s with non-compact target space. In this talk, I will review a number of such applications and describe how they can lead to mock and false modular forms at higher depth. Next, I will describe joint work with Bringmann, Kaszian, and Milas that gives a methodical exploration into modular properties of false theta functions based on concepts developed for higher depth mock modular forms. I will then give examples on how such generalized modular properties can be used to obtain Rademacher type exact formulae for the Fourier coefficients of (higher depth) false/mock modular forms. Finally, I will explain how the modular framework generalizes to a certain subclass of theta functions that are both indefinite and false using Zwegers’ mock Maass theta functions and comment on further interrelations and developments (based on joint work with Bringmann).

22 Informal Discussion Session

12:00 Lunch

23 2d CFTs, Borcherds products and hyperbolization of affine Lie algebras (Kaiwen Sun)

In 1983, Feigold and Frenkel posed a question about possible relations between affine Lie algebras, hyperbolic Kac--Moody algebras and Siegel modular forms. We give an automorphic answer to this question and its generalization. We classify hyperbolic Borcherds--Kac--Moody superalgebras whose super-denominators define reflective automorphic products of singular weight on lattices of type $2U\oplus L$. We prove that there are exactly 81 affine Lie algebras $g$ which have nice extensions to hyperbolic BKM superalgebras for which the leading Fourier--Jacobi coefficients of super-denominators coincide with the denominators of $g$. We find that 69 of them appear in Schellekens’ list of semi-simple $V_1$ structures of holomorphic CFT of central charge 24, while 8 of them correspond to the $N=1$ structures of holomorphic SCFT of central charge 12 composed of 24 chiral fermions. The 4 extra cases are related to the exceptional modular invariants from nontrivial automorphisms of fusion algebras. This is based on a joint work with Haowu Wang and Brandon Williams.

15:00 Tea and Coffee

24 Optional Discussion Session

25 Optional Colloquium (A world from a sheet of paper)

Saturday 18 February

26 Final discussion session and closing