Workshop on Black Holes:

BPS, BMS and Integrability.

September 7-11, 2020

BPS, BMS and Integrability.

September 7-11, 2020

- Sergei Alexandrov

Université de Montpellier ## Mock modularity and refinement: from BPS black holes to Vafa-Witten theory

2020/09/10, 14:00 — 15:00.The generating functions of degeneracies of D4-D2-D0 black holes in Type II

string compactifications on Calabi-Yau threefolds are examples of (higher depth)

mock modular forms. I'll explain how S-duality can be used to derive an explicit

form for their modular completions, which becomes particularly simple in

the presence of a refinement. This result turns out to have many applications

going beyond the original context. In particular, I'll show that it can be used

i) to reproduce and generalize in an easy way the known results on modular properties

of the generating functions of BPS dyons in N=4 string compactifications;

ii) to find Vafa-Witten invariants of arbitrary(!) rank on $\mathbb{CP}^2$, Hirzebruch and del Pezzo surfaces;

iii) to obtain holomorphic anomaly equations for BPS partition functions;

iv) to reveal a non-commutative structure induced by the refinement on the moduli

space of compactified theory.- Pedro Aniceto

Instituto Superior Técnico, Universidade de Lisboa ## $R^2$ corrected $AdS_2$ holography

2020/09/09, 15:15 — 16:15.We take concrete steps towards constructing the holographic $CFT_1$ dual to $AdS_2$ spacetimes. We work with two-dimensional effective actions obtained from $R^2$ corrected four-dimensional $N=2$ supergravity theories.

- Vijay Balasubramanian

University of Pennsylvania ## Black holes and complexity

2020/09/07, 15:15 — 16:15.- Alejandra Castro

University of Amsterdam ## The Holographic Landscape of Symmetric Product Orbifolds

2020/09/08, 09:00 — 10:00.I will discuss the large-N limit of two-dimensional symmetric product orbifolds with the purpose of constructing new examples of holography. I will provide two simple diagnostics of the BPS spectrum that allows us to classify the landscape of symmetric product orbifold theories. From this classification I will propose an infinite family of new holographic CFTs. The examples open a path to novel realizations of AdS$_3$/CFT$_2$.

- Atish Dabholkar

ICTP Trieste ## Three Avatars of Mock Modularity

2020/09/11, 14:00 — 15:00.Mock theta functions were introduced by Ramanujan in his famous last letter to Hardy in 1920, but were properly understood only recently with the work of Zwegers in 2002. I will describe three manifestations of this apparently exotic mathematics in three important physical contexts of holography, topology and duality, where mock modularity has come to play in important role. In particular, I will derive a holomorphic anomaly equation for the indexed partition function of a two-dimensional $CFT_2$ dual to $AdS_3$ that counts the black hole degeneracies, and for Vafa-Witten partition function for twisted four-dimensional $N=4$ super Yang-Mills theory on $CP_2$ for the gauge group $SO(3)$ that counts instantons. The holomorphic kernel of this equation is not modular but 'mock modular’ and one obtains correct modular properties only after including certain 'anomalous’ nonholomorphic boundary contributions. This phenomenon can be related to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact supersymmetric sigma model, and in a simpler context of quantum mechanics to the Atiyah-Patodi-Singer eta invariant. Mock modularity is thus essential to exhibit modular symmetries expected from the $AdS_3/CFT_2$ holographic equivalence in quantum gravity and the S-duality symmetry of four-dimensional quantum gauge theories.

- Kevin Goldstein

University of the Witwatersrand ## Applied CFT: Fire Phenomenology

2020/09/08, 10:15 — 11:15.Ecologists have observed scale invariance in the burn patterns of grassland fires noticing that the number of fires decrease linearly with the logarithm of the fire size. Looking at two and three point correlation functions allows us to characterise scale invariance in their data. We find that while fires do display scale invariance, rotational symmetry is broken. Zamolodchikov and Polchinski established that in two dimensions, subject to mild assumptions, any scale invariant, unitary quantum field theory is automatically conformally invariant. Since we are looking at a classical phenomena whose underlying dynamics we do not even expect to be unitary – grass can not “unburn” itself on short timescales – it is perhaps not surprising that scale invariance here does not extend to a larger symmetry group here. We hope that this work will provide new tools to analyse fire propagation and discriminate models.

- Daniel Grumiller

Technical University Viena ## $BMS_2$

2020/09/09, 14:00 — 15:00.Asymptotic symmetries of spacetimes with asymptotically vanishing Riemann tensor have found numerous applications, ranging from flat space holography to infrared properties of quantum field theories. I discuss the situation in two spacetime dimensions, define $BMS_2$ symmetries, identify a gravity model yielding $BMS_2$ symmetries and show how to obtain this model as a limit of the complex SYK model.

- Vishnu Jejjala

University of the Witwatersrand ## Probing the EVH limit of supersymmetric AdS black holes

2020/09/11, 10:15 — 11:15.Extremal black holes are well known to contain an $AdS_2$ factor in their near-horizon geometries. If the extremal limit is taken in conjunction with a specific vanishing horizon limit, the so-called Extremal Vanishing Horizon (EVH) limit, the $AdS_2$ factor lifts to a locally $AdS_3$ factor. We examine the EVH limit of asymptotically AdS black holes which preserve some supersymmetry. Using recent results on the large-N limit of the superconformal index of the dual $CFT_4$, we study the emergence of a $CFT_2$ in the IR of the $CFT_4$, which is the field theory dual to the emergence of the locally $AdS_3$ factor in the near-horizon. In particular, we show that the inverse Laplace transform of the superconformal index, yielding the black hole entropy, becomes equivalent to the derivation of a Cardy formula for the dual $CFT_2$.

- Abhiram Kidambi

Technical University Vienna ## Mock modular black hole entropy from $\frac{1}{2}$-BPS states

2020/09/10, 09:00 — 10:00.In this talk, we present an exact formula and construction to obtain the mock modular partition function for single-center $\frac{1}{4}$-BPS black holes in $\mathcal{N} =4, d= 4$ string compactification.

- Axel Kleinschmidt

Albert-Einstein-Institut ## 1/8-BPS couplings in effective string theory

2020/09/07, 10:15 — 11:15.The low-energy effective action of string theory on toroidal backgrounds contains an infinite sequence of higher-derivative terms, such as R4, D4R4 and D6R4. The higher the derivative order, the less strong the constraints from supersymmetry are. I will present recent results, obtained in collaboration with Guillaume Bossard and Boris Pioline, on the properties of the D6R4 correction in various dimensions. The calculation of this 1/8-BPS coupling proceeds via methods from exceptional field theory where supergravity is completed in a U-duality invariant way at loop level.

- Albrecht Klemm

Hausdorff Center for Mathematics ## Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the $Z_2$ Model

2020/09/11, 09:00 — 10:00.Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $N=4$ string-string duality is examined in a reduced rank theory on a less studied BPS sector. For this we identify a candidate partition function of ''untwisted'' quarter-BPS dyons in the heterotic $Z_2$ CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields a meromorphic Siegel modular form for the Iwahori subgroup $B(2) \subset Sp_4 (\mathbb{Z})$ which generates BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition function is shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our result also agrees with a recently conjectured formula of Bryan and Oberdieck for the partition function of primitive, untwisted DT invariants of the CHL orbifold $X=(K_3 \times T^2)/Z_2$, as demanded by string duality with type IIA theory on $X$.

- Thomas Mertens

Ghent University ## Bulk observables in JT gravity

2020/09/07, 14:00 — 15:00.Using a definition of bulk diff-invariant observables, we go into the bulk of 2d Jackiw-Teitelboim gravity. By mapping the computation to a Schwarzian path integral, we study exact bulk correlation functions and discuss their physical implications. We describe how the black hole thermal atmosphere gets modified by quantum gravitational corrections. Finally, we will discuss how higher topological effects further modify the spectral density and detector response in the Unruh heat bath.

- Sameer Murthy

King's College London ## The growth of the 1/16-BPS index in N=4 Super Yang-Mills

2020/09/07, 09:00 — 10:00.I will discuss the superconformal index that counts BPS states preserving two supercharges in $N=4$ SYM, and more generally in $N=1$ SCFTs, on $S^3$. This index is captured by a unitary matrix model with purely double trace operators in the action. The AdS/CFT correspondence predicts that the index should have exponential growth at large charges and large N, corresponding to the 1/16-BPS black hole (BH) in $AdS_5$. I will present analytical and numerical analyses of the matrix model which clearly show this expected BH growth. In particular, I will introduce a deformation of the matrix model which allows us to solve it at large-N. The deformation has interesting relations with the Bloch-Wigner dilogarithm, a function introduced by number theorists. Finally, using techniques from representation theory, I will show that the finite-N index actually interpolates between counting multi-gravitons at small charge and BHs at large charge.

- Boris Pioline

Sorbonne Université ## BPS indices, Vafa-Witten invariants and quivers

2020/09/10, 15:15 — 16:15.BPS indices counting D4-branes in a local Calabi Yau threefold $K_S$ are closely related to the Vafa-Witten invariants of the complex surface $S$. I will explain how to compute these indices from the quiver associated to an exceptional collection of coherent sheaves on $S$. The key observation is that the indices vanish in their respective attractor chamber, such that they can be computed elsewhere by applying the attractor flow tree formula. Based on work in collaboration with Guillaume Beaujard and Jan Manschot (arXiv:2004.14466).

- Martí Rosselló

Instituto Superior Técnico, Universidade de Lisboa ## Arithmetic of decay walls through continued fractions: a new exact dyon counting solution in $\mathcal{N} =4$ CHL models

2020/09/10, 10:15 — 11:15.We use continued fractions to perform a systematic and explicit characterization of the decays of two-centred dyonic black holes in $4D$ $\mathcal{N} =4$ heterotic $\mathbb{Z}_N$ CHL models. Thereby we give a new exact solution for the problem of counting decadent dyons in these models.

- Ricardo Schiappa

IST, Lisbon ## Resurgent Painlevé Equations & 2D Quantum Gravity

2020/09/09, 10:15 — 11:15.I will review recent and on-going work concerning applications of resurgence within the realm of minimal-string and JT quantum gravities in 2d.

- Miguel Tierz

Faculdade de Ciências, Universidade de Lisboa ## Phase transitions, random matrices and $T \overline{T}$ deformation

2020/09/09, 09:00 — 10:00.We introduce aspects of the theory of random matrices and its relevance in gauge theory. We review how phase transitions in gauge theories can be uncovered by studying double scaling limits of random matrix ensembles. A few examples based on simple matrix models will be given and followed by a discussion of the case of the $T \overline{T}$ deformation of both 2d Yang-Mills theory and 2d q-Yang-Mills theory.

- Max Zimet

Harvard University ## $K3$ metrics

2020/09/08, 14:00 — 15:00.I will explain a physically motivated construction of Ricci-flat $K3$ metrics, which gives the first examples of Ricci-flat metrics on compact non-toroidal Calabi-Yau manifolds. I will also relate it — both physically and mathematically — to a second such construction, which is as yet not completely explicit: the missing data is the BPS index of a little string theory on $T^2$. In particular, I will show that these discrete invariants may be extracted from the metrics produced by the first approach.