We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form Q. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by Q. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.

Japan

Meer Ashwinkumar

Matthew Dodelson

Abhiram Kidambi

United States

Jacob Leedom

Masahito Yamazaki

Chern-Simons invariants from ensemble averages

The degeneracies of single-centered dyonic   <math> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math>  $$ \frac{1}{4} $$ -BPS black holes (BH) in Type II string theory on K3×T 2 are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form Φ10. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with U-duality invariant ∆ &lt; 0 are in exact correspondence with the solution sets of the Brahmagupta-Pell equation, which implies that they are isomorphic to the group of units in the order ℤ[  <math> <msqrt> <mfenced> <mi>Δ</mi> </mfenced> </msqrt> </math>  $$ \sqrt{\left|\Delta \right|} $$ ] in the real quadratic field ℚ(  <math> <msqrt> <mfenced> <mi>Δ</mi> </mfenced> </msqrt> </math>  $$ \sqrt{\left|\Delta \right|} $$ ). We check our formula against the known numerical data arising from the Igusa cusp form, for the first 1650 polar coefficients, and find perfect agreement.

Austria

India

Abhishek Chowdhury

United Kingdom

Sameer Murthy

Belgium

Valentin Reys

Timm Wrase

Dyonic black hole degeneracies in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 string theory from Dabholkar-Harvey degeneracies

In this paper we study compactifications of the $$ \mathcal{N} $$ = 2 heterotic string on ( 3 )/ℤ with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Yau geometries are 3 fibrations that are also genus one fibered with three- sections. Moreover, we show that the intersection form on the polarization lattice of the 3 fibration has to be three times the intersection form on the Narain lattice Γ .

Andreas Banlaki

Ireland

Aradhita Chattopadhyaya

Thorsten Schimannek

Maria Schimpf

Heterotic strings on (

We provide further evidence that CY 3 manifolds are involved in an intricate way in Mathieu moonshine, i.e., their Gromov-Witten invariants are related to the expansion coefficients of the twined/twisted-twined elliptic genera of K3. We use the string duality between CHL orbifolds of heterotic string theory on K3 × T 2 and type IIA string theory on CY 3 manifolds to explicitly show this connection. We then work out two concrete examples where we exactly match the expansion coefficients on both sides of the duality.

On Mathieu moonshine and Gromov-Witten invariants

A few years ago a connection between the elliptic genus of the K3 manifold and the largest Mathieu group M 24 was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic genera and sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the elliptic genera of certain Calabi-Yau d -folds for d &gt; 3 have expansions that could potentially arise from underlying sporadic symmetry groups. We explore such potential connections by calculating twined elliptic genera for a large number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces, for a toroidal orbifold and two Gepner models.

Harald Skarke

Calabi-Yau manifolds and sporadic groups