We define a notion of ‘quantum entanglement index’ with the aim to compute it for black hole horizons in string theory at one-loop order using the stringy replica method. We consider the horizon of BTZ black holes to construct the relevant conical orbifolds, labeled by an odd integer $\textit{N}$, and compute the partition function as a function of $\textit{N}$, corresponding to the fractional indexed Rényi entropy. We show that it is free of tachyons and naturally finite both in the ultraviolet and the infrared, even though it is generically ultraviolet divergent in the field theory limit. Thus, the index provides a useful diagnostic of the entanglement structure of string theory without the need for analytic continuation in $\textit{N}$.

Italy

Atish Dabholkar

Eleanor Harris

Netherlands

Upamanyu Moitra

Quantum entanglement index in string theory

<p>We consider a scalar field coupled to massless fermions through Yukawa couplings, such as the Higgs field, in a Robertson-Walker spacetime. We compute the nonlocal quantum effective action as a functional of the background scalar field and the scale factor at one loop order by integrating the Weyl anomaly resulting from the fermions in the loop. We show that the mode expansion of the scalar field is modified by the quantum nonlocality and the power spectrum of scalar perturbations exiting the horizon exhibits a slow roll with a red tilt proportional to the square of the Yukawa coupling even though the classical spacetime is exactly de Sitter. We comment on possible implications for cosmology.</p>

André Benevides

Quantum slow roll

We consider D-branes in the Dabholkar-Park (DP) background, a 9d orientifold theory obtained by gauging symmetry in the type IIB string theory compactified on a circle. Using anomalies in the world-sheet theory, we provide physical insights into the classification of stable D-branes by relative KR-theory. The nature, such as stability, of D-branes wrapping along the compactified circle can be extracted from information about 1d Majorana fermions on the boundary of the world-sheet. These Majorana fermions need to be introduced to consistently perform the GSO projection and the orientifold. We also construct D-brane states in the DP background. The spectrum of D-branes characterized by the relative KR-theory is correctly reproduced from the D-brane states.

Japan

Hiroki Wada

Satoshi Yamaguchi

Anomalies and D-branes in the Dabholkar-Park background

We construct ℤ  N  orbifolds of the ten-dimensional heterotic string theories appropriate for implementing the stringy replica method for the calculation of quantum entanglement entropy. A novel feature for the heterotic string is that the gauge symmetry must be broken by a Wilson line to ensure modular invariance. We completely classify the patterns of symmetry breaking. We show that the tachyonic contributions in all cases can be analytically continued, with a finite answer in the domain 0 &lt; N ≤ 1, relevant for calculating entanglement entropy across the Rindler horizon. We discuss the physical implications of our results.

Heterotic strings and quantum entanglement

We consider ℤ  N  orbifolds of Type-II compactifications to four and six dimensions on several Calabi-Yau manifolds in the orbifold limit with the aim to compute the entanglement entropy. The spectrum can contain tachyons in the doubly-twisted sectors which can lead to new infrared divergences for the partition function that are not present in the orbifolds of the uncompactified ten-dimensional theory. We show that all tachyonic contributions in these models admit a resummation and analytic continuation that yields finite entropy in the physical region 0 &lt; N ≤ 1 just as in ten dimensions.

Entanglement entropy in string compactifications

We compute the exact one-loop partition function of ℤ  N  orbifolds of Euclidean BTZ black hole with the aim to compute the entanglement entropy of the black hole horizon in string theory as a function of the mass and spin of the black hole and the AdS3 radius. We analyze the tachyonic contribution to the modular integrand for the partition function known for odd integers N &gt; 1 and show that it admits an analytic continuation resulting in a finite answer for the modular integral in the physical region 0 &lt; N ≤ 1. We discuss the flat space limit and the relevance of this computation for quantum gravity near black hole horizons and holography in relation to the thermal entropy.

Quantum entanglement on black hole horizons in string theory and holography

We analyze the one-loop quantum entanglement entropy in ten-dimensional type-II string theory using the orbifold method by analytically continuing in <math><mi>N</mi></math> the genus-one partition function for string orbifolds on <math><msup><mi>R</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mi>N</mi></msub></math> conical spaces known for all odd integers <math><mi>N</mi><mo>&gt;</mo><mn>1</mn></math>. We show that the tachyonic contributions to the orbifold partition function can be appropriately summed and analytically continued to an expression that is finite in the physical region <math><mn>0</mn><mo>&lt;</mo><mi>N</mi><mo>≤</mo><mn>1</mn></math> resulting in a finite and calculable answer for the entanglement entropy. We discuss the implications of the finiteness of the entanglement entropy for the information paradox, quantum gravity, and holography.

Finite entanglement entropy in string theory

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 States

Abhiram Kidambi

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

We show that the Atiyah-Patodi-Singer η-invariant can be related to the temperature-dependent Witten index of a noncompact theory and give a new proof of the APS theorem using scattering theory. We relate the η-invariant to a Callias index and compute it using localization of a supersymmetric path integral. We show that the η-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which we compute from the noncompact path integral.

France

Diksha Jain

Arnab Rudra

APS η-invariant, path integrals, and mock modularity

A proper definition of the path integral of quantum gravity has been a long- standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cos- mological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three- point functions is well defined and crossing symmetric (numerically) for external Liouville energies corresponding to all physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy.