In the context of thermodynamics of asymptotically anti-de Sitter spaces, it is often stated that at very low temperatures, there is only one saddle point available — namely, thermal AdS — and hence this sole saddle dictates the low-temperature behavior. However, AdS-Schwarzschild black holes continue to exist at low temperatures as complex saddle points. We point out that the real part of the on-shell action of these complex black holes is $\textit{smaller}$ than that of thermal AdS at the lowest temperatures, in AdS and higher dimensions. So, naïvely, they should be the “dominant” saddles. This raises a puzzle: if these complex black holes were indeed the relevant saddle points, the physics of the bulk and that of the dual gauge theory would completely disagree at low temperatures. Using a mini-superspace approximation and contour arguments, we argue that these complex black holes do not actually contribute to the gravitational path integral, regardless of the value of their on-shell action. So the standard conclusion that thermal AdS is the correct saddle at the lowest temperatures continues to hold. We also comment on two related matters: whether the Kontsevich-Segal criterion is useful in this setting, and whether the unstable small black hole contributes to the path integral in the high-temperature phase.

India

Raghu Mahajan

Kaustubh Singhi

A brief note on complex AdS-Schwarzschild black holes

We study ZZ instanton corrections in the (2, 4k)   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 minimal superstring theory with the type 0B GSO projection, which becomes the type 0B   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 super-JT gravity in the k → ∞ limit. Each member of the (2, 4k) family of theories has two phases distinguished by the sign of the Liouville bulk cosmological constant. The worldsheet method for computing the one-loop normalization constant multiplying the instanton corrections gives an ill-defined answer in both phases. We fix these divergences using insights from string field theory and find finite, unambiguous results. Each member of the (2, 4k) family of theories is dual to a double-scaled one-matrix integral, where the double-scaling limit can be obtained starting either from a unitary matrix integral with a leading one-cut saddle point, or from a hermitian matrix integral with a leading two-cut saddle point. The matrix integral exhibits a gap-closing transition, which is the same as the double-scaled Gross-Witten-Wadia transition when k = 1. We also compute instanton corrections in the double-scaled matrix integral for all k and in both phases, and find perfect agreement with the string theory results.

United States

Vivek Chakrabhavi

Dan Eniceicu

Chitraang Murdia

Normalization of ZZ instanton amplitudes in type 0B minimal superstring theory

We compute instanton corrections to the partition function of sine-Liouville (SL) theory, which provides a worldsheet description of two-dimensional string theory in a non-trivial tachyon background. We derive these corrections using a matrix model formulation based on a chiral representation of matrix quantum mechanics and using string theory methods. In both cases we restrict to the leading and subleading orders in the string coupling expansion. Then the CFT technique is used to compute two orders of the expansion in the SL perturbation parameter λ, while the matrix model gives results which are non-perturbative in λ. The matrix model results perfectly match those of string theory in the small λ expansion. We also generalize our findings to the case of perturbation by several tachyon vertex operators carrying different momenta, and obtain interesting analytic predictions for the disk two-point and annulus one-point functions with ZZ boundary condition.

France

Sergei Alexandrov

Ashoke Sen

Instantons in sine-Liouville theory

We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact “ghost instantons”, which are extrema of the one-eigenvalue effective potential on the “unphysical sheet” of the spectral curve and have been discussed in detail recently by Mariño, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the ’t Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.

Complex eigenvalue instantons and the Fredholm determinant expansion in the Gross-Witten-Wadia model

We compute the ZZ annulus one-point function of the cosmological constant operator in non-critical string theory, regulating divergences from the boundaries of moduli space using string field theory. We identify a subtle issue in a previous analysis of these divergences, which was done in the context of the c = 1 string theory, and where it had led to a mismatch with the prediction from the dual matrix quantum mechanics. After fixing this issue, we find a precise match to the expected answer in both the c &lt; 1 and c = 1 cases. We also compute the disk two-point function, which is a quantity of the same order, and show that it too matches with the general prediction.

Pronobesh Maity

The ZZ annulus one-point function in non-critical string theory: A string field theory analysis

We compute the normalization of the general multi-instanton contribution to the partition function of (p′, p) minimal string theory and also to the dual two-matrix integral. We find perfect agreement between the two results.

Multi-instantons in minimal string theory and in matrix integrals

We use insights from string field theory to analyze and cure the divergences in the cylinder diagram in minimal string theory with both boundaries lying on a ZZ brane. We focus on theories with worldsheet matter consisting of the (2, p) minimal model plus Liouville theory, with total central charge 26, together with the usual bc-ghosts. The string field theory procedure gives a finite, purely imaginary normalization constant for non-perturbative effects in minimal string theory, or doubly non-perturbative effects in JT gravity. We find precise agreement with the prediction from the dual double-scaled one-matrix integral. We also make a few remarks about the extension of this result to the more general (p′, p) minimal string.

Normalization of ZZ instanton amplitudes in minimal string theory

We compute the sphere and disk partition functions in semiclassical Liouville and analogous quantities in double-scaled matrix integrals. The quantity sphere/disk2 is unambiguous and we find a precise numerical match between the Liouville answer and the matrix integral answer. An application is to show that the sphere partition function in JT gravity is infinite.

Douglas Stanford

Cynthia Yan

Sphere and disk partition functions in Liouville and in matrix integrals

In gauge/gravity duality, the bulk double cone geometry has been argued to account for a key feature of the spectral form factor known as the ramp. This feature is deeply associated with quantum chaos in the dual field theory. The connection with the ramp has been demonstrated in detail for two-dimensional theories of bulk gravity, but it appears natural in higher dimensions as well. In a general bulk theory the double cone might thus be expected to dominate the semiclassical bulk path integral for the boundary spectral form factor in the ramp regime. While other known spacetime wormholes have been shown to be unstable to brane nucleation when they dominate over known disconnected (factorizing) solutions, we argue below that the double cone is stable to semiclassical brane nucleation at the probe-brane level in a variety of string- and M-theory settings. Possible implications for the AdS/CFT factorization problem are briefly discussed.

Donald Marolf

United Kingdom

Jorge Santos

The double cone geometry is stable to brane nucleation

The holomorphic Coulomb gas formalism, as developed by Feigin-Fuchs, Dotsenko-Fateev and Felder, is a set of rules for computing minimal model observables using free field techniques. We attempt to derive and clarify these rules using standard techniques of quantum field theory. We begin with a careful examination of the timelike linear dilaton. Although the background charge of the model breaks the scalar field’s continuous shift symmetry, the exponential of the action remains invariant under a discrete shift because the background charge is imaginary. Gauging this symmetry makes the dilaton compact and introduces winding modes into the spectrum. One of these winding operators corresponds to the anti-holomorphic completion of the BRST current first introduced by Felder, and the full left/right cohomology of this BRST charge isolates the irreducible representations of the Virasoro algebra within the degenerate Fock space of the linear dilaton. The “supertrace” in the BRST complex reproduces the minimal model partition function and exhibits delicate cancellations between states with both momentum and winding. The model at the radius   <math> <mi>R</mi> <mo>=</mo> <msqrt> <mrow> <mi>pp</mi> <mo>′</mo> </mrow> </msqrt> </math>  $$ R=\sqrt{pp^{\prime }} $$  has two marginal operators corresponding to the Dotsenko-Fateev “screening charges”. Deforming by them, we obtain a model that might be called a “BRST quotiented compact timelike Liouville theory”. The Hamiltonian of the zero-mode quantum mechanics of this model is not Hermitian, but it is PT-symmetric and exactly solvable. Its eigenfunctions have support on an infinite number of plane waves, suggesting an infinite reduction in the number of independent states in the full quantum field theory. Applying conformal perturbation theory to the exponential interactions reproduces the Coulomb gas calculations of minimal model correlation functions. In contrast to spacelike Liouville, these “resonance correlators” are finite because the zero mode is compact. We comment on subtleties regarding the reflection operator identification, as well as naive violations of truncation in correlators with multiple reflection operators inserted. This work is part of an attempt to understand the relationship between the JT model of two dimen- sional gravity and the worldsheet description of the (2, p) minimal string as suggested by Seiberg and Stanford.