We consider a truncation of the BMN matrix model to a configuration of two fuzzy spheres, described by two coupled non-linear oscillators dependent on the mass parameter $\textit{μ}$. The classical phase diagram of the system generically ($\textit{μ}$ ≠ 0) contains three equilibrium points: two centers and a center-saddle; as $\textit{μ}$ → 0 the system exhibits a pitchfork bifurcation. We demonstrate that the system is exactly integrable in quadratures for $\textit{μ}$ = 0, while for very large values of $\textit{μ}$, it approaches another integrable point characterized by two harmonic oscillators. The classical phase space is mixed, containing both integrable islands and chaotic regions, as evidenced by the classical Lyapunov spectrum. At the quantum level, we explore indicators of early and late time chaos. The eigenvalue spacing is best described by a Brody distribution, which interpolates between Poisson and Wigner distributions; it dovetails, at the quantum level, the classical results and reemphasizes the notion that the quantum system is mixed. We also study the spectral form factor and the quantum Lyapunov exponent, as defined by out-of-time-ordered correlators. These two indicators of quantum chaos exhibit weak correlations with the Brody distribution. We speculate that the behavior of the system as $\textit{μ}$ → 0 dominates the spectral form factor and the quantum Lyapunov exponent, making these indicators of quantum chaos less effective in the context of a mixed phase space.

Mexico

Paolo Amore

Italy

United States

Leopoldo Pando Zayas

Spain

Juan Pedraza

Norma Quiroz

César Terrero-Escalante

Fuzzy spheres in stringy matrix models: quantifying chaos in a mixed phase space

The superconformal index of $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory with gauge group U($\textit{N}$) has provided powerful insights into the entropy of supersymmetric black holes in AdS $\textit{× S}$ , including some sub-leading logarithmic and non-perturbative corrections. Recently, the phase space of supersymmetric solutions has been argued to contain configurations other than the asymptotically AdS black hole. Such configurations include the so-called grey galaxies where the black hole at the center is surrounded by a gas of gravitons. By numerically evaluating the superconformal index of $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills at small values of $\textit{N}$, we detect systematic deviations from the entropy of black holes with two distinct angular momenta. We find that the giant graviton expansion of the index is a numerically efficient way of evaluating the index that complements the direct character evaluation and allows for explicit access to $\textit{N ≤}$ 15 with up to two giant gravitons in the expansion. We find it remarkable that a supersymmetric quantity in field theory, usually thought of as a rigid counting observable, indeed contains information about different phases in the space of supersymmetric solutions on the gravity side.

Evan Deddo

Wenjie Zhou

The superconformal index and black hole instabilities

We revisit the existence of monotonic quantities along renormalization group flows using only the Null Energy Condition and the Ryu-Takayanagi formula for the entanglement entropy of field theories with anti-de Sitter gravity duals. In particular, we consider flows within the same dimension and holographically reprove the c-, F -, and a-theorems in dimensions two, three, and four. We focus on the family of maximally spherical entangling surfaces, define a quasi-constant of motion corresponding to the breaking of conformal invariance, and use a properly defined distance between minimal surfaces to construct a holographic c-function that is monotonic along the flow. We then apply our method to the case of flows across dimensions: there, we reprove the monotonicity of flows from AdS D+1 to AdS3 and prove the novel case of flows from AdS5 to AdS4.

James Liu

Robert Saskowski

Explicit entropic proofs of irreversibility theorems for holographic RG flows

We study the entanglement entropy associated with a holographic RG flow from AdS7 to AdS4 × ℍ 3, where ℍ 3 is a 3-dimensional hyperbolic manifold with curvature κ. The dual six-dimensional RG flow is disconnected from Lorentz-invariant flows. In this context we address various notions of central charges and identify a monotonic candidate c-function that captures IR aspects of the flow. The UV behavior of the holographic entanglement entropy and, in particular its universal term, display an interesting dependence on the curvature, κ. We then contrast our holographic results with existing field theory computations in six dimensions and find a series of new corrections in curvature to the universal term in the entanglement entropy.

José de-la-Cruz-Moreno

Discontinuity in RG flows across dimensions: entanglement, anomaly coefficients and geometry

The low-temperature thermodynamics of near-extremal rotating black holes has recently been revisited to incorporate one-loop contributions that are dominant in this regime. We discuss these quantum corrections to the gravitational path integral for asymptotically Anti de-Sitter black holes in four and five dimensions. In four dimensions we explicitly consider Kerr-AdS4, Kerr-Newman-AdS4 and the rotating black hole in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 gauged supergravity with two scalars and two electric charges turned on. In five dimensions we explicitly address the asymptotically flat Myers-Perry black hole and the Kerr-AdS5 black hole. In every case we find that tensor modes contribute   <math> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>log</mo> <msub> <mi>T</mi> <mtext>Hawking</mtext> </msub> </math>  $$ \frac{3}{2}\log {T}_{\textrm{Hawking}} $$  to the low-temperature thermodynamics. We identify the root cause of this universality in two facts: (i) the universal presence of a SL(2, ℝ) subgroup of isometries in the near-horizon geometry and (ii) a set of cancellations in the Lichnerowicz operator. We show that these two conditions hold for near-extremal black holes in asymptotically flat and asymptotically AdS spacetimes of various dimensions.

India

Sabyasachi Maulik

South Korea

Augniva Ray

Jingchao Zhang

Universality in logarithmic temperature corrections to near-extremal rotating black hole thermodynamics in various dimensions

We consider two-point correlators of local operator insertions in a system of two Wilson-Maldacena loops in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetric Yang-Mills theory on both sides of the AdS/CFT correspondence. On the holographic side the correlator of two Wilson-Maldacena loops is given by a classical string world-sheet which in one phase connects two asymptotically AdS2 regions and in the other phase is given by two disconnected AdS2 caps; this configuration breaks supersymmetry as well as conformal invariance. We present a complete systematic account of the string world-sheet fluctuations, including the fermionic sector, and study the behavior of the holographic two-point correlators. On the field theory side we compute certain two-point correlators of local operator insertions by resumming sets of ladder diagrams. Our results demonstrate the efficacy of previously developed methods in tackling this non-conformal, non-susy regime.

Argentina

Diego Correa

Chile

Alberto Faraggi

Wolfgang Mück

Guillermo Silva

Beyond AdS2/dCFT1: insertions in two Wilson loops

In the context of gravitational theories describing renormalization group flows across dimensions via AdS/CFT, we study the role of higher-derivative corrections to Einstein gravity. We use the Null Energy Condition to derive monotonicity properties of candidate holographic central charges formed by combinations of metric functions. We also implement an entropic approach to the characterization of the four-derivative flows using the Jacobson-Myers functional and demonstrate, under reasonable conditions, monotonicity of certain terms in the entanglement entropy via the appropriate generalization of the Ryu-Takayanagi prescription. In particular, we show that any flow from a higher dimensional theory to a holographic CFT2 satisfies a type of monotonicity. We also uncover direct relations between NEC-motivated and entropic central charges.

c-functions in higher-derivative flows across dimensions

We quantitatively address the following question: for a QFT which is partially compactified, so as to realize an RG flow from a D-dimensional CFT in the UV to a d-dimensional CFT in the IR, how does the entanglement entropy of a small spherical region probing the UV physics evolve as the size of the region grows to increasingly probe IR physics? This entails a generalization of spherical regions to setups without full Lorentz symmetry, and we study the associated entanglement entropies holographically. We find a tight interplay between the topology and geometry of the compact space and the evolution of the entanglement entropy, with universal transitions from ‘cap’ through ‘bridge’ and ‘corner’ phases, whose features reflect the details of the compact space. As concrete examples we discuss twisted compactifications of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM on T 2, S 2 and hyperbolic Riemann surfaces.

United Kingdom

Christoph Uhlemann

Entanglement and topology in RG flows across dimensions: caps, bridges and corners

We consider the superconformal index of three-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetric field theories computed via localization on S 1 × S 2. We systematically develop an expansion where the ratio of the radius of S 1 to the radius of S 2 is taken very small — a Cardy-like expansion. We emphasize the sub-leading structures in this Cardy-like expansion as well as their interplay with the large-N limit for theories with gauge group of the form product of U(N) factors. We demonstrate that taking the large-N limit first leads to an expression for the effective action yielding the index that only includes terms proportional to 1/β and powers of β  i=0,1,2 where β is the ratio of radii. As we depart from the β → 0 limit for finite N, we find indications of non-perturbative contributions of the form e  −1/β . We present an explicit evaluation of the superconformal index for the ABJM theory in the large-N limit that is exact in β up to non-perturbative corrections. For the ABJM theory we explore the implications of the Cardy-like expansion for corrections to the entropy of the rotating, electricallly charged, asymptotically AdS4 dual black hole. Interestingly, we find a closed form for the entropy that includes all perturbative corrections in β and takes the form of the leading order expression but with shifted charges.

Alfredo Lezcano

Maximilian Jerdee

Cardy expansion of 3d superconformal indices and corrections to the dual black hole entropy

We explore the notion of c-functions in renormalization group flows between theories in different spacetime dimensions. We discuss functions connecting central charges of the UV and IR fixed point theories on the one hand, and functions which are monotonic along the flow on the other. First, using the geometric properties of the holographic dual RG flows across dimensions and the constraints from the null energy condition, we construct a monotonic holographic c-function and thereby establish a holographic c-theorem across dimensions. Second, we use entanglement entropies for two different types of entangling regions in a field theory along the RG flow across dimensions to construct candidate c-functions which satisfy one of the two criteria but not both. In due process we also discuss an interesting connection between corner contributions to the entanglement entropy and the topology of the compact internal space. As concrete examples for both approaches, we holographically study twisted compactifications of 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and compactifications of 6d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (2, 0) theories.

Belgium

Junho Hong

c-functions in flows across dimensions