We study the large-charge sector of $$\mathcal{N}$$ = 4 super Yang-Mills theory (SYM) with SU($\textit{N}$) gauge group by constructing a special class of half-BPS heavy operators, termed “canonical operators”. Such operators exhibit remarkable simplicity in the large-charge ’t Hooft limit, where the dimension of the operators ∆ → ∞ with $${\Delta g}_{\text{YM}}^{2}$$ held finite. Canonical operator insertions in this regime map $$\mathcal{N}$$ = 4 SYM onto the Coulomb branch, by assigning a classical profile to the scalar fields with non-vanishing values along the diagonals given by the roots of unity. We follow a semiclassical approach to study two-point, three-point and Heavy-Heavy-Light-Light (HHLL) correlators. In particular we show that HHLL correlators in the large-charge ’t Hooft limit are computed as two-point functions in a background determined by the classical profiles. We provide consistent evidence of our findings by computing the same observables via supersymmetric localization.

United Kingdom

Augustus Brown

Francesco Galvagno

Switzerland

Alba Grassi

Cristoforo Iossa

Congkao Wen

Large charge meets semiclassics in $$\mathcal{N}$$ = 4 super Yang-Mills

We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with a positive cosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant is negative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail.

United States

Gleb Aminov

Italy

Paolo Arnaudo

Giulio Bonelli

CERN

Alessandro Tanzini

Black hole perturbation theory and multiple polylogarithms

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the Ω background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.

Francesco Fucito

Jose Morales

Raffaele Savelli

Partition functions of non-Lagrangian theories from the holomorphic anomaly

We revisit the confinement/deconfinement transition in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills (SYM) theory and its relation to the Hawking-Page transition in gravity. Recently there has been substantial progress on counting the microstates of 1/16-BPS extremal black holes. However, there is presently a mismatch between the Hawking-Page transition and its avatar in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. This led to speculations about the existence of new gravitational saddles that would resolve the mismatch. Here we exhibit a phenomenon in complex matrix models which we call “delayed deconfinement.” It turns out that when the action is complex, due to destructive interference, tachyonic modes do not necessarily condense. We demonstrate this phenomenon in ordinary integrals, a simple unitary matrix model, and finally in the context of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. Delayed deconfinement implies a first-order transition, in contrast to the more familiar cases of higher-order transitions in unitary matrix models. We determine the deconfinement line and find remarkable agreement with the prediction of gravity. On the way, we derive some results about the Gross-Witten-Wadia model with complex couplings. Our techniques apply to a wide variety of (SUSY and non-SUSY) gauge theories though in this paper we only discuss the case of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM.

Spain

Christian Copetti

Zohar Komargodski

Luigi Tizzano

Delayed deconfinement and the Hawking-Page transition

We study in detail the Schrödinger equation corresponding to the four dimensional SU(2)   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

Qianyu Hao

Andrew Neitzke

Exact WKB methods in SU(2) Nf = 1

We study the correlation functions of Coulomb branch operators of four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. “Extremal” correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a “dual” description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a ’t Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the ’t Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong ’t Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.

Extremal correlators and random matrix theory

We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2, SU(2) super Yang-Mills theory, which is closely related to the modified Mathieu operator. The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus. In addition, we use the TS/ST correspondence to obtain a closed formula for the Fredholm determinant of the modified Mathieu operator. Finally, by using blowup equations, we explain the connection between this operator and the τ function of Painlevé III.

Jie Gu

Marcos Mariño

Non-perturbative approaches to the quantum Seiberg-Witten curve

We show in details that the all order genus expansion of the two-cut Hermitian cubic matrix model reproduces the perturbative expansion of the H 1 Argyres-Douglas theory coupled to the Ω background. In the self-dual limit we use the Painlevé/gauge correspondence and we show that, after summing over all instanton sectors, the two-cut cubic matrix model computes the tau function of Painlevé II without taking any double scaling limit or adding any external fields. We decode such solution within the context of transseries. Finally in the Nekrasov-Shatashvili limit we connect the H 1 and the H 0 Argyres-Douglas theories to the quantum mechanical models with cubic and double well potentials.

France

Argyres-Douglas theories, Painlevé II and quantum mechanics

We show that, in ABJ(M) theories with N = 8 $$ \mathcal{N}=8 $$ supersymmetry, the non-perturbative sector of the partition function on the three-sphere simplifies drastically. Due to this simplification, we are able to write closed form expressions for the grand potential of these theories, which determines the full large N asymptotics. Moreover, we find explicit formulae for the generating functionals of their partition functions, for all values of the rank N of the gauge group: they involve Jacobi theta functions on the spectral curve associated to the planar limit. Exact quantization conditions for the spectral problem of the Fermi gas are then obtained from the vanishing of the theta function. We also show that the partition function, as a function of N , can be extended in a natural way to an entire function on the full complex plane, and we explore some possible consequences of this fact for the quantum geometry of M-theory and for putative de Sitter extensions.

Santiago Codesido

Exact results in N = 8 $$ \mathcal{N}=8 $$ Chern-Simons-matter theories and quantum geometry

We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local ℙ 1 × ℙ 1 $$ {\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^1 $$ , which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.