Nekrasov Shatashvili Research Articles

Painlevé Kernels and Surface Defects at Strong Coupling

AbstractIt is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between $${\mathcal {N}}=2$$ N = 2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an $${{\,\mathrm{O(2)}\,}}$$ O ( 2 ) matrix model. We then show that these eigenfunctions are computed by surface defects in $${{\,\mathrm{SU(2)}\,}}$$ SU ( 2 ) super Yang–Mills in the self-dual phase of the $$\Omega $$ Ω -background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

Black hole perturbation theory meets CFT2 : Kerr-Compton amplitudes from Nekrasov-Shatashvili functions

We present a novel study of Kerr Compton amplitudes in a partial wave basis in terms of the Nekrasov-Shatashvili (NS) function of the confluent Heun equation (CHE). Remarkably, NS-functions enjoy analytic properties and symmetries that are naturally inherited by the Compton amplitudes. Based on this, we characterize the analytic dependence of the Compton phase shift in the Kerr spin parameter and provide a direct comparison to the standard post-Minkowskian (PM) perturbative approach within general relativity (GR). We also analyze the universal large frequency behavior of the relevant characteristic exponent of the CHE—also known as the renormalized angular momentum—and find agreement with numerical computations. Moreover, we discuss the analytic continuation in the harmonics quantum number ℓ of the partial wave, and show that the limit to the physical integer values commutes with the PM expansion of the observables. Finally, we obtain the contributions to the tree-level, point-particle, gravitational Compton amplitude in a covariant basis through O(aBH8), without the need to take the superextremal limit for Kerr spin. Published by the American Physical Society 2024

Absorption cross section in gravity’s rainbow from confluent Heun equation

We investigate the scattering of a massless scalar field by a charged, non-rotating black hole in the presence of gravity’s rainbow. Using the connection coefficients of the confluent Heun equation expressed in terms of the semi-classical confluent conformal blocks and the instanton part of the Nekrasov–Shatashvili free energy, we obtain an asymptotic expansion for the low-energy absorption cross section. Furthermore, we consider the resummation of the instanton contributions to the absorption cross section, which gives the area of the black hole up to a factor of (4/3)2 .

Quantum Curves, Resurgence and Exact WKB

We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of q-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.

Symmetric orbifold theories from little string residues

We study a class of Little String Theories (LSTs) of A type, described by $N$ parallel M5-branes spread out on a circle and which in the low energy regime engineer supersymmetric gauge theories with $U(N)$ gauge group. The BPS states in this setting correspond to M2-branes stretched between the M5-branes. Generalising an observation made in arXiv:1706.04425, we provide evidence that the BPS counting functions of special subsectors of the latter exhibit a Hecke structure in the Nekrasov-Shatashvili (NS) limit, i.e. the different orders in an instanton expansion of the supersymmetric gauge theory are related through the action of Hecke operators. We extract $N$ distinct such reduced BPS counting functions from the full free energy of the LST with the help of contour integrals with respect to the gauge parameters of the $U(N)$ gauge group. Physically, the states captured by these functions correspond to configurations where the same number of M2-branes is stretched between some of these neighbouring M5-branes, while the remaining M5-branes are collapsed on top of each other and a particular singular contribution is extracted. The Hecke structures suggest that these BPS states form the spectra of symmetric orbifold CFTs. We furthermore show that to leading instanton order (in the NS-limit) the reduced BPS counting functions factorise into simpler building blocks. These building blocks are the expansion coefficients of the free energy for $N=1$ and the expansion of a particular function, which governs the counting of BPS states of a single M5-brane with single M2-branes ending on it on either side. To higher orders in the instanton expansion, we observe new elements appearing in this decomposition, whose coefficients are related through a holomorphic anomaly equation.

Quantum periods and spectra in dimer models and Calabi-Yau geometries

We study a class of quantum integrable systems derived from dimer graphs and also described by local toric Calabi-Yau geometries with higher genus mirror curves, generalizing some previous works on genus one mirror curves. We compute the spectra of the quantum systems both by standard perturbation method and by Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way, we obtain some exact analytic results for the classical and quantum periods of the Calabi-Yau geometries. We also determine the differential operators of the quantum periods and compute the topological string free energy in Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other methods such as the topological vertex.

Integrability and cycles of deformed [formula omitted] gauge theory

To analyse pure N=2SU(2) gauge theory in the Nekrasov-Shatashvili (NS) limit (or deformed Seiberg-Witten (SW)), we use the Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence, and in particular its (broken) discrete symmetry in its extended version with two singular irregular points. Actually, this symmetry appears to be ‘manifestation’ of the spontaneously broken Z2 R-symmetry of the original gauge problem and the two deformed SW one-cycle periods are simply connected to the Baxter's T and Q functions, respectively, of the Liouville conformal field theory at the self-dual point. The liaison is realised via a second order differential operator which is essentially the ‘quantum’ version of the square of the SW differential. Moreover, the constraints imposed by the broken Z2 R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have their quantum counterpart in the TQ and the T periodicity relations, and integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the periods (Y(θ,±u) or their square roots, Q(θ,±u)). A latere, two efficient asymptotic expansion techniques are presented. Clearly, the whole construction is extendable to gauge theories with matter and/or higher rank groups.

Quantum periods for [formula omitted]SU(2) SQCD around the superconformal point

We study the Argyres–Douglas theories realized at the superconformal point in the Coulomb moduli space of N=2 supersymmetric SU(2) QCD with Nf=1,2,3 hypermultiplets in the Nekrasov–Shatashvili limit of the Omega-background. The Seiberg–Witten curve of the theory is quantized in this limit and the periods receive the quantum corrections. By applying the WKB method for the quantum Seiberg–Witten curve, we calculate the quantum corrections to the Seiberg–Witten periods around the superconformal point up to the fourth order in the parameter of the Omega background.

Quantum Hitchin Systems via $${\beta}$$ β -Deformed Matrix Models

We study the quantization of Hitchin systems in terms of $${\beta}$$ -deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov–Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the $${\epsilon_1}$$ -deformation of the chiral observables of the corresponding $${{\mathcal{N}=2}}$$ four dimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions.

Exact quantization conditions, toric Calabi-Yau and non-perturbative topological string

We establish the precise relation between the Nekrasov-Shatashvili (NS) quantization scheme and Grassi-Hatsuda-Mariño conjecture for the mirror curve of arbitrary toric Calabi-Yau threefold. For a mirror curve of genus g, the NS quantization scheme leads to g quantization conditions for the corresponding integrable system. The exact NS quantization conditions enjoy a self S-duality with respect to Planck constant h and can be derived from the Lockhart-Vafa partition function of non-perturbative topological string. Based on a recent observation on the correspondence between spectral theory and topological string, another quantization scheme was proposed by Grassi-Hatsuda-Mariño, in which there is a single quantization condition and the spectra are encoded in the vanishing of a quantum Riemann theta function. We demonstrate that there actually exist at least g nonequivalent quantum Riemann theta functions and the intersections of their theta divisors coincide with the spectra determined by the exact NS quantization conditions. This highly nontrivial coincidence between the two quantization schemes requires infinite constraints among the refined Gopakumar-Vafa invariants. The equivalence for mirror curves of genus one has been verified for some local del Pezzo surfaces. In this paper, we generalize the correspondence to higher genus, and analyze in detail the resolved {mathbb{C}}^3/{mathbb{Z}}_5 orbifold and several SU(N ) geometries. We also give a proof for some models at ħ = 2π/k.

Spectral determinants and quantum theta functions

It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic properties of these spectral determinants by expanding them around the point , where the quantum theta functions become conventional theta functions. We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. Our derivation requires a detailed understanding of the modular properties of topological string free energies in the Nekrasov–Shatashvili limit. We derive these properties in a diagrammatic form. Finally, we use our results to provide a new test of the duality between topological strings and spectral theory.

Nekrasov-Shatashvili limit of the 5D superconformal index

We consider the Nekrasov-Shatashvili (NS) limit of the five-dimensional superconformal index and propose a novel prescription for selecting the finite contributions. Applying the latter to various examples of U(1) theories, we find that the 5D NS index can be reproduced using recent techniques of C\'ordova and Shao, who related the 4D Schur index to the BPS spectrum of the theory on the Coulomb branch. In this picture, the 5D instanton solitons are interpreted as additional flavour nodes in an associated 5D BPS quiver.

On the large Ω-deformations in the Nekrasov-Shatashvili limit of N = 2 * $$ \mathcal{N}={2}^{*} $$ SYM

We study the multi-instanton partition functions of the $\Omega$-deformed $\mathcal N =2^{*}$ $SU(2) $ gauge theory in the Nekrasov-Shatashvili (NS) limit. They depend on the deformation parameters $\epsilon_{1}$, the scalar field expectation value $a$, and the hypermultiplet mass $m$. At fixed instanton number $k$, they are rational functions of $\epsilon_{1}, a, m$ and we look for possible regularities that admit a parametrical description in the number of instantons. In each instanton sector, the contribution to the deformed Nekrasov prepotential has poles for "large" deformation parameters. To clarify the properties of these singularities we exploit Bethe/gauge correspondence and examine the special ratios $m/\epsilon_{1}$ at which the associated spectral problem is $n$-gap. At these special points we illustrate several structural simplifications occurring in the partition functions. After discussing various tools to compute the prepotential, we analyze the non-perturbative corrections up to $k=24$ instantons and present various closed expressions for the coefficients of the singular terms. Both the regular and singular parts of the prepotential are resummed over all instantons and compared successfully with the exact prediction from the spectral theory of the Lam\'e equation, showing that the pole singularities are an artifact of the instanton expansion. The analysis is fully worked out in the 1-gap case, but the final pole cancellation is proved for a generic ratio $m/\epsilon_{1}$ relating it to the gap width of the Lam\'e equation.

Topological Strings from Quantum Mechanics

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local $${{\mathbb{P}}^2}$$ , local $${{\mathbb{P}}^1 \times {\mathbb{P}}^1}$$ and local $${{\mathbb{F}}_1}$$ . In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.

Mayer expansion of the Nekrasov prepotential: The subleading ε2-order

The Mayer cluster expansion technique is applied to the Nekrasov instanton partition function of N=2SU(Nc) super Yang–Mills. The subleading small ε2-correction to the Nekrasov–Shatashvili limiting value of the prepotential is determined by a detailed analysis of all the one-loop diagrams. Indeed, several types of contributions can be distinguished according to their origin: long range interaction or potential expansion, clusters self-energy, internal structure, one-loop cyclic diagrams, etc. The field theory result derived more efficiently in [1], under some minor technical assumptions, receives here definite confirmation thanks to several remarkable cancellations: in this way, we may infer the validity of these assumptions for further computations in the field theoretical approach.

Finite [formula omitted]-corrections to the [formula omitted] prepotential

We derive the first ε2-correction of the instanton partition functions in 4D N=2 Super Yang–Mills (SYM) to the Nekrasov–Shatashvili limit ε2→0. In the latter we recall the emergence of the famous Thermodynamic Bethe Ansatz-like equation which has been found by Mayer expansion techniques. Here we combine efficiently these to field theory arguments. In a nutshell, we find natural and resolutive the introduction of a new operator ∇ that distinguishes the singularities within and outside the integration contour of the partition function.

Instanton Effects and Quantum Spectral Curves

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.

Confinement and Mayer cluster expansions

In this paper, we study a class of grand-canonical partition functions with a kernel depending on a small parameter ϵ. This class is directly relevant to Nekrasov partition functions of 𝒩 = 2 SUSY gauge theories on the 4d Ω-background, for which ϵ is identified with one of the equivariant deformation parameter. In the Nekrasov–Shatashvili limit ϵ→0, we show that the free energy is given by an on-shell effective action. The equations of motion take the form of a TBA equation. The free energy is identified with the Yang–Yang functional of the corresponding system of Bethe roots. We further study the associated canonical model that takes the form of a generalized matrix model. Confinement of the eigenvalues by the short-range potential is observed. In the limit where this confining potential becomes weak, the collective field theory formulation is recovered. Finally, we discuss the connection with the alternative expression of instanton partition functions as sums over Young tableaux.

On a path integral representation of the Nekrasov instanton partition function and its Nekrasov–Shatashvili limit

In this work we study the Nekrasov–Shatashvili limit of the Nekrasov instanton partition function of Yang–Mills field theories with 𝒩 = 2 supersymmetry and gauge group SU(N). The theories are coupled with fundamental matter. A path integral expression of the full instanton partition function is derived. It is checked that in the Nekrasov–Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations.

On a singular Fredholm-type integral equation arising in [formula omitted] super-Yang–Mills theories

In this work we study the Nekrasov–Shatashvili limit of the Nekrasov instanton partition function of Yang–Mills field theories with N=2 supersymmetry and gauge group SU(Nc). The theories are coupled with Nf flavors of fundamental matter. The equation that determines the density of eigenvalues at the leading order in the saddle-point approximation is exactly solved when Nf=2Nc. The dominating contribution to the instanton free energy is computed. The requirement that this energy is finite imposes quantization conditions on the parameters of the theory that are in agreement with analogous conditions that have been derived in previous works. The instanton energy and thus the instanton contribution to the prepotential of the gauge theory is computed in closed form.