arXiv subjects

Sujay K. Ashok

Publications and source records attributed to Sujay K. Ashok.

At least 19 recordsLinked to original sources

arXiv2026

Universal Modular Properties of Generalized Gibbs Ensembles and Chiral Deformations

We study modular properties of conformal field theories perturbed by holomorphic fields. We prove an asymptotic formula for the modular S-transform of a generalized partition function that includes zero modes of higher spin holomorphic currents. The derivation makes use of general properties of torus correlation functions, in particular the Zhu recursion relation. The asymptotic expansion of the modular transformed partition function takes a universal form that is determined iteratively by the second order pole coefficients in the operator product expansion of the holomorphic currents. We have also found an explicit expression for the multiplicities of terms generated by the iteration. This proves and generalizes a conjecture regarding the modular transformation properties of generalized Gibbs ensembles.

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arXiv2025

Mixed Solutions to the Liouville Equation

We enlarge the set of explicit classical solutions to the Liouville equation with three singularities to the cases with mixed hyperbolic and elliptic monodromies. We analyze the large hyperbolic monodromy limit of the solutions and the farthest geodesics looping one hyperbolic singularity. These two-dimensional geometries describe a time-symmetric spatial slice of a solution to three-dimensional general relativity. The geodesics are reinterpreted as snapshots of horizons of evolving black holes. We study the spatial slice with three horizons of very heavy black holes in some detail. We use uniform saddle point integration to present the Liouville and heavy black hole geometries in terms of simpler special functions. These make a detailed analysis of mixed particle and black hole geometries possible.

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arXiv2024

Orbifolded Elliptic Genera of Non-Compact Models

We revisit the flavored elliptic genus of the N=2 superconformal cigar model and generalize the analysis of the path integral result to the case of real central charge. It gives rise to a non-holomorphic modular covariant function generalizing completed mock modular forms. We also compute the genus for angular orbifolds of the cigar and Liouville theory and decompose it in terms of discrete and continuous contributions. The orbifolded elliptic genus at fractional level is a completed mock modular form with a shadow related to U$(1)$ modular invariants at rational radius squared. We take the limit of the orbifolded genera towards a weighted ground state index and carefully interpret the contributions. We stress that the orbifold cigar and Liouville theories have a maximal and a minimal radius, respectively.

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arXiv2024

Integrable Structure of Higher Spin CFT and the ODE/IM Correspondence

We study two dimensional systems with extended conformal symmetry generated by the ${\mathcal W}_3$ algebra. These are expected to have an infinite number of commuting conserved charges, which we refer to as the quantum Boussinesq charges. We compute the eigenvalues of the quantum Boussinesq charges in both the vacuum and first excited states of the higher spin module through the ODE/IM correspondence. By studying the higher spin conformal field theory on the torus, we also calculate thermal correlators involving the energy-momentum tensor and the spin-3 current by making use of the Zhu recursion relations. By combining these results, we show that it is possible to derive the current densities, whose integrals are the quantum Boussinesq charges. We also evaluate the thermal expectation values of the conserved charges, and show that these are quasi-modular differential operators acting on the character of the higher spin module.

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arXiv2024

Thermal Correlators and Currents of the $\mathcal{W}_3$ Algebra

Two dimensional conformal field theories with the extended $\mathcal{W}_3$ symmetry algebra have an infinite number of mutually commuting conserved charges, which are referred to as the quantum Boussinesq charges. In this work we construct local operators whose zero modes are precisely these conserved charges. For this purpose we study the higher spin conformal field theory on the torus and compute thermal correlators involving the stress tensor and the spin-3 current in a higher spin module of the W3 algebra. In addition we independently obtain the excited state eigenvalues of the quantum Boussinesq charges within the higher spin module via the ODE/IM correspondence. A judicious combination of these data allows us to derive the local operators, whose integrals are the conserved charges of the integrable hierarchy.

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arXiv2023

The Operator Rings of Topological Symmetric Orbifolds and their Large N Limit

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise.

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arXiv2023

Effective Gravitational Couplings of Kaluza-Klein Gauge Theories

We study the effective gravitational couplings of four-dimensional Kaluza-Klein compactified gauge theories with eight supercharges. The class of theories we consider are the pure SU(N) Yang-Mills theories at admissible Chern-Simons levels and the conformal gauge theories with 2N fundamental flavours. The resolvent of the gauge theory plays a crucial role in the calculation of these gravitational couplings. The results obtained from the Seiberg-Witten geometry are matched against independent computations using localisation.

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arXiv2023

The Chiral Ring of a Symmetric Orbifold and its Large N Limit

We analyze the chiral operator ring of the symmetric orbifold conformal field theory on the complex two-plane. We compute the large N limit of the ring and exhibit its factorized leading order behaviour. We moreover calculate all structure constants at the subleading and sub-subleading order. These features are coded as properties of the symmetric group and we review the relevant mathematical theorems on the product of conjugacy classes in the center of the group algebra. We illustrate the efficiency of the formalism by iteratively computing broad classes of higher point extremal correlators. We point out generalizations of our simplest of models and argue that our combinatorial analysis is relevant to the organization of the large N perturbation theory of generic symmetric orbifolds.

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arXiv2022

Long Strings and Quasinormal Winding Modes

We compute the path integral for a particle on the covering group of SL(2,R) using a decomposition of the Lie algebra into adjoint orbits. We thus intuitively derive the Hilbert space of the particle on the group including discrete and continuous representations. Next, we perform a Lorentzian hyperbolic orbifold of the partition function and relate it to the Euclidean BTZ partition function. We use the particle model to inform further discussion of the spectral content of the one loop vacuum amplitude for strings on BTZ black hole backgrounds. We argue that the poles in the loop integrand code contributions of long string modes that wind the black hole. We moreover identify saddle point contributions of quasinormal winding modes.

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arXiv2022

Path Integrals on sl(2,R) Orbits

We quantise orbits of the adjoint group action on elements of the sl(2,R) Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL(2,R) on the adjoint orbits and discuss global subtleties in choices of adapted coordinate systems. Using the hyperbolic slicing of orbits, we describe the quantum mechanics of an irreducible sl(2,R) representation in a hyperbolic basis and relate the basis to the mathematics of the Mellin integral transform. We moreover discuss the representation theory of the double cover SL(2,R) of PSL(2,R) as well as that of its universal cover. Traces in the representations of these groups for both elliptic and hyperbolic elements are computed. Finally, we motivate our treatment of this elementary quantisation problem by indicating applications.

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arXiv2021

Twisted Strings in Three-dimensional Black Holes

We revisit the classical dynamics of fundamental strings in BTZ black holes with NSNS flux. We analyse probe strings in the black hole background and determine their classical energy using a Nambu-Goto action. Three-dimensional gravity has a non-local effect on the metric surrounding a massive object which can be measured by winding strings. We also study the behaviour of short and long strings in the Wess-Zumino-Witten orbifold and show that their energy matches the long string probe analysis. In the process, we fix subtleties in the Lorentzian choice of NSNS flux. In Euclidean signature, we analyse the one-loop fundamental string partition function on the BTZ black hole and interpret the off-shell degrees of freedom as those of a coset orbifold. Finally, we speculate on how to identify the quantum Lorentzian string spectrum from the Euclidean orbifold partition function.

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arXiv2021

Triangle Groups: Automorphic Forms and Nonlinear Differential Equations

We study the relations governing the ring of quasiautomorphic forms associated to triangle groups with a single cusp, thereby extending our earlier results on Hecke groups. The Eisenstein series associated to these triangle groups are shown to satisfy Ramanujan-like identities. These identities in turn allow us to associate a nonlinear differential equation to each triangle group. We show that they are solved by the quasiautomorphic weight-2 Eisenstein series associated to the triangle group and its orbit under the group action. We conclude by discussing the Painlevé property of these nonlinear differential equations.

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arXiv2021

The $AdS_3 \times S^1$ Chiral Ring

We study $AdS_3 \times S^1 \times Y$ supersymmetric string theory backgrounds with Neveu-Schwarz-Neveu-Schwarz flux that are dual to ${\cal N}=2$ superconformal theories on the boundary. We classify all worldsheet vertex operators that correspond to space-time chiral primaries. We compute space-time chiral ring structure constants for operators in the zero spectral flow sector using the operator product expansion in the worldsheet theory. We find that the structure constants take a universal form that depends only on the topological data of the ${\cal N}=2$ superconformal theory on $Y$.

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arXiv2021

Exact WKB Analysis of $\mathbb{CP}^1$ Holomorphic Blocks

We study holomorphic blocks in the three dimensional ${\mathcal N}=2$ gauge theory that describes the $\mathbb{CP}^1$ model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the ${}_1ϕ_1$ $q$-hypergeometric function. We show in detail how the $q$-Borel resummation methods reproduce the results obtained previously by using block-integral methods.

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arXiv2021

String Scale Thermal Anti-de Sitter Spaces

We study finite temperature string scale $AdS_3$ backgrounds. One background is $AdS_3 \times S^1 \times T^2$ in which the anti-de Sitter space-time and the circle are at the radius $\sqrt{α'}$. Using path integral techniques, we show that the bulk spectrum includes a continuum of states as well as Ramond-Ramond ground states that agree with those of the symmetric orbifold of the two-torus after second quantization. We also examine the one-loop free energy of the background $AdS_3 \times S^1$ at curvature radius $\sqrt{2 α'/3}$. In the space-time NSNS sector, the string theory spontaneously breaks conformal symmetry as well as R-charge conjugation symmetry. We prove that the minimum in the boundary energy is reached for a singly wound string. In the RR sector, we classify the infinite set of ground states with fractional R-charges. Moreover, we remark on the behaviour of critical temperatures as the curvature scale becomes smaller than the string scale. In an appendix, we derive the Hawking-Page transition in string theory by integrating a world sheet one-point function.

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arXiv2020

Superstrings in Thermal Anti-de Sitter Space

We revisit the calculation of the thermal free energy for string theory in three-dimensional anti-de Sitter spacetime with Neveu-Schwarz-Neveu-Schwarz flux. The path integral calculation is exploited to confirm the off-shell Hilbert space and we find that the Casimir of the discrete representations of the isometry group takes values in a half-open interval. We extend the free energy calculation to the case of superstrings, calculate the boundary toroidal twisted partition function in the Ramond-Ramond sector, and prove lower bounds on the boundary conformal dimension from the bulk perspective. We classify Ramond-Ramond ground states and construct their second quantized partition function. The partition function exhibits intriguing modular properties.

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arXiv2020

On BPS Strings in ${\mathcal N}=4$ Yang-Mills Theory

We study singular time-dependent $\frac{1}{8}$-BPS configurations in the abelian sector of ${{\mathcal N}= 4}$ supersymmetric Yang-Mills theory that represent BPS string-like defects in ${{\mathbb R}\times S^3}$ spacetime. Such BPS strings can be described as intersections of the zeros of holomorphic functions in two complex variables with a 3-sphere. We argue that these BPS strings map to $\frac{1}{8}$-BPS surface operators under the state-operator correspondence of the CFT. We show that the string defects are holographically dual to noncompact probe D3-branes in global $AdS_5\times S^5$ that share supersymmetries with a class of dual-giant gravitons. For simple configurations, we demonstrate how to define a good variational problem and propose a regularization scheme that leads to finite energy and global charges on both sides of the holographic correspondence.

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arXiv2020

Aspects of Hecke Symmetry: Anomalies, Curves, and Chazy Equations

We study various relations governing quasi-automorphic forms associated to discrete subgroups of ${\rm SL}(2,\mathbb{R}) $ called Hecke groups. We show that the Eisenstein series associated to a Hecke group ${\rm H}(m)$ satisfy a set of $m$ coupled linear differential equations, which are natural analogues of the well-known Ramanujan identities for quasi-modular forms of ${\rm SL}(2,\mathbb{Z})$. Each Hecke group is then associated to a (hyper-)elliptic curve, whose coefficients are determined by an anomaly equation. For the $m=3$ and $4$ cases, the Ramanujan identities admit a natural geometric interpretation as a Gauss-Manin connection on the parameter space of the elliptic curve. The Ramanujan identities also allow us to associate a nonlinear differential equation of order $ m $ to each Hecke group. These equations are higher-order analogues of the Chazy equation, and we show that they are solved by the quasi-automorphic Eisenstein series $E_2^{(m)}$ associated to ${\rm H}(m) $ and its orbit under the Hecke group. We conclude by demonstrating that these nonlinear equations possess the Painlevé property.

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