arXiv subjects

Suresh Govindarajan

Publications and source records attributed to Suresh Govindarajan.

At least 19 recordsLinked to original sources

arXiv2026

Conformal blocks and braiding matrices for RCFTs I: Virasoro minimal models

We develop a systematic method for computing the braiding matrices of four-point functions in RCFTs. This is implemented for the case of Virasoro minimal models. The first few terms of the conformal block series are directly computed through the Shapovalov form. The Fuchsian (BPZ) ODEs for order three and higher have accessory parameters that are fixed by this direct computation. We thus bypass the derivation from the null-state condition which gets tedious for higher-level null states. The braiding F-matrices are connection matrices between the Frobenius solutions at two singular points of the ODE. The F-matrices are computed numerically at first, agreeing with the results of Dotsenko--Fateev, which uses the Coulomb-gas formalism. We find that after a change of basis, the F-matrix is rendered unitary, which determines (products of) the three-point structure constants. We conjecture that the squares of the entries of the unitary F-matrix lie in a cyclotomic extension of the rational numbers. This enables us to convert our numerical estimates for the F-matrix into exact ones. The formalism is illustrated through numerous examples in Virasoro minimal models. We also discuss how these methods can be extended to cases involving symmetries that extend the Virasoro symmetry as well as for tenable examples that arise from the holomorphic modular bootstrap program.

hep-th↗

arXiv2026

Quasi-Characters for three-character Rational Conformal Field Theories

We revisit (3,0) and (3,3) admissible solutions obtained using the MLDE method. We show that all $(3,0)$ solutions can be written in terms of a universal formula involving the ${}_3F_2$ hypergeometric function that takes into account the monodromy at the elliptic points. We construct $(3,3)$ admissible solutions from (3,0) CFTs using a duality due to Bantay and Gannon. This enables us to compute their modular properties such as the S-matrix and the fusion rules. We find that only 7 of the 15 known (3,3) admissible solutions have proper fusion rules. Using the theory of matrix MLDE, starting with a known (3,0) and (3,3) solutions, we construct two other solutions, that are typically quasi-characters that share the same multiplier as the original solution. We then construct linear combinations that lead to new admissible solutions. We observe that admissible solutions arise as integer points that lie on a polytope. We construct all possible (3,6) and (3,9) admissible solutions that arise in this fashion. In some cases, we identify RCFT that arise from our (3,6) admissible solutions. In addition, we obtain a large family of admissible solutions with higher Wronskian index.

hep-th↗

arXiv2026

Updating the holomorphic modular bootstrap

We update the holomorphic modular bootstrap incorporating a recent result that computes the exact S-matrix within the Modular Linear Differential Equation (MLDE) setting. Further, using knowledge of the allowed exponents modulo one, we obtain admissible solutions to all MLDE's with up to six characters and Wronskian index < 6 and one accessory parameter with c_eff <= 24. We then identify which of the admissible solutions have good fusion rules -- we call such solutions tenable. When possible, we identify the CFT and in the unitary cases the MTC class they belong to.

hep-th↗

arXiv2026

Two approaches to the holomorphic modular bootstrap

The holomorphic bootstrap attempts to classify rational conformal field theories. The straight ahead approach is hard to implement when the number of characters become large. We combine all characters of an RCFT to form a vector valued modular form with multiplier. Using known results from the theory of vector valued modular forms, given a known RCFT, we obtain new vector valued modular forms that share the same multiplier as the original RCFT. By taking particular linear combinations of the new solutions, we look for and find new admissible solutions. In the well-studied two character case, we reproduce all known admissible solutions with Wronskian indices $6$ and $8$. The method is illustrated with examples with up to six characters. The method using vector valued modular forms thus provides a new approach to the holomorphic modular bootstrap.

hep-th↗

arXiv2026

S-matrices in the holomorphic modular bootstrap approach

We numerically determine the S-matrix by using connection formulae in the modular linear differential equation (MLDE) approach to the holomorphic modular bootstrap. We then determine exact formulae using the fact that entries in the $S$-matrix are integer entries in a cyclotomic extension of the field of rational numbers. This provides a method that is intrinsic to the MLDE setup and does not require inputs outside this framework. The method is illustrated with a selection of examples.

hep-th↗

arXiv2026

Numerical Computations of Entanglement Measures in Curved Space

We numerically compute the entanglement entropy and negativity for scalar fields and abelian gauge fields in a variety of situations. These extend computations of Srednicki to situations involving curved space. We discretize space in a covariant way. Finally, we compare some of our results with those obtained via the heat kernel coefficients.

hep-th↗

arXiv2025

The affine Brylinski filtration and $\mathscr{W}$-algebras

The Brylinski-Kostant filtration on a representation of a finite-dimensional semisimple Lie algebra has interpretations in terms of the algebra, geometry and combinatorics of the representation. Its extension to affine Lie algebras was first studied by Slofstra. Recent work of the present authors constructed a Poincaré-Birkhoff-Witt type basis for the dominant weight spaces of the basic representation of affine Lie algebras of type $A$, which is compatible with the affine Brylinski filtration. In this paper, we overcome the constraint of type dependence, and furnish a new, uniform proof which holds for all simply-laced affine Lie algebras.

math.RT↗

arXiv2024

Positivity of discrete information for CHL black holes

Black holes carry more information about the microstates than just the total degeneracy. As a concrete example, the Z(N)-twined helicity trace indices for 1/4-BPS black holes of the CHL models allow extracting information about the distribution of the Z(N) charges among the black hole microstates. The number of black hole microstates carrying a definite eigenvalue under the generator of the Z(N) twining group must be positive. This leads to a specific prediction for the signs of certain linear combinations of Fourier coefficients of Siegel modular forms. We explicitly test these predictions for low charges. In the D1-D5-P duality frame, we compute the appropriate hair removed partition functions and show the positivity of the appropriate Fourier coefficients for low charges. We present various consistency checks on our computations.

hep-th↗

arXiv2022

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras -- II

The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did not appear in that case. We restrict our analysis to the first N terms in the expansion as a first attempt at deconstructing the Siegel modular forms and unravelling the structure of potentially new Lie algebras that occur for N=5,6.

hep-th↗

arXiv2022

Black Hole Hair Removal For N=4 CHL Models

Although BMPV black holes in flat space and in Taub-NUT space have identical near-horizon geometries, they have different indices from the microscopic analysis. For K3 compactification of type IIB theory, Sen et al in a series of papers identified that the key to resolving this puzzle is the black hole hair modes: smooth, normalisable, bosonic and fermionic degrees of freedom living outside the horizon. In this paper, we extend their study to N = 4 CHL orbifold models. For these models, the puzzle is more challenging due to the presence of the twisted sectors. We identify hair modes in the untwisted as well as twisted sectors. We show that after removing the contributions of the hair modes from the microscopic partition functions, the 4d and 5d horizon partition functions agree. Special care is taken to present details on the smoothness analysis of hair modes for rotating black holes, thereby filling an essential gap in the literature.

hep-th↗

arXiv2021

The Brylinski filtration for affine Kac-Moody algebras and representations of $\mathcal{W}$-algebras

We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra $\mathfrak{g}$, a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of $\mathfrak{g}$ has Hilbert series coinciding with Lusztig's $t$-analogue of weight multiplicities. For the level 1 vacuum module $L(Λ_0)$ of affine Kac-Moody algebras of type $A$, we show that the Brylinski filtration may be most naturally understood in terms of (vertex algebra) representations of the corresponding $\mathcal{W}$-algebra. We show that the dominant weight spaces together form an irreducible Verma module of $\mathcal{W}$ and that the natural PBW basis of this module is compatible with the Brylinski filtration, thereby determining explicitly the subspaces of the filtration. Our basis is the analogue for the principal vertex operator realization of $L(Λ_0)$, of Feigin-Frenkel's basis of $\mathcal{W}$.

math.RT↗

arXiv2021

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

hep-th↗

arXiv2021

Mathieu Moonshine and Siegel Modular Forms

A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper, we express the product formulae for all conjugacy classes of $M_{24}$ in terms of products of standard modular forms. This provides a new proof of their modularity.

hep-th↗

arXiv2019

BKM Lie superalgebras from counting twisted CHL dyons -- II

We revisit our earlier work which lead to a periodic table of Borcherds-Kac-Moody algebras that appeared in the context of the refined generating function of quarter-BPS (dyons) in $\mathcal{N}=4$ supersymmetric four-dimensional string theory. We make new additions to the periodic table by making use of connections with generalized Mathieu moonshine as well as umbral moonshine. We show the modularity of some Siegel modular forms that appear in umbral moonshine associated with Niemeier lattices constructed from A-type root systems and further show that the same Siegel modular forms appear for generalized Mathieu moonshine in some cases. We argue for the existence of a new kind of BKM Lie superalgebras that arise from the dyon generating functions for the $\mathbb{Z}_5$ and $\mathbb{Z}_6$ CHL orbifolds.

hep-th↗

arXiv2019

Two moonshines for $L_2(11)$ but none for $M_{12}$

In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of $M_{12}$ to Jacobi forms of weight one and index zero. We construct Jacobi forms for all conjugacy classes of $M_{12}$ that are consistent with constraints from group theory as well as modularity. However, we obtain 1427 solutions that satisfy these constraints (to the order that we checked) and are unable to provide a unique Jacobi form. Nevertheless, as a consequence, we are able to provide a group theoretic proof of the evenness of the coefficients of all EOT Jacobi forms associated with conjugacy classes of $M_{12}:2 \subset M_{24}$. We show that there exists no solution where the Jacobi forms (for order 4/8 elements of $M_{12}$) transform with phases under the appropriate level. In the absence of a moonshine for $M_{12}$, we show that there exist moonshines for two distinct $L_2(11)$ sub-groups of the $M_{12}$. We construct Siegel modular forms for all $L_2(11)$ conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds-Kac-Moody Lie superalgebra.

hep-th↗

arXiv2018

Unravelling Mathieu Moonshine

The D1-D5-KK-p system naturally provides an infinite dimensional module graded by the dyonic charges whose dimensions are counted by the Igusa cusp form, Phi_{10}(Z)$. We show that the Mathieu group, M_{24}, acts on this module by recovering the Siegel modular forms that count twisted dyons as a trace over this module. This is done by recovering Borcherds product formulae for these modular forms using the M_{24} action. This establishes the correspondence (`moonshine') proposed in arXiv:0907.1410 that relates conjugacy classes of M_{24} to Siegel modular forms. This also, in a sense that we make precise, subsumes existing moonshines for M_{24} that relates its conjugacy classes to eta-products and Jacobi forms.

hep-th↗

arXiv2016

On a square-ice analogue of plane partitions

We study a one-parameter family ($\ell=1,2,3,\ldots$) of configurations that are square-ice analogues of plane partitions. Using an algorithm due to Bratley and McKay, we carry out exact enumerations in order to study their asymptotic behaviour and establish, via Monte Carlo simulations as well as explicit bounds, that the asymptotic behaviour is similar to that of plane partitions. We finally carry out a series analysis and provide independent estimates for the asymptotic behaviour.

cond-mat.stat-mech↗

arXiv2014

A superasymptotic formula for the number of plane partitions

We revisit a formula for the number of plane partitions due to Almkvist. Using the circle method, we provide modifications to his formula along with estimates of the errors. We show that the improved formula continues to be an asymptotic series. Nevertheless, an optimal truncation (i.e., superasymptotic) of the formula provides exact numbers of plane partitions for all positive integers n <6400 and numbers with estimated errors for larger values. For instance, the formula correctly reproduces 305 of the 316 digits of the numbers of plane partitions of 6999 as predicted by the estimated error. We believe that an hyperasymptotic truncation might lead to exact numbers for positive integers up to 50000.

math.NT↗