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Holographic thermal propagator from modularity

Borut Bajc, Katarina Trailović

TL;DR

The paper links the holographic thermal propagator in a 4d CFT to the Nekrasov-Shatshvili prepotential of $\mathrm{SU}(2)$ with $N_f=4$, exploiting Zamolodchikov's $q$-recursion and $S$-duality to show the instanton coefficients are quasi-modular in $q$. In the large-$a$ (low-$T$) limit, it derives a compact expression for $G_R$ and computes the low-temperature expansion analytically, finding it to be asymptotic and amenable to Padé resummation for improved finite-$T$ accuracy. The work provides a robust framework to obtain analytic low-$T$ propagators in higher-dimensional holography by organizing instanton data into Eisenstein-series-based quasi-modular forms, with potential extensions to finite energy and richer mass invariants. This advances both the technical toolkit for holographic correlators and the understanding of modular structures in NS prepotentials.

Abstract

It is known that the holographic thermal propagator in 4 spacetime dimensions can be related to the Nekrasov-Shatashvili limit of the $Ω$-deformed ${\cal N}=2$ supersymmetric $SU(2)$ Yang-Mills theory with $N_f=4$ hypermultiplets. There are two expansions involved: one is the expansion in small temperature which in the Seiberg-Witten language is equivalent to the semiclassical expansion in inverse powers of the large adjoint vev and the second is the expansion in instanton numbers. Working in the simplified case of zero energy, we find that the latter expansion gives rise to quasi-modular forms which can be resummed as functions of Eisenstein series. The so obtained series in positive powers of small temperature shows clear signs of being asymptotic.

Holographic thermal propagator from modularity

TL;DR

The paper links the holographic thermal propagator in a 4d CFT to the Nekrasov-Shatshvili prepotential of with , exploiting Zamolodchikov's -recursion and -duality to show the instanton coefficients are quasi-modular in . In the large- (low-) limit, it derives a compact expression for and computes the low-temperature expansion analytically, finding it to be asymptotic and amenable to Padé resummation for improved finite- accuracy. The work provides a robust framework to obtain analytic low- propagators in higher-dimensional holography by organizing instanton data into Eisenstein-series-based quasi-modular forms, with potential extensions to finite energy and richer mass invariants. This advances both the technical toolkit for holographic correlators and the understanding of modular structures in NS prepotentials.

Abstract

It is known that the holographic thermal propagator in 4 spacetime dimensions can be related to the Nekrasov-Shatashvili limit of the -deformed supersymmetric Yang-Mills theory with hypermultiplets. There are two expansions involved: one is the expansion in small temperature which in the Seiberg-Witten language is equivalent to the semiclassical expansion in inverse powers of the large adjoint vev and the second is the expansion in instanton numbers. Working in the simplified case of zero energy, we find that the latter expansion gives rise to quasi-modular forms which can be resummed as functions of Eisenstein series. The so obtained series in positive powers of small temperature shows clear signs of being asymptotic.

Paper Structure

This paper contains 10 sections, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Zamolodchikov's $q$-recursion
  3. The retarded Green's function of the black brane
  4. $\mathcal{S}$-duality constraints on the prepotential: quasi-modular forms
  5. Low temperature expansion of the black brane propagator
  6. Conclusion
  7. Holographic two-point functions: black brane
  8. Eisenstein series and (quasi-) modular forms
  9. $\tilde{\alpha}_{2n}$ up to $2n=18$
  10. $P_l(\nu)$ up to $l=10$

Figures (1)

  • Figure 1: For $\nu=0.5$, we compare the Padé approximant (blue) with the truncated low-temperature expansion of the propagator (dashed, green) as $T \to 0$, and with the numerical solution (dotted, red) within a finite interval. The three panels display Padé approximants of degree $(n,m) = (1,6)$, $(3,4)$, and $(5,2)$, respectively.