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Thermal field theory correlators in the large-$N$ limit and the spectral duality relation

Sašo Grozdanov, Mile Vrbica

TL;DR

The paper develops a rigorous framework for meromorphic retarded thermal correlators in large-$N$ QFTs and proves a spectral duality that tightly constrains the relationship between pole and zero spectra across double-trace deformations and Legendre transforms. It introduces product and partial-fraction representations, the thermal product formula, and the generalised Christmas-tree spectrum to capture asymptotic branch structure, enabling the reconstruction of one spectrum from its dual. The authors verify the framework with explicit analytic results in AdS$_3$/CFT$_2$ (BTZ) and numerical results in AdS$_5$/CFT$_4$ ($\, ext{N}=4$ SYM), including scalar and current correlators and the dynamics of poles under RG flows. They extend the SDR to conserved currents in CFT$_3$ and to theories related by particle–vortex duality, and they discuss the implications for pole-skipping, sum rules, and potential extensions to branch cuts and the thermal bootstrap. Overall, the work provides a robust, holographically motivated set of constraints and practical reconstruction methods for thermal spectra in large-$N$ theories, with broad applicability to UV/IR flows and dualities.

Abstract

In Ref.~\cite{Grozdanov:2024wgo}, we derived a spectral duality relation applicable to the spectra of 3$d$ conformal field theories (CFTs) and their holographically dual 4$d$ black holes. In this work, we further elaborate on the properties of this duality relation and argue that the same relation can be applied to certain pairs of thermal correlator spectra in large-$N$ quantum field theories in any number of spacetime dimensions, provided the correlators are meromorphic functions with only simple poles and satisfy the thermal product formula. We discuss a rich set of properties that such retarded two-point functions must exhibit. We then show that the spectral duality relation and its implications apply to pairs of correlators in double-trace deformed CFTs and, more generally, to correlators in theories related by the Legendre transform. We illustrate, through several examples, how the spectrum of one correlator can be reconstructed from that of its dual correlation function. Notably, this includes cases relating the thermal spectra of scalar primary operators at ultraviolet and infrared fixed points, as well as current operators in a CFT$_3$ and its particle-vortex dual.

Thermal field theory correlators in the large-$N$ limit and the spectral duality relation

TL;DR

The paper develops a rigorous framework for meromorphic retarded thermal correlators in large- QFTs and proves a spectral duality that tightly constrains the relationship between pole and zero spectra across double-trace deformations and Legendre transforms. It introduces product and partial-fraction representations, the thermal product formula, and the generalised Christmas-tree spectrum to capture asymptotic branch structure, enabling the reconstruction of one spectrum from its dual. The authors verify the framework with explicit analytic results in AdS/CFT (BTZ) and numerical results in AdS/CFT ( SYM), including scalar and current correlators and the dynamics of poles under RG flows. They extend the SDR to conserved currents in CFT and to theories related by particle–vortex duality, and they discuss the implications for pole-skipping, sum rules, and potential extensions to branch cuts and the thermal bootstrap. Overall, the work provides a robust, holographically motivated set of constraints and practical reconstruction methods for thermal spectra in large- theories, with broad applicability to UV/IR flows and dualities.

Abstract

In Ref.~\cite{Grozdanov:2024wgo}, we derived a spectral duality relation applicable to the spectra of 3 conformal field theories (CFTs) and their holographically dual 4 black holes. In this work, we further elaborate on the properties of this duality relation and argue that the same relation can be applied to certain pairs of thermal correlator spectra in large- quantum field theories in any number of spacetime dimensions, provided the correlators are meromorphic functions with only simple poles and satisfy the thermal product formula. We discuss a rich set of properties that such retarded two-point functions must exhibit. We then show that the spectral duality relation and its implications apply to pairs of correlators in double-trace deformed CFTs and, more generally, to correlators in theories related by the Legendre transform. We illustrate, through several examples, how the spectrum of one correlator can be reconstructed from that of its dual correlation function. Notably, this includes cases relating the thermal spectra of scalar primary operators at ultraviolet and infrared fixed points, as well as current operators in a CFT and its particle-vortex dual.

Paper Structure

This paper contains 32 sections, 4 theorems, 248 equations, 6 figures.

Table of Contents

  1. Introduction and statement of results
  2. Meromorphic correlators
  3. Analytic structure and decomposition
  4. Poles, zeroes, and the spectral duality relation
  5. Generalised Christmas tree configuration
  6. Comments on sum rules and pole-skipping
  7. CFTs related by double-trace deformations
  8. Double-trace deformations and the spectral duality relation
  9. The fixed-point limit and the Legendre transform
  10. Motion of the poles as a dynamical system
  11. Analytic example: 2$d$ CFT and the QNMs of the BTZ black hole
  12. Numerical example: 4$d$ $\mathcal{N}=4$ supersymmetric Yang-Mills theory
  13. Beyond the scalar operators: conserved currents in CFT$_3$
  14. Current-current correlators in 3$d$
  15. Particle-vortex duality
  16. ...and 17 more sections

Key Result

Theorem 1

Let $R_N$ be a sequence of positive real numbers such that Let $f(\omega)$ be a meromorphic function, with the set of poles denoted by $\qty{\omega_n}$, and corresponding residues denoted by $\qty{r_n}$. All the poles are assumed to be simple. Furthermore, let $M_N$ denote the maximal modulus of $f(\omega)$ on concentric circles Suppose that $f(\omega)$ is such that, for some sequence $R_N$, we

Figures (6)

  • Figure 1: A toy example of a spectrum of poles (red) and zeroes (blue). We have two branches, $\omega_{m,1}$ and $\omega_{m,2}$ with, $d_1^+=d_1^-=d_1$ and $d_2^+=d_2^-=d_2$. The zero on the imaginary line is an element of $\{\omega_{m,2}\}$, which displaces its indexing relative to its mirror counterpart. The spacings $d^\pm_{j}$ are the same for the poles and the zeroes, while the relative offsets are $\sigma_1=-2/3$ and $\sigma_2=1/3$. This gives $G(\omega)\sim \omega^{-1/3}$.
  • Figure 2: Left panel: Spectrum of the retarded scalar correlator deformed by an arbitrary double-trace coupling, expressed as the full set of solutions to Eq. \ref{['eq:BTZ_trace']} at $k=3/\beta$ and $\nu=0.58$. We define $\kappa=\frac{f_--1}{f_-+1}$, so that $\kappa=-1$ at the UV and $\kappa=1$ at the IR fixed point. Along the RG flow, some poles move downwards, while some move upwards. One pair of poles 'escapes to infinity'. This behaviour is highly sensitive to the scaling dimension of the scalar operator. Right panel: Double-trace RG flows of the same scalar operators plotted at $k=3/\beta$ for different scaling dimensions with $\nu=\qty{0.2,0.21,0.22,\ldots,0.9}$. For every scaling dimension, there is a pair of poles that 'escapes to infinity'. The 'choice' of the pair depends sensitively on the scaling dimension.
  • Figure 3: The double-trace deformed flows of the scalar spectrum between the UV to IR fixed points shown for $\nu=1/3$ (above) and $\nu=2/3$ (below). Both flows are shown for two values of momentum. Here, $\kappa$ is a rescaled coupling constant $\kappa=\frac{f_- G_-(0)-1}{f_- G_-(0)+1}=\frac{1-f_+ G_+(0)}{1+f_+ G_+(0)}$, such that $\kappa=-1$ at the UV and $\kappa=1$ at the IR fixed point. The spectra at all couplings were computed through the knowledge of the IR spectrum (see Eq. \ref{['num:swipe_IR']}), with the constant $\lambda_+$ set by the lowest-lying UV-fixed-point pole (i.e., by setting $G_+(\omega_{1,1}^-)=0$). The exception is the zero-momentum spectrum with $\nu=2/3$, where the spectra at various $\kappa$ were computed only with the knowledge of UV spectrum (see Eq. \ref{['num:swipe_UV']}), as that proved to be most numerically accurate. The fixed point spectra (circles and squares) were computed numerically by solving the bulk wave equation with Neumann or Dirichlet boundary conditions. We note that the trajectories that are closer to the complex origin are 'more blue', meaning that they are affected more by a small deformation of the UV fixed point. As we increase the coupling, the low-lying poles then relax to their IR positions much sooner than the higher-energy QNMs.
  • Figure 4: The parameter $\lambda=\lambda_+\approx 0.27$ computed with elementary symmetric polynomials for various values of $j$ (see Eq. \ref{['eq:lambda_e']}). This is compared with the same parameter as computed through $\lambda=-2i G_-'(0)/\beta G_-(0)$ (downwards triangles), or $\lambda=2 i G_+'(0)/\beta G_+(0)$ (upwards triangles), where $G_-(\omega)$ and $G_+(\omega)$ are expressed through Eqs. \ref{['num:gUV']} and \ref{['num:gIR']} respectively, with $M=50$. The former is equivalent to Eq. \ref{['eq:lambda']}, while the latter is determined from the IR spectrum and the lowest-lying UV pole through $G_+(\omega_{1,1}^-)=0$.
  • Figure 5: The convergence of the spectral duality relation \ref{['eq:SDR']} with the UV and the IR spectra computed directly from the bulk. The odd part of the truncated product \ref{['num:S']} is shown for $m_\text{max}=\qty{10,20,30,40,50}$. The function $-2\lambda \sin \beta\abs{\omega}/2$ is shown in red.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Corollary 1.1
  • Theorem 2
  • proof
  • Theorem 3
  • proof