Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High Energy Modular Bootstrap, Global Symmetries and Defects

Sridip Pal, Zhengdi Sun

TL;DR

The paper develops a Tauberian-enhanced modular bootstrap for 2D unitary CFTs with topological defect lines, proving Cardy-like growth bounds for defect and symmetry sectors. It shows that all irreps of a finite global symmetry G appear in the untwisted sector with growth ∝ d_α^2/|G| times the seed density ρ_0(Δ), and extends to twisted sectors in the non-anomalous case, as well as to continuous U(1) charges with explicit Δ-dependent bounds. In the Schwarzian (JT gravity) limit, the results reproduce the expected bulk gauge-field factor d_α^2/|G|, and the framework covers Virasoro primaries for c>1 and large-c theories. The methodology relies on projecting partition functions onto sectors, using S-transforms to relate defect and symmetry channels, and applying vector-valued Tauberian arguments to obtain sharp, order-one bounded growth across sectors. Practical implications include holographic completeness intuition and quantitative sector-wise density bounds in theories with finite and continuous symmetries, plus potential extensions to other modular-function contexts.

Abstract

We derive Cardy-like formulas for the growth of operators in different sectors of unitary $2$ dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval $[Δ-δ,Δ+δ]$ for large $Δ$ at fixed $δ$. Consequently we prove that given any unitary modular invariant $2$D CFT symmetric under finite global symmetry $G$ (acting faithfully), all the irreducible representations of $G$ appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the ''square'' of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For $c>1$, the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous $U(1)$ group.

High Energy Modular Bootstrap, Global Symmetries and Defects

TL;DR

The paper develops a Tauberian-enhanced modular bootstrap for 2D unitary CFTs with topological defect lines, proving Cardy-like growth bounds for defect and symmetry sectors. It shows that all irreps of a finite global symmetry G appear in the untwisted sector with growth ∝ d_α^2/|G| times the seed density ρ_0(Δ), and extends to twisted sectors in the non-anomalous case, as well as to continuous U(1) charges with explicit Δ-dependent bounds. In the Schwarzian (JT gravity) limit, the results reproduce the expected bulk gauge-field factor d_α^2/|G|, and the framework covers Virasoro primaries for c>1 and large-c theories. The methodology relies on projecting partition functions onto sectors, using S-transforms to relate defect and symmetry channels, and applying vector-valued Tauberian arguments to obtain sharp, order-one bounded growth across sectors. Practical implications include holographic completeness intuition and quantitative sector-wise density bounds in theories with finite and continuous symmetries, plus potential extensions to other modular-function contexts.

Abstract

We derive Cardy-like formulas for the growth of operators in different sectors of unitary dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval for large at fixed . Consequently we prove that given any unitary modular invariant D CFT symmetric under finite global symmetry (acting faithfully), all the irreducible representations of appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the ''square'' of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For , the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous group.

Paper Structure

This paper contains 17 sections, 4 theorems, 73 equations, 10 figures.

Table of Contents

  1. Summary & Discussion
  2. Results:
  3. Lightning review of Topological defect line
  4. Charting Defect Hilbert Space $\mathcal{H}_{\mathcal{L}}$ associated with TDL $\mathcal{L}$
  5. Cardy Formula for Defect Hilbert Space
  6. Charting Hilbert Space $\mathcal{H}^{\mathcal{L}}$ associated with invertible TDL $\mathcal{L}$
  7. Warm up: $G=\mathbb{Z}_2$
  8. Arbitrary finite group $G$ ala Tauberian
  9. Untwisted sector
  10. Twisted sector
  11. TDL associated with continuous symmetry $U(1)$
  12. Tauberian for Vector-valued modular function
  13. Verification
  14. Ising CFT with $\mathbb{Z}_2$ symmetry
  15. Compact Boson at $R=\frac{1}{2}$ with $U(1)$ symmetry
  16. ...and 2 more sections

Key Result

Theorem 1

Given a TDL $\mathcal{L}$, the asymptotic behavior ($\Delta\to \infty$) of the growth of states in an order one window of width $2\delta$, centered at $\Delta$ in the defect Hilbert space $\mathcal{H}_{\mathcal{L}}$ is given by where $N_0=1$ if the TDL is associated with a global symmetry, i.e. invertible one, otherwise it is taken to be a positive number as defined below and $\rho_0(\Delta)$ is

Figures (10)

  • Figure 1: The theory $A$ and $D$ are related by orbifolding by $\mathbb{Z}_2$. The theory $A$ and $F$ are related by Bosonization-Fermionization and so are $D$ and $\tilde{F}$.
  • Figure 2: $\mathcal{L}_g$ denotes the $g$ symmetry TDL; dashed line denotes the trivial line; $\mathcal{O}$ is arbitrary local operator; and $\phi_{(0,0)}$ is the operator correspond to the weight $(0,0)$ state in $\mathcal{H}_{\mathcal{L}_g}$. The existence of $\phi_{(0,0)}$ allows us to open the TDL to show that the $\mathcal{L}_g$ commutes with every local operator $\mathcal{O}$. In the main text, we drop the $g$ subscript where it is redundant.
  • Figure 3: The partition function of the defect Hilbert space (the left figure, which we will denote $Z_{\mathcal{L}}(\beta,g)$) is related to the partition function with the insertion of the corresponding charge operator (the right figure, which we will denote $Z^{\mathcal{L}}(\beta,g)$).
  • Figure 4: Here, we estimate the number of even and odd operators (under $\mathbb{Z}_2$) in Ising CFT. We plot the logarithm of the ratio of actual number of operators in the interval of size $2\delta=2.2$ and the leading prediction from Tauberian-Cardy analysis (in cyan). We see that they are well within the bounds (the black and the red line) as predicted in the main text.
  • Figure 5: Here, we estimate the number of operators in the defect Hilbert space corresponding to $\mathbb{Z}_2$ in Ising CFT. We plot the logarithm of the ratio of actual number of operators in the interval of size $2\delta=2.2$ and the leading prediction from Tauberian-Cardy analysis (in cyan). We see that they are well within the bounds (the black and the red line) as predicted in the main text.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4