The Chern-Simons Natural Boundary and Black Hole Entropy

Abstract

The method of resurgent continuation of transseries reveals a new correspondence between the $q$-series for enumerating degeneracies of quarter-BPS states in supersymmetric black holes and $\hat{Z}$ invariants of Chern-Simons theory on a class of 3 dimensional orientation-reversed manifolds.

Figures (4)

• Figure 1: Symbolic representation of the two sides of the natural boundary, related by $q\to 1/q$.
• Figure 2: This paper adds an additional link to the correspondence in the first row of Table \ref{['fig:boundarysides']}. The double sided arrow on the left indicates that the $\widehat{Z}^{[j]}$ invariants on either side are related under $q \to 1/q$. The dashed arrow on the right indicates a precise matching between the Chern-Simons $\widehat{Z}^{[j]}$ invariants and the $q$-series coefficients $h_{(M,\ell_j)}$ of the decomposition of $\mathcal{Q}_{M}$ in \ref{['eq:key']} on the black hole side, where $M=p_1p_2p_3p_4$.
• Figure 3: The $\hbar$ (left) and $q=e^{-\hbar}$ (right) complex planes. The natural boundaries of the false theta functions at $\text{Re}(\hbar)=0$, separating the left and right half planes, and $|q|=1$, bounding the unit circle, are denoted with red lines. The analytic continuation of the integrals from $\text{Re}(\hbar)<0$ to $\text{Re}(\hbar)>0$, which experiences no obstruction at the boundary, induces a continuation on the $q$-series as they cross over the unit circle from $|q|>1$ to $|q|<1$, leading to the dual$q$-series functions $\Phi(q)^\vee$.
• Figure 4: Growth of the coefficients $a_n^{(210,j)}$ for $j=1$ (left) and $j=6$ (right). The larger spread of the coefficients in the $j=6$ case is due to the relative size of the subleading corrections, which increase as $j$ increases.

Theorems & Definitions (1)

• Conjecture 3.40