Elliptic Genera and 3d Gravity

TL;DR

The work derives a universal bound on the polar coefficients of the elliptic genus for 2d (2,2) SCFTs with large-radius gravity duals by matching bulk BH entropy and enforcing a Hawking-Page transition at $eta=2 extpi$. It tests this bound across symmetric products, Calabi–Yau families, and the Monster CFT, finding that symmetric products like ${ m Sym}^N(K3)$ satisfy the bound while many other constructions exhibit growth incompatible with a weakly curved gravity dual. The analysis clarifies how growth rates of polar terms distinguish between supergravity- and string-dominated duals, and proposes a framework to quantify the prevalence of gravity-dual theories in moduli space, albeit with caveats about the choice of measure. Overall, the paper illuminates the connections between modular form data, bulk phase structure, and the nature of possible holographic duals in AdS$_3$/CFT$_2$.

Abstract

We describe general constraints on the elliptic genus of a 2d supersymmetric conformal field theory which has a gravity dual with large radius in Planck units. We give examples of theories which do and do not satisfy the bounds we derive, by describing the elliptic genera of symmetric product orbifolds of $K3$, product manifolds, certain simple families of Calabi-Yau hypersurfaces, and symmetric products of the "Monster CFT." We discuss the distinction between theories with supergravity duals and those whose duals have strings at the scale set by the AdS curvature. Under natural assumptions we attempt to quantify the fraction of (2,2) supersymmetric conformal theories which admit a weakly curved gravity description, at large central charge.

Figures (9)

• Figure 1: The tessellation by $\Gamma_\theta$ and its sub-tessellation by $\Gamma_\infty\backslash\Gamma_\theta$. The thick lines are where phase transitions in supergravity can occur.
• Figure 2: Here, we plot the polar coefficients of ${\rm Sym}^{20}(K3)$ versus polarity, and also the coefficients allowed by the bounds. We see that at this value of $c$ (=120), the bounds are satisfied by the symmetric product conformal field theory, after allowing minor shifts due to the $O(\log{m})$ correction.
• Figure 3: Here, we plot the polar coefficients of the product conformal field theory with target $K3^{20}$.
• Figure 4: Here, we plot the polar coefficients of $Z_{RR}^{d=10}$.
• Figure 5: Here, we plot the polar coefficients of $Z_{RR}^{d=20}$.