Genus Two Modular Bootstrap

Abstract

We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z3-invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find 'critical surfaces' that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.

Figures (6)

• Figure 1: Left: The 3-fold cover of the Riemann sphere with four branch points is a genus-two surface. The partition function of the CFT on the covering surface can be regarded as the four-point function of $\mathbb{Z}_3$ twist fields in the 3-fold product CFT on the sphere. Right: The genus two conformal block associated with the $\sigma_3 \bar{\sigma}_3$ OPE channel.
• Figure 2: Left: The pillow geometry is the quotient $T^2/\mathbb{Z}_2$. The four branch points on the plane $0,z,1,\infty$ are mapped to the $\mathbb{Z}_2$ fixed points $v=0,\pi,\pi(\tau+1),\pi\tau$ respectively. Right: The pillow with the $\mathbb{Z}_3$ twist fields inserted at the corners. In Section \ref{['beyondz']} we will obtain the full set of genus two modular crossing equations by inserting the stress-energy tensor or more generally arbitrary Virasoro descendants of the identity at the front center on each sheet of the 3-fold-pillow.
• Figure 3: Top: Three-dimensional plots of the domain $D^{(3)}_h$ for $c=1,4,25$. Bottom: Plots of the cross-sections of these domains for various values of $h_1$. The structure constants of primaries with twists $(\tau_1,\tau_2,\tau_3) = (2h_1,2h_2,2h_3)$ outside these critical domains are bounded by those whose twists lie within the domains.
• Figure 4: A slice of the $c=4$ critical domain $D_h^{(N)}$, which converges quickly with the truncation order $N$ of the $q$-expansion.
• Figure 5: Slices of the critical domain $D_h^{(3)}$ for $c=8$ and $c=10$ at small values of $h_1$. For $c=8$, the critical domain $D_h$ intersects the $h_1=0$ plane only along a segment of the diagonal line $h_2=h_3$. This is not the case for $c=10$.