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Register a new userA new handle on three-point coefficients: OPE asymptotics from genus two modular invariance
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We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.
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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.
In this work we apply the lightcone bootstrap to a four-point function of scalars in two-dimensional conformal field theory. We include the entire Virasoro symmetry and consider non-rational theories with a gap in the spectrum from the vacuum and no conserved currents. For those theories, we compute the large dimension limit (h/c>>1) of the OPE spectral decomposition of the Virasoro vacuum. We then propose a kernel ansatz that generalizes the spectral decomposition beyond h/c>>1. Finally, we estimate the corrections to the OPE spectral densities from the inclusion of the lightest operator in the spectrum.
The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $ttoinfty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.
We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genus-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t to infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8 {cal R}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $ln (|v|)$ up to contributions which are power-behaved in $v$ as $v to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.
We exploit a gauge invariant approach for the analysis of the equations governing the dynamics of active scalar fluctuations coupled to the fluctuations of the metric along holographic RG flows. In the present approach, a second order ODE for the active scalar emerges rather simply and makes it possible to use the Greens function method to deal with (quadratic) interaction terms. We thus fill a gap for active scalar operators, whose three-point functions have been inaccessible so far, and derive a general, explicitly Bose symmetric formula thereof. As an application we compute the relevant three-point function along the GPPZ flow and extract the irreducible trilinear couplings of the corresponding superglueballs by amputating the external legs on-shell.
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