Abstract
The SCHOK bound states that the number of marginal deformations of certain two-dimensional conformal field theories is bounded linearly from above by the number of relevant operators. In conformal field theories defined via sigma models into Calabi-Yau manifolds, relevant operators can be estimated, in the point-particle approximation, by the low-lying spectrum of the scalar Laplacian on the manifold. In the strict large volume limit, the standard asymptotic expansion of Weyl and Minakshisundaram-Pleijel diverges with the higher-order curvature invariants. We propose that it would be sufficient to find an a priori uniform bound on the trace of the heat kernel for large but finite volume. As a first step in this direction, we then study the heat trace asymptotics, as well as the actual spectrum of the scalar Laplacian, in the vicinity of a conifold singularity. The eigenfunctions can be written in terms of confluent Heun functions, the analysis of which gives evidence that regions of large curvature will not prevent the existence of a bound of this type. This is also in line with general mathematical expectations about spectral continuity for manifolds with conical singularities. A sharper version of our results could, in combination with the SCHOK bound, provide a basis for a global restriction on the dimension of the moduli space of Calabi-Yau manifolds.
Highlights
In recent years, a revival of the conformal bootstrap program has led to remarkable progress on a priori constraints on the operator content of certain types of conformal field theories
Besides its phenomenological relevance as a starting point for phenomenological finiteness in string theory, a strict upper bound on the number of marginal operators of a SCFT of central charge c = 3 would imply the existence of an upper bound on the dimension of cohomology groups of Calabi-Yau threefolds, which is a hopeful, but largely open, mathematical conjecture
The main idea of the present paper goes back to a question that arose in [11]: are there any interesting constraints on the number of relevant operators that depend on a geometric origin of the conformal field theory, but that are independent of other, topological data such as the number of marginal deformations? What sorts of constraints on the massless spectrum can be derived from these results?
Summary
In a remarkable paper [8], Hellerman and Schmidt-Colinet have shown how modular invariance of the torus partition function can be exploited for the purpose of deriving universal bounds on state degeneracies and related thermodynamic quantities in 2-dimensional conformal field theories. One notes that for sufficiently small central charge, the marginal operators contribute states with positive energy in (2.2). As is well-known, exactly marginal operators in N = 2 SCFT arise from chiral and antichiral primary fields (BPS representations of the N = 2 algebra), and, as shown in [9], make a positive contribution to the suitably weighted vanishing partition function. For central charge c = 3, negative contributions to the partition function come only from non-BPS primaries of sufficiently small conformal dimension ∆total. For the following discussion of supplemental geometric bounds, it will be convenient to summarize and remember the SCHOK bound as the statement that for fixed central charge, there exist constants C0, C1 such that in any N = 2 SCFT of that given central charge, the number Nmarginal of exactly marginal BPS operators is bounded linearly by the number of tachyons, i.e., Nmarginal < C0 + C1 · Ntachyons (2.4). We can drop C0 from the above statement without penalty
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