Steklov Approximations of Solutions of Laplacian Boundary Value Problems

Eigenfunction expansion methods have been studied in various ways to study solutions of PDEs. This talk will feature error estimates for approximation of solutions of Laplace’s equation with Dirichlet, Robin or Neumann boundary value conditions using the harmonic Steklov eigenfunctions. Based on the spectral theory of trace spaces the solutions are represented by orthogonal basis from normalized Steklov eigenfuntions. When the region is a rectangle, with explicit formulae for the Steklov eigenfunctions, both theoretical analysis and numerical experiments will introduce the efficiency and accuracy of the Steklov expansion methods in this talk

On string one-loop correction to the Einstein-Hilbert term and its implications on the Kahler potential

To compute the string one-loop correction to the Kahler potential of moduli fields of string compactifications in Einstein-frame, one must compute: the string one-loop correction to the Einstein-Hilbert action, the string one-loop correction to the moduli kinetic terms, the string one-loop correction to the definition of the holomorphic coordinates. In this note, we compute the string one-loop correction to the Einstein-Hilbert action of type II string theory compactified on orientifolds of Calabi-Yau threefolds. We find that the one-loop correction is determined by the new supersymmetric index studied by Cecotti, Fendley, Intriligator, and Vafa. As a simple application, we apply our results to estimate the size of the one-loop corrections around a conifold point in the Kahler moduli space.Comment: v1: 25 pages, 2 figures, and appendices. v2: typos fixed, numerical errors corrected, references added, discussions expanded. v3: Discussions expanded, references added, additional check of the normalization provided by comparing to the known R^4 term in 10

D-Instanton Superpotential In String Theory

We study non-perturbative superpotential generated by D(-1)-branes in type IIB compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. To compute D-instanton superpotential, we study F-theory compactification on toric complete intersection elliptic Calabi-Yau fourfolds. We take the Sen-limit, but with finite $g_s,$ in F-theory compactification with a restriction that all D7-branes are carrying SO(8) gauge groups, which we call the global Sen-limit. In the global Sen-limit, the axio-dilaton is not varying in the compactification manifold. We compute the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, and show that the Picard-Fuchs equations of the elliptic fourfolds split into that of the underlying Calabi-Yau threefolds and of the elliptic fiber. We then demonstrate that this splitting property of the Picard-Fuchs equation implies that the fourform period of the elliptic Calabi-Yau fourfolds in the global Sen-limit does not contain exponentially suppressed terms $\mathcal{O}(e^{-\pi/g_s})$. With this result, we finally show that in the global Sen-limit, superpotential of the underlying type IIB compactification does not receive D(-1)-instanton contributions.Comment: v4. 34 pages. Discussion extended. Results unchanged. Matches the Jhep versio

On one-loop corrected dilaton action in string theory

This manuscript concerns string one-loop corrections to the Kahler potential in 4d N=1 vacua of string theories, and it largely consists of two parts. In the first part, we compute the string one-loop correction to the dilaton kinetic term in heterotic string theories, and we show that the dilaton kinetic term is not renormalized at one-loop. After reviewing the well-known result on the string one-loop correction to the Kahler potential in heterotic string theories, we explain how this result can be reconciled with the known result in the literature. In the second part, we study the dilaton dependence of the one-loop corrected Kahler potential in type II string theories. To do so, we compute the string one-loop correction to the kinetic action of the 4d-dilaton in type II string theories compactified on orientifolds of Calabi-Yau threefolds. We find that the string one-loop corrected 4d dilaton kinetic term is determined by the Witten index and the new supersymmetric index of the string worldsheet CFT.Comment: v2: 59 pages. It was found that even at the four fermi level, vertex collisions can generate an enhancement by the factor of 1/delta. Such corrections are correctly included in the computation. v3: typos fixed. v4: Typos fixed, discussions expande

On the intermediate Jacobian of M5-branes

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise a method to analyze the Hodge structure (and hence the dimension of the intermediate Jacobian) of vertical divisors in these fourfolds, using only the data available from a type IIB compactification on the O3/O7 Calabi-Yau orientifold. Our method utilizes simple combinatorial formulae (that we prove) for the equivariant Hodge numbers of the Calabi-Yau orientifolds and their prime toric divisors, along with a formula for the Euler characteristic of vertical divisors in the corresponding elliptic Calabi-Yau fourfold. Our formula for the Euler characteristic includes a conjectured correction term that accounts for the contributions of pointlike terminal $\mathbb{Z}_2$ singularities corresponding to perturbative O3-planes. We check our conjecture in a number of explicit examples and find perfect agreement with the results of direct computations.Comment: 110 pages plus appendices. Reviews various aspects of toric geometry, including toric hypersurfaces and stratifications. v2: Corrected minor typos, included clarifications about families of F-theory vacua to which our results appl