Lectures on the topological recursion for Higgs bundles and quantum curves

© 2018 World Scientific Publishing Co. Pte. Ltd. This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed

Extremal Transitions in Heterotic String Theory

In this paper we study extremal transitions between heterotic string compactifications, i.e., transitions between pairs (M,V) where M is a Calabi-Yau manifold and V a gauge bundle. Bundle transitions are described using language recently espoused by Friedman, Morgan, Witten. In addition, partly as a check on our methods, we also study how small instantons are described in the same language, and also describe the sheaves corresponding to small instantons.Comment: 26 pages, LaTex, 3 figures, references adde

Compact and accurate models of large single-wall carbon-nanotube interconnects

Single-wall carbon nanotubes (SWCNTs) have been proposed for very large scale integration interconnect applications and their modeling is carried out using the multiconductor transmission line (MTL) formulation. Their time-domain analysis has some simulation issues related to the high number of SWCNTs within each bundle, which results in a highly complex model and loss of accuracy in the case of long interconnects. In recent years, several techniques have been proposed to reduce the complexity of the model whose accuracy decreases as the interconnection length increases. This paper presents a rigorous new technique to generate accurate reduced-order models of large SWCNT interconnects. The frequency response of the MTL is computed by using the spectral form of the dyadic Green's function of the 1-D propagation problem and the model complexity is reduced using rational-model identification techniques. The proposed approach is validated by numerical results involving hundreds of SWCNTs, which confirm its capability of reducing the complexity of the model, while preserving accuracy over a wide frequency range

Quantum curves for Hitchin fibrations and the Eynard-Orantin theory

We generalize the topological recursion of Eynard-Orantin (2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle $T^*C$ of an arbitrary smooth base curve $C$. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal $\hbar$-deformation family of $D$-modules over an arbitrary projective algebraic curve $C$ of genus greater than $1$, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the $SL(2,\mathbb{C})$-character variety of the fundamental group $\pi_1(C)$. We show that the semi-classical limit through the WKB approximation of these $\hbar$-deformed $D$-modules recovers the initial family of Hitchin spectral curves.Comment: 34 page

F-theory and linear sigma models

We present an explicit method for translating between the linear sigma model and the spectral cover description of SU(r) stable bundles over an elliptically fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional duality between (0,2) heterotic and F-theory compactifications. We indirectly find that much interesting heterotic information must be contained in the `spectral bundle' and in its dual description as a gauge theory on multiple F-theory 7-branes. A by-product of these efforts is a method for analyzing semistability and the splitting type of vector bundles over an elliptic curve given as the sheaf cohomology of a monad.Comment: 40 pages, no figures; minor cosmetic reorganization of section 4; reference [6] update

Mathematical Tools for Calculation of the Effective Action in Quantum Gravity

We review the status of covariant methods in quantum field theory and quantum gravity, in particular, some recent progress in the calculation of the effective action via the heat kernel method. We study the heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold without boundary. We develop a manifestly covariant method for computation of the heat kernel asymptotic expansion as well as new algebraic methods for calculation of the heat kernel for covariantly constant background, in particular, on homogeneous bundles over symmetric spaces, which enables one to compute the low-energy non-perturbative effective action.Comment: 71 pages, 2 figures, submitted for publication in the Springer book (in preparation) "Quantum Gravity", edited by B. Booss-Bavnbek, G. Esposito and M. Lesc

Asymptotics for general connections at infinity

For a standard path of connections going to a generic point at infinity in the moduli space $M_{DR}$ of connections on a compact Riemann surface, we show that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular the convex subset representing the exponential growth rate of the monodromy is a polygon, whose vertices are in a subset of points described explicitly in terms of the spectral curve. Unfortunately we don't get any information about the size of the singularities of the Laplace transform, which is why we can't get asymptotic expansions for the monodromy.Comment: My talk at the Ramis conference, Toulouse, September 200