1,537 research outputs found
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
Interplay between opers, quantum curves, WKB analysis, and Higgs bundles
Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees -module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a generalization of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show a surprising result that a PDE version of the topological recursion provides all-order WKB analysis for the Rees -modules, defined as the quantization of Hitchin spectral curves associated with meromorphic -Higgs bundles. Topological recursion is thus identified as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE topological recursion, agree for holomorphic and meromorphic -Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants. <br
Mirror curve of orbifold Hurwitz numbers
Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological recursion, which is a mirror B-model corresponding to these counting problems. We show that for the case of orbifold Hurwitz numbers, the mirror objects, i.e., the spectral curve and the differential forms on it, are constructed solely from the edge-contraction operations of the counting problem in genus and one marked point. This forms a parallelism with Gromov-Witten theory, where genus 0 Gromov-Witten invariants correspond to mirror B-model holomorphic geometry
Parity Mixed Doublets in A = 36 Nuclei
The -circular polarizations () and asymmetries
() of the parity forbidden M1 + E2 -decays: MeV) and MeV)
MeV) are investigated theoretically. We use the recently proposed
Warburton-Becker-Brown shell-model interaction. For the weak forces we discuss
comparatively different weak interaction models based on different assumptions
for evaluating the weak meson-hadron coupling constants. The results determine
a range of values from which we find the most probable values:
= for and = for .Comment: RevTeX, 17 pages; to appear in Phys. Rev.
Polaron to molecule transition in a strongly imbalanced Fermi gas
A single down spin Fermion with an attractive, zero range interaction with a
Fermi sea of up-spin Fermions forms a polaronic quasiparticle. The associated
quasiparticle weight vanishes beyond a critical strength of the attractive
interaction, where a many-body bound state is formed. From a variational
wavefunction in the molecular limit, we determine the critical value for the
polaron to molecule transition. The value agrees well with the diagrammatic
Monte Carlo results of Prokof'ev and Svistunov and is consistent with recent
rf-spectroscopy measurements of the quasiparticle weight by Schirotzek et. al.
In addition, we calculate the contact coefficient of the strongly imbalanced
gas, using the adiabatic theorem of Tan and discuss the implications of the
polaron to molecule transition for the phase diagram of the attractive Fermi
gas at finite imbalance.Comment: 10 pages, 4 figures, RevTex4, minor changes, references adde
Exploration of Finite 2D Square Grid by a Metamorphic Robotic System
We consider exploration of finite 2D square grid by a metamorphic robotic
system consisting of anonymous oblivious modules. The number of possible shapes
of a metamorphic robotic system grows as the number of modules increases. The
shape of the system serves as its memory and shows its functionality. We
consider the effect of global compass on the minimum number of modules
necessary to explore a finite 2D square grid. We show that if the modules agree
on the directions (north, south, east, and west), three modules are necessary
and sufficient for exploration from an arbitrary initial configuration,
otherwise five modules are necessary and sufficient for restricted initial
configurations
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