25 research outputs found
On the rank and the convergence rate toward the Sato-Tate measure
Anstract. Let be an abelian variety defined over a number field and let denote its SatoTate group. Under the assumption of certain standard conjectures on -functions attached to the irreducible representations of , we study the convergence rate of any virtual selfdual character of . We find that this convergence rate is dictated by several arithmetic invariants of , such as its rank or its Sato-Tate group . The results are consonant with some previous experimental observations, and we also provide additional numerical evidence consistent with them. The techniques that we use were introduced by Sarnak, in order to explain the bias in the sign of the Frobenius traces of an elliptic curve without complex multiplication defined over . We show that the same methods can be adapted to study the convergence rate of the characters of its Sato-Tate group, and that they can also be employed in the more general case of abelian varieties over number fields. A key tool in our analysis is the existence of limiting distributions for automorphic -functions, which is due to Akbary, Ng, and Shahabi
Quantum Field Theories, Isomonodromic Deformations and Matrix Models
Recent years have seen a proliferation of exact results in quantum field theories, owing mostly to supersymmetric localisation. Coupled with decades of study of dualities, this ensured the development of many novel nontrivial correspondences linking seemingly disparate parts of the mathematical landscape. Among these, the link between supersymmetric gauge theories with 8 supercharges and Painlev{\'e} equations, interpreted as the exact RG flow of their codimension 2 defects and passing through a correspondence with two-dimensional conformal field theory, was highly surprising. Similarly surprising was the realisation that three-dimensional matrix models coming from M-theory compute these solutions, and provide a non-perturbative completion of the topological string. Extending these two results is the focus of my work.
After giving a review of the basics, hopefully useful to researchers in the field also for uses besides understanding the thesis, two parts based on published and unpublished results follow. The first is focused on giving Painlev{\'e}-type equations for general groups and linear quivers, and the second on matrix models
Exact WKB and the quantum Seiberg-Witten curve for 4d pure Yang-Mills, Part I: Abelianization
We investigate the exact WKB method for the quantum Seiberg-Witten curve of
4d pure Yang-Mills, in the language of abelianization. The
relevant differential equation is a third-order equation on
with two irregular singularities. We employ the exact WKB method to study
solutions to such a third-order equation and the associated Stokes phenomena.
We also investigate the exact quantization condition for a certain spectral
problem. Moreover, exact WKB analysis leads us to consider new Darboux
coordinates on a moduli space of flat SL(3,)-connections. In
particular, in the weak coupling region we encounter coordinates of higher
length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux
coordinates are conjectured to admit asymptotic expansions given by the formal
quantum periods series; we perform numerical analysis supporting this
conjecture.Comment: 28 pages, 5 figures, 6 table
Supersymmetric Field Theories and Isomonodromic Deformations
The topic of this thesis is the recently discovered correspondence between supersymmetric gauge theories, two-dimensional conformal field theories and isomonodromic deformation problems. Its original results are organized in two parts: the first one, based on the papers [1], [2], as well as on some further unpublished results, provides the extension of the correspondence between four-dimensional class S theories and isomonodromic deformation problems to Riemann Surfaces of genus greater than zero. The second part, based on the results of [3], is instead devoted to the study of five-dimensional superconformal field theories, and their relation with q-deformed isomonodromic problems
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
Elliptic blowup equations for 6d SCFTs. Part II: Exceptional cases
The building blocks of 6d SCFTs include certain rank one theories with gauge group . In this paper, we propose a universal recursion formula for the elliptic genera of all such theories. This formula is solved from the elliptic blowup equations introduced in our previous paper. We explicitly compute the elliptic genera and refined BPS invariants, which recover all previous results from topological string theory, modular bootstrap, Hilbert series, 2d quiver gauge theories and 4d superconformal theories. We also observe an intriguing relation between the -string elliptic genus and the Schur indices of rank SCFTs, as a generalization of Lockhart-Zotto's conjecture at the rank one cases. In a subsequent paper, we deal with all other non-Higgsable clusters with matters
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal