2,444 research outputs found
Categorical Torelli theorems for Fano threefolds
The derived category Db(X) of a variety contains a lot of information about X. If
X and XⲠare Fano, then an equivalence Db(X) â Db(Xâ˛) implies that X and Xâ˛
are isomorphic. For prime Fano threefolds X (of Picard rank 1, index 1, and genus
g ⼠6) the derived category decomposes semiorthogonally as â¨Ku(X), E, OX âŠ,
where E is a certain vector bundle on X. Therefore one can ask whether less
data (in particular the Kuznetsov component Ku(X)) than the whole of Db(X)
determines X isomorphically (or at least birationally).
In this thesis, we focus on this question in the case of ordinary GushelâMukai
threefolds (genus 6 prime Fano threefolds). We show that Ku(X) determines the
birational class of X which proves a conjecture of KuznetsovâPerry in dimension
3. We also prove a refined categorical Torelli theorem for oridnary GushelâMukai
threefolds. In other words, we show that Ku(X) along with the data of the vector
bundle E is enough to determine X up to isomorphism
Non-perturbative topological string theory on compact Calabi-Yau 3-folds
We obtain analytic and numerical results for the non-perturbative amplitudes
of topological string theory on arbitrary, compact Calabi-Yau manifolds. Our
approach is based on the theory of resurgence and extends previous special
results to the more general case. In particular, we obtain explicit
trans-series solutions of the holomorphic anomaly equations. Our results
predict the all orders, large genus asymptotics of the topological string free
energies, which we test in detail against high genus perturbative series
obtained recently in the compact case. We also provide additional evidence that
the Stokes constants appearing in the resurgent structure are closely related
to integer BPS invariants.Comment: 85 pages, 16 figures, 15 table
Geproci sets and the combinatorics of skew lines in
Geproci sets of points in are sets whose general projections to
are complete intersections. The first nontrivial geproci sets
came from representation theory, as projectivizations of the root systems
and . In most currently known cases geproci sets lie on very special
unions of skew lines and are known as half grids. For this important class of
geproci sets we establish fundamental connections with combinatorics, which we
study using methods of algebraic geometry and commutative algebra. As a
motivation for studying them, we first prove Theorem A: for a nondegenerate
-geproci set with being the least degree of a space curve
containing , that if , then is a union of skew lines and is
either a grid or a half grid. We next formulate a combinatorial version of the
geproci property for half grids and prove Theorem B: combinatorial half grids
are geproci in the case of sets of points on each of skew lines when
. We then introduce a notion of combinatorics for skew lines
and apply it to the classification of single orbit combinatorial half grids of
points on each of 4 lines. We apply these results to prove Theorem C,
showing, when , that half grids of points on lines with two
transversals must be very special geometrically (if they even exist). Moreover,
in the case of skew lines having two transversals, our results provide an
algorithm for enumerating their projective equivalence classes. We conjecture
there are equivalence classes of combinatorial -half grids
in the two transversal case when is prime.Comment: 36 page
The Basso-Dixon Formula and Calabi-Yau Geometry
We analyse the family of Calabi-Yau varieties attached to four-point fishnet
integrals in two dimensions. We find that the Picard-Fuchs operators for
fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder
integrals. This implies that the periods of the Calabi-Yau varieties for
fishnet integrals can be written as determinants of periods for ladder
integrals. The representation theory of the geometric monodromy group plays an
important role in this context. We then show how the determinant form of the
periods immediately leads to the well-known Basso-Dixon formula for four-point
fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau
geometry implies that the volume is also expressible via a determinant formula
of Basso-Dixon type. Finally, we show how the fishnet integrals can be written
in terms of iterated integrals naturally attached to the Calabi-Yau varieties.Comment: 42 page
Hecke action on tamely ramified Eisenstein series over
We study the space of automorphic functions for the rational function field
tamely ramified at three places. Eisenstein series are
functions induced from the maximal torus. The space of Eisenstein series
generates a trimodule for the affine Hecke algebra. We conjecture a generators
and relations description of this module and prove the conjecture when
and
Tropical invariants for binary quintics and reduction types of Picard curves
In this paper, we express the reduction types of Picard curves in terms of
tropical invariants associated to binary quintics. These invariants are
connected to Picard modular forms using recent work by Cl{\'e}ry and van der
Geer. We furthermore give a general framework for tropical invariants
associated to group actions on arbitrary varieties. The previous problem fits
in this general framework by mapping the space of binary forms to symmetrized
versions of the Deligne--Mumford compactification . We
conjecture that the techniques introduced here can be used to find tropical
invariants for binary forms of any degree
Berry Connections for 2d Theories, Monopole Spectral Data & (Generalised) Cohomology Theories
We study Berry connections for supersymmetric ground states of 2d
GLSMs quantised on a circle, which are generalised periodic
monopoles, with the aim to provide a fruitful physical arena for mathematical
constructions related to the latter. These are difference modules encoding
monopole solutions due to Mochizuki, as well as an alternative algebraic
description of solutions in terms of vector bundles endowed with filtrations.
The simultaneous existence of these descriptions is an example of a
Riemann-Hilbert correspondence. We demonstrate how these constructions arise
naturally by studying the ground states as the cohomology of a one-parameter
family of supercharges. Through this, we show that the two sides of this
correspondence are related to two types of monopole spectral data that have a
direct interpretation in terms of the physics of the GLSM: the Cherkis-Kapustin
spectral variety (difference modules) as well as twistorial spectral data
(vector bundles with filtrations). By considering states generated by D-branes
and leveraging the difference modules, we derive novel difference equations for
brane amplitudes. We then show that in the conformal limit, these degenerate
into novel difference equations for hemisphere or vortex partition functions,
which are exactly calculable. Beautifully, when the GLSM flows to a nonlinear
sigma model with K\"ahler target , we show that the difference modules are
related to deformations of the equivariant quantum cohomology of , whereas
the vector bundles with filtrations are related to the equivariant K-theory.Comment: 52 pages + appendix, comments welcom
Reading between the rational sections: Global structures of 4d KK theories
We study how the global structure of rank-one 4d
supersymmetric field theories is encoded into global aspects of the
Seiberg-Witten elliptic fibration. Starting with the prototypical example of
the gauge theory, we distinguish between relative and
absolute Seiberg-Witten curves. For instance, we discuss in detail the three
distinct absolute curves for the and 4d
gauge theories. We propose that the -form symmetry of an absolute theory is
isomorphic to a torsion subgroup of the Mordell-Weil group of sections of the
absolute curve, while the full defect group of the theory is encoded in the
torsion sections of a so-called relative curve. We explicitly show that the
relative and absolute curves are related by isogenies (that is, homomorphisms
of elliptic curves) generated by torsion sections -- hence, gauging a one-form
symmetry corresponds to composing isogenies between Seiberg-Witten curves. We
apply this approach to Kaluza-Klein (KK) 4d theories that arise
from toroidal compactifications of 5d and 6d SCFTs to four dimensions,
uncovering an intricate pattern of 4d global structures obtained by gauging
discrete -form and/or -form symmetries. Incidentally, we propose a 6d BPS
quiver for the 6d M-string theory on .Comment: 60 pages plus appendi
Flop-flop autoequivalences and compositions of spherical twists
The main of focus of this thesis is the study of cohomological symmetries. Namely, given an algebraic variety, we study the symmetries of its derived category, which are also known as autoequivalences. The thesis is split into five chapters. In §1 we give an introduction to the material presented in the thesis, as well as a motivation as to why one might be interested in studying these topics. We encourage the reader to have a look, so as to know what is coming. In §2, we set up the preliminary notions we will need throughout the whole thesis. The arguments touched in this chapter comprise triangulated categories, dg-categories, and spherical functors. In §3, we begin to present the novel mathematics developed in this thesis. The focus of this chapter is on how to compose spherical twists around spherical functors. We describe a general recipe that takes as input two spherical functors and outputs a new spherical functor whose twist is the composition of the twists around the functors we started with, and whose cotwist is a gluing of the cotwists. We conclude the chapter by specialising the theory to the case of spherical objects and P-objects. In §4, we study autoequivalences arising from geometric correspondences. We prove that such autoequivalences have a natural representation as the inverse of the spherical twist around a spherical functor, and that in some examples this geometric spherical functor agrees with the construction described in §3. We conclude the thesis with §5, in which we present some possible future applications of this work. In doing so, we hope to stimulate further mathematical discussion around topics that the author of this thesis finds really exciting
Lectures on Generalized Symmetries
These are a set of lecture notes on generalized global symmetries in quantum
field theory. The focus is on invertible symmetries with a few comments
regarding non-invertible symmetries. The main topics covered are the basics of
higher-form symmetries and their properties including 't Hooft anomalies,
gauging and spontaneous symmetry breaking. We also introduce the useful notion
of symmetry topological field theories (SymTFTs). Furthermore, an introduction
to higher-group symmetries describing mixings of higher-form symmetries is
provided. Some advanced topics covered include the encoding of higher-form
symmetries in holography and geometric engineering constructions in string
theory. Throughout the text, all concepts are consistently illustrated using
gauge theories as examples.Comment: 138 pages, added reference
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