2,444 research outputs found

    Categorical Torelli theorems for Fano threefolds

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    The derived category Db(X) of a variety contains a lot of information about X. If X and X′ are Fano, then an equivalence Db(X) ≃ Db(X′) implies that X and X′ are isomorphic. For prime Fano threefolds X (of Picard rank 1, index 1, and genus g ≥ 6) the derived category decomposes semiorthogonally as ⟨Ku(X), E, OX ⟩, 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 Gushel–Mukai threefolds (genus 6 prime Fano threefolds). We show that Ku(X) determines the birational class of X which proves a conjecture of Kuznetsov–Perry in dimension 3. We also prove a refined categorical Torelli theorem for oridnary Gushel–Mukai 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

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    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 P3\mathbb P^3

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    Geproci sets of points in P3\mathbb P^3 are sets whose general projections to P2\mathbb P^2 are complete intersections. The first nontrivial geproci sets came from representation theory, as projectivizations of the root systems D4D_4 and F4F_4. 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 (a,b)(a,b)-geproci set ZZ with dd being the least degree of a space curve CC containing ZZ, that if d≤bd\leq b, then CC is a union of skew lines and ZZ 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 aa points on each of bb skew lines when a≥b−1≥3a\geq b-1\geq 3. We then introduce a notion of combinatorics for skew lines and apply it to the classification of single orbit combinatorial half grids of mm points on each of 4 lines. We apply these results to prove Theorem C, showing, when n≫mn\gg m, that half grids of mm points on nn 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 (m2−1)/12(m^2-1)/12 equivalence classes of combinatorial [m,4][m,4]-half grids in the two transversal case when m>2m>2 is prime.Comment: 36 page

    The Basso-Dixon Formula and Calabi-Yau Geometry

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    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 p1\mathbb{p}^1

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    We study the space of automorphic functions for the rational function field Fq(t)\mathbb{F}_q(t) 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 G=PGL(2)G=\mathrm{PGL}(2) and SL(3)\mathrm{SL}(3)

    Tropical invariants for binary quintics and reduction types of Picard curves

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    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 Mˉ0,n\bar{M}_{0,n}. We conjecture that the techniques introduced here can be used to find tropical invariants for binary forms of any degree

    Berry Connections for 2d (2,2)(2,2) Theories, Monopole Spectral Data & (Generalised) Cohomology Theories

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    We study Berry connections for supersymmetric ground states of 2d N=(2,2)\mathcal{N}=(2,2) 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 XX, we show that the difference modules are related to deformations of the equivariant quantum cohomology of XX, 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 N=2\mathcal{N}=2 KK theories

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    We study how the global structure of rank-one 4d N=2\mathcal{N}=2 supersymmetric field theories is encoded into global aspects of the Seiberg-Witten elliptic fibration. Starting with the prototypical example of the su(2)\mathfrak{su}(2) gauge theory, we distinguish between relative and absolute Seiberg-Witten curves. For instance, we discuss in detail the three distinct absolute curves for the SU(2)SU(2) and SO(3)±SO(3)_\pm 4d N=2\mathcal{N}=2 gauge theories. We propose that the 11-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 N=2\mathcal{N}=2 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 00-form and/or 11-form symmetries. Incidentally, we propose a 6d BPS quiver for the 6d M-string theory on R4×T2\mathbb{R}^4\times T^2.Comment: 60 pages plus appendi

    Flop-flop autoequivalences and compositions of spherical twists

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    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

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    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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