## Publications

42 results found

Coates T, Kasprzyk A, Veneziale S, 2023, Machine learning the dimension of a Fano variety, *Nature Communications*, Vol: 14

Fano varieties are basic building blocks in geometry – they are ‘atomic pieces’ of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a numerical fingerprint for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of a class of Fano varieties. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.

Coates T, Doran C, Kalashnikov E, 2022, Unwinding Toric Degenerations and Mirror Symmetry for Grassmannians, *FORUM OF MATHEMATICS SIGMA*, Vol: 10

Coates T, Lutz W, Shafi Q, 2022, The Abelian/Nonabelian Correspondence and Gromov-Witten Invariants of Blow-Ups, *FORUM OF MATHEMATICS SIGMA*, Vol: 10

Coates T, Kasprzyk AM, 2022, Databases of quantum periods for Fano manifolds, *SCIENTIFIC DATA*, Vol: 9

Coates T, Corti A, da Silva G, 2022, On the Topology of Fano Smoothings, *Springer Proceedings in Mathematics and Statistics*, Vol: 386, Pages: 135-156, ISSN: 2194-1009

Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a Laurent polynomial f, and that P is the Newton polytope of f. In this setting it is expected that there is a family of algebraic varieties over the unit disc with general fiber X and special fiber the toric variety defined by the spanning fan of P. Building on recent work and conjectures by Corti–Hacking–Petracci, who construct such families of varieties, we determine the topology of the general fiber from combinatorial data on P. This provides evidence for the Corti–Hacking–Petracci conjectures, and verifies that their construction is compatible with expectations from Mirror Symmetry.

Coates T, Kasprzyk AM, Pitton G,
et al., 2021, Maximally mutable laurent polynomials, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, Pages: 1-21, ISSN: 1364-5021

We introduce a class of Laurent polynomials, called maximally mutable Laurentpolynomials (MMLPs), that we believe correspond under mirror symmetry to Fanovarieties. A subclass of these, called rigid, are expected to correspond toFano varieties with terminal locally toric singularities. We prove that thereare exactly 10 mutation classes of rigid MMLPs in two variables; under mirrorsymmetry these correspond one-to-one with the 10 deformation classes of smoothdel~Pezzo surfaces. Furthermore we give a computer-assisted classification ofrigid MMLPs in three variables with reflexive Newton polytope; under mirrorsymmetry these correspond one-to-one with the 98 deformation classes ofthree-dimensional Fano manifolds with very ample anticanonical bundle. Wecompare our proposal to previous approaches to constructing mirrors to Fanovarieties, and explain why mirror symmetry in higher dimensions necessarilyinvolves varieties with terminal singularities. Every known mirror to a Fanomanifold, of any dimension, is a rigid MMLP.

Coates T, Galkin S, Kasprzyk A, et al., 2020, Quantum Periods for Certain Four-Dimensional Fano Manifolds, Publisher: TAYLOR & FRANCIS INC

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- Citations: 6

Coates T, Kasprzyk A, Galkin S,
et al., 2020, Quantum periods for certain four-dimensional fano manifolds, *Experimental Mathematics*, Vol: 29, Pages: 183-221, ISSN: 1944-950X

We collect a list of known four-dimensional Fano manifolds and compute their quantumperiods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds,and certain complete intersections in projective bundles.

Coates T, Corti A, Iritani H,
et al., 2020, Hodge-theoretic mirror symmetry for toric stacks, *Journal of Differential Geometry*, Vol: 114, Pages: 41-115, ISSN: 1945-743X

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirrordescription for the big equivariant quantum cohomology of toric Deligne-Mumfordstacks. More precisely, we prove that the big equivariant quantum D-module of atoric Deligne-Mumford stack is isomorphic to the Saito structure associated tothe mirror Landau-Ginzburg potential. We give a GKZ-style presentation of thequantum D-module, and a combinatorial description of quantum cohomology as aquantum Stanley-Reisner ring. We establish the convergence of the mirrorisomorphism and of quantum cohomology in the big and equivariant setting.

Coates T, Kasprzyk A, Prince T, 2019, Laurent inversion, *Pure and Applied Mathematics Quarterly*, Vol: 15, Pages: 1135-1179, ISSN: 1558-8599

We describe a practical and effective method for reconstructing thedeformation class of a Fano manifold X from a Laurent polynomial f thatcorresponds to X under Mirror Symmetry. We explore connections to nefpartitions, the smoothing of singular toric varieties, and the construction ofembeddings of one (possibly-singular) toric variety in another. In particular,we construct degenerations from Fano manifolds to singular toric varieties; inthe toric complete intersection case, these degenerations were constructedpreviously by Doran--Harder. We use our method to find models of orbifold delPezzo surfaces as complete intersections and degeneracy loci, and to constructa new four-dimensional Fano manifold.

Coates T, Iritani H, 2019, Gromov-witten invariants of local P^2 and modular forms, Publisher: arXiv

We construct a sheaf of Fock spaces over the moduli space of elliptic curvesE_y with Gamma_1(3)-level structure, arising from geometric quantization ofH^1(E_y), and a global section of this Fock sheaf. The global sectioncoincides, near appropriate limit points, with the Gromov-Witten potentials oflocal P^2 and of the orbifold C^3/mu_3. This proves that the Gromov-Wittenpotentials of local P^2 are quasi-modular functions for the group Gamma_1(3),as predicted by Aganagic-Bouchard-Klemm, and proves the Crepant ResolutionConjecture for [C^3/mu_3] in all genera.

Kalashnikov E, 2019, Four-dimensional Fano quiver flag zero loci, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 475, Pages: 1-23, ISSN: 1364-5021

Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov–Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.

Coates T, Corti A, Iritani H, et al., 2019, Some applications of the mirror theorem for toric stacks, Publisher: INT PRESS BOSTON, INC

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- Citations: 5

Coates T, Iritani H, 2018, A Fock sheaf for Givental quantization, *Kyoto Journal of Mathematics*, Vol: 58, Pages: 695-864, ISSN: 2156-2261

We give a global, intrinsic, and co-ordinate-free quantization formalism for Gromov{Witten invariants and their B-model counterparts, which simultaneously generalizes the quantizationformalisms described by Witten, Givental, and Aganagic{Bouchard{Klemm. Descendant potentialslive in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the(3g2)-jet condition of Eguchi{Xiong; they also satisfy a certain anomaly equation, which gen-eralizes the Holomorphic Anomaly Equation of Bershadsky{Cecotti{Ooguri{Vafa. We interpretGivental's formula for the higher-genus potentials associated to a semisimple Frobenius manifold inthis setting, showing that, in the semisimple case, there is a canonical global section of the Focksheaf. This canonical section automatically has certain modularity properties. WhenXis a varietywith semisimple quantum cohomology, a theorem of Teleman implies that the canonical sectioncoincides with the geometric descendant potential de ned by Gromov{Witten invariants ofX. Weuse our formalism to prove a higher-genus version of Ruan's Crepant Transformation Conjecture forcompact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that thetotal descendant potential for compact toric orbifoldXis a modular function for a certain group ofautoequivalences of the derived category ofX.

Coates T, Manolache C, 2018, A splitting of the virtual class for genus one stable maps

Moduli spaces of stable maps to a smooth projective variety typically haveseveral components. We express the virtual class of the moduli space of genusone stable maps to a smooth projective variety as a sum of virtual classes ofthe components. The key ingredient is a generalised functoriality result forvirtual classes. We show that the natural maps from 'ghost' components of thegenus one moduli space to moduli spaces of genus zero stable maps satisfy thestrong push forward property. As a consequence, we give a cycle-level formulawhich relates standard and reduced genus one Gromov--Witten invariants of asmooth projective Calabi--Yau theefold.

Coates TH, Iritani H, Jiang Y, 2018, The Crepant Transformation Conjecture for toric complete intersections, *Advances in Mathematics*, Vol: 329, Pages: 1002-1087, ISSN: 0001-8708

Let X and Y be K-equivalent toric Deligne–Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quantum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Givental spaces for X and Y, with a Fourier–Mukai transformation between the K-groups of X and Y, via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We impose only very weak geometric hypotheses on X and Y: they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne–Mumford stacks and toric complete intersections, and the Mellin–Barnes method for analytic continuation of hypergeometric functions.

Coates T, Corti A, Galkin S,
et al., 2016, Quantum Periods for 3-Dimensional Fano Manifolds, *Geometry & Topology*, Vol: 20, Pages: 103-256, ISSN: 1465-3060

The quantum period of a variety X is a generating function for certainGromov-Witten invariants of X which plays an important role in mirror symmetry.In this paper we compute the quantum periods of all 3-dimensional Fanomanifolds. In particular we show that 3-dimensional Fano manifolds with veryample anticanonical bundle have mirrors given by a collection of Laurentpolynomials called Minkowski polynomials. This was conjectured in joint workwith Golyshev. It suggests a new approach to the classification of Fanomanifolds: by proving an appropriate mirror theorem and then classifying Fanomirrors. Our methods are likely to be of independent interest. We rework theMori-Mukai classification of 3-dimensional Fano manifolds, showing that each ofthem can be expressed as the zero locus of a section of a homogeneous vectorbundle over a GIT quotient V/G, where G is a product of groups of the formGL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses theFano 3-fold as a toric complete intersection; in the remaining cases, itexpresses the Fano 3-fold as a tautological subvariety of a Grassmannian,partial flag manifold, or projective bundle thereon. We then compute thequantum periods using the Quantum Lefschetz Hyperplane Theorem ofCoates-Givental and the Abelian/non-Abelian correspondence ofBertram-Ciocan-Fontanine-Kim-Sabbah.

Coates T, Iritani H, Jiang Y,
et al., 2015, K-theoretic and categorical properties of toric Deligne-Mumford stacks, *Pure and Applied and Mathematics Quarterly*, Vol: 11, Pages: 239-266, ISSN: 1558-8599

We prove the following results for toric Deligne–Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant KK-theory; the equivariant Hirzebruch–Riemann–Roch theorem; the Fourier–Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.

Coates T, Corti A, Iritani H,
et al., 2015, A mirror theorem for toric stacks, *Compositio Mathematica*, Vol: 151, Pages: 1878-1912, ISSN: 0010-437X

© The Authors 2015. We prove a Givental-style mirror theorem for toric Deligne-Mumford stacks χ. This determines the genus-zero Gromov-Witten invariants of χ in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen-Ruan orbifold cohomology of χ.

Anderson G, Buck D, Coates T,
et al., 2015, Drawing in Mathematics <i>From Inverse Vision to the Liberation of Form</i>, *LEONARDO*, Vol: 48, Pages: 439-448, ISSN: 0024-094X

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- Citations: 4

Akhtar M, Coates T, Corti A,
et al., 2015, Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces, *Proceedings of the American Mathematical Society*, Vol: 144, Pages: 513-527, ISSN: 1088-6826

We state a number of conjectures that together allow one to classify a broadclass of del Pezzo surfaces with cyclic quotient singularities using mirrorsymmetry. We prove our conjectures in the simplest cases. The conjecturesrelate mutation-equivalence classes of Fano polygons with Q-Gorensteindeformation classes of del Pezzo surfaces.

Coates T, Givental A, Tseng H-H, 2015, Virasoro Constraints for Toric Bundles

We show that the Virasoro conjecture in Gromov--Witten theory holds for thethe total space of a toric bundle $E \to B$ if and only if it holds for thebase $B$. The main steps are: (i) we establish a localization formula thatexpresses Gromov--Witten invariants of $E$, equivariant with respect to thefiberwise torus action, in terms of genus-zero invariants of the toric fiberand all-genus invariants of $B$; and (ii) we pass to the non-equivariant limitin this formula, using Brown's mirror theorem for toric bundles.

Coates T, Iritani H, 2015, On the Existence of a Global Neighbourhood, *Glasgow Mathematical Journal*, Vol: 58, Pages: 717-726, ISSN: 1469-509X

Suppose that a complex manifold M is locally embedded into ahigher-dimensional neighbourhood as a submanifold. We show that, if the localneighbourhood germs are compatible in a suitable sense, then they glue togetherto give a global neighbourhood of M. As an application, we prove a globalversion of Hertling--Manin's unfolding theorem for germs of TEP structures;this has applications in the study of quantum cohomology.

Coates T, Kasprzyk A, Prince T, 2015, Laurent Inversion

There are well-understood methods, going back to Givental and Hori--Vafa,that to a Fano toric complete intersection X associate a Laurent polynomial fthat corresponds to X under mirror symmetry. We describe a technique forinverting this process, constructing the toric complete intersection X directlyfrom its Laurent polynomial mirror f. We use this technique to construct a newfour-dimensional Fano manifold.

Coates T, Kasprzyk A, Prince T, 2015, Four-dimensional Fano toric complete intersections, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 471, Pages: 20140704-20140704, ISSN: 1364-5021

We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

Coates T, Iritani H, 2014, A Fock Sheaf For Givental Quantization

We give a global, intrinsic, and co-ordinate-free quantization formalism forGromov-Witten invariants and their B-model counterparts, which simultaneouslygeneralizes the quantization formalisms described by Witten, Givental, andAganagic-Bouchard-Klemm. Descendant potentials live in a Fock sheaf, consistingof local functions on Givental's Lagrangian cone that satisfy the (3g-2)-jetcondition of Eguchi-Xiong; they also satisfy a certain anomaly equation, whichgeneralizes the Holomorphic Anomaly Equation of Bershadsky-Cecotti-Ooguri-Vafa.We interpret Givental's formula for the higher-genus potentials associated to asemisimple Frobenius manifold in this setting, showing that, in the semisimplecase, there is a canonical global section of the Fock sheaf. This canonicalsection automatically has certain modularity properties. When X is a varietywith semisimple quantum cohomology, a theorem of Teleman implies that thecanonical section coincides with the geometric descendant potential defined byGromov-Witten invariants of X. We use our formalism to prove a higher-genusversion of Ruan's Crepant Transformation Conjecture for compact toricorbifolds. When combined with our earlier joint work with Jiang, this showsthat the total descendant potential for compact toric orbifold X is a modularfunction for a certain group of autoequivalences of the derived category of X.

Coates T, Iritani H, Jiang Y, 2014, The Crepant Transformation Conjecture for Toric Complete Intersections

Let X and Y be K-equivalent toric Deligne-Mumford stacks related by a singletoric wall-crossing. We prove the Crepant Transformation Conjecture in thiscase, fully-equivariantly and in genus zero. That is, we show that theequivariant quantum connections for X and Y become gauge-equivalent afteranalytic continuation in quantum parameters. Furthermore we identify the gaugetransformation involved, which can be thought of as a linear symplectomorphismbetween the Givental spaces for X and Y, with a Fourier-Mukai transformationbetween the K-groups of X and Y, via an equivariant version of theGamma-integral structure on quantum cohomology. We prove similar results fortoric complete intersections. We impose only very weak geometric hypotheses onX and Y: they can be non-compact, for example, and need not be weak Fano orhave Gorenstein coarse moduli space. Our main tools are the Mirror Theorems fortoric Deligne-Mumford stacks and toric complete intersections, and theMellin-Barnes method for analytic continuation of hypergeometric functions.

Coates T, 2014, The Quantum Lefschetz Principle for Vector Bundles as a Map Between Givental Cones

Givental has defined a Lagrangian cone in a symplectic vector space whichencodes all genus-zero Gromov-Witten invariants of a smooth projective varietyX. Let Y be the subvariety in X given by the zero locus of a regular section ofa convex vector bundle. We review arguments of Iritani, Kim-Kresch-Pantev, andGraber, which give a very simple relationship between the Givental cone for Yand the Givental cone for Euler-twisted Gromov-Witten invariants of X. When theconvex vector bundle is the direct sum of nef line bundles, this gives asharper version of the Quantum Lefschetz Hyperplane Principle.

Coates T, Gonshaw S, Kasprzyk A,
et al., 2013, Mutations of Fake Weighted Projective Spaces, *Electronic Journal of Combinatorics*, Vol: 21, Pages: 1-27, ISSN: 1097-1440

We characterise mutations between fake weighted projective spaces, and giveexplicit formulas for how the weights and multiplicity change under mutation.In particular, we prove that multiplicity-preserving mutations between fakeweighted projective spaces are mutations over edges of the correspondingsimplices. As an application, we analyse the canonical and terminal fakeweighted projective spaces of maximal degree.

Coates T, Corti A, Galkin S, et al., 2012, Mirror Symmetry and Fano Manifolds, Vol: n/a

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