## Publications

41 results found

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: 3

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: 2

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 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.

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

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- Citations: 3

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.

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, 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, 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, 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

Coates T, Iritani H, 2012, On the Convergence of Gromov-Witten Potentials and Givental's Formula

Coates T, Gholampour A, Iritani H,
et al., 2012, The Quantum Lefschetz Hyperplane Principle Can Fail for Positive Orbifold Hypersurfaces, *Math. Res. Lett*, Vol: 19, Pages: 997-1005

We show that the Quantum Lefschetz Hyperplane Principle can fail for certainorbifold hypersurfaces and complete intersections. It can fail even fororbifold hypersurfaces defined by a section of an ample line bundle.

Akhtar M, Coates T, Galkin S,
et al., 2012, Minkowski Polynomials and Mutations, *SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS*, Vol: 8, ISSN: 1815-0659

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- Citations: 31

Coates T, Corti A, Lee Y-P,
et al., 2009, The quantum orbifold cohomology of weighted projective spaces, *ACTA MATHEMATICA*, Vol: 202, Pages: 139-193, ISSN: 0001-5962

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- Citations: 42

Coates T, 2009, On the Crepant Resolution Conjecture in the Local Case, *COMMUNICATIONS IN MATHEMATICAL PHYSICS*, Vol: 287, Pages: 1071-1108, ISSN: 0010-3616

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- Citations: 17

Coates T, Corti A, Iritani H,
et al., 2009, COMPUTING GENUS-ZERO TWISTED GROMOV-WITTEN INVARIANTS, *DUKE MATHEMATICAL JOURNAL*, Vol: 147, Pages: 377-438, ISSN: 0012-7094

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- Citations: 71

Coates T, Iritani H, Tseng H-H, 2009, Wall-crossings in toric Gromov-Witten theory I: crepant examples, *GEOMETRY & TOPOLOGY*, Vol: 13, Pages: 2675-2744, ISSN: 1465-3060

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- Citations: 42

Coates T, 2008, Givental's Lagrangian cone and S-1-equivariant Gromov-Witten theory, *MATHEMATICAL RESEARCH LETTERS*, Vol: 15, Pages: 15-31, ISSN: 1073-2780

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- Citations: 5

Coates T, Ruan Y, 2007, Quantum Cohomology and Crepant Resolutions: A Conjecture

Coates T, Corti A, Iritani H, et al., 2007, The Crepant Resolution Conjecture for Type A Surface Singularities

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