Holomorphic generating functions for invariants counting coherent sheaves on Calabi–Yau 3–folds

Joyce, Dominic¹

¹ The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, UK

Geometry & topology, Tome 11 (2007) no. 2, pp. 667-725.

Résumé

Let $X$ be a Calabi–Yau 3–fold, $\mathcal{T}=D^b\left(coh\left(X\right)\right)$ the derived category of coherent sheaves on $X$, and $Stab\left(\mathcal{T}\right)$ the complex manifold of Bridgeland stability conditions on $\mathcal{T}$. It is conjectured that one can define invariants $J^{\alpha }\left(Z,\mathcal{P}\right)\in \mathbb{Q}$ for $\left(Z,\mathcal{P}\right)\in Stab\left(\mathcal{T}\right)$ and $\alpha \in K\left(\mathcal{T}\right)$ generalizing Donaldson–Thomas invariants, which “count” $\left(Z,\mathcal{P}\right)$–semistable (complexes of) coherent sheaves on $X$, and whose transformation law under change of $\left(Z,\mathcal{P}\right)$ is known.

This paper explains how to combine such invariants $J^{\alpha }\left(Z,\mathcal{P}\right)$, if they exist, into a family of holomorphic generating functions $F^{\alpha }:Stab\left(\mathcal{T}\right)\to ℂ$ for $\alpha \in K\left(\mathcal{T}\right)$. Surprisingly, requiring the $F^{\alpha }$ to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over $Stab\left(\mathcal{T}\right)$ with values in an infinite-dimensional Lie algebra $\mathcal{ℒ}$.

The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.

Détail

DOI : 10.2140/gt.2007.11.667

Keywords: generating function, stability condition, coherent sheaf, Calabi–Yau 3–fold, Donaldson–Thomas invariant, moduli space, mirror symmetry

Bibliographie

[1] M Bershadsky, S Cecotti, H Ooguri, C Vafa, Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993) 279

[2] M Bershadsky, S Cecotti, H Ooguri, C Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994) 311

[3] T Bridgeland, in preparation

[4] T Bridgeland, Stability conditions on triangulated categories, to appear in Ann. of Math. $(2)$

[5] T Bridgeland, Derived categories of coherent sheaves, from: "Proceedings of the International Congress of Mathematicians, Vol. 2 (Madrid, 2006)" (editors M Sanz-Solé, J Soria, J Varona, J Verdera) (2007) 563

[6] S K Donaldson, R P Thomas, Gauge theory in higher dimensions, from: "The geometric universe (Oxford, 1996)", Oxford Univ. Press (1998) 31

[7] S I Gelfand, Y I Manin, Methods of homological algebra, Springer Monographs in Mathematics, Springer (2003)

[8] A Goncharov, Multiple polylogarithms and mixed Tate motives

[9] A L Gorodentsev, S A Kuleshov, A N Rudakov, $t$-stabilities and $t$-structures on triangulated categories, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004) 117

[10] D Joyce, Constructible functions on Artin stacks, J. London Math. Soc. $(2)$ 74 (2006) 583

[11] D Joyce, Motivic invariants of Artin stacks and ‘stack functions’, to appear in Q. J. Math.

[12] D Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203 (2006) 194

[13] D Joyce, Configurations in abelian categories. II. Ringel–Hall algebras, Adv. Math. 210 (2007) 635

[14] D Joyce, Configurations in abelian categories. III. Stability conditions and identities, to appear in Adv. Math.

[15] D Joyce, Configurations in abelian categories. IV. Invariants and changing stability conditions

[16] M Kontsevich, Homological algebra of mirror symmetry, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)", Birkhäuser (1995) 120

[17] D Mcduff, D Salamon, $J$-holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994)

[18] D Ramakrishnan, On the monodromy of higher logarithms, Proc. Amer. Math. Soc. 85 (1982) 596

[19] A Rudakov, Stability for an abelian category, J. Algebra 197 (1997) 231

[20] R P Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000) 367

[21] B Toën, Higher and derived stacks: a global overview

[22] B Toën, Derived Hall algebras, Duke Math. J. 135 (2006) 587

[23] E Witten, Quantum background independence in String Theory