Timetable Spring School


  Monday Tuesday Wednesday Thursday Friday
9:00-10:00 Caldararu I Caldararu II Caldararu III Caldararu IV B5
10:15-11:15 Perutz I Perutz II A5 Perutz III C3
11:30-12:00 B1 (+15) A3 B3 (+30) C1 C4 (+30)
14:30-15:30 A1 B2 / C2 /
15:30-16:30 A2 A4 / B4 /



A1a) Meazzini, Francesco: section 1 in https://arxiv.org/abs/q-alg/9709040

A1b) Poliakova, Dasha: section 3 in https://arxiv.org/abs/q-alg/9709040

A2a) Marchetti, Giovanni: L-infinity definition and equivalent coalgebra point of view https://arxiv.org/abs/math/0405485

A2b) Kaushal, Tanya: section 4 in https://arxiv.org/abs/q-alg/9709040

A3) Simi, Luca: L-infinity and formality criteria https://arxiv.org/abs/hep-th/9406095

A4) Filip, Matej: chapter 1-5 of https://arxiv.org/abs/alg-geom/9710032

A5) Arkhipov, Sergey: chapter 6-8 of https://arxiv.org/abs/alg-geom/9710032


B1) Barbieri, Anna: Classical Geometry of the Calabi-Yau moduli space (45min), chaper 2 in http://math.bu.edu/people/sili/thesis_SiLi.pdf

B2a) Buring, Ricardo: Kontsevich quantization and Feynman diagrams, section 3.1 plus background from Kontsevich's work

B2b) Tirelli, Andrea: Batalin-Vilkovisky formalism, section 3.2

B3a) Genovese, Francesco: Effective Field theory, Renormalization group flow and Chern Simons theory, section 3.3+3.4 until and including CS example 3.22

B3b) Belmans, Pieter: Quantum master equation, deformation obstruction complex and theory on the complex plane 3.4.2 to end of 3.5

B4a) Sutherland, Tom: Classical BCOV theory and its quantization, section 4

B4b) Melani, Valerio: Quantization of BCOV on the elliptic curve I, existence and uniqueness, section 5.1 until middle of page 98 (to the end of the proof of 5.12), i.e. section 5.1.1-5.3.1

B5a) Bousseau, Pierrick: Quantization of BCOV on the elliptic curve II, proving the dilaton axiom and holomorphicity, section 5.3.2-5.4

B5b) Ruddat, Helge: Proving higher genus mirror symmetry for the elliptic curve, section 6


C1) Anno, Rina: "Mumford toric degenerations", section 1.1 and 1.2 of https://arxiv.org/pdf/0808.2749.pdf

C2a) Felten, Simon: "From a toric degeneration to (B,P,phi)"

C2b) Logvinenko, Timothy: "Scattering I"

C3a) Nikolaev, Nikita: "Scattering II"

C3b) Zhi, Jin: "Overview of the Gross-Hacking-Keel construction"

C4a) Kelly, Tyler: "Broken lines"

C4b) van Garrel, Michel: "Generalized theta functions"

Andrei Caldararu: "Categorical Gromov-Witten Invariants"

Caldararu I: What are categorical Gromov-Witten invariants?

I will start my lecture by reviewing the construction of classical Gromov-Witten invariants, and working through a standard example, the g=1, n=1 GW invariants of an elliptic curve. Then I will outline how we can hope to obtain these invariants from the Fukaya category by an abstract categorical construction. In the case of an arbitrary abstract category I will discuss what properties the category should have, what replaces the quantum cohomology ring, etc. I will conclude with a discussion of potential applications of the construction of categorical Gromov-Witten invariants and a general discussion of how mirror symmetry is supposed to work.

Caldararu II: Chains on moduli space of curves, ribbon graphs, and A_infinity algebras

I will begin by reviewing the main algebraic structure on the space of chains on the moduli spaces of curves, namely that they form a Batalin-Vilkovisky algebra. We will study the Quantum Master Equation in such chains; by work of Costello its solution gives rise to the string vertices of Zwiebach and Sen. I will follow with an introduction to ribbon graphs and how they give a model for the chains on M_{g,n}. I will conclude with a quick review of A_infinity algebras and a description of the original Kontsevich partition function which pairs ribbon graphs with cyclic A_infinity algebras.

Caldararu III: Algebraic structures on the periodic cyclic complex

I will introduce the Hochschild chain complex of an A_infinity algebra, and its two main differentials, the Hochschild differential b and the Connes differential B, leading to the construction of the different variants of cyclic homology (periodic, negative, positive). Using this construction I will outline Kontsevich-Soibelman's construction of the action of the PROP of chains on M_{g,n} on the cyclic complex of an A_infinity algebra. Following Costello, part of this action allows us to regard the periodic cyclic complex as a symplectic vector space with a Lagrangian inside it. This leads to the construction of a Weyl algebra and Fock module associated with this structure. Costello constructs a deformation of this Fock module induced using a choice of string vertices. This leads to the construction of an abstract Gromov-Witten potential (a line in a Fock space).

Caldararu IV: Hodge-de Rham degeneration and applications

The periodic cyclic homology carries a natural filtration, the Hodge filtration. I will outline its construction, and explain how a choice of a splitting of this filtration allows us to regard the abstract Gromov-Witten potential previously constructed as an actual categorical Gromov-Witten potential. Everything will be put together into the calculation of the categorical B-model potential of the family of elliptic curves (joint work with Junwu Tu). I will conclude with a list of potential directions in which the whole construction can be extended, and expected applications and conjectures.

Tim Perutz: "From homological to Hodge-theoretic mirror symmetry"

(Based on work of S. Ganatra, N. Sheridan and myself, and various subsets of this trio.)

Perutz I: Fukaya categories, open-closed maps, and pairings.

I will review some of the mathematics of Fukaya categories of symplectic manifolds. These categories come with important structure: the closed-open map CO, which is a map of algebras from quantum cohomology to Hochschild cohomology of the category, and the open-closed map OC, from Hochschild homology to quantum cohomology, which is an isometry with respect to a “Mukai pairing” on Hochschild homology and a cup-product pairing on quantum cohomology. (The B-model analog of this structure was studied by Andrei Caldararu several years ago, and an abstract of OC is key to Costello’s construction, as discussed in Andrei’s lectures.) Abouzaid has shown that a subcategory generates the Fukaya category if OC is surjective on that subcategory.

Perutz II: Smoothness, automatic generation, and homological mirror symmetry.

Symplectic topologists have recently been learning a principle which Kontsevich has advocated for some time: the importance of categorical smoothness. The prototypical example of a smooth DG category is the derived category of a smooth algebraic variety. In the setting of homological mirror symmetry (HMS), the Fukaya category of the mirror symplectic manifold will then also be smooth. From smoothness many things flow: sharp versions of generation for the Fukaya category; that OC and CO are isomorphisms; that OC coincides with Costello’s abstract version, hence that his formal constructions are in fact geometric; and that HMS implies closed-string mirror symmetry.

Perutz III: The cyclic open-closed map, the categorical Gauss—Manin connection, and the mirror map.

The cyclic homology of an A-infinity category, defined over (say) a formal punctured disc, carries a Gauss—Manin connection, as constructed by Getzler. In the case of the derived category of a smooth algebraic variety over the punctured disc, this is expected (but not actually proven) to coincide with the classical GM connection in algebraic de Rham cohomology. I will explain that, in the case of a smooth Fukaya category, it coincides with the quantum differential operator, a connection in quantum cohomology. Under the assumption of HMS, the consequences are as follows: (i) that with an undetermined mirror map, the mirror map can be characterized Hodge-theoretically; and (ii) that the quantum differential equation matches the (derived) algebro-geometric GM connection. Statement (ii) is (genus 0) Hodge-theoretic mirror symmetry. It encompasses the famous rational curve-counts on the quintic 3-fold.