Advanced Topics in Quantum Mathematics

PhD course (5 ECTS), Spring 2026, Syddansk Universitet

Organizer: Guillaume Laplante-Anfossi.

The goal of these lectures is to study the algebraic construction of categorical enumerative invariants from a cyclic A-infinity category and a splitting of the Hodge filtration due to Caldararu—Costello—Tu, and its relation to topological conformal and Deligne—Mumford field theories.


You can find here the course plan and a poster.

Planning

Week 5

Week 6

With Barkan and Steinebrunner, we show that any open field theory extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of the E_1-Frobenius algebras (i.e. E_1-Calabi-Yau objects) A associated to F. In particular, we show that the open-closed bordism category is obtained by formally adjoining iterated Hochchild homology to the open bordism category. As a corollary, we determine that the moduli spaces of surfaces is the space of universal formal operations on the Hochschild homology object of E_1-Frobenius algebras. This provides a space level refinement of previous work of Costello (over Q) and Wahl (over Z).

Time permitting, I will also explain work with Andrea Bianchi on canonical local to global extensions of graph TFTs associated with E_\infty-Frobenius objects, i.e. in dimension infinity. In this case, we show that the graph cobordism category is obtained by formally adjoining to the \infty-category of graphs the factorization homology of the universal E_\infty-Frobenius algebra over any sapce X.

Week 7

The goal is to explain, following Costello's Topological conformal field theories and Calabi—Yau categories, how to construct a universal open-closed TCFT from a cyclic A-infinity algebra, and how to extract categorical enumerative invariants. This is an algebraic version of the theorem of Barkan-Steinebrunner-Zhang presented by Adela last week.


For the next 6 weeks, we will be focusing on the paper Effective Categorical Enumerative Invariants by Caldararu--Tu. Dates are tentative. In all the talks, the sign issues can safely be ignored.

Week 11

The goal is to present the construction of Caldararu-Tu, introduce some basic definitions, sketch the proof strategy and distribute the upcoming talks.

Week 12

The goal of this talk is to present the definition of a wheeled PROP, the endomorphism PROP and its totalisation, and the construction of the dg Lie algebras h_A and h_A^ associated to a cyclic A-infinity algebra A in Section 4 of Caldararu--Tu. Some good references for wheeled PROPs are the original Markl-Merkulov-Shadrin paper and Part 1 of this Kawazumi-Vespa paper.

Week 13

The goal of this talk is to present the wheeled PROP of chains on the moduli spaces of curves, its "combinatorial variant", and the canonical Maurer--Cartan elements in their totalisation Lie algebras, a.k.a string vertices. This is Section 5 of Caldararu--Tu. Some helpful references include papers by Wahl-Westerland and Egas Santander.

Week 14

The goal of this talk is to cover Section 6 of Caldararu--Tu: define the moduli spaces wheeled PROP algebra structure on the Hoschild complex of an A-infinity algebra (including a sketch of proof of this result), and use a splitting of the Hodge filtration to define categorical enumerative invariants. These are just the image of the string vertex under the algebra structure map. The subsection on Givental group action can be omitted.

Week 15

This slightly computation-heavy talk focuses on Section 7 of Caldararu--Tu. The goal is to define the trivialised version of the Lie algebra h_A and construct an explicit L-infinity quasi-isomorphism between h_A and its trivialisation using the splitting of the Hodge filtration.

Week 16

The goal of this talk is to cover the final Sections 8 and 9 of Caldararu--Tu: construct the "Koszul version" of the trivialising L-infinity morphism by introducing "partially directed graphs" and mimicking the arguments of the previous talk. This leads to an explicit "Feynman graph sum" formula for categorical enumerative invariants, which is the main contribution of the Caldararu--Tu paper. Examples of explicit computations in low genus can be given, together with ideas of potential applications.


For the end of the semester we will turn to the geometric side of the story (which was in fact the motivation all along).

Week 17

The goal of this talk is to present a construction of the closed Gromov--Witten invariants associated to a symplectic manifold, and explain the interest of defining open invariants.

Week 18

In this talk, the recent work of Hirschi-Hugtenburg on open-closed Deligne--Mumford field theories associated with a Lagrangian submanifold will be presented. This is a major step towards the general conjecture: from a symplectic manifold X, one should be able to construct an open-closed Deligne--Mumford field theory whose open part is the Fukaya category of X, and whose closed part is the Gromov--Witten theory of X. One should then be able to recover the Gromov--Witten theory of X from the datum of the Fukaya category.



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Last update January 29, 2026.