Lecture: Higher structures in geometry and moduli spaces

4 hours lecture + 2 hours exercise session

(In der Prüfungsordnung erscheint diese Vorlesung unter Modul M40: Ausgewählte Themen der Mathematik)

Dingyu Yang

Winter term 2019/2020

Lecture: Tuesdays 11:00-12:30 in room 1.012 (RUD 25) and Fridays 9:30-11:00 in room 1.012 (RUD 25)
Veranstaltungsnummer: 3314514
First lecture: 15.10.2019

Exercise session: Fridays 11:00-13:00 in room 1.012 (RUD 25)
Veranstaltungsnummer: 33145141
First exercise session: 18.10.2019


Content: The term "higher structure" refers to a phenomenon in which natural algebraic identities hold only "up to homotopy". For example, in topology, addition in the fundamental group is associative thanks to a homotopy that reparametrizes the domain when concatenating three loops. This is a special case of a product operation on the chain complex of a based loop space (or analogously on the free loop space) being only associative up to homotopy; the higher structure mainly comes from the domain reparametrization. Aside from this reparametrization issue, the operation of cutting up loops is strictly co-associative on the nose, but to make things parametrized by manifolds feasible for a homology theory, one has to achieve transversality, which makes the co-associativity identity hold only up to homotopy. Higher structures can also naturally arise in moduli spaces with corners for elliptic geometric PDEs: for example, the Floer cohomology ring acting on a Fukaya algebra at the chain level should be an E_2-algebra acting on a cyclic A-infinity algebra. The combinatorial objects underlying the infinity-structures are also very interesting: these include Stasheff associahedra, permutohedra and secondary polytopes.

Higher structures are much richer than numerical invariants and can in themselves be regarded as geometric objects, which are sometimes even conjectured to faithfully represent the original geometric objects, e.g. in mirror symmetry and noncommutative geometry. At a more basic level, they provide a framework for understanding what invariant information one has captured, and for parametrizing the inductive constructions one hopes to carry out. Understanding and constructing higher structures is an active field of research and under rapid development. I hope to cover some aspects of the above story.

The following will provide some concrete topics and key words: I hope to cover the classical higher Massey product (as a better way to capture the ring structure on cohomology), and its "quantum" analogue in the Fukaya A-infinity algebra/category, the latter being an important symplectic invariant that provides bridges to other vastly different subjects. As an A-infinity structure is the first basic infinity structure, I will provide various viewpoints on it. I hope to cover the interaction of product and coproduct in the involutive bi-Lie infinity algebra of Cieliebak-Fukaya-Latschev, which connects string topology with symplectic field theory and the higher genus Fukaya category (and all three pictures can co-exist). Supplementary to this concrete description, I also hope to cover the recent conceptual treatment by Campos-Merkulov-Willwacher of a similar topic: the Frobenius properad satisfies Koszul duality. As a last topic, when a symplectic manifold is Kähler and equipped with a holomorphic Morse function, there is also a deep conjectural infinity-structure uncovered by physicists Gaiotto-Moore-Witten and mathematicians Kapranov-Kontsevich-Soibelman on a Fukaya category.

Prerequisites and Exam: I will provide background motivations and necessary definitions gradually, with useful results carefully stated and packaged into independent units. Prerequisites should include Differential Geometry I+II and Topology I+II, but will otherwise be consciously kept to a minimum, aside from curiosity and a certain willingness to favor "global and scenery pictures" over "complete details from axioms" (the latter can still be furnished for motivated individuals with guided follow-up reading). The knowledge assumed in each lecture will be summarized at the beginning of that lecture. Hopefully, the course should appeal to an audience with a variety of different interests and tastes in geometry, topology and/or algebra. The examination format will be flexible, e.g, it may consist in explaining or following up on a specific topic or result from the lecture.

Exercise sheets: (available after lectures on Tuesdays)
Exercise 1 (pdf)
Exercise 2 (pdf)
Exercise 3 (pdf)

Table of Contents: (updated after the lectures, lecture notes are expository mostly borrowing from places freely with credits (borrowing from Banksy: everything here is recycled including ideas), typed version may appear optimistically with a lag of one week or two.)

Lecture 1 (15/10/2019): Homotopy transfer via homotopy retract from DGA. Introduction to A infinity algebra. Higher Massey products. Lecture note 1 (pdf)

Lecture 2 (18/10/2019): Loop product and coproduct on geometric chains of a loop space as an example to illustrate situations where associativity up to homotopy naturally arises. Transversality issues and reparametrization discussed. Expected Frobenius infinity compatibility not discussed yet.

Lecture 3 (22/10/2019): Universal property of tensor product. Koszul sign convention in differential graded context. Definition of A infinity algebra with correct signs. Tensor coalgebra (without unit) of the shifted graded (free) module and square 0 coderivation of degree 1 as an alternative definition, which explains the signs in the A infinity relation as an instance of the Koszul convention.

Lecture 4 (25/10/2019): Third interpretation of A infinity algebra, as a self-bracketing 0 (namely, homological) vector field on a non-commutative formal manifold in Kontsevich-Soibelman's viewpoint. A infinity morphisms (in two ways), composition being strictly associative.

Lecture 5 (29/10/2019): A infinity morphism as exponential, homotopy between two A infinity morphisms (in two ways), homological perturbation lemma, transfer by a formal diffeomorphism, quasi-isomorphisms are invertible up to homotopy. Three notions of units and equivalences.

Lecture 6 (1/11/2019): Formality. As an example to illustrate how to work with A infinity algebra and the transfer (by a formal diffeomorphism which is identity up to a certain order), we did a formality theorem due to Seidel that for an A infinity algebra (over field of characteristic 0) equipped with an Euler vector field (which is Hochschild 1-cocycle whose first term acts homologically on degree k homogeneous elements by multiplying by k). The notion of vector field here is different to that in Lecture 4, and will be made precise later. Another classical formality result is left as a guided exercise in Exercise sheet 4, to appear.

Lecture 7 (5/11/2019): Some introductory discussion on the Fukaya algebra associated to a compact relatively spin Lagrangian in a compact symplectic manifold. Curved A infinity algebra.


Lecture 1: We borrow from Part 1 of Vallette's https://arxiv.org/abs/1202.3245, but changed into cochain conventions for strict associativity of singular/de Rham cochains. (Remark: The C infinity algebra on cohomology is a complete invariant for simply connected rational homotopy type is a result of Kadeishvili in 2008, https://arxiv.org/abs/0811.1655.)

Lecture 2: Some background of string topology operations and compatibility can be found in Chas-Sullivan's https://arxiv.org/abs/9911159 and Sullivan's status on string topology circa 2007 at https://arxiv.org/abs/0710.4141. (Stasheff polytopes will be dicussed in detail later, as planer rooted trees, as Deligne-Mumford moduli for disks with boundary marked point, as the secondary polytope of a convex polygon.) Typed note for materials covered in the lecture will be provided here soon. (Lecturer's research preprint (with Manuel Rivera) contains a rigorous construction of coproduct that is closest to the geometric picture we discussed and its co-A infinity structure. Might touch upon it later after the preprint is polished and posted.)

Lecture 3: Definition of A infinity algebra can be found in Fukaya-Oh-Ohta-Ono's book 1 on Lagrangian Intersection Floer theory, and Seidel's EMS book on Fukaya categories and Picard-Lefschetz Theory. Some details of tensor coalgebra can be found in 1.2 of Loday-Vallette's Algebraic Operads. (Please read it only after attempting the last exercise in Exercise sheet 2.)

Lecture 4: For A infinity structure as a homological vector field on a non-commutative formal manifold, see Kontsevich-Soibelman's https://arxiv.org/abs/math/0606241 (which elaborates on the functor of points) as well as Cho-Lee's https://arxiv.org/abs/1002.3653.

Lecture 5: Seidel's Fukaya categories and Picard-Lefschetz Theory and Fukaya-Oh-Ohta-Ono's book 1 on Lagrangian Intersection Floer theory.

Lecture 6: Theorem 2.6 in Abouzaid-Smith's paper https://arxiv.org/abs/1311.5535.