### Report on AIM Workshop: Towards Relative Symplectic Field Theory

#### Posted by Urs Schreiber

*guest post by Jim Stasheff*

September 24 to September 28, 2007: AIM workshop
Towards Relative Symplectic Field Theory
at the CUNY Graduate Center, New York City
organized by
Kai Cieliebak, Tobias Ekholm, Yakov Eliashberg, Kenji Fukaya, Dennis Sullivan, and Michael Sullivan.

See this for more details about the scope of the workshop.

I was able to attend September 25 and 25 so this is only a partial report. It was truly a workshop. There were lectures in the morning, but ones designed to provoke for the afternoon discussion and interaction between a variety of groups represented,roughly divided into algebra (in fact infinity algebra) and geometry (specifically symplectic geometry and in low dimensions).

Note: SFT usually meant Symplectic Field Theory but occasionally it meant String Field Theory. Some of the pictures are remarkably similar.

Here a rough sketch of what I did get to hear: A sub-theme was $D^2=0$.

Fukaya had already spoken but I did hear two of his 3 Os and the third (Oh) was also present.

Ohta: Whitehead’s Theorem for $A_\infty$-algebra

meaning when a weak homotopy equivalence implies a strong homotopy equivalence. Here extended/weak/curved $A_\infty$-algebras were included and the ground field was extended to e.g. the Novikov ring. The proofs were by classical obstruction theory, so $A_n$-algebras for finite n played a role. A closed or even algorithmic formula was missing e.g. no analog of the ‘tensor trick’.

Ono: Canonical models of filtered $A_\infty$-algebras (including extended/weak/curved ones)

This involved a souped up version of Kadeishvili’s original result (and alternative proofs later) that the homology of an $A_\infty$-algebra inherits an $A_\infty$-structure. For those who speak the language, call a homotopy for a deformation retract a “Green’s operator”! Here the proofs (following Kontsevich and Soibelman?) were by induction expressed in terms of rooted planar trees with stubs and an energy function.

Discussion:

Michael Sullivan: $D^2=0$ for algebras over an operad with two binary operations, $\circ$ and $\star$, and one unary inspired by pictures from symplectic geometry. Each operation to satisfy Jacobi and !! also their sum.

Sasha Voronov: Such a combination of operations, though with different pictues and different relations, occurs in dgBV algebras. cf. Zwiebach’s String Field Theories.

9/26/07

Janko Latschev (work with Kai Cieliebak) $D^2=0$ for algebra inspired by Relative SFT.

Consider a Legendrian submanifold of dimension n in a contact manifold of Y dimension 2n+1which has no closed Reeb orbits. Reeb chords are arcs in Y with ends on the Legendrian (perhaps thought of as a brane). For n=1 and a Legendrian knot in $R^3$, the Reeb chords could be vertical segments in $R^3$. Goal: Find a Master equation and a solution in terms of moduli spaces of circles with marked points. Idea: A closed loop given as a map of a circle with twice as many marked points, alternating paths in the Legendrian with Reeb chords.

Algebra of chains on space of such: $\{ , \}$ given by glueing of such loops along a common Reeb chord (if any) which is then erased. $[ , ]$ given by cutting and reglueing of such loops at a common point in the Legendrian by the usual reinterpretation of X as $\gt \lt$ transforming to the top $V$ and the bottom $\wedge$. Each operation satisfies Jacobi and !! also their sum. In addition to the usual differential $\partial$, there is another one, $\delta$, from inserting a degenerate example of the loops being considered, the kind which spends only an instant in the Legendrian. That $D^2=0$ for $D=\partial+\delta$ follows as one would expect. That $D$ is a derivation of the total $\langle , \rangle = \{ , \} + [ , ]$ is more subtle, involving cancelation of mixed terms.

Open questions: 1. Variance under cobordism up to some kind of homotopy? 2. Special case: $n=1$, Legendrian knot (or link) in $R^3$. 3. a ‘herded’ version?? 4. augmentations? linearizations? removal of $m_0$?

Octav Cornea: Master equation for Cluster algebra

This was not so easy to summarize, but intriguing features are: Related to Morse theory, especially a le Fukaya Pictures in terms of planar rooted trees with disks at internal nodes where the disks carry geometry. Homotopy theory is difficult, even rationally so settle for homology.

Discussion: Stasheff: Old stuff perhaps knot known to all present: Kadeishvilli’s transfer of algebra structure up to $A_\infty$ Massey products are representatives Gugenheim, Huebschmann et al transfer by the tensor trick Huebschmann: $L_\infty$ analog - tensor trick doesn’t symmetrize Markl: higher homotopy insights Lie-massey brackets - cf. Retakh

Sullivan: $A_\infty$-structures and other $D^2=0$ in terms of a formal manifold and vector field Relation to correlations functions, e.g. Gromov-Witten, Donaldson

John Terilla: Master Equations as non-commutative deformations Versal solution $\leftrightarrow$ specific $A_\infty$-structure and specific quism (= quasi-iso) For $H = H(A,d)$ and a splitting $H \subset A$, work in $A[[H^{*}]]$ BV algebras and dgBV Quantum Master Equation by perturbing d to $d+\hbar \Delta$ For a solution $\Gamma$, perturb further to $d+\hbar \Delta + [\Gamma, -]$ Theorem; There exists a versal $\Gamma$ iff… Barannikov and Kontsevich: special case in the presence of a $\partial,\bar\partial Lemma$ (Compare result of Heubschmann and Stasheff) If there does NOT exist such a $\Gamma$, the obstructions can be expressed in terms of another $L_\infty$-structure on $H$

Stasheff: review of joint work with Kajiura on OCHA = open closed strong homotopy algebra - inspired by string field theory

## Re: Report on AIM Workshop: Towards Relative Symplectic Field Theory

Associahedral categories, particles and Morse functor

Authors: Jean-Yves Welschinger

(Submitted on 25 Jun 2009)

Abstract:Every smooth manifold contains particles which propagate. These form objects and morphisms of a category equipped with a functor to the category of Abelian groups, turning this into a 0+1 topological field theory. We investigate the algebraic structure of this category, intimately related to the structure of Stasheff’s polytopes, introducing the notion of associahedral categories. An associahedral category is preadditive and close to being strict monoidal. Finally, we interpret Morse-Witten theory as a contravariant functor, the Morse functor, to the homotopy category of bounded chain complexes of particles.