The n-Category Café

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