## Duke Mathematical Journal

### The symplectic arc algebra is formal

#### Abstract

We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one Hochschild cohomology class on a Floer $A_{\infty}$-algebra associated to the $(k,k)$-nilpotent slice $\mathcal{Y}_{k}$ obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification $\bar{\mathcal{Y}}_{k}$. The space $\bar{\mathcal{Y}}_{k}$ is obtained as the Hilbert scheme of a partial compactification of the $A_{2k-1}$-Milnor fiber. A sequel to this paper will prove formality of the symplectic cup and cap bimodules and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.

#### Article information

Source
Duke Math. J., Volume 165, Number 6 (2016), 985-1060.

Dates
Revised: 16 June 2015
First available in Project Euclid: 28 January 2016

https://projecteuclid.org/euclid.dmj/1453994024

Digital Object Identifier
doi:10.1215/00127094-3449459

Mathematical Reviews number (MathSciNet)
MR3486414

Zentralblatt MATH identifier
1346.53073

#### Citation

Abouzaid, Mohammed; Smith, Ivan. The symplectic arc algebra is formal. Duke Math. J. 165 (2016), no. 6, 985--1060. doi:10.1215/00127094-3449459. https://projecteuclid.org/euclid.dmj/1453994024

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