Singular Hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gerstenhaber bracket, on $\mathcal{D}^*(A,A)$. Moreover, we prove that the cohomology algebra $H^*(\mathcal{D}^*(A,A))$ is a Batalin-Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two Frobenius algebras are quasi-isomorphic as dg algebras then their Tate-Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate-Hochschild complex to string topology.Comment: 48 pages, 9 figures, Revisions made based on a referee report. To appear in Journal of Noncommutative Geometr

A∞_\infty deformations of extended Khovanov arc algebras and Stroppel's Conjecture

Extended Khovanov arc algebras $\mathrm{K}_m^n$ are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of $\mathrm{K}_m^n$ vanish in a certain range, implying that the algebras $\mathrm K_m^n$ admit no nontrivial A$_\infty$ deformations, in particular that the algebras are intrinsically formal. Whereas Stroppel's Conjecture is known to hold for the algebras $\mathrm K_m^1$ and $\mathrm K_1^n$ by work of Seidel and Thomas, we show that $\mathrm K_m^n$ does in fact admit nontrivial A$_\infty$ deformations with nonvanishing higher products for all $m, n \geq 2$. We describe both $\mathrm K_m^n$ and its Koszul dual concretely as path algebras of quivers with relations and give an explicit algebraic construction of A$_\infty$ deformations of $\mathrm K_m^n$ by using the correspondence between A$_\infty$ deformations of a Koszul algebra and filtered associative deformations of its Koszul dual. These deformations can also be viewed as A$_\infty$ deformations of Fukaya--Seidel categories associated to Hilbert schemes of surfaces based on recent work of Mak and Smith.Comment: 54 pages, 12 figures, comments are very welcom