Proof of Swiss Cheese Version of Delignes Conjecture

The interest to various versions [6], [8], [18], [20], [23], [24], [25], [27], [31], [34], [36] of the Deligne conjecture on Hochschild complex is motivated by generalizations [11], [12], [30], [33], [35] of the famous Kontsevich’s formality theorem [21]. Thus, in recent preprint [24] M. Kontsevich and Y. Soibelman proposed a proof of the chain version of Deligne’s conjecture for Hochschild complexes of an A ∞ -algebra. This is an important step in proving the formality for the homotopy calculus algebra of Hochschild (co)chains [12].

Let A be an associative algebra and C • (A, A) be the Hochschild cochain complex of A . The original version of Deligne’s conjecture says that the operad of natural operations on C • (A, A) is quasi-isomorphic to the singular chain operad of the operad E 2 of little discs [10], [26]. This statement is not very precise because there are different choices of what one may call “the operad of natural operations on C • (A, A) .” One may use the so-called minimal operad of M. Kontsevich and Y. Soibelman [23] or the operad of braces [16], [19] as in [27] and [37] or the “big operad” of M. Batanin and M. Markl [5]. Due to works of various people [5], [8], [23], [27], [32], and [37] it is now known that all these operads are quasi-isomorphic to the singular chain operad of the operad E 2 .

The topological operad E 2 of little discs admits a natural extension to a 2-colored topological operad which is called the Swiss Cheese operad SC 2 . This operad was proposed by A. Voronov in [38].

In [38] A. Voronov also described the homology operad H -• (SC 2 ) . More precisely, he showed that an algebra over the operad H -• (SC 2 ) is a pair of graded vector spaces (V 1 , V 2 ), where V 1 is a Gerstenhaber algebra1 , and V 2 is an associative algebra equipped with a module structure over the commutative algebra

satisfying the following condition

where

and for the multiplication of the corresponding elements we use either the associative algebra structure in V 2 or the commutative algebra structure in V 1 .

It is not hard to prove the following proposition:

Proposition 1.1. If A is an associative algebra and HH • (A, A) is its Hochschild cohomology then the pair (HH • (A, A), A) forms an algebra over the operad H -• (SC 2 ) .

Proof. Indeed the associative algebra structure on A is already given. HH • (A, A) is a Gerstenhaber algebra due to [14]. Finally, to define the module structure on A over the commutative algebra HH • (A, A) we use the fact that the zeroth Hochschild cohomology HH 0 (A, A) is the center Z(A) of A. Namely, we declare z • a = z a , if z ∈ HH 0 (A, A) = Z(A) , 0 , otherwise .

Equation (1.2) is nontrivial only when u i ∈ HH 0 (A, A). In this case the required condition is automatically satisfied since u i ’s are elements of the center Z(A) of A .

In this paper we prove the Swiss Cheese version of Deligne’s conjecture which extends Proposition 1.1 to the level of cochains.

To formulate this version of Deligne’s conjecture we, first, construct a 2-colored DG operad Λ of natural operations on the pair (C • (A, A); A) . Roughly speaking, this operad is generated by the insertions of a cochain into a cochain, the cup-product of cochains and the insertions of elements of the algebra A into a cochain. The precise description of Λ is given in Section 2.

The main result of this paper is the following theorem Theorem 1.2. The 2-colored DG operad Λ of natural operations on the pair (C • (A, A), A) is quasiisomorphic to the singular chain operad of Voronov’s Swiss Cheese operad SC 2 . The induced action of the homology operad H -• (SC 2 ) on the pair (HH • (A), A) recovers the one from Proposition 1.1 .

We prove this theorem using ideas from [32] and Batanin’s theorem [2] which identifies the homotopy type of Voronov’s Swiss Cheese operad with that of the symmetrization of a contractible cofibrant Swiss Cheese type 2-operad. The required facts about 2-operads are reviewed in Sections 4,5

1.1. Remarks on higher dimensional versions. Voronov’s Swiss Cheese operad admits the obvious higher dimensional analogue SC d (d ≥ 2) . This operad extends the operad of d-cubes in the same way as the operad SC 2 extends the operad of little disks. From this point of view, Theorem 1.2 is a 2-dimensional case of the following conjecture formulated by M. Kontsevich in [22]: the DG operad of natural operations on the pair “a d-algebra 2 and its Hochschild complex” is quasi-isomorphic to the singular chain operad of SC d+1 . In [22,Section 2.5] M. Kontsevich also conjectures that the Hochschild cochain complex of a d-algebra is a final object in an appropriate category of “Swiss Cheese algebras”. In our paper, this question about universality is not addressed.

In [13] J.N.K. Francis showed that an appropriate deformation

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