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Deformations of gerbes on smooth manifoldsPaul Bressler , Alexander Gorokhovsky , Ryszard Nest, and Boris Tsygan 1 IntroductionIn [5] we obtained a classification of formal deformations of a gerbe on a manifold (C 1or complex-analytic) in terms of Maurer–Cartan elements of the differential graded Liealgebra (DGLA) of Hochschild cochains twisted by the cohomology class of the gerbe.In the present paper we develop a different approach to the derivation of this classificationin the setting of C 1 manifolds, based on the differential-geometric approachof [4].The main result of the present paper is the following theorem which we prove inSection 8.Theorem 1. Suppose that X is a C 1 manifold and S is an algebroid stack on X whichis a twisted form of O X . Then, there is an equivalence of 2-groupoid valued functorsof commutative Artin C-algebrasDef X .S/ Š MC 2 .g DR .J X / ŒS /:Notations in the statement of Theorem 1 and the rest of the paper are as follows.We consider a paracompact C 1 -manifold X with the structure sheaf O X of complexvalued smooth functions. Let S be a twisted form of O X , as defined in Section 4.5.Twisted forms of O X are in bijective correspondence with OX -gerbes and are classifiedup to equivalence by H 2 .XI O / Š H 3 .XI Z/.One can formulate the formal deformation theory of algebroid stacks ([17], [16])which leads to the 2-groupoid valued functor Def X .S/ of commutative Artin C-algebras.We discuss deformations of algebroid stacks in Section 6. It is natural to expectthat the deformation theory of algebroid pre-stacks is “controlled” by a suitably constructeddifferential graded Lie algebra (DGLA) well-defined up to isomorphism in thederived category of DGLA. The content of Theorem 1 can be stated as the existence ofsuch a DGLA, namely g.J X / ŒS , which “controls” the formal deformation theory ofthe algebroid stack S in the following sense.To a nilpotent DGLA g which satisfies g i D 0 for i< 1 one can associate itsDeligne 2-groupoid which we denote MC 2 .g/, see [11], [10] and references therein.We review this construction in Section 3. Then Theorem 1 asserts equivalence of the2-groupoids Def X .S/ and MC 2 .g DR .J X / ŒS /. Supported by the Ellentuck Fund Partially supported by NSF grant DMS-0400342 Partially supported by NSF grant DMS-0605030