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20) from the left by the natural bimodule E AB given by f · a · b = f (a)b, obtain eq. (21), and reverse the argument above it. This direction of proof does not make use of generator hypothesis. 8. Since B AB ⊕ Ω ∼ = B A ⊗B AB is always the case for some B-B-bimodule Ω, the B-B-bimodules A ⊗B A and A are similar or Hequivalent under the conditions of the theorem: thus their endomorphism algebras are Morita equivalent. 1, left multiplication B → E is depth two. There is a general Galois theory of depth two extensions which in this case specializes to total algebra End B A ⊗B AB and base algebra EB ∼ = End B AB as parts of a bialgebroid.

CHENG, J. J. HECKMAN, AND C. VAFA Now we are ready to deﬁne the relevant cohomology. 26) where adj(φ(0) ) = [φ(0) , · ] is the adjoint action of the background ﬁeld φ(0) . 4) satisﬁed by the back¯ satisﬁes the Leibniz rule: ground ﬁelds A¯(0) and φ(0) . Second, D ¯ ∧ Ψ + (−1)degΨ Ψ ∧ DΨ ¯ . 27) Third, we have ¯ = ∂¯ D when acting on gauge singlets. The ﬁrst property shows it is meaningful to talk about a cohomology with respect to this operator. As we will see in more detail later, the last two properties ensure that the Yukawas are invariant under a change of zero ¯ mode wave-functions by a D-exact piece, given that some mild convergence properties are satisﬁed by these wave-functions.

3 and Mackey’s theorem. Acknowledgments The second author is grateful to David Harbater and Gestur Olafsson for discussions related to this paper. 5/S/58852, Project ”Postdoctoral programme for training scientiﬁc researchers” coﬁnanced by the European Social Found within the Sectorial Operational Program Human Resources Development 2007–2013. 36 S. BURCIU AND L. KADISON References [1] J. L. Alperin with R. B. Bell, Groups and Representations, GTM 162, Springer, New York, 1995. [2] S. Burciu, Coset decomposition for semisimple Hopf Algebras, Comm.