John Terilla

Smoothness theorem for differential BV algebras

Given a differential Batalin-Vilkovisky algebra (V, Q,�, ·), the as- sociated odd differential graded Lie algebra (V, Q + �,( , )) is always smooth formal. The quantum dgLa L~ := (V ((~)), Q + ~�,( , )) is not always smooth formal, but when it is—for example when a Q-� version of the @-@ Lemma holds—there is a weak-Frobenius manifold structure on the homology of L that is important in applications and relevant to quantum correlation func- tions. In this paper, we prove that L~ is smooth formal if and only if the spectral sequence associated to the filtration F p := ~pV ((~)) on the complex (V ((~)), Q + ~�) degenerates at E1. A priori, this degeneration means that a collection of first order obstructions vanish and we prove that it follows that all obstructions vanish. For those differential BV algebras that arise from the Hochschild complex of a Calabi-Yau category, the degeneration of this spectral sequence is an expression of the noncommutative Hodge to deRham degenera- tion, conjectured by Kontsevich and Soibelman and proved to hold in certain cases by Kaledin. The results in this paper imply that the noncommutative Hodge to de Rham degeneration conjecture is equivalent to the existence of a versal solution to the quantum master equation. At the end of the paper, some physical considerations are mentioned.

Quantizing deformation theory

We describe a step toward quantizing deformation theory. The L ∞ operad is encoded in a Hochschild cocyle o1 in a simple universal algebra (P, o0). This Hochschild cocyle can be extended naturally to a star product ‚=o0+ħo1+ħ2o2 +…. The algebraic structure encoded in * is the properad Ω(coFrob) which, conjecturally, controls a quantization of deformation theory—a theory for which Frobenius algebras replace ordinary commutative parameter rings.

Thermodynamic Interpretation of the Quantum Error Correcting Criterion

AbstractShannons fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics. For example, Schumacher proved a quantum version of Shannons 1948 classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur during the transmission, storage, and manipulation of quantum information. One can formulate a criterion of necessary and sufficient conditions for the errors to be detectable and correctable. We show that this criterion has a thermodynamical interpretation. PACS: 03.67; 05.30; 63.10