nLab curved dg-algebra

Idea

A curved dg-algebra is like a dg-algebra, but instead of the differential squaring to 0, it squares to the graded commutator with a fixed element of the algebra: its “curvature”.

This is like the covariant derivative on the sections of a vector bundle with connection satisfying $\nabla \circ \nabla = F_\nabla$, where $F_\nabla$ is the curvature 2-form of the connection (valued, here, in fiber endomorphism)s.

Curved dg-algebras appear in the description of various TQFTs.

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Related concepts

• curved L-infinity algebra

• curved A-infinity algebra

References

A basic exposition of the definition is in

• A. Polishchuk, Introduction to curved dg-algebra , notes taken in a talk (pdf)

For applications in derived categories of D-branes in Landau-Ginzburg models see

• Dmitri Orlov, Derived Categories of Singularities and D-branes in Landau-Ginzburg models , (arXiv:math.ag/0503632)

• Anatoly Preygel, Thom-Sebastiani and duality for matrix factorizations arXiv:1101.5834

An natural construction of curved dg-algebras as de Rham / Dolbeault complexes on a circle 2-bundle with connection is in

• Jonathan Block, Duality and equivalence of module categories in noncommutative geometry, pdf, in R. Bott Memorial Volume

and with more details in section 2 of

• Jonathan Block, Calder Daenzer, Mukai duality for gerbes with connection (arXiv:0803.1529v2)