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)

An early use of curved dg-algebras can be found in a 1993 paper of Positselski:

• L. E. Positselski, Nonhomogeneous quadratic duality and curvature, Funktsional. Anal. i Prilozhen. 27 (1993), no. 3, 57–66; English transl., Funct. Anal. Appl. 27 (1993), no. 3, 197–204.

Available at (https://www.mathnet.ru/links/e9c418f6c2cdbb10fafba78627743cdb/faa712.pdf)