TY - JOUR

T1 - Graded Geometry, Q-Manifolds, and Microformal Geometry

T2 - LMS/EPSRC Durham Symposium on Higher Structures in M-Theory

AU - Voronov, Theodore

PY - 2019/5/6

Y1 - 2019/5/6

N2 - We give an exposition of graded and microformal geometry, and the language of Q-manifolds. Q-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of “non-linear homological algebra”, and a powerful tool for describing algebraic and geometric structures. This language goes together with that of graded manifolds, which are supermanifolds with an extra Z-grading in the structure sheaf. “Microformal geometry” is a new notion referring to “thick” or “microformal” morphisms, which generalize ordinary smooth maps, but whose crucial feature is that the corresponding pullbacks of functions are nonlinear. In particular, “Poisson thick morphisms” of homotopy Poisson supermanifolds induce L∞-morphisms of homotopy Poisson brackets. There is a quantum version based on special type Fourier integral operators and applicable to Batalin–Vilkovisky geometry. Though the text is mainly expository, some results are new or not published previously.

AB - We give an exposition of graded and microformal geometry, and the language of Q-manifolds. Q-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of “non-linear homological algebra”, and a powerful tool for describing algebraic and geometric structures. This language goes together with that of graded manifolds, which are supermanifolds with an extra Z-grading in the structure sheaf. “Microformal geometry” is a new notion referring to “thick” or “microformal” morphisms, which generalize ordinary smooth maps, but whose crucial feature is that the corresponding pullbacks of functions are nonlinear. In particular, “Poisson thick morphisms” of homotopy Poisson supermanifolds induce L∞-morphisms of homotopy Poisson brackets. There is a quantum version based on special type Fourier integral operators and applicable to Batalin–Vilkovisky geometry. Though the text is mainly expository, some results are new or not published previously.

U2 - 10.1002/prop.201910023

DO - 10.1002/prop.201910023

M3 - Article

SN - 1521-3978

JO - Fortschritte der Physik

JF - Fortschritte der Physik

ER -