22,865 research outputs found
Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem
We develop a new approach to the conformal geometry of embedded hypersurfaces
by treating them as conformal infinities of conformally compact manifolds. This
involves the Loewner--Nirenberg-type problem of finding on the interior a
metric that is both conformally compact and of constant scalar curvature. Our
first result is an asymptotic solution to all orders. This involves log terms.
We show that the coefficient of the first of these is a new hypersurface
conformal invariant which generalises to higher dimensions the important
Willmore invariant of embedded surfaces. We call this the obstruction density.
For even dimensional hypersurfaces it is a fundamental curvature invariant. We
make the latter notion precise and show that the obstruction density and the
trace-free second fundamental form are, in a suitable sense, the only such
invariants. We also show that this obstruction to smoothness is a scalar
density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein
metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand
machinery. The solution to the constant scalar curvature problem provides a
smooth hypersurface defining density determined canonically by the embedding up
to the order of the obstruction. We give two key applications: the construction
of conformal hypersurface invariants and the construction of conformal
differential operators. In particular we present an infinite family of
conformal powers of the Laplacian determined canonically by the conformal
embedding. In general these depend non-trivially on the embedding and, in
contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even
dimensional hypersurfaces, exist to all orders. These extrinsic conformal
Laplacian powers determine an explicit holographic formula for the obstruction
density.Comment: 37 pages, LaTeX, abridged version, functionals and explicit
invariants from previous version treated in greater detail in another postin
One-dimensional Chern-Simons theory and the genus
We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras
on any one-dimensional manifold and quantize this theory using the
Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul
duality and derived geometry allow us to encode topological quantum mechanics,
a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X,
as such a Chern-Simons theory. Our main result is that the partition function
of this theory is naturally identified with the A-genus of X. From the
perspective of derived geometry, our quantization construct a volume form on
the derived loop space which can be identified with the A-class.Comment: 61 pages, figures, final versio
Derived Algebraic Geometry
This text is a survey of derived algebraic geometry. It covers a variety of
general notions and results from the subject with a view on the recent
developments at the interface with deformation quantization.Comment: Final version. To appear in EMS Surveys in Mathematical Science
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