4 research outputs found
Symmetries of the Space of Linear Symplectic Connections
There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term
Projectively equivariant quantizations over the superspace
We investigate the concept of projectively equivariant quantization in the
framework of super projective geometry. When the projective superalgebra
pgl(p+1|q) is simple, our result is similar to the classical one in the purely
even case: we prove the existence and uniqueness of the quantization except in
some critical situations. When the projective superalgebra is not simple (i.e.
in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a
one-parameter family of equivariant quantizations. We also provide explicit
formulas in terms of a generalized divergence operator acting on supersymmetric
tensor fields.Comment: 19 page
A Schwarz lemma for K\"ahler affine metrics and the canonical potential of a proper convex cone
This is an account of some aspects of the geometry of K\"ahler affine metrics
based on considering them as smooth metric measure spaces and applying the
comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a
version for K\"ahler affine metrics of Yau's Schwarz lemma for volume forms. By
a theorem of Cheng and Yau there is a canonical K\"ahler affine Einstein metric
on a proper convex domain, and the Schwarz lemma gives a direct proof of its
uniqueness up to homothety. The potential for this metric is a function
canonically associated to the cone, characterized by the property that its
level sets are hyperbolic affine spheres foliating the cone. It is shown that
for an -dimensional cone a rescaling of the canonical potential is an
-normal barrier function in the sense of interior point methods for conic
programming. It is explained also how to construct from the canonical potential
Monge-Amp\`ere metrics of both Riemannian and Lorentzian signatures, and a mean
curvature zero conical Lagrangian submanifold of the flat para-K\"ahler space.Comment: Minor corrections. References adde