Geometry and Mathematical Physics Seminar 2012/2013

Thursdays October 11 and 18: The Frank Adams Room (Room 1.212), the Alan Turing Building. 3pm

Integration of Densities over Surfaces in a (Super)space, I and II

Hovhannes Khudaverdian (University of Manchester)

A density $A=A(x, \partial x/\partial u)$ is a function which is defined on surfaces given by parametric equations $x=x(u)$ such that under a change of parameterization it is multiplied by the Jacobian: $A(x, \partial x/\partial v)=A(x, \partial x/\partial u) \det (\partial u/ \partial v)$. (More precisely, if we consider $k$-dimensional surfaces in an $n$-space, these are the $k$-densities.)

To every density $A$ one can assign a functional on surfaces $S_A[C]$, the integral of $A$ over a surface $C$.

Densities are the most general objects of integration over surfaces. We study densities and the corresponding functionals. In the case when a density $A$ is linear with respect to the tangent vectors, we come to a differential form and the corresponding functional obeys the Stokes theorem. We analyze the conditions which specify differential forms in terms of functionals and the Euler--Lagrange equations of these functionals.

We also describe these constructions in the dual language, which is useful in the case if a surface is given not by a parameterization, but by equations. (The well-known elementary example of the dual description is the flux of a vector field through a surface).

These results provide us with a background for developing integration theory for surfaces in a superspace.