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Suppose that $P$ is a partial differential operator with constant coefficients. An old result of Petrowski (see Thm. 3.2 of Hörmander's 1955 Acta Math. paper On the theory of general partial differ …

There are quite a few connections. I will mention a result of mine where the connection is explicit and essential. Fix the metric $g$. Set $m=\dim M$ and assume that ${\rm vol}_g(M)=1$. Denote by $ …

Denote by $A$ the connection and by $F_A$ its curvature. Then $$dA=F_A-A\wedge A. $$ If $A$ is in Coulomb gauge we have an additional equation $$d^*A=0. $$ The advantage is that the operator $d\op …

In many variational problems one is given an action functional $f\mapsto S[f]$, described by an integral $$ S[f]=\int_\Omega L\bigl(\;x,f(x),D f(x),\dotsc, D^k f(x)\;\bigr) dx $$ in which $\O …

For Lie algebras of matrices (which is what you really care about in Chern-Simons theory) think of $A$ as a form with matrix coefficients $$ A=\sum_i A_i dx^i, $$ where $A_i$ are $r\times r$ matrice …

You should have a look at the book by Gelfand and Vilenkin Generalized functions. Vol. 4: Applications of harmonic analysis where they describe how to construct Gaussian measures on (duals …

Have you tried the book "the Geometry of physics" by Th. Frankel?

Suppose that $A$ has discrete spectrum consisting of eigenvalues with finite multiplicities $$ 0<\lambda_1 < \lambda_2<\cdots $$ with $\lambda_n\to\infty$ as $n\to \infty$. Denote by $(\psi_n)$ …

There are different incarnation of the Maslov index. The one that I prefer is the one proposed in Arnold's paper suggested by Igor Rivin. The paper by Cappell-Lee-Miller suggested by Greg Friedman …

To find the Fourier transform of this and many other functions I enthusiastically recommend volume 1 of the magnificent treatise Generalized Functions, by Gelfand and coauthors. This monograph …

There are Frobenius algebras in the story. see http://arxiv.org/pdf/1201.1273v1.pdf

The eta invariant is typically defined for Dirac operators on odd-dimensional Riemann manifolds. They depend on the various geometric data needed to define them (metrics, connections etc) and for t …

$\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\Hom}{Hom}$ Suppose that $V$ is a finite dimensional real space equipped with an almost complex structure $J$. Let …