Abstract:
Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.

Abstract:
When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the "formal" or "Feynman-diagrammatic" path integral construction. We then outline a proof of the following claim: the formal path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences.

Abstract:
We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path \gamma connecting points q_0 and q_1 in time t, we define a formal power series V_\gamma(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_\gamma) satisfies Schr\"odinger's equation, and explain in what sense the t\to 0 limit approaches the \delta distribution. As such, our construction gives explicitly the full \hbar\to 0 asymptotics of the fundamental solution to Schr\"odinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams.

Abstract:
We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler--Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.

Abstract:
This note describes a strictly-unital $A_\infty$-category whose representations are exact triangles such that the three-fold symmetry on exact triangles is manifest on the $A_\infty$-category.

Abstract:
We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.

Abstract:
The embedding Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaningful algebraic structure on homology that does not lift in a geometrically meaningful way to the chain level.

Abstract:
The notions of "oriented" and "unitary" quantum field theory are the two possible real forms of the same complexified notion. Indeed, oriented and unitary quantum field theories are precisely the unoriented quantum field theories fibered over the two $\mathbb Z/2$-torsors over $\mathrm{Spec}(\mathbb R)$. By the same token, we show that spin and super field theories are two "bosonic forms" of the same "superified" notion. Undergirding this is a coincidence between the homotopy $0$- and $1$-types of the stable orthogonal group and the absolute Galois and "categorified absolute Galois" groups of $\mathbb R$.

Abstract:
Among its many corollaries, Poincare duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra --- a commutative Frobenius algebra in which the comultiplication has cohomological degree equal to the dimension of the manifold. We study the question of whether this structure lifts to a "homotopy" shifted commutative Frobenius algebra structure at the cochain level. To make this question nontrivial, we impose a mild locality-type condition that we call "quasilocality": strict locality at the cochain level is unreasonable, but it is reasonable to ask for homotopically-constant families of operations that become local "in the limit." To make the question concrete, we take the manifold to be the one-dimensional circle. The answer to whether a quasilocal homotopy-Frobenius algebra structure exists turns out to depend on the choice of context in which to do homotopy algebra. There are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations: one can work at "tree level," corresponding roughly to the world of operadic homotopy algebras and their homotopy modules; or one can work at "graph level," corresponding to the world of PROPs. For the tree-level version of our question, the answer is the unsurprising "Yes, such a structure exists" --- indeed, it is unique up to a contractible space of choices. But for the graph-level version, the answer is the surprising "No, such a structure does not exist." Most of the paper consists of computing explicitly this nonexistence, which is controlled by the numerical value of a certain obstruction, and we compute this value explicitly via a sequence of integrals.