Feynman geometry

In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman geometry is a geometry when the vector space of $A_\infty$ structures is infinite dimensional while the relevant operators are of trace-class. We construct families of Feynman geometries with "continuum" as their limit

Tropical Mirror Symmetry: Correlation functions

We formulate the mirror symmetry for correlation functions of tropical observables. We prove the tropical mirror correspondence for correlation functions of evaluation observables on toric space. The key point of the proof is the localization of correlation functions for mirror states in type-B higher topological quantum mechanics on trees. The correlation functions localize to the correlation functions of holomorphic functions, defined recursively in Landau-Ginzburg-Saito theory with exponential mirror superpotential and tropical good section.Comment: 77 pages, minor correction

Tropical mirror for toric surfaces

We describe the tropical mirror for complex toric surfaces. In particular we provide an explicit expression for the mirror states and show that they can be written in enumerative form. Their holomorphic germs give an explicit form of good section for Landau-Ginzburg-Saito theory. We use an explicit form of holomorphic germs to derive the divisor relation for tropical Gromov-Witten invariants. We interpret the deformation of the theory by a point observable as a blow up of a point on the toric surface. We describe the implication of such interpretation for the tropical Gromov-Witten invariants.Comment: 42 pages, minor correction

New Objects in Scattering Theory with Symmetries

We consider 1D quantum scattering problem for a Hamiltonian with symmetries. We show that the proper treatment of symmetries in the spirit of homological algebra leads to new objects, generalizing the well known T- and K-matrices. Homological treatment implies that old objects and new ones are be combined in a differential. This differential arises from homotopy transfer of induced interaction and symmetries on solutions of free equations of motion. Therefore, old and new objects satisfy remarkable quadratic equations. We construct an explicit example in SUSY QM on $S^1$ demonstrating nontriviality of the above relation

New observables in topological instantonic field theories

Instantonic theories are quantum field theories where all correlators are determined by integrals over the finite-dimensional space (space of generalized instantons). We consider novel geometrical observables in instantonic topological quantum mechanics that are strikingly different from standard evaluation observables. These observables allow jumps of special type of the trajectory (at the point of insertion of such observables). They do not (anti)commute with evaluation observables and raise the dimension of the space of allowed configurations, while the evaluation observables lower this dimension. We study these observables in geometric and operator formalisms. Simple examples are explicitly computed; they depend on linking of the points. The new "arbitrary jump" observables may be used to construct correlation functions computing e.g. the linking numbers of cycles, as we illustrate on Hopf fibration.Comment: 16 pages, accepted to Journal of Geometry and Physic

On enumerative problems for maps and quasimaps: freckles and scars

We address the question of counting maps between projective spaces such that images of cycles on the source intersect cycles on the target. In this paper we do it by embedding maps into quasimaps that form a projective space of their own. When a quasimap is not a map, it contains freckles (studied earlier) and/or scars, appearing when the complex dimension of the source is greater than one. We consider a lot of examples showing that freckle/scar calculus (using excess intersection theory) works. We also propose the "smooth conjecture" that may lead to computation of the number of maps by an integral over the space of quasimaps.Comment: 53 page