Abstract
We review some aspects of the cutting and gluing law in local quantum field theory (QFT) and study it from a new point of view. In particular, we emphasize the description of gluing by a path integral over a space of polarized boundary conditions, which are given by leaves of some Lagrangian foliation in the phase space. We think of this path integral as a non-local (d − 1)-dimensional gluing theory associated to the parent local d-dimensional QFT. This is a novel point of view paving the way for applications of the standard QFT techniques (that do not rely on locality) to the gluing theory. We describe various properties of this procedure and spell out conditions under which symmetries of the parent theory lead to symmetries of the gluing theory. The purpose of this paper is to set up a playground for the companion paper where these techniques are applied to obtain new results in supersymmetric theories.
Highlights
Where W denotes W with opposite orientation, they can be attached along W, and this topological operation has a counterpart for the quantum field theory (QFT) data.1 In topology, the gluing operation is denoted by:
We review some aspects of the cutting and gluing law in local quantum field theory (QFT) and study it from a new point of view
We prove that every symmetry of the parent theory that preserves polarization and the two boundary states descends to the symmetry of the gluing theory
Summary
Is it possible to compute the path integral in (1.7), or even make any sense of it? At this level of generality, the problem seems hopeless: just to write the integrand of (1.7) already requires solving QFTd on M with arbitrary polarized boundary conditions B. Gluing by the boundary path integral, as described above, is a purely formal procedure which ignores that the boundary conditions are quantum objects: they can receive quantum corrections, get renormalized, and undergo non-trivial RG flows The latter set of problems is very familiar already in the case of bulk path integrals: it can be concisely stated as the need for regularization in QFT. Similar analysis was performed for a free Maxwell theory (in arbitrary spacetime dimension) and 2d Yang-Mills in [28] Another case in which one can efficiently evaluate the QFTd−1 path integral is in the presence of extra symmetries, such as supersymmetry: under favorable conditions, it allows to make sense of (1.7) using the supersymmetric localization
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