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Simple structures axiomatized by almost sure theories
In this article we give a classification of the binary, simple,
-categorical structures with SU-rank 1 and trivial pregeometry. This is
done both by showing that they satisfy certain extension properties, but also
by noting that they may be approximated by the almost sure theory of some sets
of finite structures equipped with a probability measure. This study give
results about general almost sure theories, but also considers certain
attributes which, if they are almost surely true, generate almost sure theories
with very specific properties such as -stability or strong minimality.Comment: 27 page
Simple Recursion Relations for General Field Theories
On-shell methods offer an alternative definition of quantum field theory at
tree-level, replacing Feynman diagrams with recursion relations and interaction
vertices with a handful of seed scattering amplitudes. In this paper we
determine the simplest recursion relations needed to construct a general
four-dimensional quantum field theory of massless particles. For this purpose
we define a covering space of recursion relations which naturally generalizes
all existing constructions, including those of BCFW and Risager. The validity
of each recursion relation hinges on the large momentum behavior of an n-point
scattering amplitude under an m-line momentum shift, which we determine solely
from dimensional analysis, Lorentz invariance, and locality. We show that all
amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are
3-line constructible if an external particle carries spin or if the scalars in
the theory carry equal charge under a global or gauge symmetry. Remarkably,
this implies the 3-line constructibility of all gauge theories with fermions
and complex scalars in arbitrary representations, all supersymmetric theories,
and the standard model. Moreover, all amplitudes in non-renormalizable theories
without derivative interactions are constructible; with derivative
interactions, a subset of amplitudes is constructible. We illustrate our
results with examples from both renormalizable and non-renormalizable theories.
Our study demonstrates both the power and limitations of recursion relations as
a self-contained formulation of quantum field theory.Comment: 27 pages and 2 figures; v2: typos corrected to match journal versio
The structure of 2D semi-simple field theories
I classify all cohomological 2D field theories based on a semi-simple complex
Frobenius algebra A. They are controlled by a linear combination of
kappa-classes and by an extension datum to the Deligne-Mumford boundary. Their
effect on the Gromov-Witten potential is described by Givental's Fock space
formulae. This leads to the reconstruction of Gromov-Witten invariants from the
quantum cup-product at a single semi-simple point and from the first Chern
class, confirming Givental's higher-genus reconstruction conjecture. The proof
uses the Mumford conjecture proved by Madsen and Weiss.Comment: Small errors corrected in v3. Agrees with published versio
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