Constructive tensor field theory: the \({T^4_3}\) model
Constructing interacting quantum field theories in a mathematically precise sense is a challenging task. The construction consists of two main steps: a combinatorial step to prove the renormalizability of the theory followed by an analytical step to prove the summability of the corresponding perturbative series providing the \(S\)-matrix elements of the renormalized theory. In this paper the authors construct the partition function (the simplest \(S\)-matrix element) of the simplest interacting superrenormalizable tensor field theory called the \textit{\(T^4_3\) model}. More precisely, this specific theory arises as follows. For the underlying space-time one takes the 3-torus and the two fields in this theory are a section of the trivial complex Hermitian line bundle over the 3-torus and a complex conjugate section. The Lagrangian contains a usual (Laplacian) kinetic term and quartic interaction terms with a common complex coupling constant \(g\). (For the precise formulation of the theory using a Fourier transformed dual picture cf. Section 2 and in particular Equations 2.1, 2.2 and 2.3 in the paper.) The main result is that the carefully defined partition function \(Z(g)\) of the theory exists and is a complex-analytic function of \(g\) if the coupling constant belongs to a small cardioidal region about the origin of the complex plane, moreover \(\log Z(g)\) arises as the Borel sum of a perturbative power series in \(g\) (cf. Theorem 3.1 of the paper). The proof is based on a so-called multiscale loop vertex expansion (MLVE) technique.
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