Abstract
We continue the constructive program about tensor field theory through the next natural model, namely the rank five tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on \mathbf{U(1)^5}𝐔(1)5. We make a first step towards its construction by establishing its power counting, identifying the divergent graphs and performing a careful study of (a slight modification of) its RG flow. Thus we give strong evidence that this just renormalizable tensor field theory is non perturbatively asymptotically free.
Highlights
We identify in particular “cardioid-like” domains of the complex plane invariant under this modified RG flow, see Theorems 6.1 and 6.3 and Corollary 6.4
Hc means c =c Hc, ˆc is the identity on the tensor product Hc, Trc is the partial trace over Hc and 〈, 〉ˆc the scalar product restricted to Hc
T and T can be considered both as vectors in H⊗ or as diagonal operators acting on H⊗, with eigenvalues Tn and T n
Summary
Hairer [1] solved a series of stochastic differential equations such as the KPZ equation or the φ34 equation. We identify in particular “cardioid-like” domains of the complex plane invariant under this modified RG flow, see Theorems 6.1 and 6.3 and Corollary 6.4 Solving this T54 model means defining its correlation functions non perturbatively in the coupling constant g. The infinite cutoff limit is well-defined and analytic These expansions consist in partial resummations of the perturbative series, either expressed in an intermediate field representation (this is the so-called Loop Vertex Expansion [46, 47, 52]) or obtained from a specific change of the initial tensor field variables (in which case it is called Loop Vertex Representation [53,54,55]). An update of all currently known approaches to constructive tensor field theory seems necessary [56]
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