Random tensor models in the large $N$ limit: Uncoloring the colored tensor models

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a $1/N$ expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. nonsymmetric, complex tensor. Colors appear in this setting as a canonical bookkeeping device and not as a fundamental feature. In the large $N$ limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents ${\gamma }_m=1-1/m$.

Figure 1

Graphical representation of trace invariants.

Figure 2

A Feynman graph.

Figure 3

Trace invariants and gluings of simplices in $D=3$.

Figure 4

Stranded graph and a nonbipartite invariant.

Figure 5

Melonic graphs at first and second orders.

Figure 6

Graphs contributing to the 3-dipole expectation.