The O(N) vector model in the large N limit revisited: multicritical points and double scaling limit

$O(N)$ invariant vector models have been shown to possess non-trivial scaling large $N$ limits, at least perturbatively within the loop expansion, a property they share with matrix models of 2D quantum gravity. In contrast with matrix models, however, vector models can be solved in arbitrary dimensions. We present here the analysis of field theory vector models in $d$ dimensions and discuss the nature and form of the critical behaviour. The double scaling limit corresponds fo...

The 1/N expansion for the O(N) vector model in four dimensions is reconsidered. It is emphasized that the effective potential for this model becomes everywhere complex just at the critical point, which signals an unstable vacuum. Thus a critical O(N) vector model cannot be consistently defined in the 1/N expansion for four-dimensions, which makes the existence of a double-scaling limit for this theory doubtful.

In these lecture notes prepared for the 11th Taiwan Spring School, Taipei 1997}, and updated for the Saalburg summer school 1998, we review the solutions of O(N) or U(N) models in the large N limit and as 1/N expansions, in the case of vector representations. The general idea is that invariant composite fields have small fluctuations for N large. Therefore the method relies on constructing effective field theories for these composite fields after integration over the initial ...

This is a short summary of the phase structure of vector O(N) symmetric quantum field theories in a singular limit, the double scaling limit.It is motivated by the fact that summing up dynamically triangulated random surfaces using Feynman graphs of the O(N) matrix model results in a genus expansion and it provides,in some sense, a nonperturbative treatment of string theory when the double scaling limit is enforced. The main point emphasized here is that this formal singular ...

In the limit where $N\to\infty$ and the coupling constant $g \to g_{c}$ in a correlated manner, O(N) symmetric vector models represent filamentary surfaces. The purpose of these studies is to gain intuition for the long lasting search for a possible description of quantum field theory in terms of extended objects. It is shown here that a certain limiting procedure has to be followed in order to avoid several difficulties in establishing the theory at a critical negative coupl...

The multi-critical fixed points of $O(N)$ symmetric models cease to exist in the $N\to\infty$ limit, but the mechanism regulating their annihilation still presents several enigmatic aspects. Here, we explore the evolution of high-order multi-critical points in the $(d,N)$ plane and uncover a complex mosaics for their asymptotic behaviour at large $N$. This picture is confirmed by various RG approaches and constitutes a fundamental step towards the full comprehension of critic...

We study interacting critical UV regime of the long-range $O(N)$ vector model with quartic coupling. Analyzing CFT data within the scope of $\epsilon$- and $1/N$-expansion, we collect evidence for the equivalence of this model and the critical IR limit of the cubic model coupled to a generalized free field $O(N)$ vector multiplet.

Recent interest in large N matrix models in the double scaling limit raised new interest also in O(N) vector models. The limit $N \rightarrow \infty$, correlated with the limit $g \rightarrow g_c$, results in an expansion in terms of filamentary surfaces and explicit calculations can be carried out also in dimensions $d\geq 2$. It is shown here that the absence of physical massless bound states in two dimensions sets strong constraints on this limit.

We suggest a general relation between theories of infinite number of higher-spin massless gauge fields in $AdS_{d+1}$ and large $N$ conformal theories in $d$ dimensions containing $N$-component vector fields. In particular, we propose that the singlet sector of the well-known critical 3-d O(N) model with the $(\phi^a \phi^a)^2$ interaction is dual, in the large $N$ limit, to the minimal bosonic theory in $AdS_4$ containing massless gauge fields of even spin.

The large $N$ expansion plays a fundamental role in quantum and statistical field theory. We show on the example of the O$(N)$ model that at $N=\infty$, its traditional implementation misses in all dimensions below four some fixed points of the renormalization group. These new fixed points show singularities at $N=\infty$ in their effective potential that become a boundary layer at finite $N$. We show that they have a physical impact on the multicritcal physics of the $O(N$) ...