Srednicki — Quantum Field Theory · Problem 80.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · large-n-limit-matrix-fields

Problem

The large-

N

limit. Let

\lambda = cg^2

, where

c

is a number of order one. Now consider evaluating the path integral, without sources, as a function of

g

and

N

Z(g,N) = e^{iW(g,N)} = \int \mathcal{D}B\, e^{i\int d^d x\, \mathcal{L}} , \tag{80.18}

where

W(g,N)

is normalized by

W(0,N) = 0

. As usual,

W

can be expressed as a sum of connected vacuum diagrams, which we draw in the double-line notation. Consider a diagram with

V_3

three-point vertices,

V_4

four-point vertices,

E

propagators or edges, and

F

closed single-line loops or faces. a) Find the dependence on

g

and

N

of a diagram specified by the values of

V_3

V_4

E

, and

F

. b) Express

E

for a vacuum diagram in terms of

V_3

and

V_4

. c) Recall, derive, or look up the formula for the Euler character

\chi

of the two-dimensional surface of a polyhedron in terms of the values of

V \equiv V_3 + V_4

E

, and

F

. The Euler character is related to the genus

\mathcal{G}

of the surface by

\chi = 2 - 2\mathcal{G}

;

\mathcal{G}

counts the number of handles, so that a sphere has genus zero, a torus has genus one, etc. d) Consider the limit

g \to 0

and

N \to \infty

with the 't Hooft coupling

\bar{\lambda} = g^2 N

held fixed (and not necessarily small). Show that

W(\bar{\lambda}, N)

has a topological expansion of the form

W(\bar{\lambda}, N) = \sum_{\mathcal{G}=0}^{\infty} N^{2-2\mathcal{G}}\, W_{\mathcal{G}}(\bar{\lambda}) , \tag{80.19}

where

W_{\mathcal{G}}(\bar{\lambda})

is given by a sum over diagrams that form polyhedra with genus

\mathcal{G}

. In particular, the leading term,

W_0(\bar{\lambda})

, is given by a sum over diagrams with spherical topology, also known as planar diagrams.