Feynman Path Integral Formulation

92 3 Gravity in 2 + ε Dimensions
order in the ε expansion these authors then find
β(G) =ε G − (9 − c)G 2 + ... (3.112)
and therefore a non-trivial ultraviolet fixed point close to two dimensions at G c =
ε/(9−c), where c is the central charge of the superconformal matter multiplet, such
as c free scalars and spinor fields. Taken at face value, these result would suggest
that, besides ordinary Einstein gravity, also N = 1 supergravity could have a nontrivial
strong coupling phase, which appears for G > G c .
3.6 Phases of Gravity in 2+ε Dimensions
The gravitational β-function of Eqs. (3.106) and (3.110) determines the scale dependence
of Newton’s constant G for d close to two. It has the general shape shown
in Fig. 3.7. Because one is left, for the reasons described above, with a single coupling
constant in the pure gravity case, the discussion becomes in fact quite similar
to the non-linear σ-model case.
For a qualitative discussion of the physics it will be simpler in the following
to just focus on the one loop result of Eq. (3.106); the inclusion of the two-loop
correction does not alter the qualitative conclusions by much, as it has the same
sign as the lower order, one-loop term. Depending on whether one is on the right
(G > G c )orontheleft(G < G c ) of the non-trivial ultraviolet fixed point at
G c = d − 2
β 0
+ O[(d − 2) 2 ] , (3.113)
(with G c positive provided one has c < 25) the coupling will either flow to increasingly
larger values of G, or flow towards the Gaussian fixed point at G = 0, respectively.
In the following we will refer to the two phases as the strong coupling and
weak coupling phase, respectively. Perturbatively one only has control on the small
Fig. 3.7 The renormalization
group β-function for gravity
in 2 + ε dimensions. The
arrows indicate the coupling
constant flow as one approaches
increasingly larger
distance scales.
β (G)
Gc
G