Investigating the Ultraviolet Properties of Gravity with a W(20)

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

Notethattheoverallsignofthesecondterminthesecondlineof(31)isnegativewhend>2.ThisisthefamousproblemoftheunboundednessoftheEuclideanEinstein–teron,wewillneedthetraces:

tr1=d(d+1)

RµνRµν+3

d 22d 5d2+8d+4;trW=d(d 1)

Onthed-dimensionalspherewecanwrite

U=1 νµ¯µ 2δµgC RννC.R Pd 2

dR .d(d 1)

Then,usingthesecondlineof(31),wehave

Γk=(2)Z

d(d 1)R d 2

dR .

(33)

Wewillnowdiscussseparatelyvarioustypesofcuto schemes.

A.Cuto oftypeIa

Thisistheschemethatwasusedoriginallyin[8].Itisde nedbythecuto term

1√ Sk[hµν]=ghµνRk( 2)µνρσhρσ dx

dZ

Z

22=ZK tRk( )+ηRk( ).

dt(37)