1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. 1-AND 2-LOOP VARIATION FORMULAS RELATED TO RG FLOW 35

so that

(17.110)

O(,aC1l,'YC1l) (-~ 1Rml

2

e-f dμ) + O(,aC2ld2J) ( ( R + 2~f -1\7 fl^2 ) e-f dμ)

=-div (~e-f\7 IRml^2 ) dμ + 2 (a, Rc+\7\7 J) e-f dμ

- IP* - l'Vf Rml^2 e-f dμ.

We obtain

(17.111) 8(,ac1J,'YC1J)F2 (g, f) + 8(,ac2J,'Y<2J)F1 (g, f)

Note that

(17.112)

= - JM IP* - l\lf Rml^2 e-f dμ + 2 JM (a, Re +\7\7 J) e-f dμ.

-o(,B(l),')'(l))F2 (g, f) + o(,B(2),')'(2))Fl (g, f) =JM IP* - l\lf Rml^2 e-f dμ ~ 0.

REMARK 17.31. If one defines (3i~) = 9ij, '/'(o) = ~' and Fo (g, f)

• JM fe-f dμ, then

(17.113) O(,aCol,'YCol)Fl (g, f) + O(,BClldll)Fo (g, f) = 0.

PROBLEM 17.32. Do any of the above formulas fit into an infinite se-

quence of formulas? One would like to obtain a monotonicity formula ex-

tending Perelman's entropy monotonicity formula.

In particular, one may wish to consider an expression of the form

(17.114)

o (,ac1l ,')'<1l) F1 (g, f) + .A ( o (f3Cll ,')'c1J) Fz (g, f) + o (,aC2l ,')'c2J) F1 (g, f))

+ .A

2

( 8(,aCll ,')'<1l) F3 (g, f) + 8(,a<2l ,')'c2J) Fz (g, f) + o (f3C3),'YC3J )F1 (g, f))

+ ... '

where (3(k), '/'(k), Fk are suitably defined for k E N and A E IR+. However it

is not clear whether or not one should introduce new fields in addition tog

and f.
Some related calculations are in Oliynyk, Suneeta, and Woolgar [145];
see also Zamolodchikov [194] and Tseytlin [182].^13 In the physics literature
there are various 3-and 4-loop calculations.

(^13) We would like to thank C. Vafa for discussions at the California Institute of Tech-
nology during January and February of 2003. The first author would also like to thank
A. Tseytlin for discussions at Ohio State University during May of 2003.