- BUSCHER DUALITY TRANSFORMATION 49
satisfy(1.82)whereU¢~ b¢+f.
REMARK 1.82. Note that b\7 af = tt\7 af = 0.
The functions b ¢ and U¢ are called dilatons and the transformation from
b¢ to U¢ is called the dilaton shift. The Buscher duality transformation
takes solutions of the modified Ricci fl.ow with dilaton b ¢ to solutions of the
modified Ricci fl.ow with dilaton U¢.
LEMMA 1.83 (Buscher duality preserves modified Ricci tensor after dila-
ton shift).
U Ra/3 + 2 U\7 a U\7 /3 U¢ = - ~2 ( b Raf3 + 2 b\7 a b\7 /3 b cP) '
URai+2 U\7a U\7i U¢= bRai+2 b\7a b\7i b¢=0,URij+2 tt\7i tt\7j H¢= bRij+2 b\7i b\7j b¢.REMARK 1.84. If log A= -~f, then
b ~j + 2 b\7i b\7j b<P = ~j + \7i\7jf - ~\7d\7jf + 2\7i\7j b¢,
q
1
tt Rij + 2 tt\7i tt\7j U¢ = Rij - \7i \7jf - -\7d\7jf + 2\7i\7j U¢.
qSuppose gt b 9ab = -2 b Rab so that b ¢ = 0. By Exercise 1.80, taking the
limit as q--+ oo, the equation for the metric 9ij (t) on the base manifold is
8
at9ij = -2 (Rij + \7i\7jf).Note that dμ (bg) = e-f dμ/\dy^1 /\ · · · /\dyq. The effective action of (bg,b ¢)
is (e.g., see Alvarez and Kubyshin [2], equation (26))
S (bg,b cP) = fMxP ( R (bg) + 41\7 b¢12) e-2 "¢dμ eg).
=JM ( R+2flf-q; 1 l\7fl2 +41\7 b<Pl2) e-2 "¢e-fdμ,
and similarly for the dual pair (ttg,tt ¢). (Note that by taking ¢b = 0, we
have S (b g, 0) = f MxP R (b g) dμog·) Buscher duality preserves the effective
action: