ATTEMPT AT A SYNTHESIS
10.4 Extrinsic Geometry of Strings
283
In this section we shall discuss critical exponents of strings—a subject
intimately connected with their extrinsic geometry. Again, this problem
is not solved and we can only show an approach to it, and describe
several options, which we have in string theory.
The most interesting critical exponent is defined as follows. We start
from the Nambu-Goto action (or its fermionic extension):
fioA(S) = fio
9ab
daX-di,x
(10.81)
and try to choose the bare surface tension /Xq in such a way that the
physical surface tension /x can be defined by:
C i(0 = Z e x p ( - H o
Sc
e x p ( - H A ^ i„ ( Q )
(10.82)
(here C is a boundary loop and ^min(0 is an area of the minimal
surface bounded by this loop). The quantity /x(/Xq) is by definition the
physical surface tension. At the critical value of /Xo, jUocr we expect that:
( g o - go e rf
where a is the critical exponent to be determined. One can relate many
interesting quantities to this exponent.
What determines a? The answer is different for bosonic and fermionic
strings. Let us begin with the former. We have to ask ourselves at first,
whether the Nambu term in the action, which is a kind of a cosmological term is the only relevant one in the continuum limit. One would
expect that the Einstein term can be important as well, since the
Newton constant is dimensionless in two dimensions. It is commonly
known, however, that in this case the Einstein term is just the Euler
character of the manifeld which is presumed to be fixed. Nevertheless,
for string theory a dimensionless term in the action does exist. It is
formed out of the extrinsic curvature of the surface and is defined as
follows. Let us introduce the second fundamental form
given by the
equation:
d^dt,x = r/i, d,x + /Cife/i,.
(n^nj) = Sip (rti-d^x) = 0
i= l ,...,^ - 2
(10.83)