Quantum Strings and Random Surfaces
9.4 General Theory of Random Hypersurfaces
In the previous sections we discussed one dimensional curves randomly immersed into ^-dimensional Euclidean space. This problem was equivalent to the problems of Brownian motion and (after analytic continuation to Minkowski space) to the quantum theory of a free relativistic particle. It was hardly possible to get any new results in this field since it has been completely investigated for many years by classical mathematics and quantum physics. However, we have devel
oped an approach which is readily generalizable to the case of an n- dimensional hypersurfaces immersed into ^-dimensional space.
This problem is of great interest for both physics and mathematics.
At the same time, the case n > 1 is incomparably harder than that of n = 1. Some incomplete success has been achieved for n = 2 and will be discussed in later sections. Here we shall develop the general formalism for any n up to the furthest point possible at present.
Let US consider an immersion of the hypersurface described by the functions:
(9.71) The action must depend on in such a way that it is invariant under the transformation of diffeomorphisms:
or, explicitly:
= a = U ...,n
where the functions /" have to satisfy the condition:
= det > 0
(9.72)
(9.73) This claim follows from the fact that x = x{^) and x = x(/(^)) represent the same hypersurface, differently parametrized.
When we are looking for a continuous theory of surfaces, we have to start from actions containing the minimal number of derivatives. If a theory with such an action is renormalizable, then all the terms with higher derivatives are irrelevant. Sometimes in order to achieve renor- malizability one has to include higher terms. These questions we shall discuss later.
A covariant expression with the minimal number of derivatives is just the hypervolume A of our hypersurface. Therefore, the simplest action which can be constructed is the following:
5[x(0] = m"oJd"«^(^))^/^
h(i) ^ detllMOII M O = ^.x^ ^^x.
In principle, one can add some higher derivative terms like d"i R(h)h^^^
(9.74)
Si = Cj
S2 = C2 d"(^ {A(h)xyh1/ 2 (9.75) etc.
(Here R(h) is the scalar curvature computed with the metric M and A{h) is the corresponding Laplace operator). In each case special
analysis is needed in order to determine the relevance of those terms. It is reasonable, however, to begin with the simplest action (9.74) and to postpone the question whether renormalization produces higher deriv
ative terms.
The amplitude for a surface with a given boundary can be written as:
G(c{s)) =
m o 'exp - m j
m o expl -mñ 9x{^)6{d,x d,x - h j (9.76) Here the boundary of the surface x{0 is an n — 1-dimensional hyper
surface parametrized by the functions ..., i). To give a precise formulation for the boundary condition to the functional integral (9.76) one has to postulate that the topology of the (^-space is identical to the topology of the surfaces which we consider. This is the necessary condition under which we can consider as smooth functions. If the boundary in c^-space is determined from the equations
= (9.77)
then the boundary conditions for the integral (9.76) are
x(«5)) = c(s) (9.78)
The measure is, just as in the case of paths, the measure on a coset space obtained by factorising all functions jc((J) by the group of diffeomorphisms / (c^). In other words it is an invariant measure on the space of gauge orbits where the gauge transformations in this case are induced by the changes
í - > l = / ( ¿ ) . (9.79)
For the invariance of our integral we have to restrict the possible f{<0 not only by (9.73) but also by the condition that they do not move the boundary points:
/( i( s ) ) = ¿(S) (9.80)
The result of integration, G[c(s)], is invariant under the reparametriza- tion of the boundary:
G[c(5)] = G[c(a(s)] (9.81)
(where:
{a'(s^ ..., 5"'^), I 1}
is a diffeomorphism).
As we have done before, let us begin with the computation of the quantity
^ic(sX = (9.82)
which is equal to the number of immersions of a surface with fixed intrinsic metric.
The use of the Lagrange multiplier representation gives:
A + iao Jf[c, /i] = exp(
X d^x-d^x (9.83)
The norms in functional spaces defining the volume elements in this functional integral are given by;
WôxiOf = \h^^\ôx(OŸ d"i
X d"(i
(9.84)
In deriving these formulas we have used, apart from locality and general covariance, the claim that the distance in functional space has to be invariant under
x ( 0 ^ x { 0 + a(i) A“'’( 0 ^ A “*(0 + c‘'‘(0
(9.85) where a and c“'’ are arbitrary. The invariance (9.85) of the measure ensures the possibility of using the equations of motion for x and A, which are derived just by the replacements (9.85) in the action.
It appears to be convenient to decompose
(9.86) with
U i ) / “‘(0 = 0.
T h e in teg ra l (9 .8 3 ) ta k es th e form : expl n
dgX df,x + d^x ^¿jc) Jf[c, /z] =
(9.87)
In order to continue the calculation we have to conjecture (and check it later) that the correlation lengths of and are small in comparison with the size of our region. If this is true, then, as we have seen in case of paths, these quantities can be replaced by their mean values. On the grounds of general covariance we have:
<a(i)> = a + CiR(0 +
---< r \ o > = 0
(9.88iz)
where à is an unknown constant, being of the order of A" and R { ^ is the scalar curvature, computed with the metric Equation (9.88) reflects the fact that ol{^) is a scalar, while is a traceless tensor.
We shall be interested in such metrics h^b which are slowly varying on the cut-ofT scale, or, in other words with R{^) A^, and we can neglect the second term in (9.88).
After that we obtain the following expression for jr[c , K]:
jT[c, h] = exp^na d”(^
X J^ x ( i ) e x p ^ \ d ^ x • dj , x ( 9 M b )
Before going further, we have to check whether our conjectures concerning the “freezing” of the Lagrange multipliers are correct.
Let us begin with fluctuations of a. Introducing:
a ( 0 = a-(l + m ) ) (9.89)
GAUGE FIELDS AND STRINGS
where we have accounted for the fact that jS interacts with the x-field through the Lagrangian:
j
A straightforward estimate of (9.91) gives:
B(q^) = c A V 4- 0(^VA"))
(9.92)
(9.93) If the constant c in (9.93) does not vanish, then the correlation length for the j5-field, which is determined by the singularities in ^-space, is of the order of A“ ^ Therefore, in the generic case c / 0, we can neglect the influence of jS-fluctuations since:
(9.94) where A is the volume of our object. In the presence of nontrivial our computation indicates that we have a term in the induced action
-f •• (9.95)
which supresses fluctuations of Turning to the case of /-fluctuations, by similar arguments we find the most singular term in the correspond
ing action:
5 „ [/ ] = d ■ A"(dt)- ^ j d - i (9.96a)
which for d # 0 indicates the irrelevance of /-fluctuations.
Let us notice at this point that the question of whether c and d may be taken to have generic values or whether one should apply the condition that either of them be zero is far from trivial. There could exist different continuum limits for string theory, the simplest one obtained without extra conditions on c and d, while the others require fine tuning of these constants. One has to decide what kind of possible continuum limits have the desired properties and correspond to gauge theory. At present, this question is unsolved, and we shall mainly investigate the generic continuum theory, keeping in mind other options.
In the generic case we have shown that the fluctuations of the Lagrange multiplier may be dropped, and we end up with the following
QUANTUM STRINGS AND RANDOM SURFACES
expression (after trivial change of scale):
175
K t iO '] = exp A
^x((^)exp( — d,x-d^xd"n (9.966)
X(i(s)) = c(s)
The Green function for the contour c(s) is obtained by integration on
^ K t
^ab’
G[c(5)] = exp - f i
X j*^ac(i) exp^ — J d^x • d^^x (9.97)
(where fi is the critical parameter).
Since expressions like (9.97) will form the basis of our further discussion, it is worthwhile to present another derivation of it. Let us consider the integral:
j m , , txp(^-n j d " ^ - j (9.98) where QabiO is some tensor. The integral in (9.98) is supposed to be covariantly regularized. That means that we can compute it by the following procedure. First find the saddle point of the action (9.98):
4 d"i + d"i^
+ 1 j h'i\h^“g^,h‘“’dK, + g,,0h^“) d"^ = 0 (9.99) This equation gives the position of the saddle point and the value of the action:
(9.100) If we consider small fluctuations near this saddle point, we notice that they are nonpropagating owing to the absence of derivatives in (9.98).
More precisely, this means that they have a correlation length propor
tional to the inverse cut-off. Therefore their leading correction to the effective action must be local, and on the basis of general covariance must have the form (9.100). This is again the demonstration of our general rule that fields with short range correlations can be replaced by their mean values. In our case covariance dictates that these mean values are given by (9.100).
So, by the /i-integration of (9.98) we have recovered the action (9.74), provided that = d^x • df,x. This proves the equivalence of (9.97) to (9.76). Again, as in our first derivation of this equivalence, we have assumed the generic situation, i.e. that no divergent constants are zero.
Whether this continuum limit is what we are interested in must be investigated separately in each particular case. In the next section we shall show how to compute (9.97) in the case n — 2 and what kind of physics is described by it.