Gauge invariant actions for string models Page: 3 of 38
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If on the other hand we consider a string propagating in space time we see
a very different picture. The string world sheet is a two dimensional manifold,
and the simple requirement that it be smooth considerably restricts its topology.
If the manifold is to represent the propagation of a closed oriented string for
example, it is completely classified by giving the number of handles and the
number of boundaries (corresponding to incoming and outgoing strings). This
leads one to suspect that a perturbation expansion of a theory of interacting
strings would have only one diagram in each order. This is indeed correct, though
as we shall see, these unique dual diagrams are related to ordinary Feynman
diagrams in a manner reminiscent of the relation between the latter and the
diagrams of old fashioned time ordered perturbation theory..
At the present time, the only complete derivation of the dual Feynman
rules which ] am about to present is in the so called light cone gauge. A theory
of interacting strings is constructed in infinite momentum frame Hamiltonian
formalism. The perturbation series for this Hamiltonian is worked out and then
one secs how to add up the various terms in a given order to give a single dual
diagram. To be frank, a complete and rigorous derivation of the dual diagrams
h'$s only been constructed (including such details as numerical factors) through
one loop order. Lorcnts invariance becomes obvious only at the end of the calcu-
lation, and other fabulous properties of string models such as duality and general
coordinate invariance, are not obvious at all. It is tempting to regard the infinite
momentum frame siring theory as a gauge fixed version of a beautiful action
which manifests all of these wondrous properties at a glance. In later chapters
we will review the attempts that have been made to realize this dream.
The string Feynman rules are written in terms of a conformally invariant
two dimensional field theory, which is also invariant under two dimensional gen-
eral coordinate transformations. For the expansion around flat ten dimensional
space this field theory is given by a Lagrangian first written by Brink, DiVecchia,
Howe,1,1 Deser and Zumino161 and explored by Polyakov.1"’1
C -- y/ggae
dia
(1)
String S matrix elements are given in terms or expectation values of certain gen-
erally coordinate invariant, conformally invariant operators (called vertex oper-
ators) in this field theory. This gives us the tree level S matrix. As usual the
expectation values can be written as Euclidean functional integrals. The fields in
these functional integrals are defined on the plane, or equivalently, the Ricmann
sphere. L loop corrections to the amplitudes are given by the same functional
integral on a surface with I, handles. (These are the rules for closed oriented
strings. Open and non-orientable strings have slightly more complicated rules.)
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Banks, T. Gauge invariant actions for string models, article, June 1, 1986; Menlo Park, California. (https://digital.library.unt.edu/ark:/67531/metadc1066253/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.