arXiv:hep-th/9611107v1 14 Nov 1996
String Winding Modes From Charge Non-Conservation
in Compact Chern-Simons Theory
Leith Cooper,∗Ian I. Kogan†and Kai-Ming Lee‡
Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, UK
In this letter we show how string winding modes can be constructed using
topological membranes. We use the fact that monopole-instantons in compact
topologically massive gauge theory lead to charge non-conservation inside the
membrane which, in turn, enables us to construct vertex operators with differ-
ent left and right momenta. The amount of charge non-conservation inside the
membrane is interpreted as giving the momentum associated with the string
winding mode and is shown to match precisely the full mass spectrum of com-
pactified string theory.
‡email@example.com. Present address: Department of Physics, The Chinese University of
Hong Kong, Shatin, N.T., Hong Kong.
Space-time compactification is of central importance to string theory: it provides a
mechanism for reducing higher-dimensional string theory down to four dimensions and is
an alternative to the Chan-Paton method for introducing isospin. Compactification, there-
fore, is a key to performing meaningful four-dimensional string phenomenology. Space-
time compactification also leads to the appearance of certain dualities  between string
theories at strong and weak coupling and therefore any new insight into the compact-
ification mechanism may be helpful in finding a non-perturbative description of string
In this letter we examine space-time compactification in topological membrane (tm)
theory . In particular, we consider the simple example of string theory compactified on
a circle, which can be described by a compact U(1) topologically massive gauge theory
(tmgt) . The main difference between string theories with target-space containing the
(non-compact) line R1or (compact) circle S1is the existence of winding modes in the latter
case, corresponding to the string wrapping around the circle (see, for example, [4, 5]).
The major problem with introducing winding modes in tm theory is the fact that the
corresponding worldsheet vertex operators must have different left and right momenta. As
we shall discuss in detail later, worldsheet vertex operators are represented in tm thoery
by Wilson lines of charged particles propagating between the left and right boundaries
(which represent the left and right string worldsheets) of the topological membrane. The
charge along one of these Wilson lines is interpreted as giving the momentum of the
corresponding vertex operator. If the vertex operator has different left and right momenta,
then the charge along the Wilson line connecting left and right membrane boundaries
must change accordingly. So, in order to construct string winding modes, we need some
process which leads to charge non-conservation in compact tmgt. Fortunately, precisely
this type of process was discussed by Lee , who considered the effect of monopole-
instantons in compact Chern-Simons gauge theory. The presence of monopole-instantons
in compact U(1) tmgt was also discussed in  using a Hamiltonian approach, where it
was also found that monopole-instantons induce a phase transition in the bulk matching
precisely the bkt phase transition  on the string worldsheet . The purpose of this
letter is to explicitly construct the string winding modes in tm theory using the fact that
monopole-instantons lead to charge non-conservation in compact tmgt. We interpret the
amount of charge non-conservation as giving the momentum of the string winding mode
and show that the resultant spectrum matches precisely the mass spectrum for string
theory compactified on a circle. We then generalize our argument to include the case of
compactification on a D-dimensional torus.
We first recall how string scattering amplitudes can be expressed in terms of corre-
lation functions of some 2d conformal field theory. For simplicity, we consider closed
bosonic strings in the critical dimension which means that we may neglect 2d gravity.
The scattering amplitude has the general form:
i) is the vertex operator corresponding to a particle of type i with momentum
i. Averaging is done using the path integral:
where α,β = 1,2 and µ = 1,...,26. Compactification on the circle S1proceeds by
identifying X26= X26+ 2πRn where R is the radius of S1and n is the number of times
the string winds around the circle. The compact part of the string action becomes
SXY = −R2
where θ ∈ [0,2π). The resulting mass spectrum, obtained from the on-shell condition
2(L0+˜L0− 2) = 0 (see  for details), is given by
α′M2= −4 + 2(NR+ NL) + m2α′
withNL− NR= nm. (4)
NL and NR are the numbers of left- and right-moving excitations and m is an integer
describing the allowed momentum eigenvalues in the compact direction. The last two
terms in the spectrum give the contributions of the compact momentum and winding
energy to the 25-dimensional mass. It easy to see that there is a symmetry:
R ↔ α′/Randm ↔ n,(5)
which leaves the spectrum (4) invariant. This is the famous T-duality of compactified
string theory (for a review, see ).
Before proceeding to show how the compactified string spectrum arises in tm theory,
we first recall how to obtain string scattering amplitudes (1) from topological membranes
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