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arXiv:hep-th/9611107v1 14 Nov 1996

OUTP-96-68P

hep-th/yymmxxx

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

Abstract

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.

∗leith@thphys.ox.ac.uk

†kogan@thphys.ox.ac.uk

‡kmlee@sun1.phy.cuhk.edu.hk. Present address: Department of Physics, The Chinese University of

Hong Kong, Shatin, N.T., Hong Kong.

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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 [1] 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

theory.

In this letter we examine space-time compactification in topological membrane (tm)

theory [2]. 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) [3]. 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 [6], 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 [7] 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 [8] on the string worldsheet [9]. 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.

1

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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:

A(p1,...,pN) ∼

?N

i=1

?

Vi(pµ

i)

?

,(1)

where Vi(pµ

pµ

i) is the vertex operator corresponding to a particle of type i with momentum

i. Averaging is done using the path integral:

?

DXµ(σ)exp

?

−

1

4πα′

?

d2σ ηαβ∂αXµ∂βXµ

?

,(2)

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

4πα′

?

d2σ(∂αθ)2, (3)

where θ ∈ [0,2π). The resulting mass spectrum, obtained from the on-shell condition

2(L0+˜L0− 2) = 0 (see [4] for details), is given by

α′M2= −4 + 2(NR+ NL) + m2α′

R2+ n2R2

α′

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 [5]).

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

[2]. Using the normalizations of [10], it is known that the abelian tmgt

STMGT= −1

4e2

?

MFµνFµν+

k

8π

?

MǫµνλAµ∂νAλ,(6)

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defined on a three-manifold M, induces the chiral string action

S =

k

8π

?

∂M∂zθ∂¯ zθ(7)

where θ is the pure gauge part of Aµon the boundary ∂M. When M is the filled cylinder

depicted in Figure 1, its two boundaries represent the left- and right-moving sectors of the

string worldsheet. Worldsheet vertex operators can be constructed if we allow charged

matter (either bosonic or fermionic) to propagate inside the topological membrane. The

vertex operator

Vq(z, ¯ z) = Vq(z)Vq(¯ z) = eiqθ(z)eiqθ(¯ z)

(8)

can be obtained from the bulk tmgt as the open Wilson line

Wq[C] = exp

?

iq

?

CAµdxµ

?

(9)

where the contour C connects left and right boundaries. Since the vector potential be-

comes pure gauge on the boundary, it is easy to see that the Wilson line (9) coincides

on the left and right boundaries with the holomorphic and anti-holomorphic parts of the

vertex operator (8), respectively. Moreover, the charge along the Wilson trajectory is to

be interpreted as giving the momentum of the corresponding vertex operator. This sug-

gests that the N-point function (1) should be related to the correlator of N Wilson lines

in tmgt. For the simplest case, N = 2, the short-distance operator product expansion

gives

?Vq(z1)V−q(z2)? ∼ (z1− z2)−2∆,(10)

where ∆ is the scaling dimension of the chiral vertex operator Vq(z). The corresponding

three-dimensional picture is that of a charged particle-antiparticle pair propagating inside

the membrane. The two-point function (10) should be related to the correlator of two

Wilson lines in tmgt which, in the infrared (Chern-Simons) limit, is simply

?W[C1]W[C2]? = exp

?

−4πi(q2/k)γ[C1,C2]

?

(11)

where γ[C1,C2] is the linking number of the curves C1and C2. The relationship between

the two- and three-dimensional correlation functions becomes clear if we adiabatically

rotate V (z1) around V (z2). This induces a phase factor e−4πi∆in (10). A similar phase

factor appears in (11) due to the linking of the two Wilson lines (see Figure 1). Since

∆ = q2/k(12)

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1

-qq

2z

2z

1z

z

Figure 1: String scattering amplitude as the correlator of Wilson lines in the bulk tmgt.

Adiabatic rotation of the chiral vertex operators V (z1) and V (z2) corresponds to the

linking of the two Wilson lines.

is the transmuted spin of the charged particle due to its interaction with the Chern-Simons

gauge field, this establishes the equivalence between the anomalous spins in the two- and

three-dimensional theories. Note that the spectrum of anomalous dimensions can also be

obtained perturbatively from the tmgt [11]. Vertex operators representing higher-spin

states have the general form [4]:

Vq(z, ¯ z) ∼

?

j

?

∂j

zθ

?mjeiqθ(z)?

k

?

∂k

¯ zθ

?mkeiqθ(¯ z),(13)

where the number of left- and right-moving excitations are given by NL=

NR=?

we can construct the pre-exponential (spin) factors in tm theory using the boundary value

of Aµand its derivatives. For details, we refer the reader to [2].

?

jjmj and

kkmk, respectively. Since the vector potential Aµis pure gauge on the boundary,

We now turn our attention to the topological membrane description of string com-

pactification on S1. Comparing the boundary action (7) with the compact part of the

string action (3) shows that the compactification radius R is related to the Chern-Simons

coefficient k by k = 4R2/α′. Moreover, if θ ∈ [0,2π) then we must take the gauge group

in (6) to be compact U(1). As we shall see, there is a crucial difference between compact

and non-compact tmgt which enables us to construct string winding modes in the for-

4