Cosmic strings in string perturbation theory

Fortschr. Phys. 59, No. 7 – 8, 762 – 768 (2011) / DOI 10.1002/prop.201100025

Cosmic strings in string perturbation theory

Dimitri Skliros1,2,∗1 Department of Physics and Astronomy, University of Sussex, Brighton, East Sussex BN1 9QH, UK2 Cripps Center for Astronomy and Particle Theory, School of Physics and Astronomy,

University of Nottingham, University Park, Nottingham NG7 2RD, UK

Received 11 February 2011, accepted 13 February 2011Published online 7 March 2011

Key words Cosmic strings, covariant vertex operators, covariant coherent states, string wavepackets.

Fundamental cosmic superstrings are defined in string perturbation theory in terms of vertex operators,which are loosely identified with arbitrarily excited macroscopic string states. We construct explicit normalordered cosmic string vertex operators, which are given in terms of covariant coherent string states. We alsowrite down the general expression for arbitrarily excited mass-eigenstate covariant vertex operators, anddiscuss the construction of string wavepackets.

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Overview

Cosmic superstrings are macroscopic and massive oscillating lines of energy that may have been producedin string theory realizations of the early universe, e.g. at the end of D3-D3-brane inflation, in the con-text of warped throats (KLMT) [1] or large extra dimensions [2]. The role of cosmic superstrings can beplayed by the fundamental or F-strings themselves (that define the string theory perturbatively), by the dualD-strings (or higher dimensional Dp-branes with p − 1 directions compactified and 1 non-compact direc-tion), and there are also more exotic possibilities [3]. Given an initial distribution of cosmic superstrings,intercommutations/reconnections [4–6], string decay [7–9] and the possible presence of junctions [10], aswell as cosmological factors (the cosmological expansion, the distribution of matter/energy) determine thesubsequent cosmological evolution of cosmic superstrings. The upcoming gravitational wave experiments(e.g. Advanced LIGO and LISA) will be, it seems, sufficiently sensitive to detect a wide range of sig-nals from such objects [11]. The possible presence of cosmic superstrings thus provides an observationalsignature for string theory and it has therefore become of utmost importance to make sound and robustpredictions about what we expect to observe.

The vast majority of work has (i) been based entirely on classical string evolution, thus obscuring the sig-natures unique to string theory (F-strings are only defined quantum-mechanically); (ii) backreaction effectsare systematically neglected, even though it has been shown [12] that such effects can be important evenfor order of magnitude estimates. Backreaction effects are also widely believed to be responsible for thesmall scale structure of strings, which may determine the long sought after loop production scale [12–14].Furthermore, (iii) classical or supergravity computations cannot easily account for the possibility of mas-sive loops being produced, although the latter [15] may capture some strong coupling string dynamics [16].This might be expected to become significant at regions on the string where the classical evolution becomessingular (i.e. where the determinant of the embedding metric vanishes), e.g. close to cusps. The obstacles(i)-(iii) may be resolved in the context of string theory, where cosmic superstring asymptotic states are(at least perturbatively) accurately described in terms of vertex operators, V (z, z), and the correspondingcosmic superstring evolution can be rephrased in terms of S-matrix elements.

∗ E-mail: [email protected]

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Fortschr. Phys. 59, No. 7 – 8 (2011) 763

The S-matrix approach to cosmic string computations, pioneered in [5,7,17–19], naturally incorporatesbackreaction effects (as first noticed in [7]). The above references however dealt with relatively simple andunrealistic cosmic string states, and in what follows I discuss some recent developments in the constructionof arbitrarily excited cosmic strings – such a construction is expected to resolve obstacles (i)-(iii) mentionedabove, and may possibly lead to the first detections of cosmic superstrings. For further details the reader isreferred to the original articles [20] and [21].

2 General asymptotic states

Asymptotically free states in string scattering experiments are described in terms of vertex operators. Theseare local functionals on the worldsheet, that are composed of the fields present in the string theory; in thebosonic theory these are the spacetime embedding, Xμ(z, z), (with μ = 0, . . . 25) and the worldsheetmetric, ds2 = 2gzzdzdz.1

Let us consider the particular case of mass eigenstates and concentrate on the closed string. It is wellknown [23,24], but see also [25] and [20], that a complete set of covariant mass eigenstates can be generatedby acting with DDF operators, Ai−n, Ai−n, on a physical vacuum, e.g. eip·X(z,z) with p2 = 2, for positiveintegers n. Explicitly,

Ain =

∮dz ∂X i(z) einq·X(z) and Ai

n =∮

dz ∂X i(z) einq·X(z), (1)

with dz ≡ dz/(2π); these satisfy [Ain, Aj

m] = nδijδn+m,0, with qμ a null vector, q2 ≡ 0, such thatqμδi

μAin = 0, [Lm, Ai

n] = 0 (with Lm the Virasoro generators [22]) for all n, m (similarly for Ain, Lm).

If eip·X(z,z) satisfies the Virasoro constraints [22], that is L0 · eip·X ∼= eip·X and Ln>0 · eip·X ∼= 0 (andsimilarly for the antiholomorphic sector), general mass eigenstate vertex operators are given by:

V (z, z) = CU(z)U(z), (2)

which is guaranteed to be physical and covariant when [20],

U(z) = ξi...jAi−n1

. . . Aj−ng

eip·X(z)

∼= : ξi1...ig

�g/2�∑a=0

∑π∈Sg/∼

a∏�=1

δiπ(2�−1)iπ(2�) Snπ(2�−1),nπ(2�)

g∏q=2a+1

Hiπ(q)nπ(q) eik·X(z) :

(3)

with �g/2� indicating that the maximum value of a saturates the inequality a ≤ g/2. The quantities Hin(z)

and Sn,m(z) are defined in terms of elementary Schur polynomials, Sn(z),

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Hin(z) = piSn(z) +

∑∞�=1

i(�−1)! ∂�X i(z)Sn−�(z),

Sm,n(z) =∑n

r=1 rSm+r(z)Sn−r(z),

1 An arbitrary worldsheet can be parametrized by the metric ds2 = 2gzzdzdz, and because the theory is conformal [22], wecan always locally further set, say, gzz = 1/2. We use Euclidean coordinates where z = e−i(σ+iτ), z = ei(σ−iτ) [22] withσ ∈ [0, 2π), and −∞ ≤ τ ≤ ∞. Furthermore, we use units where α′ = 2

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764 D. Skliros: Cosmic strings in string perturbation theory

which in turn, writing as ≡ − 1s! inq · ∂sX(z) (and

∮0dw/w = i), read:

Sn(z) = −i

∮0

dw w−n−1 expn∑

s=1

asws, (4a)

=∑

k1+2k2+···+nkn=n

ak11

k1!. . .

aknn

kn!(4b)

with the properties Sn<0 = 0 and S0 = 1. In the second equality in (3) we have performed the normalordering [20], which is similar to that associated to the usual construction [26].

Note that kμ = pμ − Nqμ is the momentum (with N =∑

� n� the mass level) of V , so that k2 =2 − 2N . We are to require that: p2 = 2, p · q = 1, and q2 = 0. The polarization tensor ξij... (withi = 1, . . . , 24), satisfies: ξ...i...q

i = 0, and transforms under irreducible representations of SO(25) (formassive states). (There is a similar expression for U(z), with ξi...j , Ai

n and eip·X(z) replacing ξi...j , Ain and

eip·X(z) respectively.) The combinatorial normalization constant, C, contains a factor of 1√n

for every Ai−n

and factors of 1√μn,i!

, (with μn,i the multiplicity of Ai−n), with similar factors for the anti-holomorphic

sector:

C ≡(∏

r

nr

∏n,i

μn,i!)−1/2

×( ∏

s

ns

∏n,i

μn,i!)−1/2

, (5)

in which case the normalization is, on account of the onshell constraints and ξ∗ij...ξij... ≡ 1,

⟨V ′|V ⟩

= T (2π)δ(p′+ − p+)(2π)24δ24(p′ − p)δ{n′

i},{ni}, (6)

where we define∫

dX+0 = 2πδ(0) ≡ T (from the constant zero mode in X+(z, z) [22]), and it is to be un-

derstood that T → ∞. The normalization (6) holds when we choose q+ = qi = 0 (so that p+ = −1/q−),a choice that can be made while still spanning the full the phase space2. Note that the right-hand-side of(6) is independent of N . Furthermore, we may choose q+ = qi = 0 universally in a string amplitude,even though (in general) q− will differ from one state to the next.3 Finally, note also that the vertex op-erators (2) form a complete set, given that they are in one-to-one correspondence with lightcone gaugestates [22], |V 〉 = Cξij...ξkl... α

i−n1αj−n2

. . . αk−n1αl−n2

. . . |0, 0; p+, pi〉, with |0, 0; p+, pi〉 the oscillatorand momentum space vacuum state. The covariant and lightcone gauge states V and |V 〉 respectively areconjectured [20, 25] to share identical correlation functions.

The δ-function normalization (6) is not entirely satisfactory: the states V (being eigenstates of mo-mentum) will have indeterminate position and hence the physical interpretation of these states becomessomewhat unclear. This is precisely the same difficulty that appears in field theory, and the solution thereis to construct wavepackets which do have unit norm. This is of course not strictly necessary, as bothapproaches lead to (for all purposes) equivalent scattering amplitudes [27], but it is certainly at least in-structive to discuss the wavepacket construction in string theory. Consider the relativistic phase spaceintegral,

∫d26k

(2π)26 (2π)δ(k2 + 2N − 2)θ(k0). Working in lightcone coordinates k± = 1√2(k0 ± k25)

(dk− ∧ dk+ = dk0 ∧ dk25), let us redefine the integration variable:

k− = p− + N/p+, k+ = p+, ki = pi, i = 1, . . . , 24. (7)

2 Only 25 of the 2 × 26 = 52 parameters in pμ, qμ contribute to the phase space, thus enabling one to set q+ = qi = 0.3 A more extensive discussion will appear in [20]. Notice that with this choice of qμ, we have q · q′ = 0 and it is due to this fact

that [A′in , Aj

m] = nδijδn+m,0, with A′in and Ai

n the DDF operators constructed out of q′ and q respectively. In the operatorformulation, one may normalize the vertex operators by making use of this commutator.

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Fortschr. Phys. 59, No. 7 – 8 (2011) 765

This removes the N -dependence from the δ-function, δ(k2 + 2N − 2) = δ(p2 − 2), and dk− ∧ dk+ =dp− ∧ dp+. Ignoring the tachyon, so that θ(k0) = θ(p+), the Lorentz invariant phase space reads,

∫d26k

(2π)26(2π)δ(k2 + 2N − 2)θ(k0) =

∫d26p

(2π)26(2π)δ(p2 − 2)θ(p+)

=∫

R24

d24p(2π)24

∫ ∞

0

dp+

2π

12p+

,

(8)

where we have integrated out p−, so that p− = 12p+ (p2−2), the tachyon onshell condition. Next introduce

a quantity, φ(p) (defined in (10)), and construct a string wavepacket as follows (Uφ is not to be confusedwith U in (3)),

Uφ(z, z) ≡ 1√T

∫d24p

(2π)24dp+

2π

φ(p)2p+

V (z, z). (9)

One is to take the formal limit T → ∞ in scattering amplitude computations. The change of integrationvariables (7) was such that pμ coincides with the momentum of the tachyonic vacuum on which DDFoperators act, and q− = −1/p+. As the Lorentz invariant measure is independent of N , we may identifyV (z, z) with a mass eigenstate (2) or (see below) a coherent state vertex operator (11), both of which aredefined on the mass shell. On account of (6), we learn that,

⟨U

′φ|Uϕ⟩

= δ{n′i},{ni}

∫d24p

(2π)24dp+

2π

φ∗(p)ϕ(p)(2p+)2

,

∫d24p

(2π)24dp+

2π

|φ(p)|2(2p+)2

≡ 1, (10)

where the second expression is to be thought of as the definition of the function φ(p) (equivalently ϕ(p))and ensures that wavepackets have unit norm, 〈Uφ|Uφ〉 = 1.

3 Cosmic string asymptotic states

Let us now discuss how to apply the above construction to the case where the asymptotic states of interestare cosmic strings. Loosely speaking, we define:

cosmic (super)string macroscopic string state.

We could also become more specific and demand that the string state be ‘quasi-classical’, i.e. that expec-tation values should evolve classically with small uncertainties, provided these are compatible with thesymmetries of the theory. The quasi-classicality requirement may or may not be relevant for cosmic stringevolution, or it may be that string states become quasi-classical if they are macroscopic (as is often as-sumed [5]). In practice, of course, it is also necessary that this object be produced in the early universe,possibly in a phase transition after a period of inflation, as is the case for instance in the context of braneinflation scenarios of the early universe, see e.g. [28].

Appealing to our general intuition about harmonic oscillator coherent states [29], natural candidate cos-mic string states are then ‘string coherent state vertex operators’.4 The construction of covariant coherentstates in string theory has been a mystery since the 80’s [30,31], and only recently has it become clear howcoherent states can be constructed in covariant gauge [21].

4 Terminology: a better term is possibly ‘string coherent vertex operators’, or ‘coherent vertex operators’ for short.



Given that we know how to construct arbitrarily excited covariant vertex operators (Sect. 3) we proceedby analogy and write down the following candidate string coherent vertex operator,5

Vλλ(z, z) = Cλλ

∫ 2π

0

ds exp{ ∑

n>0

1n

einsλn · A−n +∑m>0

1m

e−imsλm · A−m

}eip·X(z,z) (11)

with Cλλ =[ ∫ 2π

0ds exp(

∑n>0

1n |λn|2eins + 1

n |λn|2e−ins)]−1/2

a (combinatorial) normalization con-stant. The polarization tensors, {λi

n} for n ∈ Z and i = 1, . . . , 24, are defined to satisfy, λn · q = 0,λ∗

n = λ−n, and∑∞

n=1 |λn|2 < ∞, and similarly for the anti-holomorphic sector {λin}. Given that the

DDF operators are proportional to einq−X+(z) or eimq−X+(z), with n, m ∈ Z+ (see comments below (6)),

Vλλ(z, z) is an eigenstate of pi and p+, but not of p−. The normalization is such that:

⟨V ′

λλ|Vζζ

⟩= T (2π)δ(p

′+ − p+)(2π)24δ24(p′ − p)Nλζ , (12)

with Nλζ ≡ CλλCζζ

∫ 2π

0ds exp

( ∑n>0

1n λ∗

n · ζn eins + 1n λ∗

n · ζn e−ins)

(and Nλλ = 1). These verticesform a complete set [20],

1 =1T

∫dp+

2π

d24p(2π)24

∫ (∏n,i

d2λin

2πn

)( ∏n,i

d2λin

2πn

)∣∣Vλλ

⟩⟨Vλλ

∣∣, (13)

with 1 the unit operator with respect to the Hilbert space spanned by Vλλ, so that 1 · Vλλ∼= Vλλ. On

account of (3) and (11), one can immediately write down the normal ordered coherent vertex operators.The corresponding coherent state wavepacket representation, Uφ

λλ(z, z), is obtained from (9), with

V (z, z) → Vλλ(z, z), and (10) with δ{n′i},{ni} → Nλζ . Note that Uφ

λλhas unit norm: 〈Uφ

λλ|Uφ

λλ〉 = 1.

Coherent states have classical expectation values. In the wavepacket representation, we find 〈pμ〉 ≡〈Uφ

λλ|pμ|Uφ

λλ〉 = 〈pμ〉φ − ∑

n |λn|2〈qμ〉φ, with

〈. . . 〉φ ≡∫

d24p(2π)24

dp+

2π

|φ(p)|2(2p+)2

. . . , and 〈1〉φ ≡ 1.

If φ(p) is peaked around, say, pμ0 , then 〈pμ〉φ = pμ

0 , (recall that q− = −1/p+). Note also, that L0 − L0 isthe generator of spacelike shifts, [L0 − L0, X ] = i∂σX and that vertex operators are invariant under suchshifts, (L0 − L0) · V ∼= 0. It follows that it is not possible [21, 31] for a (lightcone or covariant) state tosatisfy

⟨Xμ(z, z) − xμ

⟩=

(Xμ(z, z) − xμ

)cl, unless [21] we work in lightcone gauge and spacetime is

lightlike-compactified.6 Rather, an alternative requirement for a state with a classical interpretation is (inthe wavepacket representation):7

⟨:[Xμ(z′, z′)−xμ

][Xν(z, z)−xν

]:⟩

=∫ 2π

0

ds[X(z′eis, z′e−is)−x

]μ

cl

[X(zeis, ze−is)−x

]ν

cl(14)

where we average over σ-translations on the classical side. It is a simple exercise to verify (14) in lightconegauge: one takes X i(z, z) and X±(z, z) to be given by the standard lightcone gauge mode expansions [22].

5 The integral over s is not required in order for the Virasoro constraints to be satisfied [20], but is needed if spacetime is notcompactified in a lightlike direction. In particular, (p−L − p−R) ·V ∼= 0 only if the s-integral is included, with pμ

L =∮

dz∂Xμ

and pμR = − ∮

dz∂Xμ. Had we not integrated over s, single-valuedness of Vλλ under spacelike worldsheet translationswould require [21]: X− ∼ X− + 2π/p+.

6 This is because in lightcone gauge, the constraint analogous to (L0−L0)·V ∼= 0, reads p+(p−L − p−R)·V = (L⊥0 −L⊥

0 )·V ,and hence lightlike compactification breaks the invariance under spacelike shifts [21]. This is a gauge problem [21, 31] andsays noting about the classicality of the underlying vertex operators [21].

7 I thank to Joe Polchinski for suggesting this.


Fortschr. Phys. 59, No. 7 – 8 (2011) 767

The lightcone gauge classical evolution associated to the coherent states is then [20]:

(X i(z, z) − xi

)cl

= −i〈pi〉φ ln |z|2 + i∑n=0

1n

(λi

n z−n + λin z−n

),

(X−(z, z) − x−)

cl= −i〈p−〉φ ln |z|2 + i

∑n=0

1n

∑r∈Z

⟨ 12p+

(λn−r · λrz

−n + λn−r · λr z−n

)⟩φ,

and X+(z, z)cl = −i〈p+〉φ ln |z|2, with p− = 12p+

(p2 +

∑n(|λn|2 + |λn|2) − 2

).8 We are to enforce in

addition classical level matching [20],∑

n |λn|2 =∑

n |λn|2. It follows that by choosing λn, λn appro-priately we can construct macroscopic string states, namely cosmic strings.

A good overall consistency check is to show that the expectation value of the spin, Sμν , 9 of the covariantgauge, lightcone gauge and classical descriptions are all identical, as this would provide further supportfor the conjecture that the lightcone and covariant descriptions are identical. We would expect this to bethe case if the states (11) have a classical interpretation, and because Sμν is gauge invariant operator,[Ln, Sμν ] = 0. We indeed find [20],

〈Sij〉cov = 〈Sij〉lc =∑n>0

2n

Im(λ∗i

n λjn + λ∗i

n λjn

)= Sij

cl , (15a)

〈S−i〉cov = 〈S−i〉lc =∑m>0

∑�∈Z

⟨ 1np+

Im(λ∗

m−� · λ∗� λi

m + λ∗m−� · λ∗

� λim

)⟩φ

= S−icl , (15b)

with components involving the + direction equal to zero, and diagonal components vanishing due to anti-symmetry.

4 Conclusions

We have described the construction of arbitrarily excited mass eigenstate and coherent state vertex oper-ators in bosonic string theory and suggest that the latter may be identified with arbitrarily excited macro-scopic cosmic string loops. We discussed their covariant and lightcone gauge realization and constructedan explicit map to the corresponding classical solutions. We also discussed the construction of stringwavepackets for both mass eigenstates and coherent states. As well as being more easy to physically inter-pret, string wavepackets correspond to more realistic string states than do the corresponding momentumeigenstates, while possessing unit norm. It will be very interesting to construct the corresponding super-string vertex operators and compute scattering amplitudes associated to cosmic string states – progress inthis direction will be reported elsewhere.

Acknowledgements I primarily thank Mark Hindmarsh for earlier collaboration and valuable conversations, andDavid Bailin and Joseph Polchinski for useful suggestions. I would also like to thank the organizers of the XVIthEuropean Workshop on String Theory 2010 in Madrid, where the majority of this work was presented.

References

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dzX[μ∂Xν], with a[μν] ≡ 12(aμν − aμν).



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Cosmic strings in string perturbation theory