Exact treatment of D0 brane decay in two-dimensional string theory

Fortschr. Phys. 53, No. 5 – 6, 626 – 633 (2005) / DOI 10.1002/prop.200510230

Exact treatment of D0 brane decayin two-dimensional string theory

J. Teschner∗

Institut fur theoretische Physik, Freie Universitat Berlin, Arnimallee 14, 14195 Berlin, Germany

Received 17 January 2005, accepted 18 February 2005Published online 27 April 2005

We study the decay of the unstable D0-branes in the c = 1 noncritical string theory with the help of theduality with free fermionic field theory.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The study of noncritical string theories has recently seen a renaissance, initiated by the appearance of [1–3].One of the main reasons for the renewed interest in these string theoretical toy models is the observationthat nonperturbatively stable definitions of the relevant theories exist [4, 5], after all [6]. These stablenonperturbative definitions are furnished by certain field theories of free fermions which represent thedouble scaling limits of the relevant matrix models, see e.g. [7] for a review.

These developments also open the possibility to study certain time-dependent phenomena such as D-brane decay in an exactly soluble framework. This line of research is motivated by the desire to improve ourunderstanding of foundational aspects of string theory for which time-dependent phenomena represent amajor challenge. One may in particular hope to learn how to describe the final state of a decaying D-brane,and to what extend one may describe the process with the help of the usual perturbative approach to stringtheory.

Previous work on this subject is contained in [8,9]. The present note will briefly outline a new approachto this problem which allows us to go somewhat further and to clarify a number of aspects which have notyet been properly discussed in the literature. More details and further results will appear in a forthcomingpublication [10].

2 The c = 1 string as free fermionic field theory

We shall exclusively consider the type 0B c = 1 string theory [11] in the present paper. It was conjecturedin [4, 5] that this two-dimensional string theory has a dual description in terms of a nonrelativistic freefermionic field theory, generalizing previously discussed dualities between noncritical string theories andmatrix models [7]. In the next subsection we shall briefly review the main features of the type 0B c = 1string theory. Our discussion of the dual free fermionic field theory in the following subsection will be alittle more detailed since some elements of our approach will differ from previous treatments [12,13].

∗ E-mail: [email protected]


Fortschr. Phys. 53, No. 5 – 6 (2005) / www.fp-journal.org 627

2.1 c = 1 string

The c = 1 string theory is a two-dimensional string background with coordinates (x0, φ) ∈ R2, where x0

represents time. This background is characterized by the following expectation values for the target-spacemetric Gµν , the dilaton Φ and the tachyon field T :

Gµν = ηµν , Φ = φ, T = µeφ . (2.1)

The worldsheet-description of this theory [11] (we follow [5] in our presentation) is characterized by theworld-sheet action

S =14π

∫d2zd2θ

(DX0DX0 +DΞDΞ + 2iµ0e

Ξ)+ (ghosts), (2.2)

where (X0,Ξ) are the superfields

Ξ = φ+ iθψ + iθψ + iFθθ , X0 = x0 + +iθχ+ iθχ+ iGθθ , (2.3)

and D, D are the covariant derivatives ∂∂θ + θ∂z and ∂

∂θ+ θ∂z respectively.

The 0B, c = 1 string theory has two propagating space-time fields, the tachyon T from the NS-sectorand the RR-scalar C. The vertex operators which create the modes of these field with definite space-timeenergy ω will be denoted as Tι(ω) and Cι(ω) respectively, where ι = in/out distinguishes the asymptoticstates associated to time t = ∓∞. As a useful short-hand notation let us introduce

Tı(ω) =

{Tı(ω) if ω ≡ (ω,NS),

Cı(ω) if ω ≡ (ω,RR).(2.4)

It is reasonable to expect that the future development of the SCFT characterized by the action (2.2) willallow us to define arbitrary string scattering amplitudes in an asymptotic expansion in powers of the stringcoupling constant gs,

⟨Tout(ω1) . . . Tout(ωn)Tin(ω′

1) . . . Tin(ω′m)

⟩ gs�gs�

∞∑h=0

g2h−2s

⟨Tout(ω1) . . . Tout(ωn)Tin(ω′

1) . . . Tin(ω′m)

⟩(h),

(2.5)

The notationgs� means equality of asymptotic expansions in gs ∝ µ−1. The amplitudes 〈· · · 〉(h) associated

to Riemann surfaces with genus h are expected to have a representation as SCFT-correlation functionsintegrated over the moduli space of Riemann surfaces, as usual.

2.2 Free fermionic field theory

We will reconsider the field theory of free fermions in the inverted harmonic oscillator,

H =∫dλ Ψ†

(− ∂2

∂λ2 − 14λ2

)Ψ . (2.6)

The fermionic field operators may be expanded as

Ψ†(λ, t) =∫dω e−iωt

(F+(ω|λ)c†

+(ω) + F−(ω|λ)c†−(ω)

), (2.7)

where {Fp(ω|λ) ; ω ∈ R , p ∈ {+,−}} is a complete set of eigenfunctions for the corresponding singleparticle Hamiltonian and {cp1

(ω) , c†p2

(ω′)} = δp1p2δ(ω−ω′). The vacuum |µ 〉〉 is the Fermi sea in whichall states with energies ω < −µ are filled.


628 J. Teschner: Exact treatment of D0 brane decay

In- and out-fields can be defined as follows: Consider the asymptotics for t → ±∞ of the fermionicoperators

Ψ†[ψ | t ) ≡∫dλ ψ(λ) Ψ†(λ, t) . (2.8)

A crucial feature of the inverted harmonic oscillator potential is that the asymptotic behavior of the wave-function ψ(λ, t) for late/early times can be represented in terms of solutions of a relativistic wave-equationas follows:

ψ(λ, t) ˜t→±∞ (2π|λ|)− 12 e

i4 λ2

χ±s (t∓ x), x ≡ ln |λ|, s ≡ sgn(λ). (2.9)

It can be shown that (2.9) implies that (2.8) may be asymptotically represented as

Ψ†[ψ | t ) ∼t→±∞ Θ†

±[ψ | t ) ≡∫

dx

2πχ±

s (t∓ x)Θ†±,s(x), (2.10)

where we are using the usual summation convention and χ±s is defined in (2.9). The fields

Θ†±,s(x) =

∫R

dω e−iωx d†±,s(ω) (2.11)

are the in- and out-fields that create the asymptotic states. The relation between the oscillators d†±,s(ω)

and c†p(ω) takes the form d†

±,s(ω) = (M(ω)†)±1sp c†

p(ω), where the matrix M can be calculated from theasymptotic behavior of the eigenfunctions Fp(ω|λ) [10]. In- and out-oscillators are related to each othervia the fermionic reflection matrix R(ω) ≡ (M(ω))2, which can be represented as

R(ω) =(ρ(ω) θ(ω)θ(ω) ρ(ω)

), (2.12)

where

ρ(ω) =1√2πe− π

2 ωΓ( 1

2 − iω), θ(ω) = −ieπωρ(ω). (2.13)

We are particularly interested in the bosonic oscillators asι (ω) which are defined by bosonizing the in-

and out fields Ψ†± as

asι (ω) =

∫R

dω′ d†ι,s(ω

′)dι,s(ω + ω′) ,s ∈ {+,−} = {R,L},ι ∈ {+,−} = {out, in}. (2.14)

Natural bases for a subspace H0 ⊂ H can be defined with the help of the bosonic oscillators asι (ω) as

follows. Let us introduce the linear combinations

aı(ω) =

aNSı (ω) = eiδNS(ω)(aR

ı (ω) + aLı (ω)

)if ω ≡ (ω,NS),

aRRı (ω) = eiδRR(ω)(aR

ı (ω) − aLı (ω)

)if ω ≡ (ω,RR),

(2.15)

These linear combinations will have a particularly simple correspondence [4, 5] with the modes of thespace-time fields T and C of the c = 1 noncritical string theory,

Tı(ω) aı(ω) , (2.16)



provided the phases δNS and δRR are chosen as

eiδNS(ω) ≡ Γ(+iω)Γ(−iω)

, eiδRR(ω) ≡ Γ( 12 + iω)

Γ( 12 − iω)

. (2.17)

To each tuple = (ωn, . . . , ωn), we may then assign the state | 〉〉ι = aι(ωn) · · · aι(ω1)|µ〉〉 . The vectors| 〉〉ι will then generate (plane-wave) bases for H0 once we vary arbitrarily.

The change of basis induced bydout,s = Rss′(ω)din,s′ defines the S-matrix of c = 1 string theory [13,14].Combined with bosonization (2.14) this leads to the following representation of the perturbative scatteringof the bosonic excitations created by the oscillators aı(ω):

〈〈µ | aout(ω1) . . . aout(ωn)gs� (2.18)

gs�∞∑

m=1

∫R+

dω′1 . . . dω

′m R(m�→n)(ω1, . . . , ωn|ω′

1, . . . , ω′m) 〈〈µ | ain(ω′

1) . . . ain(ω′m),

The short-hand notation∫dω implies summation over the two possible choices ω = (ω,NS) and ω =

(ω,RR). There exists a counterpart to formula (2.18) which is valid nonperturbatively [10], in which caseone needs to take into account possible transitions between the k-vacua discussed in [14].

3 D0 branes vs. fermions

3.1 Worldsheet-description

The type 0B c = 1 noncritical string theory contains unstable D0-branes [4,5,16,17] which are analogs ofthe D0-branes in the bosonic c = 1 string theory [15]. These D0-branes are localized in the strong couplingregion φ = ∞. In order to describe their decay one may consider the boundary interaction

Sint = κ

∫∂Σdτdθ cosX0 . (3.1)

A construction for the corresponding boundary states was first proposed in [18]. These boundary states haveto be tensored with the boundary states for Super-Liouville theory which describe the D0-branes [16,17].In this way one arrives at the following result for the leading order closed string emission amplitudes [5]:

〈Tout(ω) |Bκ 〉HH ∝ eiδNS(ω)e−iω log sin2 πκµ−iω,

〈Cout(ω) |Bκ 〉HH ∝ sgn(κ) eiδRR(ω)e−iω log sin2 πκµ−iω .(3.2)

The notation |Bκ 〉HH reminds of the fact [19] that the definition of the boundary state associated to theboundary interaction (3.1) depends on a choice of integration contour, |Bκ 〉HH being the boundary stateassociated to the so-called Hartle-Hawking contour [19].

3.2 Proposed duality

The authors of [2, 4, 5] propose that the state |λ0 〉〉 which describes a fermion with a well-defined initiallocalization at λ0 may – at least to leading order in the semiclassical limit – be represented as |λ0 〉〉 Ψ†

σ0,out(τ0) |µ 〉〉, where τ0 ≡ ln |λ0|, σ0 ≡ sgn(λ0). The application of well-known bosonization formulaesuch as

Ψ†σ0,out(τ0) = : exp

(i

∫dω

ωe−iωτ0aσ0

out(ω))

: , (3.3)



would then straightforwardly lead to the expressions

〈〈µ | aNSout(ω) |λ0 〉〉 = eiδNS(ω)e−iωτ0 ,

〈〈µ | aRRout(ω) |λ0 〉〉 = σ0 e

iδRR(ω)e−iωτ0 .(3.4)

This matches the result of the worldsheet-computation (3.2) provided that the initial location λ0 of thefermion is related to the parameter κ of the unstable D0-brane via

|λ0| ∝ sin2 πκ , σ0 = sgn(κ) . (3.5)

The match of (3.4) with the worldsheet-computation for the closed string emission from a decaying D0-branerepresents evidence for the identification of the single fermion state with the D0-brane.

3.3 Fermion number conservation

We have to observe, however, that the left hand sides of (3.4) vanish identically due to fermion numberconservation. This is related to the fact that the bosonization formula (3.3) has serious infrared problems.This raises the following two questions:

(i) How to define reasonable quantities within the free fermionic field theory which match the results ofthe world-sheet computation?

(ii) What is an appropriate interpretation of these quantities in terms of D-brane decay?

To answer the first question let us observe that the Hilbert space H of the free fermionic field theorysplits into superselection sectors with definite fermion number, H =

⊕n∈Z Hn. Within each sector Hn

there exist states with energy arbitrarily close to the Fermi level µ. A useful class of examples are the states

| z 〉〉 ≡ |2�z| 12 e−iµz Ψ†(z) |µ 〉〉 , �z < 0 (3.6)

with large positive values of |�z|. By making |�z| large one gets arbitrarily small energy expectation value.Given a minimal measurable energy Emin it is impossible to distinguish states with energies smaller thanEmin from the true vacuum |µ 〉〉. For given sensitivity of our detectors we only need to make |�z| largeenough to get states which resemble the bosonic vacuum |µ 〉〉 as much as we want. We will call such statesapproximate vacua.

Reasonable transition amplitudes are the following:

〈〈 z | aNSout(ω) |λ0 〉〉 z |2�z|− 1

2 eiδNS(ω)e−i(ω−µ)τ0 ,

〈〈 z | aRRout(ω) |λ0 〉〉 z |2�z|− 1

2 σ0 eiδRR(ω)e−i(ω−µ)τ0 ,

(3.7)

where the notation z means equality up to an error controlled by |�z|−1. This clearly looks quite similarto the result of the world-sheet computation (3.2).

We propose the following interpretation in terms of 2d string theory. The identification between D0-branes and fermions is possible once certain subleties are understood. Fermion number conservation meansthat the D0-branes can never decay completely. However, the radiation observed by a closed string observerin the weak coupling region φ → −∞ will to any given accuracy look like the radiation from a decayingD0-brane.

When closed string radiation has carried away most of the initial energy of the D0-brane we still have aremnant of the D0 brane in the strong coupling region φ → +∞. However, this remnant ultimately becomesinvisible to the closed string observer in the weak coupling region φ → −∞ as its energy must drop belowthe minimal measurable energy after some finite time. The D0-brane and its associated open string sectorhave become “confined” behind the Liouville wall.



The deviation between (3.7) and (3.2) is not detectable to closed string observers. Fermion numberconservation translates into the fact that a background which contains D0-branes is not a small deformationof the usual c = 1-background. The extra factors in (3.7) will appear in any quantity corresponding to oneand the same background.

4 Exact treatment

We would like to understand the relation between single fermion states |F 〉〉,

|F 〉〉 =∫dλ ψ(λ) |λ 〉〉 , |λ 〉〉 ≡ Ψ†(λ) |µ 〉〉 ∈ H1 , (4.1)

and the D0-branes more precisely. Following the suggestion from [2] we would like to interpret amplitudessuch as

(1)out〈〈ω1, . . . , ωn |F 〉〉, (4.2)

as the amplitude for emission of closed strings from a decaying D0-brane.

4.1 Open string picture

We know that |F 〉〉 has the following equivalent descriptions:

|F 〉〉 =∫

R

dτ(χR

out(τ)Θ†out,R(τ) + χL

out(τ)Θ†out,L(τ)

) |µ 〉〉

=∫

R

dλ ψ(λ)Ψ†(λ) |µ 〉〉

=∫

R

dτ(χR

in(τ)Θ†in,R(τ) + χL

in(τ)Θ†in,L(τ)

) |µ 〉〉.

(4.3)

Let us next note the simple relation

〈〈 z | aNSout(ω1) . . . aNS

out(ωnNS) aRRout(ω

′1) . . . a

RRout(ω

′nRR

) |F 〉〉 z (4.4)

z

nNS∏r=1

eiδNS(ωr)nRR∏s=1

eiδRR(ωs)[χR

out(ω − µ) + (−1)nRR χLout(ω − µ)

],

where ω =∑nNS

i=1 ωi +∑nRR

i=1 ω′i. It remains to calculate χR/L

out for given initial wave-function ψ or χin.Given any of the wave-functions (χout, ψ, χin), we can calculate the two others by going to the energyrepresentation and using

χsout(ω) = Ms

p(ω) ψp(ω) , χsout(ω) = Rs

s′(ω) χs′in (ω). (4.5)

Viewing the fermionic field theory as a representation for the open string theory on a gas ofD0-branes [1]motivates us to call the resulting representation the “open string picture”. Quantum corrections to the D-brane dynamics are calculated in the dual open string theory before we analyze the final state in terms ofclosed string observables.



4.2 Closed string picture

We seem to stumble into a problem: The amplitudes (4.2) depend on a lot of choices, in particular onthe choice of an initial wave-function χin(τ) or ϕ(λ). On the other hand we expect that the perturbativeexpansion in powers of gs of the amplitudes for emission of closed strings from a decaying D0-brane can beconstructed with the help of standard world-sheet techniques. The result should be essentially1 unique. Thismeans that there should be preferred choices for the initial wave-function χin(τ) or ϕ(λ) which lead to amatch between free fermionic field theory and world-sheet description of 2d string theory. In the followingwe shall argue that the relevant choice is

(χL

in(τ)χR

in(τ)

)= δ(τ − τ0)

12

(1 − σ0

1 + σ0

)corresponding to |λ0 〉〉 ≡ Ψ†

σ0,in(τ0) |µ 〉〉 .

The corresponding wave-function χR/Lout (τ) can then be calculated from (4.5). It is possible to show [10] that

(2.18) implies the approximate relation

〈〈 z | aout(ω1) . . . aout(ωn)gs�z (4.6)

gs�z

∞∑m=1

∫R+

dω′1 . . . dω

′m R(m�→n)(ω1, . . . , ωn|ω′

1, . . . , ω′m) 〈〈 z | ain(ω′

1) . . . ain(ω′m),

where the notationgs�z means equality of asymptotic expansions in gs up to errors controlled by |�z|−1. By

using this relation one gets the following representation:

〈〈 z | aout(ω1) . . . aout(ωn) |λ0 〉〉 gs�z (4.7)

gs�z

∞∑m=1

∫

R

m∏r=1

dω′r 〈Tin(ω′

r) |Bλ0〉 R(m�→n)(ω1, . . . , ωn|ω′1, . . . , ω

′m)

In this form it is straightforward to derive the expansion in powers of gs. It suffices to notice that eachS-matrix element R(m�→n) has an asymptotic expansion in powers of gs. It is not hard to see that theresulting expansion has the right form to be identified with the perturbative representation for the closedstring emission amplitude that is furnished by the world-sheet representation.

We propose the following interpretation of the representation (4.7) in terms of 2d string theory. Theinitial state of the D0-brane is represented as a coherent state of closed strings on top of an unobservableD0-remnant. The resulting out-state is obtained by applying the scattering operator to the closed stringoscillators which generate the initial state.

Comparison of the representation discussed in the previous subsection with (4.7) allows one to find theconditions under which the world sheet construction of closed string emission amplitudes gives a usefulapproximation to the exact result [10].

Acknowledgements I would like to thank the organizers of the 37th International Symposium Ahrenshoop for thekind invitation. Support from the DFG by a Heisenberg-Fellowship, and from the EU via the TMR network EUCLID,contract number HPRN-CT-2002-00325, is gratefully acknowledged.

1 Possible ambiguitites like those related to the choice of integration contour in the definition of the boundary state [19] will bediscussed in [10].



References

[1] J. McGreevy and H. Verlinde, JHEP 0312, 054 (2003).[2] I. R. Klebanov, J. Maldacena, N. Seiberg, JHEP 0307, 045 (2003).[3] J. McGreevy, J. Teschner, and H. Verlinde, JHEP 0401, 039 (2004).[4] T. Takayanagi and N. Toumbas, JHEP 0307, 064 (2003).[5] M. R. Douglas, I. R. Klebanov, D. Kutasov, J. Maldacena, E. Martinec, and N. Seiberg, [arXiv:hep-th/0307195].[6] J. Polchinski, Phys. Rev. Lett. 74, 638–641 (1995).[7] P. Ginsparg and G. Moore, in: Recent Directions in Particle Theory, edited by J. Harvey and J. Polchinski,

Proceedings of the 1992 TASI (World Scientific, Singapore, 1993).[8] M. Gutperle and P. Kraus, Phys. Rev. D 69, 066005 (2004).[9] J. Ambjorn and R.A. Janik, Phys. Lett. B 584, 155–162 (2004).

[10] Paper in preparation.[11] P. Di Francesco and D. Kutasov, Nucl. Phys. B 375, 119–172 (1992).[12] G. Moore, Nucl. Phys. B 368, 557–590 (1992).[13] G. Moore, M. R. Plesser, and S. Ramgoolam, Nucl. Phys. B 377, 141–190 (1992).[14] O. DeWolfe, R. Roiban, M. Spradlin, A. Volovich, and J. Walcher, JHEP 0311, 012 (2003).[15] A. B. Zamolodchikov and A. B. Zamolodchikov, [arXiv:hep-th/0101152].[16] T. Fukuda and K. Hosomichi, Nucl. Phys. B 635, 215–254 (2002).[17] C. Ahn, C. Rim, and M. Stanishkov, Nucl. Phys. B 636, 497–513 (2002).[18] A. Sen, w J. High-Energy Phys. 0204, 048 (2002); J. High-Energy Phys. 0207, 065 (2002).[19] N. Lambert, H. Liu, and J. Maldacena, [arXiv:hep-th/0303139].


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Exact treatment of D0 brane decay in two-dimensional string theory