Closed string tachyon condensation: an overview

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Closed string tachyon condensation: an overview

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2004 Class. Quantum Grav. 21 S1539

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INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class. Quantum Grav. 21 (2004) S1539–S1565 PII: S0264-9381(04)78332-X

Closed string tachyon condensation: an overview

Matthew Headrick1, Shiraz Minwalla2 and Tadashi Takayanagi2

1 Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA2 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Received 18 March 2004Published 20 April 2004Online at stacks.iop.org/CQG/21/S1539 (DOI: 10.1088/0264-9381/21/10/027)

AbstractThese notes are an expanded version of a review paper on closed stringtachyon condensation at the RTN workshop in Copenhagen in September2003. We begin with a lightning review of open string tachyon condensation,and then proceed to review recent results on localized closed string tachyoncondensation, focusing on two simple systems, C/Zn orbifolds and twistedcircle compactifications.

PACS number: 11.25.Sq

1. Introduction

Consider a classical, nonrelativistic charged particle interacting with an electromagnetic field.Let it be localized near the origin of space by a spherically symmetric potential V (r) thattakes the form shown in figure 1. A glance at this potential carries a lot of information. Forinstance, it is clear that the particle has at least two stable equilibria (at r = a and r = c)and two unstable equilibria (at r = 0 and r = b). The spectrum of small fluctuations aboutthe unstable equilibrium r = 0 is that of an inverted harmonic oscillator, i.e. it is tachyonic.Displacing the particle away from r = 0 initiates a process of tachyon condensation, and thelong-time behaviour is rather clear. It eventually settles down to the stable equilibrium r = a,while the extra energy V (0) − V (a) is carried away by electromagnetic radiation.

In contrast to this toy model, our understanding of the global structure of the string theoryconfiguration space is rather limited. We do not know whether all of the different consistentstring theories and their various solutions fit together as extrema of some potential in a singleconfiguration space, such as the points r = 0, a, b, c. For example, can the bosonic stringbe continuously connected to the superstrings? Recent studies of tachyon condensation instring theory may be thought of as an attempt to better understand the global structure of thestring theory configuration space. These investigations study the decay of specific unstablebackgrounds of string theory. Their aim is to excite the instability about the chosen unstablebackground (the analogue of r = 0) and then follow the long-time behaviour of the ensuingmotion, to determine the nearest ‘minimum’ (the analogue of r = a).

0264-9381/04/101539+27$30.00 © 2004 IOP Publishing Ltd Printed in the UK S1539

http://stacks.iop.org/cq/21/S1539

S1540 M Headrick et al

V(r)

r0 b ca

Figure 1. A possible energy landscape as a function of position for a particle.

In practice, however, exact time dependent solutions of string theory are difficult to comeby. Furthermore, they sometimes carry more information than is of interest (such as the preciseform of the electromagnetic radiation in our toy model). Consequently most studies of tachyoncondensation circumvent the problem of solving for the exact time evolution, substituting amore tractable and possibly more interesting problem.

In our toy model, what techniques could we use to determine the endpoint of tachyoncondensation about r = 0, without solving for the full time evolution? One suggestion wouldbe to model the back-reaction of the electromagnetic radiation by a friction term in the particleevolution equation:

d2r

dt2= −V ′(r) − k

dr

dt, (1.1)

where k is a positive number. A stringy equivalent of this friction term is provided by a dilatonthat varies linearly in the time direction. Such a linear dilaton is present in Liouville theory.Consequently, some recent studies of tachyon condensation analyse the Liouville evolution ofunstable backgrounds.

Note that in our toy model the long-time behaviour of the dynamics governed by (1.1)is actually independent of k. We could therefore further simplify the analysis by taking k toinfinity, yielding the gradient flow equation

dr

dt ′= −V ′(r), (1.2)

where t = kt ′. As we argue below, the RG flow equations on the worldsheet of the string areanalogous to equation (1.2). Consequently, several recent studies analyse RG flows in orderto study tachyon condensation.

In our toy model, the energy released by the particle falling to its ground state is carriedoff to infinity by radiation. This provided a crucial simplification: it meant that the particle’slong-time behaviour was simply to find the nearest minimum of the potential. For the samereason, studies of localized tachyon condensation have generally focused on tachyons that arelocalized near some special point in space, such as an orbifold fixed point, allowing the energyreleased in the decay to radiate to infinity and leave the system in its ground state.

In this paper, we will introduce the reader to recent studies of localized closed stringtachyon condensation. We will not survey all the results that have been obtained in this area,but will instead focus on two of the simplest and best studied backgrounds of type II stringtheory: the C/Zn orbifolds, and the twisted circle compactifications. In our study of thesetwo examples, we will have occasion to employ many of the techniques (RG flows, Liouville

Closed string tachyon condensation: an overview S1541

flows, D-brane probes and supergravity dynamics) that have recently been brought to bear onthe problem of tachyon condensation in string theory. The reader who wishes to continuehis study of tachyon condensation in other examples (including non-supersymmetric C2/Zn

orbifolds) is referred to the recent review by Martinec [1] (and references therein).We begin our paper, in section 2, with a lightning review of open string tachyon

condensation on the world-volume of D-branes. Open string tachyon condensation processesshare some similarities with their closed string counterparts. However, at weak string couplingopen strings are decoupled from gravity and so open string tachyon condensation takes placeon a fixed background geometry. Consequently, it involves fewer conceptual complicationsand has been much better studied than its closed string counterpart. The review of open stringtachyon condensation in section 2 will help orient us in our later study of closed string tachyoncondensation. In section 3, we turn to the analysis of the simplest closed string tachyoncondensation process, the decay of C/Zn to flat space in type II theories, and study it using avariety of methods. In section 4, we turn to tachyon condensation in a more versatile set ofmodels, twisted circle compactifications of type II theories, which reduce in one limit to C/Zn

orbifolds and in another to type 0 string theories. Our analysis of these models will employLiouville flow.

2. Open string tachyon condensation on D-branes

2.1. Sen conjectures

Consider a Dp-brane in bosonic string theory. The spectrum of fluctuations of the D-brane isobtained by quantizing open strings, and includes a tachyon, implying the instability of thebrane in question. It is natural to ask what the endpoint of this instability might be. Addressingthis issue, Sen conjectured that (i) the endpoint of homogeneous tachyon condensation on theworld-volume of the D-brane is closed string radiation (which eventually escapes to infinity)about the closed string vacuum. Sen also proposed that (ii) the condensation of inhomogeneousmodes of the tachyon field leads to lower dimensional D-branes (plus closed string radiation)[2–4].

Sen’s conjectures have straightforward generalizations to the superstring theories. Themost familiar D-branes in those theories are supersymmetric, hence stable and withouttachyons on their world-volumes. However, open string tachyons appear on the world-volumesof brane–antibrane systems [3, 5] and the less familiar unstable (or non-BPS) D-branes [6–8].Sen’s conjectures concern the endpoint of tachyon condensation on these nonsupersymmetricbrane systems.

We first consider the unstable Dp-branes in the type II theories; these exist with the‘wrong’ dimensionality, i.e. p odd in IIA and p even in IIB. Their properties are similar tothose of D-branes in bosonic string theory; in particular they are uncharged and unstable,and possess a real scalar tachyonic field T with m2 = −1/2α′. However, in contrast to thebosonic D-branes, the tachyon effective action on unstable type II branes is invariant underT → −T , and is believed to be bounded from below. Sen conjectured that, as for bosonicbranes, homogeneous tachyon condensation on the world-volume of an unstable type II braneends simply in the closed string vacuum T = ±Tmin (with closed string radiation).

We now turn to inhomogeneous tachyon condensation in the same system. As we haveremarked above, the tachyon potential is invariant under T → −T ; as it also admits aminimum at a non-zero value of T, it has the form of a double well. Consequently, the tachyonfield admits kink solutions that asymptote to −Tmin on the left and +Tmin on the right. Senconjectured that such a kink is in fact a supersymmetric D(p − 1)-brane, which may therefore


be reached as the end product of condensation of a particular inhomogeneous mode of thetachyon.

A brane–antibrane system consists of a usual BPS Dp-brane and its antibrane (i.e. D-branewith the opposite R–R charge). The open strings between two branes (or two antibranes) aresubject to the usual GSO projections; the tachyon is projected out and the gauge field isprojected in. On the other hand, the open strings between a brane and an antibrane aresubject to the opposite GSO projection, so the tachyon field T is projected in and the gaugefield is projected out. The retained tachyon is a complex scalar field. Sen conjectured thathomogeneous condensation of this tachyon field result in the closed string vacuum (withradiation), while a vortex solution with winding number n,

T (r, θ) ∼ T0 einθ ,

∫(F (1) − F (2)) = 2πn (2.1)

is simply a supersymmetric D(p − 2)-brane, which may therefore be obtained as the endpointof inhomogeneous tachyon condensation with the appropriate boundary conditions. Sen hasfurther conjectured that a kink solution (which is clearly unstable) on this world-volumerepresents an unstable D(p − 1)-brane.

Sen’s conjectures have been verified in a number of different ways, as we will review inthe rest of this section.

2.2. The potential: cubic string field theory

An inspection of the potential as a function of configuration space provides a conceptuallysimple method to determine the endpoint of tachyon condensation. Unfortunately, stringworldsheet techniques are best suited to addressing on-shell questions; string S-matricesencode information about off-shell quantities such as potentials only indirectly. Witten’s cubicopen string field theory [9], on the other hand, is an off-shell description of open string theory,and is formulated in terms of a potential on the open string configuration space. Consequently,if we can calculate the potential of this string field theory, then inspection of it should besufficient to determine the endpoint of open string tachyon condensation. This is indeed thecase, as we review in this subsection. This approach to open string tachyon condensation waspioneered in [10, 11]; for more complete references see the reviews [12–17].

The space of classical configurations of cubic open string field theory is simply thequantum Hilbert space of the first-quantized theory. The string field theory action associatesa number to every state in the theory of 26 free bosons plus the b and c ghosts on the strip.Specifically, the action is given by [9]

S = − 1

g2o

(1

2〈�|QB�〉 +

1

3〈�|� ∗ �〉

), (2.2)

where QB is the usual worldsheet BRST operator and ∗ is a multiplication operator onworldsheet states; see the reviews referred to above for more details. Equation (2.2) enjoysinvariance under the gauge transformations � → �+QB� +�∗� −� ∗� (at the linearizedlevel this implies a familiar statement: BRST trivial states are pure gauge). The equationsof motion that follow from (2.2) are QB� + � ∗ � = 0 (at linearized order, on-shell fieldconfigurations are simply BRST closed, as we expect).

Note that every oscillator state in � is also a function of the zero-mode xµ of everycoordinate Xµ that obeys Neumann boundary conditions. Thus each open string oscillatorstate corresponds to a quantum field living on the world-volume of the brane, and one may


think of string field theory as a gauge invariant quantum field theory with infinitely many fieldspropagating on the brane world-volume. More precisely, the string field may be expanded as

� = T (xµ)c1|0〉 + u(xµ)c−1|0〉 + v(xµ)L−2c1|0〉 + · · · , (2.3)

where the lowest mode T is the open string tachyon field and u, v represent higher modes.Formally, one may obtain an effective potential V (T ) for the tachyon [18] by integrating outall massive fields. Sen’s conjecture implies that this potential takes a minimum value at onepoint T = T0, this point represents the closed string vacuum, and its depth V (0) − V (T0) isequal to the tension of the original Dp-brane.

Unfortunately, the potential in the action (2.2) is so complicated that it has not yet provedpossible to analytically locate T0 and evaluate its energy. However, this programme has beencarried through to great precision in an approximation scheme known as level truncation. Thisscheme consists of simply ignoring (i.e. setting to zero rather than integrating out) all fieldswhose worldsheet energy is larger than a certain fixed ‘level’. At the lowest non-trivial level(level 0), the scheme retains only the tachyon, setting all other fields to zero. At this order thepotential in (2.2) reduces simply to

V (T ) = 2π2

(−1

2T 2 +

1

3

T 3

r3

), r = 4

3√

3, (2.4)

where we have normalized the potential energy in units of the D-brane tension. Thisapproximated potential has a (local) minimum at T = r3; the potential evaluated at thisminimum is V (0)− V (T0) = π2

3 r6 ≈ 0.684 (compared to 1, as predicted by the Senconjecture). It is more difficult to locate the minimum and evaluate its potential at higherorders in the level truncation scheme, and the relevant computations have been performedonly numerically. The resulting values for V (0) − V (T0) appear to converge rapidly (but notmonotonically) to that predicted by the Sen conjecture: 0.986 at level 4 [11], 0.9991 at level 10[19], and 1.000 63 at level 28 [20] (for more details see the reviews [12, 17]). These numericalresults provide rather impressive validation of the Sen conjecture, and demonstrate that cubicstring field theory, which was designed to reproduce perturbative string amplitudes, alsocontains non-perturbative information about string theory. They also testify to the validityof the level truncation approximation; even though it is a natural scheme for neglectingpresumably irrelevant highly massive modes, it important to keep in mind that there is noproof that it should converge to the correct answer.

In a similar fashion, it has proved possible to describe an inhomogeneous solution,depending on a single direction on the brane world-volume, within the level truncationapproximation. According to the Sen conjecture such a lump represents a D(p − 1)-brane,and the numerical computation of the energies of these lump solutions again agrees rather wellwith this conjecture (see [21–23]; for more complete references see the review [12]).

The Sen conjectures regarding tachyon condensation on the world-volume of unstableD-branes in type II theory have also been successfully tested [24, 25] within the level truncationapproximation scheme using Berkovits’ supersymmetric cubic string field theory [26, 27]. Thesituation with the superstring is qualitatively similar to that with the bosonic string; we referthe reader to the reviews mentioned before for more details.

2.3. RG flows and boundary string field theory

We now turn to a rather different technique for studying tachyon condensation, involving theanalysis of RG flows on the worldsheet of the string. We will begin this subsection witha review of some features of worldsheet RG flows for closed as well as open strings, and


their relevance to tachyon condensation. Readers who are interested in detail and completereferences should refer to [28].

Consider a type II closed string background of the form CFT1 + CFT2, where CFT2 isany unitary conformal field theory with c = d, and CFT1 is the free sigma model on R9−d,1

with fields Xa (a = 0, . . . , 9 − d). It is not difficult to see that the operator spectrum of CFT2

determines the stability of this background under time evolution. In fact, a conformal operatorO of dimension (δ, δ) in CFT2 may be dressed by a momentum factor from CFT1 to yield amarginal operator in the full CFT: O = eiPXO where α′P 2/2 + δ = 1. Thus the worldsheetoperator O corresponds to a spacetime fluctuation with squared mass M2 = 2(δ−1)/α′ and sorepresents a tachyon or instability when O is a relevant operator. We conclude that the existenceof relevant operators in the spectrum of CFT2 implies an instability of the corresponding stringbackground.

The discussion of the previous paragraph is easily generalized to open strings and boundaryRG flows. A boundary operator of dimension δ in CFT2 corresponds to a spacetime particleof squared mass

(δ − 1

2

)/α′; once again the existence of a relevant boundary operator implies

the instability of the spacetime theory.It is tempting to go beyond such an ‘infinitesimal’ statement and to conjecture a relation

between the full dynamical evolution in string theory and renormalization group flows onthe worldsheet. Clearly, the two sides have many features in common. A worldsheet RGflow away from an unstable string background ends at an infrared conformal field theory thatmay generically be expected to be stable. Similarly, after the dust has settled, the dynamicalprocess of tachyon condensation is generically expected to decay into a stable solution ofstring theory. We will now argue that, in fact, boundary RG flow equations are a rather preciseanalogue of gradient flow equations for the toy model introduced in the introduction, and soshould accurately capture the late-time behaviour in the tachyon condensation process. Laterin this section, we will make a similar argument for bulk RG flows.

In order to draw the analogy between RG flows and the gradient flow equation (1.2), wewish to demonstrate that spacetime energy decreases along worldsheet RG flows. Recall thatZamolodchikov’s famous c-theorem [29] for bulk RG flows asserts that the C function, anoff-shell generalization of the central charge, is a non-increasing function of RG scale. Asimilar theorem is believed to be true for boundary RG flows. Consider a boundary conformalfield theory perturbed by λiOi , where Oi is a basis of boundary operators. Within boundaryperturbation theory, it has been shown that there exists a function (sometimes called theboundary entropy) g(λ) that [30] (see also [31, 32]):

1. decreases along renormalization group flows;

2. is proportional to the disc partition function at fixed points.

Now the disc partition function has a simple spacetime interpretation: it is minus thespacetime energy of the open string sector of the theory, as may be verified by computingthe one-point function of the dilaton on the disc [33]. From all of this, we see that (up to aproportionality constant) the g-function is an off-shell generalization of spacetime energy, andthat spacetime energy decreases along boundary worldsheet flows. Consequently RG flowsare analogous to (1.2) and may be used to study the tachyon condensation process.

We will now review the RG flow that describes the decay of a D1-brane into a D0-brane on the circle in bosonic string theory. A D1-brane on a circle of radius R has energyR/α′gs. On the other hand, the energy of a D0-brane on the circle is given by 1/

√α′gs .

Note the energy of a single D1-brane is larger than the energy of an array of n D0-branes,provided R2 > n2α′. Now consider the theory on the world-volume of the wrapped D1-brane.


The boundary operator

T (X) = λ cos

(nX

R

), (2.5)

where X is the compact direction, is relevant provided α′(n/R)2 < 1, i.e. if and only if thewrapped D1-brane is more massive than an array of n D0-branes. This leads us to conjecturethat the RG flow induced by this operator (when it is relevant) ends up in an array of nD0-branes. This conjecture is rather intuitive; if the operator in (2.5) is relevant, λ increasesas RG flow proceeds, increasingly localizing open strings to the n minima of the potential.It is certainly plausible that as λ → ∞ the endpoint of the RG flow is simply an array of nD0-branes located at these n minima. Indeed this conjecture can be rigorously verified, sincethe perturbation (2.5) about the D1 background is integrable [28]. Consequently, it is possibleto follow this RG flow exactly, and in particular to compute the boundary entropy as a functionof the flow. In particular, the ratio of g at the UV fixed point to g at the IR fixed point is givenby

gUV

gIR= R

n√

α′ = ED1

nED0, (2.6)

in exact agreement with the conjectured endpoint of this RG flow.This analysis may partially be extended to the superstring. Consider a D1-brane and

anti-D1-brane wrapping a circle of radius R. Turn on a Wilson line∫

dx1 A1 = π on thebrane, setting the gauge potential on the antibrane to zero. As we have described above, thespectrum of brane–antibrane strings includes a complex tachyon. Since the tachyon field ischarged under the relative gauge field, the gauge configuration described above ensures thatT is expanded in half-integer modes on the circle. The lightest mode of the tachyon hasmomentum 1/2R; its effective mass is given by

m2 = 1

4R2− 1

2α′ . (2.7)

This is negative, and so the corresponding boundary operator is relevant, provided

2R

α′gs>

√2√

α′gs

, (2.8)

i.e. as long as the energy of the brane–antibrane pair is larger than the energy of an unstableD0-brane. We are thus led to conjecture that the endpoint of the corresponding renormalizationgroup flow is a single unstable zero brane sitting on the circle. Note that a tachyon with halfa unit of momentum around the circle is basically a kink, so this result is consistent with theSen conjecture.

It has, unfortunately, not yet proved possible to rigorously verify this conjecture. Note thatthe D1-brane boundary conformal field theory perturbed by the tachyon boundary operator isgiven by

Sb =∫

dτ dθ(Dθ + T (X)) + h.c., (2.9)

where = η + Fθ represents a (complex) boundary fermionic superfield [28, 34]. Thefields η and η are called boundary fermions, and represent the 2 × 2 Chan–Paton matrix uponquantization [34]. Note also that this term is consistent with the fact that the tachyon fieldbelongs to the GSO-odd sector in open string. The bosonic field F is an auxiliary field. In theparticular case of interest,

T (X) = λ cosX

2R. (2.10)


Rigorous verification of the Sen conjecture, in this case, would require some exact results onthe IR behaviour (for instance the IR value of the g-function) of (2.9), (2.10).

As we described above, the g-function in the middle of RG flow can be considered asan off-shell energy in open string theory. This statement might lead us to search for an off-shell formulation of string field theory, whose action is some generalization of the g-function.Although we will touch on this topic only very briefly, indeed such a formulation has beenfound, and is known as boundary string field theory, or BSFT [31, 35–37] (originally calledbackground independent string field theory [38, 39]; see the reviews [12, 40] for more details).One convenient definition3 [31] of the BSFT action S is given by

∂S

∂λj= βiGij . (2.11)

The parameters λj are the coefficients of possible boundary perturbations as before4 (e.g.T (X) in (2.5)), whose beta functions are given by βj , and Gij is a boundary theory analogueof Zamolodchikov’s metric. S may be argued to be a generalization of the g-function; inparticular it may be shown that S evaluates on-shell to the disc worldsheet partition function.Indeed, in the case of the superstring it turns out that this statement is true even off-shell; asimple solution to (2.11) is obtained by setting S = Z where Z is the worldsheet disc partitionfunction, even off-shell [34, 36, 37, 41].

Boundary string field theory has been employed to a good effect in the study of openstring tachyon condensation. For instance, it has been shown that the profile of the tachyonpotential on an unstable D-brane in BSFT is given by V (T ) ∝ e−T 2/4. The closed stringvacuum corresponds to |T | = ∞ (note that this T is related by a nonlinear field redefinition tothe field T that appears in the tachyon potential (2.4) of cubic string field theory).

We will not explain these results in greater detail in this paper, but will only pause to pointout that inverting the relation (2.11), and applying the definition of the beta function, yieldsthe gradient flow equation

∂λi

∂(− ln )= −Gij ∂S

∂λj, (2.12)

where is the renormalization scale. Hence the analogy between RG flow and (1.2) is clearestin the language of boundary string field theory.

2.4. Time evolution: rolling tachyons

Quite remarkably, it is possible to solve exactly for the classical time evolution of a decayingD-brane. The solution is called the rolling tachyon [42, 43] or S-brane [44, 45]. Thehomogeneous decay can simply be described by perturbing the D-brane conformal fieldtheory by the boundary operator [46–48]5

Sb = λ

∫∂�

dτ eX0. (2.13)

(This is also known as a half S-brane.) To understand why this is a conformal field theory, wefirst note that (2.13) may be obtained from the BCFT [42]

Sb = λ′∫

∂�

dτ cosh(X0(τ )) (2.14)

3 This is equivalent to the original definition of the background independent string field theory [38] via a redefinitionof parameters λi .4 In actual computations, we cannot consider irrelevant perturbations because they are not renormalizable. Eventhough this is the most serious unsolved problem in BSFT, we can usually get the correct results on tachyoncondensation exactly without taking them into account.5 Following the general convention, we have set α′ = 1 in this subsection.


(known as a full S-brane) by taking the limit λ′ → 0, X0 → ∞ with λ′ eX0held fixed. Now

upon analytically continuing X = iX0, (2.14) turns into the marginal boundary perturbation(2.5). This perturbation is in fact exactly marginal [4, 49, 50], as can be rotated into a Wilsonline by using the exact level 1 SU(2) current algebra present in the open string theory atR = n = 1. We conclude that the deformation (2.13) is exactly marginal, and represents aboundary conformal field theory, i.e. an exact time dependent solution of classical open stringtheory. The physical interpretation of this solution is clear: at large negative times (2.13)represents the D-brane perturbed very slightly by a tachyon operator. This operator thenproceeds to grow in time according to its equation of motion. Thus this solution correspondsto tachyon condensation.

Now that we have the exact solution to the time decay of the tachyon, it might seem thatwe should easily be able to determine its long-time behaviour. In fact this issue turns out to berather subtle. Equation (2.13) represents a boundary conformal field theory, and so a theoryof open strings. How is this consistent with the Sen conjecture, which holds that the finalstate must contain only closed strings? An important clue is obtained from computation of theenergy emitted in closed string radiation in the background (2.13). The presence of a D-branewith rolling tachyons leads to a linear source, a tadpole, for each closed string mode [51, 52],and the amount of radiation can be computed by the one-point function of the closed stringvertex operator on the disc. The amplitude to emit a state of energy E is proportional to [53]⟨

eiEX0 ⟩disc = e−iE log λ π

sinh(πE). (2.15)

In the high-energy region, the square of the amplitude (2.15) decays exponentially like e−2πE .Consequently, the D-brane decay process populates the closed string states in a thermal fashion,at an effective temperature that is twice the Hagedorn temperature. However, not all closedstring states are thus excited; the D-brane couples only to left–right symmetric states, andthe density of excited states scales like ρ(E) ∼ e2πE , the square root of Hagedorn growth.Note that the exponential growth in the density of states exactly cancels the exponentialsuppression in the squared amplitude. Consequently, the amount of energy emitted into closedstrings, at tree level, is controlled by subleading powers of E in the density of states. Detailedcalculations demonstrate that the energy of emitted radiation from unstable Dp-brane scaleslike

∫dE E−p/2, and diverges for p = 0, 1, 2. The energy radiated is formally finite for p > 2

[53]; however this finiteness is only power-law and probably disappears at higher loop.Let us focus on the case p = 0. The computation reviewed above strongly suggests that

the D0-brane dumps all of its energy into closed string radiation in a time of order the stringscale. The divergence in the computation appears simply to reflect the fact that the energy of aD0-brane scales such as 1/gs and so diverges in perturbation theory. However, the divergencealso formally reflects a breakdown in perturbation theory, so it is unclear if more can be saidabout the process.

Does the uncontrolled production of closed strings ensure that the boundary conformalfield theory (2.13) breaks down at a time of order string scale? Naively that would appear to bethe case. However, Sen has recently conjectured instead that the production of closed stringshas a dual description in terms of open strings [54–56] (see also related arguments [45, 57]).This conjecture appears to be supported by computations in two-dimensional string theory.However, aspects of it are controversial and the last word on the subject is perhaps yet to besaid.

Several other issues surrounding this time evolution process remain unclear. Theseinclude questions about the endpoint of the process with high-dimensional branes and theinterpretation of similar computations in linear dilaton backgrounds [58]. We will not attempta discussion of these cutting-edge issues here.


2.5. Open versus closed string tachyon condensation

In the last four subsections, we saw that the problem of open string tachyon condensationon D-branes has been successfully addressed using several different techniques. Let us nowrecall the various approaches used, and ask if they generalize to closed string theories.

In subsection 2.2, we reviewed the direct construction of the open string tachyon potentialusing string field theory. It seems likely that any attempt to effect similar constructions inclosed string field theory will be considerably more difficult. The reasons for this expectationare conceptual as well as technical.

At the classical level, open string field theory is ‘merely’ a field theory, albeit a verycomplicated one involving an infinite number of fields. In particular, all dynamics takesplace in a static background, and every configuration may unambiguously be assigned anenergy. Closed string theories are always theories of gravity and spacetime is dynamicalin such theories. Consequently the notion of energy is rather subtle. In particular, severalrecent examples of closed string tachyon condensation involve processes that modify theasymptotics of spacetime; in such cases it is not clear whether it is possible, even in principle,to compare the energies of the initial and final solutions. A satisfactory potential function onthe space of closed string configurations would be a very exciting object indeed. However,it seems likely that several conceptual barriers must be overcome before such an object isfound.

At the technical level, the existing formulations of closed string field theories are morecomplicated than open string field theories. In particular, these theories are non-polynomial,and it is unclear if any simple approximation scheme, such as level truncation, will workin closed string field theory. In a very recent study, however, Okawa and Zwiebach reportexciting progress in this direction [59].

The direct study of the dynamical decay of closed string tachyons depends on findingexact solutions, which is usually difficult. This approach may also run into conceptual hurdles.As we described above, it proved rather difficult to extract the qualitative properties of thelong-time behaviour of open string tachyon condensation from the exact solution, even thoughwe had a good guess for the answer. This process may be even more difficult in the case ofclosed strings. Nonetheless exact time dependent solutions, when available, are exciting, anddeserve to be studied in detail [60, 61].

Of the techniques used successfully to study open string tachyon condensation, the onethat generalizes most simply to closed strings is that of worldsheet RG flows. As we reviewedabove, instabilities of a spacetime theory imply the existence of a relevant operator, andhence an instability of RG flow, on the worldsheet of the string. The problem of followingthis RG flow to the IR is mathematically well posed. However its physical relevance to theproblem of tachyon condensation may at first seem questionable. In the case of open stringssimilar concerns were addressed by demonstrating that the worldsheet RG flow equations areanalogous to the gradient flow equation (1.2); in particular spacetime energy decreases alongRG flows. It turns out that a similar (though weaker) result is true of bulk RG flows. Inparticular, one may show [62] that whenever an RG flow connects two spaces that are closeenough to each other so that their energies may be compared, the ADM energy of the IRsolution is lower than that of the UV solution. Thus spacetime energy decreases along bulkworldsheet RG flows—for all flows for which this statement may be sensibly formulated.Consequently it appears that, like their boundary counterparts, bulk RG flows may also bethought of as analogues of the gradient flows (1.2).

As we will review in this paper, RG flows have been widely used to study closed stringtachyon condensation. Another technique that has been used is the so-called Liouville flow


[63], which describes a continuous path (presumably from higher to lower ‘energy’) throughthe space of string configurations.

Finally, before we see explicit examples, we would like to alert readers to one important(and traditional) problem encountered when applying RG flow arguments to bulk closed stringtachyon condensation. According to the c-theorem [29], the central charge will decrease underthe RG flow. If we assume that its endpoint (IR fixed point) is a critical string, then we have acontradiction because the matter central charge is always fixed to c = 15 by the ghost anomalycancellation. One way to solve this problem is to consider a non-critical string theory (see[63, 64] for earlier discussions of related issues in c � 1 string theory). Then we need aLiouville-like field and this will lead to a spontaneous breaking of the rotational symmetry.This conceptual problem remains to be solved. On the other hand, in the localized closedstring tachyon condensation which we will consider below there is no such problem. This isbecause the induced RG flow is also localized at the origin and will not change the centralcharge which is a bulk quantity (see [62, 65, 66] for more details).

3. Decay of C/Zn

3.1. Introduction

The C/Zn orbifold of type II string theory is an ideal laboratory for studying closed stringtachyon condensation. Its tachyons are localized on a co-dimension-2 surface, the orbifoldfixed plane, far from which supersymmetry is restored. Furthermore, it is connected by smoothdeformations to a supersymmetric vacuum, namely type II string theory in flat space. Finally,and perhaps most importantly, it is extremely simple. Because of these features, C/Zn hasbecome a paradigm for the study of closed string tachyons ever since it was studied in theseminal paper [65].

The model [67] is constructed by orbifolding a flat target space C by a Zn subgroup ofits U(1) rotation group. Due to the existence of fermions in the system, this U(1) is a doublecover of the usual SO(2) rotation group: the rotation angle ranges from 0 to 4π . Thereforethe generator of the Zn subgroup is e4π iJ/n, where J is the angular momentum in this plane.If n is even, then this subgroup includes e2π iJ = (−1)F , where F is the fermion number, andamong the twisted sectors are type 0 fields, including a bulk tachyon. To avoid a discussionof bulk tachyons, we will therefore take n odd. In this case, it may be more useful to think ofthe element (−1)F e2π iJ/n as the group’s generator.

The seven other spatial dimensions may be flat or compactified; they will play no role inour discussion. In particular, in discussing the time evolution below, we will always have inmind that the tachyon field is rolling homogeneously in these other directions.

Geometrically, the C/Zn orbifold is a cone, rather like the space transverse to a cosmicstring. ADM mass for co-dimension-2 objects is measured by the deficit angle at infinity;in this case the mass (or, more precisely, tension, since the orbifold is extended in sevendimensions) is (see e.g. [62, 68])

MC/Zn= 2π

κ2

(1 − 1

n

). (3.1)

Just as a cosmic string is a soliton of some matter-gravity equations of motion, the orbifoldshould be thought of a stringy soliton, a static solution to the full string equations of motionwith a localized concentration of energy at the fixed point. The dependence of the mass (3.1)on the gravitational coupling shows that the orbifold, such as, for example, the NS5-brane, isa closed string soliton. Unlike the NS5-brane, however, it is non-BPS: no spacetime spinorsare invariant under the rotation, so the spacetime supersymmetry is completely broken.


The orbifold has n − 1 twisted sectors, each of which can be labelled by its charge0 < k < n under the ‘quantum’ Z′

n symmetry group [69] (which should not be confused withthe original Zn group, under which all surviving states are invariant). The kth twisted sectoris defined by the worldsheet boundary conditions

Z(τ, σ + 2π) = e2π ik/nZ(τ, σ ), (3.2)

for the boson, and similar ones for the fermions. The twisted strings must stretch in order tomove away from the fixed point, so they do not have zero-modes for moving in the orbifolddirections. Thus the twisted string fields live ‘on’ the fixed point, in the same sense that openstring fields live on the D-brane world-volume. Since the spectrum of untwisted strings issupersymmetric, the supersymmetry breaking is localized near the fixed point, and away fromit the theory is locally type II strings in flat space.

The tachyonic modes arise only in twisted sectors, and are due to a shift in the zero-pointenergy and a change in the GSO projection. The total zero-point energy is easily computed tobe 1

2 (k/n − 1). The GSO projection is the usual one (−1)FL = 1 for even k and the oppositeone (−1)FL = −1 for odd k, so the lowest allowed level in sector k is the ground state |0〉when k is even and the first excited state ψ−1/2+k/n|0〉 when k is odd. We find that the lightestmode in each sector has a mass given by

m2k = − 2

α′

1 − k

n, k odd

k

n, k even.

(3.3)

Because the tachyons are localized near the orbifold fixed point, when they condensethe effect should initially only be felt in that region. There are various scenarios one couldconsider: a throat or a baby universe could be created, or a tear or bubble of nothing couldopen up. A less dramatic possibility is that the topology of the spacetime does not change,but instead its geometry simply smooths out, with the curvature singularity at the fixed pointreplaced by a smooth cap. If this cap then expands into the surrounding space, diluting thecurvature and eventually growing to infinite size, then we would say that the endpoint of thetachyon condensation is flat space (although technically that endpoint would not be reachedin any finite time). This last scenario is closely analogous to what happens when tachyons onD-branes condense: at the endpoint supersymmetry is restored, and those open strings endingon the decaying branes (the analogues of the twisted strings) are lifted out of the perturbativespectrum, while any bulk open strings (the analogues of the untwisted strings) persist.

The last scenario was first conjectured to be the correct one by the authors of [65]. Theygave substantial evidence in its favour, and quite a bit more has accumulated since then. Thepurpose of this section gives a brief overview of some of this evidence, most of which fallsin two of the categories of analysis described in section 2, namely worldsheet RG flow andtime evolution. The strongest evidence in favour of the conjecture is an exact constructionof the RG trajectory of the C/Zn orbifold theory perturbed by the most relevant tachyonvertex operator, which has as its IR fixed point the sigma model onto the plane [70]. Thisconstruction is described in subsection 3.2. An exact solution for the flow can also be obtainedin the sigma model limit, which gives intuition about how the flow proceeds at late stageswhen the geometry becomes smooth and the curvatures are small in string units [62]. Thissolution is described in subsection 3.3.

As explained in section 2, however, in the closed string context the relation betweenworldsheet RG flow and the actual dynamical tachyon condensation process is conjectural,so the above results on RG flow cannot be taken as sure indicators of the endpoint of thecondensation. Unfortunately, there are fewer results about the time evolution of the system,


which is likely to be quite complicated. Although the technology does not yet exist to followthe dynamics on time scales comparable to the string length, it is possible to gain some supportfor the conjecture by looking at its behaviour on shorter and longer time scales. When thetachyon has a very small expectation value, we can see how the geometry is deformed as seenby a D-brane probe; the result [65, 66] is that the tip of the cone is indeed smoothed out, andthe size of the cap grows with the tachyon expectation value. This analysis is described insubsection 3.4. Furthermore, if a tear or other singularity does not form in the space, thenthe curvature scales will eventually become large enough for supergravity to be valid. At thatpoint, as explained in subsection 3.5, we can confidently predict the future evolution of thesystem, and see that a bubble of flat space expanding at the speed of light is its inevitablelate-time behaviour [71, 72].

Although we will not review it here, there has also been important recent progress towardsfinding an effective potential for the tachyons. One such potential was proposed in [73] on thebasis of the so-called t t∗ geometry for the chiral ring of the system. A level truncation schemeclosely analogous to that described in subsection 2.2 was applied to the non-polynomial closedstring field theory to find critical points of the tachyon potential for C/Zn orbifolds of thebosonic string [59]. Finally, an off-shell effective potential was deduced on the basis ofscattering amplitudes for light tachyons in C/Zn at large n [74] (see also [75]).

The evidence presented here is strong but not conclusive in favour of the APS conjecture,and it is important to note that alternative proposals exist. In particular, the authors of [66]have argued on the basis of an analysis of the theory’s chiral ring that, at least under worldsheetRG flow, the spacetime does in fact undergo a topology change, becoming a disjoint union offlat space (with type II strings) and several cones (with type 0 strings).

3.2. RG flow: GLSM analysis

Although the C/Zn orbifold breaks spacetime supersymmetry, it preserves N = (2, 2)

supersymmetry on the worldsheet. Furthermore, the most relevant operator in each twistedsector is a chiral primary operator. We can therefore use the power of supersymmetry tostudy worldsheet RG flows seeded by these vertex operators. This is what Vafa did in hisconstruction [70] of these RG flows using a gauged linear sigma model (GLSM), which wewill review in this subsection. He also showed using Hori–Vafa mirror symmetry [76] that theprocess has an elegant dual description in terms of a Landau–Ginzburg theory, which we willnot review here.

The coupling constant e for a gauge theory in two dimensions has units of mass, sothe theory is free in the ultraviolet. Typically, one uses a GLSM to construct an interestingconformal field theory as its infrared fixed point [77]. In this case, we will construct an entireRG flow, not just a single fixed point, and the strategy is rather clever. We will choose aGLSM whose infrared fixed point is C, so we have a flow from a free gauge theory in the UVto a free sigma model in the IR. Moreover, by construction this flow will pass close to theunstable C/Zn fixed point, and by taking an appropriate limit of the parameters in the GSLMwe can make it pass right through that fixed point. In this limit, the flow actually breaks intotwo consecutive flows: the first from the free gauge theory to C/Zn, and the second (whichis the one of interest to us) from C/Zn to C. Finally, we will show that in the initial stagesof this second flow the deviations from the C/Zn CFT are precisely those we would get byperturbing it by the most relevant vertex operator, namely the chiral primary operator in thefirst twisted sector. This proves that the infrared physics of the C/Zn theory perturbed by thisoperator is indeed the free sigma model.


The GLSM we need is very simple. It has a U(1) gauge group and two chiral superfields �1

and �−n.6 The subscripts indicate their respective charges, so that under gauge transformationsthey transform as

�1 → eiα�1, �−n → e−inα�−n. (3.4)

The gauge field vµ is the lowest component of a vector superfield V , and the field strength v01

resides in the super-field strength �, which is a twisted chiral superfield. The action is

S = 1

2π

∫d2σ

[∫d4θ

(�1 eV �1 + �−n e−nV �−n − 1

2e2|�|2

)− Re

∫d2θ t�

]. (3.5)

The complex coupling constant t = r+iθ parametrizes both the Fayet–Ilipoulos term r∫

d4θ V

and the theta term θv01.The quantum theory generated by (3.5) is super-renormalizable; in fact, thanks to the

high degree of supersymmetry, all correlators are rendered finite after a simple one-looprenormalization of the FI term:

r( ) = −(n − 1) ln

(

0

), (3.6)

where 0 is defined to be the energy scale at which r vanishes.An important technical point is that we will need to be able to impose GSO projections on

the CFTs we find at low energies. We will now describe an exact Z2 symmetry of the GLSMthat flows at low energies to (−1)FL (similar remarks apply to (−1)FR ), allowing us to imposea GSO projection on the GLSM. Classically, the gauge theory has a discrete chiral symmetrygiven by ψ− → −ψ−, vµ → −vµ, λ+ → −λ+, where ψ and λ are the spinors in the chiral andvector superfields respectively, and the subscript refers to the chirality of the spinor (this anelement of the left-moving R-symmetry group). This change of variables induces a Jacobian(−1)(n−1)

∫d2σv01/2π in the path integral, which is trivial if and only if n is odd. The oddness of

n thus ensures that (−1)FL is non-anomalous in the quantum theory.Now let us consider the low-energy dynamics of (3.5). In Wess–Zumino gauge, the terms

involving the auxiliary gauge field D are

1

2π

∫d2σ

(D2

2e2+ D(|ϕ1|2 − n|ϕ−n|2 − r)

), (3.7)

where ϕ1, ϕ−n are the lowest components of �1,�−n respectively. At energies small comparedto e we may restrict attention to fluctuations on the manifold of supersymmetric, zero-energyconfigurations satisfying

−D

e2= |ϕ1|2 − n|ϕ−n|2 − r = 0. (3.8)

The moduli space of classical vacua is parametrized by ϕ1 and ϕ−n constrained by (3.8), modulogauge transformations: the dynamics is that of a one complex-dimensional supersymmetricsigma model.

According to (3.4), at low energies the FI parameter r becomes large. Since, according to(3.8), |ϕ1|2 must be positive, we can use the gauge freedom to make ϕ1 real and positive andthen use (3.8) to solve for ϕ1: ϕ1 =

√n|ϕ−n|2 + r . We then plug this solution into the kinetic

terms,1

2π

∫d2σ(−Dµϕ1Dµϕ1 − Dµϕ−nDµϕ−n), (3.9)

6 To follow the argument as presented here it is not necessary to be familiar with the details of N = (2, 2)

supersymmetry, such as the definitions of the various types of superfields, although certain points will have to betaken on faith. The necessary background material is explained in [76, 77].


and classically integrate out the gauge boson vµ. (When r is large ϕ1 is also large, so the gaugeboson is very massive, and since the gauge theory is free in the UV the classical approximationis valid in this limit.) We find that the dynamics in the limit r → ∞ is governed by the flatsigma model S = − 1

2π

∫d2σ∂µϕ−n∂

µϕ−n.On the other hand, if 0 � e then there exists an intermediate range of energy scales

0 � � e for which (3.8) is valid but r is large and negative. In this case, it is ϕ−n that wecan make real and positive and integrate out along with the gauge field. Now ϕ1 parametrizesthe target space with a flat metric. Note, however, that our gauge choice leaves unfixed aresidual Zn group of gauge transformations, generated by

ϕ1 → e2π i/nϕ1. (3.10)

Consequently the theory in this range of energy scales is nothing but the C/Zn conformalfield theory. In order to make this statement exact, we need to take the limit r → −∞,while keeping the energies small compared to e. Thus in the limit e/ 0 → ∞, the RG flowseparates into two stages: in the first the theory flows from a free gauge theory to C/Zn, andin the second from C/Zn to C.

In order to make the story complete, we need to show that this second flow is preciselythe one that is seeded by a tachyon vertex operator of the C/Zn sigma model, and in particularby V1, the lowest dimension operator in the first twisted sector. We first note that, sincethe flow preserves supersymmetry, it must be seeded by one of the chiral primary operators.To find out which one, let us see how the physics as described by the GLSM changes asr increases from minus infinity. When it is large and negative but finite, the gauge theoryadmits fractional instantons [77], vortices in which the phase of ϕ−n winds at infinity on the(Euclidean) worldsheet. In such a configuration, we cannot (as we did in the r → −∞limit above) make ϕ−n everywhere real and positive. The path integral includes a sum oversectors containing arbitrary numbers of instantons and anti-instantons, and integrals over theirpositions. Now, due to the configuration of the gauge field, the phase of ϕ1 jumps by 2π/n

upon circling a fractional instanton. Therefore the effect of an instanton on a correlator isreproduced in the C/Zn sigma model by the insertion of V1, which contains a twist fieldthat enforces precisely this behaviour, i.e. (3.2) (with k = 1). The sum over instantons andanti-instantons that occurs in the GLSM path integral is reproduced by perturbing the C/Zn

sigma model action by∫d2σ e(r+iθ)/nV1 + c.c. (3.11)

The coefficient of V1 is the GLSM action of a single instanton.By replacing the chiral superfield �1 in the GLSM by one of any odd charge k less than

n, it is easy to show that the RG flow seeded by

TkVk + c.c. (3.12)

has as its IR limit the lower-order orbifold C/Zk . Note that the restriction to odd k, required inorder to be able to impose a GSO projection, implies no loss of generality, since V ∗

k = Vn−k .

3.3. RG flow: gravity regime

The argument of the previous subsection was effective in showing that the endpoint of RGflow seeded by the tachyon vertex operator V1 is flat space, but the question of how the floweffected the transformation of C/Zn into C remained a bit obscure. At early RG ‘times’, theC/Zn CFT is perturbed by the tachyon vertex operator, which is a twist field and so non-localin spacetime, so we should not expect any geometric picture. In the late stages of the flow, in


order to find out what is happening to the geometry we should carefully integrate out the gaugefield at large but finite values of r. It turns out that the metric for ϕ−n is actually flat only in theregion |ϕ−n|2 � r , and remains conical for |ϕ−n|2 � r (see appendix B of [78] for a detaileddiscussion). As r grows according to (3.6), this flat region grows in size, so that from the pointof view of an observer at any fixed distance from the origin the space does indeed eventuallybecome flat (although strictly speaking the asymptotic geometry never changes). This is asatisfying picture because it respects the target space quasi-locality property that worldsheetRG flow inherits from string theory.

At late RG ‘times’, we therefore have a nonlinear sigma model whose target spacegeometry is a cone with an expanding smooth cap. In this regime, the correct tool forquantitatively following the flow is not the GLSM, but rather the one-loop sigma-model betafunctions. The beta function equation for the metric7 is the Ricci flow equation:

dGµν

dλ= −α′Rµν + ∇µξν + ∇νξµ, (3.13)

where λ = −ln is the RG ‘time’. The vector ξ is arbitrary, representing the freedom tomake continuous changes of target space coordinates along the flow.

In principle solving this equation would require knowing the initial conditions, that is thegeometry of the target space at the value of λ where string effects first become negligible.However, since Ricci flow is diffusive in character, we would not actually expect the late-timebehaviour to depend very much on the details of those initial conditions. For example, theheat equation has the property that for any initial conditions (with suitable asymptotics), aftera sufficiently long time the distribution will be approximately Gaussian. Why a Gaussianrather than some other function? Because a Gaussian is the only function that retains itsshape under diffusion—over time it simply becomes a broader and flatter Gaussian. We cansimilarly expect that, regardless of the details of the initial conditions, the target space willafter some time evolve into a geometry that changes under Ricci flow (3.13) only by an overallscaling (it is important that the asymptotic geometry is self-similar, otherwise this would notbe possible).

There is in fact a unique smooth, rotationally symmetric two-dimensional geometry thatis asymptotically conical with a given deficit angle, and that changes under Ricci flow only byan overall rescaling. It is [62]

ds2 = λ

(f 2 dr2 +

r2

n2dθ2

), ξ = 1

2rf dr, (3.14)

or by making the λ-dependent change of coordinates r = ρ/√

λ,

ds2 = f 2 dρ2 +ρ2

n2dθ2, ξ = 1

2λρf (1 − f ) dρ. (3.15)

The function f interpolates smoothly between f = 1/n at r = 0 and f = 1 at r = ∞:

f =[

1 + W

((n − 1) exp

(n − 1 − r2

2α′

))]−1

, (3.16)

where W is the product log function (the inverse function of x ex).The RG flow parameter λ in this solution ranges only from 0 to ∞. The limits λ → 0

and λ → ∞ are best taken at fixed ρ (rather than fixed r), giving respectively the cone andthe plane. For finite λ, as shown in figure 2, the geometry is conical at infinity but smoothat the origin, with typical radius of curvature

√λα′. Thus the curvature which was initially

7 The B-field must be trivial because we are in two dimensions. The dilaton may be doing something non-trivial, butthat will not affect the flow of the geometry.


Figure 2. Cross section of the cone (3.14), (3.15) in the case n = 3.

concentrated at the origin eventually diffuses over an infinite area. Note that we can only trustthis solution for λ � 1, when the curvature is small enough for equation (3.13) to be valid, sothere is no significance to the fact that we cannot continue this solution to negative λ.

In the previous subsection, we saw that the flow seeded by the tachyon vertex operatorVk leads to the lower-order orbifold C/Zk . The solution described in this subsection can begeneralized to describe such a flow simply by replacing n by n/k in the definition of f (3.16).

3.4. Time evolution: substringy regime

We now turn to the classical spacetime dynamics of tachyon condensation in C/Zn. Inprinciple, the proper framework for studying these dynamics is closed string field theory. Thissubject is still in its infancy, however, and a direct attack remains out of reach. Instead, weshall find two opposite limits for which present tools can help us infer important qualitativefeatures of the dynamics. In this subsection, we will discuss the very early dynamics, beforenonlinear effects kick in, and in the following subsection the very late dynamics, after stringyeffects are washed away.

The classical dynamics of closed string fields does not depend on the string coupling,which appears in the action only in an overall prefactor. The supergravity actions are familiarexamples of this, but it is true also for closed string tachyons such as the ones we are studying:

ST = 1

2κ2

∫d8x

[−

∑k

(∂µTk∂

µTk + m2kT

2k

)+ Lint

]. (3.17)

Here Lint includes the tachyons’ interactions with each other as well as with the other closedstring fields. The coefficients of the terms in Lint (as well as m2

k) are of order 1 in string units.Therefore the linearized equation of motion for Tk , whose solution is

Tk(x0) = Tk(0) cosh(|mk|x0) +

1

|mk| Tk(0) sinh(|mk|x0), (3.18)

is valid as long as T 2k � 1 and α′T 2

k � 1. This will be true for a time that is logarithmic inTk(0) and Tk(0), so if we tune the initial values and velocities to be very small, then (3.18)will be valid over many string times.

Simply writing down the solution (3.18), however, is not very satisfying. We really wantto know what is happening to the spacetime while the tachyon field is growing according to(3.18). How is its geometry changing? This is not a sensible question from the point of viewof the string: the worldsheet theory is the C/Zn orbifold theory perturbed by the marginaloperators Tk(x

0)Vk , which is simply not a sigma model. Luckily, there is another object inthe theory we can use to probe the geometry: the D-brane. D-branes are very massive yetelementary and therefore good at probing very short spacetime distances [79]. The theory on


U(1)1

...

U(1)2

U(1)3

U(1)4

U(1)nZ1,2

Z2,3

Z3,4

...

Zn,1

Figure 3. Quiver diagram for the scalars of the C/Zn probe theory.

a D-brane localized in a space such as C/Zn reduces at low energies to a moduli space andthe geometry of that moduli space is how the brane ‘sees’ the space. In the case of C/Zn,the moduli space is simply C/Zn itself, as we will show below. However, when the tachyonfield is turned on, the theory gets perturbed and as a result the moduli space is deformed. Wewill see that it remains asymptotically conical, but with the singularity replaced by a smoothcap whose area is proportional to the value of the tachyon field. We will merely sketch thecalculations here; details can be found in the papers [65, 66].

The theory on a D-brane in C/Zn is the theory on n D-branes in C, projected down to theZn-invariant states (there are no twisted sectors, because we are dealing with open strings).In the massless sector, the result is a U(1)n GLSM with n complex transverse scalars andn fermions. (If the brane has other transverse directions besides the C/Zn ones, then therewill also be other transverse scalars, but we are not interested in those.) Each scalar is abi-fundamental under a different pair of U(1) s. The pattern of charges is best summarizedby the quiver diagram shown in figure 3. Each node j represents a different U(1), whosegauge field we will denote Ajj . Each arrow, connecting a node j to a neighbouring one j + 1,represents a scalar Zj,j+1, which has charge +1 under U(1)j and −1 under U(1)j+1. Thefermions are also bi-fundamentals, but with a different quiver diagram.

Unlike the theories we studied in subsection 3.2, this GLSM is non-supersymmetric:supersymmetry has been broken by the orbifold projection, just as it was in the closed stringtheory. However, its classical action is still determined by supersymmetry, since it is simply anorbifolded version of a supersymmetric theory (and there are no twisted sectors). In particular,in addition to their gauge interactions, the scalars have a D-term potential:

V = 1

2

∑j

(|Zj,j+1|2 − |Zj−1,j |2)2. (3.19)

To explain the derivation of the moduli space from this potential, we can do no better than toquote the original reference [65] (subsection 2.2):

The vanishing of the potential (3.19) implies that the magnitude Zj,j+1 is independentof j . Of the nU(1) symmetries, the diagonal decouples. The remaining n − 1gauge symmetries can be used to set the phases of the Zj,j+1 equal as well, sothat the common value Zj,j+1 = Z parametrizes the branch. The branch is thus two-dimensional, as it should be for the interpretation of a probe. The gauge choice leavesunfixed a Zn gauge symmetry, whose generator is exp

(−2π i∑

j jQj/n). This

identifies Z → e2π i/nZ, so the probe moduli space is indeed the C/Zn spacetime.


The authors then proceed to check that the metric on this moduli space, obtained byintegrating out the massive gauge bosons, is indeed that of C/Zn.

Singularities in moduli spaces are typically associated with points of enhanced gaugesymmetry and this case is no exception: at the origin all of the Zj,j+1 vanish simultaneouslyand the U(1)n−1 gauge symmetry is restored. The Zj,j+1 are related to each other by theZ′

n quantum symmetry of the orbifold, which manifests itself in the probe theory as theglobal symmetry generated by the permutation j → j + 1 (mod n), i.e. the discrete rotationalsymmetry of the quiver diagram (figure 3). The Z′

n symmetry is thus responsible for the factthat the Zj,j+1 all vanish at the same point on the moduli space, and therefore for the conicalsingularity appearing at that point. By this logic, we would expect that turning on tachyonsand breaking the symmetry should make the singularity go away.

To test this conclusion, we should see how the operator∑

k TkVk added to the worldsheetLagrangian changes the open string dynamics on the D-brane, in particular the low-energygauge theory. This was done by the authors of [66], who calculated the disc diagram with twoscalar vertex operators on the boundary and a tachyon vertex operator in the interior. Theyfound that the scalar potential (3.19) is changed to quadratic order by the addition of a term

�V = −∑

j

ζj (|Zj,j+1|2 − |Zj−1,j |2), (3.20)

where ζj are the discrete Fourier transform of the tachyon vevs Tk ,8

ζj = C

2i

n−1∑k=1

e−2π ijk/n(−1)kTk, (3.21)

with C is a real constant. Adding (3.20) to (3.19) deforms the ‘D-term constraint’ to

|Zj,j+1|2 − |Zj,j−1|2 = ζj . (3.22)

In particular, for generic tachyon vevs all the Zj,j+1 must be different. If only one of them canvanish at a time, then there will be no point of enhanced symmetry on the moduli space andtherefore no singularity. Indeed, the authors of [65] showed this to be the case by integratingout the gauge bosons and finding the metric

ds2 = n(r) dr2 +r2

n(r)dθ2. (3.23)

Here n(r) is a function that depends on the ζj parameters; when they vanish it equals nidentically, but for generic values of the ζj it interpolates smoothly between n(0) = 1 andn(∞) = n. The curvature radius of the smooth cap goes as the square root of the ζj parameters,and hence of the value of the tachyon fields.

3.5. Time evolution: gravity regime

What happens during the stringy phase of the decay, when the field T and its velocity T arestring scale? Lacking the tools for a quantitative analysis in this regime, let us see how farwe can get by general reasoning. First we recall that, from the spacetime point of view,the orbifold fixed point is a localized positive-energy object, a soliton of the closed string

8 The reader may object that we are treating the Tk as constants, whereas in fact they are operators depending onx0. In this sense, the calculation that follows should be taken as providing heuristic support for the smoothing outof the cone, since in practice the ζj parameters will be varying in time too fast for the moduli space approximationto be valid. One way around this problem is to give the tachyons not a time- but a space-dependence, for exampleTk(x

1) = Tk(0) cos(|mk |x1). A D-brane localized in the x1 direction would be able to resolve this x1 dependence,and its moduli space would be a fibration over the x1 direction of the one we find for the orbifold directions.


field theory. Furthermore, since gravity does not act as a binding force on co-dimension-2objects, it is not surprising that this configuration should be unstable (besides the fact that it isnon-BPS). When the soliton decays, nothing stops its energy from diffusing outwards (barringa cataclysmic event such as a the formation of a tear in spacetime). The energy density is atfirst so high that we may expect it to be transformed into very massive strings, as occurs inthe decay of D-branes discussed in subsection 2.4. These strings will eventually decay intomassless ones that fly away from the origin at the speed of light. When this happens, and oncethe energy densities and curvatures are below the string scale, then the low-energy effectivetheory—supergravity—becomes an appropriate description. We re-emerge into the light, aregime where a quantitative analysis is once again possible.

In this subsection, we will study the supergravity equations of motion for this system, andfind that they are surprisingly tractable, particularly considering the fact that we are dealingwith a time-dependent, non-supersymmetric process. The discussion summarizes the paper[72], which should be consulted for details; see also [71]. We first show that the dynamicsreduces to a massless scalar, the dilaton, minimally coupled to 2 + 1 gravity. Recalling thatgravitons cannot go on-shell and carry energy in 2+1 dimensions, we see that the dilaton playsan indispensible role, namely to carry the energy released in the decay off to infinity. Usuallyin general relativity such coupled matter-gravity equations of motion are very difficult to solvedue to the problem of gravitational back-reaction. However, we will see in this case by usingthe rotational symmetry and making a judicious choice of coordinate system that the equationof motion for the dilaton decouples from the back-reaction of the geometry: it is the same asthat for a dilaton in Minkowski space. This allows the equations to be solved in two steps:first, the dilaton equation of motion is straightforwardly solved, then the back-reaction of thegeometry is found by solving a first-order equation for a metric coefficient that is sourcedby the dilaton’s energy. Among other things, this method guarantees that under non-singularinitial conditions the energy will indeed radiate to infinity and no singularity can form.

Let us begin by showing that the dilaton and metric are the only fields we need to consider.Since the tachyons are all in the NS–NS sector, we can consistently truncate to that sector(remember, we are considering a classical process). We can furthermore consistently truncatethe dynamics to 2 + 1 dimensions, namely the two of the orbifold plus time [65]. (The CFTdescribing the extra seven dimensions is coupled to the (2 + 1)-dimensional CFT neither inthe initial configuration nor by the tachyon vertex operator, and therefore decouples entirely.)Finally, the equation of motion for the B-field requires its field strength to be a constantmultiple of the volume form on the unreduced 2 + 1 dimensions. This constant vanishes in theinitial orbifold configuration, and must therefore vanish everywhere.

The Einstein equation can be written as

Rµν = 4∂µ�∂ν� (3.24)

(we are working in the Einstein frame). By rotational invariance the θ–θ component of thisequation is Rθθ = 0 and one can show that the general solution is a metric of the form

ds2 = e2σ(t,r)(−dt2 + dr2) + r2 dθ2. (3.25)

The static orbifold is described by a constant value of σ :

σ = ln n (C/Zn). (3.26)

The great advantage of the metric (3.25) is that the unknown function σ drops out of thedilaton’s equation of motion (assuming again that it does not depend on θ ); just as in flatspace, we have (

∂2t − 1

r∂rr∂r

)� = 0. (3.27)


15 10 5 0 5 10 15r

15 10 5 0 5 10 15r

15 10 5 0 5 10 15r

15 10 5 0 5 10 15r

Figure 4. Numerical simulation of decay of C/Z3 to the plane, using the source function (3.29).Each of the four figures shows a constant t slice, at t = −3 (top left), t = 3 (top right), t = 9(bottom left), and t = 15 (bottom right). For each t the field configuration �(t, r) is plotted abovea cross section of the geometry.

This linear equation can be straightforwardly solved using standard methods. One implicationof it is that, regardless of what happens during the decay, the dilaton will at late times return toits original value. Given a solution to (3.27), the back-reaction of the dilaton’s energy on thegeometry can be determined by the remaining components of the Einstein equation (3.24):

∂tσ = 4r∂t�∂r�, ∂rσ = 2r((∂t�)2 + (∂r�)2). (3.28)

As an example, figure 4 shows a numerical solution (obtained using Mathematica) for thedecay of C/Z3 to the plane. The decay occurs at t = 0 and, in order to simulate the energybeing dumped from the defect into the dilaton, a source term was added to the right-hand sideof (3.27): (

∂2t − 1

r∂rr∂r

)� = β e−t2−r2

, (3.29)

where the normalization β = 1.2 was set by requiring σ to decrease by ln 3. We see that theoutgoing dilaton wave carries with it the cone’s curvature, leaving behind an expanding flatdisc. Note that the geometry at late times is qualitatively different from that at late ‘times’ in


the worldsheet RG flow (figure 2), which is much rounder, reflecting the difference betweenthe first- and second-order equations governing the two different types of evolution.

4. Decay of twisted circles

4.1. Introduction

From the example of the C/Zn it might appear that in order to localize closed string tachyonsit is necessary to have some singularity in the spacetime, a defect on which the tachyons live.In fact this is not the case, as shown by the twisted circle compactifications that are the subjectof this section. They are a simple generalization of the C/Zn orbifold in which the action ofthe rotation generator on the plane is made free by combining it with a translation in a thirddirection, the ‘circle’ or y direction:

e2π iζJ e2π iRpy . (4.1)

This operator acting on C × R generates a Z (not Zn) group, so there is no quantizationcondition on the the rotation parameter ζ , which can vary continuously from 0 to 2. Spacetimesupersymmetry is broken as long as ζ �= 0. The special values ζ = 0 and ζ = 1 correspondrespectively to normal circle and Scherk–Schwarz (or ‘thermal circle’ or ‘interpolatingorbifold’) compactifications, times the plane.

Dimensionally reducing these models along the U(1) isometry generated by ζJ + Rpy

yields a ‘Melvin’ configuration of the Kaluza–Klein gauge field; T-dualizing along it yields aco-dimension-2 flux-brane of the NS–NS 3-form flux.

The spectrum of type II strings on the twisted circle is straightforward to derive [80–82](see [83, 84] for a brief review). The lightest twisted field has a mass given by

m2 = R2

α′2 − 2ζ ′

α′ , (4.2)

where ζ ′ ≡ 1 − |ζ − 1|. Thus the system is perturbatively unstable if and only if

ζ ′ >R2

2α′ . (4.3)

Even when the theory is perturbatively stable, however, as long as supersymmetryis broken it will have a non-perturbative instability. For example, the Scherk–Schwarzcompactification (ζ = 1) is well known to decay via a bounce to nothing at all [85]. Thenon-perturbative decay of the Melvin by nucleation of branes has also been studied [86–88];there is disagreement in the literature as to the final results.

When the twist parameter is rational, ζ = 2m/n, the twisted circle can be thought of as aZn orbifold of C × S1

nR (where S1nR denotes a normal circle of radius nR). This is convenient,

since it means we have only a finite number (n − 1) of twisted sectors. If n is even (andtherefore m is odd) then the orbifold group includes (−1)F eπ inRpy . The strings twisted by thiselement see the space as a Scherk–Schwarz circle times a plane, and are therefore delocalized,just as in the case of C/Zn at even n. We will therefore take n odd; the result is that stringsin all twisted sectors are localized, since they must stretch in order to move away from theorigin of the plane. For simplicity, we will restrict ourselves further in this section to the casem = (n + 1)/2, i.e. to the orbifold generated by

(−1)F e2π iJ/n e2π iRpy . (4.4)

This orbifold has a couple of interesting limits. In the limit n → ∞ with R fixed, it goesover to the Scherk–Schwarz circle times the plane. On the other hand, in the limit R → 0 with


n fixed, it goes over to our familiar C/Zn. The way to see this is to note that the original circle(of radius nR) is becoming very small. A T-duality yields a circle of radius α′/nR → ∞times the plane. Only strings with zero winding, i.e. py = 0, survive, so the orbifold action(4.4) is reduced to (−1)F e2π iJ/n, and we have C/Zn × R. Finally, if we simultaneously taken → ∞ and R → 0 holding nR fixed, we end up, after a T-duality with respect to J/n + Rpy ,with an H flux-brane in type 0 theory.

It is reasonable to conjecture by analogy with the C/Zn system that the condensation of thetachyons in this orbifold leads to the orbifold being undone, leaving the C × S1

nR background.The main evidence for this conjecture comes not from time evolution or worldsheet RG flow,but from a process that can be considered somewhere in between the two, namely Liouville flow[63]. Following [84], in the next subsection we will construct a CFT with a four-dimensionaltarget space, one of whose dimensions is a Liouville direction. (See also [89, 90] for differentapproaches to the same problem.) The other three directions are fibred over this Liouvilledirection in such a way that in one limit—the ‘UV’ end—they form the twisted circle, and atthe other limit—the ‘IR’ end—they are simply C × S1

nR .

4.2. GLSM construction of the Liouville flow

The Liouville flow CFT is straightforwardly constructed using the GLSM techniqe. We willsketch it here; for details, see [84]. The GLSM we need is almost identical to the one weused in subsection 3.2; it has a U(1) gauge group and two chiral superfields �1 and �−n withcharges 1 and −n respectively. However, in addition it has an axionic field P = P1 + iP2. Thetarget space for this field is a cylinder, P ∼ P + 2π i, and under a gauge transformation withgauge parameter eiα it is translated rather than rotated:

P → P + iα. (4.5)

The action for this system is

S = 1

2π

∫d2σ d4θ

[�1 eV �1 + �−n e−nV �−n +

k

4(P + P + V )2 − 1

2e2|�|2

]. (4.6)

As in the GLSM of subsection 5.1, the low-energy dynamics is described by a sigma modelonto the manifold of supersymmetric zero-energy configurations, modulo gauge equivalences.The D-term condition is

−D

e2= |ϕ1|2 − n|ϕ−n|2 + kP1 = 0. (4.7)

Here k is the coefficient of the kinetic term of the P field. Note that P1 plays the role of adynamical FI term. As in subsection 3.2, in order to impose the type II GSO projection on thelow-energy theory we should mod this GLSM out by an appropriate Z2 R-symmetry.

At positive values of P1, ϕ−n must get a vev, so the gauge group is Higgsed down toZn. Since ϕ1 parametrizes C and P2 parametrizes an S1 of radius

√kα′, the target space

is the twisted circle(C × S1

nR

)/Zn, with R = √

α′k/n. That the metric is indeed that of(C × S1

nR

)/Zn can be checked as usual by integrating out the gauge bosons [78]. This metric

will only be valid at large P1, since then the gauge bosons are very massive and quantumeffects are suppressed. On the other hand, when P1 is negative it is ϕ1 that gets a vev, andthe gauge group is completely broken. The target space here is simply C × S1

nR , with this Cparametrized by ϕ−n. Again, the metric must be checked.

In [84], it was argued that P1 may be regarded as a Liouville or worldsheet scale direction,with P1 → ∞ corresponding to the UV and P1 → −∞ to the IR. Consequently, the changein the transverse directions as P1 goes from infinity to minus infinity mimics their flow as


the energy scale goes from the UV to the IR. Another way of thinking about it is that thecoordinate −P1 can be regarded as the real time t after a Wick rotation. We conclude that theresult of tachyon condensation in the twisted circle is that the Zn orbifold is lifted, and thetheory decays into a supersymmetric circle compactification.

What does this result say about the various limits described in the last subsection? Ifwe take n → ∞ with R fixed, then we would say that a Scherk–Schwarz circle decaysperturbatively to non-compact space. Taking R → 0 holding n fixed, we find that C/Zn

also decays to flat space, in agreement with the findings of the previous section. Finally,taking n → ∞ and R → 0 holding nR fixed, we find that the type 0 flux-brane decays toa supersymmetric circle in type II theory (more precisely, 0A would decay to IIB and 0Bto IIA).

Acknowledgments

We would like to thank all of our collaborators and several of our colleagues for many usefuldiscussions over the span of several years on the topics reviewed in this paper. The workof MH was supported by a Pappalardo Fellowship from MIT and by funds provided by theUS Department of Energy under cooperative research agreement DF-FC02-94ER40818. Thework of SM was supported in part by DOE grant DE-FG03-91ER40654, in part by NSF grantnumber PHY-0239626 and in part by a Sloan Fellowship. TT was supported in part by DOEgrant DE-FG03-91ER40654.

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Closed string tachyon condensation: an overview