LLiving with the Wrong Sign
Patrick Cooper, Sergei Dubovsky, and Ali Mohsen
Center for Cosmology and Particle Physics,Department of Physics, New York UniversityNew York, NY, 10003, USA
Abstract
We describe a UV complete asymptotically fragile Lorentz-invariant theory exhibit-ing superluminal signal propagation. Its low energy effective action contains “wrong”sign higher dimensional operators. Nevertheless, the theory gives rise to an S-matrix,which is defined at all energies. As expected for a non-local theory, the correspond-ing scattering amplitudes are not exponentially bounded on the physical sheet, butotherwise are healthy. We study some of the physical consequences of this S-matrix. a r X i v : . [ h e p - t h ] D ec Introduction
A number of interesting apparently consistent low energy effective theories giving rise to longdistance modifications of gravity has been constructed over the past decade [1–5]. Explainingthe observed acceleration of the Universe following this path, which was one of the majormotivations for these developments, remains a long shot. Nevertheless, these models are ofdefinite interest from both theoretical and phenomenological points of view. In particular,they give rise to a number of striking observational signatures, including anomalous preces-sion of the Moon perihelion [6, 7], a strong monochromatic gravitational line from a massivegraviton [8], hairy supermassive black holes [9], an exotic cosmic microwave background B -mode spectrum [10], or a violation of the equivalence principle for compact astrophysicalobjects [11].On the theoretical side, all these models are effective field theories with a very lowcutoff scale and it remains to be seen whether any of them can be UV completed intoa microscopic theory with acceptable physical properties. There is a common underlyingphysical reason why finding a UV completion for these models is hard. One of the prices (orgains, depending on a viewpoint) to pay for affecting gravity at long distance scales is thepossibility of superluminal signal propagation. This results in a tension between locality andcausality making it impossible to construct a conventional UV completion [12]. The tensionis especially severe in long distance modifications of gravity possessing a Poincar´e invariantground state, as required for applying the arguments of [12]. A somewhat related tension inLorentz-violating scenarios arises in the presence of black hole horizons [13].The simplest toy model, where the arguments of [12] apply, describes a single Goldstoneboson X with an effective Lagrangian of the form L = 12 ( ∂X ) + c Λ ( ∂X ) + . . . . (1)It is straightforward to construct a UV completion for this model if the constant c is positive(we work in the mostly “ − ” signature). On the other hand, despite its benign appearance,this simple effective field theory cannot be UV completed into a consistent local microscopicquantum theory for negative values of c . The physical reason for this is the existence ofclassical backgrounds (such as X ∝ t ) exhibiting superluminal propagation of small pertur-bations.At this stage two viewpoints are possible. The conservative conclusion is that this prob-lem signals an incurable pathology and effective field theories with wrong sign operatorsshould be discarded. However, an adventurous person may argue that superluminality sim-ply indicates that one should look for a UV completion beyond the realm of conventionalquantum field theories. Of course, to justify the second viewpoint one needs to construct anexample of such a UV completion, so that it becomes possible to judge whether its physicalproperties are acceptable. The goal of this note is to provide a first example of this kind andto study its physical properties. 1 disclaimer is in order. All of our construction operates in two space-time dimensions.Two-dimensional theories are special in many respects. In particular, unlike in higher dimen-sional theories, in two dimensions superluminality does not allow one to construct smoothbackgrounds exhibiting closed time-like curves. Somewhat related to this, in two dimensionsit is possible to introduce an alternative causal structure, which is compatible with Lorentzsymmetry, yet allows for superluminal (in fact, instantaneous) signal propagation. Basedon this observation examples of UV complete Lorentz invariant superluminal theories havealready been constructed in the past [14, 15]. However, the setup considered here relies onpeculiarities of two-dimensional physics to a much lesser extent and the possibility of itsextension to higher dimensional models appears more likely. Our construction is based on recent progress achieved in understanding the dynamics of longstrings [16, 17]. One starts with a classical Nambu–Goto action, describing ( D −
2) scalarfields X i (transverse perturbations of a string), S NG = − (cid:96) − s (cid:90) d σ (cid:112) − det ( η ab − (cid:96) s ∂ a X i ∂ b X i ) , (2)where (cid:96) − s is the string tension. At the quantum level this may be considered as an effectivetheory similar to the Goldstone Lagrangian (1). This theory is naively non-renormalizable.Famously, however, it can be consistently quantized when the number of flavors is equal to24, giving rise to a critical bosonic string [18].The traditional treatment of string dynamics focuses on the properties of short (open orclosed) strings, which is natural if the goal is to provide a target space-time interpretation.However, for the purpose of this paper we are mainly interested in a purely two-dimensionalinterpretation of (2). From this perspective the most natural and simplest set of observablesto consider are the S -matrix elements for the scattering of worldsheet perturbations of aninfinitely long string. Geometrically these represent wiggles (or phonons) propagating alongthe string. As explained in [17], in this language the critical string can be defined by its exact S -matrix. The theory is integrable and reflectionless, so that the two-particle scattering ispurely elastic and is characterized by the phase shift of the form e iδ ( s ) = e is(cid:96) s / , (3)where s is the Mandelstam variable. Phase shifts for multiparticle processes are obtained bysumming pairwise the phase shifts of all the colliding particles.Despite its simplicity, the phase shift (3) exhibits a number of remarkable properties, andcorresponds to an integrable theory of gravity rather than to a conventional field theory. Inparticular, the UV asymptotics of this theory are not controlled by a conventional fixed point.2his is clear from the form of the phase shift (3) which retains a non-trivial dependence onthe microscopic length scale (cid:96) s at arbitrarily high energies. Clearly, this is incompatible withscale invariant behavior in the UV. This gives rise to a number of unconventional features,and this type of asymptotic behavior was called “asymptotic fragility”.An asymptotically fragile phase shift (3) satisfies all the properties expected from ahealthy S -matrix. In particular, it is analytic and polynomially bounded everywhere on thephysical sheet , which is usually taken as a definition of locality in the S -matrix language.On the other hand, it exhibits an essential singularity at s → ∞ , which prevents one fromdefining local off-shell observables, corresponding to (3). This is one of the reasons whythis theory is gravitational, rather than a conventional field theory. Another gravitationalproperty of the model is the universal time delay proportional to the center of mass energyof the collision, δt del = 12 E(cid:96) s , which may be considered as an integrable precursor of black hole formation and evaporation.Perhaps the most direct proof that the phase shift (3) indeed defines a critical stringcomes from calculating the finite volume spectrum of the theory using the ThermodynamicBethe Ansatz technique [19, 20]. This way one exactly reproduces the spectrum of a criticalbosonic string.As a further check, it is straightforward to see that the phase shift (3) agrees with theperturbative tree-level and one-loop amplitudes following from the Nambu–Goto action (2)at D = 26. However, the action (2) is non-renormalizable and presumably at higher looporder has to be supplemented with an infinite set of scheme-dependent counterterms toreproduce the phase shift (3).Note, that the phase shift (3) defines a consistent relativistic two-dimensional theory foran arbitrary number of flavors. However, for D (cid:54) = 26 the one-loop amplitude following fromthe Nambu–Goto action differs from (3) by a rational annihilation term (the Polchinski–Strominger interaction [22]). To reproduce (3) for D (cid:54) = 26 the action (2) needs to besupplemented with a counterterm, which cancels this effect. This counterterm is perfectlyconsistent from two-dimensional perspective, but incompatible with non-linearly realizedtarget space Poincar´e symmetry.A further discussion of this family of integrable models can be found in [16, 17]. Here,instead, let us move directly to the main point and describe the superluminal setup.The basic idea is very simple and motivated by the following observation. Note that,unlike what one may be used to in more sophisticated quantum theories, the perturbativelow energy expansion for the phase shift (3) is not asymptotic. It has an infinite radius ofconvergence and the phase shift is given simply by the sum of all perturbative terms. Thissuggests that flipping the sign of the coupling constant, (cid:96) s , may also result in a well-definedtheory. D = 3 is another interesting exceptional case, c.f. [21]. δ n of the form e iδ n ( s ) = e − is(cid:96) s / , (4)where, as before, we assume (cid:96) s >
0. When expanded at low energies this scattering amplitudeviolates the positivity condition of [12]. It can be reproduced from the action (2), with (cid:96) s → − (cid:96) s , and supplemented with the same set of scheme-dependent counterterms asrequired to reproduce the conventional phase shift (3). Geometrically this action describesan infinitely long string in a space-time with ( D −
1) time coordinates and a single spatialcoordinate. The target space interpretation of such a system is highly obscure, but for ourpurposes we do not need it and will consider the system from a purely two-dimensional pointof view.In agreement with general arguments of [12] this theory is even more non-local thanthe worldsheet theory of a conventional string (3). Namely, not only does the phase shift(4) exhibit an essential singularity at the infinity, it is also polynomially unbounded on thephysical sheet. This represents an interesting step forward compared to the usual situationwith superluminal effective theories. Conventionally, these theories are non-renormalizable,and one simply concludes that superluminality indicates the absence of the UV completion.Here, the theory is non-renormalizable, but we managed to construct finite on-shell scatteringamplitudes. We still are not able to determine local off-shell observables, but the samesituation holds for the conventional string worldsheet theory, which is known nevertheless tobe an interesting and healthy physical system. So an interesting question arises to comparethe two models and to characterize in what sense the “wrong” sign theory is pathological ascompared to its “right” sign cousin, if this is really the case.In the rest of the paper we will make several steps in this direction, by probing some ofthe physics following from the superluminal phase shift (4). At the technical level all of ourcalculations are close counterparts of the corresponding steps in the analysis of the “right”sign theory (3). Essentially, they can be performed by flipping the sign of (cid:96) s in the formulasof [17]. Nevertheless, as we will see, the resulting physics turns out to be quite different. Let us start by illustrating superluminal properties of the phase shift (4) as seen in theclassical limit. For simplicity, let us consider a single flavor case, so that the correspondingclassical action is S = (cid:96) − s (cid:90) dτ dσ (cid:112) (cid:96) s ( ∂X ) . (5)In agreement with the discussion in the Introduction , superluminality is manifest when con-sidering a classical background of the form X cl = vτ(cid:96) s . π around this background takes theform S = (cid:90) dτ dσ (cid:18) v ) / ( ∂π ) − v v ) / ( ∂ τ π ) (cid:19) . This gives rise to a linear dispersion relation ω = c s k with a superluminal velocity c s = √ v . Related to this superluminality the phase shift (4) gives rise to a time advance δt ad = 12 E(cid:96) s (6)for scattering processes around the trivial background X = 0. It is instructive to see howthis time advance comes out at the classical level. For this purpose let us consider a purelyleft-moving field configuration, which is always a classical solution for the action (5), X cl = X ( τ + σ ) . Understanding the scattering of a small right-moving perturbation off this backgroundamounts to studying the null geodesics propagating in the metric induced by the classi-cal solution ds = (cid:0) η ab + (cid:96) s ∂ a X∂ b X (cid:1) dσ a dσ b = (1 + (cid:96) s X (cid:48) ) dτ + 2 (cid:96) s X (cid:48) dσdτ − (1 − (cid:96) s X (cid:48) ) dσ . The null geodesic equation is given by dτdσ = − (cid:96) s X (cid:48) + 1 (cid:96) s X (cid:48) ± , where the upper sign corresponds to a right moving excitation which experiences a non-trivialtime shift. We see this is a time advance since τ (cid:48) <
1. Note that τ (cid:48) can even become negativefor large enough X (cid:48) . In this case the right-mover moves “back in time” at intermediate times,which simply indicates that τ is not a good Cauchy time for such a background. The timeadvance is given by δt ad = (cid:90) ∞−∞ dσ (1 − τ (cid:48) ) = (cid:90) ∞−∞ dσ (cid:96) s X (cid:48) (cid:96) s X (cid:48) + 1= (cid:90) ∞−∞ dσ + (cid:96) s X (cid:48) ( σ + ) = (cid:96) s ∆ E, (7)where ∆ E is the energy of the classical solution relative to the vacuum. This classical timeadvance exactly agrees with (6), following from the exact quantum S -matrix, after one takesinto account that the expression (6) calculates the time shift in the rest frame of the collidingwave packets, as opposed to the “lab frame” time advance (7).5 Thermodynamics and Finite Volume Spectrum
An important piece of physical information, which becomes accessible when the exact S -matrix is known, is the finite temperature equation of state of the system, i.e. the freeenergy density as a function of temperature, f ( T ). For an integrable theory it can beextracted using the (ground state) Thermodynamic Bethe Ansatz (TBA) [19]. Moreover, fora Lorentz invariant theory, free energy determines also the vacuum Casimir energy E ( R )on a circle of circumference R through E ( R ) = Rf ( R − ) . The derivation of the ground state TBA equations for a superluminal phase shift (4) iscompletely parallel to the one presented in [17] for a conventional string. As expected, theresult is different only by a flip of a sign in front of (cid:96) s . Namely, the free energy density isgiven by f ( R − ) = − (cid:96) − s + 12 πR D − (cid:88) j =1 (cid:90) ∞ dp (cid:48) ln (cid:16) − e − R(cid:15) jL ( p (cid:48) ) (cid:17) + 12 πR D − (cid:88) j =1 (cid:90) ∞ dp (cid:48) ln (cid:16) − e − R(cid:15) jR ( p (cid:48) ) (cid:17) , where the “pseudoenergies” (cid:15) L,R are determined from a system of integral equations of theform (cid:15) iL ( p ) = p (cid:32) − (cid:96) s πR D − (cid:88) j =1 (cid:90) ∞ dp (cid:48) ln (cid:16) − e − R(cid:15) jR ( p (cid:48) ) (cid:17)(cid:33) (cid:15) iR ( p ) = p (cid:32) − (cid:96) s πR D − (cid:88) j =1 (cid:90) ∞ dp (cid:48) ln (cid:16) − e − R(cid:15) jL ( p (cid:48) ) (cid:17)(cid:33) . Just like for a conventional string, these equations are straightforward to solve analytically.By picking up the solution which approaches the free theory in the limit (cid:96) s →
0, we obtain E ( R ) = Rf ( R − ) = − (cid:115) R l s + 4 πl s D − . Not surprisingly, this expression closely resembles the ground state energy of a bosonic string.One major difference however, is that the free energy is real at all temperatures. This meansthat the Hagedorn behavior is not present in this theory. In a sense, the thermodynamics ofa superluminal theory is less pathological than that of a conventional string.Related to this the dependence of the energy density on the pressure, ρ = p l s p , (8)does not exhibit a singularity, which was present for a conventional string.6
10 20 30 40 50 N ˜ N N crit E ( N, ˜ N ) =0 Figure 1: Above is a plot of the region where energy becomes imaginary. Blue andgreen are at D = 2, R/l s = 10 and 20 respectively. Red and Yellow are at D = 62, R/l s = 10 and 20 respectively. The allowed region is the left of each curve.The absence of the Hagedorn behavior is straightforward to understand. From a twodimensional perspective the Hagedorn behavior arises as a consequence of a large bindingenergy following from the phase shift (3). This results in the fast growth of the densityof states implying the divergence of the heat capacity in the thermodynamic limit of thetheory. By flipping the sign of (cid:96) s we replaced attraction with repulsion, which eliminatedthe Hagedorn behavior.To confirm this interpretation let us take a look at the spectrum of the excited states ofthe superluminal theory in a finite volume. Following the derivation in [17] we obtain E ( N, ˜ N ) = − (cid:32) π ( N − ˜ N ) R + R l s − πl s (cid:18) N + ˜ N − D − (cid:19)(cid:33) / , (9)where the positive integers N, ˜ N count the total left- and right-moving KK momentum of astate (in units of 2 π/R ). For a conventional bosonic string these would be called the levelsof a state. It is straightforward to see now that the two-particle binding energy is positive∆ E = E (1 , − E (0 , − E (1 , − E (0 , > i.e. , the interaction is repulsive.A further inspection of the spectrum (9) reveals, however, that in spite of the absenceof the Hagedorn behavior for the ground state, this spectrum exhibits an even more bizarre7roperty. Namely, for any value of R there are infinitely many states with imaginary energies.We illustrated this behavior in Fig.1, where we have shown the region of positive and negative E in the ( N, ˜ N ) plane for several values of D and R/(cid:96) s . This property indicate that thesuperluminal theory cannot be put in a finite volume, at least in a conventional way.Pathological at first sight, this behavior is straightforward to understand. Indeed, as aconsequence of the time advance (7) two points at the same constant time surface becomecausally connected if the “string” between them is excited to sufficiently high energy. Identi-fication of these two points is clearly a bad idea then, because it results in a closed time-likecurve. From (7) we can estimate that this happen when (cid:96) s ∆ E (cid:39) π(cid:96) s NR becomes of order R . This nicely matches what we observe in Fig.1, namely the E = 0 curvewith D = 2 (which corresponds to the classical answer for the spectrum) touches the N -axisat N crit = R π(cid:96) s . We see that the proliferation of states with imaginary energies in the finite volume cannot beconsidered as a pathology of the theory, but rather is a sign that we attempted to performan illegal action. It would be interesting to find whether the model can nevertheless beconsistently put in a finite volume. Classically, this would require identification of pointson Cauchy slices, which are background dependent and in general do not correspond toconstant time surfaces. TBA, at least in its standard form, is not an appropriate tool insuch a situation, so currently we do not know how to proceed in the quantum theory. It isalso puzzling that the TBA result exhibits an island of apparently healthy E > N > N crit (see Fig.1). Classically these contain a closed time-like curve,so that their physical interpretation is obscure.
Another interesting property of the conventional worldsheet theory is the presence of cos-mological backgrounds [17]. Let us see what these look like in the superluminal theory. Forsimplicity, we restrict to a single field case. In a (1 + 1)-dimensional world isotropy is not aconstraint, so one is looking for homogeneous solutions. Imposing an invariance under coor-dinate translations, σ → σ + const leaves one with only trivial vacuum solutions. Instead,following [17], one can look for homogeneous solutions where the isometry is generated byboosts. In fact, there is a larger set of solutions, labeled by a continuous parameter γ , suchthat the isometry is generated by the combination of a boost and a shift of the field of theform X ( σ + , σ − ) → X ((1 + (cid:15) ) σ + , (1 + (cid:15) ) − σ − ) − (cid:15)γ , (10)8here σ ± = τ ± σ . A general field configuration invariant under (10) can be presented inthe form (cid:96) s X ( σ + σ − ) = f ( σ + σ − ) + γ σ + σ − . (11)Then the field equation becomes ∂ α α ∂ α f (cid:113) α ( ∂ α f ) − γ α = 0 , (12)where α = σ + σ − . The general solution then takes the following form f ( α ) = L log( √ L − α + (cid:112) γ − α ) + γ log αγ √ L − α + L (cid:112) γ − α + C , (13)where C and L are the integration constants. The constant C can be, so that the field X remains real. To get an insight into the physics of these solutions it is instructive to inspectthe induced metric g ab = η ab + (cid:96) s ∂ a X∂ b X , which determines the propagation of small perturbations. One observes that this family ofsolutions splits into two disconnected physical branches, A and B , where the field takes realvalues. The A -branch covers the region4 α > L , γ , where one can introduce coordinates ( ρ, λ ) defined as σ ± = ρe ± λ . In these coordinates the induced metric becomes ds A = 16 ρ − L γ ρ (4 ρ − L ) dρ + 2 Lγ (cid:112) ρ − γ ρ (cid:112) ρ − L dρdλ − (4 ρ − γ ) dλ By making a shift λ → λ + f ( ρ ) we can get rid of the off-diagonal term in the metric, sothat it takes the Friedmann–Robertson–Walker form ds A = 16 ρ dρ ρ − L − (4 ρ − γ ) dλ = dτ − ( τ + L − γ ) dλ , (14)where at the last step we introduced a new time coordinate τ = (cid:112) ρ − L .The B -branch of solutions covers the region4 α < L , γ . ds B = ( γ − L + r ) dλ − dr , where r = (cid:112) L − ρ .We conclude that the A -branch describes a cosmological solution and the B -branch cor-responds to a static space-time. For γ < L the cosmology is non-singular, while the staticsolution exhibits a curvature singularity. For larger values of γ the singularity moves onthe cosmological branch. This does not appear very different from what one finds for aconventional string. The solutions take the same form there with the role of time and spaceinterchanged. There, for γ < L one finds a smooth static geometry and for γ > L anon-singular bouncing cosmology. To summarize, it is fair to admit that our results are inconclusive at this point. We did notmanage to identify a clean pathology associated with a superluminal sign. Definitely, thistheory allows one to calculate a smaller set of observables than a conventional renormaliz-able quantum field theory, but the same holds also for the conventional string world-sheettheory. Nevertheless, the latter definitely gives rise to a rich and healthy system. The ver-dict for the former is not out yet. Both theories exhibit a number of gravitational features.Gravitational theories are expected to be less predictive than quantum field theories by notallowing one to calculate local off-shell observables. Perhaps the most interesting lesson fromthe construction presented here, is that it raises the question where is the proper place tostop on a slippery road between conventional UV complete quantum field theories and non-renormalizable effective theories. The former allow sharp prediction of both on-shell andoff-shell observables. The latter do not allow sharp predictions at all in the mathematicalsense, but often are quite adequate from the practical point of view. Gravitational theorieslive in the middle.
We thank Raphael Flauger, Victor Gorbenko and Sergey Sibiryakov for numerous helpfuldiscussions. This work is partially supported by the NSF grants PHY-1068438 andPHY-1316452.
References [1] G. Dvali, G. Gabadadze, and M. Porrati, “4-D gravity on a brane in 5-D Minkowskispace,”
Phys.Lett.
B485 (2000) 208–214, hep-th/0005016 .102] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, “Ghost condensationand a consistent infrared modification of gravity,”
JHEP (2004) 074, hep-th/0312099 .[3] S. Dubovsky, “Phases of massive gravity,”
JHEP (2004) 076, hep-th/0409124 .[4] A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification ofgravity,”
Phys.Rev.
D79 (2009) 064036, .[5] C. de Rham, G. Gabadadze, and A. J. Tolley, “Resummation of Massive Gravity,”
Phys.Rev.Lett. (2011) 231101, .[6] G. Dvali, A. Gruzinov, and M. Zaldarriaga, “The Accelerated universe and the moon,”
Phys.Rev.
D68 (2003) 024012, hep-ph/0212069 .[7] A. Lue and G. Starkman, “Gravitational leakage into extra dimensions: Probing darkenergy using local gravity,”
Phys.Rev.
D67 (2003) 064002, astro-ph/0212083 .[8] S. Dubovsky, P. Tinyakov, and I. Tkachev, “Massive graviton as a testable cold darkmatter candidate,”
Phys.Rev.Lett. (2005) 181102, hep-th/0411158 .[9] S. Dubovsky, P. Tinyakov, and M. Zaldarriaga, “Bumpy black holes from spontaneousLorentz violation,” JHEP (2007) 083, .[10] S. Dubovsky, R. Flauger, A. Starobinsky, and I. Tkachev, “Signatures of a GravitonMass in the Cosmic Microwave Background,”
Phys.Rev.
D81 (2010) 023523, .[11] L. Hui and A. Nicolis, “Proposal for an Observational Test of the VainshteinMechanism,”
Phys.Rev.Lett. (2012) 051304, .[12] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, “Causality,analyticity and an IR obstruction to UV completion,”
JHEP (2006) 014, hep-th/0602178 .[13] S. Dubovsky and S. Sibiryakov, “Spontaneous breaking of Lorentz invariance, blackholes and perpetuum mobile of the 2nd kind,”
Phys.Lett.
B638 (2006) 509–514, hep-th/0603158 .[14] S. Dubovsky and S. Sibiryakov, “Superluminal Travel Made Possible (in twodimensions),”
JHEP (2008) 092, .[15] S. Dubovsky and V. Gorbenko, “Superluminal Travel, UV/IR Mixing and Turbulencein the Lineland,”
Phys.Rev.
D84 (2011) 105039, .1116] S. Dubovsky, R. Flauger, and V. Gorbenko, “Effective String Theory Revisited,”
JHEP (2012) 044, .[17] S. Dubovsky, R. Flauger, and V. Gorbenko, “Solving the Simplest Theory of QuantumGravity,”
JHEP (2012) 133, .[18] P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, “Quantum dynamics of amassless relativistic string,”
Nucl.Phys.
B56 (1973) 109–135.[19] A. Zamolodchikov, “Thermodynamic Bethe Ansatz in relativistic models. Scalingthree state Potts and Lee-Yang models,”
Nucl.Phys.
B342 (1990) 695–720.[20] P. Dorey and R. Tateo, “Excited states by analytic continuation of TBA equations,”
Nucl.Phys.
B482 (1996) 639–659, hep-th/9607167 .[21] L. Mezincescu and P. K. Townsend, “Anyons from Strings,”
Phys.Rev.Lett. (2010)191601, .[22] J. Polchinski and A. Strominger, “Effective string theory,”
Phys.Rev.Lett.67