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String Winding in a Black Hole Geometry.

arXiv:hep-th/9108010v1 21 Aug 1991Technion-PH-23-91String Winding in a Black Hole Geometry.Mordechai Spiegelglas *Physics DepartmentTechnion, Haifa 32000, Israel.U(1) zero modes in the SL(2, IR)k/U(1) and SU(2)k/U(1) conformal coset theories,are investigated in conjunction with the string black hole solution. The angular variablein the Euclidean version, is found to have a double set of winding.

Region III is shownto be SU(2)k/U(1) where the doubling accounts for the cut sructure of the parafermionicamplitudes and fits nicely across the horizon and singularity. The implications for stringthermodynamics and identical particles correlations are discussed.July 1991*Lady Davis Fellow at the Technion.Bitnet: phr74ms@technion.

Recently, a particularly interesting string solution was found [1], [2], [3]. In [3] thesolution for the graviton-dilaton field equations, which shares many features with a 3+1dimensional black hole, was also found as the conformal coset model SL(2, IR)/U(1).

Du-ality, relating region I ending at the horizon to region V starting at the singularity, becamepromptly a particularly intriguing feature of this model [4], [5]. The later, also sets theformalism and notation used here.We are going to discuss the U(1) winding in this string solution.

In the Euclideanversion, the line element for the cigar shaped region I is given byds2 = dr2 + tanh2 r dθ2. (1)We will find that θ has actually a double set of winding.

This is in contrast with the usualcase in conformal field theory, where right-moving modes and left-moving modes share thewinding and the primary operators those imply1. Heuristically, the source for doubling isthe topological U(1)/U(1) theory [7], [8] hiding in the SL(2, IR)/U(1) coset constructionas argued at [9], [10] (where the current-algebra H ⊂G, was guaged to form G/H bycomplexifying H and then employing complex BRST to cancel the H propagating degreesof freedom).

The U(1)/U(1) is classically the theory of flat U(1) gauge connections, orextra winding for an angular variable θ.Before elaborating on widing doubling, we would like to present a somewhat broadercontext which makes flat gauge conections particularly interesting around black hole [3],a goemetry which may have access to some non-perturbative aspects of string theory.Independently of the coupling constant it has a horizon and a singularity, with gravitationalinteractions becoming strong at the later. This non-perturbative flavor of the black holesolution is a motivation to the search for string features on which it differs from flat space-time.

We will find that Euclidean winding is different around the black hole than in theflat space-time case.Winding, has important roles in string theory. It is responsible for extra gauge sym-metries of string origin [11].

It is also important in string thermodynamics. Strings intemperatue 1/β, are formulated by curling one (Euclidean space) dimension, to radius β[12], [13].

Increasing the temperture by shrinking β, a string winding state, could have itsM 2 = −C + t β2 becoming tachyonic (as the second winding term cannot cancel the first1 Doubling is thus related to ideas presented in [6] about orbifolds in complexified space-timeand the separation of left and right movers.1

tachyonic zero point term anymore). This was argued to indicate a transition temperaturefor the string, namely Hagedorn temperature, where the number of excited states winsover their Boltzman supression and the free energy is dominated by highly excited stringstates [14], [15].

The implications of the winding doubling for string thermodynamics willbe discussed subsequently.An additional string aspect which seems different between the black hole and flatspace-time, is the way the Hilbert space is constrained to a ghost-free positive normedphysical spectrum [16]. The well known mechanism of BRST cohomology does not seemto help, at least in its straightforward version, which requires a flat time-like direction [17],[18].

There are suggestions, based on the SL(2, IR)k current algebra, giving negative-normstate free spectrum for the black-hole string solution [19]. However, their compatibilitywith modular invariance has yet to be settled, along with finding the symmetry principleunderlying them.A different way out is a non-linear physical state condition, possibly an interactingstring field equation like QΨ + Ψ ⋆Ψ = 0, for an appropriately defined string field theory(with product ⋆), instead of the condition QΨ = 0 solved by BRST cohomology.

A non-linear condition of this kind could fit with the non-perturbative nature of the black holesolution. Regretably, the relevant version of string field theory is not clear yet.

Nonetheless,we could entertain the formal similarity between this physical state condition and a flatgauge connection. Although flat gauge configurations cannot be suggeted so far as a usefulfield theory approximation to the relevant string field theory2, we will proceed to studythem.Let us take stock of angular variables in the coset model SL(2, IR)k/U(1).

It is con-venient to specify the action of the U(1) gauge transformations in the WZW model [20] inthe Euler notation for g(z, z) ∈SL(2, IR)g I, euc. (z, z) = ei2θLσ2e12 r σ1ei2θRσ2.

(2)The left handed (holomporhic) transformations act on θL while the right handed (anti-holomporhic) ones act on θR. The choice of the relative sign of the right handed gauge2In string field theory Q is constructed out of Virasoro generators T(z) and ghosts b(z),c(z).

There is, however, the Sugawara formula gives T(z) as a bilinear of SL(2, IR)k currentsparallelled by b(z) expressed as a product of SL(2, IR)k ghost and a current [7]. With some luck,the similarity between the physical state condition, the highest weight condition with respect tothe U(1) gauged in SL(2, IR)k and flat gauge condition may be more than formal.2

action, gives vectorial or axial U(1) gauge transformations [21] and its results are discussedbelow. Anyway, gauging a U(1) on the world-sheet will leave us with one angular field,θ(z, z) and some doubts counting zero modes associated with it.

The Euclidean region Ican be found by gauging the vectorial U(1) generated by the Pauli matrix σ2. A convenientgauge choice [3] is taking g(z, z) symmetric, leading to the line element (1).

One could tryother gauge choices, like taking g(z, z) traceless, which yields a different angular variable˜θ (and the line element ds2 = dr2 + coth2 r d˜θ2). In both cases the coset has a singleangular variable.

θ is locally related to ˜θ by a U(1) gauge transformation. If, however, theworld-sheet is topologically non-trivial, this would not mean that the two are equivalent.It would rather mean that the ambiguity in the choice of an angular field, is given by anextra U(1) flat connection as a global set of degrees of freedom.In our discussion we will often use space-time (or Euclidean space) arguments for theblack hole solution, which are usually justified semi-classically at infinite level k which isweak WZW coupling constant.

Since, to give c = 26, k is small3, it is lucky that the twodimensional nature of the solution opens a better justification to space-time arguments.They turn out to be exactly applicable for the “tachyon” states, which dominate thissolution [5]. We will check the effects of winding doubling in the parafermionic conformalfield theory and substantiate them independently of space-time arguments (This check isencouraging for bolder applications of space-time arguments).So far we were disucssing the Euclidean region I formed gauging the vectorial U(1)subalgebra generated by σ2.

Let us see how similar gauge conditions look through regionsIII and V. We will get the Euclidean picture of these regions starting from the Minkowskiversion. For region I the later can be found writingg I, mink.

= e12 tLσ3e12 r σ1e12 tRσ3. (3)and gauging the U(1) generated by σ3.The gauge where g is symmetric gives theMinkowski line element ds2 = dr2 −tanh2 r dt2.The other regions follow in Kruskalcoordinates u, v [3], where they can be put together.

In region I, u =12 sinh r et, v =−12 sinh r e−t. The line element isds2 =du dv1 −uv.

(4)3 Rather than the geometrically suggested c ∼2, see [22] for another way to get c = 26.3

Region I is uv < 0, region III 0 < uv < 1, bounded between the horizon uv = 0 andthe singularity uv = 1 and region V is past the later. Kruskal description follows fromSL(2, IR) when parametrizingg =au−vb,ab + uv = 1 .

(5)We continue this description to region III, where u = 12 sin r et, v = 12 sin r e−t and ds2 =dr2 −tan2 r dt2. SL(2, IR) is now parametrizedg III, mink.

= e12 tLσ3ei2 r σ2e12 tRσ3. (6)Wick rotation in region I turns (3) into (2), changing the gauged U(1) subgroup eiασ3into eασ2 (acting vectorially as a similarity transformation) and the line element intods2 = dr2 + tanh2 r dθ2.

In region III (6) chages tog III, euc. = ei2θLσ3ei2 r σ2ei2θRσ3(7)(σ3 cannot turn into σ2 already used by r) giving g III, euc.

∈SU(2)! The line element isds2 = dr2+ tan2 r dθ2; as a check, near the horizon r = 0, it agrees with ds2 for the sphere4S2 = SU(2)/U(1).

SU(2)k/U(1)k in the Euclidean region III, will lead us to parafermions[23] and their duality, after further examination of the Euclidean solution and its winding.As a further check for the Euclidean regions we have found and in order to establish theway they are attached to each other, we will find a path in SL(2, IR) through all of them.This is done by imposing the condition5 θR = 0, (a2+v2 = b2+u2 in the parametrization of(5) ) in addition to the gauge conditions. These read u + v = 0 (g I symmetric) in region I,(u+v)2 + (a+b)2 = 4 in III (g ∈SL(2, IR) always satisfies (u+v)2 + (a+b)2 ≥4, saturatedat the horizon1 00 1and singularity 0 1−1 0.) and a + b = 0, in V (dual to u + v = 0,).

Thispath is parametrized bygρ =cosh ρsinh ρsinh ρcosh ρin I,gω =cos ωsin ω−sin ωcos ωin IIIand gσ =sinh σcosh σ−cosh σ−sinh σin V.(8)4 The difference further on, shows that cosetting in current algebra does not give the cosetmanifold, but is rather done by integrating out the gauge field. The bilinear form in the gaugefield A is thus responsible for the singularity, demonstrating the importance of string effects inthis context.5 providing a space-time description for left movers on the world-sheet, which is sometimesinteresting by itself [6].4

gω belongs to SU(2) as well (covering the intersection of the two groups).Now we can look closely for winding in the Euclidean solution. It is interestinglycontrasted with a cylinder, which is the Euclidean thermal version of flat Minkowski space[12], whose set of winding is topologically stable.

For special values of the radius namely ktimes the self-dual radius, they lead to k + 1 conformal blocks6. The Euclidean black holeis a bit different.

The cigar shaped region I by itself, would drive us into the conclusionthat winding is not topologically stable here. This is not the case, since we have alreadyestablished that region III is attached to region I at r = 0, by threading a path betweenthem.

Argued differently, the horizon, a Minkowski light-cone in Kruskal coordinate isWick rotated, giving r = 0, which is the point common to the regions. In fact, we aregoing to argue that the winding is actually doubled, by closely studying the parafermionictheory in region III and latter continuing it to the other regions.The study of parafermionic conformal field theories [23] was motivated by criticaltwo-dimensional Zk clock models in Statistical Mechanics [24].

They provide one of thebest studied coset models, SU(2)k/U(1)k. In addition to the role of a useful laboratoryfor U(1) cosetting they also happened to serve as the centerpiece for the Euclidean stringblack hole solution. Most notable in these models, is the wealth of observables (∼k3).We will concentrate in this note on the set of order operators σl and the set of disorderoperators µl.

The µl’s are known to be non-local with respect to the σl’s and correlationfunctions containing both have cuts [25]. σ’s and µ’s follow from gauging different U(1)subalgebras of SU(2)k, the vectorial (electric) U(1) gives the σ’s and the axial (magnetic)U(1) gives the µ’s.

In other words, they result from quantizing different sets of zero modes(windings) in the theory thereby demonstrating doubling.To see that this is indeed the case we note that σ’s and µ’s are related by Kramers-Wannier duality [26]. This duality, which relates magnetic cosetting to electric cosetting,works the same way as the duality found in the string black hole solution [4] [5].

For morecareful examination, we observe that the cut structure in the parafermionic amplitudes isthe inverse of the cut structure between electric and magnetic U(1) vertices. In chapter5 of [23] the SU(2)k holomorphic current J3(z) = ∂zφ(z), is taken to be the holomorphic6 In addition to giving the primary operators, they also have effects, like holomorphic factor-ization in the τ dependence of the torus partition function, whose subtlety in our case we will seesoon.5

U(1) current, φ(z) being the holomorphic part of the scalar field Φ(z, z). By formula (5.10)in [23]σl(z, z) : exp i l√kφ(z) + φ(z): = σl(z, z) : exp i l√kΦ(z, z):(9)gives a cutless SU(2)k correlator withµn(z, z) : exp i n√kφ(z) −φ(z): = µn(z, z) : exp i n√k˜Φ(z, z): .

(10)Thus, all the parafermionic Zk cut structure is between the electric Φ(z, z) and magnetic˜Φ(z, z) = φ(z) −φ(z) vertices. A bit more algebraic version of the description suggestedhere for σ and µ is found introducing the rational torus U(1)k ⊂SU(2)k by which wecoset each time.

Then σ are characterized as the operators which are local with respectto J3(z) and J+(z)k ¯J+(z)k. µ are rather local with respect to J+(z)k ¯J−(z)k. Doublingof winding is noted realizing that different sets of winding give the σ’s and the µ’s uponquantization, since they came about, cosetting different U(1)’s. All the operators in theparafermionic theory will be discussed in this laguage elsewhere.The cut structure in parafermions provides an important property shared with non-critical strings, or Liouville theory and hence an additional relationship [27] between theblack hole solution and non-critical strings.

If we consider the σ’s as operators in the bulkof the system, then the µ’s seem to introduce boundaries in the form of cut lines betweenthe various µ’s. It is also natural to specify modified boundary conditions on the cylider(or torus) as though a cut is running between two µ’s on the boundary[28] [29].

Thisresults in more modular invariant combinations, under a subgroup of the mapping classgroup7. This distinction between operators, naturally specified on points and operatorsassociated with boundaries was made by Seiberg and Moore[30] [31] (the later were asso-ciated with normalizable wave functions) in the context of two dimensional gravity.

In ourcase operators and states are dual.Further understanding of the parafermionic theories as coset model is required in orderto get semiclassical intuition. Large k is currently under investigation and seems hopeful,since self dual Zk clock models fall withing a Kosterlitz-Thouless phase for k > 4 and7 Along with the ambiguity in specifying the chiral algebra and the cuts in the amplitudes,this makes the definition of conformal blocks tricky.

For example, the Ising model has 3 blocksunder the Virasoro algebra but only 2 blocks when the chiral algebra consists of a free Majoranafermion. In this case, the modular invariance is under the group generated by S and T 2.

Thesesubtleties in defining conformal blocks are typical to doubling of winding.6

should not be too sensitive to 1/k corrections (as well as to moving offcriticallity, possiblyby Ginzburg-Landau formulation. A further speculation will be that lattice models coulddo as well as their coninuum limits, in accord with discrete or topological approaches to2-d gravity.).

It should also be mentioned that parafermionic models continues smoothlyaccross the horizon to region I with Φ as θ. Instead of warrying about continuation acrossthe singularity, we will simply use the parafermionic duality to see that ¯Φ continues to ˜θ,further empasizing that Kramers-Wannier duality is equivalent to the duality found in theblack hole solution[4] [5].The importance of winding for string thermodynamics was already mentioned above.ℓfold Euclidean winding states becoming massless, was also interpreted in string thermo-dynamics [32].

Much the same way that by becoming massless, a minimal winding statesignifies that higher string excitations start to dominate the string free energy; these ℓfoldwindings become massless when highly excited ℓidentical strings states get to dominate theensemble of ℓidentical string excitations. This interpretation was shown to be consistentwith Bose-Einstein as well as Fermi-Dirac quantum statistics, in the bosonic and fermionicstring theories.

It is tempting to abstract from this theromdynamical argument that theEuclidean winding states are related to quantum correlations between identical particles.By that we would learn that identical particles correlations could behave differently in theblack hole case. One could speculate that the doubling of winding states, would naivelypresent itself as a Z2 additional quantum number (“color”).

Along with implications toblack-hole thermodynamics, this should be examined more carefully.A calculation of the partition function on the torus or other amplitudes will be veryhelpful to clarify this issues as well as the physical spectrum. It will also be interesting torelate the arguments found here to the extra twisted states found at [33] in the SL(2, IR)theory.Acknowledgments: I am grateful to J. Avron, M. Dine, F. Englert, Y. Feinberg, M.Marinov, B. Reznik, N. Rosen, J. Sonnenschein and S. Yankielowicz for valuable discus-sions.

S. Yankielowicz has convinced me how tricky the no-ghost situation is and cosequenlyfocused my interest on the black hole solution. This work was supported in part by USIsrael Binational Science Foundation (BSF), by the Israeli Academy of Sciences and bythe Technion V.P.R.

Fund.7

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