Physical states of the string in a black hole background

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 25A (1992) 104-108 North-Holland

PIIYSICAL STATES OF TIIE STRING IN A BLACK IIOLE BACKGROUND* Jacques Distler JosepA itear~ La6oralories, Princeton University, Princeton, NJ 0854~ USA

Philip Nelson Ph~Jicm Dep6rtmenl, Unlversity o.t Pen~syh,ania, Philadelphia, PA 19104 USA i review the construction of the physical spetZrmn for the "black hole" solution recently proposed by E. W i t t e n as ma exact string b a d , ground with a 2-dlmenslonal target space. T h e s pe c t rum contains some of the states found in studies

of d -- 1 noncritical string theory but, in addition, has new states not previously found in the d -- 1 noncritical string. Along the way ! will note a remarkable %tringy" symnlctry of the s pe c t r u m relating massive states to m a s s l e u ones and comment on the relation between rids theory mid the d = 1 noncritical striltg theory.

The subject of this talk is discussed in great detail in the recent prepriut [1]. For these Proceedings, i will try to give an overview of that work [3] and make some comments on future directions. The interpretation of certain noucompact coset models as string black holes in a twodimensional spacetime was first given by Wittcn [2]. The S L ( 2 , ~ ) / S O ( 1 , 1 ) coset model corresponds to a Minkowski signature black bole, while the SL(2,I~)/U(I) coset corresponds to what might be called the Euclidean black hole. Since string theory is supposed to provide a theory of quantum gravity, it is interesting to probe how the exact string theory in these backgrounds compares with the geometrical picture one obtains from the classical a-model. More generally, one is interested in probing the consistency of string propagation in curved spacetime backgrounds. We are fairly used to considering background in which the spatial di*

This work was supp~,~.ed by N S F grants PIIY88-

57200 a n d PHYSO-19754.

rections are curved. From the world sheet point of view, we replace some of the 26 free bosons wlfich describe the string in fiat space with a more general honlinear a-model, or more abstractly, by some unitary conformal field theory with the same central charge, in fiat spacetime, the time direction is represented by a wrongsignature free bosoa. This is a nonunitary (albeit free) CFT. When we look at curved spacetimes, we nmst therefore consider more general nonunitary CFT's. To obtain a consistent quantum mechanics out of such a theory, we need some generalization of the No Ghost theorem [5-7] which, in flat spacethne, guarantees the existence of a consistent truncation of the theory to a space of only positive norm states. The 2-d black hole is the perhaps tile simplest string theory in which one can explore the generalization of this to nonflat spacetimes. The modern approach to the no-ghost theorem is to define the BRS operator Q associated to the gauge synnnetry of the problem. One then passes frmn the full (indefinite) Ililbert space to

J. Distler and P. Nelson / Physica| Stales of the String ... the (hopefully positive definite) cohon,ology of Q. In the present ease of SL(2,~)/U(1) cosct conformal field theories, there are two alternative viewpoints one can take: (i) We first construct the coset C F T by imposing the the U(I) highest weight condition and then couple the resulting tl:eory to gravity. The BRS operator t h a t results is

Q = ~ e T ~°'a + bcOe J which is associated to gauging tile diffeomo~phism symmetry. (ii) Alternatively, we can take seriously the realization of the cosct model as a 9asfed W Z W model. Therefore we are gauging both diffeomorphisms and the U(I). These two are, in fact equivalent. One can view the first as computing an iterated eohomology, H~2(H~uo~). Tile latter amounts to computing the cohomology of the "total" BItS charge QT = Q+Quo). It turns out t h a t the spectral sequence of the double complex (where tile bigradlug is by diffeomorphism- and U(1)-ghost uumbers) degenerates at the E2 term which shows t h a t the cohomology of QT is isomorphic to the iterated cohomology. Unfortunately, unlike the case of e~mpact SU(2), we d o n ' t really know how to construct these SL(2, R) eoset models as full-fledged string theories. Indoed, by identifying which current algebra representations give rise to nontrivial BItS eohomology, we are really taking; tile first step in t h a t direction, llaving id, atilied tile relevant coset modules, we need a prescription for putting tcgether left- and right-moving degrees of freedom (more on this later) and finally for assembling the whole shebang into a modular invariant theory. In particular, it is cru¢;M t h a t the operator algebra close, hi the ease of compact Sb(2), tile fusion rule are a t r u r cared version of tile usual rules for tile addition

105

of angular momenta. The representations with spins j < k/2 close upon then~selves. This unfortunately depends rather delicately on the l,ull-vcctor structure of the .S'U(2) current algebra representations, and seem~ mllikely to ~:old ill tile SL(2, ~) ease. We'll take the conservative approach and consider all j . Since this is a so.ncwhat confusing point, I'd like to emph:~size the difference between the BIts approach (or any other approach) to the no-ghost theorem and what is commonly done in constructing mlitary conformal I~eld theories d In BPZ. When one starts with a positivescmidefinite llilbert space, one can construct Ly hand a positive-definite il;Ibert space as the quotient by tile subspace of null vectors, in an iadefitlite Ililbert space, this simple proceedure is not possible. Instead, we must ]irst restrict to tile (positive-semidefinite) subspaee of BRSclosed states before we can take the quotient. Without Io~ of generality, we may consider representations of tile current algebra built upon irreducible representations of global SL(2, ~) at the base. These are classified by tile spectrum of m eigenvalues at the base. If m is unbounded, as occnrs for generic values of j, m then we label the resulting coset nlo,lule C. if m is bounded from below at tile base, we have a lowest weight representatiou, and we call distinguish two subcases: 1) the nlininluul value o f m is j + l , in which case we label the resulting coset n . o d u l e / ) + , or 2) the minimum value of m is - j , in which case, we label the coset module ~ + . Similarly, for highest weight representations, m is bounded from above at tbe base, and these give rise to coset module ~ - and ~ - . Finally, starting with representations in which m at the base is bounded from both above and below, we obtain the eoset modules II. Tile BRS cohomology that one finds in these module~ con,~.~ in two basic flavours. First are

108

.I. Distler and P. Nelson / Physical States of the String...

states which ec~,ae i~om tile base of the current algebra representation. These have the form (en are the ghost oscillators) ~'llj, m )

with m = ±~0 + i/2).

These states have zero oscillator number and are the analogues of the tachyon states in the d = 1 noncritical string. Of course, "tachyou" is something of a misnomer. In two spaeetime dimensions, such states are massless. The states at higher oscillator number come paired. For each state in the cohomology at ghost number one, there is a state at ghost number zero. if ther~ are two states in the eohomology at a given mass level, then there is another state in tile cohomology at ghost num[.er two. The situation is summarized in table 1. All of this was for tire Euclidean SL(2, ~)/U(1) coset. The ease of the Mhlkow~kiau SL(2,1~)/ S O ( l , 1 ) eoset differs ill that there ~e wish to diagonalize one of the noncompact generators. T h a t would l o t he necessary were we only concerned with constructing states at ma~s level ~ero (as was tile focus in [8]). The massless states are aimihilated by the positive frequency modes of all of the currents. ~lb get the whole parafermion module, we need to construct states annihilated by the positive frequency modes of the ~ ( 1,1) generator, but nol necessarily by the other two currents, llence it is essential that the ~0(1,1) generator he diagoualizeable. In an indefinite llilbert space, tile familiar statement that a llerm~tian operator has only real ei/;envalues does not hold. Nevertheless, for an irreducible representation, the eigenvalues, p of the z0(1,1) generator cannot be arbitrary complex numbers. Rather, we must have Ira(u) ~ ½Z.

Ill fact, the spectrum t h a t we find has either p pure real or pure imaginary. The base states ( " t a c h y o n s ' ) have p E l~ and j = - ½ 4 - i p / 3 . A priori, we also ]lave the analogues of a subset of the discrete states found above with j real a n d p = im E -i ~ . Quit possibly, however, it may be consistent to truncate | h e physical s p e c t r u m to real It, ill which case, the only discrete states would have p = 0. As already alluded to, the tachyons one finds here are in precise correspondence with those that one liuds ill tile d = 1 nonentleai string. If we want a precise dictionary, we need only compare the dispersion rela lions in tile two theories. One finds t h a t the correct identification is p~. =

~--~-(m o~"~),

p~ = 2vf~j

The relatiou between the discrete states of the two models is a little more complicated. Using the above dictionary, the discrete states of Liouville [9] at ghost number 1 (the canonical ghost number) occur at j = ½(u + v - 2) and m = ½(u - v) for u,v positive integers. From the above table, we see t h a t these are in precise correspondence with the discrete states from the modules "D+ , D - , e, except t h a t for u, v both even, there are two states in the coset module (one from/9 + and one from D " ). The states from tile modules ~ : seem to have no correspondents in ihe Liouville theory. The discrepancy hetweeu the two theories may not be as large as it appears a t first. As discussed in [1], there is a stringy isomorphism which relates /~je to Tl~:+lls, so we are perhaps double counting if we i'.~~ .~de both. This still does not explain the double degeneracy for u v even. It also spercads ~ cloud over the guiding principle of our "dictionary". The isomorphism relates states of d ~ e r e n t j, m and different oscillator numbers, hence these are somewhat ambiguous q u a n t u m nlmd~pr~ f~r hh~|~n¢ elaine ~t" t h , o m i t l h ~ . P ~

J.

Distler emd P. Nelson / Physical States of the String...

107

Table 1 Representation

C

z3~5vV/3+

V+ /)+

u

j 1) -~(2s + 4, - 5)

~(~+,-

¼(z, + ~, - a) ¼(s + 2 , -

3) ~(2s + 4 r - s) 5) ~(2s + 2~ - 3) ~(s + :~,. - a) l ( 2 s + 4 r - 5) ½(s+,2)

'

m ~(s - r)

-~(2s-

4 , ' - l)

- ~ ( ~ s - 2,.+ ~(s-2,.+ ~(2~ - 4,. ~(~s - 4,. -

dim I1~ 1 0

l) 0 1) 1)

1 0 l o

~ ( 2 s - 2 , ' + 1)

1

0

o

- ~ ( 2 s - - 4 , ' - l)

-~(s-2,.+

1

:~(s - ,-)

1

The correspondence we have been trying to es.. tablish is perhaps best viewed ~s a senficlassical (large k) one. The existence of the discrete states in the coset model is an inherently stringy phenomenon. One can show t h a t for k > 9/4, all but two of the discrete states di.~appear. The existence of an infinite number of liscrete states is an artifact of k = 9/4 (that is, of having a two dimensional target space for the .~triug). Reasoning valid at large k should not be expected to reproduce them accurately. The situation is somewhat remimscent of the familiar R --* 1 / R duality tymmetry. Semicl~sical reasoning is certainly suspect at small R. The novel feature here is t h a t the new symmetry does not commute with oscillator number (as R -* I / R does), and titus exchanges states of different spins! Probably, tho existence of such a symmetry is, as is much else in this subject, a peculiarity of 2 dimensional target spaces. Nevertheless, it would be interestit,g to find other (higher dimensional) examples of this phenomenon. I should mention one final point ou the relation between the "black hole" and the d = 1 noncritical string (with a compact free boson X).

dim lI~ 1 1 0 1 0 1 0 1 0 0

dim It~ C 9 0 0 0 0 0 0 1

Ill both ca.ses, one should properly allow for the existence of "winding states" [8]. T h a t is, when we put together left- and right-reeving degrees of freedom, the allowed values of m, rh should be correlated

m = ½(" + ~u'),

,~, = " ( - . + k . ' )

for n, W E E. Note t h a t this is ill sharp constrast to the ca.se of simply connected current algebras such as SU(2), where m - ,h E Z. The proof of all of these statements [1] relies, ill part, on the free-boson representation of the coset r.,odules # 1 Since it is somewhat perilous to take the frec-boson representation 1oo seriously, we should clearly isolate what we actually used it for. In fact, we used the free-boson representation to obtain two pieces of information: first, we used it to find tire cohomology of the coset module (~; second~ we used it to establish the "stringy isomorphism" between the coset modules __D~ and --~- qs" Either of these could have been established, perhaps with greater difficuity, using other methods. #| For some other recent applications of the free field representation in nus context, see [IUJ.

108

J. Dist/cr and P. Ne/son / Physical States of the String ...

H a v i n g c o m p u t e d the physical s p e c t r m n , we can return to t h e no-ghost theorem. Strictly speaking, t h e no-ghost t h e o r e m does not require the all physical s t a t e s have positive norm. T h e r e i~ a limited scope for h a v i n g discrete s t a t e s with negative norm. Indeed, there are such s t a t e s even in t h e critical bosonic s t r i n g in 2(3 dimensional fiat space, llowever, in o r d e r not to screw up the u n i t a r i t y of physical amplitudes, these s t a t e s m u s t be delicately paired with s t a t e s of positive n o r m with t i ~ s a m e q u a n t u m nuntbers. T h i s holds true in the critical bosonic string, and it holds true in t h e Minkowski SL(2, P ) / S O ( I , I) coset. Propagating states, on the o t h e r hand, m n s t be positive norm. T h i s also holds true, alt h o u g h since t h e only p r o p a g a t i n g s t a t e is t h e " t a c h y o n ' , it w~s never seriously in doubt. S t r i n g propagation in this two dimensional b a c k g r o u n d passes this m o s t basic o f consistency checks. Clearly, one would like to go beyoud this a n d consider some higher dimensional examples, where the no-ghost t h e o r e m would b e realized ia a less " t r i v i a r ~ fasition, it. i~ .,lso clear t h a t u , e would like to do b e t t e r titan we have, mid actually construct the full conformal field the~rie~

associated to these noncompact coset models. A~ w~ i~ave seen, s o m e of the ntost novel features of t h e black hole, like those of the noncritical string, are mtlikely to persist in higher dimensions. It will be interesting to see w h a t new feature arise instead.

References [I) J. Distler and P. Nelson, "New Discrete States of Strings Near a Black Ilole", Princeton preprint PUPT-1262 (1991). [2] E. Witten, "On string theory and black holes", IASSNS-IIEP-91/12. (3] See also: L Bars, "String propagation on black holes", USC

Fl~eprint USC-9-11E-B3 (1991) for a somewhat different view on this subject. [4] For eadier work on this class o~"theories, s*~e e.g.: J. Balog, L. O'P,~aifcartalgh, P. Forgacs and A. Wipf, "Conslstency of string propagation on curved spacetimes: an SU(1.1) example" Nucl. Phys. B32~ (1989) 225' P. Petropoulos, "Comments on SU(I, 1) string theory", Phys. Lett. 236B (1990) 151 i. B.xrs and D. Nemeschansky, "S;llng propag&tlon in backgrounds with curved sp~ce-time", Nud. Phys. A348 (1~1) 89. [5) P. Goddard and C. Thorn, "Compatibility of the dual pomeron with unitarity and the absence of ghosts in the dual resonance m~Jdel", Phys. Lett. 4OB (1972) 235; R. Brower, "Spe,:trum-g~nerating algebra and noghost themCm for the dual model", Phys. Bey. D 6 (1972) 1655. [6] M. Kato m~d If,. Ogawa, "Covariant quantizaticm of sn'ing based on BRS invarlance", rqucl. Phys. B212 (1983) 4.|3. [7] I. Frcnkel, II, Gai|and and G. Zuckerm~m, "Semiildiafite cohomology and string theory", Proc. Nat. Acad. Sci. USA 83 (1986) 8442. (~q] IL Dijkgraaf, iL Verlinde v.nd E. Veriinde, "String p!~i,dgalion ill a Idack hole background", Princeton preprhU PUPT-1252 (1991). [9] A. Polyako", Mod. Phys. Lett. A6 (1991) 635; D. Gl~ss and 1. Klebanov, "S=1 at c=1", Nud. Phys. B359 (1991) 3. (10] E. Martinet mid S. Shatashvili, ':Black hole physics and Liouville theory", Chicago preprint EFI-91-22 (19m); M. Bershadsky and D. Kut~ov, "Conmlent on gauged WZW theory", Princeton preprint PUPT1201 (1~1).