$P$-brane solutions of low-energy string actions have traditionally provided
the first evidence for the existence of string dualities, in which fundamental
and solitonic $p$-branes are identified with perturbative and non-perturbative
BPS states. In this talk we discuss the composite nature of solutions, which
allows for the interpretation of general solutions as bound states or
intersections of maximally supersymmetric fundamental constituents. This
feature lies at the heart of the recent success of string theory in reproducing
the Beckenstein-Hawking black hole entropy formula.Comment: 8 pages, harvmac, talk given at Strings '9

Khuri, Ramzi R.

English

arXiv

$P$-brane solutions of low-energy string actions have traditionally provided the first evidence for the existence of string dualities, in which fundamental and solitonic $p$-branes are identified with perturbative and non-perturbative BPS states. In this talk we discuss the composite nature of solutions, which allows for the interpretation of general solutions as bound states or intersections of maximally supersymmetric fundamental constituents. This feature lies at the heart of the recent success of string theory in reproducing the Beckenstein-Hawking black hole entropy formula.$P$-brane solutions of low-energy string actions have traditionally provided the first evidence for the existence of string dualities, in which fundamental and solitonic $p$-branes are identified with perturbative and non-perturbative BPS states. In this talk we discuss the composite nature of solutions, which allows for the interpretation of general solutions as bound states or intersections of maximally supersymmetric fundamental constituents. This feature lies at the heart of the recent success of string theory in reproducing the Beckenstein-Hawking black hole entropy formula

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arXiv:hep-th/9609094v1  11 Sep 1996CERN-TH/96-235McGill/96-32hep-th/9609094Supersymmetry, Duality and Bound StatesRamzi R. Khuri1Theory Division, CERN, CH-1211, Geneva 23, SwitzerlandandPhysics Department, McGill University, Montreal, PQ, H3A 2T8 CanadaP -brane solutions of low-energy string actions have traditionally provided the firstevidence for the existence of string dualities, in which fundamental and solitonic p-branesare identified with perturbative and non-perturbative BPS states. In this talk we discussthe composite nature of solutions, which allows for the interpretation of general solutionsas bound states or intersections of maximally supersymmetric fundamental constituents.This feature lies at the heart of the recent success of string theory in reproducing theBeckenstein-Hawking black hole entropy formula.CERN-TH/96-235McGill/96-32September 19961 Talk given at “Strings ’96”, Institute for Theoretical Physics, University of California atSanta Barbara, July 15-20, 1996. Supported by a World Laboratory Fellowship.In the last few years, it has become increasingly clear that p-branes, i.e. (p+1)-dimensional extended objects that arise as fundamental or solitonic solutions of stringtheory, play an important role in establishing the various string dualities and ultimatelyin the non-perturbative formulation of an underlying theory containing strings and highermembranes.Under a given duality map, the fundamental p-brane of a given string or higher mem-brane theory tranforms into a solitonic solution of the dual theory, with the correspondinginterchange of singular/non-singular backgrounds, perturbative/non-perturbative statesand even spacetime/worldsheet loops (see [1] and references therein).An interesting example of the implications of duality is that of six-dimensional het-erotic string/string duality, which, when reduced to four dimensions, leads to a duality ofdualities, namely, the interchange of the spacetime strong/weak coupling S-duality withthe worldsheet target space T -duality [2]. As a result, the conjectured S-duality followsfrom the established T -duality provided four-dimensional string/string duality is shown.More recently, interest has focused on duality between different string theories, inparticular, heterotic/type IIA string/string duality in D = 6 [3,4]. In this case, one has amore straightforward strong/coupling duality relating the two theories with gIIA = 1/ghet.The presence of certain p-branes, however, coupled with the assumption of a givenduality, can point to gaps in the formulation of string theory. In particular, it was shownin [5] that, in the context of heterotic/type duality, aside from the usual interchange offundamental and solitonic strings, there exist membrane solutions which are non-singular(solitonic) in the heterotic theory but singular in the type IIA theory. Such solutionswere then interpreted as fundamental membranes in type IIA, since they could neither beignored as purely singular configurations nor simply counted in the solitonic spectrum ofeither theory. The authors of [5] then concluded that the formulation of type IIA theory asa theory of strings alone was incomplete. The subsequent discovery of the role of D-branesas carriers of Ramond-Ramond (RR) charge that should be coupled to type II theories [6](see also [7]) effectively resolved this problem, since the fundamental membrane solutionof [5] could now have the interpretation of a D-brane.A picture then emerges in which fundamental and solitonic p-branes carrying Neveu-Schwarz-Neveu-Schwarz charge correspond to perturbative and non-perturbative BPSstates of string theory, while p-branes carrying RR charge correspond to D-brane BPSstates which must be coupled to the perturbative spectrum. From both the p-brane andBPS state points of view, the mass of a fundamental solution (state) is independent of1the string coupling g, while the mass of a solitonic solution (state) scales as 1/g2. Bycontrast, the mass of a D-brane scales as 1/g [8,9]. This intermediate behaviour betweenfundamental and solitonic states is an inherently stringy departure from the usual 1/g2solitonic scaling found in field theory.Recent activity has also focused on the existence of an eleven-dimensional theory, theso-called M -theory [4,10–12], whose low-energy limit is eleven-dimensional supergravityand whose construction should lead to the establishment of the various string/string dual-ities [2,3,4]. While we still do not know exactly whatM -theory is, we know thatM -theorycontains membranes and fivebranes. From the point of view of p-brane solutions, these arerepresented by a fundamental membrane [13] and a solitonic fivebrane [14].An example of how p-brane solutions can provide evidence for a duality is the follow-ing. In [15], the conjecture was made that the effective theory of heterotic string theorycompactified on K3 × S1 is dual to eleven-dimensional supergravity compactified on aCalabi-Yau threefold. Point-like (electric) solutions are obtained in D = 5 by wrappingthe membrane fromM -theory around two-cycles in the Calabi-Yau space, while string-like(magnetic) solutions in D = 5 arise by wrapping the fivebrane around four-cycles. Forthe specific Calabi-Yau manifold X24(1, 1, 2, 8, 12) with h(1,1) = 3 and h(2,1) = 243, thesepoint and string solutions/states can be matched with perturbative and non-perturbativesolutions/states of heterotic string theory compactified on K3× S1 [16].The heterotic perturbative solutions (or states) include the fundamental string andelectrically charged point-like H-monopole [17] and Kaluza-Klein [18] solutions, the lattertwo being obtained by wrapping the string around S1. The fundamental string can beidentified with one of the three states arising from the M -theory fivebrane, while theelectric H-monopole and Kaluza-Klein monopole can be identified with two of the threestates arising from the M -theory membrane.The non-perturbative heterotic solutions/states consist of the point-like solution ob-tained from the heterotic fivebrane wrapped around K3×S1 and the magnetically chargedstring-like H-monopole and Kaluza-Klein solutions obtained by wrapping the fivebranearound K3 only. Here the point-like state can be identified with one of the three statesarising from the M -theory membrane, while the magnetic H-monopole and Kaluza-Kleinstates can be identified with two of the three states arising from the M -theory fivebrane.The “basic” fundamental and solitonic p-branes preserve half the spacetime supersym-metries, and arise as extremal limits of more general, non-supersymmetric black p-brane2solutions of string theory. It turns out, however, that the low-energy supergravity equa-tions of motion possess a feature that allows for the immediate construction of compositesolutions from the basic ones. Consider, for example, the double-instanton string solutionof heterotic string theory [19]φ = φ1 + φ2,gmn = e2φ1δmn m,n = 2, 3, 4, 5,gij = e2φ2δij i, j = 6, 7, 8, 9,gµν = ηµν µ, ν = 0, 1,Hmnp = ±2ǫmnpq∂qφ m, n, p, q = 2, 3, 4, 5,Hijk = ±2ǫijkl∂kφ i, j, k, l = 6, 7, 8, 9(1)with e−2φ1 1e2φ1 = e−2φ2 1e2φ2 = 0, where 1 and 2 are the Laplacians in the (2345)and (6789) spaces, respectively. For constant chiral spinors ǫ = ǫ2 ⊗ η4 ⊗ η′4, (1) solves thesupersymmetry equations with zero background fermi fields. Due to the presence of twoindependent instantons [17] in the generalized curvature containing HMNP as a torsion,the chiralities of the spinors ǫ2, η4 and η′4 are correlated by (1 ∓ γ3)ǫ2 = (1 ∓ γ5)η4 =(1∓ γ5)η′4 = 0, so that 1/4 of the spacetime supersymmetries is preserved.For either φ1 = 0 or φ2 = 0 we recover the solitonic fivebrane solution of [20] whichpreserves 1/2 the spacetime supersymmetries, since only one instanton is present. In thisrespect, this string is the composite of two independent fivebranes intersecting along thestring. This feature can in fact be generalized to arbitrary p-brane solutions [21–24],whereby a given p-brane soliton can be interpreted as the intersection of one or moremaximally supersymmetric basic fundamental or solitonic p-branes. From the viewpointof M -theory, the statement then translates into saying that all p-brane solutions arise asintersections of membrane and fivebranes [21].Compositeness was also seen in the specific context of string-like solutions of toroidallycompactified D = 4 heterotic string theory [25], where each solution could be understoodin terms of four independent harmonic functions. The existence of these latter solutionsalso pointed to an interesting interplay between supersymmetry and duality. In partic-ular, the different supersymmetry breaking patterns of the string-like solutions conformto the different large duality groups (containing both S and T duality) of the variouscompactifications [25].3A related composite picture of p-brane solutions is the bound states picture [26–31].For the simplest case of extremal black hole solutions in D = 4, consider the Einstein-Maxwell scalar actionS =∫d4x(R − 12(∂φ)2 − 14e−aφF 22), (2)where a is an arbitrary parameter. It turns out that for the specific values of a =√3, 1, 1/√3 and 0, supersymmetric extreme black holes arising from string compactifi-cations were found. Moreover, from both the spacetime solutions [27] and supersymme-tries [28] point of view, these four solutions can be interpreted as bound states of 1, 2, 3and 4 distinct a =√3 black holes, respectively, the latter corresponding to maximallysupersymmetric (N = 4 supersymmetry in an N = 8, D = 4 theory) Kaluza-Klein [18]or H-monopole [17] solutions with flat metric on moduli space [32]. For example, thea = 0 Reissner-Nordstro¨m black hole arises as a bound state of two T -dual pairs of elec-tric/magnetic a =√3 black holes, each pair producing an a = 1 black hole. Again, thisfeature extend quite naturally to arbitrary supersymmetric p-branes in any dimension, aswell as to non-extremal, non-supersymmetric p-branes [24].This compositeness feature lies at the heart of the recent success of string theoryin reproducing the Beckenstein-Hawking formula for the entropy of black holes [33–35].In particular, for a given p-brane arising as a bound state or as intersections of basicconstituent p-branes of charge Q1,Q2,. . .,Qn, the area law for the entropy yieldsS = A/4G = 2π√Q1Q2 . . .Qn, (3)which agrees, for large Qi, with the microscopic entropy formula obtained by countingstring states. This was first seen for five-dimensional extremal black holes in [33] andsubsequently for four-dimensional extremal black holes in [34]. Analogous results for near-extremal black holes were obtained in [35], which seems to indicate that this sort of factor-ization is not a property of supersymmetry alone, although it is only for supersymmetricsolutions that one can invoke non-renormalization theorems to protect the counting ofstates in going from the perturbative state-counting picture to the non-perturbative blackhole picture.The question then arises as to whether a similar property holds for arbitrary non-extremal black holes. Such a formula was, in fact, found from the p-brane picture in[35], where, however, it was noted that the corresponding D-brane counting argument4was unknown. More recently, it was argued in [36] that the counting arguments relatingperturbative states to black holes break down for “fat” black holes, i.e. large black holesfar from extremality (such as the Schwarzschild black hole). 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Supersymmetry, Duality and Bound States

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