View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server M-THEORY AND N=2 STRINGS EMIL MARTINEC Enrico Fermi Institute and Dept. of Physics 5640 South El lis Ave., Chicago, IL 60637-1433 Abstract. N=2 heterotic strings may provide a windowinto the physics of M-theory radically di erent than that found via the other sup ersymmetric string theories. In addition to their sup ersymmetric structure, these strings carry a four-dimensional self-dual structure, and app ear to be completely integrable systems with a stringy density of states. These lectures give an overview of N=2 heterotic strings, as well as a brief discussion of p ossible applications of b oth ordinary and heterotic N=2 strings to D-branes and matrix theory. 1. Intro duction A few years ago, if asked to describ e string theory, the average practi- tioner would have classi ed its di erent manifestations according to their various worldsheet gauge principles. On the 1+1 dimensional worldsheet, there can be (p; q ) sup ersymmetries that square to translations along the (left,right)-handed light cone; one says that the worldsheet has (p; q ) gauged sup ersymmetry. The b osonic string has no sup ersymmetry; p = q = 0. The sup ersymmetric string theories have, say, q = 1. Thus typ e I IA/B string theory has (1,1) sup ersymmetry. The typ e I/IA strings are the orbifold of these byworldsheet parity, and the heterotic strings are in the class (0,1). Remarkably, we now understand that all the sup ersymmetric string theo- ries { typ e I IA/B, typ e I, and heterotic { app ear to describ e asymptotic expansions of a single nonp erturbative master theory: M-theory. This the- ory has many miraculous duality prop erties that are now only b eginning to b e unravelled; other lecturers at this scho ol will review the current state of a airs. In these lectures, I will giveanoverview of a relatively unexplored corner of string theory, namely the N=2 strings [1, 2, 3] (more sp eci cally strings with (2,2) or (2,1) gauged worldsheet sup ersymmetry). 2 EMIL MARTINEC The driving force b ehind the recent uni cation has b een the recognition of the fundamental imp ortance of the spacetime (as opp osed to worldsheet) sup ersymmetry algebra. The small (BPS) representations of the sup ersym- metry algebra form a quasi-top ological sector of the theory. By tracking this BPS sp ectrum across mo duli space, one can deduce the interconnec- tions of the various string limits. The issue we wish to address is the role of the heterotic (2,1) string, which is also a stringy realization of spacetime sup ersymmetry and therefore ought to play a role somehow. As we will see [4, 5], many of the basic ob jects of M-theory are realized in the (2,1) string. Self-duality and integrability are further features of the (2,1) string, arising from the chiral sector with N=2 worldsheet sup ersymmetry. We will see that the chiral critical dimensions of the (2,1) string are d = 4 (2 space, 2 time) for the N=2 sector, and d = 12 (10 space, 2 time) for the N=1 sector. To see the relation to M-theory, consider standard heterotic target space geometry. Here this is a 2+2 dimensional base manifold for the dimensions common to b oth chiralities, with the additional left-movers parametrizing an eight-dimensional torus of stringy dimensions b ered over it (see gure 1). T 8 M2,2 Figure 1. (2,1) heterotic geometry. The geometrical elds are the graviton h, antisymmetric tensor eld b, gravitino , and the gauge connection " on the b er. These elds are further restricted by the extra constraints of N=2 lo cal sup ersymmetry on the worldsheet, giving rise to prep otentials h + b ! I @ a a a " ! I @ ' ! I @ : (1) M-Theory and N=2 Strings 3 Here I is an almost complex structure on the base space, which is required by the lo cal N=2 worldsheet sup ersymmetry. Under linearized gauge trans- formations, these restricted elds transform as (h + b)=@ ! a " = @ ! ' = @ ! : (2) In other words, the remnant elds are Nambu-Goldstone elds of sp on- taneously broken symmetries (spacetime antisymmetric tensor eld gauge transformations, translations, sup ersymmetries) on a D-brane. More pre- cisely, one has in a complex co ordinate basis " = i(@ @ )' +(@ +@) = @(+i')+@( i'), where the (real) gauge symmetry is =.However, one can go to a holomorphic basis by complexifying the gauge group G; then ' is a co ordinate on G =G whose dynamics is determined by holomorphic C gauge symmetry much as in the 2d WZW mo del [6, 7, 8]. This virtually guarantees us a connection to brane physics, since brane dynamics is almost by de nition given by the nonlinear Lagrangian of the spacetime symme- tries broken by the brane. A picture of this asp ect of (2,1) string theory is shown in gure 2. The (2,1) string worldsheet maps into the worldvolume of a D-brane, which is itself emb edded in spacetime. The quanta of this brane are the (2,1) strings themselves. Since the transverse uctuations of the brane can b e traced to those of the b er connection " @', while the longitudinal directions are those of the base space, we see that spacetime itself is the total space of the heterotic geometry of gure 1. Figure 2. Chain of brane embeddings implied by (2,1) string states. The dimension of the brane is determined by the numb er of indep endent ik x momentum comp onents in string vertex op erators O e . When the (2,1) 2;2 8 string target space is IR T , this kinematics is 1+1 or 2+1 dimensional; 4 EMIL MARTINEC when the spatial dimensions are further compacti ed, the kinematics is more or less 9+1 dimensional { the target is a kind of ninebrane. As mentioned ab ove, an additional geometric structure is self-duality. The almost complex structure I is one of a triplet of such structures pre- served by the target space geometry (these almost complex structures are not integrable, since (@ @ )I = db = torsion). Thus one can bring to b ear the machinery of twistor theory to characterize the classical solution space. One of the characteristic prop erties of self-dual gravity is the symmetry group of area preserving di eomorphisms SDi SU (1), suggesting a 2 connection to matrix theory [9]. There are indeed a number of intriguing analogies among the matrix mo del of M-theory, (2,1) strings, and other matrix mo dels: 1. The (BFSS) matrix mo del of M-theory [9] realizes the membrane as a collective phenomenon of D0-brane sup ergravitons, i.e. as a state in the large N collective eld theory. In a sense, the BFSS matrix mo del gives a map from a single noncommutative torus into `spacetime'. In the graviton limit, the matrices approximately commute; in the membrane limit, the commutators are large. Similarly, the (2,1) string describ es a map from a single brane into spacetime. 2. Other examples where large N collective eld theory generates string theory as an asymptotic expansion around a particular master eld include: (a) The 1+1d noncritical string based on the original matrix mo del of [10]. (b) 2d Yang-Mills [11]. (c) 2+1d SU(k) Chern-Simons theory as k; N !1 [12]. (d) 2+2d self-dual gravity [13],[1]. This connection will be outlined in section 2 b elow. 3. (2,1) string p erturbation theory is an expansion around the in nite 2 tension limit; T g . In other words, it is an expansion in low brane str energy relative to the brane tension. This is similar to 1+1d noncritical string theory, where the e ective expansion parameter is the energy of excitations relative to some scale (in that case the cosmological constant of Liouville theory, or equivalently the fermi energy of the matrix partons). Thus one sees common themes cropping up in diverse settings. The fo cus of these lectures will be the structure of N=2 strings, and their p ossible relation to matrix dynamics. We b egin in section 2 with the somewhat simpler (2,2) string, in order to intro duce the novel features of lo cal N=2 worldsheet sup ersymmetry and its asso ciated self-duality struc- ture. Included are an overview of the relation b etween the self-dual gravity M-Theory and N=2 Strings 5 of the (2,2) string and SDi dynamics; as well as an aside on the D-brane 2 sp ectrum of (2,2) strings, which leads to yet another interesting connection to matrix theory. Section 3 intro duces (2,1) strings { their worldsheet gauge algebra, sp ectrum, and simple compacti cations. The target space e ective action for (2,1) strings is derived in section 4; parallels with matrix theory are explored in section 5. 2. (2,2) Strings The gauged N=2 sup ersymmetry on the string worldsheet consists of cur- rents T (the stress tensor), G (the two sup ercurrents), and J (a U(1) R-symmetry current). Their algebra is schematically 3 G [T; G ] 2 [T; J ] J + [G ;G ] T + J (3) [J;G ] G : In the conformal gauge, the contributions to the Virasoro central charge from the Faddeev-Pop ov ghosts is 26+211 2=6, so that the critical dimension is d =2=3 c = 4.