D-Branes, Guage-String Duality and Noncommutative Theories [thesis]

Free download. Book file PDF easily for everyone and every device. You can download and read online D-Branes, Guage-String Duality and Noncommutative Theories [thesis] file PDF Book only if you are registered here. And also you can download or read online all Book PDF file that related with D-Branes, Guage-String Duality and Noncommutative Theories [thesis] book. Happy reading D-Branes, Guage-String Duality and Noncommutative Theories [thesis] Bookeveryone. Download file Free Book PDF D-Branes, Guage-String Duality and Noncommutative Theories [thesis] at Complete PDF Library. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Here is The CompletePDF Book Library. It's free to register here to get Book file PDF D-Branes, Guage-String Duality and Noncommutative Theories [thesis] Pocket Guide.

Schubert and A. Regelskis and A. A 45 Babichenko and A. Ohlsson-Sax, B. Borsato, O. Ohlsson-Sax, A. Sfondrini, B. Sax, A. D 88 Itsios, K. Sfetsos, K. Siampos and A. B 64 4 Pittelli, A. Torrielli and M. A 47 [In J. Hoare, A. Pittelli and A. Nieri, S. Pasquetti, F.

Passerini and A. Rolph and A. D 91 Prinsloo, V. D 92 Proceedings 1. A 18 2. Bigarini, J. Nishimura, Y. Susaki, A. Torrielli and J. A 23 Moriyama, V. A: Math. Beisert et al. To appear in Phys. Theses 1. Supervisor Prof C. Becchi, co-supervisor Dr N. Supervisor Prof Bassetto, collaborator Dr G. Sfetsos; deputy A. James Goodwin on a week Summer project Sponsored Dr. Award presented at IGST Yang-Mills-Chern-Simons theory with four massless bi-fundamental matter multiplets and their complex conjugates , and two massless adjoint matter multiplets corresponding to the motion of the D3-branes in the directions 34 common to the two 5-branes.

Equivalently, there must exist an SO 3 R R-symmetry corresponding to the possibility of having the same SO 3 rotations in the and subspaces. It was realised in [ 12 ] that under T-duality in the x 6 direction and uplifting the configuration to M-theory, the brane construction gets mapped to N M2-branes probing some configuration of KK-monopoles. Flowing to the IR in the dual gravitational picture is equivalent to probing the near horizon of these geometries, which includes the expected AdS 4 factor times a quotient of the 7-sphere.

There are several topics not included in previous sections that are also relevant to the subjects covered in this review. There are many results in this subject, nicely reviewed in [ ]. It is particularly relevant to stress the work done in formulating the M5-brane equations of motion covariantly [ , ] and their use to identify supersymmetric world volume solitons [ , ], and in pointing out the relation between superembeddings and non-linear realisations of supersymmetry [ 5 ].

MKK-monopoles and other exotic brane actions : This review was focused on the dynamics of D-branes and M-branes. It is well known that string and M theory have other extended objects, such as KK-monopoles or NS5-branes. There is a nice discussion regarding the identification of the degrees of freedom living on these branes in [ ].

Subsequently, effective actions were written down to describe the dynamics of its bosonic sectors in [ 83 , 80 , , ]. In particular, it was realised that gauged sigma models are able to encapsulate the right properties for KK monopoles. The results obtained in these references are consistent with the action of T-duality and S-duality.

Of course, it would be very interesting to include fermions in these actions and achieve kappa symmetry invariance. Blackfolds : The blackfold approach is suitable to describe the effective world volume dynamics of branes, still in the probe approximation, having a thermal population of excitations. In some sense, it describes the dynamics of these objects on length scales larger than the brane thickness. This formalism was originally developed in [ , ] and extended and embedded in string theory in [ ].

It was applied to the study of hot BIons in [ , ], emphasising the physical features not captured by the standard Dirac-Born-Infeld action, and to blackfolds in AdS [ 20 ]. Non-relativistic kappa invariant actions : All the branes described in this review are relativistic. It is natural to study their non-relativistic limits, both for its own sake, but also as an attempt to identify new sectors of string theory that may be solvable. The latter is the direction originally pursued in [ , ] by considering closed strings in Minkowski. At the level of brane effective actions in Minkoswki, non-relativistic diffeomorphism and kappa symmetry invariant versions of them were obtained in [ ] for D0-branes, fundamental strings and M2-branes, and later extended to general D p -branes in [ ].

The consistency of these non-relativistic actions under the action of duality transformations was checked in [ ]. Multiple M5-branes : It is a very interesting problem to find the non-abelian formulation of the 2,0 tensor multiplet describing the dynamics of N M5-branes. Following similar ideas to the ones used in the construction of the multiple M2-brane action using 3-algebras, some non-abelian representation of the 2,0 tensor supermultiplet was found in [ ]. Their formulation includes a non-abelian analogue of the auxiliary scalar field appearing in the PST formulation of the abelian M5-brane.

Closure of the superalgebra provides a set of equations of motion and constraints. Some further work along this direction can be found in [ ]. Some of the BPS equations derived from this analysis were argued to be naturally reinterpreted in loop space [ ].

15. Physics of D-branes, Part III

There has been a different approach to the problem involving non-commutative versions of 3-algebras [ ], but it seems fair to claim that this remains a very important open problem for the field. This was developed further in [ , ]. The calculations of beta functions in general nonlinear sigma models were done in [ 17 , ]. For a general discussion of string theory in curved backgrounds see [ ] or the discussions in books [ , ]. Polyakov used the formulation of classical string theory in terms of an auxiliary world sheet metric [ , ] to develop the modern approach to the path integral formulation of string theory in [ , ].

It remains to be seen whether this is the case. The existence of kappa symmetry as a fermionic gauge symmetry was first pointed out in superparticle actions in [ , , , ]. Though the term kappa symmetry was not used in these references, since it was later coined by Townsend, the importance of WZ terms for its existence is already stated in these original works.

For a proper definition of these superfields, see Appendix A. See Appendix A. See [ , ] for reviews and textbooks on what an effective field theory is and what the principles behind them are. I will introduce this notion more thoroughly in Section 3. The first examples of this phenomena were reported by Nambu [ ] and Goldstone [ ].

For earlier reviews on D-brane effective actions and on M-brane interactions, see [ ] and [ ], respectively. This is the correct way to compute the energy momentum tensor due to the coupling of branes to gravity. The importance of these assumptions will be stressed when discussing the regime of validity of brane effective actions in Section 3. There actually exist further gravitational interaction terms necessary for the cancellation of anomalies [ ], but we will always omit them in our discussions concerning D-brane effective actions.

Relevant work on the subject includes [ 24 , 77 , 16 , 75 ]. Following the same philosophy as for their bosonic truncations, this functional dependence can be derived from the double dimensional reduction of the supersymmetric M2-brane action to be discussed in Section 3. This also provides a derivation of the WZ couplings to be constructed in this subsection. Of course, this consideration would only apply to the D2-brane, but T-duality would allow one to extend this conclusion for any D p -brane [ , ].

The derivation of the property is made more manifest in formalisms in which the tension is generated dynamically by the addition of an auxiliary volume density [ 86 , , 94 ]. I will prove this explicitly in Section 5. All our charge conjugation matrices are antisymmetric and unitary, i. For earlier work, see [ 4 ], which extended the original Volkov-Akulov approach in [ ]. There exists some similar phenomena on the M5-brane dynamics with the self-dual 3-form field strength. See [ 74 ] for a discussion on the emergence of noncommutative gauge theories when the self-dual 3-form field strength is close to its critical value.

There are several claims in the literature advocating that extra massless degrees of freedom emerge in brane effective actions when the latter probe black holes very close to their horizons. See [ , , , ] for interesting work in this direction. BPS stands for Bogomolny, Prasad and Sommerfield and their work on stable solitonic configurations [ , ]. For a complete and detailed discussion of the supersymmetric and kappa invariant D-brane Hamiltonian formalism, see [ 94 ], which extends the bosonic results in [ , ] and the type IIB superMinkowski ones in [ ].

Here I follow [ 94 ] even though the analysis is restricted to the bosonic sector. There are many papers studying the dynamics of BIons, including [ , 31 , ] and [ ], where the solution to the Born-Infeld action reviewed here is proven to solve the equations of motion derived from higher-order corrections to the effective action. This equation has a huge history in mathematical physics. For a self-contained presentation on all the mathematical developments regarding this equation, see [ ]. For generalisations to higher dimensions, see [ , ].

Free D Branes, Guage String Duality And Noncommutative Theories [Thesis] 2004

Strictly speaking, if the supertube cross-section is open, they can carry D2-brane charge. The arguments given above only apply to closed cross-sections. The reader is encouraged to read the precise original discussion in [ ] concerning this point and the bounds on angular momentum derived from it. For a list of reviews on this subject, see [ , , 65 , , 36 , ]. By arbitrary, it is meant a general curve that is not self-intersecting and whose tangent vector never vanishes.

I do not write this term explicitly here because it will not couple to our D3-brane probes. Dual giant gravitons are spherical rotating D3-branes in which the 3-sphere wrapped by the brane is in AdS 5. See [ ] for a proper construction of these configurations and some of its properties. These configurations appeared in [ 64 , 71 ], extending earlier seminal work [ , 53 ]. What is meant here by maximally entropic is that, given a large black hole, there may be more than one possible deconstruction of the total charge in terms of constituents with different charge composition.

By maximally entropic I mean the choice of charge deconstruction whose moduli space of configurations carries the largest contribution to the entropy of the system. There is a lot of work in this direction. For a review on the emergence of geometry and gravity in matrix models, in particular in the context of the IKKT matrix conjecture [ ], see [ ]. For more recent discussions, see [ ]. Using T-duality arguments this would also include acceleration and higher-derivative corrections in the scalar sector X m describing the excitations of the D-brane along the transverse dimensions.

For a complete list of references, see [ 12 ]. The equivalence of the equations of motion obtained in the PST-formalism and the ones developed in the superembedding formalism was proven in [ 46 ]. Some of the material covered in this review is based on the PhD thesis World Volume Approach to String Theory defended by the author under the supervision of Prof. Gomis at the University of Barcelona in May JS would like to thank his PhD advisor J.

Gomis for introducing him to this subject, to P. Townsend for sharing his extensive knowledge on many topics reviewed here, and to F. Brandt, K. Kamimura, O. Lunin, D. Mateos, A. Ramallo and J. Skip to main content Skip to sections. Advertisement Hide. Download PDF. Living Reviews in Relativity December , Cite as. Open Access. First Online: 27 February The purpose of this review is to describe the kinematical properties characterising the super-symmetric gauge theories emerging as brane effective field theories in string and M-theory, and some of their important applications.

In particular, I will focus on D-branes, M2-branes and M5-branes. Open image in new window. Figure 1 Layout of the main relations covered in this review. These effective theories depend on the number of branes in the system and the geometry they probe. When a single brane is involved in the dynamics, these theories are abelian and there exists a spacetime covariant and manifestly supersymmetric formulation, extending the Green-Schwarz worldsheet one for the superstring. The main concepts I want to stress in this part are a the identification of their dynamical degrees of freedom, providing a geometrical interpretation when available,.

The covariant abelian brane actions provide a generalisation of the standard charged particle effective actions describing geodesic motion to branes propagating on arbitrary on-shell super-gravity backgrounds. In the second part of this review, I describe some of their important applications. These will be split into two categories: supersymmetric world volume solitons and dynamical aspects of the brane probe approximation. Solitons will allow me to a stress the technical importance of kappa symmetry in determining these configurations, linking Hamiltonian methods with supersymmetry algebra considerations,.

The purpose of this section is to briefly review the Green-Schwarz GS formulation of the superstring. This is not done in a self-contained way, but rather as a very swift presentation of the features that will turn out to be universal in the formulation of brane effective actions. There exist two distinct formulations for the super string: 1. Just as point particles can be charged under gauge fields, strings can be charged under 2-forms. For completeness, let me stress that at the classical level, the dynamics of the background fields couplings is not specified.

Quantum mechanically, the consistency of the interacting theory defined in Eq. This is illustrated in Figure 2. Figure 2 Different superstring formulations require curved backgrounds to be on-shell. Instead of providing the answer directly, it is instructive to go over the explicit construction, following [ ]. The current standard resolution to this situation is the addition of an extra term to the action while still preserving supersymmetry. This extra term can be viewed as an extension of the bosonic WZ coupling 4 , a point I shall return to when geometrically reinterpreting the action so obtained [ ].

The missing ingredient in the above discussion is the existence of an additional fermionic gauge symmetry, kappa symmetry , responsible for the removal of half of the fermionic degrees of freedom. Curved background extension : One of the spins of the superspace reinterpretation in Eq. More specifically, I will be concerned with the kinematical properties characterising S brane when the latter describes a single brane, though in Section 7 , the extension to many branes will also be briefly discussed. From the perspective of full string theory, it is important to establish the regime in which the full dynamics is governed by S brane.

This requires one to freeze the gravitational sector to its classical on-shell description and to neglect its backreaction into spacetime. To argue this, analyse the field content of these vector supermultiplets. This point agrees with the open string conformal field theory description of D branes.

Summary : All half-BPS D p -branes, M2-branes and M5-branes are described at low energies by effective actions written in terms of supermultiplets in the corresponding world-volume dimension. The number of on-shell bosonic degrees of freedom is 8. Effective actions satisfying these two symmetry requirements involve the addition of both extra, non-physical, bosonic and fermionic degrees of freedom. To preserve their non-physical nature, these supersymmetric brane effective actions must be invariant under additional gauge symmetries world volume diffeomorphisms, to gauge away the extra scalars, kappa symmetry, to gauge away the extra fermions.

Before discussing the general strategy, let me introduce the on-shell bosonic configurations to be analysed below. All of them are described by a non-trivial metric and a gauge field carrying the appropriate brane charge.

  • mathematics and statistics online.
  • Albatros D.III.
  • Food and Nutritional Toxicology.
  • The Symmetrical English;
  • Scotland: Mapping the Nation.
  • Brane Effective Actions, Kappa-Symmetry and Applications.

Let me first sketch the argument behind the generation of massless modes in supergravity theories, where all relevant symmetries are gauge, before discussing the specific details below. Scalar modes : These are the most intuitive geometrically. They correspond to the breaking of translations along the transverse directions to the brane. The relevant gauge symmetry is clearly a diffeomorphism. Notice the dependence on the harmonic function guarantees the appropriate behaviour at infinity, for any s.

Dynamical fields transform under diffeomorphisms through Lie derivatives. Fermionic modes : These must correspond to the breaking of supersymmetry. Table 4 Summary of supergravity Goldstone modes. Vector modes : The spectrum of open strings with Dirichlet boundary conditions includes a vector field. Since the origin of such massless degrees of freedom must be the breaking of some abelian supergravity gauge symmetry, it must be the case that the degree form of the gauge parameter must coincide with the one-form nature of the gauge field.

Tensor modes : The presence of five transverse scalars to the M5-brane and the requirement of world volume supersymmetry in six dimensions allowed us to identify the presence of a two-form potential with self-dual field strength. Bosonic M2-brane : In the absence of world volume gauge field excitations, all brane effective actions must satisfy two physical requirements 1. The relevance of the minimal charge coupling can be understood by considering the full effective action involving both brane and gravitational degrees of freedom Restricting ourselves to the kinetic term for the target space gauge field, i.

Since M2-branes do not involve any gauge field degree of freedom, the above discussion covers all its bosonic degrees of freedom. Bosonic D-branes : Due to the perturbative description in terms of open strings [ ], D-brane effective actions can, in principle, be determined by explicit calculation of appropriate open string disk amplitudes. Let me first discuss the dependence on gauge fields in these actions.

Since in the absence of world-volume gauge-field excitations, D-brane actions should reduce to Eq. The DBI action is a natural extension of the NG action for branes, but it does not capture all the relevant physics, even in the absence of acceleration terms, since it misses important background couplings, responsible for the WZ terms appearing for strings and M2-branes.

Let me stress the two main issues separately: 1. The functional dependence on the gauge field V 1 in a general closed string background. D-branes are hypersurfaces where open strings can end. Thus, open strings do have endpoints. The coupling to the dilaton. The D-brane effective action is an open string tree level action, i.

The WZ couplings. Thus, their effective actions should include a minimal coupling to the pullback of such form. Such coupling would not be invariant under the target space gauge transformations Bosonic covariant M5-brane : The bosonic M5-brane degrees of freedom involve scalar fields and a world volume 2-form with self-dual field strength.

The former are expected to be described by similar arguments to the ones presented above. The situation with the latter is more problematic given the tension between Lorentz covariance and the self-duality constraint. This problem has a fairly long history, starting with electromagnetic duality and the Dirac monopole problem in Maxwell theory, see [ ] and references therein, and more recently, in connection with the formulation of supergravity theories such as type IIB, with the self-duality of the field strength of the RR 4-form gauge potential.

There are several solutions in the literature based on different formalisms: 1. In this review, I follow the PST formalism. The specific dualities I will be appealing to are the strong coupling limit of type IIA string theory, its relation to M-theory and the action of T-duality on type II string theories and D-branes. Figure 3 summarises the set of relations between the brane tensions discussed in this review under these symmetries. Figure 3 Set of half-BPS branes discussed in this review, their tensions and some of their connections under T-duality and the strongly-coupled limit of type IIA.

Let me focus on the bulk transformation. T-duality is a perturbative string theory duality [ ]. But it is known [ ] that there is just such a unique supergravity theory. This process is illustrated in the diagram of Figure 4. These are the T-duality rules. Connection to the string worldsheet : Consider the propagation of an M2-brane in an dimensional background of the form The next step is to describe the world volume dualisation and the origin of the U 1 gauge symmetry on the D2 brane effective action [ ].

The second question can be addressed by an analysis of the D-brane action bosonic symmetries. I argued that the realisation of T-duality on the brane action requires one to study its double-dimensional reduction. Let me comment on Eq. But in its gauge-fixed functionally-truncated version, it transforms like a world volume scalar. Having clarified the origin of symmetries in the T-dual picture, let me analyse the functional dependence of the effective action. This can be achieved by adding and subtracting the relevant pullback quantities.

Here I follow similar techniques to the ones developed in [ , ]. The expert reader may have noticed that the RR T-duality rules do not coincide with the ones appearing in [ ]. The reason behind this is the freedom to redefine the fields in our theory. In particular, there exist different choices for the RR potentials, depending on their transformation properties under S-duality. In the study of global supersymmetric field theories, one learns the superfield formalism is the most manifest way of writing interacting manifestly-supersymmetric Lagrangians [ ].

The main difference in the GS formulation of brane effective actions is that it is spacetime itself that must be formulated in a manifestly supersymmetric way. There are two crucial points to appreciate for our purposes 1. Both these points were already encountered in our review of the GS formulation for the superstring. The same features will hold for all brane effective actions discussed below. After all, both strings and branes are different objects in the same theory. Consequently, any manifestly spacetime supersymmetric and covariant formulation should refer to the same superspace.

This is not what occurs in standard superspace formulations of supersymmetric field theories. There exists a classically equivalent formulation to the GS one, the superembedding formulation that extends both the spacetime and the world volume to supermanifolds. Though I will briefly mention this alternative and powerful formulation in Section 8 , I refer readers to [ ]. Figure 5 Kappa symmetry and world volume diffeomorphisms allow one to couple the brane degrees of freedom to the superfield components of supergravity in a manifestly covariant and supersymmetric way.

Let me first set my conventions. Let me start the discussion with the DBI piece of the action. This involves couplings to the NS-NS bulk sector, a sector that is also probed by the superstring. The above defines a cohomological problem whose solution is not guaranteed to be kappa invariant. Since our goal is to construct an action invariant under both symmetries, let me first formulate the requirements due to the second invariance. Second, kappa symmetry must be able to remove half of the fermionic degrees of freedom. This fact can be used to conveniently parameterise the kappa invariance of the total Lagrangian.

The explicit construction of these objects was given in [ 9 ]. Here, I simply summarise their results. Two observations are in order: 1. Let me summarise the global and gauge symmetry structure of the full action. Its symmetry structure is analogous to the one described for D-branes. The structure of the transformations is universal, but the details of the kappa symmetry matrix depend on the specific theory, as described below. A second universal feature, associated with the projection nature of kappa symmetry transformations, i. The effective action describing a single M2-brane in an arbitrary dimensional background is formally the same as in Eq.

The action is manifestly 3d-diffeomorphism invariant. It was shown to be kappa invariant under the transformations , without any gauge field, whenever the background superfields satisfy the constraints reviewed in Appendix A. Besides its defining projective nature when acting on fermions, i. The main purpose of this section is to discuss the global symmetries of brane effective actions, the algebra they close and to emphasise the interpretation of some of the conserved charges appearing in these algebras before and after gauge fixing of the world volume diffeomorphisms and kappa symmetry.

To prove that background symmetries give rise to brane global symmetries, one must first properly define the notion of superisometry of a supergravity background.

  1. Primary Research Area;
  2. mathematics and statistics online!
  3. Suggest Documents;
  4. All the Way to Berlin: A Paratrooper at War in Europe.
  5. Gauge Theories on Deformed Spaces;
  6. It is by now well known that such super algebras contain more bosonic charges than the ones geometrically realised as super isometries. There are several ways of reaching this conclusion: 1. One can decompose this representation into irreducible representations of the bosonic spacetime isometry group. Let me review how these structures emerge in both supergravity and brane effective actions. The above is merely based on group theory considerations that may or may not be realised in a given physical theory. The latter is invariant both under supersymmetry and bulk translations.

    Quasi-invariance of the WZ term will be responsible for the generation of extra terms in the calculation of the Poisson bracket of these charges [ ]. This was confirmed for the case at hand in [ ], where the M5-brane superalgebra was explicitly computed. The link between both superalgebras is achieved through the gauge fixing of world volume diffeomorphisms and kappa symmetry, the gauge symmetries responsible for the covariance of the original brane action in the GS formalism.

    For infinite branes, this choice is valid globally and does describe a vacuum configuration. On general grounds, there must exist sixteen linear combinations of supersymmetries being linearly realised, whereas the sixteen remaining will be spontaneously broken by the brane. Another natural choice corresponds to picking the projector describing the preserved supersymmetries by the brane from the spacetime perspective.

    Having established the relation between spacetime and world volume symmetries, it is natural to close our discussion by revisiting the superalgebra closed by the linearly realised world volume super symmetries, once both diffeomorphisms and kappa symmetry have been fixed. Since space-time superalgebras included extra bosonic charges due to the quasi-invariance of the brane WZ action, the same will be true for their gauge fixed actions.

    Consider the M2-brane discussed above. Dynamically, all brane effective actions reviewed previously, describe the propagation of a brane in a fixed on-shell spacetime background solving the classical supergravity equations of motion. The purpose of this section is to spell out more precisely the conditions that make the above requirements not sufficient. As in any effective field theory action, one must check the validity of the assumptions made in their derivation.

    In our discussions, this includes 1. Validity of the background description : Whenever the supergravity approximation is not reliable, the brane description will also break down. Similar considerations apply to dimensional supergravity. Thus, kappa symmetric invariant D-brane effective actions ignore corrections in derivatives of this field strength, i. Interestingly, these corrections map to acceleration and higher-order derivative corrections in the scalar fields X m under T-duality, see Eq. Thus, there exists the further requirement that all dynamical fields in brane effective actions are slowly varying.

    One such realisation is in terms of classical bosonic on-shell configurations. As it often occurs with supersymmetric configurations, instead of focusing on the integration of the equations of motion, I will focus on the conditions ensuring preservation of supersymmetry and on their physical interpretation. Figure 6 Set of relations involving kappa symmetry, spacetime supersymmetry algebras, their bounds and their realisation as field theory BPS bounds in terms of brane solitons using the Hamiltonian formulation of brane effective actions.

    The same question for brane effective actions is treated in a conceptually analogous way. We are interested in deriving a general condition for any bosonic configuration to preserve supersymmetry. We must work in the subspace of field configurations being both physical and bosonic [ 85 ]. Because of this, I find it convenient to break the general argument into two steps.

    Invariance under kappa symmetry. Invariance under supersymmetry. This matrix encodes information 1. Just as in supergravity, any solution to Eq. More precisely, 1. The existence of energy bounds in supersymmetric theories can already be derived from purely superalgebra considerations. For example, consider the M-algebra If the system is invariant under time translations, energy will be preserved, and it can be computed using the Hamiltonian formalism, for example.

    Depending on the amount and nature of the charges turned on by the configuration, the general functional dependence of the bound changes. Depending on whether these commute or anticommute, the bound satisfied by the energy P 0 changes, see for example a discussion on this point in [ ]. Thus, one expects to be able to decompose the Hamiltonian density for these configurations as sums of the other charges and positive definite extra terms such that when they vanish, the bound is saturated.

    As in any Hamiltonian formulation 31 , the first step consists in breaking covariance to allow a proper treatment of time evolution. The Hamiltonian formulation for the M2-brane can be viewed as a particular case of the analysis provided above, but in the absence of gauge fields. It was originally studied in [ 88 ]. Callan and J. Curtis G. Callan, Jr.

    “ Schools ”

    Kiritsis pages, 22 figures, to be published in book form by Leuven University Press. Schwarz 44 pages; Lectures presented at the St. Stefan Forste pages. West pages, 2 figures. Miao Li 76 pages. Alvarez, P. Meessen 87 pages, 4 figures , JHEP 02 Cumrun Vafa 53 pages. Gomez, R. Hernandez pages, 6 figures. Dijkgraaf pages, 31 figures. Aharony, S. Gubser, J. Maldacena, H. Ooguri, Y. Oz pages, 42 post-script figures , Phys. Igor R.

    Jacob Sonnenschein 47 pages, 13 figs; Lectures presented at Santiago de Compostela Christopher P. Herzog, Igor R. Klebanov, Peter Ouyang 40 pages; based on I. Gordon W. Semenoff, K. Eric D'Hoker, Daniel Z. Freedman pages; 14 figures. Kostas Skenderis 42 pages. Marcos Marino 69 pages, 13 figures; to appear in the Proceedings of the meeting of theoretical physicists and mathematicians held at Strasbourg, May Belitsky, S. Vandoren, P. Caselle 80 pages, 6 figures, lectures delivered at the School of Theoretical Physics, Parma, , Int. A15 Duff pages.

    Jens L. Petersen 71 pages, lecture notes , Int. Michael R. Douglas, S.

    Ralph Blumenhagen

    Klebanov 20 pages; based on the talk at Orbis Scientiae '98, Ft. Lauderdale, FL, December Leonard Susskind 19 pages. Bytsenko and F. Williams, World Scientific, Gary T.


    Horowitz 13 pages. Talk given at Strings '99, Potsdam, Germany , Class. Horowitz, Saul A. Teukolsky 13 pages. Steven B. Andy Strominger 70 pages, 18 figures. Thomas Banks 65 pages, 9 figures , Nucl. Hawking 62 pages with 51 figures. Amanda W. Peet 53 pages , Class. Juan M. Maldacena 80 pages, 8 figures; Princeton University Ph. Thesis, June Justin R. David, Gautam Mandal, Spenta R. Wadia pages, 4 figures; Written for Physics Reports.

    Robert C. Sumit R. Das, Samir D. Mathur 49 pages, 4 figures , Ann. Peet 80 pages, 8 figures. Spenta R. Wadia 58 pages. Emil J. Martinec 29 pages, 12 figures; lectures at Cargese Renata Kallosh 34 pages; published in the Proceedings of a conference held in Corfu, Greece in September Kostas Skenderis 44p; Erice lecture notes , Lect. Notes Phys. Maldacena 13 pages, 2 figures , Nucl.

    Horowitz 33 pages. Leonard Susskind, John Uglum 20 pages, 6 figures , Nucl. Tseytlin 37 pages, contribution to Yuri Golfand memorial volume. Konstantin G. Duff 76 pages, based on lectures given at the T. Douglas J. Smith 84 pages. Donam Youm pages , Phys. Vazquez-Poritz pages, 15 figures, Penn Ph.

    Thesis, advisor: Mirjam Cvetic. Townsend Cargese Lectures. John March-Russell 12 pages ,. Antonio Padilla PhD. Thesis, supervisor: Ruth Gregory. Arkani-Hamed, S. Dimopoulos, and G. Dvali 51 pages, Phys. D59 Dominic Joyce 56 pages. Steven S. Duff 15 pages, talk at the Supergravity 25 conference, Stony Brook, December Paul S. Aspinwall 82 pages, 8 figures, TASI Shamit Kachru 16 pages, 6 figures. Lectures delivered at 33rd Karpacz Winter School , Nucl. Aspinwall pages, 11 figures, TASI Sean M. Nick E.