E11, Brane Dynamics and Duality Symmetries

Following  we use the non-linear realisation of the semi-direct product of E11 and its vector representation to construct brane dynamics. The brane moves through a spacetime which arises in the non-linear realisation from the vector representation and it contains the usual embedding coordinates as well as the world volume fields. The resulting equations of motion are first order in derivatives and can be thought of as duality relations. Each brane carries the full E11 symmetry and so the Cremmer-Julia duality symmetries. We apply this theory to find the dynamics of the IIA and IIB strings, the M2 and M5 branes, the IIB D3 brane as well as the one and two branes in seven dimensions.

Related papers:

title: Introduction to Strings and Branes
abstract: Preface 1. The point particle 2. The classical bosonic string 3. The quantum bosonic string 4. The light-cone approach 5. Clifford algebras and spinors 6. The classical superstring 7. The quantum superstring 8. Conformal symmetry and two-dimensional field theory 9. Conformal symmetry and string theory 10. String compactification and the heterotic string 11. The physical states and the no ghost theorem 12. Gauge covariant string theory 13. Supergravity theories in 4, 10 and 11 dimensions 14. Brane dynamics 15. D-branes 16. String theory and Lie algebras 17. Symmetries of string theory 18. String interactions Appendices Index.

title: Charge multiplets and masses for E11
abstract: The particle, string and membrane charge multiplets are derived in detail from the decomposition of the l1 (charge) representation of E11 in three, four, five, six, seven and eight spacetime dimensions. A tension formula relating weights of the l1 (charge) representation of E11 to the fundamental objects of M-theory and string theory is presented. The reliability of the formula is tested by reproducing the tensions of the content of the charge multiplets. The formula reproduces the masses for the pp-wave, M2, M5 and the KK-monopole from the low level content of the l1 representation of E11. Furthermore the tensions of all the Dp-branes of IIA and IIB theories are found in the relevant decomposition of the l1 representation, with the string coupling constant and α ′ appearing with the expected powers. The formula leads to a classification of all the exotic, KK-brane charges of M-theory.

title: Exotic E(11) branes as composite gravitational solutions
abstract: A two-parameter group element is presented that interpolates between M-brane solutions. The group element is used to interpret a number of exotic branes related to the generators of the adjoint representation of E11 as non-marginal half-BPS bound states of M-branes. It is conjectured that the adjoint representation of E11 contains only generators related to bound states of fundamental M-branes which, in the limit, may be understood as membrane molecules.

title: Bound states of string theory and beyond
abstract: A bstractAll bound states of fundamental strings, D-branes and NS-branes of string theory, both type-IIA and type-IIB, which may be described by a null geodesic motion on the coset G/K(G) where G is a group of type An, Dn or En embedded within E11 are presented.

title: Hidden superconformal symmetry in M-theory
abstract: The bosonic sectors of the eleven-dimensional and IIA supergravity theories are derived as non-linear realisations. The underlying group includes the conformal group, the general linear group and as well as automorphisms of the supersymmetry algebra. We discuss the supersymmetric extension and argue that Osp(1/64) is a symmetry of M-theory. We also derive the effective action of the closed bosonic string as a non-linear realisation.

title: Generalised space-time and duality
abstract: In this Letter we consider the previously proposed generalised space–time and investigate the structure of the field theory upon which it is based. In particular, we derive a SO ( D , D ) formulation of the bosonic string as a non-linear realisation at lowest levels of E 11 ⊗ s l 1 where l 1 is the first fundamental representation. We give a Hamiltonian formulation of this theory and carry out its quantisation. We argue that the choice of representation of the quantum theory breaks the manifest SO ( D , D ) symmetry but that the symmetry is manifest in a non-commutative field theory. We discuss the implications for the conjectured E 11 symmetry and the role of the l 1 representation.

title: Duality rotations in string theory
abstract: Starting from a σ-model with target space coordinates x μ ( μ = 1,…, n ), we introduce a dual σ-model with coordinates y α ( α = 1,…, n ) for which the roles of field equations and Bianchi identities are interchanged. The hidden symmetries are then made manifest by considering a target space of double the dimension with coordinates Z M = ( x μ , y α ).

title: Duality rotations in membrane theory
abstract: In analogy with a previous treatment of strings, it is shown that membrane theories exhibit global noncompact symmetries which have their origin in duality transformations on the three-dimensional worldvolume which rotate field equations into Bianchi identities. However, in contrast to the string, the worldvolume metric also transforms under duality by a conformal factor. In this way the Cremmer-Julia hidden symmetries of supergravity are seen to be a consequence of supermembrane duality. Moreover, the string duality follows from that of the membrane by simultaneous dimensional reduction. Generalization to higher-dimensional objects is straightforward.

title: Generalised vielbeins and non-linear realisations
abstract: A bstractWe briefly review why the non-linear realisation of the semi-direct product of a group with one of its representations leads to a field theory defined on a generalised space-time equipped with a generalised vielbein. We give formulae, which only involve matrix multiplication, for the generalised vielbein, the Cartan forms and their transformations. We consider the generalised space-time introduced in 2003 in the context of the non-linear realisation of the semi-direct product of E11 and its first fundamental representation. For this latter theory we give explicit expressions for the generalised vielbein up to and including the levels associated with the dual graviton in four, five and eleven dimensions and for the IIB theory in ten dimensions. We also compute the generalised vielbein, up to the analogous level, for the non-linear realisation of the semi-direct product of very extended SL(2) with its first fundamental representation, which is a theory associated with gravity in four dimensions.

title: E-11, generalised space-time and IIA string theory
abstract: As advocated in  we construct the non-linear realisation of the semi-direct product of E 11 and its first fundamental representation at lowest level from the IIA viewpoint. We find a theory that is SO ( 10 , 10 ) ⊗ GL ( 1 ) invariant and contains the fields of gravity, a two form and a dilaton but which depend on coordinates which belong to the vector representation of SO ( 10 , 10 ) . The resulting Lagrangian agrees that of recent work on the so-called doubled field theory. However, the construction given in this Letter is straightforward and systematic. It also reveals the relevant underlying symmetries and opens the way to include the Ramond–Ramond, and higher level, fields together with additional coordinates of the generalised space–time.

title: Superspace duality in low-energy superstrings.
abstract: We extend spacetime duality to superspace, including fermions in the low-energy limits of superstrings. The tangent space is a curved, extended superspace. The geometry is based on an enlarged coordinate space where the vanishing of the d'Alembertian is as fundamental as the vanishing of the curl of a gradient.

title: Properties of the Eleven-Dimensional Supermembrane Theory
abstract: We study in detail the structure of the Lorentz covariant, spacetime supersymmetric 11-dimensional supermembrane theory. We show that for a flat spacetime background, the spacetime supersymmetry becomes an N=8 world volume (rigid) supersymmetry in a â€œphysicalâ€� gauge; we also present the field equations and transformation rules in a â€œlightconeâ€� gauge. We semiclassically quantize the closed torodial supermembrane on a spacetime (Minkowki)4 A— (flat 7-torus), and review some mathematical results that are relevant for path integral quantization.

title: Introduction to Strings and Branes
abstract: Preface 1. The point particle 2. The classical bosonic string 3. The quantum bosonic string 4. The light-cone approach 5. Clifford algebras and spinors 6. The classical superstring 7. The quantum superstring 8. Conformal symmetry and two-dimensional field theory 9. Conformal symmetry and string theory 10. String compactification and the heterotic string 11. The physical states and the no ghost theorem 12. Gauge covariant string theory 13. Supergravity theories in 4, 10 and 11 dimensions 14. Brane dynamics 15. D-branes 16. String theory and Lie algebras 17. Symmetries of string theory 18. String interactions Appendices Index.

title: M2- and M5-branes in E11 Current Algebra Formulation of M-theory
abstract: Equations of motion for M2- and M5-branes are written down in the $E_{11}$ current algebra formulation of M-theory. These branes correspond to currents of the second and the fifth rank antisymmetric tensors in the $E_{11}$ representation, whereas the electric and magnetic fields (coupled to M2- and M5-branes) correspond to currents of the third and the sixth rank antisymmetric tensors, respectively. We show that these equations of motion have solutions in terms of the coordinates on M2- and M5-branes. We also discuss the geometric equations, and show that there are static solutions when M2- or M5-brane exists alone and also when M5-brane wraps around M2-brane. This situation is realized because our Einstein-like equation contains an extra term which can be interpreted as gravitational energy contributing to the curvature, thus avoiding the usual intersection rule.