Axel Kleinschmidt

The gauge structure of generalised diffeomorphisms

A bstractWe investigate the generalised diffeomorphisms in M-theory, which are gauge transformations unifying diffeomorphisms and tensor gauge transformations. After giving an En(n)-covariant description of the gauge transformations and their commutators, we show that the gauge algebra is infinitely reducible, i.e., the tower of ghosts for ghosts is infinite. The Jacobiator of generalised diffeomorphisms gives such a reducibility transformation. We give a concrete description of the ghost structure, and demonstrate that the infinite sums give the correct (regularised) number of degrees of freedom. The ghost towers belong to the sequences of representations previously observed appearing in tensor hierarchies and Borcherds algebras. All calculations rely on the section condition, which we reformulate as a linear condition on the cotangent directions. The analysis holds for n < 8. At n = 8, where the dual gravity field becomes relevant, the natural guess for the gauge parameter and its reducibility still yields the correct counting of gauge parameters.

A dynamical inconsistency of Horava gravity

The dynamical consistency of the nonprojectable version of Hoifmmode check{r}else v{r}fi{}ava gravity is investigated by focusing on the asymptotically flat case. It is argued that for generic solutions of the constraint equations the lapse must vanish asymptotically. We then consider particular values of the coupling constants for which the equations are tractable and in that case we prove that the lapse must vanish everywherechar22{}and not only at infinity. Put differently, the Hamiltonian constraints are generically all second-class. We then argue that the same feature holds for generic values of the couplings, thus revealing a physical inconsistency of the theory. In order to cure this pathology, one might want to introduce further constraints but the resulting theory would then lose much of the appeal of the original proposal by Hoifmmode check{r}else v{r}fi{}ava. We also show that there is no contradiction with the time-reparametrization invariance of the action, as this invariance is shown to be a so-called ``trivial gauge symmetry in Hoifmmode check{r}else v{r}fi{}ava gravity, hence with no associated first-class constraints.

Representations of g(+++) and the role of space-time

We consider the decomposition of the adjoint and fundamental representations of very extended Kac-Moody algebras +++ with respect to their regular A type subalgebra which, in the corresponding non-linear realisation, is associated with gravity. We find that for many very extended algebras almost all the A type representations that occur in the decomposition of the fundamental representations also occur in the adjoint representation of +++. In particular, for E8+++, this applies to all its fundamental representations. However, there are some important examples, such as AN−3+++, where this is not true and indeed the adjoint representation contains no generator that can be identified with a space-time translation. We comment on the significance of these results for how space-time can occur in the non-linear realisation based on +++. Finally we show that there is a correspondence between the A representations that occur in the fundamental representation associated with the very extended node and the adjoint representation of +++ which is consistent with the interpretation of the former as charges associated with brane solutions.

E10 and SO(9, 9) invariant supergravity

We show that (massive) D = 10 type IIA supergravity possesses a hidden rigid D9 ≡ SO(9,9) symmetry and a hidden local SO(9) × SO(9) symmetry upon dimensional reduction to one (timelike) dimension. We explicitly construct the associated locally supersymmetric Lagrangian in one dimension, and show that its bosonic sector, including the mass term, can be equivalently described by a truncation of an E10/K(E10) non-linear σ-model to the level l ≤ 2 sector in a decomposition of E10 under its D9 subalgebra. This decomposition is presented up to level l = 10, and the even and odd level sectors are identified tentatively with the Neveu‐ Schwarz and Ramond sectors, respectively. Further truncation to the level l = 0 sector yields a model related to the reduction of D = 10 type I supergravity. The hyperbolic Kac‐Moody algebra DE10, associated to the latter, is shown to be a proper subalgebra of E10, in accord with the embedding of type I into type IIA supergravity. The corresponding decomposition of DE10 under D9 is presented up to level l = 5.

IIB supergravity and E10

We analyse the geodesic E10/K(E10) σ-model in a level decomposition w.r.t. the A8 × A1 subalgebra of E10, adapted to the bosonic sector of type IIB supergravity, whose SL(2, R) symmetry is identified with the A1 factor. The bosonic supergravity equations of motion, when restricted to zeroth and first order spatial gradients, are shown to match with the σ-model equations of motion up to level l = 4. Remarkably, the self-duality of the five-form field strength is implied by E10 and the matching.

Counting supersymmetric branes

Maximal supergravity solutions are revisited and classified, with particular emphasis on objects of co-dimension at most two. This class of solutions includes branes whose tension scales with xxxx. We present a group theory derivation of the counting of these objects based on the corresponding tensor hierarchies derived from E11 and discrete T- and U-duality transformations. This provides a rationale for the wrapping rules that were recently discussed for σu2009≤u20093 in the literature and extends them. Explicit supergravity solutions that give rise to co-dimension two branes are constructed and analysed.

Supersymmetric quantum cosmological billiards

D=11 supergravity near a spacelike singularity admits a cosmological billiard description based on the hyperbolic Kac-Moody group E{sub 10}. The quantization of this system via the supersymmetry constraint is shown to lead to wave functions involving automorphic (Maass wave) forms under the modular group W{sup +}(E{sub 10}) congruent with PSL{sub 2}(O) with Dirichlet boundary conditions on the billiard domain. A general inequality for the Laplace eigenvalues of these automorphic forms implies that the wave function of the Universe is generically complex and always tends to zero when approaching the initial singularity. We discuss possible implications of this result for the question of singularity resolution in quantum cosmology and comment on the differences with other approaches.

An E9 multiplet of BPS states

We construct an infinite E9 multiplet of BPS states for 11D supergravity. For each positive real root of E9 we obtain a BPS solution of 11D supergravity, or of its exotic counterparts, depending on two non-compact transverse space variables. All these solutions are related by U-dualities realised via E9 Weyl transformations in the regular embedding E9 ⊂ E10 ⊂ E11. In this way we recover the basic BPS solutions, namely the KK-wave, the M2 brane, the M5 brane and the KK6-monopole, as well as other solutions admitting eight longitudinal space dimensions. A novel technique of combining Weyl reflexions with compensating transformations allows the construction of many new BPS solutions, each of which can be mapped to a solution of a dual effective action of gravity coupled to a certain higher rank tensor field not contained in 11D supergravity. For real roots of E10 which are not roots of E9, we obtain additional BPS solutions transcending 11D supergravity (as exemplified by the lowest level solution corresponding to the M9 brane). The relation between the dual formulation and the one in terms of the original 11D supergravity fields has significance beyond the realm of BPS solutions. We establish the link with the Geroch group of general relativity, and explain how the E9 duality transformations generalize the standard Hodge dualities to an infinite set of non-closing dualities.

Eisenstein series for infinite-dimensional U-duality groups

A bstractWe consider Eisenstein series appearing as coefficients of curvature corrections in the low-energy expansion of type II string theory four-graviton scattering amplitudes. We define these Eisenstein series over all groups in the En series of string duality groups, and in particular for the infinite-dimensional Kac-Moody groups E9, E10 and E11. We show that, remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series. This resonates with the idea that the constant term of the Eisenstein series encodes perturbative string corrections in BPS-protected sectors allowing only a finite number of corrections. We underpin our findings with an extensive discussion of physical degeneration limits in D <u20093 space-time dimensions.

Gradient representations and affine structures in AEn

We study the indefinite Kac–Moody algebras AEn, arising in the reduction of Einsteins theory from (n + 1) spacetime dimensions to one (time) dimension, and their distinguished maximal regular subalgebras and A(1)n−2. The interplay between these two subalgebras is used, for n = 3, to determine the commutation relations of the gradient generators within AE3. The low-level truncation of the geodesic σ-model over the coset space AEn/K(AEn) is shown to map to a suitably truncated version of the SL(n)/SO(n) nonlinear σ-model resulting from the reduction Einsteins equations in (n + 1) dimensions to (1 + 1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space , where is the (extended) Heisenberg group, and its maximal compact subgroup. We clarify the relation between and the corresponding subgroup of the Geroch group.