Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

We formulate an extension of Maldacena’s AdS/CFT conjectures to the case of branes located at singular points in the ambient transverse space. For singularities which occur at finite distance in the moduli space of M or F theory models with spacetime-filling branes, the conjectures identify the worldvolume theory on the p-branes with a compactification of M or IIB theory on AdSp+2 × HD−p−2. We show how the singularity determines the horizon H, and demonstrate the relationship between global symmetries on the worldvolume and gauge symmetries in the AdS model. As a first application, we study some singularities relevant to the D3-branes required in four-dimensional F -theory. For these we are able to explicitly derive the low-energy field theory on the worldvolume and compare its properties to predictions from the dual AdS model. In particular, we examine the baryon spectra of the models and the fate of the Abelian factors in the gauge group. October 1998 ∗ On leave from Department of Particle Physics, Weizmann Institute of Science, Rehovot, Israel Spacetime-filling branes have emerged as an essential feature of string and M-theory compactifications in at least three contexts: (1) new branches of the heterotic string in six dimensions with “extra” tensor multiplets, which can be represented by a Hořava– Witten-type compactification of M-theory on (S/Z2) × K3 but with extra spacetimefilling M5-branes representing the extra tensor multiplets [1,2,3]; (2) F -theory models in four dimensions (which can be regarded as compactifications of the IIB string with D7branes included) which in general require spacetime-filling D3-branes to cancel a tadpole anomaly [4]; and (3) M-theory models in three dimensions, which require spacetime-filling M2-branes to cancel a similar tadpole anomaly [4]. In each of these cases, the spacetimefilling brane meets the compactifying space at a single point, and the string or M-theory remains finite near the brane. Remarkably, this short list of branes (M5, D3, and M2) is precisely the list of branes for which a certain scaling limit is expected to lead to a “boundary” conformal field theory in the recent AdS/CFT conjectures [5,6,7]. In fact, the scaling limit can be taken even when the space transverse to the branes is curved, as in the compactification scenarios above. The details of the metric far from the location y0 of the brane in the transverse space become irrelevant; for the purposes of studying the scaling limit, the metric on the compactifying space can be approximated by some metric on its tangent space Ty0 at y0. In the scaling limit, the rescaled supergravity metric approaches a metric of the form AdSp+2 × S in which the anti-de Sitter space has been formed out of the worldvolume of the brane and the radial direction within Ty0 , and S k is the unit sphere within Ty0 . Maldacena’s conjecture proposes that the M or string theory on this space AdSp+2 × S, with N units of flux of the supergravity k-form field strength through S, is dual to a specific conformal field theory on the boundary of AdSp+2. The conjecture applies to the large N limit when a large number of these branes have been brought together; in the compactification context, fairly large values of N can be obtained by bringing together all available branes in a given model. Virtually all points in the compactifying space have identical behavior in this scaling limit. That situation changes, however, if we consider a compactifying space which itself 1 This last requirement excludes consideration of F -theory models in eight dimensions with the D7-brane being spacetime-filling. 2 The exception is the four-dimensional F -theory models, where points located along the D7branes behave differently; in particular, the string coupling becomes infinite at such points. The behavior of D3-branes at such points has recently been determined [8,9,10], and we will not

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1999/0003/0001/ATMP-1999-0003-0001-a001.pdf

Non-spherical horizons, I

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler–Einstein metric.

/pdf/sasaki-einstein-manifolds-and-volume-minimisation-klv3fs597v.pdf

Sasaki-Einstein manifolds and volume minimisation

We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics.

Stable forms and special metrics

Quantum coherence and geometric quantum discord

We give a description of all complete simply connected Riemannian manifolds carrying real Killing spinors. Furthermore, we present a construction method for manifolds with the exceptional holonomy groupsG
2 and Spin(7).

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253076

Real Killing spinors and holonomy

We study the question to what extend classical Hodge–deRham theoryfor harmonic diﬀerential forms carries over to harmonic spinors. Despitesome special phenomena in very low dimensions and despite the Atiyah–Singer index theorem which provides a link between harmonic spinors andthe topology of the underlying manifold it turns out that in many dimen-sions harmonic spinors are not topologically obstructed. In this respectharmonic spinors behave very diﬀerently from harmonic diﬀerential forms.We also discuss parallel spinors and Killing spinors.PACS numbers: 02.40.Ma, 02.40.Vh

On harmonic spinors

/pdf/wave-equations-on-lorentzian-manifolds-and-quantization-3mb6769q9r.pdf

Wave Equations on Lorentzian Manifolds and Quantization

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.

Generalized cylinders in semi-Riemannian and spin geometry

We derive upper eigenvalue estimates for generalized Dirac operators on closed Riemannian manifolds. In the case of the classical Dirac operator the estimates on the first eigenvalues are sharp for spheres of constant curvature.

Christian Bär

Papers

Real Killing spinors and holonomy

On harmonic spinors

Wave Equations on Lorentzian Manifolds and Quantization

Generalized cylinders in semi-Riemannian and spin geometry

Lower eigenvalue estimates for Dirac operators