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Register a new userSome examples of projective and $c$--projective compactifications of Einstein metrics
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We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective compactification of neutral signature Einstein metrics canonically defined on certain rank--$n$ affine bundles $M$ over $n$-dimensional manifolds endowed with projective structures.
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We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [ abla])$. We show that there is a canonical Einstein of metric on an $R^*$--bundle over $T^*N$, with a connection whose curvature is the pull--back of the natural symplectic structure from $T^*N$.
We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart-Duval lifts.
After a brief exposition of the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections, we consider the spherical and cylindrical reductions of these theories to two-dimensional dilaton-vecton gravity (DVG) field theories. The distinctive feature of these theories is the presence of a massive/tachyonic vector field (vecton) with essentially nonlinear coupling to the dilaton gravity. In the massless limit, the classical DVG theory can be exactly solved for a rather general coupling depending only on the field tensor and the dilaton. We show that the vecton field can be consistently replaced by a new effectively massive scalar field (scalaron) with an unusual coupling to dilaton gravity (DSG). Then we concentrate on considering the DVG models derived by reductions of D=3 and D=4 affine theories. In particular, we introduce the most general cylindrical reductions that are often ignored. The main subject of our study is the static solutions with horizons. We formulate the general conditions for the existence of the regular horizons and find the solutions of the static DVG/DSG near the horizons in the form of locally convergent power - series expansion. For an arbitrary regular horizon, we find a local generalization of the Szekeres - Kruskal coordinates. Finally, we consider one-dimensional integrable and nonintegrable DSG theories with one scalar. We analyze simplest models having three or two integrals of motion, respectively, and introduce the idea of a `topological portrait giving a unified qualitative description of static and cosmological solutions of some simple DSG models.
We derive a formula for D3-brane charge on a compact spacetime, which includes torsion corrections to the tadpole cancellation condition. We use this to classify D-branes and RR fluxes in type II string theory on RP^3xRP^{2k+1}xS^{6-2k} with torsion H-flux and to demonstrate the conjectured T-duality to S^3xS^{2k+1}xS^{6-2k} with no flux. When k=1, H eq 0 and so the K-theory that classifies fluxes is twisted. When k=2 the square of the H-flux yields an S-dual Freed-Witten anomaly which is canceled by a D3-brane insertion that ruins the K-theory classification. When k=3 the cube of H is nontrivial and so the D3 insertion may itself be inconsistent and the compactification unphysical. Along the way we provide a physical interpretation for the AHSS in terms of boundaries of branes within branes.
Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koisos Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koisos. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step in the proof is that when R-plus is negative definite, the Hessian of S is strictly positive modulo gauge.
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