Products Of Riemann Surfaces Research Articles

Sasakian geometry on sphere bundles

The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety N with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki [48] for K-contact manifolds to the Sasaki case. This construction depends on the choice of d+1 integral Kähler classes [ωj] on N that are not necessarily colinear in the Kähler cone. We show that the colinear case is equivalent to a subclass of a different join construction originally described in [7,11], applied to the spherical case by the authors in [19,20] when d=1, and known as cone decomposable [13]. The non-colinear case gives rise to infinite families of new inequivalent cone indecomposable Sasaki contact CR structures on certain sphere bundles. We prove that the Sasaki cone for some of these structures contains an open set of extremal Sasaki metrics and, for certain specialized cases, the regular ray within this cone is shown to have constant scalar curvature. We also compute the cohomology groups of all such sphere bundles over a product of Riemann surfaces.

Cubulable Kähler groups

We prove that a K\"ahler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly, we prove that a closed aspherical K\"ahler manifold with a cubulable fundamental group has a finite cover which is biholomorphic to a topologically trivial principal torus bundle over a product of Riemann surfaces. Along the way, we prove a factorization result for essential actions of K\"ahler groups on irreducible, locally finite CAT(0) cubical complexes, under the assumption that there is no fixed point in the visual boundary.

6D attractors and black hole microstates

We find a family of AdS2×ℳ4 supersymmetric solutions of the six-dimensional F(4) gauged supergravity coupled to one vector multiplet that arises as a low energy description of massive type IIA supergravity on (warped) AdS6 × S4. ℳ4 is either a Kähler-Einstein manifold or a product of two Riemann surfaces with a constant curvature metric. These solutions correspond to the near-horizon region of a family of static magnetically charged black holes. In the case where ℳ4 is a product of Riemann surfaces, we successfully compare their entropy to a microscopic counting based on the recently computed topologically twisted index of the five-dimensional mathcal{N} = 1 USp(2N) theory with Nf fundamental flavors and an antisymmetric matter field. Furthermore, our results suggest that the near-horizon regions exhibit an attractor mechanism for the scalars in the matter coupled F(4) gauged supergravity, and we give a proposal for it.

The fiber product of Riemann surfaces: a Kleinian group point of view

Let P 1 : S 1 → S and P 2 : S 2 → S be non-constant holomorpic maps between closed Riemann surfaces. Associated to the previous data is the fiber product \({S_{1} \times_{S} S_{2}=\{(x,y) \in S_{1} \times S_{2}: P_{1}(x)=P_{2}(y)\}}\) . This is a compact space which, in general, fails to be a Riemann surface at some (possible empty) finite set of points \({F \subset S_{1} \times_{S} S_{2}}\) . One has that S 1 × S S 2 − F is a finite collection of analytically finite Riemann surfaces. By filling out all the punctures of these analytically finite Riemann surfaces, we obtain a finite collection of closed Riemann surfaces; whose disjoint union is the normal fiber product \({\widetilde{S_{1}\times_{S} S_{2}}}\) . In this paper we prove that the connected components of \({\widetilde{S_{1}\times_{S} S_{2}}}\) of lowest genus are conformally equivalents if they have genus different from one (isogenous if the genus is one). A description of these lowest genus components are provided in terms of certain class of Kleinian groups; B-groups.

Convergence of adiabatic family of anti-self-dual connections on products of Riemann surfaces

We prove a convergence theorem for a sequence of anti-self-dual connections on a family of products of two Riemann surfaces, where the metric of one factor shrinks, establishing the conjecture of Bershadsky et al. [Topological reduction of 4D SYM to 2D sigma-models,” Nucl. Phys. B 448, 166 (1995)].

A new handle on de sitter compactifications

We construct a large new class of de Sitter (and anti de Sitter) vacua of critical string theory from flux compactifications on products of Riemann surfaces. In the construction, the leading effects stabilizing the moduli are perturbative. We show that these effects self-consistently dominate over standard estimates for further $\alpha^\prime$ and quantum corrections, via tuning available from large flux and brane quantum numbers.

Semi-Infinite Realization of Unitary Representations of the N=2 Algebra and Related Constructions

In the examples of the N=2 super-Virasoro algebra and the affine \(\widehat{s\ell }\left( 2 \right)\) algebra, we investigate the construction of unitary representations of infinite-dimensional algebras in terms of “collective excitations” over a filled Dirac sea of fermionic or bosonic operators satisfying a generalized exclusion principle and represented by semi-infinite forms in the modes of one of the generators. We develop the methods for investigating properties of semi-infinite spaces (polynomial realization of the dual space) and for constructing the appropriate algebra action on these spaces (a filtration by subspaces similar to Demazure modules). We also consider relations of the semi-infinite realizations to the Rogers–Ramanujan-type identities, to the expression of coinvariants through meromorphic functions on products of Riemann surfaces with a prescribed behavior on multiple diagonals, and to some combinatorial facts; we also consider the relation between modular functors and fusion rules for the N=2 and \(\widehat{s\ell }\left( 2 \right)\) theories.

Carleman approximation on products of riemann surfaces

In [19] Scheinberg generalized a celebrated theorem of Carleman by showing that a continous function on R n can be approximated with arbitrary speed by entire ffunctions of n complex variables. Alexander showed in [1] that an unbounded smooth curve in C n is such a set of Carleman approximation. Frih and Gauthier in [8] studied Carleman approximation by entire functions in Cn on products of curves in C n, n1+…+n>k =n. In the present paper, we show that a product set in a product of Riemann surfaces is a Carleman set if and only of each projection is a Carleman set, thus generalizing the results of [18] and [19].