Moduli space holography and the finiteness of flux vacua

A holographic perspective to study and characterize field spaces that arise in string compactifications is suggested. A concrete correspondence is developed by studying two-dimensional moduli spaces in supersymmetric string compactifications. It is proposed that there exist theories on the boundaries of each moduli space, whose crucial data are given by a Hilbert space, an Sl(2, ℂ)-algebra, and two special operators. This boundary data is motivated by asymptotic Hodge theory and the fact that the physical metric on the moduli space of Calabi-Yau manifolds asymptotes near any infinite distance boundary to a Poincaré metric with Sl(2, ℝ) isometry. The crucial part of the bulk theory on the moduli space is a sigma model for group-valued matter fields. It is discussed how this might be coupled to a two-dimensional gravity theory. The classical bulk-boundary matching is then given by the proof of the famous Sl(2) orbit theorem of Hodge theory, which is reformulated in a more physical language. Applying this correspondence to the flux landscape in Calabi-Yau fourfold compactifications it is shown that there are no infinite tails of self-dual flux vacua near any co-dimension one boundary. This finiteness result is a consequence of the constraints on the near boundary expansion of the bulk solutions that match to the boundary data. It is also pointed out that there is a striking connection of the finiteness result for supersymmetric flux vacua and the Hodge conjecture.

Finiteness and infiniteness results for Torelli groups of (hyper-)Kähler manifolds

The Torelli group $$\mathcal T(X)$$ T ( X ) of a closed smooth manifold X is the subgroup of the mapping class group $$\pi _0(\mathrm {Diff}^+(X))$$ π 0 ( Diff + ( X ) ) consisting of elements which act trivially on the integral cohomology of X. In this note we give counterexamples to Theorem 3.4 by Verbitsky (Duke Math J 162(15):2929–2986, 2013) which states that the Torelli group of simply connected Kähler manifolds of complex dimension $$\ge 3$$ ≥ 3 is finite. This is done by constructing under some mild conditions homomorphisms $$J: \mathcal T(X) \rightarrow H^3(X;\mathbb Q)$$ J : T ( X ) → H 3 ( X ; Q ) and showing that for certain Kähler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in (Verbitsky in Duke Math J 162(15):2929–2986, 2013) where Verbitsky claims that the Torelli group of hyperkähler manifolds are finite. These examples are detected by the action of diffeomorphsims on $$\pi _4(X)$$ π 4 ( X ) . Finally we confirm the finiteness result for the special case of the hyperkähler manifold $$K^{[2]}$$ K [ 2 ] .

Positive solutions to Schrödinger equations and geometric applications

A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

Finiteness Properties of Pseudo-Hyperbolic Varieties

Motivated by Lang–Vojta’s conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik’s theorem for dynamical systems of infinite order with properties of Prokhorov–Shramov’s notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again not only Amerik’s theorem and Prokhorov–Shramov’s notion of quasi-minimal model but also Weil’s regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi–Ochiai’s finiteness result for varieties of general type and thereby generalize Noguchi’s theorem (formerly Lang’s conjecture).

Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

Let G be a connected complex simple Lie group, and let {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of {\widehat{G}^{\mathrm{d}}} come from {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of {\widehat{F}_{4}^{\mathrm{d}}}.

FRIEZE PATTERNS WITH COEFFICIENTS

Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.

Unitary representations with non-zero Dirac cohomology for complex E 6

This paper classifies the equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology for complex {E_{6}} . This is achieved by using our finiteness result, and by improving the computing method. According to a conjecture of Barbasch and Pandžić, our classification should also be helpful for understanding the entire unitary dual of complex {E_{6}} .

FINITENESS OF LOG MINIMAL MODELS AND NEF CURVES ON -FOLDS IN CHARACTERISTIC

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.