A gravitino distance conjecture

We conjecture that in a consistent supergravity theory with non-vanishing gravitino mass, the limit m3/2 → 0 is at infinite distance. In particular one can write Mtower ~ $$ {m}_{3/2}^{\delta } $$ m 3 / 2 δ so that as the gravitino mass goes to zero, a tower of KK states as well as emergent strings becomes tensionless. This conjecture may be motivated from the Weak Gravity Conjecture as applied to strings and membranes and implies in turn the AdS Distance Conjecture. We test this proposal in classical 4d type IIA orientifold vacua in which one obtains a range of values $$ \frac{1}{3} $$ 1 3 ≤ δ ≤ 1. The parameter δ is related to the scale decoupling exponent in AdS vacua and to the α exponent in the Swampland Distance Conjecture for the type IIA complex structure. We present a general analysis of the gravitino mass in the limits of moduli space in terms of limiting Mixed Hodge Structures and study in some detail the case of two-moduli F-theory settings. Moreover, we obtain general lower bounds δ ≥$$ \frac{1}{3},\frac{1}{4} $$ 1 3 , 1 4 for Calabi-Yau threefolds and fourfolds, respectively. The conjecture has important phenomenological implications. In particular we argue that low-energy supersymmetry of order 1 TeV is only obtained if there is a tower of KK states at an intermediate scale, of order 108 GeV. One also has an upper bound for the Hubble constant upon inflation H ≲ $$ {m}_{3/2}^{\delta }{M}_{\mathrm{P}}^{\left(1-\delta \right)} $$ m 3 / 2 δ M P 1 − δ .

Hodge-Deligne polynomials of character varieties of free abelian groups

Let F F be a finite group and X X be a complex quasi-projective F F -variety. For r ∈ N r\in {\mathbb{N}} , we consider the mixed Hodge-Deligne polynomials of quotients X r / F {X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F , where F F acts diagonally, and compute them for certain classes of varieties X X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X X is the maximal torus of an affine reductive group G G , and F F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E E -polynomials of (the distinguished component of) G G -character varieties of free abelian groups. In the cases G = G L ( n , C ) G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and S L ( n , C ) SL\left(n,{\mathbb{C}}\hspace{-0.1em}) , we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

The Geometric P=W Conjecture in the Painlevé Cases via Plumbing Calculus

We use plumbing calculus to prove the homotopy commutativity assertion of the Geometric $P=W$ conjecture in all Painlevé cases. We discuss the resulting Mixed Hodge structures on Dolbeault and Betti moduli spaces.

MIXED HODGE STRUCTURES WITH MODULUS

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

Ghost classes in $${\mathbb {Q}}$$-rank two orthogonal Shimura varieties

In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature $$(1, n-1)$$ ( 1 , n - 1 ) . For the values of $$n = 4, 5$$ n = 4 , 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.

Variation of Mixed Hodge Structures Associated to an Equisingular One-dimensional Family of Calabi-Yau Threefolds

We study the variations of mixed Hodge structures (VMHS) associated with a pencil ${\mathcal{X}}$ of equisingular hypersurfaces of degree $d$ in $\mathbb{P}^{4}$ with only ordinary double points as singularities, as well as the variations of Hodge structures (VHS) associated with the desingularization of this family $\widetilde{{\mathcal{X}}}$ . The notion of a set of singular points being in homologically good position is introduced, and, by requiring that the subset of nodes in (algebraic) general position is also in homologically good position, we can extend Griffiths’ description of the $F^{2}$ -term of the Hodge filtration of the desingularization to this case, where we can also determine the possible limiting mixed Hodge structures (LMHS). The particular pencil ${\mathcal{X}}$ of quintic hypersurfaces with 100 singular double points with 86 of them in (algebraic) general position that served as the starting point for this paper is treated with particular attention.