Intersection Pairing Research Articles

Unified invariant of knots from homological braid action on Verma modules

AbstractWe re‐build the quantum unified invariant of knots from braid groups' action on tensors of Verma modules. It is a two variables series having the particularity of interpolating both families of colored Jones polynomials and ADO polynomials, that is, semisimple and non‐semisimple invariants of knots constructed from quantum . We prove this last fact in our context that re‐proves (a generalization of) the famous Melvin–Morton–Rozansky conjecture first proved by Bar‐Natan and Garoufalidis. We find a symmetry of nicely generalizing the well‐known one of the Alexander polynomial, ADO polynomials also inherit this symmetry. It implies that quantum non‐semisimple invariants are not detecting knots' orientation. Using the homological definition of Verma modules we express as a generating sum of intersection pairing between fixed Lagrangians of configuration spaces of disks. Finally, we give a formula for using a generalized notion of determinant, that provides one for the ADO family. It generalizes that for the Alexander invariant.

Colored Jones polynomials and abelianized Lefschetz numbers

We show that colored Jones polynomials of the closure of a braid compute weighted sums of abelianized Lefschetz numbers associated with homeomorphisms on configuration spaces on punctured disks defined from the braid. The sum is over number of configuration points. Then we interpret this sum in terms of Poincaré–Lefschetz duality intersection pairing between homology classes.

Chiral Matter Multiplicities and Resolution-Independent Structure in 4D F-Theory Models

Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic Calabi–Yau fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models with simple, simply-laced gauge groups and generic matter content, as well as models with U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {U}}(1)$$\\end{document} gauge factors. We explicitly analyze resolutions of these F-theory models in which the elliptic fiber is realized as a cubic in P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document} over an arbitrary (e.g., not necessarily toric) smooth base, and confirm the independence of the intersection pairing of the vertical part of the middle cohomology for the resolutions we study. In each model, we find that vertical flux backgrounds can produce nonzero multiplicities for a spanning set of anomaly-free chiral matter field combinations, suggesting that F-theory geometry imposes no additional linear constraints on allowed matter representations beyond those implied by 4D anomaly cancellation.

Witten–Reshetikhin–Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces

In this paper, we construct a topological model for the Witten–Reshetikhin–Turaev invariants for 3 -manifolds coming from the quantum group U_q(\mathrm{sl}(2)) , as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level \mathcal{N}\in \mathbb{N} , we show that the level \mathcal{N} WRT invariant for a 3 -manifold is a state sum of Lagrangian intersections in a covering of a fixed configuration space in the punctured disc. This model brings a new perspective on the structure of the level \mathcal{N} Witten–Reshetikhin–Turaev invariant, showing that it is completely encoded by the intersection points between certain Lagrangian submanifolds in a fixed configuration space, with additional gradings which come from a particular choice of a local system. This formula provides a new framework for investigating the open question about categorifications of the WRT invariants.

Moment maps and cohomology of non-reductive quotients

Let H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document} be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document}. Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q$\\end{document} of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}, we define a notion of moment map for the action of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}, and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} and express the rational cohomology ring of X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} in terms of the rational cohomology ring of the GIT quotient X//TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/T^{H}$\\end{document}, where TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$T^{H}$\\end{document} is a maximal torus in H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}. We relate intersection pairings on X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} to intersection pairings on X//TH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/T^{H}$\\end{document}, obtaining a residue formula for these pairings on X//H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/H$\\end{document} analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree.

Examples of diffeomorphic complete intersections with different Hodge numbers

In this paper, we give three pairs of complex 3-dim complete intersections and a pair of complex 5-dim complete intersections, and every pair of them is diffeomorphic but with different Hodge numbers. Moreover, the diffeomorphic complex 3-dim complete intersections have different Chern numbers c13,c1c2.

Finite generation of nilpotent quotients of fundamental groups of punctured spectra

In SGA 2, Grothendieck conjectures that the étale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses p-adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology.

Manin–Drinfeld cycles and derivatives of $L$-functions

We study algebraic cycles in the moduli space of \operatorname{PGL}_2 -shtukas, arising from the diagonal torus. Our main result shows that their intersection pairing with the Heegner–Drinfeld cycle is the product of the r -th central derivative of an automorphic L -function L(\pi,s) and Waldspurger's toric period integral. When L(\pi,1/2) \neq 0 , this gives a new geometric interpretation for the Taylor series expansion. When L(\pi,1/2) = 0 , the pairing vanishes, suggesting higher order analogues of the vanishing of cusps in the modular Jacobian, as well as other new phenomena. Our proof sheds new light on the algebraic correspondence introduced by Yun and Zhang, which is the geometric incarnation of “differentiating the L -function”. We realize it as the Lie algebra action of e+f \in \mathfrak{sl}_2 on (\mathbb{Q}_\ell^2)^{\otimes 2d} . The comparison of relative trace formulas needed to prove our formula is then a consequence of Schur–Weyl duality.

A topological model for the coloured Alexander invariants

Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group Uq(sl(2)) at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for these invariants, showing that they can be obtained as graded intersection pairings between homology classes in a covering of the configuration space in the punctured disc.

Disoriented homology and double branched covers

AbstractThis paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball.To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon–Litherland to the general setting.Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the n-ball, which generalizes previous results of Akbulut–Kirby and others.

Subarea Partition Based on Correlation Analysis with Edge-Elimination Strategy Using Automatic License Plate Recognition Data

To partition an urban network into several subareas (i.e., subarea partition) is a vital step for regional coordinated signal control. The correlation between intersections must be analyzed for achieving reliable subarea partition results. However, because of the incompleteness of spatial–temporal information in traffic data, previous studies merely explored the relationship between any intersections. Subarea partition considering the correlation of any pair of intersections remains a challenge in a large-scale network. This paper proposes a subarea partition method that integrates a novel correlation-degree model and the Newman fast algorithm with an edge-elimination strategy using automatic license plate recognition (ALPR) data. First, vehicle trips are extracted and a correlation-degree model is developed for measuring the relationship of any pair of intersections. Second, an edge-elimination strategy is proposed to generate candidate subarea partition solutions under conditions with different proportions of correlated edges. Finally, an optimal solution of subarea partition is identified by the ratios of ideal connected intersections to total intersections of different partition solutions’ correlation index (CI). The proposed method was implemented in a real-world urban network in Kunshan, China. The results show that the optimal partition solution can be obtained when the top 33% of correlated edges are maintained, and the ratio of ideal connected intersections’ CI is 72.35% with most of the intersections being connected, which demonstrates the rationality of the proposed partition method in large-scale urban networks.

A topological model for the coloured Jones polynomials

In this paper we will present a homological model for Coloured Jones Polynomials. For each colour N in {mathbb {N}}, we will describe the invariant J_N(L,q) as a graded intersection pairing of certain homology classes in a covering of a configuration space on the punctured disc. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups.

Numerical equivalence of ℝ-divisors and Shioda–Tate formula for arithmetic varieties

Abstract Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K {X_{K}} . We give a formula that relates the dimension of the first Arakelov–Chow vector space of X with the Mordell–Weil rank of the Albanese variety of X K {X_{K}} and the rank of the Néron–Severi group of X K {X_{K}} . This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ {\mathbb{R}} -divisors on X are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming a conjecture by H. Gillet and C. Soulé.

The degree of irrationality of most abelian surfaces is 4

The degree of irrationality of a smooth projective variety $X$ is the minimal degree of a dominant rational map $X\dashrightarrow \mathbb{P}^{\dim X}$. We show that if an abelian surface $A$ over $\mathbb{C}$ is such that the image of the intersection pairing $\text{Sym}^2NS(A)\to \mathbb{Z}$ does not contain $12$, then it has degree of irrationality $4$. In particular, a very general $(1,d)$-polarized abelian surface has degree of irrationality $4$ provided that $d\nmid 6$. This answers two questions of Yoshihara by providing the first examples of abelian surfaces with degree of irrationality greater than $3$ and showing that the degree of irrationality is not isogeny-invariant for abelian surfaces.

Intersection homology duality and pairings: singular, PL and sheaf-theoretic

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky-MacPherson PL intersection product. We also show that the de Rham isomorphism of Brasselet, Hector, and Saralegi preserves product structures.

Duals of Feynman integrals. Part I. Differential equations

We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.

Intersection Pairings for Higher Laminations

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface. We give a geometric interpretation of these intersection pairings in the case that $G=SL_n$. In particular, we show that they can be computed as the length of minimal weighted networks in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [LO] The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, Konig's theorem, and the Kuhn-Munkres algorithm.

Pullbacks of 𝜅 classes on \overline{ℳ}_{0,𝓃}

The moduli space M ¯ 0 , n \overline {\mathcal {M}}_{0,n} carries a codimension- d d Chow class κ d \kappa _{d} . We consider the subspace K n d \mathcal {K}^{d}_{n} of A d ( M ¯ 0 , n , Q ) A^d(\overline {\mathcal {M}}_{0,n},\mathbb {Q}) spanned by pullbacks of κ d \kappa _d via forgetful maps. We find a permutation basis for K n d \mathcal {K}^{d}_{n} , and describe its annihilator under the intersection pairing in terms of d d -dimensional boundary strata. As an application, we give a new permutation basis of the divisor class group of M ¯ 0 , n \overline {\mathcal {M}}_{0,n} .

Wrońskian algebra and Broadhurst–Roberts quadratic relations

Through algebraic manipulations on Wro\'nskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wro\'nskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst--Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne's conjectures for critical values of motivic $L$-functions.

The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the N\'eron-Tate height pairing. More recently, Moriwaki and Yuan-Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow's $K/k$-trace functor. As an application of the Hodge Index Theorem, we also prove a rigidity theorem for the set of canonical height zero points of polarized algebraic dynamical systems over function fields. For function fields over finite fields, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, Baker-DeMarco, and Yuan-Zhang.