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Motivated by our attempt to recast Cartans work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem. From this point of view, the novelty of this paper is that we study forms with coefficients. However, the main contribution of this paper is conceptual: the finding of the relationship between multiplicative forms and Cartans work, which provides a completely new approach to integrability theorems for multiplicative forms. Back to Cartan, the multiplicative point of view shows that, modulo Lies functor, the Cartan Pfaffian system (itself a multiplicative form with coefficients!) is the same thing as the classical Spencer operator.
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We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn-Drummond [4] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connections in the tangent bundle, that we call infinitesimally multiplicative (IM) connections and we prove an integration theorem for IM connections. Finally, we present a few toy examples. Along the way, we discuss fiber-wise linear connections in the tangent bundle of the total space of a vector bundle.
A differential 1-form $alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $alpha wedge (dalpha)^n eq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $alpha (O) = 0$, is called a textit{singular contact form} at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $omega$ in dimension $2n$, i.e. differential 1-forms $gamma$ which vanish at a point and such that $dgamma = omega$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin cite{Lychagin-1stOrder1975}, Webster cite{Webster-1stOrder1987} and Zhitomirskii cite{Zhito-1Form1986,Zhito-1Form1992}. We make use of both the classical normalization techniques and the toric approach to the normalization problem for dynamical systems cite{Zung_Birkhoff2005, Zung_Integrable2016, Zung_AA2018}.
It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) mapsto d(A ;B)in[0,infty]$ over pairs of measurable subsets of $Ri^d$. Then [ |(phi_A,S_tphi_B)|leq e^{-d(A;B)^2(4t)^{-1}}|phi_A|_2|phi_B|_2 ] for all $t>0$ and all $phi_Ain L_2(A)$, $phi_Bin L_2(B)$. Moreover $S_tL_2(A)subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=infty$ where $A^c$ denotes the complement of $A$.
Extracting parton distribution functions (PDFs) from lattice QCD calculation of quasi-PDFs has been actively pursued in recent years. We extend our proof of the multiplicative renormalizability of quasi-quark operators in Ref. [1] to quasi-gluon operators, and demonstrated that quasi-gluon operators could be multiplicatively renormalized to all orders in perturbation theory, without mixing with other operators. We find that using a gauge-invariant UV regulator is essential for achieving this proof. With the multiplicative renormalizability of both quasi-quark and quasi-gluon operators, and QCD collinear factorization of hadronic matrix elements of there operators into PDFs, extracting PDFs from lattice QCD calculated hadronic matrix elements of quasi-parton operators could have a solid theoretical foundation.
These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomology, the multiplicativity of the relative Chern characters, and the Riemann-Roch relation between the relative Chern character of the Bott symbol and of the relative Thom class.
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