GENERALIZED EULER CHARACTERISTICS, GRAPH HYPERSURFACES, AND FEYNMAN PERIODS

We generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.

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Quantum Field Theory over $\mathbb{F}_q$

The Electronic Journal of Combinatorics ◽

10.37236/589 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 10

Author(s):

Oliver Schnetz

Keyword(s):

Field Theory ◽

Perturbation Series ◽

The Other ◽

Field Theories ◽

Roots Of Unity ◽

Quantum Field Theories ◽

Quantum Field ◽

Feynman Rules ◽

Graph Hypersurfaces ◽

Prime 2

We consider the number $\bar N(q)$ of points in the projective complement of graph hypersurfaces over $\mathbb{F}_q$ and show that the smallest graphs with non-polynomial $\bar N(q)$ have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class $\bar N(q)$ depends on the number of cube roots of unity in $\mathbb{F}_q$. At graphs with 16 edges we find examples where $\bar N(q)$ is given by a polynomial in $q$ plus $q^2$ times the number of points in the projective complement of a singular K3 in $\mathbb{P}^3$. In the second part of the paper we show that applying momentum space Feynman-rules over $\mathbb{F}_q$ lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.

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A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-01-02876-8 ◽

2001 ◽

Vol 353 (11) ◽

pp. 4405-4427 ◽

Cited By ~ 16

Author(s):

I. P. Goulden ◽

J. L. Harer ◽

D. M. Jackson

Keyword(s):

Moduli Spaces ◽

Algebraic Curves ◽

Euler Characteristics

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Euler characteristics for Gaussian fields on manifolds

10.1214/aop/1048516527 ◽

2003 ◽

Vol 31 (2) ◽

pp. 533-563 ◽

Cited By ~ 50

Author(s):

Jonathan E. Taylor ◽

Robert J. Adler

Keyword(s):

Gaussian Fields ◽

Euler Characteristics

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Euler characteristics and their congruences in the positive rank setting

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000429 ◽

2020 ◽

pp. 1-18

Author(s):

Anwesh Ray ◽

R. Sujatha

Keyword(s):

Elliptic Curves ◽

Euler Characteristic ◽

Selmer Groups ◽

Euler Characteristics ◽

Rank Zero ◽

Homology Groups ◽

Main Conjecture ◽

Congruence Relations ◽

Higher Rank ◽

Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

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Elementary amenable groups and 4-manifolds with Euler characteristic 0

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700032638 ◽

1991 ◽

Vol 50 (1) ◽

pp. 160-170 ◽

Cited By ~ 19

Author(s):

Jonathan A. Hillman

Keyword(s):

Normal Subgroup ◽

Fundamental Group ◽

Euler Characteristic ◽

Fundamental Groups ◽

Normal Subgroups ◽

Euler Characteristics ◽

Amenable Groups ◽

Amenable Subgroup ◽

Finite Subgroups ◽

Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

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