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Journal articles on the topic 'Equivariant index'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Wang, Yong. "The Noncommutative Infinitesimal Equivariant Index Formula." Journal of K-Theory 14, no. 1 (July 3, 2014): 73–102. http://dx.doi.org/10.1017/is014006002jkt268.

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AbstractIn this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regularity and give an explicit formula for them. We also establish an infinitesimal equivariant family index formula and introduce the infinitesimal equivariant eta forms as well as compare them with the equivariant eta forms.

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2

Garza, Gabriel López, and Slawomir Rybicki. "Equivariant bifurcation index." Nonlinear Analysis: Theory, Methods & Applications 73, no. 9 (November 2010): 2779–91. http://dx.doi.org/10.1016/j.na.2010.06.001.

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3

Gołębiewska, Anna, and Sławomir Rybicki. "Equivariant Conley index versus degree for equivariant gradient maps." Discrete and Continuous Dynamical Systems - Series S 6, no. 4 (December 2012): 985–97. http://dx.doi.org/10.3934/dcdss.2013.6.985.

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4

Wruck, Philipp. "Genericity in equivariant dynamical systems and equivariant Fuller index theory." Dynamical Systems 29, no. 3 (April 14, 2014): 399–423. http://dx.doi.org/10.1080/14689367.2014.903588.

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5

MARZANTOWICZ, WACLAW, and CARLOS PRIETO. "THE UNSTABLE EQUIVARIANT FIXED POINT INDEX AND THE EQUIVARIANT DEGREE." Journal of the London Mathematical Society 69, no. 01 (January 28, 2004): 214–30. http://dx.doi.org/10.1112/s0024610703004721.

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6

Ono, Kaoru. "Equivariant index of Dirac operators." Tohoku Mathematical Journal 42, no. 3 (1990): 319–32. http://dx.doi.org/10.2748/tmj/1178227613.

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7

Vergne, Michele. "Equivariant index formulas for orbifolds." Duke Mathematical Journal 82, no. 3 (March 1996): 637–52. http://dx.doi.org/10.1215/s0012-7094-96-08226-5.

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8

Wang, Yong. "Volterra calculus, local equivariant family index theorem and equivariant eta forms." Asian Journal of Mathematics 20, no. 4 (2016): 759–84. http://dx.doi.org/10.4310/ajm.2016.v20.n4.a8.

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9

Troitskiĭ, E. V. "THE EQUIVARIANT INDEX OFC*-ELLIPTIC OPERATORS." Mathematics of the USSR-Izvestiya 29, no. 1 (February 28, 1987): 207–24. http://dx.doi.org/10.1070/im1987v029n01abeh000967.

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10

Dzedzej, Zdzisław. "Fixed orbit index for equivariant maps." Nonlinear Analysis: Theory, Methods & Applications 47, no. 4 (August 2001): 2835–40. http://dx.doi.org/10.1016/s0362-546x(01)00402-3.

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11

Bunke, Ulrich. "Orbifold index and equivariant K-homology." Mathematische Annalen 339, no. 1 (May 5, 2007): 175–94. http://dx.doi.org/10.1007/s00208-007-0111-5.

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12

Prokhorenkov, Igor, and Ken Richardson. "Witten deformation and the equivariant index." Annals of Global Analysis and Geometry 34, no. 3 (July 22, 2008): 301–27. http://dx.doi.org/10.1007/s10455-008-9113-0.

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13

De Concini, C., C. Procesi, and M. Vergne. "Box splines and the equivariant index theorem." Journal of the Institute of Mathematics of Jussieu 12, no. 3 (June 1, 2012): 503–44. http://dx.doi.org/10.1017/s1474748012000734.

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AbstractIn this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

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14

Hirano, Norimichi, and Sławomir Rybicki. "Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation." Abstract and Applied Analysis 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/560975.

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We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.

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15

Erbe, L. H., K. Gęba, and W. Krawcewicz. "Equivariant Fixed Point Index and the Period-Doubling Cascades." Canadian Journal of Mathematics 43, no. 4 (August 1, 1991): 738–47. http://dx.doi.org/10.4153/cjm-1991-042-4.

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Properties of fixed points of equivariant maps have been studied by several authors including A. Dold (cf. [2], 1982), H. Ulrich (cf. [9], 1988), A. Marzantowicz (cf. [7], 1975) and others. Closely related is the work of R. Rubinsztein (cf. [8], 1976) in which he investigated homotopy classes of equivariant maps between spheres. There have been many attempts to introduce and effectively apply these concepts to nonlinear problems. In particular we mention the work of E. Dancer (cf. [1], 1982) in which some applications to nonlinear problems are given.

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16

Berline, Nicole, and Michele Vergne. "The Equivariant Index and Kirillov's Character Formula." American Journal of Mathematics 107, no. 5 (October 1985): 1159. http://dx.doi.org/10.2307/2374350.

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17

Ferrario, Davide L. "A fixed point index for equivariant maps." Topological Methods in Nonlinear Analysis 13, no. 2 (June 1, 1999): 313. http://dx.doi.org/10.12775/tmna.1999.017.

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18

Gong, Donggeng. "Higher equivariant index theorem for homogeneous spaces." Manuscripta Mathematica 86, no. 1 (December 1995): 239–52. http://dx.doi.org/10.1007/bf02567992.

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19

Hochs, Peter, and Yanli Song. "An equivariant index for proper actions I." Journal of Functional Analysis 272, no. 2 (January 2017): 661–704. http://dx.doi.org/10.1016/j.jfa.2016.08.024.

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20

Hochs, Peter, and Hang Wang. "An equivariant orbifold index for proper actions." Journal of Geometry and Physics 154 (August 2020): 103710. http://dx.doi.org/10.1016/j.geomphys.2020.103710.

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21

Emerson, Heath, and Ralf Meyer. "Equivariant embedding theorems and topological index maps." Advances in Mathematics 225, no. 5 (December 2010): 2840–82. http://dx.doi.org/10.1016/j.aim.2010.05.011.

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22

Fitzpatrick, Sean. "An equivariant index formula in contact geometry." Mathematical Research Letters 16, no. 3 (2009): 375–94. http://dx.doi.org/10.4310/mrl.2009.v16.n3.a1.

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23

KATZ, GABRIEL. "THE BURNSIDE RING-VALUED MORSE FORMULA FOR VECTOR FIELDS ON MANIFOLDS WITH BOUNDARY." Journal of Topology and Analysis 01, no. 01 (March 2009): 13–27. http://dx.doi.org/10.1142/s1793525309000059.

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Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula [Formula: see text] which takes its values in A(G). Here Ind G(v) denotes the equivariant index of the field v, [Formula: see text] the v-induced Morse stratification (see [10]) of the boundary ∂X, and [Formula: see text] the class of the (n - k)-manifold [Formula: see text] in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝn defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas ([3, 4]).

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24

FORSYTH, IAIN, and ADAM RENNIE. "FACTORISATION OF EQUIVARIANT SPECTRAL TRIPLES IN UNBOUNDED -THEORY." Journal of the Australian Mathematical Society 107, no. 02 (December 21, 2018): 145–80. http://dx.doi.org/10.1017/s1446788718000423.

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We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$ -deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

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25

Perrot, Denis. "The equivariant index theorem in entire cyclic cohomology." Journal of K-Theory 3, no. 2 (May 28, 2008): 261–307. http://dx.doi.org/10.1017/is008004027jkt047.

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AbstractLet G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.

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26

CAVE, CHRIS, and DENNIS DREESEN. "EQUIVARIANT COMPRESSION OF CERTAIN DIRECT LIMIT GROUPS AND AMALGAMATED FREE PRODUCTS." Glasgow Mathematical Journal 58, no. 3 (June 10, 2016): 739–52. http://dx.doi.org/10.1017/s0017089516000082.

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AbstractWe give a means of estimating the equivariant compression of a group G in terms of properties of open subgroups Gi ⊂ G whose direct limit is G. Quantifying a result by Gal, we also study the behaviour of the equivariant compression under amalgamated free products G1∗HG2 where H is of finite index in both G1 and G2.

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27

SPERA, MAURO. "THE RIEMANN ZETA FUNCTION AS AN EQUIVARIANT DIRAC INDEX." International Journal of Geometric Methods in Modern Physics 09, no. 08 (October 29, 2012): 1250071. http://dx.doi.org/10.1142/s0219887812500715.

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In this note an interpretation of Riemann's zeta function is provided in terms of an ℝ-equivariant L2-index of a Dirac–Ramond type operator, akin to the one on (mean zero) loops in flat space constructed by the present author and T. Wurzbacher. We build on the formal similarity between Euler's partitio numerorum function (the S1-equivariant L2-index of the loop space Dirac–Ramond operator) and Riemann's zeta function. Also, a Lefschetz–Atiyah–Bott interpretation of the result together with a generalization to M. Lapidus' fractal membranes are also discussed. A fermionic Bost–Connes type statistical mechanical model is presented as well, exhibiting a "phase transition at (inverse) temperature β = 1", which also holds for some "well-behaved" g-prime systems in the sense of Hilberdink–Lapidus.

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28

Wang, Yong. "The noncommutative infinitesimal equivariant index formula, Part II." Journal of Noncommutative Geometry 10, no. 1 (2016): 375–400. http://dx.doi.org/10.4171/jncg/236.

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29

Takata, Doman. "An analytic $LT$-equivariant index and noncommutative geometry." Journal of Noncommutative Geometry 13, no. 2 (July 17, 2019): 553–86. http://dx.doi.org/10.4171/jncg/330.

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30

Masuda, Mikiya. "Unitary toric manifolds, multi-fans and equivariant index." Tohoku Mathematical Journal 51, no. 2 (1999): 237–65. http://dx.doi.org/10.2748/tmj/1178224815.

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31

Dzedzej, Zdzisław, and Grzegorz Graff. "Fixed point index for $G$-equivariant multivalued maps." Topological Methods in Nonlinear Analysis 8, no. 1 (September 1, 1996): 179. http://dx.doi.org/10.12775/tmna.1996.027.

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32

Charton, Isabelle. "Hamiltonian S1-spaces with large equivariant pseudo-index." Journal of Geometry and Physics 147 (January 2020): 103521. http://dx.doi.org/10.1016/j.geomphys.2019.103521.

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33

Mathai, R. B. Melrose, and I. M. Singer. "Equivariant and fractional index of projective elliptic operators." Journal of Differential Geometry 78, no. 3 (March 2008): 465–73. http://dx.doi.org/10.4310/jdg/1207834552.

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34

Fu, Baohua, and Pedro Montero. "Equivariant compactifications of vector groups with high index." Comptes Rendus Mathematique 357, no. 5 (May 2019): 455–61. http://dx.doi.org/10.1016/j.crma.2019.05.002.

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35

Bao, Kai Hua, Jian Wang, and Yong Wang. "The equivariant family index theorem in odd dimensions." Acta Mathematica Sinica, English Series 31, no. 7 (June 15, 2015): 1149–62. http://dx.doi.org/10.1007/s10114-015-3637-6.

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36

Paterson, Alan L. T. "The Equivariant Analytic Index for Proper Groupoid Actions." K-Theory 32, no. 3 (July 2004): 193–230. http://dx.doi.org/10.1007/s10977-004-0839-6.

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37

Wang, Yong. "The equivariant noncommutative Atiyah–Patodi–Singer index theorem." K-Theory 37, no. 3 (March 2006): 213–33. http://dx.doi.org/10.1007/s10977-006-0024-1.

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38

Bao, Kaihua, Jian Wang, and Yong Wang. "A Local Equivariant Index Theorem for Sub-Signature Operators." Journal of Nonlinear Mathematical Physics 28, no. 3 (2021): 309. http://dx.doi.org/10.2991/jnmp.k.210427.001.

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39

Dylawerski, Grzegorz, Kazimierz Gęba, Jerzy Jodel, and Wacław Marzantowicz. "An $S^1$-equivariant degree and the Fuller index." Annales Polonici Mathematici 52, no. 3 (1991): 243–80. http://dx.doi.org/10.4064/ap-52-3-243-280.

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40

Braverman, Maxim. "Index Theorem for Equivariant Dirac Operators on Noncompact Manifolds." K-Theory 27, no. 1 (September 2002): 61–101. http://dx.doi.org/10.1023/a:1020842205711.

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41

Ize, Jorge, and Alfonso Vignoli. "Equivariant degree for Abelian actions. Part II: Index computations." Topological Methods in Nonlinear Analysis 7, no. 2 (June 1, 1996): 369. http://dx.doi.org/10.12775/tmna.1996.017.

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42

Takata, Doman. "LT-equivariant index from the viewpoint of KK-theory." Journal of Geometry and Physics 150 (April 2020): 103591. http://dx.doi.org/10.1016/j.geomphys.2019.103591.

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43

Troitsky, E. V. "The index of equivariant elliptic operators over C-algebras." Annals of Global Analysis and Geometry 5, no. 1 (1987): 3–22. http://dx.doi.org/10.1007/bf00140752.

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44

Bismut, Jean-Michel. "Index theorem and equivariant cohomology on the loop space." Communications in Mathematical Physics 98, no. 2 (June 1985): 213–37. http://dx.doi.org/10.1007/bf01220509.

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45

Shan, Lin. "An equivariant higher index theory and nonpositively curved manifolds." Journal of Functional Analysis 255, no. 6 (September 2008): 1480–96. http://dx.doi.org/10.1016/j.jfa.2008.06.025.

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46

Gołȩbiewska, Anna, Marta Kowalczyk, Sławomir Rybicki, and Piotr Stefaniak. "Periodic solutions to symmetric Newtonian systems in neighborhoods of orbits of equilibria." Electronic Research Archive 30, no. 5 (2022): 1691–707. http://dx.doi.org/10.3934/era.2022085.

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<abstract><p>The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the classical Lyapunov center theorem to the case of symmetric potentials with orbits of non-isolated critical points. Our tool is an equivariant version of the Conley index. To compare the indices we compute cohomological dimensions of some orbit spaces.</p></abstract>

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47

Fujita, Hajime. "$S^1$-equivariant local index and transverse index for non-compact symplectic manifolds." Mathematical Research Letters 23, no. 5 (2016): 1351–67. http://dx.doi.org/10.4310/mrl.2016.v23.n5.a5.

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48

Hochs, Peter, and Yanli Song. "An equivariant index for proper actions II: Properties and applications." Journal of Noncommutative Geometry 12, no. 1 (March 23, 2018): 157–93. http://dx.doi.org/10.4171/jncg/273.

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49

Berline, Nicole, and Michèle Vergne. "A computation of the equivariant index of the Dirac operator." Bulletin de la Société mathématique de France 79 (1985): 305–45. http://dx.doi.org/10.24033/bsmf.2036.

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50

Gukov, Sergei, Du Pei, Wenbin Yan, and Ke Ye. "Equivariant Verlinde Algebra from Superconformal Index and Argyres–Seiberg Duality." Communications in Mathematical Physics 357, no. 3 (January 11, 2018): 1215–51. http://dx.doi.org/10.1007/s00220-017-3074-8.

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