DOCSLIB.ORG

Topological Modular Forms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topological Modular Forms Mathematical Surveys and Monographs Volume 201 Topological Modular Forms Christopher L. Douglas John Francis André G. Henriques Michael A. Hill Editors American Mathematical Society Topological Modular Forms http://dx.doi.org/10.1090/surv/201 Mathematical Surveys and Monographs Volume 201 Topological Modular Forms Christopher L. Douglas John Francis André G. Henriques Michael A. Hill Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer 2010 Mathematics Subject Classification. Primary 55N34, 55N35, 55N22, 57R77, 55T05, 55T15, 18G40, 14H52, 14D23, 13D03. For additional information and updates on this book, visit www.ams.org/bookpages/surv-201 Library of Congress Cataloging-in-Publication Data Topological modular forms / Christopher L. Douglas, John Francis, Andr´e G. Henriques, Michael A. Hill, editors. pages cm. — (Mathematical surveys and monographs ; volume 201) Includes bibliographical references. ISBN 978-1-4704-1884-7 (alk. paper) 1. Topological fields. 2. Algebraic fields. 3. Curves, Elliptic. I. Douglas, Christopher L., editor. II. Francis, John, 1982– editor. III. Henriques, Andr´e G. (Andr´e Gil), 1977– editor. IV. Hill, Michael A. (Michael Anthony), editor. QA611.T656 2014 514.23—dc23 2014029076 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 Contents Preface and Acknowledgments viii Introduction xi Part I 1 1. Elliptic genera and elliptic cohomology Corbett Redden 3 2. Elliptic curves and modular forms Carl Mautner 17 3. The moduli stack of elliptic curves Andre´ G. Henriques 25 4. The Landweber exact functor theorem Henning Hohnhold 35 5. Sheaves in homotopy theory Christopher L. Douglas 47 6. Bousfield localization and the Hasse square Tilman Bauer 79 7. The local structure of the moduli stack of formal groups Jacob Lurie 89 8. Goerss–Hopkins obstruction theory Vigleik Angeltveit 93 9. From spectra to stacks after Michael J. Hopkins 99 10. The string orientation after Michael J. Hopkins 109 11. The sheaf of E∞-ring spectra after Michael J. Hopkins 125 12. The construction of tmf Mark Behrens 131 13. The homotopy groups of tmf and of its localizations Andre´ G. Henriques 189 v vi CONTENTS Part II 207 Elliptic curves and stable homotopy I Michael J. Hopkins and Haynes R. Miller 209 From elliptic curves to homotopy theory Michael J. Hopkins and Mark Mahowald 261 K(1)-local E∞-ring spectra Michael J. Hopkins 287 Glossary 303 4 8 4 8 4 4 84 8 2 4 4 4 84 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 4 2 2 8 2 2 2 4 2 2 2 8 2 2 2 2 2 2 2 2 22 22 22 222 222 222 4 8 4 4 8 8 2 4 4 4 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 4 8 8 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 8 8 8 8 2 4 4 4 4 4 84 8 8 8 2 4 4 4 4 4 8 8 2 22 22 22 22 22 22 4 22 22 22 22 22 22 22 2 2 2 2 2 2 2 2 2 22 22 22 22 2 22 22 22 22 22 22 22 22 22 22 22 22 2 2 2 22 22 2 22 22 22 Preface This is a book about the theory of topological modular forms. It is also a record of the efforts of a group of graduate students to learn that theory at the 2007 Talbot Workshop, and so a book born of and steeped in the Talbot vision. In the fall of 2003, Mike Hopkins taught a course at MIT about tmf . Our gen- eration of Cambridge algebraic topologists, having survived and thrived in Haynes Miller’s Kan seminar, found in Mike’s class our next, and really our last common, mathematical crucible. The course hacked through the theory of algebraic modu- lar forms, formal groups, multiplicative stable homotopy theory, stacks, even more stacks, moduli stacks of elliptic curves, Bousfield localization, Morava K-andE- theory, the arithmetic and Hasse squares, Andr´e–Quillen cohomology, obstruction theory for moduli of associative and commutative ring spectra—by this point we were having dreams, or maybe nightmares, about the spiral exact sequence. In the middle of the course, we all flew over to M¨unster for a week-long workshop on tmf with lectures by Mike, Haynes, Matt Ando, Charles Rezk, and Paul Goerss. A transatlantic mix of students spent the late afternoons coaxing and cramming the knowledge in at a cafe off Steinfurter Strasse; there we devised a plan to reconvene and sketched a vision of what would become the Talbot Workshops: a gathering for graduate students, focused on a single topic of contemporary research interest, lectured by graduate students and guided by a single faculty mentor, having talks in the morning and in the evening and every afternoon free for discussion and outdoor activities, with participants sleeping and lecturing and cooking together under the same roof. We pitched it to Mike and Haynes and they agreed to back a ragtag summit. Talbot was born. Three years later, in 2007, we decided to bring Talbot home with a workshop on tmf , mentored by Mike Hopkins. Mike stopped by Staples on his way to the workshop and picked up a big red “That was easy” button. Throughout the work- shop, whenever he or anyone else completed a particularly epic spectral sequence computation or stacky decomposition, he’d hit the button and a scratchy electronic voice would remind us, “That was easy!” It became the workshop joke (for much of it was evidently not easy) and mantra (for shifting perspective, whether to mul- tiplicative stable homotopy or to stacky language or to a suitable localization, did make the intractable seem possible). This book is a record and expansion of the material covered in the Talbot 2007 workshop. Though the authors of the various chapters have brought their own expositional perspectives to bear (particularly heroically in the case of Mark Behrens), the contemporary material in this book is due to Mike Hopkins, Haynes Miller, and Paul Goerss, with contributions by Mark Mahowald, Matt Ando, and Charles Rezk. viii Acknowledgments We thank the participants of Talbot 2007 for their dedication and enthusi- asm during the workshop: Ricardo Andrade, Vigleik Angeltveit, Tilman Bauer, Mark Behrens, Thomas Bitoun, Andrew Blumberg, Ulrik Buchholtz, Scott Car- nahan, John Duncan, Matthew Gelvin, Teena Gerhardt, Veronique Godin, Owen Gwilliam, Henning Hohnhold, Valentina Joukhovitski, Jacob Lurie, Carl Mautner, Justin Noel, Corbett Redden, Nick Rozenblyum, and Samuel W¨uthrich. Special thanks to Corbett, Carl, Henning, Tilman, Jacob, and Vigleik for writing up their talks from the workshop. And super-special thanks to Mark for his enormous effort writing up his perspective on the construction of tmf , and for extensive support and assistance throughout the development of the book. Our appreciation and thanks also to Tilman, to Mark, and to Niko Naumann for substantial contributions to and help with the glossary, and to Paul Goerss, Charles Rezk, and David Gepner for suggestions and corrections. Our appreciation goes to Nora Ganter, who in 2003 ran a student seminar on tmf —it was our first sustained exposure to the subject; Nora also prepared the orig- inal literature list for the Talbot 2007 program. We also acknowledge the students who wrote up the ‘Course notes for elliptic cohomology’ based on Mike Hopkins’ 1995 course, and the students who wrote up ‘Complex oriented cohomology the- ories and the language of stacks’ based on Mike’s 1999 course—both documents, distributed through the topology underground, were helpful for us in the years leading up to and at Talbot 2007. The book would not have happened without Talbot, and the Talbot workshops would not have happened without the support of MIT and the NSF. Christopher Stark at the NSF was instrumental in us securing the first Talbot grant, DMS- 0512714, which funded the workshops from 2005 til 2008. MIT provided the facili- ties for the first workshop, at the university retreat Talbot House, and has provided continual logistical, administrative, and technical support for the workshops ever since.
Load more

Recommended publications
  • Persistent Obstruction Theory for a Model Category of Measures with Applications to Data Merging
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 8, Pages 1–38 (February 2, 2021) https://doi.org/10.1090/btran/56 PERSISTENT OBSTRUCTION THEORY FOR A MODEL CATEGORY OF MEASURES WITH APPLICATIONS TO DATA MERGING ABRAHAM D. SMITH, PAUL BENDICH, AND JOHN HARER Abstract. Collections of measures on compact metric spaces form a model category (“data complexes”), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on a product space that marginalizes to any measures on pairs of its factors. The homotopy and homology for this category allow measurement of obstructions to finding measures on larger and larger product spaces. The obstruction theory is compatible with a fibrant filtration built from the Wasserstein distance on measures. Despite the abstract tools, this is motivated by a widespread problem in data science. Data complexes provide a mathematical foundation for semi- automated data-alignment tools that are common in commercial database software. Practically speaking, the theory shows that database JOIN oper- ations are subject to genuine topological obstructions. Those obstructions can be detected by an obstruction cocycle and can be resolved by moving through a filtration. Thus, any collection of databases has a persistence level, which measures the difficulty of JOINing those databases. Because of its general formulation, this persistent obstruction theory also encompasses multi-modal data fusion problems, some forms of Bayesian inference, and probability cou- plings. 1. Introduction We begin this paper with an abstraction of a problem familiar to any large enterprise. Imagine that the branch offices within the enterprise have access to many data sources.
    [Show full text]
  • Some Stuff About the String Gerbe
    Some Stuff about the String Gerbe Danny Stevenson Department of Mathematics, University of California, Riverside Riverside, CA 92521, USA email: [email protected] March 14, 2006 1 Introduction Let G be a compact, simple and simply connected Lie group and let π : P → M be a principal G bundle over a smooth manifold M. Let ν denote the universally transgressive generator of H3(G; Z) = Z and let c ∈ H4(BG; Z) = H3(G; Z) be the transgression of ν. By regarding H4(BG; Z) as a lattice in (S2g∗)G we will allow ourselves to confuse c and the basic inner product h , i on g. Recall [23] that h ,√i is the Killing form on g, normalised so that the longest root θ has length 2. In the physics literature, M is said to be string, or admit a string structure, if a certain characteristic class in H3(LM; Z) vanishes (here LM denotes the free loop space of M). This characteristic class is the obstruction to lifting the structure group of the principal LG- bundle LP → LM to LGc — the Kac-Moody group. As has been observed by several authors [12, 13, 19] the obstruction in H3(LM; Z) is closely related to the characteristic class c ∈ H4(M; Z): if M is 2-connected a lift of the structure group to LGc exists precisely when the map c: M → K(Z, 4) is null- homotopic. As is well known, if G = Spin(n) then 2c = p1. This obstruction problem on LM can be phrased in the language of homotopy theory down on M.
    [Show full text]
  • Obstructions, Extensions and Reductions. Some Applications Of
    Obstructions, Extensions and Reductions. Some applications of Cohomology ∗ Luis J. Boya Departamento de F´ısica Te´orica, Universidad de Zaragoza. E-50009 Zaragoza, Spain [email protected] October 22, 2018 Abstract After introducing some cohomology classes as obstructions to ori- entation and spin structures etc., we explain some applications of co- homology to physical problems, in especial to reduced holonomy in M- and F -theories. 1 Orientation For a topological space X, the important objects are the homology groups, H∗(X, A), with coefficients A generally in Z, the integers. A bundle ξ : E(M, F ) is an extension E with fiber F (acted upon by a group G) over an space M, noted ξ : F → E → M and it is itself a Cech˘ cohomology element, arXiv:math-ph/0310010v1 8 Oct 2003 ξ ∈ Hˆ 1(M,G). The important objects here the characteristic cohomology classes c(ξ) ∈ H∗(M, A). Let M be a manifold of dimension n. Consider a frame e in a patch U ⊂ M, i.e. n independent vector fields at any point in U. Two frames e, e′ in U define a unique element g of the general linear group GL(n, R) by e′ = g · e, as GL acts freely in {e}. An orientation in M is a global class of frames, two frames e (in U) and e′ (in U ′) being in the same class if det g > 0 where e′ = g · e in the overlap of two patches. A manifold is orientable if it is ∗To be published in the Proceedings of: SYMMETRIES AND GRAVITY IN FIELD THEORY.
    [Show full text]
  • On Modular Forms for the Paramodular Group
    On Modular Forms for the Paramodular Group Brooks Roberts and Ralf Schmidt Contents 1 Definitions 3 2 Linear independence at different levels 6 3 The level raising operators 8 4 Oldforms and newforms 13 5 Saito–Kurokawa liftings 16 6 Two theorems 23 Appendix 26 References 28 Introduction Let F be a p–adic field, and let G be the algebraic F –group GSp(4). In our paper [RS] we presented a conjectural theory of local newforms for irreducible, admissible, generic representations of G(F ) with trivial central character. The main feature of this theory is that it considers fixed vectors under the paramodular groups K(pn), a certain family of compact-open subgroups. The group K(p0) is equal to the standard maximal compact subgroup G(o), where o is the ring of integers of F . In fact, K(p0) and K(p1) represent the two conjugacy classes of maximal compact subgroups of G(F ). In general K(pn) can be conjugated into K(p0) if n is even, and into K(p1) if n is odd. Our theory is analogous to Casselman’s well-known theory for representations of GL(2,F ); see [Cas]. The main conjecture made in [RS] states that for each irreducible, admissible, generic representation (π, V ) of PGSp(4,F ) there exists an n such that the space V (n) of K(pn) invariant vectors is non-zero; if n0 is the minimal such n then dimC(V (n0)) = 1; and the Novodvorski zeta integral of a suitably normalized vector in V (n0) computes the L–factor L(s, π) (for this last statement we assume that V is the Whittaker model of π).
    [Show full text]
  • PIECEWISE LINEAR TOPOLOGY Contents 1. Introduction 2 2. Basic
    PIECEWISE LINEAR TOPOLOGY J. L. BRYANT Contents 1. Introduction 2 2. Basic Definitions and Terminology. 2 3. Regular Neighborhoods 9 4. General Position 16 5. Embeddings, Engulfing 19 6. Handle Theory 24 7. Isotopies, Unknotting 30 8. Approximations, Controlled Isotopies 31 9. Triangulations of Manifolds 33 References 35 1 2 J. L. BRYANT 1. Introduction The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic ac- counts of the subject may be found in [2] and [63]. Whitehead’s important paper [63] contains the foundation of the geometric and algebraic theory of simplicial com- plexes that we use today. More recent sources, such as [30], [50], and [66], together with [17] and [37], provide a fairly complete development of PL theory up through the early 1970’s. This chapter will present an overview of the subject, drawing heavily upon these sources as well as others with the goal of unifying various topics found there as well as in other parts of the literature. We shall try to give enough in the way of proofs to provide the reader with a flavor of some of the techniques of the subject, while deferring the more intricate details to the literature. Our discussion will generally avoid problems associated with embedding and isotopy in codimension 2. The reader is referred to [12] for a survey of results in this very important area. 2. Basic Definitions and Terminology. Simplexes. A simplex of dimension p (a p-simplex) σ is the convex closure of a n set of (p+1) geometrically independent points {v0, .
    [Show full text]
  • A Construction of String 2-Group Modelsusing a Transgression
    A Construction of String 2-Group Models using a Transgression-Regression Technique Konrad Waldorf Fakult¨at f¨ur Mathematik Universit¨at Regensburg Universit¨atsstraße 31 D–93053 Regensburg [email protected] Dedicated to Steven Rosenberg on the occasion of his 60th birthday Abstract In this note we present a new construction of the string group that ends optionally in two different contexts: strict diffeological 2-groups or finite-dimensional Lie 2-groups. It is canonical in the sense that no choices are involved; all the data is written down and can be looked up (at least somewhere). The basis of our construction is the basic gerbe of Gaw¸edzki-Reis and Meinrenken. The main new insight is that under a transgression- regression procedure, the basic gerbe picks up a multiplicative structure coming from the Mickelsson product over the loop group. The conclusion of the construction is a relation between multiplicative gerbes and 2-group extensions for which we use recent work of Schommer-Pries. MSC 2010: Primary 22E67; secondary 53C08, 81T30, 58H05 arXiv:1201.5052v2 [math.DG] 4 May 2012 1 Introduction The string group String(n) is a topological group defined up to homotopy equivalence as the 3-connected cover of Spin(n), for n =3 or n> 4. Concrete models for String(n) have been provided by Stolz [Sto96] and Stolz-Teichner [ST04]. In order to understand, e.g. the differential geometry of String(n), the so-called “string geometry”, it is necessary to have models in better categories than topological groups. Its 3-connectedness implies that String(n) is a K(Z, 2)-fibration over Spin(n), so that it cannot be a (finite-dimensional) Lie group.
    [Show full text]
  • Math 527 - Homotopy Theory Obstruction Theory
    Math 527 - Homotopy Theory Obstruction theory Martin Frankland April 5, 2013 In our discussion of obstruction theory via the skeletal filtration, we left several claims as exercises. The goal of these notes is to fill in two of those gaps. 1 Setup Let us recall the setup, adopting a notation similar to that of May x 18.5. Let (X; A) be a relative CW complex with n-skeleton Xn, and let Y be a simple space. Given two maps fn; gn : Xn ! Y which agree on Xn−1, we defined a difference cochain n d(fn; gn) 2 C (X; A; πn(Y )) whose value on each n-cell was defined using the following \double cone construction". Definition 1.1. Let H; H0 : Dn ! Y be two maps that agree on the boundary @Dn ∼= Sn−1. The difference construction of H and H0 is the map 0 n ∼ n n H [ H : S = D [Sn−1 D ! Y: Here, the two terms Dn are viewed as the upper and lower hemispheres of Sn respectively. 1 2 The two claims In this section, we state two claims and reduce their proof to the case of spheres and discs. Proposition 2.1. Given two maps fn; gn : Xn ! Y which agree on Xn−1, we have fn ' gn rel Xn−1 if and only if d(fn; gn) = 0 holds. n n−1 Proof. For each n-cell eα of X n A, consider its attaching map 'α : S ! Xn−1 and charac- n n−1 teristic map Φα :(D ;S ) ! (Xn;Xn−1).
    [Show full text]
  • Congruences Between Modular Forms
    CONGRUENCES BETWEEN MODULAR FORMS FRANK CALEGARI Contents 1. Basics 1 1.1. Introduction 1 1.2. What is a modular form? 4 1.3. The q-expansion priniciple 14 1.4. Hecke operators 14 1.5. The Frobenius morphism 18 1.6. The Hasse invariant 18 1.7. The Cartier operator on curves 19 1.8. Lifting the Hasse invariant 20 2. p-adic modular forms 20 2.1. p-adic modular forms: The Serre approach 20 2.2. The ordinary projection 24 2.3. Why p-adic modular forms are not good enough 25 3. The canonical subgroup 26 3.1. Canonical subgroups for general p 28 3.2. The curves Xrig[r] 29 3.3. The reason everything works 31 3.4. Overconvergent p-adic modular forms 33 3.5. Compact operators and spectral expansions 33 3.6. Classical Forms 35 3.7. The characteristic power series 36 3.8. The Spectral conjecture 36 3.9. The invariant pairing 38 3.10. A special case of the spectral conjecture 39 3.11. Some heuristics 40 4. Examples 41 4.1. An example: N = 1 and p = 2; the Watson approach 41 4.2. An example: N = 1 and p = 2; the Coleman approach 42 4.3. An example: the coefficients of c(n) modulo powers of p 43 4.4. An example: convergence slower than O(pn) 44 4.5. Forms of half integral weight 45 4.6. An example: congruences for p(n) modulo powers of p 45 4.7. An example: congruences for the partition function modulo powers of 5 47 4.8.
    [Show full text]
  • Arxiv:2101.06841V2 [Math.AT] 1 Apr 2021
    C2-EQUIVARIANT TOPOLOGICAL MODULAR FORMS DEXTER CHUA Abstract. We compute the homotopy groups of the C2 fixed points of equi- variant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a TMF-module, it is isomorphic to the tensor product of TMF with an explicit finite cell complex. Contents 1. Introduction 1 2. Equivariant elliptic cohomology 5 3. The E2 page of the DSS 9 4. Differentials in the DSS 13 5. Identification of the last factor 25 6. Further questions 29 Appendix A. Connective C2-equivariant tmf 30 Appendix B. Sage script 35 References 38 1. Introduction Topological K-theory is one of the first examples of generalized cohomology theories. It admits a natural equivariant analogue — for a G-space X, the group 0 KOG(X) is the Grothendieck group of G-equivariant vector bundles over X. In 0 particular, KOG(∗) = Rep(G) is the representation ring of G. As in the case of non-equivariant K-theory, this extends to a G-equivariant cohomology theory KOG, and is represented by a genuine G-spectrum. We shall call this G-spectrum KO, omitting the subscript, as we prefer to think of this as a global equivariant spectrum — one defined for all compact Lie groups. The G-fixed points of this, written KOBG, is a spectrum analogue of the representation ring, BG 0 BG −n with π0KO = KOG(∗) = Rep(G) (more generally, πnKO = KOG (∗)). These fixed point spectra are readily computable as KO-modules. For example, KOBC2 = KO _ KO; KOBC3 = KO _ KU: arXiv:2101.06841v2 [math.AT] 1 Apr 2021 This corresponds to the fact that C2 has two real characters, while C3 has a real character plus a complex conjugate pair.
    [Show full text]
  • The Rings of Hilbert Modular Forms for Q( 29) and Q( 37)
    p p THE RINGS OF HILBERT MODULAR FORMS FOR Q( 29) AND Q( 37) BRANDON WILLIAMS Abstract. We use Borcherds products and their restrictions to Hirzebruch-Zagier curvesp to deter- minep generators and relations for the graded rings of Hilbert modular forms for the fields Q( 29) and Q( 37). These seem to be the first cases where the graded ring can be computed despite obstructions to the existence of Borcherds products with arbitrary divisors. 1. Introduction The problem considered in this note is to determine generators and relations for the graded rings M∗(ΓK ) of Hilbert modular forms over real quadraticp fields K. Therep is some history to this.p The structure has been worked out in thep cases Q( 5) (Gundlach, [12]), Q( 2) (Hammond, [13]), Q( 13) (van der Geer and Zagier, [10]), Q( 17) (Hermann, [14] determinedp the structure of the subring of symmetric forms; the full ring appears in section 6.5 of [18]), and Q( 6) (van derp Geer,p [8]). Sectionp 6.2 of [18] gives a more detailed overview. Simpler derivations for the fields Q( 5); Q( 13); Q( 17) were given by Mayer [18] using a reduction process against Borcherds products with simple zeros on the diagonal, and the work here extends this idea. p We only consider fields K = Q( p) of prime discriminant p ≡ 1 (4) and class number one as this makes a number of aspects of the theory simpler (but there are ultimately workarounds for both of these assumptions). The first few such primes are p = 5; 13; 17; and the corresponding graded rings have the form p M∗(Γ ) = [X2;X5;X6;X15]=(R30); Q( 5) C p M∗(Γ ) = [X2;X3;X4;X5;X6;X8;X9]=(R9;R10;R12;R18); Q( 13) C p (1) (2) (1) (2) (1) (2) M∗(Γ ) = [X2;X3;X ;X ;X ;X ;X ;X ;X7;X8;X9]=I; Q( 17) C 4 4 5 5 6 6 (1) (2) I = (R9;R10;R11;R12 ;R12 ;R13;R14;R18); (i) (i) where Xk;Xk denote modular forms of weight k and where Rk;Rk denote relations that are homo- geneous of weight k, see [18].
    [Show full text]
  • On the String 2-Group
    On the String 2-Group Urs Schreiber∗ May 3, 2006 Oberseminar Topologie Universit¨atBonn, May 2, 2006 Abstract We try to convey the main idea for 1. • what the String-group StringG is • and how it is the nerve of a 2-group StrG; 2. as well as • what a StrG-2-bundle is • and how it is “the same” as a StringG-bundle. The first point is due to [10, 9], which will be reviewed in section 2. The second point has been addressed in [8] using the language of bundle gerbes. In section 3 we review this, using a 2-functorial language which is natural with respect to the 2-group nature of StrG. A commented bibliography is given in section 1. Contents 1 Motivation 2 2 The String 2-Group 3 2.1 2-Groups . 4 2.2 StringG ................................ 6 3 String 2-Bundles 9 3.1 2-Bundles . 9 3.2 Transition Bundles . 15 4 Outlook: String Connections 19 ∗e-mail: urs.schreiber at math.uni-hamburg.de 1 1 Motivation The main motivation for the following discussion has its origin in theoretical physics. Elementary particles with spin are described by sections of spin bundles. From the physical point of view, the necessity of a spin structure on spacetime may be deduced from a certain global anomaly for the path integral of a single, point-like, fermion. The path integral (albeit a somewhat heuristic device) can be regarded as a single valued function on the space of configurations of the particle, only if the (first and) second Stiefel-Whitney class of spacetime vanishes.
    [Show full text]
  • Arxiv:1908.02868V2 [Math.AT] 30 Dec 2020
    A DE RHAM MODEL FOR COMPLEX ANALYTIC EQUIVARIANT ELLIPTIC COHOMOLOGY DANIEL BERWICK-EVANS AND ARNAV TRIPATHY Abstract. We construct a cocycle model for complex analytic equivariant elliptic co- homology that refines Grojnowski's theory when the group is connected and Devoto's when the group is finite. We then construct Mathai{Quillen type cocycles for equivari- ant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique complex analytic equivariant refinement of Hopkins' \theorem of the cube" construction of the MString-orientation of elliptic cohomology. Contents 1. Introduction 1 2. G-bundles on elliptic curves 5 3. A cocycle model for equivariant elliptic cohomology 13 4. Comparing with Grojnowski's equivariant elliptic cohomology 19 5. Comparing with Devoto's equivariant elliptic cohomology 21 6. Loop group representations and cocycle representatives of Thom classes 22 7. Equivariant orientations and the theorem of the cube 33 Appendix A. Background 38 References 42 1. Introduction Equivariant K-theory facilitates a rich interplay between representation theory and topology. For example, universal Thom classes come from representations of spin groups; power operations are controlled by the representation theory of symmetric groups; and the equivariant index theorem permits geometric constructions of representations of Lie groups. Equivariant elliptic cohomology is expected to lead to an even deeper symbiosis be- tween representation theory and topology. First evidence appears in the visionary work of Grojnowski [Gro07] and Devoto [Dev96]. Grojnowski's complex analytic equivariant elliptic arXiv:1908.02868v2 [math.AT] 30 Dec 2020 cohomology (defined for connected Lie groups) makes contact with positive energy repre- sentations of loop groups [And00, Gan14].
    [Show full text]