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23
Conformal Field Theory and Elliptic Cohomology
"... The purpose of the present paper is to address an old question (posed by Segal [37]) to find a geometric construction of elliptic cohomology. This question has recently become much more pressing due to the work of Mike Hopkins and Haynes Miller [19], who constructed exactly the “right”, or universal ..."
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Cited by 45 (11 self)
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The purpose of the present paper is to address an old question (posed by Segal [37]) to find a geometric construction of elliptic cohomology. This question has recently become much more pressing due to the work of Mike Hopkins and Haynes Miller [19], who constructed exactly the “right”, or universal, elliptic cohomology,
Closed and open conformal field theories and their anomalies
 Comm. Math. Phys
"... Abstract. We describe a formalism allowing a completely mathematical rigorous approach to closed and open conformal field theories with general anomaly. We also propose a way of formalizing modular functors with positive and negative parts, and outline some connections with other topics, in particul ..."
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Cited by 26 (5 self)
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Abstract. We describe a formalism allowing a completely mathematical rigorous approach to closed and open conformal field theories with general anomaly. We also propose a way of formalizing modular functors with positive and negative parts, and outline some connections with other topics, in particular elliptic cohomology. 1.
Elliptic cohomology
 In preparation
"... This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of deri ..."
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Cited by 17 (1 self)
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This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of derived algebraic geometry. We apply this theory in §3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In §4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In §5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [21]. In carrying out the work described in this paper, I have benefitted from the ideas of many people. I
(Pre)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws
, 2004
"... In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. ..."
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Cited by 17 (1 self)
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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.
Twists of Ktheory and TMF
 In Superstrings, geometry, topology, and C∗algebras
, 2010
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Generalized Schubert Calculus
"... Dedicated to C.S. Seshadri on the occasion of his 80th birthday In this paper we study the Tequivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the BottSamelson classes. After s ..."
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Cited by 9 (2 self)
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Dedicated to C.S. Seshadri on the occasion of his 80th birthday In this paper we study the Tequivariant generalized cohomology of flag varieties using two models, the Borel model and the moment graph model. We study the differences between the Schubert classes and the BottSamelson classes. After setup of the general framework we compute, for classes of Schubert varieties of complex dimension � 3 in rank 2 (including A2,), moment graph representatives, PieriChevalley formulas and products of Schubert classes. These computations generalize the computations in equivariant Ktheory for rank 2 cases which are given in GriffethRam [GR]. B2, G2 and A (1)
CIRCLEEQUIVARIANT CLASSIFYING SPACES AND THE RATIONAL EQUIVARIANT SIGMA GENUS
"... Abstract. We analyze the circleequivariant spectrum MStringC which is the equivariant analogue of the cobordism spectrum MU〈6 〉 of stably almost complex manifolds with c1 = c2 = 0. In [Gre05], the second author showed how to construct the ring Tspectrum EC representing the Tequivariant elliptic c ..."
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Cited by 2 (2 self)
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Abstract. We analyze the circleequivariant spectrum MStringC which is the equivariant analogue of the cobordism spectrum MU〈6 〉 of stably almost complex manifolds with c1 = c2 = 0. In [Gre05], the second author showed how to construct the ring Tspectrum EC representing the Tequivariant elliptic cohomology associated to a rational elliptic curve C. In the case that C is a complex elliptic curve, we construct a map of ring Tspectra MStringC → EC which is the rational equivariant analogue of the sigma orientation of [AHS01]. Our method gives a proof of a conjecture of the first author in [And03b]. Contents
Topological modular forms with level structure
, 2013
"... The cohomology theory known as Tmf, for “topological modular forms,” is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic cur ..."
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The cohomology theory known as Tmf, for “topological modular forms,” is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of Ktheory. In particular, this allows us to construct a connective spectrum tmf0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a sheaf of locally evenperiodic elliptic cohomology theories, equipped with highly structured multiplication, on the logétale site of the moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf with level structure. 1