## Elliptic cohomology

Venue: | In preparation |

Citations: | 7 - 1 self |

### BibTeX

@INPROCEEDINGS{Lurie_ellipticcohomology,

author = {Jacob Lurie},

title = {Elliptic cohomology},

booktitle = {In preparation},

year = {}

}

### OpenURL

### Abstract

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 2-equivariance. 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

### Citations

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(Show Context)
Citation Context ...) as the total space over a certain S 1 -gerbe over Spin(V ). This has the advantage of being a “finite dimensional” object that can be studied using ideas from differential geometry (see for example =-=[10]-=-). Let M be a smooth manifold of dimension n. Choosing a Riemannian metric on M, we can reduce the structure group of the tangent bundle of M to O(V ). An orientation (spin structure, string structure... |

191 | Symmetric spectra
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(Show Context)
Citation Context ... E∞-ring. We will not give a definition here, though we will give a brief outline of the theory in §2.1. For definitions and further details we refer the reader to the literature (for example [12] or =-=[17]-=-). Returning to the subject of elliptic cohomology, we can now consider the diagram {E∞ − Rings } � � � � � � � � � � {φ : Spec R → M1,1} � {Multiplicative Cohomology Theories}. Once again, our object... |

173 |
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Citation Context ...ved algebraic geometry. For this, we will assume that the reader is familiar with the language of toric varieties (for a very readable account of the theory of toric varieties, we refer the reader to =-=[15]-=-). Fix an E∞-ring R. Let Λ be a lattice (that is, a free Z-module of finite rank) and let F = {σα}α∈A be a rational polyhedral fan in Z. For each α ∈ A, let σ∨ α ⊆ Λ∨ denote the dual cone to σα, regar... |

144 |
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Citation Context ... Hom(MP(∗), R) of commutative ring homomorphisms from MP(∗) into R, and the set of power series f(u, v) with coefficients in R that satisfy the three identities asserted above. We refer the reader to =-=[1]-=- for a proof of Quillen’s theorem and further discussion. The construction that associates the formal group G = Spf A(CP ∞ ) to an even periodic cohomology theory A has turned out to be a very powerfu... |

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Citation Context ...imply an E∞-ring. We will not give a definition here, though we will give a brief outline of the theory in §2.1. For definitions and further details we refer the reader to the literature (for example =-=[12]-=- or [17]). Returning to the subject of elliptic cohomology, we can now consider the diagram {E∞ − Rings } � � � � � � � � � � {φ : Spec R → M1,1} � {Multiplicative Cohomology Theories}. Once again, ou... |

81 |
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Citation Context ...of G to the associated bundle on the classifying space. This map is never an isomorphism unless G is trivial. However, it is not far from being an isomorphism: by the Atiyah-Segal completion theorem (=-=[6]-=-), φ identifies K(BG) with the completion of Rep(G) with respect to a certain ideal (the ideal consisting of virtual representations of dimension zero). Let us consider the situation in more detail fo... |

64 | Elliptic spectra, the Witten genus and the theorem of the cube
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(Show Context)
Citation Context ...ntially the same thing as an elliptic curve.8 Jacob Lurie Remark 1.12. Our notion of elliptic cohomology theory is essentially the same as the notion of an elliptic spectrum as defined in defined in =-=[4]-=-. In view of Remark 1.11, there is a plentiful supply of elliptic cohomology theories: roughly speaking, there is an elliptic cohomology theory for every elliptic curve. Of course, we have a similar s... |

55 |
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(Show Context)
Citation Context ...n M has a canonical orientation with respect to elliptic cohomology. The relationship between elliptic cohomology orientations of a manifold and string structures goes back to the work of Witten (see =-=[30]-=-). Heuristically, the elliptic cohomology of a space M can be thought of as the S 1 -equivariant K-theory of the loop space LM. To obtain an elliptic cohomology orientation of M, one wants to write do... |

42 | Generalized group characters and complex oriented cohomology theories
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(Show Context)
Citation Context ...p. Suppose further that Proposition 3.2 remains valid in this case. Then it is possible to prove that assumption (5) holds, using the method of complex-oriented descent (as explained, for example, in =-=[16]-=-). • When G is an oriented elliptic curve and G is a connected compact Lie group, the theory of G-equivariant A-cohomology described by Proposition 3.3 is closely related to interesting geometry, such... |

38 | Principal G-bundles over elliptic curves
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(Show Context)
Citation Context ...nected, MG is a derived scheme whose underlying classical scheme can be identified with the moduli space for regular Galg-bundles on E; here Galg is the reductive algebraic group associated to G (see =-=[14]-=- for a discussion of regular Galg-bundles on elliptic curves over the complex numbers). Remark 5.1. It is possible to describe M SU(n) and M U(n) more explicitly, in the language of derived algebraic ... |

37 | Twisted K-theory and loop group representations. Available on archive as math.AT/0312155
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(Show Context)
Citation Context ...s of ˜LG + with the global sections of the theta bundle over the underlying ordinary scheme of MG, is proven in [3]. Remark 5.5. Theorem 5.1 is related to the work of Freed, Hopkins and Telemann (see =-=[13]-=-), which identifies the K-theory of loop group representations with the twisted G-equivariant K-theory of the group G itself. 5.3 The String Orientation Let V be a finite dimensional real vector space... |

37 | Conformal field theory and elliptic cohomology
- Hu, Kriz
(Show Context)
Citation Context ...how in [28] that supersymmetry predicts that the coefficients in the q-expansion of the partition function of such a theory should be integral. Alternative speculations on the problem can be found in =-=[18]-=- and [7], among other places. Our moduli-theoretic interpretation of elliptic cohomology has the advantage of being well-suited to proving comparison results with other theories. Suppose that we are g... |

36 |
Axiomatic approach to homology theory
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(Show Context)
Citation Context ... n (X) = H n (X; Z). These invariants have a number of good properties, which are neatly summarized by the following axioms (which, in a slightly modified form, are due to Eilenberg and Steenrod: see =-=[11]-=-): (1) For each n ∈ Z, A n is a contravariant functor from the category of pairs of topological spaces (Y ⊆ X) to abelian groups. (We recover the absolute cohomology groups A n (X) by taking Y = ∅.) (... |

32 | Two-vector bundles and forms of elliptic cohomology. Topology, geometry, and quantum field theory
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(Show Context)
Citation Context ...8] that supersymmetry predicts that the coefficients in the q-expansion of the partition function of such a theory should be integral. Alternative speculations on the problem can be found in [18] and =-=[7]-=-, among other places. Our moduli-theoretic interpretation of elliptic cohomology has the advantage of being well-suited to proving comparison results with other theories. Suppose that we are given som... |

30 |
What is an elliptic object? In Topology, geometry and quantum field theory, volume 308
- Stolz, Teichner
- 2004
(Show Context)
Citation Context ...ted methods, one canA Survey of Elliptic Cohomology 37 realize String(V ) as a topological group. An explicit realization of String(V ) as an (infinite dimensional) topological group is described in =-=[28]-=-. An alternative point of view is to consider String(V ) as the total space over a certain S 1 -gerbe over Spin(V ). This has the advantage of being a “finite dimensional” object that can be studied u... |

29 |
Notes on the Hopkins-Miller Theorem. Homotopy Theory via Algebraic Geometry and Group Representation Theory
- Rezk
(Show Context)
Citation Context ...g Oκ. One can show that this mapping property characterizes the Lubin-Tate spectrum E2 associated to the formal group Ê0, which is known to have all of the desired properties. (We refer the reader to =-=[26]-=- for a proof of the Hopkins-Miller theorem on Lubin-Tate spectra, which is very close to establishing the universal property that we need here.) If the elliptic curve E0 → Spec k is not supersingular,... |

27 |
Fonctions thêta et théorème du cube
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(Show Context)
Citation Context ...X; Z). Then Ll is a line bundle on the product E ×Spec A E. The symmetry and bi-additivity of the cup product operation translate into the assertion that Ll is a symmetric biextension of E by Gm (see =-=[9]-=- for a discussion in the classical setting). This gives an identification of E with the dual elliptic curve E ∨ . We will not describe the construction of 2-equivariant elliptic cohomology here. Howev... |

24 |
Homological Properties of Comodules over MU∗(MU) and BP∗(BP
- Landweber
(Show Context)
Citation Context ...lat. If φ is flat, then we say that the formal group G is Landweber-exact. Using the structure theory of formal groups, Landweber has given a criterion for a formal group G to be Landweber-exact (see =-=[19]-=-). We will not review Landweber’s theorem here. However, we remark that Landweber’s criterion is purely algebraic, easy to check, and is quite often satisfied. Remark 1.8. In the case where G is the f... |

20 |
On the periodicity theorem for complex vector bundles
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- 1964
(Show Context)
Citation Context ...is called the Bott element, because multiplication by β induces isomorphisms K n (X) → K n−2 (X) for every space X and every integer n: this is the content of the famous Bott periodicity theorem (see =-=[5]-=-). The following definition abstracts some of the pleasant properties of complex K-theory. Definition 1.1. Let A be a multiplicative cohomology theory. We will say that A is even if A i (∗) = 0 whenev... |

20 |
Elliptic cohomology (after Landweber-Stong, Ochanine
- Segal
- 1988
(Show Context)
Citation Context ...uivariant elliptic cohomology is related to the theory of loop group representaions. Graeme Segal has suggested that elliptic cohomology should bear some relationship to Euclidean field theories (see =-=[27]-=-). Building on his ideas, Stolz and Teichner have proposed that the classifying space for elliptic cohomology might be interpreted as a moduli space for supersymmetric quantum field theories. To suppo... |

14 | Power operations in elliptic cohomology and representations of loop groups
- Ando
(Show Context)
Citation Context ... analogue of this result, which identifies the K-group of positive energy representations of ˜LG + with the global sections of the theta bundle over the underlying ordinary scheme of MG, is proven in =-=[3]-=-. Remark 5.5. Theorem 5.1 is related to the work of Freed, Hopkins and Telemann (see [13]), which identifies the K-theory of loop group representations with the twisted G-equivariant K-theory of the g... |

13 |
The Morava K-theories of Eilenberg-Mac Lane spaces and the ConnerFloyd conjecture
- Ravenel, Wilson
(Show Context)
Citation Context ...) even, periodic cohomology theory. This cohomology theory is called Morava K-theory and denoted by K(n). The Morava K-theory of Eilenberg-Mac Lane spaces has been computed by Ravenel and Wilson (see =-=[25]-=-). In particular, they show that for m > n, the natural map K(n) ∗ (∗) → K(n) ∗ (K(Z/pZ, m)) is an isomorphism. In other words, the Morava K-theory K(n) cannot tell the difference between the space K(... |

11 |
A variant of E.H. Brown’s representability theorem. Topology 10
- Adams
- 1971
(Show Context)
Citation Context ... presheaf that takes values in the category of cohomology theories. To remedy this difficulty, it is necessary to represent our cohomology theories. According to Brown’s representability theorem (see =-=[2]-=-), any cohomology theory A has a representing space Z, so that there is a functorial identification A(X) ≃ [X, Z] of the A-cohomology of every cell complex X with the set [X, Z] of homotopy classes of... |

9 | From HAG to DAG: derived moduli stacks. Axiomatic, enriched, and motivic homotopy theory - Toen, Vezzosi |

5 |
Loop groups. Oxford Science Publications
- Pressley, Segal
- 1986
(Show Context)
Citation Context ...roup G, but the latter is related to the representation theory of the loop group LG. We begin with a quick review of the theory of loop groups: for a more extensive discussion, we refer the reader to =-=[24]-=-. Fix a connected compact Lie group G, which for simplicity we will assume to be simple. Then the group H 4 (BG; Z) is canonically isomorphic to Z. Let us fix a nonnegative integer l, which we identif... |

2 |
Derived algebraic geometry. Unpublished MIT
- Lurie
- 2004
(Show Context)
Citation Context ... R. One builds ˜ R as the inverse limit of a convergent tower of approximations, which are constructed using condition (3). For details, and a discussion of the meaning of (3), we refer the reader to =-=[20]-=-. We wish to apply Proposition 4.1 to the functor A ↦→ E ′ (A). Condition (1) is clear: on ordinary commutative rings, E ′ is represented by the classical moduli stack M1,1. The remaining conditions a... |

1 |
Topological Automorphic Forms. Available on the mathematics archive as math.AT/0702719
- Behrens, Lawson
(Show Context)
Citation Context ...ssarily arise from elliptic curves. For example, one can produce “derived versions” of certain Shimura varieties, at least p-adically, using the same methods. For more details, we refer the reader to =-=[8]-=-. 4.3 Elliptic Cohomology near ∞ Classically, an elliptic curve E → Spec C is determined, up to noncanonical isomorphism, by its j-invariant j(E) ∈ C. Consequently, the moduli space of elliptic curves... |

1 | Higher topos theory. To be published by the Princeton Univesity Press; presently available for download at http://www.math.harvard.edu/ lurie - Lurie |