## Conformal Field Theory and Elliptic Cohomology

Citations: | 37 - 9 self |

### BibTeX

@MISC{Hu_conformalfield,

author = {P. Hu and I. Kriz},

title = {Conformal Field Theory and Elliptic Cohomology},

year = {}

}

### OpenURL

### Abstract

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,

### Citations

337 |
Lectures on Theta
- Mumford
- 1983
(Show Context)
Citation Context ...� � 1 1 0 1 and the subgroup Γ(m)SL2Z of matrices congruent to Id mod m, for m ∈ N. To recall the modularity properties of the function (144) we review some basic facts about θ-functions of lattices (=-=[30, 38, 31]-=-). This is a very special case of the theory of Siegel modular forms [30], in fact, in some sense, the “trivial” case, i.e. one which reduces to the classical context. By a theta series associated wit... |

270 |
J.Lepowsky and A.Meurman, Vertex operator algebras and the monster
- Frenkel
- 1988
(Show Context)
Citation Context ...erds’ proof [2] of the Moonshine conjectures [5] on the modularity of the Thompson series, which are character series of a certain graded representation of the Monster called the Moonshine module V y =-=[13]-=-. (More precisely, the statement of the conjectures is that the Thompson series are Hauptmoduln. Much work has been done on this and related topics; see e.g. [21,25,32,42–44] for reference.) The Thomp... |

228 |
N.: Classical and quantum conformal field theory
- Moore, Seiberg
- 1989
(Show Context)
Citation Context ...mean a partial CFTsELLIPTIC COHOMOLOGY 19 with 1-dimensional modular functor on DΓ (resp. CFT with 1-dimensional modular functor on D + Γ .) Remark 1: From a physical point of view, Moore and Seiberg =-=[29]-=- proved a result which, in the present language, says roughly that every 1-CFT gives rise to a CFT. However, this result has not yet been checked in the rigorous mathematical framework, and so, at the... |

225 |
Fusion rules and modular transformations in 2D conformal field theories
- Verlinde
- 1988
(Show Context)
Citation Context ... is that we would need a discussion of the labels. The situation is actually quite simple: The labels correspond to elements of L 0 =L where L 0 is the dual lattice, and the ‘‘Verlinde algebra’’ (cf. =-=[45]-=-, also [36]) is just the groupalgebra CL 0 =LŠ: However, since the rest of this paper avoids labels, we do not want to introduce them here, and will therefore discuss the CFT associated with a genera... |

168 | 1963]: Functorial Semantics of Algebraic Theories - Lawvere |

135 |
Modular invariance of characters of vertex operator algebras
- Zhu
(Show Context)
Citation Context ...o the form (42). Thus, a directed 0-CFT with some mild assumptions does always give rise to a VOA. Under such assumptions, similarly a directed 1-CFT gives rise to a rational VOA in the sense of [9], =-=[47]-=-. One important point in this discussion is the ‘central charge’. A VOA is, in particular, a representation of the universal central extension of the Lie algebra VectðS 1 Þ C of polynomial complex vec... |

132 |
Vertex operator algebras and
- Frenkel, Lepowsky, et al.
- 1988
(Show Context)
Citation Context ...orcherds’ proof [2] of the Moonshine conjectures on the modularity of the Thompson series, which are character series of a certain graded representation of the Monster called the Moonshine module V ♮ =-=[13]-=-. (More precisely, the statement of the conjectures is that the Thompson series are Hauptmoduln. Much work has been done on this and related topics; see e.g. [32, 23, 25, 42, 43, 44] for reference.) T... |

116 | Twisted representations of vertex operator algebras, preprint q-alg/9509005
- Dong, Li, et al.
(Show Context)
Citation Context ...ded to the form (42). Thus, a directed 0-CFT with some mild assumptions does always give rise to a VOA. Under such assumptions, similarly a directed 1-CFT gives rise to a rational VOA in the sense of =-=[9]-=-, [47]. One important point in this discussion is the ‘central charge’. A VOA is, in particular, a representation of the universal central extension of the Lie algebra VectðS 1 Þ C of polynomial compl... |

114 | Monstrous moonshine and monstrous Lie superalgebras
- Borcherds
- 1992
(Show Context)
Citation Context ...e written down, and finding simply a candidate which would satisfy them all is a non-trivial goal. First of all, elliptic cohomology should be related to, and ideally better explain, Borcherds’ proof =-=[2]-=- of the Moonshine conjectures [5] on the modularity of the Thompson series, which are character series of a certain graded representation of the Monster called the Moonshine module V y [13]. (More pre... |

97 |
Vertex algebras associated with even lattices
- Dong
- 1993
(Show Context)
Citation Context ...ow follows from Properties 1 and 2 (see the beginning of this section). It remains to be shown that the 0-CFT we have defined extends to a 1-CFT, but in view of [47], this follows from the results of =-=[8]-=-; some discussion of modularity of the lattice 0-CFTs will be carried out below in Section 6 in connection with y-series.ARTICLE IN PRESS P. Hu, I. Kriz / Advances in Mathematics 189 (2004) 325–412 3... |

94 | Non-Monstrous Moonshine
- Norton
- 1996
(Show Context)
Citation Context ...y a candidate which would satisfy them all is a non-trivial goal. First of all, elliptic cohomology should be related to, and ideally better explain, Borcherds’ proof [2] of the Moonshine conjectures =-=[5]-=- on the modularity of the Thompson series, which are character series of a certain graded representation of the Monster called the Moonshine module V y [13]. (More precisely, the statement of the conj... |

93 | Vertex Algebras and Algebraic Curves
- Frenkel, Ben-Zvi
- 2001
(Show Context)
Citation Context ...en the concepts we introduced and vertex operator algebras (VOAs), which is the sameARTICLE IN PRESS P. Hu, I. Kriz / Advances in Mathematics 189 (2004) 325–412 347 as conformal vertex algebras, see =-=[11,12]-=-. First of all, the nature of the relationshipis that both concepts are different mathematical models of related physical structures; neither structure is in general known to imply presence of the oth... |

88 |
A classification of irreducible prehomogeneous vector spaces and their relative invariants
- Sato, Kimura
- 1977
(Show Context)
Citation Context ...t all of them. For example, the smallest non-trivial irreducible representation of E(8) is the adjoint representation, so there clearly are no representations V where V − {0} would be transitive (cf. =-=[35]-=-). However, all canonical complexifications of compact Lie groups are rational varieties (by BN-decomposition). We claim that for any rational smooth variety Z over C, and any complex algebraic curve ... |

78 |
Two-dimensional conformal geometry and vertex operator algebras
- Huang
- 1997
(Show Context)
Citation Context ...rm vertex operator algebra refers to the presence of a certain element oAV2 such that the vertex operator associated with o encodes (infinitesimal) boundary reparametrizations. We refer the reader to =-=[2,11,22]-=- for details. Anyway, in this case, we may refer to the directed 0-CFT H as a Hilbert completion of the V(O)A V; however, note that the Hilbert structure on H plays a marginal role, and, in fact, can ... |

76 |
Sphere packings, lattices and groups. Third edition. Grundlehren der Mathematischen Wissenschaften, 290
- Conway, Sloane
- 1999
(Show Context)
Citation Context ...lattice. One can detect images of elements in πiB(Co0 ⋉ ΛC/Λ ′ )Q in K∗[[q]] using the θ-function of Λ similarly as we did above for E8. For example, using the standard coordinate frame u1, ..., u24 (=-=[6]-=-), we can investigate the coefficient of (174) θΛ(τ, u) at If we put 24 � j=1 uj. g2 = 1 ⎛ ⎝1 + 240 12 � n≥0 n 3 q n 1 − q n ⎞ ⎠ , g3 = 1 6 3 ⎛ ⎝−1 + 504 � n≥0 n 5 q n 1 − q n ⎞ ⎠ (omitting some 2π po... |

69 |
A Concise Course in Algebraic Topology
- May
- 1999
(Show Context)
Citation Context ...ltiplication by (120) defines a self-map S 2k S N Eþ-S N Eþ: ð122Þ Then (121) is defined as the telescope of the map (122). (A reader in need of a quick introduction to homotopy theory is referred to =-=[28]-=-.) For motivation, we look at the following classical example. Consider the inclusion Then h specifies a homotopy class h : S 2 CP 1 CCP N : bAp2S N CP N þ : Since SNCPN þ is an EN ring spectrum, b ... |

62 |
Loop groups, Oxford Mathematical Monographs
- Pressley, Segal
- 1986
(Show Context)
Citation Context ...e give more advanced examples. We give a map BG → BellHG for any simply connected simply laced group G where HG is the conformal field theory on the basic level 1 representation of the loop group LG (=-=[33]-=-). For G = E8, this affords a choice of an element ω whose image in K∗[[q]] is the discriminant form ∆ (at least up to a multiplicative constant, i.e. localized, in the sense of homotopy theory, away ... |

62 |
A course in arithmetic, Graduate Texts
- Serre
- 1973
(Show Context)
Citation Context ...the matrix 1 1 0 1 and the subgroup GðmÞSL2Z of matrices congruent to Id mod m; for mAN: To recall the modularity properties of function (144) we review some basic facts about y-functions of lattices =-=[30,31,38]-=-. This is a very special case of the theory of Siegel modular forms [30], in fact, in some sense, the ‘‘trivial’’ case, i.e. one which reduces to the classical context. By a theta series associated wi... |

54 |
String theory on Calabi-Yau manifolds
- Greene
(Show Context)
Citation Context ...ics, a notion of moduli space of CFTs; one can start trying to understand such moduli spaces by means of local deformation of conformal field theories, using truly marginal operators, or (1;1)-fields =-=[16]-=-. Note that the type of conformal field theories H we are considering is chiral (holomorphic), and hence their moduli space would be discrete.ARTICLE IN PRESS P. Hu, I. Kriz / Advances in Mathematics... |

52 |
The De of Conformal Field Theory", preprint
- Segal
- 1989
(Show Context)
Citation Context ...ucture which is also the first rigorous mathematical encoding of the physical concept of (chiral perturbative) conformal field theory (CFT). On the other hand, a more ‘‘maximalistic’’ approach to CFT =-=[36]-=- builds in modularity in the form of correlation functions on elliptic curves. This is not directly visible to VOAs, and the proof [2] takes a different route. But the modularity of CFT suggests tryin... |

36 |
The friendly giant
- Griess
- 1982
(Show Context)
Citation Context ...ch work has been done on this and related topics; see e.g. [32, 23, 25, 42, 43, 44] for reference.) The Thompson series are characters, i.e. trace series, of a certain graded module of the Monster F1 =-=[17]-=-, known as the Moonshine module [13] V ♮ = � n≥−1 Vnq n . Thus, if one wants to work in homotopy theory, one can interpret V ♮ as a map (1) BF1 → K[[q]][q −1 ] where K is K-theory. This might suggest ... |

35 |
Representations of Kac-Moody algebras and dual resonance models
- Frenkel
- 1985
(Show Context)
Citation Context ...en the concepts we introduced and vertex operator algebras (VOAs), which is the sameARTICLE IN PRESS P. Hu, I. Kriz / Advances in Mathematics 189 (2004) 325–412 347 as conformal vertex algebras, see =-=[11,12]-=-. First of all, the nature of the relationshipis that both concepts are different mathematical models of related physical structures; neither structure is in general known to imply presence of the oth... |

33 |
A nonmeromorphic extension of the moonshine module vertex operator algebra, in: Moonshine, the Monster and related topics
- Huang
- 1994
(Show Context)
Citation Context ...g 0-CFT operators of H y arise by means of averaging with respect to y0; composition and reversals of orientation of boundary components. For a good discussion of this from the VOA point of view, see =-=[21]-=-. To describe HT; one must discuss the twisted loopgroupT 1=2 S1 : This is the group of (analytic) maps such that f ð0Þ 1 f ð1Þ consider the group T 1=2 S1 ARTICLE IN PRESS P. Hu, I. Kriz / Advances... |

30 |
Vorlesungen über allgemeine Funktionentheorie
- Hurwitz
- 1964
(Show Context)
Citation Context ...t there exists a harmonic function where 0, 1 are regular points, hi : X → [0, 1] (84) (hi) −1 (0) = ci, (85) (hi) −1 (1) = � Such function is a solution to the Dirichlet problem (see Hurwitz-Courant =-=[24]-=-, 8.7). Now if we denote by Hol, Harm the sheaves of holomorphic and real harmonic functions, and by R, Z the constant sheaves, we have a short exact sequence j�=i (86) 0 → R → Hol → Harm → 0 (The sec... |

29 |
Compatibility of the dual pomeron with unitarity and the absence of ghosts in the dual resonance model, Phys
- Goddard, Thorn
- 1972
(Show Context)
Citation Context ...ory (182) (see [36], Section 9), and factoring out the kernel of <?, ? > on P . (In fact, it is desirable to consider all of this in real form, see [2] and Section 3 above.) The no ghost theorem [12],=-=[15]-=- asserts that if we consider the bigrading of M inherited from VII1,1, then the (m, n)-graded piece is R 2 if m = n = 0 Vmn otherwise where Vm is the weight m + 1 piece of V ♮ . Further, M is a Lie al... |

23 | Elliptic cohomology. Seminaire Bourbaki 1987-88 no - Segal - 1988 |

21 |
Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke
- Huang, Lepowsky
- 1999
(Show Context)
Citation Context ...tempting to go even further and consider the case when K would be a groupoid; such structures would be useful in encoding conformal field - like theories corresponding to vertex intertwining algebras =-=[21]-=-. However, note that then sets over K form a 2-category. The appropriate fully lax analogue of (14) gives, as sections over an object B of B, the set of all gerbes on a covering space of B. Gerbes, ho... |

14 |
Disorder fields in twodimensional conformal quantum field theory and N = 2 extended supersymmetry
- Zamolodchikov, Fateev
- 1986
(Show Context)
Citation Context ...cs. We drop, therefore, at least for the purpose of this Remark, the rigorous standards applied elsewhere in this paper, and use the language of physics with liberty. Recall (cf. Zamolodchikov-Fateev =-=[46]-=-) that there is, at least in physics, a notion of moduli space of CFT’s; one can start trying to understand such moduli spaces by means of local deformation of conformal field theories, using truly ma... |

14 | Power operations in elliptic cohomology and representations of loop groups
- Ando
(Show Context)
Citation Context ...ory, one can interpret V y as a map BF1-KqŠŠq 1 Š; ð1Þ where K is K-theory. This might suggest KqŠŠq 1 Š as the first candidate for elliptic cohomology. This approach was indeed pursued by Ando =-=[1]-=-, and leads to some valuable conclusions; in particular, KqŠŠq 1 Š is the ‘‘homotopical counterpart’’ of the Tate curve. However, we want to go further: the coefficient ring of the Tate curve does ... |

12 |
More on moonshine, in: Computational Group Theory
- Norton
- 1984
(Show Context)
Citation Context ...e considered as representations of the Monster. This works because the Monster’s action on V y carries through the no ghost theorem. In [2], one then uses (183) to prove Norton’s replication formulae =-=[32]-=-, which constitute the main stepin the Moonshine conjecture proof. More explicitly, one remarks that LðUÞ exp X c i ! ðUÞ=i ; ð184Þ where c i is the Adams operations on a G-representation U (here G ... |

11 | On the relationship between Monstrous Moonshine and the uniqueness of the Moonshine module - Tuite - 1995 |

10 | Monstrous moonshine from orbifolds - Tuite - 1992 |

6 |
The structure of 24 dimensional manifolds having normal bundles which lift to BO[8]. Recent Progress in Homotopy Theory
- Mahowald, Hopkins
(Show Context)
Citation Context ...eory K[[q]][q −1 ] and Adams operations. However, it would be nice to phrase these ideas is in terms of the power operations in elliptic cohomology considered by Ando [1]. We also refer the reader to =-=[18]-=- for more background and further considerations. 8. Appendix: Some homotopy theory The main point of this section is to give a Proof of Proposition 8: Let D = {z ∈ C| ||z|| ≤ 1}. We shall define a spa... |

5 |
On spin and modularity in conformal field theory
- Kriz
(Show Context)
Citation Context ...is situation. We divide this task as follows: In Section 2, we give the complete axiomatic definition of chiral CFT (and related notions needed). This greatly exceeds the rather limited step taken in =-=[26]-=-. The completely rigorous axioms are quite complicated, and involve substantially the language of 2-categories and stacks. In Section 3, we give, also in substantial detail, the construction of 1CFT’s... |

5 | Handbook of categorical algebra 1–2 - Borceux |

5 |
Algebraic cobordism and
- Snaith
- 1979
(Show Context)
Citation Context ... formally inverting, in a suitable sense, certain elements oAp BellH: Evidence in favor of such approach is given by a well known result in K-theory, where inverting the Bott class in SNCPN þ gives K =-=[39,40]-=-. With this approach to E; the construction of the map g of (2) becomes a non-trivial problem, more difficult than with homotopy theory-based definitions of E: The definition of E and construction of ... |

5 | Generalised moonshine and abelian orbifold constructions - Tuite |

4 |
Localized stable homotopy of some classifying spaces
- Snaith
- 1981
(Show Context)
Citation Context ... formally inverting, in a suitable sense, certain elements oAp BellH: Evidence in favor of such approach is given by a well known result in K-theory, where inverting the Bott class in SNCPN þ gives K =-=[39,40]-=-. With this approach to E; the construction of the map g of (2) becomes a non-trivial problem, more difficult than with homotopy theory-based definitions of E: The definition of E and construction of ... |

3 | W.F.Lawvere: Functorial semantics of algebraic theories - A |

3 |
Le symbole modéré, Inst
- Deligne
- 1991
(Show Context)
Citation Context ...s not necessary: it is merely important that some examples exist, which we have already shown. Because of this, we will proceed in somewhat less detail than above. We shall use some ideas implicit in =-=[7]-=-. First of all, we will restrict ourselves to the 0-CFT structure on H y : Second, we will only construct the Hilbert version of the twisted module HT where H is the lattice 1-CFT associated with an e... |

2 |
operations in elliptic cohomology and representations of loop groups
- Power
- 2000
(Show Context)
Citation Context ..., one can interpret V ♮ as a map (1) BF1 → K[[q]][q −1 ] where K is K-theory. This might suggest K[[q]][q −1 ] as the first candidate for elliptic cohomology. This approach was indeed pursued by Ando =-=[1]-=-, and leads to some valuable conclusions; in particular, K[[q]][q −1 ] is the “homotopical counterpart” of the Tate curve. However, we want to go further: the coefficient ring of the Tate curve does n... |

2 | Realizations of the Monster Lie algebra
- Jurisich, Lepowsky, et al.
- 1995
(Show Context)
Citation Context ...nster called the Moonshine module V ♮ [13]. (More precisely, the statement of the conjectures is that the Thompson series are Hauptmoduln. Much work has been done on this and related topics; see e.g. =-=[32, 23, 25, 42, 43, 44]-=- for reference.) The Thompson series are characters, i.e. trace series, of a certain graded module of the Monster F1 [17], known as the Moonshine module [13] V ♮ = � n≥−1 Vnq n . Thus, if one wants to... |

2 |
The Friendly Giant
- Jr
- 1982
(Show Context)
Citation Context ...duln. Much work has been done on this and related topics; see e.g. [21,25,32,42–44] for reference.) The Thompson series are characters, i.e. trace series, of a certain graded module of the Monster F1 =-=[17]-=-, known as the Moonshine module [13] V y M Vnq n : nX 1 Thus, if one wants to work in homotopy theory, one can interpret V y as a map BF1-KqŠŠq 1 Š; ð1Þ where K is K-theory. This might suggest K... |

2 |
D-brane cohomology and elliptic cohomology
- Hu, Kriz
(Show Context)
Citation Context ...ably one needs a better additive theory than the ‘‘free’’ theory (suspension spectrum). On some of these points, there have been recent clues. For example, a candidate additive theory is suggested in =-=[20]-=- via 2-vector spaces, and this theory makes contact with Rognes’ K-theory of K-theory [34]. A fascinating program has been also recently revealed by Stolz and Teichner, whereby the elliptic cohomology... |

2 |
Elliptic cohomology. The University Series in Mathematics
- Thomas
- 1999
(Show Context)
Citation Context ...tðHÞÞ-BellH: This is at present only a conjecture, but we will give some examples where it is true (see Sections 6 and 7 below). Now E should be ‘‘elliptic cohomology’’ in the sense of [37] (see also =-=[41]-=-). Our approach to the map b of (2) is to define a choice of E based on BellH: In fact, it is a ‘‘free construction’’, obtained by taking the suspension spectrum and formally inverting, in a suitable ... |

1 | symbole modére, Inst - Le - 1991 |

1 |
C.S.Lam: Lattices and θ-function identities II: Theta series
- Gannon
(Show Context)
Citation Context ...Γ8 ⊗ C] of the theta function θΓ8 (τ, u) ∈ Sym[Γ8 ⊗ C][[q]]. It is quite remarkable that in each dimension, most of these coefficients must be linearly dependent by (171). Now concretely, recall from =-=[14]-=- that for a suitable basis x1, ..., x8 of Γ8 ⊗ C, θΓ8 (τ, u) = 1 2 ( 8 � i=1 θ1(τ, xi) + 8 � i=1 θ2(τ, xi) + 8 � i=1 θ3(τ, xi) + 8 � i=1 θ4(τ, xi))s62 P. HU AND I. KRIZ where θi, i = 1, ..., 4 are the... |

1 |
I.Kriz: D-brane cohomology anfd elliptic cohomology
- Hu
(Show Context)
Citation Context ...umably one needs a better additive theory than the “free” theory (suspension spectrum). On some of these points, there have been recent clues. For example, a candidate additive theory is suggested in =-=[20]-=- via 2-vector spaces, and this theory makes contact with Rognes’ K-theory of K-theory [34]. A fascinating program has been also recently revealed by S.Stolz and P.Teichner, whereby the elliptic cohomo... |

1 |
Y.Z.Huang: A nonmeromorphic extension of the Moonshine module vertex operator algebra, Moonshine, the Monster, and related topics (South
- Soc
- 1994
(Show Context)
Citation Context ...nster called the Moonshine module V ♮ [13]. (More precisely, the statement of the conjectures is that the Thompson series are Hauptmoduln. Much work has been done on this and related topics; see e.g. =-=[32, 23, 25, 42, 43, 44]-=- for reference.) The Thompson series are characters, i.e. trace series, of a certain graded module of the Monster F1 [17], known as the Moonshine module [13] V ♮ = � n≥−1 Vnq n . Thus, if one wants to... |

1 |
V.P.Snaith: Algebraic cobordism and
- Soc
- 1979
(Show Context)
Citation Context ...ally inverting, in a suitable sense, certain elements ω ∈ π∗BellH. Evidence in favor of such approach is given by a well known result in K-theory, where inverting the Bott class in Σ ∞ CP ∞ + gives K =-=[39]-=-, [40]. With this approach to E, the construction of the map γ of (2) becomes a non-trivial problem, more difficult than with homotopy theory-based definitions of E. The definition of E and constructi... |