Questions tagged [elliptic-cohomology]

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6 votes
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How exactly does the Kreck-Stolz description of elliptic homology match the one by Totaro?

In Kreck, Matthias; Stolz, Stephan, $\mathbf H\mathbf P^2$-bundles and elliptic homology, Acta Math. 171, No. 2, 231-261 (1993). ZBL0851.55007. the $n$th elliptic homology group of a space $X$ is ...
მამუკა ჯიბლაძე's user avatar
6 votes
1 answer
253 views

What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?

H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi: ...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...
Matthew Niemiro's user avatar
6 votes
0 answers
212 views

Borel vs genuine equivariant cohomology in quantum field theory

A lot of important work in quantum field theory involves Borel equivariant cohomology of certain geometric objects, usually with the goal of computing integrals over some complicated moduli stack. In ...
Doron Grossman-Naples's user avatar
2 votes
0 answers
133 views

Localization for generalized Borel cohomology

For both equivariant de Rham cohomology and equivariant K-theory (in the "naive" or Borel sense), we have localization formulae which allow us to compute this cohomology in terms of the ...
Doron Grossman-Naples's user avatar
4 votes
0 answers
152 views

Rigidity of the TMF-valued equivariant elliptic genus

Let me preface this question by saying that I wrote it at least in part to understand its statement. As such, I hope that the reader will excuse any mistakes. $\DeclareMathOperator{\ind}{ind}\...
Bertram Arnold's user avatar
21 votes
1 answer
2k views

Why does elliptic cohomology fail to be unique up to contractible choice?

It is often stated that the derived moduli stack of oriented elliptic curves $\mathsf{M}^\mathrm{or}_\mathrm{ell}$ is the unique lift of the classical moduli stack of elliptic curves satisfying some ...
Jack Davies's user avatar
6 votes
0 answers
144 views

Which one is the long version of the Segal Bourbaki seminar article that Nora Ganter refers to on her TMF literature list?

I mean the extremely useful literature list compiled by Nora Ganter. One of the entries there is Segal: Bourbaki and the long version of the Bourbaki article What I know is the version on numdam (...
მამუკა ჯიბლაძე's user avatar
5 votes
0 answers
67 views

Does the theorem that genera vanishing on even-dim complex projective bundles are elliptic also apply for integral-valued genera?

Ochanine proved in this paper that for genera taking values in $\mathbb{Q}$-algebras, vanishing on even-dimensional projective bundles is equivalent to being an elliptic genus (i.e. a specialization ...
xir's user avatar
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7 votes
1 answer
1k views

What's special about elliptic cohomology?

Apologies for any basic mistakes in this question; I'm a beginner to this theory and don't have anyone at my institution to consult for advice. What I mean is, if you take an elliptic curve $E$ over $...
xir's user avatar
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11 votes
1 answer
336 views

(Pre)orientation vs. formal completion

Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \...
A Rock and a Hard Place's user avatar
20 votes
2 answers
2k views

elliptic curves and group cohomology

Recently, I've been trying to understand Jacob Lurie's 2-equivariant elliptic cohomology a bit better than I had in the past. From what I can tell, the fragment of the story that only deals with ...
André Henriques's user avatar
18 votes
0 answers
327 views

"High-concept" explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
user avatar
16 votes
0 answers
1k views

How is an $S^1$-equivariant elliptic cohomology theory affected as we continuously vary the underlying elliptic curve?

Grojnowski constructs a $S^1$-equivariant cohomology theory over a complex elliptic curve $E$, designed to trivially satisfy: $$E^*_{S^1}(pt) = E$$ The functor $E^*_{S^1}(-)$ takes in a space $X$ ...
Catherine Ray's user avatar
13 votes
1 answer
820 views

Can we construct a Baas-Sullivan presentation of TMF?

Quick Review: The Baas-Sullivan construction cones off generators $\alpha_1, ..., \alpha_n \in \pi_*(MU)$ from $MU$ to get a new spectrum $MU/(\alpha_1, ..., \alpha_n)$, which is isomorphic to some ...
Catherine Ray's user avatar
19 votes
1 answer
1k views

What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$. Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ (...
Catherine Ray's user avatar
7 votes
1 answer
661 views

What is an example of a formal group law that is Landweber-exact but not flat?

Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v_1x^{p^1} + ... + v_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact ...
Catherine Ray's user avatar
14 votes
2 answers
938 views

What's an example of 2 elliptic curves with the same ground ring s.t. their associated cohomology theories detect different things?

My understanding is that a complex-oriented spectrum is a ring spectrum $E$ with a map $MU \to E$. Analogously, a ring with a formal group law is a ring $R$ with a map $L \to R$ (where $L$ is the ...
Catherine Ray's user avatar
9 votes
1 answer
571 views

Must we know $MU^*(X)$ in order to compute $Ell^*(X)$?

Let $Ell^*(X)$ be the elliptic cohomology theory (associated to a given elliptic curve $E$) of a nice space $X$. Recall the Landweber-Ravenel-Stong construction: $MU^*(X) \otimes_{MU^*} R \simeq Ell^...
Catherine Ray's user avatar
9 votes
0 answers
534 views

Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...
Qiaochu Yuan's user avatar
35 votes
2 answers
3k views

What do loop groups and von Neumann algebras have to do with elliptic cohomology?

Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways: Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of ...
Akhil Mathew's user avatar
  • 25.3k
17 votes
1 answer
686 views

Character of parity-twisted supersymmetric VOA module -- question inspired by the Stolz-Teichner program

I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader: Topological modular forms ($TMF$) is a generalized cohomology theory whose ...
André Henriques's user avatar
9 votes
2 answers
732 views

Elliptic genus for manifolds with boundary

Let M be a closed spin manifold of dimension $d$. One form of the elliptic genus of $M$ is $$ F(q)=q^{-d/8} \hat A(M) {\rm ch} \otimes_{k=1/2,3/2,\cdots} \Lambda_{q^k}T \otimes_{\ell=1}^\infty S_{q^\...
Jeff Harvey's user avatar
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17 votes
2 answers
2k views

Virasoro action on the elliptic cohomology

I'm trying to understand better the mathematical notion of elliptic cohomology. Note that I only know the physics definition of the elliptic genus given in Witten's paper. Let $X$ be a Calabi-Yau ...
Yuji Tachikawa's user avatar
15 votes
3 answers
2k views

complex cobordism from formal group laws?

Reading Ravenel's "green book", I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. ...
Thomas Riepe's user avatar
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22 votes
3 answers
2k views

What is a TMF in topology?

What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
Ilya Nikokoshev's user avatar