Questions tagged [elliptic-cohomology]
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25 questions
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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 ...
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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 ...
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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 ...
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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 ...
- 1,969
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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}\...
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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 ...
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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 (...
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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 ...
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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 $...
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(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 \...
- 2,011
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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 ...
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"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 ...
user82367
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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 $X$ ...
- 3,539
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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 ...
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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 $L \to R$ (...
- 3,539
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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 ...
- 3,539
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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 ...
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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^...
- 3,539
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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 ...
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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 ...
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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 ...
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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^\...
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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 ...
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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. ...
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What is a TMF in topology?
What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
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