I am writing out a proof that, for a good orbifold $X \!\sslash\! G$, Huan’s orbifold loop groupoid $\Lambda^{Huan}_{S^1}( X \!\sslash\! G)$ (Huan 18’s Def. 2.14 on p. 9, see Dove 19’s last line on p. 62) is equivalent to the homotopy quotient of the cofree BZ-action “restricted” along the shape unit of $S^1$:

$\Lambda^{Huan}_{S^1}\big( \mathcal{X} \big) \;\simeq\; \big[ \mathbf{B} \mathbb{Z}, \, \mathcal{X} \big] \sslash S^1 \,.$This may not be surprising, once one knows what all the ingredients “mean” – but it is a little fiddly to prove. Is this known, in some form? I don’t expect there to be a citable reference, since the existing literature is rather remote from making such statements. But maybe this has been secretly known to some expert reading here? I guess somebody like Charles Rezk might have thought about this.

]]>Huan’s “quasi-elliptic cohomology” stands out in making the definition that one might hope for from our perspective of double dimensional reduction via cyclic loop spaces: It is simply the orbifold K-theory of (what is, I think) the stacky cyclic loop space construction applied to the given domain orbifold $\mathcal{X}$ and then restricted to constant loops.

$\mathcal{L}^{const}(\mathcal{X})\sslash S^1 \longrightarrow \mathcal{L}(\mathcal{X})\sslash S^1 \,.$Huan states this most explicitly on p. 1 of his more recent “Quasi-theories”, but, up to improved notation, this is the very construction back from his thesis.

Now, I haven’t penetrated his computations yet (nor really tried to yet, just diving in now) but I am guessing the power series nature of the resulting cohomology theory is implied from this via the cyclification of any constant loop contributing $\ast \sslash S^1 \simeq B S^1$ and using $KU(B S^1) \;\simeq\; \mathbb{Z}[ [ q ] ]$.

Now I am thinking:

the restriction to constant loops may neatly be implemented by computing the cyclic loop space not with geometric loops $S^1$, but just with their shape $B \mathbb{Z}$, hence as base change (here) not along $\ast \to \mathbf{B} S^1$, but along its shape $\ast \to B S^1 \simeq \mathbf{B}^2 \mathbb{Z}$.

While this also changes the homotopy quotient of the resulting loop stack from $(-) \sslash S^1$ to $(-)\sslash (B \mathbb{Z})$, it should still yield the same cohomology theory, since $KU(B S^1) \simeq KU(B^2 \mathbb{Z})$ is sensitive only to the shape of the loops, not their geometry.

In summary, I am thinking that one more streamlining of the definition of the quasi-elliptic cohomology of an orbifold $\mathcal{X}$ would be to define it as

$KU_{orb} \big( [B \mathbb{Z}, \mathcal{X}] \sslash (B \mathbb{Z}) \big) \,,$where in the argument we have the mapping stack in $SmoothGroupoids_\infty$ (smooth $\infty$-stacks) and $B \mathbb{Z} \,\in\, Groups(Groupoids_\infty) \overset{Disc}{\hookrightarrow} Groups(SmoothGroupoids_\infty)$.

]]>removed most of the list of references, and replaced instead by `!include`

-ing the list from *elliptic cohomology – references*

added these pointers:

On equivariant elliptic cohomology of quiver varieties in relation to the AGT correspondence:

Mina Aganagic, Andrei Okounkov,

*Elliptic stable envelopes*(arXiv:1604.00423)Andrei Okounkov,

*Inductive construction of stable envelopes and applications, I. Actions of tori. Elliptic cohomology and K-theory*(arXiv:2007.09094)

following the analogous non-elliptic discussion in:

- Davesh Maulik, Andrei Okounkov,
*Quantum Groups and Quantum Cohomology*, Asterisque 408(2019) (arXiv:1211.1287, ISBN: 978-2-85629-900-5)

Review in:

- Andrey Smirnov,
*Stable envelopes for $A_n$, $\widehat A_n$-quiver varieties*, 2019 (pdf)

I have made a subsection for *References – As ad-equivariant Tate K-theory*

Hi Rune,

thanks for adding!

Maybe you see this message here and maybe you can give me a hint on the following:

I am intrigued by Cor. 3.2.5 in in Luecke 19, which is a kind of elliptic completion of the FHT theorem (I’ll write $(-)^{ʃ S^1}$ for forming inertia groupoids of topological groupoids):

$\widehat K^\tau_{S^1} \Big( (\ast \sslash G)^{ʃ S^1} \Big) \;\; \simeq \;\; \widehat R^\tau_{pos}( S^1 \ltimes L G )$This seems quite fundamental – is this known from within some other machinery, such as Gepner’s?

But the real question I have is this:

Let $X$ be a smooth Spin $2k$-fold, regarded as a topological stack and as such equipped with a tangent bundle classifier

$X \overset{ \vdash T X }{\longrightarrow} \big( \ast \sslash Spin(2k) \big)$But then we immediately have the induced morphism of inertia stacks

$\big( X^{ʃ S^1} \big) \overset{(\vdash T X)^{ʃ S^1} }{\longrightarrow} \big( \ast \sslash Spin(2k) \big)^{ʃ S^1}$and hence the induced pullback map in Tate K-theory

$\widehat K^\tau_{S^1}(X) \;\longleftarrow\; \widehat K^\tau_{S^1} \Big( (\ast \sslash G)^{ʃ S^1} \Big) \;\; \simeq \;\; \widehat R^\tau_{pos}( S^1 \ltimes L G ) \,.$But with the above elliptic FHT-like equivalence, recalled on the right, this natural construction turns positive energy loop group reps into Tate K-theory classes on the spin-manifold $X$.

Just of this type is the Witten genus in the guise of Brylinski 90.

Is this a coincidence? Or does this natural operation reconstruct the actual Brylinski-Witten genus?

]]>Added reference to Gepner-Meier, On equivariant topological modular forms (arXiv:2004.10254).

Rune Haugseng

]]>cross-linked now with *twisted ad-equivariant Tate K-theory*

and the case of finite groups:

- Thomas Dove,
*Twisted Equivariant Tate K-Theory*(arXiv:1912.02374)

added:

following

- Nitu Kitchloo, Jack Morava, Section 5 of:
*Thom Prospectra for Loopgroup representations*(arXiv:math/0404541)

added this pointer:

- Kiran Luecke,
*Completed K-theory and Equivariant Elliptic Cohomology*(arXiv:1904.00085)

I see there’s a relevant paper recently out:

- Daniel Berwick-Evans, Arnav Tripathy,
*A geometric model for complex analytic equivariant elliptic cohomology*, (arXiv:1805.04146)

I’ll add that to the references.

]]>I have forwarded that question to MO here.

]]>I am wondering if that theorem 5.2 is restricted to the case $G = \mathrm{Spin}$. That is the case needed for the discussion that follows right afterwards, but the statement of theorem 5.2 itself would at least make verbatim sense for any $G \to O$.

]]>Made a note at *equivariant elliptic cohomology* on theorem 5.2 in Lurie’s “Survey”, now here.

Given an E-∞ ring $A$ with a derived elliptic curve $\Sigma \to Spec(A)$ there are a priori two different $A$-∞-line bundles on $B Spin$.

On the one hand there is the bundle classified by

$J_A \;\colon\; B Spin \stackrel{}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S}) \longrightarrow B GL_1(A) \,,$where $\mathbb{S}$ is the sphere spectrum, $GL_1(-)$ the ∞-group of units-construction and $J$ the J-homomorphism. (This is what appears as $\mathcal{A}_s$ in Lurie, middle of p.38). Notice that by (Ando-Blumberg-Gepner 10, section 8), for the case $A =$ tmf this is equivalently the $A$-∞-line bundle associated to the universal Chern-Simons line 3-bundle

$A(\tfrac{1}{2}p_1) \;\colon\; B Spin \stackrel{\tfrac{1}{2}p_1}{\longrightarrow} B^4 \mathbb{Z} \stackrel{\tilde \sigma}{\longrightarrow} B GL_1(A) \,,$where $\tfrac{1}{2}p_1$ is the first fractional Pontryagin class and $\tilde \sigma$ is an adjunct of the string orientation of tmf.

In addition, by equivariant elliptic cohomology there is the theta line-bundle

$\theta \;\colon\; Loc_{Spin}(\Sigma) \longrightarrow \mathbf{B} \mathbb{G}_m$on the derived moduli stack of flat connections $Loc_{Spin}(\Sigma)$ (where in (Lurie) $Loc_{Spin}(\Sigma)$ is denoted $M_{Spin}$). Evaluating this bundle on global points yields the $A$-∞-line bundle

$\Gamma_{Spec(A)}(\theta) \;\colon\; \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma)) \longrightarrow B GL_1(A) \,.$So there are a priori two $A$-$\infty$-oine bundles on bare homotopy types here. But (by 2-equivariance, Lurie, bottom of p. 38) there is a canonical map between their base space

$\phi \;\colon\; B Spin \longrightarrow \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma))$and hence the pullback of $\Gamma_{Spec(A)}(Loc_{Spin}(\Sigma))$ yields another $A$-line bundle $\phi^\ast \Gamma_{Spec(A)}(\theta)$ over $B Spin$.

These are equivalent

$J_A \simeq \phi^\ast \Gamma_{Spec(A)}(\theta) \,.$This is (Lurie, theorem 5.2).

]]>I agree that Lurie’s survey doesn’t make “global” especially explicit; I don’t know much more than what is in there. But Theorem 3.21 describes an equivariant cohomology theory globally for abelian groups, and Proposition 3.26 extends it to an equivariant cohomology theory global for all compact Lie groups.

]]>Why does the idea of ’global cohesion’ not appear on the page? Is there an issue to do with stability, or is it already implicitly there?

]]>Of course David Gepner’s formulation (briefly summarized in the entry) is manifestly “global”.

I am fully certain that Jacob Lurie knows the most optimal and general formulation of equivariant elliptic cohomology. But his Survey remains pretty vague in some key aspects.

]]>Charles: or it would be if there were more than the sketch in the survey? Maybe you have seen more details.

]]>David:

Could there be a global equivariant elliptic cohomology?

The theory that Lurie describes is by construction a globally equivariant theory.

]]>To reflect this in the entry I have now added the following further paragraphs to the end of the Idea-section (see there for working links and possible more recent edits):

Now with equivariant $tmf$ identified with the quantization of Chern-Simons theory in dimension 2 this way (the modular functor together with its pre-quantum origin via geometric quantization), the physical desireability of local quantum field theory (“extended TQFT”) suggests to ask for a refinement of this also to dimensions 1 and 0, such that the higher dimensional data arises by “tracing”/transgression. There is such a local prequantum field theory refinement of 3d Chern-Simons theory, governed in dimension 0 by the universal Chern-Simons circle 3-bundle regarded as a prequantum 3-bundle. Indeed, the transgression of that to the moduli space of flat connections is precisely the prequantum bundle over $M_G$ that appears in the above discussion (e.g. FSS 12, FSS 13).

Now that universal Chern-Simons circle 3-bundle in turn is modulated by the geometric refinement of the universal second Chern class/first fractional Pontryagin class given by a map of smooth infinity-stacks of the form $\mathbf{B}G \to \mathbf{B}^3 U(1)$. This exhibits a homomorphism of smooth infinity-group $G \to \mathbf{B}^2 U(1)$ (to the circle 3-group) and so one might wonder if there is a way to “globalize” the equivariance of equivariant elliptic cohomology (in the sense of “global equivariant homotopy theory”) such that it may be evaluated also on 3-groups such as $\mathbf{B}^2 U(1)$ and such that the homomorphism above then

inducesthe previous 1-equivariant data by transgression.Such a “localization” of equivariant elliptic cohomology seems to be just what is being vaguely hinted at in (Lurie, section 5.1) under the name “2-equivariant elliptic cohomology”, we discuss this in more detail below.

Hence we arrive at a refinement of the above

slogan:

*2-Equivariant $tmf$ over the point should be 3d Chern-Simons theory in dimension 2 (hence the modular functor) localized to dimensions 1 and 0 as a local prequantum field theory and including its higher geometric quantization in these dimensions 0,1, and 2*.

David wrote above:

Could there be a global equivariant elliptic cohomology?

So something at least closely related seems to be what in Jacob Lurie’s “Survey” is called “2-equivariant elliptic cohomology”.

What I understand of this I had made a note of at *equivariant elliptic cohomology – 2-equivariance*. (I profited from discussing all this with Joost Nuiten, but if any of the following is to be criticized, the blame is all on me.)

Basically it seems that the idea is just that of extending pre-quantum 3d Chern-Simons theory all the way to the point by realizing all its assignsments as being transgressions of the universal Chern-Simons circle 3-bundle, the way we have described in terms of higher differential geometry (not in some flavor of algebraic geometry as maybe eventually one should in order to capture the worldvolume singularities and insertions properly) in *Extended higher cup-product Chern-Simons theories (schreiber)* and in *A higher stacky perspective on Chern-Simons theory (schreiber)*.

Namely, given also what I just wrote in the previous comment here, for $\Sigma$ a given elliptic curve we may think of $G$-equivariant elliptic cohomology as assigning to the point a line bundle roughly (I’ll be glossing ehre over differential refinements, for ease of discussion) over the mapping space $Maps(\Sigma, \mathbf{B}G)$. That line bundle in turn is the transgression of a circle 3-bundle modulated by a map $\mathbf{B}G \to \mathbf{B}^3 U(1)$. The codomain here is still of the form $\mathbf{B}K$ for K a smooth $\infty$-group and that morphism exhibits an $\infty$-group homomorphism $G\to K$. So one may think about forming $Maps(\Sigma, \mathbf{B}^3 U(1))$ and relating that (“globalizing” the equivariance) to the previous mapping space. Indeed, as we discuss in some differential geometric incarnation of these matters in the above articles, that’s a way to think of that transgressed line bundle.

So I think “2-equivariant elliptic cohomology”/”2-equivariant tmf” is meant to be this kind of enlargement of the story where we “globalize the equivariance” to include at least all possible (suitable) gauge groups $G$ and the coefficient 3-group $\mathbf{B}^2 U(1)$ of their localized Chern-Simons action functionals.

In view of my previous comment I believe all this is really to be thought as just another perspective on precisely the issue of quantization of “local” (“extended”, “multi-tiered”) Chern-Simons theories.

]]>Before I reply to David, I’ll announce another addition that I just made to the entry:

Where it previously just said:

In other words, universal $G$-equivariant elliptic cohomology (meaning: we vary over the moduli space of elliptic curves), hence

$G$-equivariant tmfof the point, is essentially the modular functor of 3d Chern-Simons theory. This last statement appears as (Lurie 09, remark 5.2).

I have now expanded after that as follows:

]]>But observe that actually it is a bit more: a modular functor assigns just an abstract vector space to a surface, which however is meant to be obtained by the process of quantization of 3d Chern-Simons theory, explicitly as the space of holomorphic sections of the prequantum line bundle (over phase space, which here is the moduli space of flat connections $M_G$ on the given elliptic curve). Equivariant elliptic cohomology/tmf actually remembers this quantization process and not just the resulting space of quantum states in that it actually assigns to an elliptic curve $C$ and suitable Lie group $G$ that prequantum line bundle over the moduli space of elliptic curves (or equivalently its sheaf of sections). Notice that this pre-quantum information is criucial for deep aspects in the context of 3d Chern-Simons theory and the 2d Wess-Zumino-Witten model: the holographic relation that identifies the latter as the boundary field theory of the former (explicitly so by the FRS-theorem on rational 2d CFT) needs as input not just the quantized Chern-Simons 3d TQFT, which will assign an “abstract” vector space to a surface, but needs to know how this space arose via quantization by choosing polarizations in the form of conformal structures on the elliptic curves, such as to be actually identified with a space of conformal blocks. (In the context of the Reshetikhin-Turaev construction of the Chern-Simons 3d TQFT this information is in a choice of equivalence of the given modular tensor category with the category of representations of a rational vertex operator algebra).

In summary we have as a slogan that:

$G$-Equivariant $tmf$ over the point is essentially an incarnation of the pre-quantum modular functorof 3d G-Chern-Simons theory over genus-1 surfaces/elliptic curves_ , together with the quantization-process of that to the actual modular functor_ .

Could there be a *global* equivariant elliptic cohomology?