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stub for equivariant elliptic cohomology, for the moment just to record the references given there
added to equivariant elliptic cohomology a few words on the main statements in David Gepner’s old thesis. (this may need a bit of fine-tuning, handle with care for the moment)
Is it really correct to call $Psh_\infty(Orb(G))$ the (∞,1)-topos of $G$-∞-actions? I recall seeing this on other nLab pages as well, but it doesn’t seem consistent with the definition at ∞-action.
Yes, that’s one part that needs fine-tuning. I keep being confused about this point. I’ll see that I’ll do something about it…
Okay, I have edited a bit. Hopefully I am asymptoting to something sensible…
It still seems wrong to call $\mathrm{Psh}_\infty(\mathrm{Orb}(G))$ the $\infty$-category of $G$-actions, given how “action” is described on its own page. If it were up to me, I would call it the $\infty$-category of “$G$-equivariance”, and would reserve “$G$-action” for $\mathrm{Psh}_\infty(G)$.
Also, the inclusion $L\mathrm{Top}_G\to \mathrm{Psh}_\infty(\mathrm{Orb}(G))$ of $G$-CW complexes is an equivalence of ($\infty$,1)-categories.
A related question: what does it classify? We know $\mathrm{Psh}_\infty(G)$ is the classifying $\infty$-topos for $G$-torsors. I don’t know a good name for whatever it is $\mathrm{Psh}_\infty(\mathrm{Orb}(G))$ is a classifying $\infty$-topos of. (Geometric morphisms $f\colon \mathcal{E}\to \mathrm{Psh}_\infty(\mathrm{Orb}(G))$ correspond to functors $F\colon \mathrm{Orb}(G)\to \mathcal{E}$ such that (i) $F(G/G)\approx*$, and (ii) $\coprod_{HgK\in H\backslash G/K} F(G/D_g) \approx F(G/H)\times F(G/K)$, where $D_g=H\cap gKg^{-1}$.)
Could you say what the “category of G-actions” is in theorem 1 of David Gepner’s thesis, which the theorem says is fully embedded into $PSh_\infty(Orb(G))$?
It’s not clear how to interpret that theorem (which is perhaps the same as Proposition 7 in 2.3.1). If I read it as a theorem about $(\infty,1)$-categories, we are never told how to interpret $\mathcal{S}^G$ as an $(\infty,1)$-category ($\mathcal{S}^G$ is the category of spaces with $G$-actions, but what are the weak equivalences?)
I also see that David has no hypotheses on $G$ here (it is a general topological group). For compact $G$, the relevant inclusion (with the right weak equivalences) should be an equivalence.
started adding some remarks at Properties on how conformal blocks and loop group representations appear in equivariant elliptic cohomology. effectively a higher cohesive comment on Lurie’s higher algebraic indications in section 5 of the “Survey”
(to be expanded…)
So here $\mathbf{B} String$ has an implicit dependence on a group $G$? I’m not sure I ever realised that. It only appears right at the end of the string 2-group page here
$\mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G) \,,$I see, I have now added a clarifying comment on that right at the beginning of string 2-group.
In the little time that I find these days, with Joost Nuiten we are exploring some aspects of equivariant elliptic cohomology. One impression that became stronger as we did so is that there is a beautiful conceptual story here while however much of the literature feels conceptually a bit mysterious and sometimes without that story seems unnecessarily technically baroque.
Today I couldn’t stop myself any longer and tried my hands on what is meant (but possibly fails) to be a bit of an explanation of what’s conceptually going on. I wrote that into
While I think this improves over what was there before (hence: that it is better than nothing :-) there will be plenty of room to improve this further. But maybe it’s a start. Currently (but that will change, better see the entry) it reads as reproduced now:
{#Idea}
As usual in equivariant cohomology, there is a “naive” version and refinements thereof, and typically it is these refinements that one is really interested in. The usual motivation of these from algebraic topology/homotopy theory are indicated below in
Despite that motivation, the precise nature of the resulting “genuine” equivariant elliptic cohomology may tend to seem a bit mysterious and also a bit baroque in its technical ingredients, some of which may appear a bit unexpected in the literature. A clear conceptual picture of what equivariant elliptic cohomology is about is obtained by regarding it as encoding aspects of low dimemsional quantum field theory and worldsheet string theory; this is indicated further below in
{#MotivationFromAlgebraicTopology}
Given any cohomology theory $E$ which may be evaluated on arbitrary topological spaces, then for $G$ a compact Lie group the “naive” $G$-equivariant E-cohomology of the point is the $E$-cohomology of the classifying space $B G$ of $G$ (which is equivalently the delooping
$B G \simeq \ast //G$of $G$ regarded as an ∞-group, see at ∞-action for how that encodes actions on structures above it):
$E_G^\bullet(\ast)_{naive} \coloneqq E^\bullet(B G) \,.$In a discussion in the context of geometric homotopy theory it is clear what is “naive” about this definition: since $G$ has geometric structure of which $B G$ remembers only the underlying bare homotopy type, one would instead want to use the something like the smooth stack $\mathbf{B}G$ (the moduli stack of $G$-principal bundle), then somehow make good sense of $\mathbf{E}^\bullet(\mathbf{B}G)$ where now $\mathbf{E}$ is some sheaf of spectra and then declare this to be the actual $G$-equivariant $E$-cohomology.
The traditional argument however proceeds as follows: if $E$ is a complex oriented cohomology theory then (essentially by definition) for $G = U(1)$ the circle group then $E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$ is a formal power series which one may think of as the algebra of functions on the formal neighbourhood of a point in some larger space $M_{S^1}$.
For instance in the simpler case of equivariant K-theory this has long been well understood: here the genuine $U(1)$-equivariant cohomology of the point is the representation ring $K_{U(1)}(\ast) \simeq \mathbb{Z}[ [ t^{-1}, t] ]$ which happens to be the algebra of functions on the multiplicative group; while by complex orientation the naive equivariant cohomology $K^\bullet(B U(1)) \simeq \mathbb{Z}[ [t] ]$ is equivalently the algebra of functions on (just) the formal multiplicative group.
Based on this one may want to consider an $E$-∞-line bundle over the full space $M_{S^1}$ and take the genuine $E$-equivariant cohomology to be the global sections of that. (Specifically in elliptic cohomology that space $M_{S^1}$ is equivalent to the elliptic curve $C$ that gives the theory its name, but in some sense discussed below the spaces $M_{S^1}$ and $C$ arise conceptually differently and it is a fairly deep coincidence that they are in fact equivalent, which one may want to remember.)
In this way equivariant elliptic cohomology was defined in (Grojnowski 94, Ginzburg-Kapranov-Vasserot 95, ), see also (Ando 00, sections II.8, II.9).
More generally then genuine $G$-equivariant elliptic cohomology should assign to every $G$-action on some space $X$ a sheaf $\mathcal{F}$ of algebras over the $G$-equivariant cohomology of the point, and then the $G$-equivariant elliptic cohomology of $X$ should be the global sections of this.
While this can be made to work, it remains maybe unclear what these spaces $M_G$ “mean” and what makes them related to equivariance and elliptic cohomology. Specifically, $M_G$ turns out to be essentially the moduli space of flat connections ($G$-principal connections) on the given elliptic curve (see remark \ref{ModuliSpaceOfFlatConnections} below), which suggests strong relations to Chern-Weil theory that are not apparent here. That is considerably clarified by regarding elliptic cohomology as the coefficients for cohomological quantization of 3d and 2d quantum field theory, to which we now turn.
[ continued in next comment… ]
[ … continuation from previous comment ]
{#InterpretationInQuantumFieldTheory}
We now try to give a maybe more conceptual explanation of what genuine equivariant (and twisted) elliptic cohomology is about, when regarded over all elliptic curves (hencde: “genuine equivariant twisted tmf”).
The conceptual role of plain elliptic cohomology (not equivariant) was considerably clarified when (Witten 87) identified the elliptic genus (an element in the elliptic cohomology of a point) with the (large volume limit of) the partition function of a 2d superconformal field theory – the worldsheet quantum field theory of the “superstring” – where the worldsheet Riemann surface of the string is identified with the given elliptic curve.
If the superstring here is specifically the heterotic string then its dynamics and hence its partition function depends in general not just on the target spacetime $X$ (of which it yields the elliptic genus) but also on a background gauge field for some gauge group $G$, underlying which is a $G$-principal bundle over that spacetime. In (Kefeng Liu, 95) a succinct description of these “twisted” elliptic genera, twisted by a $G$-principal bundle, was given in terms of Kac-Weyl characters of associated loop group bundles. In (Distler-Sharpe 07) the chiral WZW-model part of the heterotic string 2d SCFT which emobodies the effect of this background gauge bundle was realized geometrically as a bundle of parameterized WZW models over $X$, and (Ando 07) highlighted (see Distler-Sharpe 07, section 8.5) that this provides the string theoretic interpretation of (Kefeng Liu, 95), in particular (Ando 07) indicates that the corresponding twisted Witten genus lands in $G$-equivariant elliptic cohomology.
Now in the special case that $X$ here is the point, then any parameterized WZW model over $X$ is just the plain single WZW model, while the plain Witten genus of $X$ vanishes. So in this case the interpretation of (Ando 07) says that the partition function of the $G$-WZW model should be an element in the $G$-equivariant elliptic cohomology of the point. But that partition function is an element in the space of conformal blocks of the WZW-model over a torus worldsheet, hence over a complex elliptic curve. Therefore the $G$-equivariant elliptic cohomology of the point should accomodate the conformal blocks of the WZW model over the given elliptic curve. (See also below at Properties – Relation to conformal blocks).
Next, by the holographic principle of the 3dCS/2dWZW-correspondence, the space of conformal blocks of the WZW model on a surface is identified with the space of quantum states of Chern-Simons theory over that surface. This in turn, by the general rules of geometric quantization and specifically by the discussion at quantization of 3d Chern-Simons theory, is the space of holomorphic sections of a prequantum line bundle over the moduli space of flat connections ($G$-principal connections) $M_G$ over the given elliptic curve. And that is indeed what $G$-equivariant elliptic cohomology assigns to the point.
In other words, universal $G$-equivariant elliptic cohomology (meaning: we vary over the moduli space of elliptic curves), hence $G$-equivariant tmf of the point, is essentially the modular functor of 3d Chern-Simons theory. This last statement appears as (Lurie 09, remark 5.2). By the above reasoning via (Ando 07) and using the 3dCS/2dWZW holographic correspondence we also have the interpretation of $G$-equivariant tmf (universal $G$-equivariant elliptic cohomology) over a more general space $X$: the space of conformal blocks of a bundle of parameterized WZW models over $X$, regarded pointwise as the gauge coupling part of the twisted Witten genus.
Here all the statements on the QFT/string theory side involve a parameter called the “level”, which is the characteristic class of the universal Chern-Simons circle 3-bundle that is the prequantum 3-bundle governing the 3d Chern-Simons theory (whose transgression to the moduli space of flat connections is the “theta”-prequantum line bundle there). On the cohomological side this corresponds to a twist of the cohomology theory.
A formal systematic discussion of this story in cohomological quantization is going to be in (Nuiten-S.). It essentially amounts to the discussion of diagram (0.0.4 b)).
[ end of reproduced text ]
Could there be a global equivariant elliptic cohomology?
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 tmf of 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 functor of 3d G-Chern-Simons theory over genus-1 surfaces/elliptic curves_ , together with the quantization-process of that to the actual modular functor_ .
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.
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 induces the 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:
David:
Could there be a global equivariant elliptic cohomology?
The theory that Lurie describes is by construction a globally equivariant theory.
Charles: or it would be if there were more than the sketch in the survey? Maybe you have seen more details.
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.
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?
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.
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 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 $G \to O$.
I have forwarded that question to MO here.
I see there’s a relevant paper recently out:
I’ll add that to the references.
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