group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Equivariant elliptic cohomology is, or is supposed to be, an equivariant version of elliptic cohomology, hence a higher chromatic analogue of equivariant K-theory.
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 traditional 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 dimensional quantum field theory and worldsheet string theory; this is indicated further below in
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
of $G$ regarded as an ∞-group, see at ∞-action for how that encodes actions on structures above it):
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 the algebra of 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 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.
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 (hence: “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 accommodate 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).
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). (Beware that, while this is true over the complex numers,as discussed here, it is at least subtle in the algebro geometric context of elliptic cohomology, see Jacob Lurie’s MO comment here). 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 crucial 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:
Moreover, 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.
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:
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).
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
Let $G$ be a compact Lie group. Write $T \hookrightarrow G$ for its maximal torus and $W$ for its Weyl group.
Let $E \in CRing_\infty$ be an elliptic E-∞ ring spectrum with elliptic curve $A \to Spec E$.
Write
for the derived scheme formed from the character group of the maximal torus mapped into the given elliptic curve.
This $A_G$ is the moduli scheme of semistable $G$-principal bundles over the dual elliptic curve $A^\vee$ (Ginzburg-Kapranov-Vasserot 95, (1.4.5)).
For geometry over the complex numbers and $A = \mathbb{C}/\tau$ a 2-torus, the scheme $A_G$ is the moduli space of flat connections on $A$, by the discussion at moduli space of connections – flat connections over a torus.
Wtite $L Top_G$ for the collection of G-CW complexes. Write $Orb(G)$ for the orbit category of $G$.
By Elmendorf's theorem we have a equivalence of (∞,1)-categories
Let the global points of the elliptic curve $A$ over $Spec E$ be equipped with an orientation in the sense of a non-degenerate ∞-group homomorphism of the form
Induced form this (…) is an essential geometric morphism
to the slice (∞,1)-topos over $A_G$.
Let
be the left adjoint to the (∞,1)-Yoneda embedding as discussed at function algebras on ∞-stacks.
The composite
takes a space with $G$-action to its $G$-equivariant elliptic cohomology spectrum.
under construction, tentative
In (Lurie, section 5.1) is a vague mentioning of a more general perspective, where one evaluates elliptic cohomology not just on action groupoids of a group, such as $B G$ but also on homotopy quotients of 2-groups, such as notably the string 2-group, and how that gives a more conceptual picture.
The following are some remarks on how to possibly realize this and at the same time refine it to geometric cohomology (differential cohomology). Tentative. Handle with care.
So let $G$ be a simple, simply connected compact Lie group.
Regard $\mathbf{B}G$ in Smooth∞Grpd = $Sh_\infty(SmthMfd)$. Then by the discussion at Lie group cohomology we have
The ∞-group extension classified by $k \in \mathbb{Z} \in \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}\mathbb{C}^\times)$ is the string 2-group at level $k$
This cocycle has a differential cohomology-refinement to the universal Chern-Simons 3-connection
Now given a torus $E = T^2$, regarded, for the moment, as a smooth manifold, we have the transgression of the definition cocycle
which now defines a $\mathbb{C}^\times$ bundle with connection on the moduli stack of connections on $E$. We can restrict to the moduli stack of flat connections, the phase space of $G$-Chern-Simons theory. This is the Hitchin connection.
Consider then a collection of tori $E$ parameterized trivially over some parameter space $B$.
Then the above yields
hence yields a $[B,\mathbb{C}^\times]$-bundle over the moduli space of $B$-collections of flat connections on $E$.
Now we want to consider this for the case that $B$ is a space in spectral geometry.
To that end, pass to the larger (∞,1)-topos of smooth E-∞ groupoids over the complex numbers.
Let $\mathbb{G}_m$ there denote the object which to a pair consisting of a smooth manifold $U$ and an E-∞ ring $R$ assigns
hence the tensor product of the ∞-group of units of $R$ with the underlying abelian group of smooth functions on $X$ with values in $\mathbb{C}^\times$.
Let then $A \in CAlg_\infty$ be an E-∞ ring, and take now $B = Spec(A)$. Write
for a $B$-collection of tori, now taken to be an elliptic curve over $Spec(A)$.
Since for a torus its fundamental group is isomorphic to its character group (via the canonical non-degenrarate bilinear form on both), we take the fundamental groupoid $\Pi(E)$ now to be
Then since $\mathbb{C}^\times = \mathbb{G}_m$ is the multiplicative group in this context, we have now (and there is a subtlety here…) that maps
are equivalently elements in the ∞-group of units of $A$. So we should get an $A$-(∞,1)-module bundle modulated by
Forming its space of co-sections yields, by the discussion at Thom spectrum, the $\chi$-twisted A-cohomology spectrum
And that should be the $G$-“equivariant” elliptic cohomology of the point. Actually the motivic quantization of $G$-Chern-Simons theory.
(…)
For $G$ a compact, simple and simply connected Lie group, consider the string 2-group ∞-group extension
The corresponding higher moduli stacks of flat ∞-connections on an elliptic curve $T$ form the ∞-group extension
Now passing to the 0-truncation turns the bottom piece into the moduli space of flat connections on the torus, which is $A_G$, def. , remark .
By the discussion at smooth higher holonomy the 0-truncation of the top left piece is $U(1)$, so under 0-truncation we should get a $U(1)$-principal bundle
This state of affairs is hinted at in (Lurie, section 5.1).
More in detail, notice that the string 2-group extension is modulated by a map
and the above circle-bundle is modulated by the transgression of that
(By the discussion at dcct.)
By the general discussion at quantization of Chern-Simons theory and the holograpic CS-WZW correspondence, the space of sections of this line bundle is the space of conformal blocks of the Wess-Zumino-Witten model on $T$.
(This statement also appears as (Lurie, remark 5.2)).
When restricting the above general construction to the Tate curve, then the confromal blocks become loop group representations (when over the complex numebers, at least) (Ando00, theorem 10.10).
In terms of differential geometry (dcct) consider the map
which locally sends a group elemnent $g$ to the constant principal connection on the circle with $g$ as its holonomy.
This induces an inclusion
and pulling the above WZW circle bundle back along this inclusion yields the bundle on the loop group which is the prequantum bundle whose geometric quantization yields the loop group representations of positive energy.
Algebraically, this corresponds to evaluating equivariant elliptic cohomology on the Tate curve, this is (Lurie, theorem 5.1).
In the full derived algebraic geometry the space of sections of the line bundle on the moduli space has the structure of a $K((q))$-∞-module, hence of an actual spectrum (Lurie, below remark 5.4).
(…)
Given an E-∞ ring $A$ with an oriented 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
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
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
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
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 spaces
Heuristically, this is the map that includes the trivial $Spin$-local system and its gauge transformations into the (points of the) moduli stack of all local systems.
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
This is (Lurie, theorem 5.2).
related but different: modular equivariant elliptic cohomology
In
a set of axioms was proposed that an equivariant elliptic cohomology theory should satisfy. See also
In
the proposal of Ginzburg-Kapranov-Vasserot 95 was formalized in terms of geometric morphisms of (infinity,1)-toposes.
See also
Ioanid Rosu, Equivariant Elliptic Cohomology and Rigidity, American Journal of Mathematics, Vol. 123, No. 4, Aug., 2001 (JSTOR)
Jacob Lurie, section 5.1 of A Survey of Elliptic Cohomology, in Algebraic Topology Abel Symposia Volume 4, 2009, pp 219-277
Daniel Berwick-Evans, Arnav Tripathy?, A geometric model for complex analytic equivariant elliptic cohomology, (arXiv:1805.04146)
See also the references at equivariant cohomology – References – Complex oriented cohomology theories.
That equivariant elliptic cohomology is related to representations of loop groups as equivariant K-theory is related to the representation theory of the underlying groups had long been conjectured. The idea appears in
based on
38(1):17–32, 1976/77.
took shape in
Matthew Ando, The sigma orientation for analytic circle equivariant
elliptic cohomology_ . Geom. Topol., 7:91–153, 2003 (atrXiv:math/0201092)
Matthew Ando, Power operations in elliptic cohomology and representations of loop groups Transactions of the American
and was then further refined in section 5.2 of (Lurie) and in (Gepner 05).
More is in
Relation to the Witten genus partition function of superstrings is discussed in
and specifically in the context of parameterized WZW models in
referring to
Discussion of this in cohomological quantization is in
based on
A proposal for realizing this via (2,1)-dimensional Euclidean field theories and tmf is in
Last revised on May 19, 2019 at 11:39:53. See the history of this page for a list of all contributions to it.