nLab equivariant elliptic cohomology





Special and general types

Special notions


Extra structure





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

Motivation from algebraic topology

Given any cohomology theory EE which may be evaluated on arbitrary topological spaces, then for GG a compact Lie group the “naive” GG-equivariant E-cohomology of the point is the EE-cohomology of the classifying space BGB G of GG (which is equivalently the delooping

BG*//G B G \simeq \ast //G

of GG regarded as an ∞-group, see at ∞-action for how that encodes actions on structures above it):

E G (*) naiveE (BG). 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 GG has geometric structure of which BGB G remembers only the underlying bare homotopy type, one would instead want to use the something like the smooth stack BG\mathbf{B}G (the moduli stack of GG-principal bundle), then somehow make good sense of E (BG)\mathbf{E}^\bullet(\mathbf{B}G) where now E\mathbf{E} is some sheaf of spectra and then declare this to be the actual GG-equivariant EE-cohomology.

The traditional argument however proceeds as follows: if EE is a complex oriented cohomology theory then (essentially by definition) for G=U(1)G = U(1) the circle group then E (BU(1))E (*)[[c 1 E]]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 1M_{S^1}.

For instance in the simpler case of equivariant K-theory this has long been well understood: here the genuine U(1)U(1)-equivariant cohomology of the point is the representation ring K U(1)(*)[[t 1,t]]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 (BU(1))[[t]]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 EE-∞-line bundle over the full space M S 1M_{S^1} and take the genuine EE-equivariant cohomology to be the global sections of that. (Specifically in elliptic cohomology that space M S 1M_{S^1} is equivalent to the elliptic curve CC that gives the theory its name, but in some sense discussed below the spaces M S 1M_{S^1} and CC 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 GG-equivariant elliptic cohomology should assign to every GG-action on some space XX a sheaf \mathcal{F} of algebras over the GG-equivariant cohomology of the point, and then the GG-equivariant elliptic cohomology of XX should be the global sections of this.

While this can be made to work, it remains maybe unclear what these spaces M GM_G “mean” and what makes them related to equivariance and elliptic cohomology. Specifically, M GM_G turns out to be essentially the moduli space of flat connections (GG-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.

Interpretation in Quantum field theory/String theory

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 XX (of which it yields the elliptic genus) but also on a background gauge field for some gauge group GG, underlying which is a GG-principal bundle over that spacetime. In (Kefeng Liu, 95) a succinct description of these “twisted” elliptic genera, twisted by a GG-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 XX, 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 GG-equivariant elliptic cohomology.

Now in the special case that XX here is the point, then any parameterized WZW model over XX is just the plain single WZW model, while the plain Witten genus of XX vanishes. So in this case the interpretation of (Ando 07) says that the partition function of the GG-WZW model should be an element in the GG-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 GG-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 (GG-principal connections) M GM_G over the given elliptic curve. And that is indeed what GG-equivariant elliptic cohomology assigns to the point.

In other words, universal GG-equivariant elliptic cohomology (meaning: we vary over the moduli space of elliptic curves), hence GG-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 GM_G on the given elliptic curve). (Beware that, while this is true over the complex numbers, 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 CC and suitable Lie group GG 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 GG-equivariant tmf (universal GG-equivariant elliptic cohomology) over a more general space XX: the space of conformal blocks of a bundle of parameterized WZW models over XX, 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 tmftmf 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 GM_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 BGB 3U(1)\mathbf{B}G \to \mathbf{B}^3 U(1). This exhibits a homomorphism of smooth infinity-group GB 2U(1)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 B 2U(1)\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 nn-dimensional XX

nnCalabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
n=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian


1-Equivariant elliptic cohomology

Let GG be a compact Lie group. Write TGT \hookrightarrow G for its maximal torus and WW for its Weyl group.

Let ECRing E \in CRing_\infty be an elliptic E-∞ ring spectrum with elliptic curve ASpecEA \to Spec E.



A G[[T,𝕋],A]/WSch /SpecE A_G \coloneqq \left[\left[T,\mathbb{T}\right],A\right]/W \;\; \in Sch_{/Spec E}

for the derived scheme formed from the character group of the maximal torus mapped into the given elliptic curve.


This A GA_G is the moduli scheme of semistable GG-principal bundles over the dual elliptic curve A A^\vee (Ginzburg-Kapranov-Vasserot 95, (1.4.5)).


For geometry over the complex numbers and A=/τA = \mathbb{C}/\tau a 2-torus, the scheme A GA_G is the moduli space of flat connections on AA, by the discussion at moduli space of connections – flat connections over a torus.

Wtite LTop GL Top_G for the collection of G-CW complexes. Write Orb(G)Orb(G) for the orbit category of GG.

By Elmendorf's theorem we have a equivalence of (∞,1)-categories

LTop GPSh (Orb(G)). L Top_G \stackrel{\simeq}{\longrightarrow} PSh_\infty(Orb(G)) \,.

Let the global points of the elliptic curve AA over SpecESpec E be equipped with an orientation in the sense of a non-degenerate ∞-group homomorphism of the form

BU(1)A(SpecE) B U(1) \longrightarrow A(Spec E)

Induced form this (…) is an essential geometric morphism

PSh (Orb(G))() GASh (Aff E) /A G PSh_\infty(Orb(G)) \stackrel{\overset{(-)\otimes_G A}{\longrightarrow}}{\stackrel{\overset{}{\leftarrow}}{\underset{}{\longrightarrow}}} Sh_\infty(Aff_E)_{/A_G}

to the slice (∞,1)-topos over A GA_G.

(Gepner 05, theorem 3)



𝒪:Sh (Aff E)Aff EEAlg op \mathcal{O} \;\colon\; Sh_\infty(Aff_E) \longrightarrow Aff_E \simeq E Alg^{op}

be the left adjoint to the (∞,1)-Yoneda embedding as discussed at function algebras on ∞-stacks.


The composite

LTop GPSh (Orb(G))() GASh (Aff E) /A G𝒪EAlg op L Top_G \hookrightarrow PSh_\infty(Orb(G)) \stackrel{(-)\otimes_G A}{\longrightarrow} Sh_\infty(Aff_E)_{/A_G} \stackrel{\mathcal{O}}{\longrightarrow} E Alg^{op}

takes a space with GG-action to its GG-equivariant elliptic cohomology spectrum.

(Gepner 05, theorem 4)

2-Equivariant elliptic cohomology

under construction, tentative

Lurie, section 5.1 makes a vague mentioning of a more general perspective, where one evaluates elliptic cohomology not just on action groupoids of a group, such as BGB G but also on homotopy quotients by actions 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 GG be a simple, simply connected compact Lie group.

Regard BG\mathbf{B}G in Smooth∞Grpd = Sh (SmthMfd)Sh_\infty(SmthMfd). Then by the discussion at Lie group cohomology we have:

π 0H(BG,B ×)H(BG,K(,4)). \pi_0 \mathbf{H}\big( \mathbf{B}G ,\, \mathbf{B}\mathbb{C}^\times \big) \simeq H\big( B G ,\, K(\mathbb{Z},4) \big) \simeq \mathbb{Z} \,.

The ∞-group extension classified by kπ 0H(BG,B ×)k \in \mathbb{Z} \in \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}\mathbb{C}^\times) is the string 2-group at level kk

B × BString k(G) BG kc B 3 × \array{ \mathbf{B}\mathbb{C}^\times &\longrightarrow& \mathbf{B}String_k(G) \\ && \downarrow \\ && \mathbf{B}G &\stackrel{k\mathbf{c}}{\longrightarrow}& \mathbf{B}^3 \mathbb{C}^\times }

This cocycle has a differential cohomology-refinement to the universal Chern-Simons 3-connection

kL:BG connB 3 conn × k \mathbf{L} \;\colon\; \mathbf{B}G_{conn} \longrightarrow \mathbf{B}^3 \mathbb{C}^\times_{conn}

Now given a torus E=T 2E = T^2, regarded, for the moment, as a smooth manifold, we have the transgression of the defining cocycle

exp(i EkL):[E,BG conn][E,kL][E,B 3 conn ×]exp(i E())B conn × \exp\left( \tfrac{i}{\hbar} \textstyle{\int}_{E} k \mathbf{L} \right) \;\colon\; [E, \mathbf{B}G_{conn}] \stackrel{[E, k \mathbf{L}]}{\longrightarrow} [E, \mathbf{B}^3 \mathbb{C}^\times_{conn}] \stackrel{\exp\left(\tfrac{i}{\hbar} \int_{E}(-)\right)}{\longrightarrow} \mathbf{B}\mathbb{C}^\times_{conn}

which now defines a ×\mathbb{C}^\times bundle with connection on the moduli stack of connections on EE. We can restrict to the moduli stack of flat connections, the phase space of GG-Chern-Simons theory. This is the Hitchin connection.

Consider then a collection of tori EE parameterized trivially over some parameter space BB.

E×BB. E \times B \to B \,.

Then the above yields

[(ΠE)×B,BG]B[B, ×] \big[ (\Pi E) \times B, \mathbf{B}G \big] \stackrel{}{\longrightarrow} \mathbf{B}[B,\mathbb{C}^\times]

hence yields a [B, ×][B,\mathbb{C}^\times]-bundle over the moduli space of BB-collections of flat connections on EE.

Now we want to consider this for the case that BB is a space in spectral geometry.

To that end, pass to the larger (∞,1)-topos of smooth E-∞ groupoids over the complex numbers.

Let 𝔾 m\mathbb{G}_m there denote the object which to a pair consisting of a smooth manifold UU and an E-∞ ring RR assigns

𝔾 m:(U,R)GL 1(R)C (U, ×) \mathbb{G}_m \;\colon\; (U, R) \mapsto GL_1(R) \otimes C^\infty(U,\mathbb{C}^\times)

hence the tensor product of the ∞-group of units of RR with the underlying abelian group of smooth functions on XX with values in ×\mathbb{C}^\times.

Let then ACAlg A \in CAlg_\infty be an E-∞ ring, and take now B=Spec(A)B = Spec(A). Write

ESpec(A) E \to Spec(A)

for a BB-collection of tori, now taken to be an elliptic curve over Spec(A)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 Π(E)\Pi(E) now to be

B[E,𝔾 m]. \mathbf{B}[E, \mathbb{G}_m] \,.

Then since ×=𝔾 m\mathbb{C}^\times = \mathbb{G}_m is the multiplicative group in this context, we have now (and there is a subtlety here…) that maps

Spec(A)𝔾 m Spec(A) \longrightarrow \mathbb{G}_m

are equivalently elements in the ∞-group of units of AA. So we should get an AA-(∞,1)-module bundle modulated by

χ:[B[E,𝔾 m],BG]BGL 1(A). \chi \,\colon\, \big[ \mathbf{B}[E,\mathbb{G}_m], \mathbf{B}G \big] \longrightarrow \mathbf{B}GL_1(A) \,.

Forming its space of co-sections yields, by the discussion at Thom spectrum, the χ\chi-twisted A-cohomology spectrum

A χ([B[E,𝔾 m],BG]). A^\chi \big( [ \mathbf{B}[E,\mathbb{G}_m], \mathbf{B}G ] \big) \,.

And that should be the GG-“equivariant” elliptic cohomology of the point. Actually the motivic quantization of GG-Chern-Simons theory.



Relation to conformal blocks of the WZW model

For GG a compact, simple and simply connected Lie group, consider the string 2-group ∞-group extension

B 2U(1) BString BG. \array{ \mathbf{B}^2 U(1) &\to& \mathbf{B}String \\ && \downarrow \\ && \mathbf{B}G } \,.

The corresponding higher moduli stacks of flat ∞-connections on an elliptic curve TT form the ∞-group extension

[Π(T),B 2U(1)] [Π(T),BString] [Π(T),BG]. \array{ [\Pi(T),\mathbf{B}^2 U(1)] &\to& [\Pi(T),\mathbf{B}String] \\ && \downarrow \\ && [\Pi(T), \mathbf{B}G] } \,.

Now passing to the 0-truncation turns the bottom piece into the moduli space of flat connections on the torus, which is A GA_G, def. , remark .

By the discussion at smooth higher holonomy the 0-truncation of the top left piece is U(1)U(1), so under 0-truncation we should get a U(1)U(1)-principal bundle

U(1) τ 0[Π(T),BString] A G. \array{ U(1) &\longrightarrow& \tau_0 [\Pi(T),\mathbf{B}String] \\ && \downarrow \\ && A_G } \,.

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

c:BG connB 3U(1) conn \mathbf{c} \;\colon\; \mathbf{B}G_{conn} \longrightarrow \mathbf{B}^3 U(1)_{conn}

and the above circle-bundle is modulated by the transgression of that

exp(i Tc):[T,BG conn][T,c][T,B 3U(1) conn]exp(i())BU(1) conn. \exp\left( \tfrac{i}{\hbar} \int_{T} \mathbf{c} \right) \;\colon\; [T, \mathbf{B}G_{conn}] \stackrel{[T,\mathbf{c}]}{\longrightarrow} [T, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(\tfrac{i}{\hbar} (-) )}{\longrightarrow} \mathbf{B}U(1)_{conn} \,.

(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 TT.

(This statement also appears as (Lurie, remark 5.2)).

Relation to loop group representations

When restricting the above general construction to the Tate curve, then the conformal blocks become loop group representations (when over the complex numbers, at least, Ando00, theorem 10.10).

In terms of differential geometry (dcct) consider the map

G[S 1,BG conn] G \longrightarrow [S^1, \mathbf{B}G_{conn}]

which locally sends a group elemnent gg to the constant principal connection on the circle with gg as its holonomy.

This induces an inclusion

[S 1,G][S 1,[S 1,BG conn]][T,BG conn] [S^1, G] \hookrightarrow [S^1, [S^1, \mathbf{B}G_{conn}]] \simeq [T, \mathbf{B}G_{conn}]

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))K((q))-∞-module, hence of an actual spectrum (Lurie, below remark 5.4).


Relation to the Chern-Simons \infty-line bundle on BG\mathbf{B}G

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

On the one hand there is the bundle classified by

J A:BSpinBOJBGL 1(𝕊)BGL 1(A), J_A \;\colon\; B Spin \stackrel{}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S}) \longrightarrow B GL_1(A) \,,

where 𝕊\mathbb{S} is the sphere spectrum, GL 1()GL_1(-) the ∞-group of units-construction and JJ the J-homomorphism. (This is what appears as 𝒜 s\mathcal{A}_s in Lurie, middle of p.38). Notice that by (Ando-Blumberg-Gepner 10, section 8), for the case A=A = tmf this is equivalently the AA-∞-line bundle associated to the universal Chern-Simons line 3-bundle

A(12p 1):BSpin12p 1B 4σ˜BGL 1(A), 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 12p 1\tfrac{1}{2}p_1 is the first fractional Pontryagin class and σ˜\tilde \sigma is an adjunct of the string orientation of tmf.

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

θ:Loc Spin(Σ)B𝔾 m \theta \;\colon\; Loc_{Spin}(\Sigma) \longrightarrow \mathbf{B} \mathbb{G}_m

on the derived moduli stack of flat connections Loc Spin(Σ)Loc_{Spin}(\Sigma) (where in (Lurie) Loc Spin(Σ)Loc_{Spin}(\Sigma) is denoted M SpinM_{Spin}). Evaluating this bundle on global points yields the AA-∞-line bundle

Γ Spec(A)(θ):Γ Spec(A)(Loc Spin(Σ))BGL 1(A). \Gamma_{Spec(A)}(\theta) \;\colon\; \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma)) \longrightarrow B GL_1(A) \,.

So there are a priori two AA-\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

ϕ:BSpinΓ Spec(A)(Loc Spin(Σ)). \phi \;\colon\; B Spin \longrightarrow \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma)) \,.

Heuristically, this is the map that includes the trivial SpinSpin-local system and its gauge transformations into the (points of the) moduli stack of all local systems.

Hence the pullback of Γ Spec(A)(Loc Spin(Σ))\Gamma_{Spec(A)}(Loc_{Spin}(\Sigma)) yields another AA-line bundle ϕ *Γ Spec(A)(θ)\phi^\ast \Gamma_{Spec(A)}(\theta) over BSpinB Spin.

These are equivalent

J Aϕ *Γ Spec(A)(θ). J_A \simeq \phi^\ast \Gamma_{Spec(A)}(\theta) \,.

This is Lurie, theorem 5.2.

Loc Spin(*) ϕ Loc Spin(*) Loc Spin(Σ) J A θ BGL 1(A) \array{ && Loc_{Spin}(\ast) \\ & \swarrow && \searrow^{\mathrlap{\phi}} \\ Loc_{Spin}(\ast) && \swArrow_{\simeq} && Loc_{Spin}(\Sigma) \\ & {}_{\mathllap{J_A}}\searrow && \swarrow_{\mathrlap{\theta}} \\ && B GL_1(A) }


Elliptic cohomology


The concept of elliptic cohomology originates around:

and in the universal guise of topological modular forms in:

  • Michael Hopkins, Algebraic topology and modular forms in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (arXiv:math/0212397)


Textbook accounts:

Equivariant elliptic cohomology

On equivariant elliptic cohomology and positive energy representations of loop groups:

Relation to Kac-Weyl characters of loop group representations

The case of twisted ad-equivariant Tate K-theory:

See also:

Via derived E E_\infty-geometry

Formulation of (equivariant) elliptic cohomology in derived algebraic geometry/E-∞ geometry (derived elliptic curves):

Elliptic genera


The general concept of elliptic genus originates with:

Early development:


  • Peter Landweber, Elliptic genera: An introductory overview In: P. Landweber (eds.) Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, vol 1326. Springer (1988) (doi:10.1007/BFb0078036)

  • Kefeng Liu, Modular forms and topology, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (pdf, pdf, doi:10.1090/conm/193)

  • Serge Ochanine, What is… an elliptic genus?, Notices of the AMS, volume 56, number 6 (2009) (pdf)

The Stolz conjecture on the Witten genus:

The Jacobi form-property of the Witten genus:

  • Matthew Ando, Christopher French, Nora Ganter, The Jacobi orientation and the two-variable elliptic genus, Algebraic and Geometric Topology 8 (2008) p. 493-539 (pdf)

The identification of elliptic genera, via fiber integration/Pontrjagin-Thom collapse, as complex orientations of elliptic cohomology (sigma-orientation/string-orientation of tmf/spin-orientation of Tate K-theory):

For the Ochanine genus:

Equivariant elliptic genera

Genera in equivariant elliptic cohomology and the rigidity theorem for equivariant elliptic genera:

The statement, with a string theory-motivated plausibility argument, is due to Witten 87.

The first proof was given in:

Reviewed in:

  • Raoul Bott, On the Fixed Point Formula and the Rigidity Theorems of Witten, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) Nonperturbative Quantum Field Theory. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) (doi:10.1007/978-1-4613-0729-7_2)

Further proofs and constructions:

On manifolds with SU(2)-action:

Twisted elliptic genera

Discussion of elliptic genera twisted by a gauge bundle, i.e. for string^c structure):

Elliptic genera as super pp-brane partition functions

The interpretation of elliptic genera (especially the Witten genus) as the partition function of a 2d superconformal field theory (or Landau-Ginzburg model) – and especially of the heterotic string (“H-string”) or type II superstring worldsheet theory has precursors in

and then strictly originates with:

Review in:

With emphasis on orbifold CFTs:


Via super vertex operator algebra

Formulation via super vertex operator algebras:

and for the topologically twisted 2d (2,0)-superconformal QFT (the heterotic string with enhanced supersymmetry) via sheaves of vertex operator algebras in

based on chiral differential operators:

In relation to error-correcting codes:

  • Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes [[arXiv:2308.12592]]
Via Dirac-Ramond operators on free loop space

Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:

Via conformal nets

Tentative formulation via conformal nets:

Conjectural interpretation in tmf-cohomology

The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space XX are the generalized cohomology of XX with coefficients in the spectrum of topological modular forms (tmf):

and the more explicit suggestion that, under this identification, the Chern-Dold character from tmf to modular forms, sends a 2d SCFT to its partition function/elliptic genus/supersymmetric index:

This perspective is also picked up in Gukov, Pei, Putrov & Vafa 18.

Discussion of the 2d SCFTs (namely supersymmetric SU(2)-WZW-models) conjecturally corresponding, under this conjectural identification, to the elements of /24\mathbb{Z}/24 \simeq tmf 3(*)=π 3(tmf) tmf^{-3}(\ast) = \pi_3(tmf) \simeq π 3(𝕊)\pi_3(\mathbb{S}) (the third stable homotopy group of spheres):

Discussion properly via (2,1)-dimensional Euclidean field theory:

See also:

Occurrences in string theory

H-string elliptic genus

Further on the elliptic genus of the heterotic string being the Witten genus:

The interpretation of equivariant elliptic genera as partition functions of parametrized WZW models in heterotic string theory:

Proposals on physics aspects of lifting the Witten genus to topological modular forms:

M5-brane elliptic genus

On the M5-brane elliptic genus:

A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with

Discussion of the resulting elliptic genus (2d SCFT partition function) originates with:

Further discussion in:

M-string elliptic genus

On the elliptic genus of M-strings inside M5-branes:

E-string elliptic genus

On the elliptic genus of E-strings as wrapped M5-branes:

On the elliptic genus of E-strings as M2-branes ending on M5-branes:

Of quiver varieties

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

following the analogous non-elliptic discussion in:

Review in:

  • Andrey Smirnov, Stable envelopes for A nA_n, A^ n\widehat A_n-quiver varieties, 2019 (pdf)

Last revised on March 8, 2024 at 17:24:44. See the history of this page for a list of all contributions to it.