group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic 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 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 $C$ 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 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
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
The concept of elliptic cohomology originates around:
Peter Landweber, Elliptic Cohomology and Modular Forms, in Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics Volume 1326, 1988, pp 55-68 (doi:10.1007/BFb0078038)
Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)
and in the universal guise of topological modular forms in:
Surveys:
Matthew Greenberg, Constructing elliptic cohomology, McGill University 2002 (oclc:898194373, pdf)
Paul Goerss, Topological modular forms (after Hopkins, Miller, and Lurie), Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 (arXiv:0910.5130, numdam:AST_2010__332__221_0)
Jacob Lurie, A Survey of Elliptic Cohomology, in: Algebraic Topology, Abel Symposia Volume 4, 2009, pp 219-277 (pdf, doi:10.1007/978-3-642-01200-6_9)
Charles Rezk, Elliptic cohomology and elliptic curves, Felix Klein Lectures, Bonn 2015 (web, pdf, pdf)
Textbook accounts:
Charles Thomas, Elliptic cohomology, Kluwer Academic, 2002 (doi:10.1007/b115001, pdf)
Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.), Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)
On equivariant elliptic cohomology and positive energy representations of loop groups:
Eduard Looijenga, Root systems and elliptic curves, Invent. Math.,
38(1):17–32, 1976/77 (doi:10.1007/BF01390167)
Ian Grojnowski, Delocalised equivariant elliptic cohomology (1994), in: Elliptic cohomology, Volume 342 of London Math. Soc. Lecture Note Ser., pages 114–121. Cambridge Univ. Press, Cambridge, 2007 (pdf, doi:10.1017/CBO9780511721489.007)
Victor Ginzburg, Mikhail Kapranov, Eric Vasserot, Elliptic Algebras and Equivariant Elliptic Cohomology (arXiv:q-alg/9505012)
Matthew Ando, Power operations in elliptic cohomology and representations of loop groups, Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (jstor:221905, pdf)
David Gepner, Homotopy topoi and equivariant elliptic cohomology, University of Illinois at Urbana-Champaign, 2005 (pdf)
David Gepner, Lennart Meier, On equivariant topological modular forms, (arXiv:2004.10254)
Michèle Vergne, Bouquets revisited and equivariant elliptic cohomology, International Journal of Mathematics 2021 (arXiv:2005.00312, doi:10.1142/S0129167X21400127)
Relation to Kac-Weyl characters of loop group representations
Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990 (doi:10.1016/0040-9383(90)90016-D)
Nora Ganter, The elliptic Weyl character formula, Compositio Mathematica, Vol 150, Issue 7 (2014), pp 1196-1234 (arXiv:1206.0528)
The case of twisted ad-equivariant Tate K-theory:
Nora Ganter, Section 3.1 in: Stringy power operations in Tate K-theory (arXiv:math/0701565)
Nora Ganter, Power operations in orbifold Tate K-theory, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. (arXiv:1301.2754, euclid:hha/1383943680)
Zhen Huan, Quasi-Elliptic Cohomology I, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007)
Zhen Huan, Quasi-theories (arXiv:1809.06651)
Kiran Luecke, Completed K-theory and Equivariant Elliptic Cohomology (arXiv:1904.00085)
Thomas Dove, Twisted Equivariant Tate K-Theory (arXiv:1912.02374)
See also:
Formulation of (equivariant) elliptic cohomology in derived algebraic geometry/E-∞ geometry (derived elliptic curves):
Paul Goerss, Michael Hopkins, Moduli spaces of commutative ring spectra, in Structured ring spectra, London
Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. (pdf, doi:10.1017/CBO9780511529955.009)
Paul Goerss, Michael Hopkins, Moduli problems for structured ring spectra (pdf)
Jacob Lurie, Elliptic Cohomology I: Spectral abelian varieties, 2018. 141pp (pdf)
Jacob Lurie, Elliptic Cohomology II: Orientations, 2018. 288pp (pdf)
Jacob Lurie, Elliptic Cohomology III: Tempered Cohomology, 2019. 286pp (pdf)
Jacob Lurie, Elliptic Cohomology IV: Equivariant elliptic cohomology, to appear.
The general concept of elliptic genus originates with:
Early development:
Don Zagier, Note on the Landweber-Stong elliptic genus 1986 (pdf, edoc:744944)
D.V. Chudnovsky, G.V. Chudnovsky, Elliptic modular functions and elliptic genera, Topology, Volume 27, Issue 2, 1988, Pages 163–170 (doi:10.1016/0040-9383(88)90035-3)
Mark Hovey, Spin Bordism and Elliptic Homology, Math Z 219, 163–170 (1995) (doi:10.1007/BF02572356)
Matthias Kreck, Stephan Stolz, $HP^2$-bundles and elliptic homology, Acta Math, 171 (1993) 231-261 (pdf, euclid:acta/1485890737)
Review:
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:
Stephan Stolz, A conjecture concerning positive Ricci curvature and the Witten genus, Mathematische Annalen Volume 304, Number 1 (1996) (doi:10.1007/BF01446319)
Anand Dessai, Some geometric properties of the Witten genus, in: Christian Ausoni, Kathryn Hess, Jérôme Scherer (eds.) Alpine Perspectives on Algebraic Topology, Contemporary Mathematics 504 (2009) (pdf, pdf,
The Jacobi form-property of the Witten genus:
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):
Michael Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 (pdf)
Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850 (doi:10.1007/s002220100175, pdf)
Matthew Ando, Michael Hopkins, Neil Strickland, The sigma orientation is an H-infinity map, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 (arXiv:math/0204053, doi:10.1353/ajm.2004.0008)
Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf, pdf)
For the Ochanine genus:
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:
Clifford Taubes, $S^1$-actions and elliptic genera, Comm. Math. Phys. Volume 122, Number 3 (1989), 455-526 (euclid:cmp/1104178471)
Raoul Bott, Clifford Taubes, On the Rigidity Theorems of Witten, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 (doi:10.2307/1990915)
Reviewed in:
Further proofs and constructions:
Friedrich Hirzebruch, Elliptic Genera of Level $N$ for Complex Manifolds, In: Bleuler K., Werner M. (eds) Differential Geometrical Methods in Theoretical Physics NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 250. Springer (1988) (doi:10.1007/978-94-015-7809-7_3)
I. M. Krichever, Generalized elliptic genera and Baker-Akhiezer functions, Mathematical Notes of the Academy of Sciences of the USSR 47, 132–142 (1990) (doi:10.1007/BF01156822)
Kefeng Liu, On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 343-396 (euclid:jdg/1214456221)
Kefeng Liu, On elliptic genera and theta-functions, Topology Volume 35, Issue 3, July 1996, Pages 617-640 (doi:10.1016/0040-9383(95)00042-9)
Anand Dessai, Rainer Jung, On the Rigidity Theorem for Elliptic Genera, Transactions of the American Mathematical Society Vol. 350, No. 10 (Oct., 1998), pp. 4195-4220 (26 pages) (jstor:117694)
Ioanid Rosu, Equivariant Elliptic Cohomology and Rigidity, American Journal of Mathematics 123 (2001), 647-677 (arXiv:math/9912089)
Matthew Ando, John Greenlees, Circle-equivariant classifying spaces and the rational equivariant sigma genus, Math. Z. 269, 1021–1104 (2011) (arXiv:0705.2687, doi:10.1007/s00209-010-0773-7)
On manifolds with SU(2)-action:
Discussion of elliptic genera twisted by a gauge bundle, i.e. for string^c structure):
Matthew Ando, Maria Basterra, The Witten genus and equivariant elliptic cohomology, Math Z 240, 787–822 (2002) (arXiv:math/0008192, doi:10.1007/s002090100399)
Matthew Ando, The sigma orientation for analytic circle equivariant elliptic cohomology, Geom. Topol., 7:91–153, 2003 (arXiv:math/0201092, euclid:gt/1513883094)
Matthew Ando, Andrew Blumberg, David Gepner, Section 11 of Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^*$-algebras, Proceedings of Symposia in Pure Mathematics, vol 81, American Mathematical Society, 2010 (arXiv:1002.3004)
Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, Journal of Differential Geometry 88.1 (2011): 1-39. (arXiv:1003.2325)
Jianqing Yu, Bo Liu, On the Witten Rigidity Theorem for $String^c$ Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)
Fei Han, Varghese Mathai, Projective elliptic genera and elliptic pseudodifferential genera, Adv. Math. 358 (2019) 106860 (arXiv:1903.07035)
Haibao Duan, Fei Han, Ruizhi Huang, $String^c$ Structures and Modular Invariants, Trans. AMS 2020 (arXiv:1905.02093)
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
A. N. Schellekens, Nicholas P. Warner, Anomalies and modular invariance in string theory, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)
A. N. Schellekens, Nicholas P. Warner, Anomalies, characters and strings, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)
Wolfgang Lerche, Bengt Nilsson, A. N. Schellekens, Nicholas P. Warner, Anomaly cancelling terms from the elliptic genus, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)
and then strictly originates with:
Edward Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536. (euclid:cmp/1104117076)
Edward Witten, On the Landau-Ginzburg Description of $N=2$ Minimal Models, Int. J. Mod. Phys.A9:4783-4800,1994 (arXiv:hep-th/9304026)
Toshiya Kawai, Yasuhiko Yamada, Sung-Kil Yang, Elliptic Genera and $N=2$ Superconformal Field Theory, Nucl. Phys. B414:191-212, 1994 (arXiv:hep-th/9306096, doi:10.1016/0550-3213(94)90428-6)
Sujay K. Ashok, Jan Troost, A Twisted Non-compact Elliptic Genus, JHEP 1103:067, 2011 (arXiv:1101.1059)
Matthew Ando, Eric Sharpe, Elliptic genera of Landau-Ginzburg models over nontrivial spaces, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 (arXiv:0905.1285)
Review in:
Miranda Cheng, (Mock) Modular Forms in String Theory and Moonshine, lecture notes 2016 (pdf)
Katrin Wendland, Section 2.4 in: Snapshots of Conformal Field Theory, in: Mathematical Aspects of Quantum Field Theories Mathematical Physics Studies. Springer 2015 (arXiv:1404.3108, doi:10.1007/978-3-319-09949-1_4)
Formulation via super vertex operator algebras:
Hirotaka Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras, Springer 1999 (doi:10.1007/BFb0092541)
Chongying Dong, Kefeng Liu, Xiaonan Ma, Elliptic genus and vertex operator algebras, Algebr. Geom. Topol. 1 (2001) 743-762 (arXiv:math/0201135, doi:10.2140/agt.2001.1.743)
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:
Tentative interpretation as indices of Dirac-Ramond operators as would-be Dirac operators on smooth loop space:
Edward Witten, The Index Of The Dirac Operator In Loop Space, Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (spire:245523, doi:10.1007/BFb0078045)
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, The Dirac-Ramond operator in string theory and loop space index theorems, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: Nonperturbative methods in field theory, 1987 (doi"10.1016/0920-5632(87)90110-1)
Orlando Alvarez, T. P. Killingback, Michelangelo Mangano, Paul Windey, String theory and loop space index theorems, Comm. Math. Phys., 111(1):1–10, 1987 (euclid:cmp/1104159462)
Gregory Landweber, Dirac operators on loop space, PhD thesis (Harvard 1999) (pdf)
Orlando Alvarez, Paul Windey, Analytic index for a family of Dirac-Ramond operators, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 (arXiv:0904.4748)
Tentative formulation via conformal nets:
The resulting suggestion that, roughly, deformation-classes (concordance classes) of 2d SCFTs with target space $X$ are the generalized cohomology of $X$ 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 $\mathbb{Z}/24$ $\simeq$ $tmf^{-3}(\ast) = \pi_3(tmf)$ $\simeq$ $\pi_3(\mathbb{S})$ (the third stable homotopy group of spheres):
Davide Gaiotto, Theo Johnson-Freyd, Edward Witten, p. 17 of: A Note On Some Minimally Supersymmetric Models In Two Dimensions, (arXiv:1902.10249) in S. Novikov et al. Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry, Proc. Symposia Pure Math., 103(2), 2021 (ISBN: 978-1-4704-5592-7)
Davide Gaiotto, Theo Johnson-Freyd, Mock modularity and a secondary elliptic genus (arXiv:1904.05788)
Theo Johnson-Freyd, Topological Mathieu Moonshine (arXiv:2006.02922)
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:
Jacques Distler, Eric Sharpe, section 8.5 of Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14:335-398, 2010 (arXiv:hep-th/0701244)
Matthew Ando, Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe, talk 2007 (lecture notes pdf)
Speculations on physics aspects of lifting the Witten genus to topological modular forms:
Yuji Tachikawa, Topological modular forms and the absence of a heterotic global anomaly (arXiv:2103.12211)
Yuji Tachikawa, Mayuko Yamashita, Topological modular forms and the absence of all heterotic global anomalies (arXiv:2108.13542)
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:
Davide Gaiotto, Andrew Strominger, Xi Yin, The M5-Brane Elliptic Genus: Modularity and BPS States, JHEP 0708:070, 2007 (hep-th/0607010)
Davide Gaiotto, Xi Yin, Examples of M5-Brane Elliptic Genera, JHEP 0711:004, 2007 (arXiv:hep-th/0702012)
Further discussion in:
Murad Alim, Babak Haghighat, Michael Hecht, Albrecht Klemm, Marco Rauch, Thomas Wotschke, Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes (arXiv:1012.1608)
Sergei Gukov, Du Pei, Pavel Putrov, Cumrun Vafa, 4-manifolds and topological modular forms (arXiv:1811.07884, spire:1704312)
On the elliptic genus of M-strings inside M5-branes:
Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and $\mathcal{N}=4$ String Amplitudes, Fortschritte der PhysikVolume 62, Issue 3 (arXiv:1310.1325)
Stefan Hohenegger, Amer Iqbal, Soo-Jong Rey, M String, Monopole String and Modular Forms, Phys. Rev. D 92, 066005 (2015) (arXiv:1503.06983)
M. Nouman Muteeb, Domain walls and M2-branes partition functions: M-theory and ABJM Theory (arXiv:2010.04233)
On the elliptic genus of E-strings as wrapped M5-branes:
J. A. Minahan, D. Nemeschansky, Cumrun Vafa, N. P. Warner, E-Strings and $N=4$ Topological Yang-Mills Theories, Nucl. Phys. B527 (1998) 581-623 (arXiv:hep-th/9802168)
Wenhe Cai, Min-xin Huang, Kaiwen Sun, On the Elliptic Genus of Three E-strings and Heterotic Strings, J. High Energ. Phys. 2015, 79 (2015). (arXiv:1411.2801, doi:10.1007/JHEP01(2015)079)
On the elliptic genus of E-strings as M2-branes ending on M5-branes:
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:
Review in: