geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
The universal elliptic cohomology cohomology called tmf may be realized (after localization at some primes) as the homotopy fixed points of an ∞-action of the modular group modulo a congruence subgroup in direct analogy to how KO-theory arises as the $\mathbb{Z}_2$-homotopy fixed points of KU-theory (Mahowald-Rezk 09, Lawson-Naumann 12). This construction extends to equip tmf for level structure with – almost – the structure of a naive G-spectrum (hence a Bredon equivariant cohomology theory) for pieces of the modular group (Hill-Lawson 13, theorem 9.1).
The way this works is roughly indicated in the following table (Lawson-Naumann 12, Hill-Lawson 13):
Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:
covering | by of level-n structures (modular curve) | ||||||||
$\ast = Spec(\mathbb{Z})$ | $\to$ | $Spec(\mathbb{Z}[ [q] ])$ | $\to$ | $\mathcal{M}_{\overline{ell}}[n]$ | |||||
structure group of covering | $\downarrow^{\mathbb{Z}/2\mathbb{Z}}$ | $\downarrow^{\mathbb{Z}/2\mathbb{Z}}$ | $\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group) | ||||||
moduli stack | $\mathcal{M}_{1dTori}$ | $\hookrightarrow$ | $\mathcal{M}_{Tate}$ | $\hookrightarrow$ | $\mathcal{M}_{\overline{ell}}$ (M_ell) | $\hookrightarrow$ | $\mathcal{M}_{cub}$ | $\to$ | $\mathcal{M}_{fg}$ (M_fg) |
of | 1d tori | Tate curves | elliptic curves | cubic curves | 1d commutative formal groups | ||||
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$ | KU | $KU[ [q] ]$ | elliptic spectrum | complex oriented cohomology theory | |||||
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf | (KO $\hookrightarrow$ KU) = KR-theory | Tate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$) | (Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology) | tmf | $\mathbb{S}$ |
For K-theory this relation induces the $\mathbb{Z}_2$-equivariant cohomology-version of KU which is KR-theory. In direct analogy to this one may hence consider $SL_2(\mathbb{Z}/n\mathbb{Z})$-equivariant versions of tmf (localized at some primes). This maybe deserves to be called modular equivariant elliptic cohomology.
This hasn’t been studied much yet (or not at all), but there is some motivation for this from string theory (see below) and in this context a relevance of modular equivariant elliptic cohomology theory has been conjectured in (Kriz-Sati 05, p.3, p. 17, 18).
A relevant role of some modular equivariant elliptic cohomology theory in string theory has been conjectured in (Kriz-Sati 05, p. 3, p. 17, 18):
First of all, $\mathbb{Z}_2$-equivariant topological K-theory, hence KR-theory, is what controls the D-brane charges in general orientifold vacua of type II superstring theory (hence also of type I superstring theory, which is thereby a special case).
Next, there are various hints (Kriz-Sati 04a, Kriz-Sati 04b Kriz-Sati 05, Sati 05) that lifting string theory to “M-theory” involves replacing K-theory by elliptic cohomology. Not the least of these is the String orientation of tmf (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10), which shows that where the partition function of a superparticle (such as that at the end of the type II superstring ending on a D-brane) is given by the Todd genus in KU-theory, so the partition function of the superstring itself (possibly itself being the boundary of the M2-brane ending on a O9 plane (heterotic string) or on an M5-brane (self-dual “M-string”)) is given by the refined Witten genus in elliptic cohomology/tmf (thus yielding “O9-plane charge” and M5-brane charge in elliptic cohomology (Sati 10)).
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
Finally, the M-theory-lift of type II string theory is (or is naturally identified as) F-theory, which describes the axio-dilaton of the type II string vacua including the modular S-duality/U-duality acting on this as an elliptic fibration over spacetime (whose monodromy “homology invariant” is hence an $SL_2(\mathbb{Z})$-local system). It is hence natural to suspect that the combined worldsheet/target space $\mathbb{Z}_2$-equivariance of orientifold type II superstring backgrounds which is captured by KR-theory lifts in F-theory to some combined worldsheet/target space modular group-action. This is excatly what would be captured by modular equivariant $tmf$ as indicated above (and as realized by theorem 1 below), and and it is what is conjectured in (Kriz-Sati 05, p.3) (there the group $SL_2(\mathbb{Z}/2\mathbb{Z})$ is mentioned, the explicit construction below does capture this but also includes equivariance under all other $SL_2(\mathbb{Z}/n\mathbb{Z})$).
So by extrapolation from the case of orientifolds, where the target space $\mathbb{Z}_2$-involution (the “real space”-structure) is accompanied by a $\mathbb{Z}_2$-action on the worldsheet (the worldsheet parity operator), this would suggest that in modular equivariant F-theory S-duality operations on the target space background would be accompanied by certain modular action on the worldsheet.
Mathematically this is precisely what happens in the equivariant version of tmf established by theorem 1 below, which by prop. 1 has the modular group action on the spectrum induced from the canonical action of the modular group on moduli of elliptic curves (hence: genus-1 worldsheets).
Physically such an effect seems not to have been discussed much, but at least the following is knonw: first of all, under S-duality the worldsheet theory of the type IIB superstring certainly is affected: the superstring here is generally a (p,q)-string, being a bound state of $p$ actual fundamental strings (F1-branes) with $q$ D1-branes, and the S-duality modular group $SL_2(\mathbb{Z})$ does act canonically on these pairs of integers. Now, that this operation has to be accompanied with a worldsheet conformal transformation as the above mathematical story would suggest has been highlighted for instance in (Bandos 00).
Generally, notice that the elliptic genus for type II superstrings lands in modular forms for the congruence subgroup $\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z})$ inside the full modular group (see at Witten genus – Modularity – For type II superstring), the subgroup which fixes one of the NS-R spin structures. Therefore if this elliptic genus does have a “topological” (homotopy theoretic) lift to elliptic cohomology (as is known for the heterotic string via the string orientation of tmf) then not to plain TMF, but to $TMF_0(2) \simeq \Gamma(\mathcal{M}_{ell}(2)_0,\mathcal{O}^{top})$ (this is implified for instance in Stojanoska 11, remark 6.2), where $\mathcal{M}_{ell}(2)_0$ is the moduli stack of elliptic curves with level structure for $\Gamma_0(2)$, see at tmf0(2).
Write $\hat {\mathbb{Z}}$ for the profinite completion of the integers.
Write
for the general linear group in dimension 2 with coefficients in $\hat{\mathbb{Z}}$.
Notice that this is the profinite group obtained as the limit over all general linear groups with coefficients in the cyclic groups (e.g.Greicius 09, (1.1))
Write
for the canonical projection from (the $\mathbb{Z}_2$-central extension $GL_2(\mathbb{Z}) \to PSL_2(\mathbb{Z})$ of) the modular group.
For $n \in \mathbb{N}$ any natural number write
for the corresponding projection to coefficients in the cyclic group of order $n$.
Notice that for $\Gamma \hookrightarrow SL_2(\mathbb{Z}/n\mathbb{Z})$ then its preimage under $p_n$ is a “profinite congruence subgroup”.
The following is a variant of the orbit category of $G = GL_2(\hat {\mathbb{Z}})$ which remembers the stage $n$ and consists only of orbits of the form of cosets by such congruence subgroups.
Write $\widetilde{Orb}_{GL_2(\hat{\mathbb{Z}})}$ for the category whose
objects are pairs $(n,\Gamma)$ with $n \in \mathbb{N}$ a natural number and $\Gamma \hookrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})$ a subgroup;
morphisms are given by
This is (Hill-Lawson 13, def. 3.15).
The construction for each $(n,\Gamma) \in \widetilde{Orb}_{GL_2(\hat{\mathbb{Z}})}$ of the compactified moduli stack $\mathcal{M}_{\overline{ell}}(\Gamma)$ over $Spec(\mathbb{Z}[\frac{1}{n}])$ of elliptic curves with level structure determined by $\Gamma$ (a modular curve) extends to a (lax) 2-functor
from the levelled orbit category of def. 1 to the 2-category of Deligne-Mumford stacks, such that
$\mathcal{M}_{\overline{ell}}(1,1) \simeq \mathcal{M}_{\overline{ell}}$ is the standard compactified moduli stack of elliptic curves over $Spec(\mathbb{Z})$
for each morphism $(n_1,\Gamma_1)\to (n_2,\Gamma_2)$ the induced morphism
is a log-etale morphism covering;
for each $n$ and each normal subgroup inclusion $K \hookrightarrow \Gamma \hookrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})$ the induced map exhibits the homotopy quotient projection by $\Gamma/K$
This is (Hill-Lawson 13, prop. 3.16, prop. 3.17).
It is possible to extend the Goerss-Hopkins-Miller theorem to the compactified moduli stacks of elliptic curves with level-n structure $\mathcal{M}_{\overline{ell}}[n]$ in prop. 1, such that taking global sections produces an (∞,1)-presheaf on the levelled orbit category of def. 1 with values in E-∞ rings
which is such that
for $n = 1$ (where $SL_2(\mathbb{Z}/\mathbb{Z}) = 1$ and hence $\Gamma = 1$ necessarily) one recovers
$Tmf(1,1)\simeq$ Tmf;
the morphism induced by any morphism of the form $(n k ,P_k(\Gamma))\to (n,\Gamma)$ is $k$-localization;
for any $n \in \mathbb{N}$ and every normal subgroup $K \hookrightarrow \Gamma \hookrightarrow GL_2(\mathbb{Z}/n\mathbb{Z})$, we have the $(\Gamma/K)$-homotopy fixed points of $Tmf(n,\Gamma)$ (induced by action of $\Gamma/K$ on $\mathcal{M}_{\overline{ell}}(\Gamma)$ given by prop. 1):
This is (Hill-Lawson 13, theorem 9.1).
The system of spectra in theorem 1 is essentially a spectrum with G-action (see there) for $G$ the “profinite modular group” $GL_2(\hat {\mathbb{Z}})$, except that the parameterization is not quite over the orbit category of this $G$, but just to the subcategory on objects which are coset spaces just by congruence subgroups and subject to that divisibility constraint on the $n$s, the “levelling”. So $Tmf(-)$ defines a “levelled” kind of genuine $GL_2(\hat {\mathbb{Z}})$-equivariant cohomology version of Tmf.
The following proposition gives one way how the modular equivariance of tmf as in theorem 1 restricts to the $\mathbb{Z}_2$-equivariance of KU (hence KR-theory, which is known to be the precise form of type II string theory orientifolds).
First observe (see also (Mahowald-Rezk 09, section 2)) that for level 3 structure we have congruence subgroups
where the first inclusion is a normal subgroup of index 2.
The inclusion of the nodal elliptic curve with its $\mathbb{Z}/2\mathbb{Z}$-worth of automorphisms (the inversion involution) as the cusp of the compactified moduli stack of elliptic curves
over $Spec(\mathbb{Z}[\tfrac{1}{3}])$ yields under theorem 1 a diagram of the form
The spectrum
(first considered in (Mahowald-Rezk 09), see at congruence subgroup for the notation) is complex oriented (Hill-Lawson 13, p.5) (in contrast to Tmf $\simeq Tmf(1,1)$). This is one more way in which the inclusion
related but different: equivariant elliptic cohomology
Mark Mahowald Charles Rezk, Topological modular forms of level 3, Pure Appl. Math. Quar. 5 (2009) 853-872 (pdf)
Vesna Stojanoska, Duality for Topological Modular Forms, Documenta Math. 17 (2012), 271–311 (arXiv:1105.3968)
Tyler Lawson, Niko Naumann, Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2, Int. Math. Res. Not. (2013) (arXiv:1203.1696)
Michael Hill, Tyler Lawson, Topological modular forms with level structure (arXiv:1312.7394)
Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850
Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)
Igor Kriz, Hisham Sati, M Theory, Type IIA Superstrings, and Elliptic Cohomology, Adv.Theor.Math.Phys. 8 (2004) 345-395 (arXiv:hep-th/0404013)
Igor Kriz, Hisham Sati, Type IIB String Theory, S-Duality, and Generalized Cohomology, Nucl.Phys. B715 (2005) 639-664 (arXiv:hep-th/0410293)
Igor Kriz, Hisham Sati, Type II string theory and modularity, JHEP 0508 (2005) 038 (arXiv:hep-th/0501060)
Igor Kriz, Hao Xing, On effective F-theory action in type IIA compactifications (arXiv:hep-th/0511011)
Hisham Sati, The Elliptic curves in gauge theory, string theory, and cohomology, JHEP 0603 (2006) 096 (arXiv:hep-th/0511087)
Hisham Sati, Geometric and topological structures related to M-branes , part I, Proc. Symp. Pure Math. 81 (2010), 181-236 arXiv:1001.5020
Igor Bandos, Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane, Nucl.Phys.B599:197-227,2001 (arXiv:hep-th/0008249)
Aaron Greicius, Elliptic curves with surjective adelic Galois representations (arXiv:0901.2513)