The topic of topological modular forms is a very broad one, and a single blog post cannot do justice to the whole theory. In this section, I’ll try to answer the question as follows: is a higher analog of –theory (or rather, connective -theory).

**1. What is ?**

The spectrum of (real) -theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex -theory. As a ring spectrum, is complex orientable, and it corresponds to the formal group : the formal multiplicative group. Along with , the formal multiplicative group is one of the few “tautological” formal groups, and it is not surprising that -theory has a “tautological” formal group because the Chern classes of a line bundle (over a topological space ) in -theory are defined by

that is, one uses the class of the line bundle itself in (modulo a normalization) to define.

The formal multiplicative group has the property that it is *Landweber-exact*: that is, the map classifying ,

from to the moduli stack of formal groups , is a flat morphism.

(According to a theorem of Landweber, reinterpreted by Hopkins, the flatness of such a morphism is a condition that certain sequences be regular.) Now Landweber developed this theory to show that the -homology of any spectrum could be determined in terms of the more primordial homology theory of complex bordism. Namely, Landweber’s criterion showed that one has a natural isomorphism (due initially to Conner and Floyd by different methods)

for any spectrum . The isomorphism is based upon a natural map ; the map can be described by recalling that is (by a fundamental theorem of Quillen) the Lazard ring that classifies formal group laws, and the map

classifies the formal group law over (i.e., a twisted ). The theorem is remarkable in that is very far from being flat over . Nonetheless, the flatness over turns out to be enough.

In other words, Landweber’s theorem enables one to conclude that, even without any mention of geometric objects like vector bundles, one could still talk about -theory. One could construct it purely homotopy-theoretically, starting with the Thom spectrum and the formal multiplicative group, and then using (1) as the *definition.* That’s a very powerful approach to defining new homology theories, such as elliptic homology theories and the Morava -theory (or Lubin-Tate) spectra that play an important role in homotopy theory but have no known geometric description.

Of course, what we’ve constructed so far is complex -theory, which is a good bit simpler than -theory, and it’s not yet clear how one might construct -theory purely homotopy-theoretically without use of vector bundles or classifying spaces. However, it turns out that there is a sort of “Galois descent” procedure that determines -theory in terms of -theory. Namely, one has a -action on -theory, given by the Adams operation

Classically, corresponds to complex conjugation of complex vector bundles. It is also possible to begin homotopy-theoretically: it corresponds to the automorphism of given by inversion. Given this -action, one has

that is, -theory can be recovered as the homotopy fixed point spectrum of -acting on . Geometrically, the above equivalence comes from the fact that to give a real vector bundle on a topological space is equivalent to giving a complex vector bundle together with a self-conjugate identification which is “coherent.” One can think of this as Galois descent from to : the category of -vector spaces is the homotopy fixed points of the -action on the category of -vector spaces given by complex conjugation. That is, one has

If one works with formal groups, and builds -theory from without mention of vector bundles, one can still construct , but the result is only a -action in the homotopy category: it fails to be homotopy coherent. More technology is required to show that one has a -action on the spectrum . These problems become much more difficult for , and a sophisticated obstruction theory was developed by Goerss, Hopkins, Miller, and others to solve such questions (and in the category of -ring spectra). In other words, one needs much more homotopical structure than the -action on in the homotopy category of spectra (let alone in the category of homology theories) to form homotopy limit constructions such as homotopy fixed points.

Nonetheless, these problems are solvable within the world of homotopy theory, and one can construct the -action on -theory such that is a perfectly respectable definition; moreover, one can compute the homotopy groups of -theory via a homotopy fixed-point spectral sequence and recover the classical eight-fold periodicity.

One can think of -theory as arising from the algebraic group , via its formal group. The existence of -theory, in this language, arises from the fact that is not “uniquely pinned down:” it has an automorphism, given by . As a result, there are one-dimensional tori that are not isomorphic to but become after an étale base change; for every -torsor over a scheme one can construct a *non-split torus* over , which is an algebraic group over that étale locally (on ) becomes . In this way, as -theory comes from , -theory comes from the “universal one-dimensional torus”—but the fact that there is no such universal one-dimensional torus (except in a stacky sense) means that -theory itself is only “locally” complex orientable.

Remarkably, -theory has an entirely equivalent but seemingly different definition as the Grothendieck group of vector bundles on a topological space , a description that cements the connection with topics in geometry such as the Atiyah-Singer index theorem. The interaction between the homotopy-theoretic and geometric sides has been very fruitful, leading for instance to deep results on the existence of positively curved metrics on smooth manifolds.

**2. Elliptic cohomology**

The spectrum of **topological modular forms** is based on a more sophisticated version of the ideas of the previous section, and it takes place solely in the world of homotopy theory. (While much desired, a geometric description of is unknown.) Instead of working with the formal group associated to the multiplicative group , one uses the only other type of one-dimensional group scheme: elliptic curves. Given an elliptic curve over a ring , one can construct a formal group over (by formally completing along the zero section), and one can try to realize the formal group via a complex-orientable ring spectrum.

According to the Landweber exact functor theorem, there is a regularity condition on (and the formal group over ) that is necessary to realize a complex-orientable spectrum with formal group . Namely, the map

that classifies the formal group , should be flat. Although is very far from being a scheme or even an Artin stack, it is a sort of infinite-dimensional stack (it is a homotopy inverse limit of Artin stacks), and one can talk about flatness over it. For example, the map

from the moduli stack of elliptic curves to the moduli stack of formal groups, is a flat affine morphism of stacks. That means that for any ring and any formal group over , there is a flat -algebra which classifies “elliptic curves over together with an isomorphism of .”

In particular, given any elliptic curve over , classified by a map , we conclude that if the classifying map is flat, then the map classifying the elliptic curve’s formal group is also flat, and we can use the Landweber exact functor theorem to build a homology theory — in fact, a complex-orientable ring spectrum, with formal group . In particular, one gets a **presheaf of homology theories on ;** these homology theories are called **elliptic homology theories.**

The idea of is that there should be a homology theory corresponding to the “universal” elliptic curve. Since elliptic curves have automorphisms, the “universal” elliptic curve really lives over a stack, —so the idea is to take the limit of these elliptic homology theories over all elliptic curves over affine schemes. In other words, one should take the global sections of the presheaf of elliptic homology theories on the flat site of .

Unfortunately, one can’t just take a limit of homology theories (or even objects in the homotopy category of spectra, which is a little stronger due to the existence of phantom maps): the category of homology theories is too poorly behaved. In order to form , as the limit of all these elliptic homology theories, one needs to strictify the diagram: one needs to find a *strictly commuting* diagram of such elliptic spectra in some model category of spectra. Alternatively, one can use -categories, and talk about diagrams there: the language of -categories efficiently the notion of a “homotopy coherent” diagram. In other words, we need a *homotopy coherent* functor

from the flat site of affine schemes over to spectra (either as an -category or in a model category), such that when applied to a flat morphism produces the elliptic homology theory associated to the elliptic curve over classified by . In fact, since is a Deligne-Mumford stack, it would be sufficient to do this for étale : the definition of a Deligne-Mumford stack is essentially that there is a covering by étale affines.

**3. -rings**

But that’s exactly what Goerss, Hopkins, and Miller were able to produce. Their key idea is to solve the lifting problem, not in spectra, but in the much more rigid category of -ring spectra. An -ring spectra is, to begin with, a homotopy commutative ring spectrum (which any Landweber exact homology theory gives rise to). However, it’s much better: the multiplication on an -ring spectra is not just homotopy commutative, but it is coherently commutative up to all possible higher homotopies. (The comes from “everything,” for “homotopy everything” ring spectrum.) In a sufficiently nice model category, such as symmetric spectra, an -ring spectrum can be modeled by a commutative algebra object in the model category itself.

In practice, this means that it is possible to do a certain amount of algebra with an -ring spectrum. For example, given an -ring spectrum , one has a category of -modules. An -module is a spectrum together with a multiplication

satisfying the associativity axioms of a module up to coherent homotopy; without the coherence, it would not be a well-behaved category. This “category” of -modules is really a “homotopy theory;” it is a well-behaved stable -category (which can also be presented via model categories). Given an ordinary commutative ring , the Eilenberg-MacLane spectrum is an -ring, and the category of modules over is equivalent to the derived category of -modules (or rather, its -categorical enhancement).

**4. Topological modular forms**

The main theorem of Goerss, Hopkins, and Miller is that there is in fact a functor

which assigns to each affine étale (classifying an elliptic curve ) an elliptic cohomology theory with formal group . In other words, it is a lift from the presheaf of homology theories to a presheaf of spectra—in fact, of -rings. In fact, this presheaf satisfies the homotopical analog of descent, and it defines a sheaf of -algebras on the (affine) étale site of .

The category (or rather, -category) of -rings has sufficient structure to support a good theory of (homotopy) limits and colimits, and one can then as corresponding to the “universal” elliptic curve, via

in other words, one takes the global sections of this sheaf of -rings. The result is an -ring , which is not an elliptic cohomology theory, since is not affine. In fact, the homotopy groups of are quite complicated, with considerable torsion at the primes and . They contain a mix of the stable homotopy groups of spheres and the ring of integral modular forms; that is, one has maps

The computation of is done via a descent spectral sequence. Namely, the observation is that one has a natural cosimplicial resolution of : take an affine étale *cover* , and then form the **cobar construction**

which is a cosimplicial resolution of . This is part of the definition of a sheaf in homotopy theory — one gets cosimplicial resolutions instead of equalizer diagrams. In any event, one has a homotopy spectral sequence for this cosimplicial resolution, and one can identify the page with

where is the sheafification of the presheaf of the th homotopy groups of the sheaf of spectra on . However, we can identify this sheaf: is by definition itself. The higher homotopy groups of a given elliptic spectrum are given by the tensor products of the cotangent sheaf — the dual to the Lie algebra. One has

so that the elliptic spectra are constructed as even periodic spectra. The existence of global periodic phenomena in stable homotopy theory (for instance, of “periodic” self-maps of finite cell complexes ) is one of the reasons that it’s useful to make these ring spectra periodic, to detect them. (It is also forced if you want a Landweber-exact spectrum.)

Anyway, it turns out that this spectral sequence is computable, and the cohomology of the moduli stack of elliptic curves can be completely written down. Even the differentials can be determined with some trickery.

July 14, 2013 at 4:15 am

Are there spectra that sit in relation to Tmf as KU sits in relation to KO (integrally)?

July 14, 2013 at 9:47 am

I believe not, because the moduli stack of elliptic curves is simply connected over the integers. (So this already can’t happen with itself.)

For higher chromatic levels, I believe that there is no known integral model at all.

January 20, 2015 at 1:21 pm

Taking the taylor series expansion of an elliptic curve $C$ about the origin, we get a formal group law with coefficient ring $C_E$.

“an elliptic curve is a one-dimensional algebraic group $C$ over a ring $R$”

What is $R$? Is it $C_E$ or is it like $y^2=4x^3+ax+b$; $a,b \in R$?

January 21, 2015 at 1:34 am

Aaron Mazel-Gee was kind enough to explain to me today R is the coefficient ring in the latter sense. Apologies for the silly question!