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: {\mathrm{tmf}} is a higher analog of {KO}–theory (or rather, connective {KO}-theory).

1. What is {\mathrm{tmf}}?

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

\displaystyle c_1( \mathcal{L}) = [\mathcal{L}] - [\mathbf{1}];

that is, one uses the class of the line bundle {\mathcal{L}} itself in {K^0(X)} (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying {\hat{\mathbb{G}_m}},

\displaystyle \mathrm{Spec} \mathbb{Z} \rightarrow M_{FG},

from {\mathrm{Spec} \mathbb{Z}} to the moduli stack of formal groups {M_{FG}}, 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 {K}-homology of any spectrum {X} could be determined in terms of the more primordial homology theory {MU} of complex bordism. Namely, Landweber’s criterion showed that one has a natural isomorphism (due initially to Conner and Floyd by different methods)

\displaystyle MU_*(X) \otimes_{MU_*} \mathbb{Z}[t, t^{-1}] \simeq K_*(X), \ \ \ \ \ (1)

for any spectrum {X}. The isomorphism is based upon a natural map {MU_* \rightarrow K_* \simeq \mathbb{Z}[t^{\pm 1}]}; the map can be described by recalling that {MU_*} is (by a fundamental theorem of Quillen) the Lazard ring that classifies formal group laws, and the map

\displaystyle MU_* \rightarrow K_*,

classifies the formal group law {x + y + txy} over {\mathbb{Z}[t, t^{-1}]} (i.e., a twisted {\widehat{\mathbb{G}_m}}). The theorem is remarkable in that {K_*} is very far from being flat over {MU_*}. Nonetheless, the flatness over {M_{FG}} 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 {K}-theory. One could construct it purely homotopy-theoretically, starting with the Thom spectrum {MU} 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 {E}-theory (or Lubin-Tate) spectra {E_n} that play an important role in homotopy theory but have no known geometric description.

Of course, what we’ve constructed so far is complex {K}-theory, which is a good bit simpler than {KO}-theory, and it’s not yet clear how one might construct {KO}-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 {KO}-theory in terms of {K}-theory. Namely, one has a {\mathbb{Z}/2}-action on {K}-theory, given by the Adams operation

\displaystyle \Psi^{-1}: K \rightarrow K.

Classically, {\Psi^{-1}} corresponds to complex conjugation of complex vector bundles. It is also possible to begin {\Psi^{-1}} homotopy-theoretically: it corresponds to the automorphism of {\widehat{\mathbb{G}_m}} given by inversion. Given this {\mathbb{Z}/2}-action, one has

\displaystyle KO \simeq K^{h \mathbb{Z}/2};

that is, {KO}-theory can be recovered as the homotopy fixed point spectrum of {\mathbb{Z}/2}-acting on {K}. Geometrically, the above equivalence comes from the fact that to give a real vector bundle on a topological space {X} is equivalent to giving a complex vector bundle {V \rightarrow X} together with a self-conjugate identification {\iota: V \simeq \overline{V}} which is “coherent.” One can think of this as Galois descent from {\mathbb{C}} to {\mathbb{R}}: the category of {\mathbb{R}}-vector spaces is the homotopy fixed points of the {\mathbb{Z}/2}-action on the category of {\mathbb{C}}-vector spaces given by complex conjugation. That is, one has

\displaystyle \mathrm{Vect}_{\mathbb{R}} \simeq \mathrm{Vect}_{\mathbb{C}}^{h \mathbb{Z}/2}.

If one works with formal groups, and builds {K}-theory from {MU} without mention of vector bundles, one can still construct {\Psi^{-1}}, but the result is only a {\mathbb{Z}/2}-action in the homotopy category: it fails to be homotopy coherent. More technology is required to show that one has a {\mathbb{Z}/2}-action on the spectrum {K}. These problems become much more difficult for {\mathrm{TMF}}, and a sophisticated obstruction theory was developed by Goerss, Hopkins, Miller, and others to solve such questions (and in the category of {E_\infty}-ring spectra). In other words, one needs much more homotopical structure than the {\mathbb{Z}/2}-action on {K} 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 {\mathbb{Z}/2}-action on {K}-theory such that {KO \stackrel{\mathrm{def}}{=} K^{h \mathbb{Z}/2}} is a perfectly respectable definition; moreover, one can compute the homotopy groups of {KO}-theory via a homotopy fixed-point spectral sequence and recover the classical eight-fold periodicity.

One can think of {K}-theory as arising from the algebraic group {\mathbb{G}_m}, via its formal group. The existence of {KO}-theory, in this language, arises from the fact that {\mathbb{G}_m} is not “uniquely pinned down:” it has an automorphism, given by {x \mapsto x^{-1}}. As a result, there are one-dimensional tori that are not isomorphic to {\mathbb{G}_m} but become {\mathbb{G}_m} after an étale base change; for every {\mathbb{Z}/2}-torsor over a scheme {X} one can construct a non-split torus over {X}, which is an algebraic group over {X} that étale locally (on {X}) becomes {\mathbb{G}_m}. In this way, as {K}-theory comes from {\mathbb{G}_m}, {KO}-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 {KO}-theory itself is only “locally” complex orientable.

Remarkably, {KO}-theory has an entirely equivalent but seemingly different definition as the Grothendieck group of vector bundles on a topological space {X}, 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 {\mathrm{TMF}} 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 {\mathrm{TMF}} is unknown.) Instead of working with the formal group { \widehat{\mathbb{G}_m} } associated to the multiplicative group {\mathbb{G}_m}, one uses the only other type of one-dimensional group scheme: elliptic curves. Given an elliptic curve {C} over a ring {R}, one can construct a formal group over {R} (by formally completing {C} 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 {R} (and the formal group {\hat{C}} over {R}) that is necessary to realize a complex-orientable spectrum with formal group {\hat{C}}. Namely, the map

\displaystyle \mathrm{Spec} R \rightarrow M_{FG},

that classifies the formal group {\hat{C}}, should be flat. Although {M_{FG}} 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

\displaystyle M_{ell} \rightarrow M_{FG},

from the moduli stack {M_{ell}} of elliptic curves to the moduli stack of formal groups, is a flat affine morphism of stacks. That means that for any ring {R} and any formal group {\mathfrak{X}} over {R}, there is a flat {R}-algebra {R'} which classifies “elliptic curves {C} over {R} together with an isomorphism of {\hat{C} \simeq \mathfrak{X}}.”

In particular, given any elliptic curve {C} over {R}, classified by a map {\mathrm{Spec} R \rightarrow M_{ell}}, we conclude that if the classifying map is flat, then the map {\mathrm{Spec} R \rightarrow M_{FG}} 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 {\hat{C}}. In particular, one gets a presheaf of homology theories on {M_{ell}}; these homology theories are called elliptic homology theories.

The idea of {\mathrm{TMF}} 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, {M_{ell}}—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 {M_{ell}}.

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 {\mathrm{TMF}}, 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 {\infty}-categories, and talk about diagrams there: the language of {\infty}-categories efficiently the notion of a “homotopy coherent” diagram. In other words, we need a homotopy coherent functor

\displaystyle \mathcal{O}^{\mathrm{top}}: \left(\mathrm{Aff}^{flat}_{/M_{ell}}\right)^{op} \rightarrow \mathrm{Sp} ,

from the flat site of affine schemes over {M_{ell}} to spectra (either as an {\infty}-category or in a model category), such that {\mathcal{O}^{\mathrm{top}}} when applied to a flat morphism {f: \mathrm{Spec} R \rightarrow M_{ell}} produces the elliptic homology theory associated to the elliptic curve over {R} classified by {f}. In fact, since {M_{ell}} is a Deligne-Mumford stack, it would be sufficient to do this for étale {f: \mathrm{Spec} R \rightarrow M_{ell}}: the definition of a Deligne-Mumford stack is essentially that there is a covering by étale affines.

3. {E_\infty}-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 {E_\infty}-ring spectra. An {E_\infty}-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 {E_\infty}-ring spectra is not just homotopy commutative, but it is coherently commutative up to all possible higher homotopies. (The {E} comes from “everything,” for “homotopy everything” ring spectrum.) In a sufficiently nice model category, such as symmetric spectra, an {E_\infty}-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 {E_\infty}-ring spectrum. For example, given an {E_\infty}-ring spectrum {R}, one has a category of {R}-modules. An {R}-module is a spectrum {M} together with a multiplication

\displaystyle R \wedge M \rightarrow M,

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 {R}-modules is really a “homotopy theory;” it is a well-behaved stable {\infty}-category (which can also be presented via model categories). Given an ordinary commutative ring {A}, the Eilenberg-MacLane spectrum {HA} is an {E_\infty}-ring, and the category of modules over {HA} is equivalent to the derived category of {A}-modules (or rather, its {\infty}-categorical enhancement).

4. Topological modular forms

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

\displaystyle \mathcal{O}^{\mathrm{top}}: \left(\mathrm{Aff}^{et}_{/M_{ell}}\right)^{op} \rightarrow \left\{E_\infty\mathrm{-rings}\right\},

which assigns to each affine étale {\mathrm{Spec} R \rightarrow M_{ell}} (classifying an elliptic curve {C \rightarrow \mathrm{Spec} R}) an elliptic cohomology theory with formal group {\hat{C}}. In other words, it is a lift from the presheaf of homology theories to a presheaf of spectra—in fact, of {E_\infty}-rings. In fact, this presheaf satisfies the homotopical analog of descent, and it defines a sheaf of {E_\infty}-algebras on the (affine) étale site of {M_{ell}}.

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

\displaystyle \mathrm{TMF} = \mathrm{holim}_{\mathrm{Spec} R \rightarrow M_{ell}} \mathcal{O}^{\mathrm{top}}( \mathrm{Spec} R);

in other words, one takes the global sections of this sheaf of {E_\infty}-rings. The result is an {E_\infty}-ring {\mathrm{TMF}}, which is not an elliptic cohomology theory, since {M_{ell}} is not affine. In fact, the homotopy groups of {\mathrm{TMF}} are quite complicated, with considerable torsion at the primes {2} and {3}. They contain a mix of the stable homotopy groups of spheres and the ring of integral modular forms; that is, one has maps

\displaystyle \pi_* S^0 \rightarrow \pi_* \mathrm{TMF} \rightarrow MF_*[\Delta^{-1}] = \mathbb{Z}[c_4, c_6, \Delta]/(c_4^3 - c_6^2 = 1728\Delta) [\Delta^{-1}],

The computation of {\pi_* \mathrm{TMF}} is done via a descent spectral sequence. Namely, the observation is that one has a natural cosimplicial resolution of {\mathrm{TMF}}: take an affine étale cover {\mathrm{Spec} R \rightarrow M_{ell}}, and then form the cobar construction

\displaystyle \mathrm{TMF} \rightarrow \{ \mathcal{O}^{\mathrm{top}}(\mathrm{Spec} R ) \rightrightarrows \mathcal{O}^{\mathrm{top}}( \mathrm{Spec} R \times_{M_{ell}} \mathrm{Spec} R) \dots \},

which is a cosimplicial resolution of {\mathrm{TMF}}. 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 {E_2} page with

\displaystyle H^i( M_{ell}, \pi_j \mathcal{O}^{\mathrm{top}}) \implies \pi_{j-i} \mathrm{TMF},

where {\pi_j \mathcal{O}^{\mathrm{top}}} is the sheafification of the presheaf of the {j}th homotopy groups of the sheaf of spectra on {M_{ell}}. However, we can identify this sheaf: {\pi_0 \mathcal{O}^{\mathrm{top}}( \mathrm{Spec} R \rightarrow M_{ell})} is by definition {R} itself. The higher homotopy groups of a given elliptic spectrum {\mathcal{O}^{\mathrm{top}}(\mathrm{Spec} R \rightarrow M_{ell})} are given by the tensor products of the cotangent sheaf {\omega} — the dual to the Lie algebra. One has

\displaystyle \pi_j ( \mathcal{O}^{\mathrm{top}}( \mathrm{Spec} R \rightarrow M_{ell})) \simeq \begin{cases} \omega^j & j \ \text{even} \\ 0 & \text{otherwise} \end{cases},

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 {\Sigma^k X \rightarrow X}) 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.