$M\text{String}$, Topological Modular Forms, and the Witten Genus

This is the third in a series about maps from some kind of Thom spectrum to a K-theory of vector bundles. However, we depart from the target being K-theory to replacing it with a different spectrum which is still periodic. Let’s begin with defining a string manifold. Suppose $M$ is a spin $n$-manifold; i.e. $w_2(M)=0$ and we have chosen a choice of lift of the orthonormal frame bundle which is a spin bundle. Then in this case, the 1st Pontryagin class is canonically twice some other class which we denote $\frac{1}{2}p_1(M)$. It’s possible this class is 0 or that it is 2-torsion and so $p_1(M)=0$. In any case, a string manifold has a lift to the $\text{String}(n)$ group which trivializes this class. There are several ways to think of this group. Topologically, it fits into a Whitehead tower $…\to \text{Fivebrane}(n) \to \text{String}(n) \to \text{Spin}(n) \to SO(n) \to O(n)$ where as we move left, the groups become increasingly connected.

From $O(n)$ to $SO(n)$, it becomes connected, then to the spin group it becomes simply connected. All finite dimensional Lie groups already have $\pi_2=0$ so the next step is for $\text{String}(n)$ to have $\pi_3=0$. For $\text{Fivebrane}(n)$, it has $\pi_i=0$ for $i=0,…,7$. Now, all finite dimensional compact Lie groups have nontrivial $\pi_3$ so $\text{String}(n)$ must be infinite-dimensional. From a physics point of view, a string manifold is such that in addition to being able to define holonomy along paths, we can also define holonomies for surfaces going between strings which we should think of as elements of the free loop space $\Lambda M := \text{Maps}(S^1,M)$. There is a short exact sequence of topological groups $0 \to K(\mathbb{Z},2) \to \text{String}(n) \to \text{Spin}(n) \to 0$ where $K(\mathbb{Z},2)$ is the homotopy fiber here and is an Eilenberg–MacLane space and often represented by $\mathbb{CP}^\infty$. Not only does it admit the structure of a topological group, it is abelian.

Since we have the short exact sequence of topological groups, we have a fibration. One can show that $\pi_3(\text{Spin}(n))$ is $0$ for $n=2$, $\mathbb{Z}\oplus \mathbb{Z}$ for $n=4$ and $\mathbb{Z}$ for $n=3, n \geq 5$. Using the long exact sequence for fibrations, we find a section of it as $…\to \pi_3(K(\mathbb{Z},2)) \to \pi_3(\text{String}(n)) \to \pi_3(\text{Spin}(n)) \to \pi_2(K(\mathbb{Z},2)) \to \pi_2(\text{String}(n)) \to …$ which becomes $0 \to \pi_3(\text{String}(n)) \to \mathbb{Z} \to \mathbb{Z} \to 0$ for $n=3$ or $n \geq 5$. Since $\mathbb{Z} \to \mathbb{Z}$ surjects, and $\pi_3(\text{String}(n)) \to \mathbb{Z}$ is injective, it must be that $\pi_3(\text{String}(n))=0$. When $n=2$, then $\pi_3(\text{String}(2))=0$ as well. For $n=4$, the homotopy fiber is $K(\mathbb{Z},2)\times K(\mathbb{Z},2)$.

The Witten Genus

Now, if a string manifold is supposed to allow us to define holonomy on the free loop space, then a natural operator we would want to define is something like a Dirac operator on the free loop space where the notion of a connection needs to make sense. But before getting there, what even should the tangent space $T_\gamma \Lambda M$ to a loop $\gamma:S^1 \to M$ be? Well, the ways we can deform this loop are using vector fields in $M$ along the loop. In other words, $T_\gamma \Lambda M \cong \Gamma(\gamma^* TM)$. If $\gamma \equiv p \in M$ is a constant loop, then $\gamma^*TM$ is trivial and splits as $S^1 \times T_pM$. Sections of a trivial bundle $S^1 \times T_pM \to S^1$ are simply maps $S^1 \to T_p M$; i.e. $T_p\Lambda M \cong \Lambda(T_p M)$! Since $T_p M$ is a vector space, the periodic maps into the vector space have a Fourier expansion using functions of the form $e^{i\pi k}$. Let $M$ denote the subspace of all constant loops in $\Lambda M$. Then $T(\Lambda M)|_M = TM \oplus \bigoplus_{k>0} q^k (TM \otimes \mathbb{C})$.

So now, there’s a clear $S^1$ action on $\Lambda M$ which is that we can rotate loops and the constant loops are the fixed point set. If we define a Dirac operator on the free loop space which is $S^1$-equivariant, we would like a way to make sense of its signature. Because we can decompose the tangent space into infinitely many finite bundles, it makes sense to try and use the usual notion of signature on finite dimensions and patch it together using $q$, into a power series. So without much details, let’s just say that it is possible to define a signature and index for the Dirac operator on the free loop space that takes values in $\mathbb{C}[[q]]$ which is power series in $q$. Edward Witten introduced this notion of the index which is now called the Witten genus $\varphi_W$.

Now, it could be defined for any kind of manifold, not necessarily a string manifold but when the manifold is string and of dimension $4k$, then it turns out the Witten genus is not an arbitrary power series but in fact, a modular form of weight $2k$ with integral Fourier expansion.

Example: Suppose $M$ is a string $24$-dim manifold. The vector space of modular forms of weight 12 is spanned by powers of two Eisenstein series which we denote as $E^3_4$ and $E^2_6$. We could also write $\Delta = \frac{1}{1728}(E^3_4 - E^2_6)$ and this vector space of weight 12 modular forms is spanned also by $\Delta$ and $E^3_4 -744\Delta$. The Witten genus would be a linear combination of these modular forms. Why did we choose this second basis? The reason is that there is an integral modular function $j(\tau) = \dfrac{E^3_4}{\Delta} = \frac{1}{q} + 744 + 196,884q + …$ where $j:\mathbb{H}/SL_2(\mathbb{Z}) \to \mathbb{CP}^1$. Here, the domain is the upper half plane modulo the group action. This $j$-invariant precisely classifies all complex elliptic curves. But on the other hand, it was noted by John McKay that the coefficients of $j-744$ are very interesting. Here are the first four: $j-744 = q^{-1} + 196,884q + 21,493,760q^2 + 864,299,970q^3+…$ These coefficients are special sums of the dimensions of the irreducible representations of the Monster group (the largest sporadic finite simple group). The connection between the Monster and $j$-invariant is the subject of Monstrous Moonshine. The more precise conjecture was given by John Conway and Simon Norton: there exists an infinite dimensional graded representation of the Monster group, whose graded traces $T_g$ are the expansions of precisely the functions of a very specific type. In particular, if $G_g \subset SL_2(\mathbb{R})$ fixes $T_g$, then quotienting the upper half plane by $G_g$ gives a sphere with finitely many points removed and $T_g$ generates the field of meromorphic functions on this sphere. The conjecture was proven by Richard Borcherds who was awarded the Fields Medal in 1992 for this work.

Alright, but back to the Witten genus. We would like it to have a property similar to the $\widehat{A}$-genus being a spin cobordism invariant or the Todd genus being a complex cobordism invariant (don’t forget this is in the context of stable tangential structures). Namely, we want the Witten genus $\varphi_W$ to be a string cobordism invariant. Even better, we would like it come from topology; i.e. a map of ring spectra $M \text{String} \to R$ where $R$ is some ring spectrum whose homotopy groups should be related to modular forms.

Topological Modular Forms

Some of this is taken from Tyler Lawson’s answer. By work of Mike Hopkins, Matthew Ando, Charles Rezk and Neil Strickland, the Witten genus can indeed be lifted to topology. It is just what we asked for above: there is a map from the string bordism spectrum to a ring spectrum called $\text{tmf}$ (this map is a so-called orientation) such that the Witten genus is recovered as the composition of the induced map on the homotopy groups of these spectra and a map of the homotopy groups of $\text{tmf}$ to modular forms. The orientation of $\text{tmf}$ is in analogy with the Atiyah–Bott–Shapiro map from the spin bordism spectrum to classical K-theory.

So what is this $\text{tmf}$? One feature that distinguishes $\text{tmf}$ is the fact that its coefficient ring, $\text{tmf}^0(pt)$ is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring. The spectrum of topological modular forms is constructed as the global sections of a sheaf of $\mathbb{E}_\infty$ ring spectra on the moduli stack of (generalized) elliptic curves. The original construction due to Hopkins, Miller, and Goerss uses obstruction theory and shows that a particular lift can be done and is unique.

A second construction is due to Jacob Lurie. He described the moduli problem $\text{tmf}$ represents and applied general representability theory to show existence: just as the moduli stack of elliptic curves represents the functor that assigns to a ring the category of elliptic curves over the ring, the stack together with the sheaf of $\mathbb{E}_\infty$ ring spectra represents the functor that assigns to an $\mathbb{E}_\infty$ ring its category of oriented derived elliptic curves, appropriately interpreted. These constructions work over the moduli stack of smooth elliptic curves, and they also work for the Deligne-Mumford compactification of this moduli stack, in which elliptic curves with nodal singularities are included. So $\text{TMF}$ is the spectrum that results from the global sections over the moduli stack of smooth curves, and $\text{tmf}$ is the spectrum arising as the global sections of the Deligne–Mumford compactification. An interesting fact is that $\text{TMF}$ has periodicity $24^2=576$ and $\text{tmf}$ is the topological connective cover of the periodic spectrum $\text{TMF}$. A connective ring spectrum does not have nontrivial homotopy groups in negative degrees so $\text{tmf}$ is unlike $KO$ or $KU$ as those are periodic and not connective but at least it has the close cousin $\text{TMF}$.

Now, the orientation which I’ll also denote with $\varphi_W:M \text{String} \to \text{tmf}$ is a map of ring spectra. When we apply the homotopy functor $\pi_*$ above, the image lands in $\pi_*(\text{tmf})$. Now, the Witten genus is supposed to live in the graded ring of modular forms $MF_*$ and it does not equal $\pi_*(\text{tmf})$. However, if we invert the primes 2 and 3 of $\pi_*(\text{tmf})$, then it is isomorphic to $MF_*$ which can be described thusly: $\delta \in MF_2, \varepsilon \in M_4, MF_* \cong \mathbb{Z}[\frac{1}{2},\delta,\varepsilon]$. What does it mean that after inverting, we have an isomorphism?

We’ll say that the map $\pi_*(\text{tmf}) \to MF_*$ is a rational isomorphism (so isomorphism after tensoring by $\mathbb{Q}$), but it’s actually not a surjection without inverting 2 and 3. As a result, there are certain values that the Witten genus does not take, just as the $\widehat{A}$-genus of a Spin manifold of dimension $8k+4$ must be an even integer (which implies Rokhlin’s theorem). Some examples: $E_6$ is not in the image but $2E_6$ is, which forces the Witten genus of 12-dimensional string manifolds to have even integers in their power series expansion; similarly $\Delta$ is not in the image, but $24\Delta$ and $\Delta^{24}$ both are. Since the modular forms have a basis which includes $\Delta$, then we see that the coefficient for $\Delta$ should be divisible by 24 similar to how the signature of smooth spin 4-manifolds is divisible by 16 (more on this in a moment).

The map $\pi_*(\text{tmf}) \to MF_*$ is also not an injection; there are many torsion classes and classes in odd degrees which are annihilated by the map. These actually provide bordism invariants of string manifolds that aren’t detected by the Witten genus, but are morally connected in some sense because they can be described cohomologically via universal congruences of elliptic genera. For example, the Witten genus of framed $S^1$ and $S^3$ is 0 but they have some refined invariants in $\pi_*(\text{tmf})$ which are torsion.

NB: We have a map $\Omega^{SO}_* \to \mathbb{Z}[\frac{1}{2},\delta,\varepsilon]$ of elliptic genera and for different values of $\delta,\varepsilon$, we can get familiar genera like the signature or $\widehat{A}$-genus. If we take the forgetful map $\Omega^{\text{Spin}}_* \to \Omega^{SO}_*$ and compose with the one above, we find that the image actually lands in $\mathbb{Z}[8\delta,\varepsilon]$. I don’t think this quite shows Rokhlin’s theorem that smooth spin 4-manifolds have signature divisible by 16 but maybe it suggests it.