Characteristic classes in $TMF$ of level $\Gamma_{1}(3)$

Characteristic classes are cohomology classes which are naturally associated to principal $G$-bundles on topological spaces. Examples include Stiefel-Whitney classes for $O(n)$-bundles or $KO$-Pontryagin classes for $SO(n)$-bundles. If the cohomology theory is oriented with respect to $G$-bordism, the characteristic classes can be evaluated on fundamental classes of manifolds with $G$-structure and yield characteristic numbers. It is well known that oriented bordism is determined by Stiefel-Whitney numbers and rational Pontryagin numbers, whereas spin bordism is determined by Stiefel-Whitney numbers and $KO$-Pontryagin numbers.

In this paper we consider characteristic classes for spin and string bundles. Here, $String$ is the 6-connected cover of the $Spin$-group. The cohomology theory is $TMF_{1}(3)$, the cohomology of topological modular forms of level $\Gamma_{1}(3)$. It is possible that string bordism is determined by Stiefel-Whitney and $TMF$-characteristic numbers for some level structures. At the prime 2, the connective version of $TMF$ should split off of the Thom spectrum $MString$ and there is some evidence that another summand is given by the 16th suspension of the connective cover of $TMF_{0}(3)$ (compare [MR09]). However, in order to get a map, a better understanding of the characteristic classes is necessary. (The relation between $TMF_{1}(3)$ and $TMF_{0}(3)$ is described in section 2.)

The homology groups $E_{*}BU\langle 6\rangle$ for complex oriented theories $E$ have been described in [AHS01] in terms of cubical structures on the formal group of $E$. Here, $BU\langle 6\rangle$ is the complex analogue of $B{\mathit{S}tring}$. One may hope that the real case $E_{*}B{\mathit{S}tring}$ admits a similar description in terms of “real” cubical structures. In [LO] the relation between cubical structures and the results of the current article has been further investigated.

The cohomology groups $E^{*}B{\mathit{S}tring}$ for various complex oriented versions of $TMF$ have been considered in [Lau99]. When $2$ is inverted these groups have been computed by the topological $q$-expansion principle. The difficulty appears at the prime $2$ since here a splitting result for the string bordism spectrum along the lines of Hovey-Ravenel [HR95] is missing. This paper deals with this remaining prime.

We first look at characteristic classes for spin bundles and show the following $q$-expansion principle.

The horizontal map is Miller’s elliptic character which corresponds to the evaluation at the Tate curve on the moduli stack of elliptic curves ([Mil89].) On coefficients this map is just the traditional q-expansion map for modular forms. The theory $K_{Tate}$ is the power series ring $K[1/3]((q))$ of $K$-theories as in [Lau99]. The right vertical map is the Chern character and the left vertical map is the Dold character, that is, the map to rational cohomology induced by the exponential of the formal group law (see ibid.)

The theorem determines the ring of $TMF_{1}(3)$-characteristic classes for spin bundles as follows. An element of $K_{Tate}^{*}B{\mathit{S}pin}$ is a $K_{Tate}$-characteristic class for spin bundles, that is, a formal series of virtual vector bundles which is naturally defined for spin bundles. If its Chern character is invariant under the appropriate Möbius transformations then it gives rise to a unique $TMF_{1}(3)$-characteristic class. This property allows the construction of many natural classes. An application to representations of loop groups will be given later.

The proof of the theorem is given in Section 3. Locally at the prime 2 the cohomology theory $TMF_{1}(3)$ is a generalized Johnson-Wilson spectrum $E(2)$ by Proposition 2.4. Hence it relates to the Morava $K$-theory $K(2)$ by a Bockstein spectral sequence. A splitting principle which is based on the Kitchloo-Laures computation [KL02] of the Morava $K(2)$-homology of $B{\mathit{S}pin}$ forces the universal coefficient spectral sequence to collapse. Its $E_{2}$-term is studied via a chromatic resolution. Note that this part of the work does not involve the theory of elliptic curves but only the study of formal group laws of height at most 2. However, the statement of the theorem is a transchromatic property which finds a natural formulation in the $q$-expansion of modular forms once the generalized $E(2)$-theory has been chosen to be $TMF_{1}(3)$. The pullback property is reduced via the universal coefficient isomorphisms to the classical q-expansion principle.

The theorem allows us to the construct Pontryagin classes for $TMF_{1}(3)$. The pullback diagram relates these to the classical Pontryagin classes of $K$-theory and rational Pontryagin classes in a nice way.

Next we turn to the calculation of the characteristic classes for $B{\mathit{S}tring}$. Here it turns out that the $q$-expansion principle only holds for odd primes. A string characteristic class may not even be determined by its characters. However, we have the following result.

Theorems 1.2 and 1.3 solve the problem stated in [KLW04a]: they give explicit generators for the cohomology rings of $B{\mathit{S}pin}$ and $B{\mathit{S}tring}$. Their proofs are given in Section 4. Theorem 1.3 relies on a description of the Morava $K(2)$-homology of $B{\mathit{S}tring}$ given by Kitchloo-Laures-Wilson in [KLW04a, KLW04b]. We remark that the completed $TMF_{1}(3)$ can be described as a fixed point spectrum of Lubin-Tate theory (see [MR09].)

As a first consequence we give a proof of the stable version of a conjecture by Brylinski which was stated in [Bry90]. The precise formulation is given in the last section. For sufficiently large $n$ divisible by 3 as in Theorem 5.2, there is a group homomorphism

from a stable group of positive energy representations V of the free loop group ${L}Spin$ to the $TMF$-cohomology with level $n$-structure. Here, the congruence group is larger than the one considered before since the character of the representation is only known to be invariant under the action of $\Gamma(n)$ by a theorem of Kac and Wakimoto [KW88, Theorem A].

We will describe the map $\varphi$ in terms of its elliptic character. Suppose $P$ is a $String$-principal bundle over $X$. Let $\tilde{L}Spin$ be the universal central extension of $L{\mathit{S}pin}$. Then by [Wit88], $LX$ carries a $\tilde{L}Spin$-principal bundle $\tilde{L}P$ whose associated $L{\mathit{S}pin}$-bundle is $LP$. In particular, this holds for the universal $String$-bundle $E{\mathit{S}tring}$ over $B{\mathit{S}tring}$. The elliptic character of $\varphi$ gives the bundle

(when $V$ is suitably normalized with a character of the rotation circle). In this formula, the right hand side is considered as a formal power series of virtual bundles by decomposing the bundle as a representation of the circle group which reparameterizes the loops. The evaluation of this class on the fundamental class of a string manifold is the formal index of the Dirac operator on $LM$ with coefficient in the bundle associated to the representation $V$ (see [Bry90]).

In this section we describe the spectrum of topological modular forms with level structures. Most of the results are well known and are collected for later reference. We start more generally than actually needed in order to put the main theorem in a broader picture.

Let ${\mathcal{M}}$ denote the stack of smooth elliptic curves. A morphism $f:S\longrightarrow{\mathcal{M}}$ determines an elliptic curve $C_{f}$ over the base scheme $S$ (see Deligne and Rapoport [DR73].) The cotangent spaces of the identity section of each elliptic curve give rise to the line bundle $\omega$ of invariant differentials on ${\mathcal{M}}$. A section of $\omega^{\otimes k}$ is called a modular form of weight $k$. We write $M_{k}$ for the group of modular forms of weight $k$.

Let $C$ be an elliptic curve over a ring $R$. Let $C[n]$ denote the kernel of the self-map which multiplies by $n$ on $C$. If $n$ is invertible in $R$, then étale-locally, $C[n]$ is of the form ${\mathbb{Z}}/n\times{\mathbb{Z}}/n$. A choice of an isomorphism is called a $\Gamma(n)$-structure on $C$. A monomorphism ${\mathbb{Z}}/n\longrightarrow C[n]$ is a $\Gamma_{1}(n)$-structure. It corresponds to a choice of a point of exact order $n$. A choice of a subgroup scheme of $C[n]$ which is isomorphic to ${\mathbb{Z}}/n$ is called a $\Gamma_{0}(n)$-structure. There are maps of associated moduli stacks with $n$ inverted

and the sections of the associated line bundles are called modular forms with the corresponding level structures. For example, a $\Gamma_{1}(n)$-modular form $f$ of weight $k$ associates an element of the ring $R$ to each triple $(C/R,\omega,P)$ where $C$ is an elliptic curve over $R=R[1/n]$, $\omega$ is a translation invariant nowhere vanishing differential and $P$ is a point of exact order $n$. This association should only depend on the isomorphism class of the triple, it should be invariant under base change, and it should satisfy

for all $a\in R^{\times}$.

The case $n=3$ can be made more explicit. Locally, for any such triple $(C,\omega,P)$ the curve can uniquely be written in the form

in a way that $P$ is the origin $(0,0)$ and $\omega$ has the standard form $\omega=dx/(2y+a_{1}x+a_{3})$. A proof of this fact can be found in [MR09, 3.2]. It means that there is universal triple $(C,\omega,(0,0))$ over the ring ${\mathbb{Z}}[1/3,a_{1},a_{3},\Delta^{-1}]$ with the property that locally any other triple is obtained by base change. Hence a $\Gamma_{1}(3)$-modular form is determined by its value on the universal triple and gives an element in this ring. Moreover, any element in the ring gives a modular form with level structure. We get

One should mention that all moduli problems for $\Gamma_{1}(n)$ and $\Gamma(n)$-structures with $n\geq 3$ are representable this way but $\Gamma_{0}(n)$ is not representable. A good reference for these classical results on modular forms and level structures is the book [KM85].

There is a derived version of these concepts as follows.

The spectrum $TMF_{1}(3)$ can be described much more elementarily if one is only interested in its associated cohomology theory. Since the moduli problem is representable, its homotopy coincides with the ring $M_{\Gamma_{1}(3)}$ by property (iv) as we now explain. (The ring $M_{\Gamma_{1}(3)}$ has been determined in (5).) When we choose a coordinate on the formal group of the curve (4) we obtain a formal group law. These are classified by a map from the Lazard ring $L$, which coincides with the homotopy groups of complex bordism $MU$. For instance, the 2-typicalization of the standard coordinate has Hazewinkel generators (see [Lau04, Lemma 1]) at $p=2$

The Hazewinkel generators form a regular sequence and hence satisfy the Landweber exactness conditions. Thus for finite complexes $X$ we have natural isomorphisms

(In the older literature for example [Bry90, Fra92, Bak94] this theory carried the names $Ell^{\Gamma_{1}(3)}$ or $E^{\Gamma_{1}(3)}$.)

Let $w_{k}\in\pi_{*}MU$ denote any element which projects to a generator of the indecomposables in $\pi_{2(p^{k}-1)}BP$. Recall from [Mat, Definition 3.9] that an $MU$-module spectrum $M$ is a form of $BP\langle n\rangle$ if there are $w_{n+1},w_{n+2},\ldots$ such that

In the sequel, we will call these theories generalized $BP\langle n\rangle$ and will use the same notation.

Theorem 1.1 involves the elliptic character map. This map originates from the Tate curve

where

with the Eisenstein series $E_{4}$ and $E_{6}$. The series $B$ and $C$ are integral power series in $q$. The evaluation of an ordinary modular form on the Tate curve with its canonical differential corresponds to its $q$-expansion. The formal group associated to the Tate curve is the multiplicative formal group. In order to get a $\Gamma_{1}(n)$-structure, that is, a point of order $n$, one can use the extension of scalars ${\mathbb{Z}}((q))\rightarrow{\mathbb{Z}}((q));\;q\mapsto q^{n}$. The resulting curve is usually denoted by $Tate(q^{n})$ and its multiplicative reduction furnishes the Miller character map [Mil89][Lau99]

In homotopy this map is the classical $q$-expansion.

In this section we will show the universal coefficient isomorphism for the $TMF_{1}(3)$-cohomology of the space $B{\mathit{S}pin}$ and the pullback diagram of Theorem 1.1. The main ingredient is the Morava $K(2)$-homology which has been computed by Kitchloo-Laures in [KL02]. We can use this result to obtain information about the $E(k,n)$-cohomology by methods of Ravenel-Wilson-Yagita. A chromatic argument enables us to compute the universal coefficients spectral sequence for $TMF_{1}(3)^{*}B{\mathit{S}pin}$. We will show that it collapses at the $E_{2}$-term and obtain Theorem 1.1 from the classical $q$-expansion principle. Some of the results of this section apply to other situations and hence are formulated more generally than actually needed.

From now on we fix a prime $p$. In a first step we do not want to deal with $\lim^{1}$-questions and hence work with $p$-completed spectra. Let $E(n)$ be a $p$-completed generalized Johnson-Wilson spectrum. Let $I_{k}$ be the invariant prime ideal $(p,v_{1},\ldots,v_{k-1})$ and let $E(k,n)$ be the spectrum

By definition we have $E(n,n)\cong K(n)$, $E(0,n)=E(n)$ and there are cofibre sequences

In the following, for a fixed $0<k\leq n$, let $X$ be a space with even $E(k,n)$-cohomology. The exact sequence induced by (11)

tells us that each element in $E(k-1,n)^{odd}X$ is infinitely divisible by $v_{k-1}$. The following result applies:

Next we look at the universal coefficient spectral sequence for $E=E(n)$

We now specify to the case $X=B{\mathit{S}pin}$ and prove a splitting principle using the following computation.