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A Survey of Elliptic Cohomology - compactifying the derived moduli stack

Abstract We sketch how to compactify $M^{\mathrm{Der}}$ such that the underlying scheme is the Deligne-Mumford compactification of $M_{1,1}$.

This is a sub-entry of

• A Survey of Elliptic Cohomology

see there for background and context.

Here are the entries on the previous sessions:

• A Survey of Elliptic Cohomology - cohomology theories

• A Survey of Elliptic Cohomology - formal groups and cohomology

• A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

• A Survey of Elliptic Cohomology - elliptic curves

• A Survey of Elliptic Cohomology - equivariant cohomology

• A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations

• A Survey of Elliptic Cohomology - A-equivariant cohomology

• A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves

• A Survey of Elliptic Cohomology - towards a proof

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Introduction

Let $M^{\mathrm{Der}}$ be the derived Deligne-Mumford moduli stack of oriented elliptic curves. Recall that the underlying Deligne-Mumford stack ${\pi }_0{M}^{\mathrm{Der}}$ is $M_{1,1}$ the classical Deligne-Mumford stack which is the fine moduli stack of elliptic curves. We would like to construct a derived Deligne-Mumford stack $\overline{M^{\mathrm{Der}}}$ such that ${\pi }_0{M}^{\mathrm{Der}}$ is the classical compactification $\overline{M_{1,1}}$.

Recall that one can define the $E_{\mathrm{\infty }}$-ring $\mathrm{tmf}[{\Delta }^{-1}]$ as the global sections of ${OM}_{M^{\mathrm{Der}}}$. If we take global sections of $O_{\overline{M^{\mathrm{Der}}}}$ then we get the $E_{\mathrm{\infty }}$-ring $\mathrm{tmf}$.

The Tate Curve

Let us focus on elliptic curves over $ℂ$. The coarse moduli space of such elliptic curves is again $ℂ$ with the classifying map given by the $j$-invariant. $ℂ$ is not compact, but we can compactify the moduli space by allowing curves with nodal singularities (generalized elliptic curves).

For each $q\in ℂ,lvertqrvert<1$, there is an elliptic curve over $ℂ$ defined by a Weierstrass equation

$$y^2+\mathrm{xy}=x^3+a_4(q)x+a_6(q).$$y^2 + xy = x^3 +a_4(q) x + a_6 (q) .

If $0<lvertqrvert<1$ the elliptic curve is isomorphic to ${ℂ}^{*}/q^{\mathbb{Z}}$ as a Riemann surface with group structure induced from ${ℂ}^{*}$ (equivalently, this curve corresponds to $ℂ/{\Lambda }_{\tau }$ where $e^{2\pi i\tau }=q$). As a function of $q$, $a_4$ and $a_6$ are analytic over the open disk and their power series at $q=0$ have integral coefficients hence the Weierstrass equation defines an elliptic curve $T$ over $\mathbb{Z}[[q]]$. We really would like to think of $T$ as an elliptic curve over $\mathbb{Z}((q)):=\mathbb{Z}[[q]][q^{-1}]$.

It should be noted that this construction (which goes back to Tate) can be extended to more general fields.

The Tate curve defines a cohomology theory $K_{\mathrm{Tate}}$ (an elliptic spectrum). As a cohomology theory $K_{\mathrm{Tate}}$ is just $K$-theory tensored with $\mathbb{Z}((q))$.

Annular Field Theories

As shown in Pokman Cheung’s thesis, the Tate curve has a connection to supersymmetric field theories as defined by Stolz and Teichner. The main result of is that a subspace of $2\mid 1$ field theories (annular theories) is the $0\mathrm{th}$-space of the spectrum $K_{\mathrm{Tate}}$.

Let $\mathrm{SAB}$ be the subcategory of $2\mid 1$-EB whose morphisms are annuli. Note that $\mathrm{SAB}$ does not contain tori. $\mathrm{SAB}$ in essence is completely determined by the supergroup ${ℝ}^{2\mid 1}$. Let ${\mathrm{AFT}}_n$ be the space of natural transformations analogous to the definition of Stolz and Teichner.

Theorem (Theorem 2.2.2 of Cheung). For each $n\in \mathbb{Z}$, we have ${\mathrm{AFT}}_n\simeq (K_{\mathrm{Tate}}{)}_n.$

Proof. Let $A_{l,\tau }$ be the cylinder obtained by identifying non-horizontal sides of the parallelogram in the upper-half plane spanned by $l$ and $l\tau$ for $l\in {ℝ}_{+}$ and $\tau \in h$.

Let $E$ be a degree 0 field theory. The family of cylinders $\{A_{l,\tau }\}$ has the properties that $A_{l,\tau +1}=A_{l,\tau }$ and $A_{l,\tau }\circ A_{l,\tau \prime }=A_{l,\tau +\tau \prime }$. For a fixed $l\in {ℝ}_{+}$, $\{E(A_{l,\tau }),\tau \in h\}$ is a commuting family of trace class (and hence compact) operators depending only on $\tau$ modulo $\mathbb{Z}$. Therefore, by writing $q=e^{2\pi i\tau }$ we can write $E(A_{l,\tau })=q^L{q}^{\overline{L}}$, where $L,\overline{L}$ are unbounded operators with discrete spectrum.

Lemma. The spectrum of $L-\overline{L}\subseteq \mathbb{Z}$. Also, $\overline{L}=G^2$, where $G$ is an odd operator.

Proof of Lemma. The spectral argument follows from having an $S^1$ action. To see the second claim note that for fixed $l$, ${ℝ}_{+}^{2\mid 1}/l\mathbb{Z}$ is a super Lie semi-group and the functor $E$ gives a representation ${ℝ}_{+}^{2\mid 1}\to \mathrm{End}(E(S_l^1))$ which is compatible with the super-semigroup law on ${ℝ}^{2\mid 1}$ given by

$$(z_1,{\overline{z}}_1,{\theta }_1),(z_2,{\overline{z}}_2,{\theta }_2)\mapsto (z_1+z_2,{\overline{z}}_1+{\overline{z}}_2+{\theta }_1{\theta }_2,{\theta }_1+{\theta }_2).$$(z_1 , \overline{z}_1 , \theta_1) , (z_2 , \overline{z}_2 , \theta_2) \mapsto (z_1 + z_2 , \overline{z}_1 +\overline{z}_2 + \theta_1 \theta_2 , \theta_1 + \theta_2) .

Now $\mathrm{Lie}({ℝ}_{+}^{2\mid 1})=⟨{\partial }_z,{\partial }_{\overline{z}},Q⟩$, where $Q={\partial }_{\theta }+\theta {\partial }_{\overline{z}}$. Under $E$ the vector fields map to $L,\overline{L},$ and $G$ respectively. Further, $[Q,Q]=2Q^2=2{\partial }_{\overline{z}},$ hence $G^2=\overline{L}$.

We see that a degree 0 theory is determined by a pair of operators $(L,G)$. By analyzing the spectral decomposition of the pair $(G,L-G^2)$ one can construct a weak equivalence of categories $\mathrm{AFT}\to V$ (a homotopy equivalence of geometric realizations), where $V$ is a certain category of Clifford-modules. Following work of Segal it is proved that $\mid V\mid \simeq (K_{\mathrm{Tate}}{)}_0$. The nonzero degrees follow a similar line of reasoning.

$\overline{M^{\mathrm{Der}}}$

We can extend the standard toric variety construction to derived schemes by replacing $ℂ$ by a fixed $E_{\mathrm{\infty }}$-ring $R$. That is given a fan $F=\{U_{\alpha }\}$ we can build a derived scheme $X_F$ where $X_F=varinjlim\{U_{\alpha }\}$ and $U_{\alpha }=\mathrm{Spec}R[S_{{\sigma }_{\alpha }}]$ is an affine derived scheme. We will define the (derived) Tate curve as the formal completion of the quotient of a toric variety.

Let $F_0=\{\{0\},{\mathbb{Z}}_{\ge 0}\}$, so $X_{F_0}=\mathrm{Spec}R[{\mathbb{Z}}_{\ge 0}]=\mathrm{Spec}R[q]$. Also, let $F=⟨{\sigma }_n{⟩}_{n\in \mathbb{Z}}$, where

$${\sigma }_n=\{(a,b)\in \mathbb{Z}\times \mathbb{Z}\mid \mathrm{na}\le b\le (n+1)a\}.$$\sigma_n = \{ (a,b) \in \mathbb{Z} \times \mathbb{Z} \; | \; na \le b \le (n+1) a\}.

Note that by projection onto the first factor we have a map of fans $F\to F_0$ and consequently an induced map on varieties $f:X_F\to \mathrm{Spec}R[q]$.

Consider $f:X_F\to \mathrm{Spec}R[q]$, one can show that

1. $f^{-1}(q)\cong G_m$ for $q\ne 0$;
2. $f^{-1}(0)$ is an infinite chain of rational curves, each intersecting the next in a node.

Consider the automorphism $\tau :\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\times \mathbb{Z}$ defined by $\tau (a,b)=(a,b+a)$. Note that $\tau ({\sigma }_n)=\tau ({\sigma }_{n+1})$, so $\tau$ preserves the fan $F$ and consequently is an automorphism of the resulting toric variety $X_F$ which we also denote by $\tau$. Then

1. $\tau$ acts on $f^{-1}(q)$ by multiplication by $q$, for $q\ne 0$;
2. $\tau$ acts freely on $f^{-1}$.

Now define ${\hat{X}}_F$ to be the formal completion of $X_F$ along $f^{-1}(0)$. Similarly, define $R[[q]]$ as the formal completion of $R[q]$ along $q=0$. One can show that ${\tau }^{\mathbb{Z}}$ which is the multiplicative group generated by $\tau$ acts freely on ${\hat{X}}_F$.

Define $\hat{T}$ to be the formal (derived) scheme ${\hat{X}}_F/{\tau }^{\mathbb{Z}}$.

Theorem (Lurie/Grothendieck). The formal derived scheme $\hat{T}$ is the completion of a unique derived scheme over $\mathrm{Spec}R[[q]]$. That is, there exists a unique derived scheme $T\to \mathrm{Spec}R[[q]]$ such that $\hat{T}=\mathrm{Spf}T$.

We call the derived scheme $T$ in the theorem above the Tate curve. It is a fact that the restriction of $T$ to the punctured formal disk $\mathrm{Spec}R((q))$ is an elliptic curve isomorphic to $G_m/q^{\mathbb{Z}}$.

We then see that an orientation of $T$ is equivalent to an orientation of $G_m$ which is equivalent to working over the complex $K$-theory spectrum $K$. Therefore, the oriented Tate curve is equivalent to a map

$$T:\mathrm{Spec}K((q))\to M^{\mathrm{Der}}.$$T: \mathrm{Spec} \; K( (q) ) \to \mathbf{M}^{Der}.

Now note that the involution $\alpha :\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\times \mathbb{Z}$ defined by $\alpha (a,b)=(a,-b)$ preserves the fan $F$ and $(\tau \circ \alpha {)}^2=1$. We allow $\alpha$ to be complex conjugation on $K((q))$ thought of as a group action of $\{\pm 1\}$, so we have a map $\mathrm{Spec}K((q))/\{\pm 1\}\to M^{\mathrm{Der}}$.

We can define a new derived Deligne-Mumford stack by forming an appropriate pushout square.

Finally, one can show that the underlying scheme ${\pi }_0\overline{M^{\mathrm{Der}}}$ is $\overline{M_{1,1}}$.

There are many subtleties associated with $\overline{M^{\mathrm{Der}}}$. For instance, we would like to glue the universal curve $E$ over $M^{\mathrm{Der}}$ with $T$ to obtain a universal elliptic curve $\overline{E}$ over $\overline{M^{\mathrm{Der}}}$, however the result is only a generalized elliptic curve; it is not a derived group scheme over $\overline{M^{\mathrm{Der}}}$ as it only has a group structure over the smooth locus of the map $\overline{E}\to \overline{M^{\mathrm{Der}}}$. Lurie asserts it is possible to construct the necessary geometric objects over $\overline{M^{\mathrm{Der}}}$ (I guess this will show up in DAG VII or VIII). The global sections of the structure sheaf thus constructed is the spectrum $\mathrm{tmf}$.