# nLab A Survey of Elliptic Cohomology - compactifying the derived moduli stack

higher algebra

universal algebra

## Theorems

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

see there for background and context.

Here are the entries on the previous sessions:

# Compactifying ${M}^{\mathrm{Der}}$

## 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 ${E}_{\infty }$-ring $\mathrm{tmf}\left[{\Delta }^{-1}\right]$ as the global sections of ${OM}_{{M}^{\mathrm{Der}}}$. If we take global sections of ${O}_{\overline{{M}^{\mathrm{Der}}}}$ then we get the ${E}_{\infty }$-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 $q\in ℂ,\phantom{\rule{thickmathspace}{0ex}}lvertqrvert<1$, there is an elliptic curve over $ℂ$ defined by a Weierstrass equation

${y}^{2}+\mathrm{xy}={x}^{3}+{a}_{4}\left(q\right)x+{a}_{6}\left(q\right).$y^2 + xy = x^3 +a_4(q) x + a_6 (q) .

If $0 the elliptic curve is isomorphic to ${ℂ}^{*}/{q}^{ℤ}$ 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 $ℤ\left[\left[q\right]\right]$. We really would like to think of $T$ as an elliptic curve over $ℤ\left(\left(q\right)\right):=ℤ\left[\left[q\right]\right]\left[{q}^{-1}\right]$.

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 $ℤ\left(\left(q\right)\right)$.

## 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 $n\in ℤ$, we have ${\mathrm{AFT}}_{n}\simeq \left({K}_{\mathrm{Tate}}{\right)}_{n}.$

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 $\left\{{A}_{l,\tau }\right\}$ 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 $l\in {ℝ}_{+}$, $\left\{E\left({A}_{l,\tau }\right),\tau \in h\right\}$ is a commuting family of trace class (and hence compact) operators depending only on $\tau$ modulo $ℤ$. Therefore, by writing $q={e}^{2\pi i\tau }$ we can write $E\left({A}_{l,\tau }\right)={q}^{L}{q}^{\overline{L}}$, where $L,\overline{L}$ are unbounded operators with discrete spectrum.

Lemma. The spectrum of $L-\overline{L}\subseteq ℤ$. 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 $l$, ${ℝ}_{+}^{2\mid 1}/lℤ$ is a super Lie semi-group and the functor $E$ gives a representation ${ℝ}_{+}^{2\mid 1}\to \mathrm{End}\left(E\left({S}_{l}^{1}\right)\right)$ which is compatible with the super-semigroup law on ${ℝ}^{2\mid 1}$ given by

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

We see that a degree 0 theory is determined by a pair of operators $\left(L,G\right)$. By analyzing the spectral decomposition of the pair $\left(G,L-{G}^{2}\right)$ 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 $\mid V\mid \simeq \left({K}_{\mathrm{Tate}}{\right)}_{0}$. 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 ${E}_{\infty }$-ring $R$. That is given a fan $F=\left\{{U}_{\alpha }\right\}$ we can build a derived scheme ${X}_{F}$ where ${X}_{F}=varinjlim\left\{{U}_{\alpha }\right\}$ and ${U}_{\alpha }=\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}R\left[{S}_{{\sigma }_{\alpha }}\right]$ is an affine derived scheme. We will define the (derived) Tate curve as the formal completion of the quotient of a toric variety.

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

${\sigma }_{n}=\left\{\left(a,b\right)\in ℤ×ℤ\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}\mathrm{na}\le b\le \left(n+1\right)a\right\}.$\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 $f:{X}_{F}\to \mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}R\left[q\right]$.

Consider $f:{X}_{F}\to \mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}R\left[q\right]$, one can show that

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

Consider the automorphism $\tau :ℤ×ℤ\to ℤ×ℤ$ defined by $\tau \left(a,b\right)=\left(a,b+a\right)$. Note that $\tau \left({\sigma }_{n}\right)=\tau \left({\sigma }_{n+1}\right)$, 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 ${f}^{-1}\left(q\right)$ by multiplication by $q$, for $q\ne 0$;
2. $\tau$ acts freely on ${f}^{-1}$.

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

Define $\stackrel{^}{T}$ to be the formal (derived) scheme ${\stackrel{^}{X}}_{F}/{\tau }^{ℤ}$.

Theorem (Lurie/Grothendieck). The formal derived scheme $\stackrel{^}{T}$ is the completion of a unique derived scheme over $\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}R\left[\left[q\right]\right]$. That is, there exists a unique derived scheme $T\to \mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}R\left[\left[q\right]\right]$ such that $\stackrel{^}{T}=\mathrm{Spf}\phantom{\rule{thickmathspace}{0ex}}T$.

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 $\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}R\left(\left(q\right)\right)$ is an elliptic curve isomorphic to ${G}_{m}/{q}^{ℤ}$.

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

$T:\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}K\left(\left(q\right)\right)\to {M}^{\mathrm{Der}}.$T: \mathrm{Spec} \; K( (q) ) \to \mathbf{M}^{Der}.

Now note that the involution $\alpha :ℤ×ℤ\to ℤ×ℤ$ defined by $\alpha \left(a,b\right)=\left(a,-b\right)$ preserves the fan $F$ and $\left(\tau \circ \alpha {\right)}^{2}=1$. We allow $\alpha$ to be complex conjugation on $K\left(\left(q\right)\right)$ thought of as a group action of $\left\{±1\right\}$, so we have a map $\mathrm{Spec}\phantom{\rule{thickmathspace}{0ex}}K\left(\left(q\right)\right)/\left\{±1\right\}\to {M}^{\mathrm{Der}}$.

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 $\mathrm{tmf}$.

Created on December 14, 2009 15:17:14 by Ryan Grady (195.37.209.182)