A Survey of Elliptic Cohomology - compactifying the derived moduli stack

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

This is a sub-entry of

see there for background and context.

Here are the entries on the previous sessions:

Compactifying M DerM^{Der}


Let M Der\mathbf{M}^{Der} be the derived Deligne-Mumford moduli stack of oriented elliptic curves. Recall that the underlying Deligne-Mumford stack π 0M Der\pi_0 \mathbf{M}^{Der} is M 1,1\mathbf{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 M Der¯\overline{\mathbf{M}^{Der}} such that π 0M Der\pi_0 \mathbf{M}^{Der} is the classical compactification M 1,1¯\overline{\mathbf{M}_{1,1}}.

Recall that one can define the E E_\infty-ring tmf[Δ 1]tmf[\Delta^{-1}] as the global sections of OM M Der\OM_{\mathbf{M}^{Der}}. If we take global sections of O M Der¯\mathbf{O}_{\overline{\mathbf{M}^{Der}}} then we get the E E_\infty-ring tmftmf.

The Tate Curve

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

For each q,lvertqrvert<1q \in \mathbb{C}, \; \lvert q \rvert \lt 1, there is an elliptic curve over \mathbb{C} defined by a Weierstrass equation

y 2+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<10 \lt \lvert q \rvert \lt 1 the elliptic curve is isomorphic to */q \mathbb{C}^* /q^\mathbb{Z} as a Riemann surface with group structure induced from *\mathbb{C}^* (equivalently, this curve corresponds to /Λ τ\mathbb{C}/ \Lambda_\tau where e 2πiτ=qe^{2\pi i \tau} = q). As a function of qq, a 4a_4 and a 6a_6 are analytic over the open disk and their power series at q=0q=0 have integral coefficients hence the Weierstrass equation defines an elliptic curve TT over [[q]]\mathbb{Z}[ [q] ]. We really would like to think of TT as an elliptic curve over ((q)):=[[q]][q 1]\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 TateK_{Tate} (an elliptic spectrum). As a cohomology theory K TateK_{Tate} is just KK-theory tensored with ((q))\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 212|1 field theories (annular theories) is the 0th0th-space of the spectrum K TateK_{Tate}.

Let SABSAB be the subcategory of 212|1-EB whose morphisms are annuli. Note that SABSAB does not contain tori. SABSAB in essence is completely determined by the supergroup 21\mathbb{R}^{2|1}. Let AFT nAFT_n be the space of natural transformations analogous to the definition of Stolz and Teichner.

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

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

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

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

Proof of Lemma. The spectral argument follows from having an S 1S^1 action. To see the second claim note that for fixed ll, + 21/l\mathbb{R}^{2|1}_+ /l \mathbb{Z} is a super Lie semi-group and the functor EE gives a representation + 21End(E(S l 1))\mathbb{R}^{2|1}_+ \to \mathrm{End} (E(S^1_l)) which is compatible with the super-semigroup law on 21\mathbb{R}^{2|1} given by

(z 1,z¯ 1,θ 1),(z 2,z¯ 2,θ 2)(z 1+z 2,z¯ 1+z¯ 2+θ 1θ 2,θ 1+θ 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 Lie( + 21)= z, z¯,Q\mathrm{Lie} (\mathbb{R}^{2|1}_+ ) = \langle \partial_z , \partial_{\overline{z}} , Q \rangle, where Q= θ+θ z¯Q = \partial_\theta + \theta \partial_{\overline{z}}. Under EE the vector fields map to L,L¯,L, \overline{L}, and GG respectively. Further, [Q,Q]=2Q 2=2 z¯,[Q,Q] = 2Q^2 = 2 \partial_{\overline{z}}, hence G 2=L¯G^2 = \overline{L}.

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

M Der¯\overline{\mathbf{M}^{Der}}

We can extend the standard toric variety construction to derived schemes by replacing \mathbb{C} by a fixed E E_\infty-ring RR. That is given a fan F={U α}F= \{U_\alpha \} we can build a derived scheme X FX_F where X F=varinjlim{U α}X_F = \varinjlim \{U_\alpha\} and U α=SpecR[S σ α]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}, 0}F_0 = \{ \{0\} , \mathbb{Z}_{\ge 0} \}, so X F 0=SpecR[ 0]=SpecR[q]X_{F_0} = \mathrm{Spec} R [\mathbb{Z}_{\ge 0}] = \mathrm{Spec} R[q]. Also, let F=σ n nF= \langle \sigma_n \rangle_{n \in \mathbb{Z}}, where

σ n={(a,b)×nab(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 FF 0F \to F_0 and consequently an induced map on varieties f:X FSpecR[q]f: X_F \to \mathrm{Spec} \; R[q].

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

  1. f 1(q)G mf^{-1} (q) \cong \mathbf{G}_m for q0q \neq 0;
  2. f 1(0)f^{-1} (0) is an infinite chain of rational curves, each intersecting the next in a node.

Consider the automorphism τ:××\tau : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z} defined by τ(a,b)=(a,b+a)\tau (a,b) = (a, b+a). Note that τ(σ n)=τ(σ n+1)\tau (\sigma_n ) = \tau (\sigma_{n+1}), so τ\tau preserves the fan FF and consequently is an automorphism of the resulting toric variety X FX_F which we also denote by τ\tau. Then

  1. τ\tau acts on f 1(q)f^{-1} (q) by multiplication by qq, for q0q \neq 0;
  2. τ\tau acts freely on f 1f^{-1}.

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

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

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

We call the derived scheme TT in the theorem above the Tate curve. It is a fact that the restriction of TT to the punctured formal disk SpecR((q))\mathrm{Spec} \; R ( (q) ) is an elliptic curve isomorphic to G m/q \mathbf{G}_m / q^\mathbb{Z}.

We then see that an orientation of TT is equivalent to an orientation of G m\mathbf{G}_m which is equivalent to working over the complex KK-theory spectrum KK. Therefore, the oriented Tate curve is equivalent to a map

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

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

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

Finally, one can show that the underlying scheme π 0M Der¯\pi_0 \overline{\mathbf{M}^{Der}} is M 1,1¯\overline{\mathbf{M}_{1,1}}.

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

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