Abstract We sketch how to compactify such that the underlying scheme is the Deligne-Mumford compactification of .
This is a sub-entry of
see there for background and context.
Here are the entries on the previous sessions:
Let be the derived Deligne-Mumford moduli stack of oriented elliptic curves. Recall that the underlying Deligne-Mumford stack is the classical Deligne-Mumford stack which is the fine moduli stack of elliptic curves. We would like to construct a derived Deligne-Mumford stack such that is the classical compactification .
Recall that one can define the -ring as the global sections of . If we take global sections of then we get the -ring .
Let us focus on elliptic curves over . The coarse moduli space of such elliptic curves is again with the classifying map given by the -invariant. is not compact, but we can compactify the moduli space by allowing curves with nodal singularities (generalized elliptic curves).
For each , there is an elliptic curve over defined by a Weierstrass equation
If the elliptic curve is isomorphic to as a Riemann surface with group structure induced from (equivalently, this curve corresponds to where ). As a function of , and are analytic over the open disk and their power series at have integral coefficients hence the Weierstrass equation defines an elliptic curve over . We really would like to think of as an elliptic curve over .
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 (an elliptic spectrum). As a cohomology theory is just -theory tensored with .
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 field theories (annular theories) is the -space of the spectrum .
Let be the subcategory of -EB whose morphisms are annuli. Note that does not contain tori. in essence is completely determined by the supergroup . Let be the space of natural transformations analogous to the definition of Stolz and Teichner.
Theorem (Theorem 2.2.2 of Cheung). For each , we have
Proof. Let be the cylinder obtained by identifying non-horizontal sides of the parallelogram in the upper-half plane spanned by and for and .
Let be a degree 0 field theory. The family of cylinders has the properties that and . For a fixed , is a commuting family of trace class (and hence compact) operators depending only on modulo . Therefore, by writing we can write , where are unbounded operators with discrete spectrum.
Lemma. The spectrum of . Also, , where is an odd operator.
Proof of Lemma. The spectral argument follows from having an action. To see the second claim note that for fixed , is a super Lie semi-group and the functor gives a representation which is compatible with the super-semigroup law on given by
Now , where . Under the vector fields map to and respectively. Further, hence .
We see that a degree 0 theory is determined by a pair of operators . By analyzing the spectral decomposition of the pair one can construct a weak equivalence of categories (a homotopy equivalence of geometric realizations), where is a certain category of Clifford-modules. Following work of Segal it is proved that . The nonzero degrees follow a similar line of reasoning.
We can extend the standard toric variety construction to derived schemes by replacing by a fixed -ring . That is given a fan we can build a derived scheme where and is an affine derived scheme. We will define the (derived) Tate curve as the formal completion of the quotient of a toric variety.
Let , so . Also, let , where
Note that by projection onto the first factor we have a map of fans and consequently an induced map on varieties .
Consider , one can show that 1. for ; 1. is an infinite chain of rational curves, each intersecting the next in a node.
Consider the automorphism defined by . Note that , so preserves the fan and consequently is an automorphism of the resulting toric variety which we also denote by . Then 1. acts on by multiplication by , for ; 1. acts freely on .
Now define to be the formal completion of along . Similarly, define as the formal completion of along . One can show that which is the multiplicative group generated by acts freely on .
Define to be the formal (derived) scheme .
Theorem (Lurie/Grothendieck). The formal derived scheme is the completion of a unique derived scheme over . That is, there exists a unique derived scheme such that .
We call the derived scheme in the theorem above the Tate curve. It is a fact that the restriction of to the punctured formal disk is an elliptic curve isomorphic to .
We then see that an orientation of is equivalent to an orientation of which is equivalent to working over the complex -theory spectrum . Therefore, the oriented Tate curve is equivalent to a map
Now note that the involution defined by preserves the fan and . We allow to be complex conjugation on thought of as a group action of , so we have a map .
We can define a new derived Deligne-Mumford stack by forming an appropriate pushout square.
Finally, one can show that the underlying scheme is .
There are many subtleties associated with . For instance, we would like to glue the universal curve over with to obtain a universal elliptic curve over , however the result is only a generalized elliptic curve; it is not a derived group scheme over as it only has a group structure over the smooth locus of the map . Lurie asserts it is possible to construct the necessary geometric objects over (I guess this will show up in DAG VII or VIII). The global sections of the structure sheaf thus constructed is the spectrum .