# Jonathan Rosenberg, Twisted derived equivalences and string theory dualities with B-field (Wednesday, July 20)

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Type II string theory: spacetime is $X = \RR^{10-2n} \times M$ where $M$ is compact Kähler with a Ricci-flat metric and trivial canonical (the Calabi-Yau condition), not necessarily simply connected.
For us: $M$ is smooth projective, $X$ is quasi-projective.
The physics is controlled by D-branes (D for Dirichlet), either complex submanifolds in IIB or isotropic submanifolds for the symplectic structure in IIA.
More generally, IIA branes are coherent sheaves, and IIA branes are classified by the Fukaya category.
In type IIB, there is interest in $\D^b \Coh(X)$, since complexes can capture joining branes, separating, collapsing into other configurations, etc.
There is a coarser topological classification given by topological \(\K\dash\)theory (Minasian-Moore, Witten). The D-branes give classes in $\KU^0(M)$ in IIB and $\KU^{-1}(M)$ for IIA.
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Today: orientifold string theories.
Classically, strings moving in time yield maps $\Sigma\to X$ where $\Sigma$ is the string worldsheet and $X$ is spacetime.
This is a $\sigma\dash$model.
Orientifold theory: ask for strings to be equivariant with respect to involutions $\iota$ and worldsheet parity operators -- if $\iota = \id_X$, this recovers type I string theory (forgetting string orientation).
In this theory, D-brane charges live in real \(\K\dash\)theory $\K R(X, \iota)$, which reduces to $\KO(X)$ when $\iota = \id_X$.
2015: joint work classifies orientifold string theories associated to (anti)holomorphic involutions on elliptic curves: there are 10.
One case corresponds to Karoubi and Donovan's twisted \(\K\dash\)theory.
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Can view an elliptic curve with antiholomorphic involution as a smooth genus one curve, possibly with no real points.
Karoubi and Weibel (2003) show that $\K R$ of the complex curve is related to $\K^\alg$ of the real variety.
Suggests that duality of orientifold string theories may be reflected in a equivalence of derived categories for *real* (rather than complex) curves.
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Derived equivalence: prototypical example is $\D(X) \cong \D(Y)$ for $X,Y$ smooth varieties over $k$ via Mukai duality.
Formulated in terms of the Poincaré sheaf on $X\times \hat{X}$ for $X$ an abelian variety and $\hat{X}$ its dual.
Integration over the fiber is really derived pushforward, so define the transform as
\[
\phi(\mcf) = \RR \pi_{2, *} (\mcl \Ltensor \pi_{1}^* \mcf)
.\]
AVs are simple examples of CYs?

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Elliptic curves are self-dual, so it works here, which suggests it may also work fiberwise for elliptic fibrations.
Căldăraru considers elliptic fibrations $f: X\to Z$ with no section.
The role of $\hat X$ is the relative Jacobian $\hat f: J\to Z$, the relative module space of semistable sheaves of rank 1 and degree zero on the fibers of $f$.
Then $\hat f$ is an elliptic fibration with section, but there's no universal sheaf to play the role of the Poincaré sheaf $\mcl$.
Two reasons for this:

- $J$ may have singularities. Not so serious, can resolve and replace $J$ by a small resolution
- May exist locally but not globally.

It turns out that $\mcl$ exists as a twisted sheaf: the transition functions may only agree up to a 2-cycle, defining a class $\alpha$ in the Brauer group $H_\et^2(J; \GG_m)$.
A similar recipe goes through: pull up to $X\fiberprod{Z} J$, tensor by the $\alpha\inv\dash$twisted sheaf $\mcl$, and push down to $X$.
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:::{.theorem title="JR 2017"}
View a complex elliptic curve $\bar{E}$ defined over $\bar{\RR} = \CC$ with a free antiholomorphic involution, regarded as $C(\CC)$ with its $\Gal(\CC/\RR)$ action for $C$ a smooth projective curve with $g=1$ defined over $\RR$ and $C(\RR) = \emptyset$.
Think of $C \to \spec \RR$ as an elliptic fibration without a section, and the relative Jacobian as $E\to \spec \RR$.
There is a Mukai equivalence $\D(C) \iso \D(E, \alpha)$ where $\alpha\in \Br(E)/\Br(\RR)$ is the unique nonzero class.
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This is an AG version of $T\dash$duality of orientifold theories, and $\alpha$ corresponds to the $B\dash$field.
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## Generalizations

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Generalizing from genus 1 curves to K3s: let $E_1, E_2$ be elliptic curves over $\CC$ with real $j\dash$invariant, so they admit antiholomorphic involutions.
Cook up a real structure on $X = (E_1\times E_2)/\mu_2$ and hence on its blowup $\hat X$ which is a Kummer K3.
Projection gives an elliptic fibration without a section.
The relative Jacobian gives a dual $K3$ fibred over $\PP^1$ with a canonical section and the same derived equivalence as in the theorem.
Can interpret as duality of orientifold type IIB string theories on K3s.
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Generalizing to genus 1 curves over general fields: let $\characteristic k = 0$ and $C\slice k$ be a smooth connected projective curve with $g=1$ and let $E$ be its Jacobian.
Then $J$ is elliptic over $k$ and $C$ is an $E\dash$torsor, so defines a class in $\delta(C) \in H^1_\et(\spec k, E(\bar k))$.
Can how a $d_2$ vanishes in the Leray spectral sequence 
\[
E_2^{p, q} H^p_\et(\spec k; \RR^q \pi_* \GG_m) \abuts H_\et^{p+q}(E; \GG_m)
.\]
Use Tsen's theorem to get vanishing.
Then $\Br(E) \cong \Br(E, \spec k) \oplus \Br(k)$ where the first term is the *relative Brauer group*.
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