# John Lott, Chern-Simons, Differential \(\K\dash\)theory, Operator Theory 

:::{.remark}
Chern character form: $M$ smooth, $E$ a Hermitian vector bundle, $\nabla$ a compatible connection, get a closed form
\[
\ch \nabla = \Tr e^{-\nabla^2} \in \Omega^{\text{even}}(M)
.\]
For two connections $\nabla_i$, take a homotopy $\nabla_s = s\nabla_1 + (1-s)\nabla_0$, the Chern-Simons form is
\[
\CS(\nabla_0, \nabla_1)\da \int_0^1 
\Tr\qty{ \dd{\nabla_s}{s} e^{-\nabla_s^2} } \ds
.\]
Alternative construction of Chern character and Chern-Simons form due to Quillen: instead of interpolating between $\nabla_0, \nabla_1$, interpolate between $\matt{\nabla_0}{0}{0}{\nabla_1}$ and $\infty$.
:::

:::{.remark}
Recall 
\[
\K^0 = { \ZZ\adjoin{ \ts{0\to A\to B \to C\to 0}} \over  \gens{[B] = [A] + [C]}}
.\]
Differential \(\check\K\dash\)theory: for $\K^0$, quadruples $(E, h^E, \nabla^E, \omega)$ with $E\in \VectBundle\slice M$, $h^E$ a Hermitian metric, $\nabla$ a Hermitian connection, $\omega$ an auxiliary form.
Quotient by relations $\mce_2 = \mce_1 + \mce_3$ when the $E_i$ form a SES of vector bundles, $\omega_2 = \omega_1 + \omega_3 + \eps$ where $\eps$ is a correction term.
Forget everything but the vector bundle to get a map $\check \K^0 \to \K^0$ and a Chern character $\check \K^0(M) \to \Omega_K^{\text{even}}(M)$.
Kernel is $\K^{-1}$ and this forms a SES, so this mixes $\K$ and $\Omega$.
:::

:::{.remark}
Integration over the fiber in \(\K\dash\)theory: index maps $\K(M) \to \K(B)$, see Atiyah-Singer index formula.
Manifests as an equality of numerical indices.
Joint work with Dan Freed establishes a similar result for $\check K$ using local index theory.
Can compute certain $\eta$ invariants without using analysis.
:::

:::{.remark}
Idea: \(\K\dash\)theory of finite dimensional vector bundles with connections.
For today, how to generalize to infinite dimensions with super connections.
Twisted \(\K\dash\)theory: use elements in $H^3$.
Problem: can't define $\Tr e^{-A^2}$ for $A$ an operator, expanding and taking the 0th term already yields $\infty-\infty$!
Fix: super connections, a sum $A = \sum A_{[i]}$, and $\Tr_s$ a super trace.
Idea: $A_{[i]}$ are in $\Omega^i(M; ?)$ with restrictions.
See $C_2\dash$graded Hilbert bundles over a manifold $M$.
:::

:::{.remark}
What should the structure group $G$ be..?
For fibers $H$, should have a subgroup $G\leq \U(H)$ (the unitary transformations).
Needs to be general enough to include $\Diff(Z)$ for $Z$ the fibers.
Construct using a type of Dirac operator, pseudo differential operators.
Define structure group as even unitary transformations intersected with a certain space of 0th order pseudo differential operators $\mathrm{op}^0$.
See Bismut-Cheeger $\eta$ form, $\eta(A, \infty)$ which "interpolates to $\infty$" as before.
:::

:::{.remark}
Generators for $\check \K^0$: triples $(\mch, A, \omega)$ with $\mch\to M$ a $C_2\dash$graded Hilbert bundle, $A$ a super connection, plus conditions.
Turns out to be isomorphic to standard $\check \K$ from before.
Unclear if the Hopkins-Singer model is isomorphic.
Standing assumptions: compact fibers, fiberwise tangent bundle is $\Spinc$.
Need a horizontal distribution.
Define an index as the image of a map $\check \K^0(M)\to \check \K^0(B)$ constructed by pushforwards.
:::

:::{.remark}
Twisted \(\K\dash\)theory: $H^3(M; \ZZ)$ classifies $\U_1$ gerbes on $M$.
Gerbes: $\mcu\covers M$, line bundles $\mcl_{ab}$ on double intersections, on triples $\mcl_{ab}\tensor \mcl_{bc} \iso \mcl_{ac}$, and on quadruples a cocycle condition.
Can define $\U_1$ connections on a gerbe.
Twisted Hilbert bundles: maps $\phi_{ab}: \mch_a \tensor \mcl_{ab}\to \mch_b$.
Define super connection on the open cover, plus compatibility on double overlaps to define globally.
Closed: $(d + H\wedge)\ch(A) = 0$.
Take same generators, quotient by new relations.
Theorem: this new twisted $\check \K$ only depends on the class of the gerbe in $H^3$, and is independent of choices for connective structures and curving.
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:::{.remark}
Open question: showing this is isomorphic to other models.
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