# nLab Oberwolfach Workshop, June 2009 -- Wednesday, June 10

Here are notes by Urs Schreiber for Wednesday, June 10, from Oberwolfach.

## Alexander Kahle: superconnections and index theory

• 1) superconnections

• 2) index theory

• 3) sketch some proofs

### 1) superconnections

definition A superconnection ${\nabla }_{s}$ on a ${ℤ}_{2}$-graded vector bundle $V\to M$ is an odd derivation on ${\Omega }^{•}\left(M,V\right)$

superconnections form an affine space modeled on ${\Omega }^{•}\left(M,\mathrm{End}\left(V\right){\right)}^{\mathrm{odd}}$

$\mathrm{End}\left(V\right)$End(V)
${\nabla }_{s}={\omega }_{0}+\nabla +{\omega }_{2}+{\omega }_{3}$\nabla_s = \omega_0 + \nabla + \omega_2 + \omega_3

class in $K$-theory given by a map $V\stackrel{f}{\to }W$

unitary superconnection on ${ℤ}_{2}$-graded unitary bundles $V$ with map as a above look like

${\nabla }_{s}=\left(\begin{array}{cc}& {f}^{*}\\ f& \end{array}\right)+\nabla$\nabla_s = \left( \array{ & f^* \\ f & } \right) + \nabla

Chern character by the usual formulas

$\mathrm{ch}\left({\nabla }_{s}\right):=\mathrm{sTr}{e}^{{\nabla }^{2}}$ch(\nabla_s) := sTr e^{\nabla^2}

### 2) index theory

definition Let $M$ be smooth Riemannian and $\mathrm{Spin}$, The Dirac operator associated to $\left(V\to M,{\nabla }_{s}\right)$ is defined by

This is

• an elliptic operator;

• of the form

$D\left({\nabla }_{s}\right)=\left(\begin{array}{cc}& D\prime \left({\nabla }_{s}\right)\\ D\prime \left({\nabla }_{s}\right)\end{array}\right)$D(\nabla_s) = \left( \array{ & D'(\nabla_s) \\ D'(\nabla_s) } \right)
• theorem (corollary of Atiyah-singer index theory)

$\mathrm{index}\left(D\left({\nabla }_{s}\right)\right)=\mathrm{index}\left(D\left(\nabla \right)\right)={\int }_{M}\stackrel{^}{A}\left({\Omega }^{m}\right)\mathrm{ch}\left({\nabla }_{s}\right)$index(D(\nabla_s)) = index(D(\nabla)) = \int_M \hat A(\Omega^m) ch(\nabla_s)

so superconnections don’t give new topological data: they are geometric objects with the same underlying topology as ordinary connections but refined “geometry”

recall that Atiyah-Singer says that

$\mathrm{Tr}\mathrm{exp}\left(-tD\left({\nabla }_{s}{\right)}^{2}\right)=\mathrm{index}\left(D\left({\nabla }_{s}\right)\right)$Tr \exp(-t D(\nabla_s)^2 ) = index(D(\nabla_s))

the heat semi-group is smoothing, therefore it is represented by a kernel

$\mathrm{exp}\left(-tD\left({\nabla }_{s}{\right)}^{2}\right)\psi \left(x\right)={\int }_{M}{p}_{t}\left(x,y\right)\psi \left(y\right)dy$\exp(-t D(\nabla_s)^2) \psi(x) = \int_M p_t(x,y) \psi(y) d y
$\mathrm{Tr}\mathrm{exp}\left(-tD\left({\nabla }_{s}{\right)}^{2}\right)={\int }_{M}\mathrm{Tr}{p}_{t}\left(x,x\right)d\mathrm{vol}$Tr \exp(-t D(\nabla_s)^2) = \int_M Tr p_t(x,x) d vol

the following expected formula which holds for ordinary connections (due to Ezra Getzler) no longer holds directly for superconnections

$\underset{t\to 0}{\mathrm{lim}}\mathrm{Tr}{p}_{t}\left(x,x\right)d\mathrm{vol}\ne \left(2\pi i{\right)}^{-n/2}\left[\stackrel{^}{A}\left({\Omega }^{m}\right)\mathrm{ch}\left({\nabla }_{s}\right){\right]}_{n}$\lim_{t \to 0} Tr p_t(x,x) d vol \neq (2 \pi i)^{-n/2} [ \hat A(\Omega^m) ch(\nabla_s) ]_n

here $n=\mathrm{dim}X$ is the dimension of the manifold

problem is that components in a superconnections scale in a different

to make it true, we need to rescale

${\nabla }_{s}^{t}:=\mid t{\mid }^{-1/2}{\omega }_{0}+\nabla +\mid t{\mid }^{1/2}{\omega }_{2}+\cdots$\nabla_s^t := |t|^{-1/2} \omega_0 + \nabla + |t|^{1/2} \omega_2 + \cdots

A Riemannian map is a triple $\left(\pi ,g,P\right)$

$\pi :M\to B$

a family with fibers close Spin manifolds, ${g}^{M/B}$ a metric onm the fibers,

$p:T\left(M\right)\to T\left(M/B\right)$p : T(M) \to T(M/B)
$\begin{array}{c}V,{\nabla }_{s}\\ ↓\\ M\\ {↓}^{\pi }\\ B\end{array}$\array{ V, \nabla_s \\ \downarrow \\ M \\ \downarrow^\pi \\ B }

${\pi }_{*}\left(V\right)$ : a fibre at $y\in B$ is

${\Gamma }_{y}\left({S}^{M/B}\otimes V\right)$

due to Bismut we get from a connection on the top a superconnecction on the bottom (which is one of the main original motivations to be interested in superconnection in the first place), which we tweak here a bit to get a superconnection on $B$ from a superconnection on $V$

${\pi }_{!}{\nabla }_{s}={\pi }_{!}\nabla +{\pi }_{!}\omega$\pi_! \nabla_s = \pi_! \nabla + \pi_! \omega

with ${\nabla }_{s}=\nabla +\omega$

$\left[{\pi }_{!}{\omega }_{!}{\right]}_{\omega }\left({\xi }_{1},\cdots ,{\xi }_{i}\right)={c}^{M/B}\left(2\left({\stackrel{˜}{\xi }}_{1}\right),2\left({\stackrel{˜}{\xi }}_{2}\right)\cdots 2\left({\stackrel{˜}{\xi }}_{k}\right)\right)$[\pi_! \omega_!]_{\omega}(\xi_1, \cdots, \xi_i) = c^{M/B}(2 (\tilde \xi_1), 2(\tilde \xi_2) \cdots 2(\tilde \xi_k))
${\pi }^{r}=\left(\pi ,r{g}^{M/B},P\right)$\pi^r = (\pi, r g^{M/B}, P)
$\underset{t\to 0}{\mathrm{lim}}\mathrm{ch}\left({\pi }_{!}^{t}{\nabla }_{s}\right)=\left(2\pi i{\right)}^{\mathrm{dim}M/B}{\pi }_{*}\left[\stackrel{^}{A}\left({\Omega }^{M/B}\mathrm{ch}\left({\nabla }_{s}\right)\right)\right]$\lim_{t \to 0} ch(\pi_!^t \nabla_s) = (2 \pi i)^{dim M/B} \pi_* [ \hat A(\Omega^{M/B} ch(\nabla_s)) ]

the scalings are related by

${\pi }_{!}^{t}\left({\nabla }_{s}\right)=\left[{\pi }_{!}{\nabla }_{s}^{1/t}{\right]}^{t}$\pi_!^t(\nabla_s) = [\pi_! \nabla_s^{1/t}]^t

### determinant line bundles

(…skipping a bunch of remarks…)

### 3) sketch of some proofs

(no time, as expected)

### $\infty$-operads

Baronikov-Kontsevich passage

## Gabriel Drummond-Cole; $\infty$-operads, ${\mathrm{BV}}_{\infty }$ and ${\mathrm{HyperComm}}_{\infty }$

(was hard to take typed notes of this otherwise pretty cool talk, does anyone have handwriitten notes?)

## Scott Wilson: Categorical algebra, mapping spaces and applications

(for closely related blog entry see

)

outline

• language for some elementary algebraic topology

• application to generalizatons of Hochschild complexes

• Examples

• invariants on mapping spaces

• contributions related to def of Laplacian

def/lema

A commutative associative differential graded algebra is (equivalently given by) a strict monoidal functor

$\left(\mathrm{FinSet},\coprod \right)\to \left(\mathrm{ChainComplexes},\otimes \right)$(FinSet, \coprod) \to (ChainComplexes, \otimes)

generalize this

def a partial DGA is a monoidal functor with coherence map given by weak equivalence in the model structure

$A:\left(\mathrm{FinSet},\coprod \right)\to \left(\mathrm{ChainComplexes},\otimes \right)$A : (FinSet, \coprod) \to (ChainComplexes, \otimes)

i.e. there exists a natural weak equivalence

$A\left(j\bigsqcup k\right)\stackrel{T}{\to }A\left(j\right)\otimes A\left(k\right)$A(j \sqcup k) \stackrel{T}{\to} A(j) \otimes A(k)

that respects the obvious coherence properties

generalized

• 1) co-algebras

• 3) note that ${\mathrm{FinSet}}_{*}$ (pointed finite sets) is a module over $\mathrm{FinSet}$, so generalize to modules, comodules, etc.

Then weak partial algebras can be functorially replaced by ${E}_{\infty }$-algebras

example

$X$ be a space $j\stackrel{f}{\to }k$

${X}^{j}=\mathrm{Map}\left(j,X\right)←\mathrm{Map}\left(k,X\right)={X}^{k}$X^j = Map(j,X) \leftarrow Map(k,X) = X^k

pass to the chains version of this

${\mathrm{Ch}}_{*}\left({X}^{j}\right)←{\mathrm{Ch}}_{*}\left({X}^{k}\right)$Ch_*(X^j) \leftarrow Ch_*(X^k)
${\mathrm{Ch}}^{*}\left({X}^{j}\right)\to {\mathrm{Ch}}^{*}\left({X}^{k}\right)$Ch^*(X^j) \to Ch^*(X^k)

by Kuenneth formula we have a chain equivalence

${C}_{*}\left({X}^{j}\right)\otimes {C}_{*}\left({X}^{k}\right)\to {C}_{*}\left({X}^{j+k}\right)$C_*(X^j) \otimes C_*(X^k) \to C_*(X^{j+k})

and similarly for cochains.

so this gives two things:

• a partial coalgebra on ${C}_{*}\left(X\right)$

• a partial algebra on ${C}^{*}\left(X\right)$

Let $Y$ be any finite simplicial space. A partial algebra

$\Delta \stackrel{gama}{\to }\mathrm{FinSet}\stackrel{A}{\to }\mathrm{ChainCompl}$\Delta \stackrel{\gama}{\to} FinSet \stackrel{A}{\to} ChainCompl

simplicial object in $\mathrm{ChainCompl}$, so total complex

${\mathrm{CH}}^{\gamma }\left(A\right)$CH^\gamma(A)

meaning generalization of Hochschild complex

• this is joint work with Tradler and Zanelli (spelling? probably wrong)

goes back to Pitashvili and more recently Gregory Ginot

For $A=\Omega \left(X\right)$, then ${\mathrm{CH}}^{\gamma }\left(A\right)$ computes cohomology of ${X}^{\gamma }$, if $X$ is sufficiently connected

example

let $A$ be a strict algebra, and $\gamma =Y={S}^{1}$ then

${\mathrm{CH}}^{{S}^{1}}\left(A\right)=\prod _{n\ge 0}A\otimes {A}^{\otimes n}$CH^{S^1}(A) = \prod_{n \geq 0} A \otimes A^{\otimes n}

is the Hochschild complex

there is also a shuffle product in the game, so this implies there is an exponential map

calculate:

$\mathrm{exp}\left(\mid \otimes x\right)=\mid +\mid \otimes x+\mid \otimes x\otimes x+\mid \otimes x\otimes x\otimes x+\cdots +$\exp(| \otimes x) = | + | \otimes x + | \otimes x \otimes x + | \otimes x \otimes x \otimes x + \cdots +
$D\mathrm{exp}\left(1\otimes x\right)=\left(1\otimes dx+x\cdot x\right)\cdot {e}^{1\otimes x}$D \exp(1 \otimes x) = (1 \otimes d x + x \cdot x) \cdot e^{1 \otimes x}

then: if $dx+x\cdot x=0$ then $D{e}^{1\otimes x}=0$

this reminds us of curvature and connection

this can be taken further

let $A={\Omega }^{•}\left(M\right)$ be differential forms on $M$

$\begin{array}{ccc}& & {\mathrm{CH}}^{{S}^{1}}\left(A\right)\left(\simeq \Omega \left({M}^{{S}^{1}}\right)\right)\\ & ↗& ↓\\ K\left(M\right)& \stackrel{\mathrm{ch}}{\to }& \Omega \left(M\right)\end{array}$\array{ && CH^{S^1}(A) (\simeq \Omega(M^{S^1})) \\ &\nearrow & \downarrow \\ K(M)&\stackrel{ch}{\to}&\Omega(M) }

commutes (due to some people)

example 2

$Y=I$ (the interval)

then ${\mathrm{CH}}^{I}\left(A\right)$ is the 2-sided bar construction

more generally $\mathrm{CH}\left(A,M,N\right)={\prod }_{n\ge 0}M\otimes {A}^{\otimes n}\otimes N$

with $M$ and $N$ $A$-modules sitting on the end of the interval

consider the case $A={\Omega }^{•}\left(\mathrm{Riemannian}\mathrm{manifold}\right)$ and $M=A$ and $N=\left({\Omega }^{•}\left(...\right),{d}^{*},\left(x\in A\right)\cdot \left(y\in N\right)={\star }^{-1}\left(x\wedge \star y\right)\right)\right)$

(the operatoin on $N$ here is the intersection product of forms)

Let $D$ be differential on ${\mathrm{CH}}^{I}$

let $D$ be differential on ${\mathrm{CH}}^{I}$ for normal structure, and and ${D}^{*}$ for $A,M,N$ as just described.

Set

$\Delta =\left[D,{D}^{*}\right]$\Delta = [D, D^*]

then acting with this $\Delta$ on something produces interesting non-linear differential equations related to Witten’t Morse-theory deformation of susy quantum mechanics and to Navier-Stokes’ equations in fluid dynamics…

Revised on August 6, 2009 14:14:10 by Urs Schreiber (134.100.222.156)