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 $\mathbb{Z}_2$-graded vector bundle $V \to M$ is an odd derivation on $\Omega^\bullet(M,V)$

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

$$End(V)$$

$$\nabla_s = \omega_0 + \nabla + \omega_2 + \omega_3$$

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

unitary superconnection on $\mathbb{Z}_2$-graded unitary bundles $V$ with map as a above look like

$$\nabla_s = \left( \array{ & f^* \\ f & } \right) + \nabla$$

Chern character by the usual formulas

$$ch(\nabla_s) := sTr e^{\nabla^2}$$

2) index theory

definition Let $M$ be smooth Riemannian and $Spin$, The Dirac operator associated to $(V \to M, \nabla_s)$ is defined by

• • $$D(\nabla_s) : \Gamma(S \otimes V) \stackrel{\nabla_s \otimes 1 \oplus 1 \otimes \nabla_s}{\to} \Omega^\bullet(M, S \otimes V) \stackrel{c(.)}{\to} \Gamma(S \otimes V)$$

This is

• an elliptic operator;

• formally self adjoint

• of the form

  $$D(\nabla_s) = \left( \array{ & D'(\nabla_s) \\ D'(\nabla_s) } \right)$$

• theorem (corollary of Atiyah-singer index theory)

  $$index(D(\nabla_s)) = index(D(\nabla)) = \int_M \hat A(\Omega^m) ch(\nabla_s)$$

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

$$Tr \exp(-t D(\nabla_s)^2 ) = index(D(\nabla_s))$$

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

$$\exp(-t D(\nabla_s)^2) \psi(x) = \int_M p_t(x,y) \psi(y) d y$$

$$Tr \exp(-t D(\nabla_s)^2) = \int_M Tr p_t(x,x) d vol$$

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

$$\lim_{t \to 0} Tr p_t(x,x) d vol \neq (2 \pi i)^{-n/2} [ \hat A(\Omega^m) ch(\nabla_s) ]_n$$

here $n = dim X$ 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

$$\nabla_s^t := |t|^{-1/2} \omega_0 + \nabla + |t|^{1/2} \omega_2 + \cdots$$

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

$\pi : M \to B$

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

$$p : T(M) \to T(M/B)$$

$$\array{ V, \nabla_s \\ \downarrow \\ M \\ \downarrow^\pi \\ B }$$

$\pi_* (V)$ : a fibre at $y \in B$ is

$\Gamma_y(S^{M/B} \otimes V)$

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 $V$

$$\pi_! \nabla_s = \pi_! \nabla + \pi_! \omega$$

with $\nabla_s = \nabla + \omega$

$$[\pi_! \omega_!]_{\omega}(\xi_1, \cdots, \xi_i) = c^{M/B}(2 (\tilde \xi_1), 2(\tilde \xi_2) \cdots 2(\tilde \xi_k))$$

$$\pi^r = (\pi, r g^{M/B}, P)$$

$$\lim_{t \to 0} ch(\pi_!^t \nabla_s) = (2 \pi i)^{dim M/B} \pi_* [ \hat A(\Omega^{M/B} ch(\nabla_s)) ]$$

the scalings are related by

$$\pi_!^t(\nabla_s) = [\pi_! \nabla_s^{1/t}]^t$$

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, $BV_\infty$ and $HyperComm_\infty$

(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

• Higher Hochschild Cohomology and Differential Forms on Mapping Spaces

)

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

$$(FinSet, \coprod) \to (ChainComplexes, \otimes)$$

generalize this

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

$$A : (FinSet, \coprod) \to (ChainComplexes, \otimes)$$

i.e. there exists a natural weak equivalence

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

that respects the obvious coherence properties

generalized

• 1) co-algebras

• 2) any operad

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

Then weak partial algebras can be functorially replaced by $E_\infty$-algebras

example

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

$$X^j = Map(j,X) \leftarrow Map(k,X) = X^k$$

pass to the chains version of this

$$Ch_*(X^j) \leftarrow Ch_*(X^k)$$

$$Ch^*(X^j) \to Ch^*(X^k)$$

by Kuenneth formula we have a chain equivalence

$$C_*(X^j) \otimes C_*(X^k) \to C_*(X^{j+k})$$

and similarly for cochains.

so this gives two things:

• a partial coalgebra on $C_*(X)$

• a partial algebra on $C^*(X)$

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

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

simplicial object in $ChainCompl$, so total complex

$$CH^\gamma(A)$$

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 $A = \Omega(X)$, then $CH^\gamma(A)$ computes cohomology of $X^\gamma$, if $X$ is sufficiently connected

example

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

$$CH^{S^1}(A) = \prod_{n \geq 0} A \otimes A^{\otimes n}$$

is the Hochschild complex

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

calculate:

$$\exp(| \otimes x) = | + | \otimes x + | \otimes x \otimes x + | \otimes x \otimes x \otimes x + \cdots +$$

$$D \exp(1 \otimes x) = (1 \otimes d x + x \cdot x) \cdot e^{1 \otimes x}$$

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

this reminds us of curvature and connection

this can be taken further

let $A = \Omega^\bullet(M)$ be differential forms on $M$

$$\array{ && CH^{S^1}(A) (\simeq \Omega(M^{S^1})) \\ &\nearrow & \downarrow \\ K(M)&\stackrel{ch}{\to}&\Omega(M) }$$

commutes (due to some people)

example 2

$Y = I$ (the interval)

then $CH^I(A)$ is the 2-sided bar construction

more generally $CH(A, M, N) = \prod_{n \geq 0} M \otimes A^{\otimes n} \otimes N$

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

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

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

Let $D$ be differential on $CH^I$

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

Set

$$\Delta = [D, D^*]$$

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…

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