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 s\nabla_s on a 2\mathbb{Z}_2-graded vector bundle VMV \to M is an odd derivation on Ω (M,V)\Omega^\bullet(M,V)

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

End(V) End(V)
s=ω 0++ω 2+ω 3 \nabla_s = \omega_0 + \nabla + \omega_2 + \omega_3

class in KK-theory given by a map VfWV \stackrel{f}{\to} W

unitary superconnection on 2\mathbb{Z}_2-graded unitary bundles VV with map as a above look like

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

Chern character by the usual formulas

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

2) index theory

definition Let MM be smooth Riemannian and SpinSpin, The Dirac operator associated to (VM, s)(V \to M, \nabla_s) is defined by

    • D( s):Γ(SV) s11 sΩ (M,SV)c(.)Γ(SV) 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( s)=( D( s) D( s)) D(\nabla_s) = \left( \array{ & D'(\nabla_s) \\ D'(\nabla_s) } \right)
  • theorem (corollary of Atiyah-singer index theory)

    index(D( s))=index(D())= MA^(Ω m)ch( s) 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

Trexp(tD( s) 2)=index(D( s)) 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(tD( s) 2)ψ(x)= Mp t(x,y)ψ(y)dy \exp(-t D(\nabla_s)^2) \psi(x) = \int_M p_t(x,y) \psi(y) d y
Trexp(tD( s) 2)= MTrp t(x,x)dvol 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 t0Trp t(x,x)dvol(2πi) n/2[A^(Ω m)ch( s)] 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=dimXn = 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

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

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

π:MB \pi : M \to B

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

p:T(M)T(M/B) p : T(M) \to T(M/B)
V, s M π B \array{ V, \nabla_s \\ \downarrow \\ M \\ \downarrow^\pi \\ B }

π *(V)\pi_* (V) : a fibre at yBy \in B is

Γ y(S M/BV)\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 BB from a superconnection on VV

π ! s=π !+π !ω \pi_! \nabla_s = \pi_! \nabla + \pi_! \omega

with s=+ω\nabla_s = \nabla + \omega

[π !ω !] ω(ξ 1,,ξ i)=c M/B(2(ξ˜ 1),2(ξ˜ 2)2(ξ˜ k)) [\pi_! \omega_!]_{\omega}(\xi_1, \cdots, \xi_i) = c^{M/B}(2 (\tilde \xi_1), 2(\tilde \xi_2) \cdots 2(\tilde \xi_k))
π r=(π,rg M/B,P) \pi^r = (\pi, r g^{M/B}, P)
lim t0ch(π ! t s)=(2πi) dimM/Bπ *[A^(Ω M/Bch( s))] \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

π ! t( s)=[π ! s 1/t] 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, BV BV_\infty and HyperComm 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

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

i.e. there exists a natural weak equivalence

A(jk)TA(j)A(k) 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 *FinSet_* (pointed finite sets) is a module over FinSetFinSet, so generalize to modules, comodules, etc.

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

example

XX be a space jfkj \stackrel{f}{\to} k

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

pass to the chains version of this

Ch *(X j)Ch *(X k) Ch_*(X^j) \leftarrow Ch_*(X^k)
Ch *(X j)Ch *(X k) Ch^*(X^j) \to Ch^*(X^k)

by Kuenneth formula we have a chain equivalence

C *(X j)C *(X k)C *(X j+k) 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)C_*(X)

  • a partial algebra on C *(X)C^*(X)

Let YY be any finite simplicial space. A partial algebra

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

simplicial object in ChainComplChainCompl, so total complex

CH γ(A) 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=Ω(X)A = \Omega(X), then CH γ(A)CH^\gamma(A) computes cohomology of X γX^\gamma , if XX is sufficiently connected

example

let AA be a strict algebra, and γ=Y=S 1\gamma = Y = S^1 then

CH S 1(A)= n0AA 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:

exp(|x)=|+|x+|xx+|xxx++ \exp(| \otimes x) = | + | \otimes x + | \otimes x \otimes x + | \otimes x \otimes x \otimes x + \cdots +
Dexp(1x)=(1dx+xx)e 1x D \exp(1 \otimes x) = (1 \otimes d x + x \cdot x) \cdot e^{1 \otimes x}

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

this reminds us of curvature and connection

this can be taken further

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

CH S 1(A)(Ω(M S 1)) K(M) ch Ω(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=IY = I (the interval)

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

more generally CH(A,M,N)= n0MA nNCH(A, M, N) = \prod_{n \geq 0} M \otimes A^{\otimes n} \otimes N

with MM and NN AA-modules sitting on the end of the interval

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

(the operatoin on NN here is the intersection product of forms)

Let DD be differential on CH ICH^I

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

Set

Δ=[D,D *] \Delta = [D, D^*]

then acting with this Δ\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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Revised on August 6, 2009 14:14:10 by Urs Schreiber (134.100.222.156)