Paths, loops and fusion
Seminar on Loops and Gerbes
Richard Melrose
Department of Mathematics
Massachusetts Institute of Technology
April 20, 2016
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
1 / 16
Outline
I will briefly describe the loop space of a manifold, concentrating on
finite-energy (H 1 ) paths and loops.
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
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Interval and circle
Consider as basic parameter space the interval [0, π].
We can realize the circle as the ‘fusion’ of two intervals
S = [0, π] ∪ [0, π]0 /{0 = 00 , π = π 0 }.
Here we think of the second variable as reversed so
[π, 2π] 3 t 7−→ t 0 = 2π − t ∈ [0, π]0 .
This recovers the circle S = [0, 2π]/(0 = 2π) as a Lipschitz
manifold – not quite smooth (altough smoothable)!
π
00
0
π0
‘Fusion’ of two intervals
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Functions of one variable
Consider the Hilbert spaces of ‘finite-energy’ (real-valued)
functions
L2 [0, π] ⊃ H 1 [0, π], L2 (S) ⊃ H 1 (S), 0 ≤ s ≤ 1.
(1)
For periodic functions these are determined by their Fourier
expansions
X
X
v=
uk eikx in L2 (S), v ∈ H 1 ⇐⇒
(1 + |k |2 )|uk |2 < ∞,
k ∈Z
0
1
C (S) ⊃ H (S) is a C
k
∞
(2)
ring.
We can define H 1 [0, π] as the image under restriction
Then fusion of the intervals gives
H 1 (S) = {(u, u 0 ) ∈ (H 1 [0, π])2 ; u(0) = u 0 (π), u(π) = u 0 (0)}
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
(3)
3 / 16
Geodesic balls in the manifold
Let M be a compact manifold; choose a Riemann metric on it.
For 0< below the injectivity radius the balls
{B(m, )}m∈M give a good cover of M with a finite subcover.
The exponential map is a diffeomorphism onto the geodesic ball
expm {w ∈ Tm M; |w|g < } −→ B(m, )
The transition maps for this cover are diffeomorphisms of open
subsets of vector spaces expm exp−1
m0 .
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Path space and tubes
The finite-energy path space of a compact manifold is
PE M = PM = {λ : [0, π] −→ M; λ is in H 1 in local coordinates.}
Pulling back λ∗ TM is an H 1 bundle (trivial for oriented M) on [0, π].
The geodesic tubes give a good coordinate cover of PM :
U(λ) = {λ0 ∈ PM; λ0 (t) ∈ B(λ(t), ), ∀ t ∈ [0, π]}
exp−1
λ(∗)
←→ {τ ∈ H 1 ([0, π]; λ∗ TM; sup |τ | < }.
The coordinate transition maps exp−1
λ2 (∗) ◦ expλ1 (∗) are then smooth
∞
making PM into a C Hilbert manifold (it is better than this).
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Path fibration
If {ti }N
i=1 ⊂ [0, π] is a finite set of distinct points then
PM −→ M N , λ 7−→ {λ(ti )}
is a fibre bundle.
In particular it is open and surjective and maps tubes to products
of balls
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Paths to loops
The most important case of this fibration is to the two end-points,
PM −→ M 2 , by default PM is interpreted as this bundle.
The importance comes from the basic ‘fusion’ construction
identifying the fibre product with loop space
P [2] M = LE M = LM
(4)
using the earlier identification of the circle with two paths (the
second reversed). We write γ = j(λ1 , λ2 ).
The discussion above of tubes as a coordinate patches for PM
extends to LM (and higher fibre products) again giving good open
covers.
Abstractly the tubes on LM are fibre products of those on PM.
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Fusion
The fundamental notion of the fusion product was introduced by
Stolz and Teichner.
Fusion arises from the three ‘simplicial maps’ (dropping a factor)
p
i
P [3] M −→
P [2] M = LM, i = 1, 2, 3.
δ2
(5)
δ3
δ1
Richard Melrose ( Department of Mathematics Massachusetts Institute
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Fusion product
These maps generate a differential on functions, for instance
f : LM −→ U(1) =⇒ δ ∗ f = (δ1∗ f )(δ2∗ f )−1 (δ3∗ f ) : P [3] M −→ U(1).
(6)
There are higher differentials with δ 2 = 1.
Take a smooth U(1) principal bundle with connection ∇ over M :
U(1)
(7)
L
M.
The holomony of L, ∇ is a function f : LM −→ U(1) with δ ∗ f = 1.
This is the basic example of (geometric) transgression.
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
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Figure-of-eight
The three-point fibration, including both end-points and central
point will be denoted P τ M −→ M 3 .
There are again three simplicial maps (which involve more
reparametrization) forming ‘figure-of-eight’ configurations
P τ [2] M
qi
/ LM
(8)
qc
qa
qb
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Fusive structure
Again there is a differential (but no higher ones it seems) on
functions on
f 7−→ δ8 f = (qa∗ f )(qb∗ f )(qc∗ f )−1 : P τ [2] M −→ U(1).
(9)
Again the holomony of a circle bundle satisfies δ8 f = 1.
In fact this characterizes holonomy functions up to homotopy
(through such functions).
Theorem (Kottke-M.)
The abelian group of homotopy classes of continuous functions
f : LM −→ U(1) which are fusive in the sense that δ ∗ f = 1, δ8 f = 1 is
naturally H 2 (M; Z).
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
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Regression
To complete the proof of this result one needs a construction of a
circle bundle from f as ‘descent data’.
Take the trivial bundle PM × U(1).
Identify points by (λ1 , z1 ) ∼ (λ2 , z2 ) if z1 = f (γ(λ1 , λ2 ))z2 .
Fusion makes this an equivalence relation, giving a principal U(1)
bundle L̃f over M 2 .
The figure-or-eight condition shows that under the simplicial maps
πi : M 3 −→ M 2 , π1∗ L̃f ⊗ (π2∗ L̃f )−1 ⊗ π3∗ L̃f ' U(1) is trivial over M 3 .
It follows that L̃f = π1∗ Lf ⊗ (π2∗ Lf )−1 where Lf is defined up to
equivalence.
Of course, the claim is that trangression and regression are
inverses of each other.
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Transgression/Regression pairs
Chris will talke about (iterated) transgression of integral
cohomology in the form of Čech cohomology.
These trasgressed objects realize higher gerbes – Vishesh will at
least start a discussion of this.
In particular bundle gerbes (in the sense of Murray) transgress to
fusive circle bundles over LM.
Stolz and Teicher introduced fusion (figure-of-eight is nominally
older!) to complete a program started by Atiyah and Witten
Theorem (Stolz-Teichner)
Spin structures on (oriented) manifolds, M, correspond precisely to
‘fusion loop-orientations’ on LM.
Loop-orientation here corresponds to continuous, i.e. locally
constant, maps e : LTM −→ Z2 which take both signs on each
fibre of LTM −→ LM. Fusion implies the figure-of-eight condition.
Richard Melrose ( Department of Mathematics Massachusetts Institute
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April 20, 2016
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Loop-Spin structures
The ‘higher’ version of this theorem is relevant for the
Dirac-Ramond operator:
Theorem (Waldorf, Kottke-M.)
String structures on a spin manifold M are in 1-1 correspondence with
Loop-spin structures on LM.
This can be explained in due course.
The upshot is that the string condition, 12 p1 (M) = 0, where
1
4
2 p1 ∈ H (M; Z) is the Spin-Pontryagin class, is equivalent to the
existence of a principal bundle with structure group the central
extension of the loop group on Spin(n) over LM which is fusive in
an appropriate sense.
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
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Bott-Virasoro
There are two important, indeed I believe fundamental, notions
that I have left out so far. One is equivarience under
reparameterization.
If Dff+ (S) is the group of orientation-preserving diffeomorphisms
of the circle then there is an action of Dff+ (S) on the loop space(s)
by reparameterization.
For finite energy loops this extends to an action of the group of
orientation-preserving Lipschitz diffeomorphisms.
For loop groups and principal bundles with structure group which
is a central extension of a loop group one expect the action of the
central extension of Dff+ , the Bott-Virasoro group.
Richard Melrose ( Department of Mathematics Massachusetts Institute
Loopsof Technology )
April 20, 2016
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Litheness
Litheness corresponds to ‘pointwise’ regularity of derivatives.
The tangent space Tγ LM ' (H 1 (S))n , essentially the linear model.
Riesz’ Representation Theorem then identifies the derivative of a
function with an element df (γ) ∈ Tγ∗ LM = (H 1 (S))n .
However this identification uses the inner product on H 1 and in
terms of the integral (L2 ) pairing this means df (γ) ∈ (H −1 (S))n .
This is an essential part of the difficulty in defining the
Dirac-Ramond operator.
Litheness (valid for LM but meaningless for general Hilbert
manifolds) means that the condition df (γ) ∈ (H 1 (S))n with respect
to the L2 pairing is meaningful (not automatic, hence the notion of
lithe structures).
This also plays an essential role in resolving the conflict arising in
constructions using the fibre-product to regain
reparameterization-invariance.
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April 20, 2016
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