Non-Abelian Poincaré lemma and Lie algebroids

31 Pages·2010·0.4 MB·English

by  Theodore Voronov

Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Non-Abelian Poincar´e lemma and Lie algebroids
Theodore Voronov
UniversityofManchester,Manchester,UK
Lyon, 5 – 10 April 2010
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
A familiar statement
Let ω ∈ Ω1(M,g) be a Lie-algebra-valued 1-form satisfying
dω+ω2 = 0. (1)
(Here ω2 = 1[ω,ω].) Then, if M is simply-connected, ω is a
2
“pure gauge”, i.e., there is a function g ∈ C∞(M,G) with values
in a Lie group G such that
ω = −dgg−1. (2)
Geometric interpretation: ∇ = d+ω is a connection (for which
ω is a connection 1-form); equation (1) is the zero curvature
condition; then equation (2) means that
d+ω ∼ d
(“trivial connection”), i.e., d+ω = g◦d◦g−1.
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Comparison with the Abelian case
If g is Abelian (e.g., g = R), the Maurer–Cartan equation
dω+ω2 = 0 becomes just dω = 0; also, −dgg−1 = −dlng, so
the above statement is nothing but the special case of the
Poincar´e lemma for 1-forms: every closed 1-form on a
simply-connected manifold is exact:
dω = 0 ⇒ ω = df (for a 1-form ω ∈ Ω1(M)).
Recall that, in general, the Poincar´e lemma says that on a
contractible manifold, every closed form is exact plus constant:
dω = 0 ⇒ ω = dσ+C (for an inhomogeneous form ω ∈ Ω(M)).
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
A “Non-Abelian Poincar´e lemma”
What is a non-Abelian analog of the Poincar´e lemma for
arbitrary forms?
Setup: let M be a (super)manifold and g be a Lie superalgebra
with a corresponding Lie supergroup G. Let ω ∈ Ωodd(M,g) be
an odd g-valued form on M (odd — w.r.t. total parity) satisfying
the Maurer–Cartan equation
dω+ω2 = 0.
Theorem (“Non-Abelian Poincar´e lemma”)
If M is contractible, then there is an even G-valued form
g ∈ Ωeven(M,G) such that
ω = gCg−1−dgg−1,
for some odd constant C ∈ g.
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Remarks to the Non-Abelian Poincar´e lemma
Remarks:
It is not essential that M is a supermanifold, but it is
essential that g = g ⊕g is a Lie superalgebra.
¯0 ¯1
For an ordinary manifold M, the form ω may be
inhomogeneous, ω = ω +ω +ω +..., so that
0 1 2
ω ∈ Ω2k+1 take values in g while ω ∈ Ω2k take values
2k+1 ¯0 2k
in g .
¯1
In general, ω is an odd pseudodifferential form, i.e.,
ω ∈ C∞(ΠTM,Πg).
An (even) G-valued form on M is by definition a map
g ∈ C∞(ΠTM,G). On ordinary M, g = g +..., where
0
g ∈ C∞(M,G ).
0 0
An odd constant C ∈ g looks innocent (or unimportant),
¯1
but it is exactly C that is essential for applications.
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Plan of the talk
1 Introduction
2 Non-Abelian homotopy formula
3 Application to Lie algebroids
4 “Non-linear” Lie algebroids
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Multiplicative push-forward
Let M be a supermanifold. Consider ω ∈ Ωodd(M×I,g) where g
is a Lie superalgebra. (Here I = [0,1].) Consider the projection
p: M×I → M. We may write ω = ω +dtω . We define
0 1
Texpp : Ωodd(M×I,g) → Ωeven(M,G)
*
by the formula
(cid:90) 1
Texpp : ω (cid:55)→ g = g(1,0) = Texp (−ω).
*
0
In general, g(t ,t ) = Texp(cid:82)t1(−ω) is the multiplicative integral
1 0 t0
along the fibers, which can be defined as the solution at time
t = t of the differential equation dg = −ω g such that g = 1 at
1 dt 1
t = t .
0
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Multiplicative ‘fiberwise Stokes formula’
Let ∆g := −dgg−1 be the Darboux derivative (the ordered
logarithmic derivative). Denote by curv: ω (cid:55)→ dω+ω2 the
‘curvature operator’ on odd g-valued forms.
Theorem
The following commutation formula holds:
∆◦Texpp +p ◦Adg(1,t)◦curv = p(cid:48)
* ∗ ∗
where p(cid:48) : ω (cid:55)→ ω | −Adgω | is the (twisted) integral
∗ 0 t=1 0 t=0
over the fiberwise boundary.
(It is an analog of the formula d◦p ±p ◦d=p(cid:48) for the Abelian
∗ ∗ ∗
case and arbitrary fiber bundles: ∆Texp∼d and curv∼d.)
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
... and more explicitly
If g = Texpp ω = g(1,0) and g(t ,t ) = Texp(cid:82)t1(−ω) as above,
* 1 0 t0
and we denote Ω := curvω = dω+ω2, then the commutation
formula reads
(cid:90) 1
−dgg−1+ g(1,t)Ωg(1,t)−1 = ω | −g(cid:0)ω | (cid:1)g−1.
0 t=1 0 t=0
0
Geometric interpretation: d+ω is like Quillen’s superconnection, and
g(t ,t ) is the corresponding ‘parallel transport’ (along the fibers of
1 0
the projection M×I→M). (Helps to reconstruct the formula.)
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids
Introduction
Non-Abelianhomotopyformula
ApplicationtoLiealgebroids
“Non-linear” Liealgebroids
Summary
Non-Abelian chain homotopy
Consider a homotopy F: M×I → N, so that F(x,t) = f (x).
t
Theorem
For an arbitrary odd g-valued form ω ∈ Ωodd(N,g), the
pull-backs along homotopic maps f and f are related by a
0 1
(non-linear) formula
(cid:90) 1
f∗ω−gf∗ωg−1 = −dgg−1+ g(1,t)F∗(dω+ω2)g(1,t)−1,
1 0
0
where g = g(1,0) = Texp(cid:82)1(−F∗ω) and
0
g(1,t) = Texp(cid:82)1(−F∗ω).
t
For a proof, apply the commutation formula above to F∗ω.
TheodoreVoronov Non-AbelianPoincar´elemmaandLiealgebroids