31 Pages·2010·0.4 MB·English

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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

A. Kotov, Th. Strobl. Characteristic classes associated to Q-bundles. arXiv:0711.4106. K. C. H. Mackenzie. General theory of Lie groupoids and Lie algebroids. Cambridge University Press, Cambridge, 2005. Th. Voronov. Graded manifolds and Drinfeld doubles for Lie bialgebroids. Contemp. Math., 315

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