Dimensional Reduction of Invariant Fields and Differential Operators. I. Reduction of Invariant Fields

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1 Communications in Mathematical Physics manuscript No. will be inserted by the editor) Dimensional Reduction of Invariant Fields and Differential Operators. I. Reduction of Invariant Fields Petko A. Nikolov 1, Nikola P. Petrov 2 1 Department of Theoretical Physics, Faculty of Physics, University of Sofia, 5 James Bourchier Blvd, 1164 Sofia, Bulgaria. pnikolov@phys.uni-sofia.bg 2 Department of Mathematics, University of Oklahoma, 601 Elm Avenue, Norman, OK 73019, USA. npetrov@ou.edu Received: 25 August 2008 Abstract: Problems related to symmetries and dimensional reduction extra dimensions ) are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group reducing group ) is compact, and its orbits the external dimensions ) are compact, and this is essential in models where the volume of the orbits is related to physical quantities. But this case is only a part of the natural problems related to dimensional reduction. In the present paper, we consider an action of a generally non-compact) Lie group on a vector bundle, construct a formalism of reduced bundles for description of all invariant sections of the original bundle, and study the algebraic structures that occur in the reduced bundle. We show how some classical results are connected to this formalism. We illustrate that in the case of a non-compact reducing group it is possible to obtain a non-standard non-canonical ) reduction, and construct an explicit obstruction for canonical reduction in terms of cohomology of groups. We consider in detail the reduction of tangent and cotangent bundles, and show that, in general, the duality between the two is violated in the process of reduction. 1. Introduction In this paper we present a geometric methodology for dimensional reduction of objects invariant under an action of a Lie group. While some techniques for symmetry reduction exist in the literature, in many of them the geometric aspects are not quite explicit, or they treat particular problems say, Marsden-Weinstein- Meyer reduction in mechanics), our approach is quite general. Clearly, the price we pay for the generality of our approach is that sometimes the information we obtain is not as detailed as from the particular methods designed for attacking one particular problem.

2 2 Petko A. Nikolov, Nikola P. Petrov Here is our general setup. We consider a Lie group G acting by bundle morphisms on a vector bundle ξ over a finite-dimensional base B all objects and mappings are assumed to be C ). Let C ξ) G be the set of all smooth sections of ξ that are invariant with respect to the action of G. We propose a natural geometric procedure for constructing a reduced bundle ξ G, the set of whose sections, C ξ G ), is in a bijective correspondence with the G-invariant sections of ξ, C ξ) G. If some general assumptions on the group actions Conditions A and B in Section 2.2) are satisfied, we give an explicit construction of the reduced bundle. The reduced bundle is constructed from local charts, so that we do not require that ξ or ξ G be globally trivial. We prove that the arbitrariness in the choice of local charts in the construction of ξ G does not affect the global object. In the process of dimensional reduction of short exact sequences of vector bundles we need to use some facts from theory of cohomology of groups briefly reviewed in the paper). It turns out that sometimes the reduced bundle has a simple structure or, as we say, in this case the dimensional reduction is canonical the rigorous definition is given in Section 3.3). The obstruction to canonical reduction is given explicitly in terms of certain cohomology groups. For an example of a non-canonical reduction see Section 3.3. We pay special attention to the case of dimensional reduction of the tangent and cotangent bundles and their tensor products. In the case of a non-compact group G, the reductions of the tangent and cotangent bundles may exhibit some interesting phenomena we give an explicit example in Section 4.5). In a second part of the paper in preparation), we describe a technique for dimensional reduction of invariant differential operators based on the jet bundle description. Within the jet bundle language the geometry of dimensional reduction becomes very explicit, and the operations on differential operators are reduced to simple algebraic manipulations. We have not attempted to survey the vast literature related to reduction for two reasons firstly, the amount of literature makes it impossible for us to give proper credit, secondly, we are still trying to understand the relation between our method and the methods of other authors, and hope to discuss these in future publications. We only point out how some classical examples of dimensional reduction are incorporated in the techniques developed in this paper. We hope that our methods shed new light on the problem of symmetry reduction in many contexts. In particular, we believe that they would be useful in the study of the new algebraic structures occurring in the process of dimensional reduction problems of Kaluza-Klein type), as well as physical applications in the case of non-canonical reduction of invariant tensor fields. Some of the techniques developed in this paper and its sequel have been used in our papers [1 4], but in this paper we consider the problem in full generality. Throughout the paper we assume that all manifolds, bundles, and maps are smooth C ), even if this is not said explicitly. The paper is organized as follows. Section 2 is devoted to a detailed explanation of the concept of a reduced vector bundle and the conditions we impose on the actions of the Lie group on the bundles in order for our construction to work. In Section 3 we introduce some facts from cohomology of groups and their use in the reduction procedure, and in Section 4 we apply this procedure to the case of reduction of the tangent and cotangent bundles and their tensor products.

3 Dimensional Reduction I 3 2. Reduced vector bundles 2.1. Local chart description of vector bundles. In this section we give the basic definitions and set up the notations concerning the description of a vector bundle in coordinate charts, referring the reader to [5, Section I.3] for more details. Let ξ = E, π, B) be a finite-dimensional vector bundle over the finite-dimensional manifold B. Let ξ b := π 1 b) be the fiber of ξ over the point b B. We will often need to restrict the base of a vector bundle to some submanifold. Let C B be a submanifold of the base B of ξ, and let i : C B be the natural embedding. Then by ξ C, or sometimes ξ C for clarity, we will denote the bundle i ξ induced by i; in other words, ξ C = E, π, C) where E := π 1 C), and π is the restriction of π to E. Let {U α } α A be an open cover of the manifold B, and σ α : U α Ũα be diffeomorphisms from U α to some manifolds Ũα. Whenever U α U β, we set Ũ α,β := σ α U α U β ) Ũα 1) and f αβ := σ α σ 1 β : Ũβ,α Ũα,β. 2) In this case we say that the manifold B is obtained from the manifolds {Ũα} α A through gluing or clutching in the terminology of [5]) them by the diffeomorphisms f αβ. For each α A, let ξ α := E α, π α, Ũα) be a vector bundle over Ũα, and ξ α,β := ξ α euα,β. Let for each pair of indices α and β in A for which U α U β, there exists a vector bundle isomorphism φ αβ := F αβ, f αβ ) : ξ β,α ξ α,β 3) over f αβ where F αβ : π 1 β Ũβ,α) πα 1 Ũα,β) and π α F αβ = f αβ π β ). Let the transition isomorphism φ αβ satisfy the cocycle conditions φ αβ φ βγ = φ αγ 4) whenever U α U β U γ. We shall call the bundles ξ α coordinate bundles. The transition isomorphisms φ αβ glue these coordinate bundles into a vector bundle over B. More precisely, there exists a unique up to isomorphism) vector bundle ξ over B and isomorphisms φ α : ξ α ξ Uα such that the diagram ξ α,β φ αβ ξ β,α φ α ξ Uα U β φ β 5) commutes see Theorem 3.2 in Chapter I of [5]). Clearly, ξ and the coordinate bundles ξ α have the same standard fiber. In these notations, defining a section ψ C ξ) is equivalent to defining sections ψ α C ξ α ) for all α A such that ψ α are compatible with the transition isomorphisms: ψ α = φ αβ ψ β ) := F αβ ψ β f 1 αβ. 6)

4 4 Petko A. Nikolov, Nikola P. Petrov Similarly, one can define other geometric structures on ξ like metric, connection, etc. they have to be defined in every coordinate bundle ξ α and compatible with the transition isomorphisms φ αβ. A common example of gluing is when the manifolds Ũα coincide with U α, and ξ α are the trivial bundles π α : U α R n U α. If U α U β and b, u) U α U β ) R n, then φ αβ b, u) = b, g αβ b)u), where g αβ : U α U β GLn, R) satisfy the cocycle conditions g αβ b) g βγ b) = g αγ b) for all b U α U β Action of Lie groups on vector bundles. Let G be a Lie group not necessarily compact) that acts from the left on the vector bundle ξ = E, π, B) by vector bundle morphisms. We denote this action by T, t), where T : G E E, t : G B B, 7) so that for each g G the following diagram commutes: E Tg E π tg B Bπ i.e., t g is the action of T g on E projected to the base B), and T g : ξ b ξ tgb) is a linear isomorphism for any b B. The action T, t) of G on ξ induces a natural action of G on the set C ξ) of all sections of ξ by 8) gψ) = T g ψ t 1 g, g G. 9) If we think of C ξ) as a vector space where the multiplication by a number and the addition are defined pointwise), then 9) determines an infinite-dimensional linear representation of G. We say that a section ψ C ξ) is G-invariant or G-equivariant if gψ) = ψ, or, in more detail, ψt g b)) = T g ψb)) for all g G and b B. 10) It would be more precise to say that a section is invariant with respect to the action T, t) of G, but we often use G-invariant for brevity. Let C ξ) G stand for the set of all G-invariant sections of ξ. Clearly, C ξ) G is a linear subspace of C ξ). Obviously, the problem of complete description of C ξ) G for a general smooth action of a Lie group G is very complicated and little can be said. Even in the degenerate case when B consists of only one point b and T is a representation of G in the only fiber ξ b, the problem is not easy it reduces to description of all invariants of the finite-dimensional representations of G. To simplify the problem of the description of C ξ) G, we will impose two conditions which will be assumed to hold throughout the rest of the paper.

5 Dimensional Reduction I 5 Condition A. All orbits of the action t : G B B of G on the base B of ξ are of the same type, and the quotient space B/G is a manifold. Moreover, the orbits of the action t form a locally trivial G-bundle B, p, N), where p : B N := B/G 11) is the natural projection. Let us discuss Condition A in more detail. First, note that N is just a short notation for the base B/G of the bundle 11). The manifold N which will be the base of the reduced bundle in Section 2.3 does not have a canonical realization, so one of the tasks in constructing the reduced bundle will be to glue N out of some concrete submanifolds of B. Let H G be a closed subgroup of G, and G/H be the space of left cosets of H in G; the canonical left action of G on G/H is g, [g 1 ]) g[g 1 ] := [gg 1 ]. The group of G-equivariant automorphisms of G/H, Aut G G/H) = { f : G/H G/H : fg[g 1 ]) = gf[g 1 ]) }, is isomorphic to NH)/H, where NH) = { g G : ghg 1 H } is the normalizer of H in G. The isomorphism NH)/H Aut G G/H) is given by NH)/H Aut G G/H) : [n] f [n], f [n] [g]) := [gn 1 ]. 12) Condition A requires that there exist a closed subgroup H of G such that all orbits of the action t of G on B are homogeneous G-spaces that are G- isomorphic to G/H in the sense that, for any x N, if by this isomorphism b [g 1 ], then t g b) g[g 1 ] for any g G). Moreover, Condition A demands that B, p, N) be a G-bundle that is locally trivial in the following sense: each point x N has a neighborhood U N for which there exists a diffeomorphism Φ : p 1 U) U G/H) satisfying Φt g b)) = pb), gπ 2 Φb))), 13) where b p 1 U), g G, and π 2 : U G/H) G/H : v, [g 1 ]) [g 1 ] is the canonical projection. If {U α } α A is a fine enough cover of N so that the diffeomorphisms Φ α : p 1 U α ) U α G/H) satisfy 13) for each α A, then on U α U β the following condition holds: Φ α Φ 1 β x, [g]) = x, φ αβx)[g])), where φ αβ : U α U β Aut G G/H) are the transition isomorphisms. The bundle 11) can be defined as a bundle with fiber the homogeneous space G/H and transition isomorphisms φ αβ x) Aut G G/H), x U α U β. Via the isomorphism 12), φ αβ x) can be considered as an element of the group NH)/H as well as a transformation NH)/H NH)/H defined as a left multiplication: φ αβ x), [n]) φ αβ x)[n]. Because of this, the transition functions φ αβ define a principal bundle P NH)/H, N) with structure group NH)/H and base N. Then the bundle 11) is associated with the principal bundle P NH)/H, N) through the action of NH)/H on G/H defined by 12).

6 6 Petko A. Nikolov, Nikola P. Petrov Remark 1. Here we briefly define some objects that will be needed in Section 2.3. A diffeomorphism f : B B is vertical with respect to the projection p) if p f = p. Let Diff v B) stand for the group of all vertical diffeomorphisms. The following two infinite-dimensional groups of vertical diffeomorphisms occur naturally. The first one is the group of vertical automorphisms of the bundle B, p, N) sometimes called the group of local gauge transformations), G vert. = { f Diff v B) : f t g = t g f g G }. The second important subgroup of Diff v B) is the group of local actions of G, G loc. act. = { f Diff v B) : x N g x G s.t. fb) = t gx b), b p 1 x) } where N = B/G). In other words, the restriction of f G loc. act. to each fiber p 1 x) coincides with the action of some element g x G. Each map χ : N G defines an element f χ G loc. act. by f χ b) = t χpb)) b), b B. 14) The elements of G loc. act. are in a natural bijective correspondence with the sections of a bundle of groups with fiber NH)/H, associated with the principal bundle P NH)/H, N) via the action Inn of NH)/H on NH)/H by inner automorphisms: Inn [n] [n 1 ]) = [nn 1 n 1 ]. Such maps will be used in the process of gluing the reduced bundle in Section 2.3. Remark 2. Condition A is satisfied when G is compact according to the Slice Theorem [6, Section 4.4]. Another result showing that Condition A is commonly encountered is the Principal Orbits Theorem [6, Theorem 4.27]. This theorem states that if G is a compact Lie group and B/G is connected, then there exists a maximum orbit type, G/H, in B, and the union of all orbits of type G/H called principal orbits) is open and dense in B. Condition B stated below) concerns the action T : G E E of G on the total space of ξ. For each point x N, p 1 x) B is an orbit of the action t, and, hence, is a homogeneous G-space which is G-isomorphic to G/H for some closed subgroup H G. In Condition B we want to impose restrictions on the structure of the vector bundles ξ p 1 x), x N. Firstly, we will require that all G-bundles ξ p 1 x), x N, be G-isomorphic to some typical G-bundle ζ := E, π, G/H) over G/H endowed with a left action T, t ) of G. Let U be a small enough open subset of N, and ξ p 1 U) := ξ p 1 U)= π 1 p 1 U). Since p 1 U) is foliated by the orbits of the action t, we demand that the G-vector bundle ξ p 1 U) be modeled after the direct product U ζ. In more detail, let be a bundle with projection U ζ := U E, Id π, U G/H)) 15) Id π )v, w) = v, π w)) for v, w) U E. The bundle U ζ is a G-vector bundle with action T, t ) of G defined by T g v, w) = v, T gw)) for v, w) U E.

7 Dimensional Reduction I 7 Now Condition B can be stated as follows. Condition B. We assume that all G-vector bundles ξ p 1 x), x N, are isomorphic to one another and to the typical G-vector bundle ζ = E, π, G/H) endowed with a left action T, t ) of G. The collection of vector bundles { } ξ p 1 x) forms a locally trivial G-vector x N bundle over N in the sense that each point x N has a neighborhood U N such that ξ p 1 U) = π 1 p 1 U) is isomorphic as a G-vector bundle to U ζ 15). In other words, there exists a vector bundle isomorphism Ψ : π 1 p 1 U) U E satisfying ΨT g e)) = x, T gw)), where e π 1 p 1 U) and Ψe) = x, w) U E with x = p πe). In the construction of the reduced bundle in the following section, we need the following Definition 1. For b B, let G b := { g G : t g b) = b } G be the stationary or isotropy) group of b with respect to the action t of G on B. Define the stationary subspace of ξ b with respect to the linear representation T of G b in the fiber ξ b ) as st ξ b := { u ξ b : T g u) = u g G b } ξ b. Condition B guarantees that the family of vector spaces st ξ b ξ b form a smooth vector subbundle of ξ which we denote by st ξ and call the stationary subbundle of ξ with respect to the action T, t) of G on ξ). Definition 2. We say that the vector bundle ξ with action T, t) of G on it is a reducible G-vector bundle if the actions t and T satisfy Conditions A and B. Remark 3. Condition B implies Condition A, but we considered Condition A independently because of its importance. Remark 4. If x µ, z a ) are local coordinates in ξ, the action T g, t g ) of g G has the form T g x µ, z a ) = t g x) µ, u a bg, x) z b ), where ug, x) GLn, R) satisfies ug 1 g 2, x) = ug 1, t g2 x)) ug 2, x) for all g 1, g 2 G. The action 9) of G on the sections ψ C ξ) becomes gψ) a x) = u a bg, t g 1x)) ψ b t g 1x)), and the invariance condition 10) reads ψ a t g x)) = u a bg, x) ψ b x). Remark 5. Readable references for more details about the concepts considered here are the book [6] and the paper [7].

8 8 Petko A. Nikolov, Nikola P. Petrov σ x) α U ~ α B ξ α U ~ β t χαβ x) ξ β σ x) ~ α U α B σ x) β U ~ β σ x) β t χαβ x) Fig. 1. Left: Constructing the base N = B/G of the reduced bundle ξ G by gluing it from the manifolds { U e α} α A ; U e α = σ αu α) is transversal to the orbits of t g the orbits of t g are drawn with dashed lines). Right: Constructing the reduced bundle ξ G by gluing it from the coordinate bundles {ξ α} α A ; each ξ α is a vector bundle with base U e α and fiber ξ α) σαx) = st ξ σαx) over σ αx) e U α Reduced vector bundles: construction. Let ξ be a reducible G-vector bundle, i.e., the action T, t) 7) of G on ξ satisfies Conditions A and B from Section 2.2. Then the space C ξ) G of all G-invariant sections of ξ has the structure of the space of all sections in some bundle ξ G which we will call the reduced vector bundle. In other words, one can construct and a natural bijective correspondence θ : C ξ G ) C ξ) G 16) between all sections of ξ G and all G-invariant sections of ξ. Below we describe the explicit construction of ξ G and θ. We start with an explicit construction of the base N = B/G of the reduced bundle by gluing it as explained in Section 2.1) from explicitly defined submanifolds of B. Let {U α } α A be a fine enough open cover of N. For each α A we choose a smooth) local section σ α of the bundle 11) whose graph Ũ α := σ α U α ) 17) is transversal to the fibers of the bundle 11) recall that the fibers of 11) are the orbits of the action t of G on B). Clearly, σ α : U α Ũα are diffeomorphisms, and the maps σ α σ 1 β : Ũβ,α Ũα,β glue the manifold N from the manifolds {Ũα} α A ; here, as before, Ũα,β := σ α U α U β ) cf. 1) and 2)). This construction is pictorially represented in the left part of Fig. 1. Now we will construct the coordinate bundles ξ α over Ũα) of the reduced bundle ξ G. First of all, it is easy to see from 10) that if ψ is a G-invariant section of ξ, then ψb) st ξ b, so that C ξ) G is a subset of C st ξ). We define the coordinate bundles ξ α := st ξ euα 18) as the restrictions of the base of the stationary bundle st ξ to the manifolds Ũα; see the right part of Fig. 1.

9 Dimensional Reduction I 9 The isomorphisms φ αβ 3) gluing the family {ξ α } α A into the reduced bundle ξ G are constructed as follows. Let α, β) be a pair of indices for which U α U β, and define ξ α,β := st ξ euα,β. Let be a map that satisfies χ αβ : U α U β G 19) t χαβ x) σβ x) ) = σ α x) for all x U α U β, 20) as shown in Fig. 1. Similarly to 14), define the local action f χαβ of G on p 1 U α U β ) by f χαβ : p 1 U α U β ) p 1 U α U β ) : b f χαβ b) := t χαβ pb))b). 21) The requirement 20) guarantees that f χαβ Ũβ,α) = Ũα,β. Next, define the action F χαβ of G on π 1 p 1 U α U β ) ) by F χαβ : π 1 p 1 U α U β ) ) π 1 p 1 U α U β ) ) : e F χαβ e) := T χαβ pπe)))e). 22) Clearly, the actions f χαβ and F χαβ are compatible: π F χαβ = f χαβ π. Therefore the pair ) F χαβ, f χαβ defines an isomorphism φαβ : ξ β,α ξ α,β : if e ξ β,α = st ξ euβ,α and x = p πe) U α U β, then φ αβ e) = T χαβ x)e) st ξ σαx). 23) The isomorphism φ αβ does not depend on the arbitrariness in the choice of χ αβ in 19) and is uniquely defined for each pair of indices α, β) for which U α U β. Indeed, if χ αβ : U α U β G is another map satisfying 20), then T χ αβ x) = T χαβ x) T χ 1 αβ x) χ αβ x), but χ 1 αβ x) χ αβ x) belongs to the stationary group G σβ x), hence the operator T χ 1 αβ x) χ αβ x) is the identity when acting on st ξ σβ x). It is easy to check that isomorphisms φ αβ satisfy the cocycle conditions 4), hence they glue the reduced bundle ξ G from the coordinate bundles {ξ α } α A. The construction of ξ G makes the correspondence 16) explicit. A section S C ξ G ) of the reduced bundle corresponds to a family of sections S α C ξ α ) compatible with the transition isomorphisms φ αβ : S α = φ αβ S β ) 24) using the notation of 6)). There exists a unique G-invariant section ψ = θs) C ξ) G whose values over Ũα coincide with the values of S α, i.e., ψσ α x)) = S α σ α x)) for all α A and x Ũα. Namely, for each b B, we define ψb) := T g S α σ α x))), 25) where x = pb) U α for some α A, and g G is such that t g σ α x)) = b. Since S α x) st ξ σαx), the value of ψb) does not depend on the arbitrariness in the choice of g; moreover, 23) and 24) imply that if x = pb) U α U β, the

10 10 Petko A. Nikolov, Nikola P. Petrov value of ψb) obtained as in 25) but by using S β instead of S α would give the same value. Conversely, given ψ C ξ) G, its values belong to the stationary subbundle st ξ and, therefore, the restrictions S α C st ξ euα ) = C ξ α ), defined by S α σ α x)) := ψσ α x)), 26) are sections of the coordinate bundles ξ α. Thanks to the G-invariance of ψ 10), S α satisfy 24), so they determine a section S = θ 1 ψ) C ξ G ). We summarize the above construction in the following Theorem 1. Let ξ be a G-reducible vector bundle, i.e., the action T, t) of the Lie group G on ξ satisfy Conditions A and B from Section 2.2. Then there exists a bijective correspondence θ 16) between all sections C ξ G ) of a vector bundle ξ G called the reduced vector bundle) and all G-invariant sections C ξ) G of ξ. The base of ξ G is the quotient space B/G. Let {U α } α A be an open cover of B/G, and, for each α A, σ α : U α B be a local section of the bundle 11) whose graph, Ũα 17), is transversal to the orbits of t in B. Then the restrictions st ξ euα are the coordinate bundles ξ α from which the reduced bundle ξ G is glued via the isomorphisms φ αβ 23). The bijection θ 16) is given explicitly by 24), 25), and 26) Reduced vector bundles: algebraic properties. In this section we will prove several simple lemmata needed in Section 4.2. Lemma 1. Let ξ 1 and ξ 2 be reducible G-vector bundles over the same base B with the same action t of G on B. Then the Whitney sum ξ = ξ 1 ξ 2 is a reducible G-vector bundle, and ξ G = ξ 1 ξ 2 ) G = ξ G 1 ξ G 2. 27) Proof. If T 1, t) and T 2, t) be corresponding actions, then the natural action T 1 T 2, t) of G on ξ 1 ξ 2 is defined by T 1 T 2 ) g a 1 a 2 ) := T 1,g a 1 ) T 2,g a 2 ) ξ 1 ξ 2 ) tgb), where a 1 a 2 ξ 1 ξ 2 ) b = ξ 1,b ξ 2,b. Clearly, st ξ 1 ξ 2 ) = st ξ 1 st ξ 2, so the coordinate bundles of ξ 1 ξ 2 ) G have the form ξ 1 ξ 2 ) α = ξ 1,α ξ 2,α. The transition isomorphisms φ αβ constructed through the action of G on ξ by 19), 20), 23) preserve the direct sum and, therefore, endow the reduced bundle ξ G with the structure of a Whitney sum of ξ G 1 and ξ G 2. Lemma 2. Let ξ 1 and ξ 2 be reducible G-vector bundles over the same base B with the same action t of G on B, and let the action t be free. Then the tensor product ξ = ξ 1 ξ 2 is a reducible G-vector bundle, and ξ G = ξ 1 ξ 2 ) G = ξ G 1 ξ G 2. 28) Proof. The proof is analogous to the proof of Lemma 1. The requirement for the action t to be free guarantees that st ξ 1 ξ 2 ) = st ξ 1 st ξ 2 because the fact that G b is trivial for all b B guarantees that st ξ 1 = ξ 1 and st ξ 2 = ξ 2.

11 Dimensional Reduction I 11 In general, if the action t of G on B is not free, one can only claim that st ξ 1 st ξ 2 st ξ 1 ξ 2 ). The simplest example to keep in mind is when the common base of the vector bundles ξ 1 and ξ 2 is a single point, the group G is the multiplicative group of all non-zero numbers, and its actions T 1 and T 2 on ξ 1 and ξ 2 are given respectively by T 1,g a 1 ) := g a 1 and T 2,g a 2 ) := 1 g a 2 where g R \ {0}, and the dot stands for multiplication). Then, clearly, st ξ 1 and st ξ 2 consist of the zero vectors only, while st ξ 1 ξ 2 ) = ξ 1 ξ 2. If the action t of G on B is not free, the representation of G b in the fiber st ξ 1,b st ξ 2,b is a tensor product of the representations T 1 and T 2, and finding all vectors fixed with respect to this representation becomes a problem of Clebsch- Gordan type of finding all stationary vectors in st ξ 1 ξ 2 ) b. An useful particular case of Lemma 2 is the following Lemma 3. Let ξ 1 and ξ 2 be reducible G-vector bundles over the same base B and with the same action t of G on B. Let the action of G on ξ 2 be such that st ξ 2 = ξ 2. Then ξ = ξ 1 ξ 2 is a reducible G-vector bundle, and ξ G = ξ 1 ξ 2 ) G = ξ G 1 ξ G 2. 29) 3. Short exact sequences and cohomology of groups 3.1. Splittings of short exact sequences. In this section we collect some elementary facts concerning short exact sequences of vector spaces and G-modules which will be needed later. Let 0 L 0 i L j L ) be a short exact sequence of vector spaces i.e., i and j are linear maps satisfying ker i = {0}, im j = L 1, and im i = ker j). The middle term L is isomorphic to L 0 L 1, but this isomorphism is not defined naturally. The choice of such an isomorphism called a splitting of the short exact sequence 30) is equivalent to defining a subspace of L that is transversal to il 0 ) L. This can be achieved by specifying a linear map S Hom L 1, L) satisfying j S = Id L1 or, equivalently, a linear map F Hom L, L 0 ) that satisfies F i = Id L0 ; since we will refer to these properties often, we collect them here: j S = Id L1, F i = Id L0. 31) If 31) are satisfied, ker F and im S are transversal to il 0 ). The isomorphism L = L 0 L 1 is given by the pair of embeddings i, S) of L 0 and L 1 into L or, equivalently, by the pair of projections F, j) from L onto L 0 and L 1 ; the isomorphism is given by L = il 0 ) SL 1 ) = ker j ker F. Clearly, the maps S and F define the same splitting of 30) if and only if ker F = im S; in this case, if S is known, the map F is given by F = i 1 Id S j). On the other hand, given S, we have Sv 1 ) = Id i F )v), where v L is such that jv) = v 1. Although giving either S or F defines a splitting completely, for convenience we will often say splitting F, S) meaning that F and S are a pair of maps corresponding to the same splitting. A splitting F, S) of 30) gives us the following decomposition of the identity in L as a sum of projection operators onto the subspaces il 0 ) and SL 1 ): Id L = i F + S j. 32)

12 12 Petko A. Nikolov, Nikola P. Petrov If S 1, S 2, F 1, and F 2 define splittings of 30), then S 2 = S 1 + i s, F 2 = F 1 + f j, 33) where s and f are linear maps from L 1 to L 0. In other words, the set of all splittings of the short exact sequence 30) is an affine space with linear group Hom L 1, L 0 ). One can easily see that if S 1, F 1 ) and S 2, F 2 ) are two splittings, then the maps s and f in 33) are related by s = f. Let D 0, D and D 1 be representations of the group G in L 0, L and L 1, respectively, and let the operators i and j in 30) be intertwining, i.e., Dg) i = i D 0 g), j Dg) = D 1 g) j for all g G. 34) In this case we will say that 30) is an intertwining short exact sequence of G- modules in another terminology, the short exact sequence 30) is G-equivariant). If the short exact sequence 30) is intertwining, then the subspace il 0 ) is invariant with respect of the representation D; the subspace SL 1 ), however, is not D-invariant in general. For a splitting F, S) we can use 31), 34), and the exactness of 30) to obtain Dg) = i F + S j) Dg) i F + S j) = i D 0 g) F + i F Dg) S j + S D 1 g) j. Each vector v L can be represented as a sum of a vector in il 0 ) and a vector in SL 1 ): v = iv 0 ) + Sv 1 ), where v 0 = F v) L 0 and v 1 = jv) L 1. If we write this symbolically as v = v 0 v 1 ) T, we can write the operator Dg) End L as ) ) ) v0 D00 g) D Dg) = 01 g) v0, v 1 0 D 11 g) v 1 where we have introduced the components of the representation D: D 00 g) = F Dg) i = D 0 g), D 01 g) = F Dg) S Hom L 1, L 0 ), D 11 g) = j Dg) S = D 1 g). Note that D 10 g) := j Dg) i = 0 for all g G. The operator D 01 g) satisfies the relation D 01 g 1 g 2 ) = D 00 g 1 ) D 01 g 2 ) + D 01 g 1 ) D 11 g 2 ) for all g 1, g 2 G Facts from cohomology of groups. In this section we collect some facts from cohomology of groups needed for the procedure of dimensional reduction. For details we refer the reader to [8] or, for physics-motivated exposition, to [9]. Definition 3. Let G be a topological group and D be a continuous linear representation of G in the generally infinite-dimensional) vector space L. The set C n G, L) of all n-chains consists of the continuous functions c : G G G L

13 Dimensional Reduction I 13 n copies of G), and C 0 G, L) = L. The coboundary operator δ n) : C n G, L) C n+1 G, L) is defined by ) δ n) c g 1, g 2,..., g n+1 ) := Dg 1 ) cg 2,..., g n+1 ) + n 1) i cg 1,..., g i 1, g i g i+1, g i+2,..., g n+1 ) i=1 + 1) n+1 cg 1,..., g n ). 35) The elements of ker δ n) and im δ n) are called cocycles and coboundaries, respectively. The nth cohomology group is defined as H n G, L) := ker δ n) /im δ n 1) for n N, and H 0 G, L) := ker δ 0). The fact that the cohomology groups are well-defined is based on the fact proved, e.g., in [9, Theorem 5.1.1]) that the coboundary operator satisfies δ n+1) δ n) = 0. Here are the explicit expressions for n = 0 and n = 1: ) δ 0) c g) = Dg)c c, ) δ 1) c g 1, g 2 ) = Dg 1 ) cg 2 ) cg 1 g 2 ) + cg 1 ). If c H 0 G, L), from δ 0) c = 0 we obtain Dg)c c = 0 for all g G, which means that c st L. This leads us to the important observation that H 0 G, L) = st L. 36) Now we return to the problem of splitting short exact sequences. Let F, S) be a splitting of the intertwining short exact sequence 30). Define the map k n : C n G, L 1 ) C n+1 G, L 0 ) by k n c)g 1,..., g n+1 ) := F Dg 1 ) S cg 2,..., g n+1 )) = D 01 g 1 ) c g 2,..., g n+1 ). 37) Lemma 4. The map k n 37) anticommutes with the coboundary operator in the sense that δ n+1) L 0 k n = k n+1 δ n) L 1, 38) and, hence, defines a map between the cohomology groups k n : H n G, L 1 ) H n+1 G, L 0 ) 39) for which we use the same notation as for the map 37)). The map k n in 39) does not depend on the splitting F, S) of the short exact sequence 30).

14 14 Petko A. Nikolov, Nikola P. Petrov Proof. The basic step in the proof of 38) is the following observation based on the decomposition 32) and the identities 31) and 34)): F Dg 1 g 2 ) S = F Dg 1 ) Dg 2 ) S = F Dg 1 ) i F + S j) Dg 2 ) S = D 0 g 1 ) F Dg 2 ) S + F Dg 1 ) S D 1 g 2 ) S. Using this to rewrite the term underlined in the equations below, we obtain for c C n G, L 1 ) ) δ n+1) L 0 k n c) g 1,..., g n+2 ) = D 0 g 1 )k n c)g 2,..., g n+2 ) 1) i k n c)g 1,..., g i g i+1,..., g n+2 ) n+1 + i=1 + 1) n+2 k n c)g 1,..., g n+1 ) = D 0 g 1 ) F Dg 2 ) Scg 3,..., g n+2 )) F Dg 1 g 2 ) Scg 3,..., g n+2 )) 1) i F Dg 1 ) Scg 2,..., g i g i+1,..., g n+2 )) n+1 + i=2 + 1) n+2 F Dg 1 ) Scg 2,..., g n+1 )) = F Dg 1 ) S D 1 g 2 )cg 3,..., g n+2 ) 1) i F Dg 1 ) Scg 2,..., g i g i+1,..., g n+2 )) n+1 + i=2 + 1) n+2 F Dg 1 ) Scg 2,..., g n+1 )) = F Dg 1 ) S D 1 g 2 )cg 3,..., g n+2 ) = F Dg 1 ) S = k n+1 δ n) L 1 c) 1) i 1 cg 2,..., g i g i+1,..., g n+2 ) n+1 + i=2 ) + 1) n+1 cg 2,..., g n+1 ) ) δ n) L 1 c)g 2,..., g n+2 ) ) g 1,..., g n+2 ). It remains to prove that the map 39) is independent of the splitting of the intertwining short exact sequence 30). Let F, S) be a splitting of 30) and k n be the corresponding map 37). Let F, S), where S = S +i s and F = F s j for some s Hom L 1, L 0 ), be another splitting of 30), and k n be the corresponding map 37). Directly from the definitions we obtain, for c C n G, L 1 ), ) δ n) L 0 s c) s δ n) L 1 c) g 1,..., g n+1 ) = D 0 g 1 ) s s D 1 g 1 ) ) cg 2,..., g n+1 )).

15 Dimensional Reduction I 15 Using this fact, the identities 31) and 34) and the exactness of 30), we obtain ) ) k n k n )c g 1,..., g n+1 ) = D 0 g 1 ) s s D 1 g 1 ) cg 2,..., g n+1 )) ) = δ n) L 0 s c) s δ n) L 1 c) g 1,..., g n+1 ). 40) Now let c C n G, L 1 ) with δ n) L 1 c = 0, and let [c] = c + δ n 1) L 1 C n 1 G, L 1 ) ) be its equivalence class in H n G, L 1 ). Then 40) implies immediately that k n k n )c) im δ n) L 0, i.e., that k n [c]) = k n [c]). In order to formulate the fundamental theorem below, we need the following definition. Definition 4. If D 1 and D 2 are representations of the groups G 1 and G 2 in the vector spaces L 1 and L 2, then a morphism between these two representations is defined as a pair ψ, φ), where ψ : G 1 G 2 is a morphism of groups and φ : L 1 L 2 is a linear map satisfying φ D 1 g) = D 2 ψg)) φ for all g G 1. 41) If ψ is an isomorphism, then ψ, φ) induces a map ψ, φ) n : C n G 1, L 1 ) C n G 2, L 2 ) by ψ, φ) n c) g 1,..., g n ) := φ cψ 1 g 1 ),..., ψ 1 g n )) ), 42) where g 1,..., g n G 2. The map ψ, φ) n commutes with the coboundary operator in the sense that ψ, φ) n+1 δ n) L 1 = δ n) L 2 ψ, φ) n which follows immediately from the definitions) and, thus, determines a map ψ, φ) n : H n G 1, L 1 ) H n G 2, L 2 ). 43) The conditions 34) for the short exact sequence 30) to be intertwining are a particular case of 41) for G 1 = G 2 =: G and ψ = Id G. Let i n := Id, i) n : H n G, L 0 ) H n G, L), j n := Id, j) n : H n G, L) H n G, L 1 ), 44) k n := Id, k) n : H n G, L 1 ) H n+1 G, L 0 ). be the maps induced by the morphisms Id, i), Id, j) and Id, k) between the corresponding representations, as in 43). A classic result in cohomology theory is the following [10, Chapter 1] Theorem 2. If 30) is an intertwining with respect to the representations of G short exact sequence of vector spaces, then the sequence 0 H 0 G, L 0 ) i0 H 0 G, L) j0 H 0 G, L 1 ) k0 H 1 G, L 0 ) i1 H 1 G, L) j1 45) is exact.

16 16 Petko A. Nikolov, Nikola P. Petrov Let D be a representation of the group G in the vector space L. Let for g G, let Inn g be the inner automorphism Inn g : G G : Inn g g 1 ) := gg 1 g 1. 46) Then the pair Inn g, Dg)) is an automorphism of the representation D in the sense of the definition 41)), and one can construct the maps Inn g, Dg)) n : C n G, L) C n G, L) as in 42), and the corresponding maps 43) from the cohomology group H n G, L) to itself. The following lemma holds: Lemma 5. The map Inn g, Dg)) n : H n G, L) H n G, L) induced by Inn g, Dg)) is trivial for any g G. Proof. For g G, define the maps Ω n i g) : Cn G, L) C n 1 G, L) by Ω n i g)c) ) g 1,..., g n 1 ) := cg 1,..., g i, g, Inn g g i+1 ),..., Inn g g n 1 )), and n 1 Ω n g) := 1) i Ωi n g) : C n G, L) C n 1 G, L). i=0 Then a long but straightforward computation gives that for c C n G, L), Inng, Dg) ) c) c = δ n 1) Ω n g)c) ), from which the Lemma follows Dimensional reduction of short exact sequences of vector bundles. Let ξ i := E i, π i, B) where i stands for 0, 1, or nothing) be reducible G-vector bundles over the same base B, with the same action t of the Lie group G on B. Let actions T i, t) of G on ξ i be such that the short exact sequence 0 ξ 0 i ξ j ξ ) be intertwining, i.e., the vector bundle morphisms i : E 0 E and j : E E 1 commute with the corresponding actions of G: T g i = i T 0, g, j T g = T 1, g j for all g G. 48) The main problem we will study in this section is whether an exact sequence like 47) holds for the reduced bundles ξ G 0, ξ G, ξ G 1. Recalling the construction of the reduced bundles from Section 2.3, it is clear that the restrictions ξ i euα of the bundles ξ i to the manifolds Ũα B see 17)) constitute an intertwining short exact sequence 0 ξ 0 euα i ξ euα j ξ 1 euα 0. 49) The coordinate bundles ξ i,α are obtained from ξ i euα by restricting each fiber to the stationary subspace in it: ξ i, α = st ξ i euα. Therefore, we have to study the behavior of the short exact sequence 49) under the process of restricting the bundles ξ i euα to their stationary subbundles st ξ i euα ξ i euα. For each x U α we

17 Dimensional Reduction I 17 can write the following intertwining short exact sequence of finite-dimensional representations of the stationary group G σαx) in the vector spaces ξ i, σαx) 0 ξ 0, σαx) i ξ σαx) j ξ 1, σαx) 0. 50) Now the cohomological interpretation 36) of the stationary subspaces plays a crucial role: we can think of the coordinate bundle ξ i,α as a bundle over Ũα = σ α U α ) with fiber st ξ i, σαx) = H 0 G σαx), ξ i, σαx)) for any x U α. According to Theorem 2, when we restrict each bundle in 50) to its stationary subbundle, we obtain the long exact sequence of vector spaces 0 st ξ 0, σαx) i 0 st ξσαx) j 0 st ξ1, σαx) k 0 H 1 G σαx), ξ 0, σαx)) i1. 51) The disjoint union of H 1 G σαx), ξ 0, σαx)) for x U α, with an appropriate equivalence relation, constitutes a possibly infinite-dimensional) vector bundle with base Ũα and fiber H 1 G σαx), ξ 0, σαx)); we introduce the short notation H 1 G, ξ 0 ) α for this bundle. Similar facts hold for all the terms in 51), for which we introduce similar notations. Our goal now is to glue the coordinate bundles H n G, ξ i ) α into a bundle H n G, ξ i ) over N = B/G; to this end, we have to construct transition isomorphisms between H n G, ξ i ) α and H n G, ξ i ) β satisfying the cocycle conditions 4). Let φ i, αβ be the transition isomorphisms gluing the coordinate bundles ξ i,α 18) into the reduced bundle ξi G see 19), 20), 21), 22), 23)). Here we explain how to use them in a natural way in order to construct isomorphisms φ n i, αβ gluing the bundles H n G, ξ i ) out of the coordinate bundles H n G, ξ i ) α for all n N. Let α, β) A A be a pair of indices for which U α U β, and let x U α U β. Let χ αβ be the map defined in 19) and 20), and let f χαβ and F χαβ be the maps defined in 21), 22) and used in the process of gluing the reduced bundle ξ G from the coordinate bundles ξ α. Recall that the fiber ξ i,σαx) of the coordinate bundle ξ i,α over the point σ α x) Ũα is the fiber st ξ i,σαx) of the stationary bundle st ξ i over σ α x), and similarly for ξ i,σαx). Let T σ βx) i : G σβ x) ξ i,σβ x) ξ i,σβ x), 52) T σαx) i : G σαx) ξ i,σαx) ξ i,σαx) 53) be the representations of the stationary groups G σβ x) and G σαx) in these fibers. Recalling Definition 4, we see that the maps and Inn χαβ x) : G σβ x) G σαx) : h Inn χαβ x)h = χ αβ x) h χ αβ x) 1 T i, χαβ x) : ξ i,σβ x) ξ i,σαx) form a pair Inn χαβ x), T i, χαβ x)) that is an isomorphism between the representation 52) of G σβ x) in ξ i,σβ x) and the representation 53) of G σαx) in ξ i,σαx): T i, χαβ x) T σ βx) i, g = T σαx) i, Inn χαβ x)g) T i, χ αβ x) for all g G σβ x)

18 18 Petko A. Nikolov, Nikola P. Petrov cf. 41)). Therefore, the maps ) Inn χαβ x), T i, χαβ x) defined by 42) and 43) n are well-defined maps between the corresponding cohomology groups: Innχαβx), T i, χαβx)) : Hn G n σβ x), ξ i,σβ x)) H n ) G σαx), ξ i,σαx), 54) or, in the short notation introduced above, Innχαβx), T i, χαβx)) n : Hn G, ξ i ) β, σβ x) H n G, ξ i ) α, σαx). The map 54) is defined uniquely, although χ αβ is not unique. Recall that χ αβ is determined by the condition t χαβ x) σβ x) ) = σ α x), and, therefore, is defined up to a multiplication on the right by g G σβ x) and a multiplication on the left by g G σαx). Lemma 5, however, guarantees that the maps ) Inng, T i, g : Hn G n σβ x), ξ i,σβ x)) H n ) G σβ x), ξ i,σβ x) and Inng, T i, g ) : Hn G n σαx), ξ i,σαx)) H n ) G σαx), ξ i,σαx) are trivial for any g G σβ x) and g G σαx). Therefore the map 54) does not depend on the arbitrariness in the choice of the map χ αβ. In the case n = 0, this is simply the fact that the map T i,χα,β x) : st ξ i,σβ x) st ξ i,σαx) is independent of the arbitrariness in the choice of χ αβ, as explained after equation 23).) Letting x vary over U α U β, and recalling that H n G, ξ i ) α,β stands for the restriction H n G, ξ i ) α euα,β of H n G, ξ i ) α over the manifold Ũα,β 1), we obtain that the maps ) Inn χαβ x), T i, χαβ x), t ) n χ αβ x) define the desired isomorphisms φ n i, αβ : H n G, ξ i ) β,α H n G, ξ i ) α,β. The maps φ n i, αβ glue the coordinate bundles {Hn G, ξ i ) α } α A into the bundle H n G, ξ i ). Since φ n i, αβ are constructed through the actions of G on ξ i, they satisfy the cocycle conditions 4) required in the gluing procedure. The only thing that remains to be proved is that the isomorphisms φ n i, αβ are compatible with the long exact sequences 51) of the coordinate bundles, i.e., that the diagram 0 st ξ0, β,α i 0 st ξβ,α j 0 st ξ1, β,α k 0 H 1 G, ξ 0 ) β,α i 1 H 1 G, ξ) β,α j 1 φ 0, αβ φ αβ φ 1, αβ φ 1 0, αβ φ 1 αβ 55) 0 st ξ0, α,β i0 st ξα,β j0 k0 st ξ1, α,β H 1 G, ξ 0 ) α,β i 1 j1 H 1 G, ξ) α,β is commutative. To give an idea of how the proof goes, let us check a part of the diagram including the connecting morphism k n. Consider the two short exact sequences the horizontal arrows to the right) in the diagram 0 i ξ 0,b F b T 0,g 0 ξ 0,tgb) i F tg b) ξ b ξ tgb) j ξ 1,b 0 S b T g j S tg b) T 1,g ξ 1,tgb) 0 56)

19 Dimensional Reduction I 19 If we choose a splitting F b, S b ) of the first short exact sequence, then thanks to 48) we obtain that the isomorphisms T i,g : ξ i,b ξ i,tgb) determine a splitting F tgb), S tgb)) of the second short exact sequence that satisfies T 0,g F b = F tgb) T g, T g S b = S tgb) T 1,g. 57) Let [c] H n G b, ξ 1,b ), g G, g 1 G tgb),..., g n+1 G tgb). Using 42), 37), 46), and 57), we obtain Inng, T 0,g ) n+1 k n c) ) g 1,..., g n+1 ) = T 0,g k n c)inn g 1g 1 ),..., Inn g 1g n+1 )) = T 0,g F b T Inng 1 g 1) S b cinng 1g 2 ),..., Inn g 1g n+1 )) ) = F tgb) T g T g 1 T g1 T g S b cinng 1g 2 ),..., Inn g 1g n+1 )) ) = F tgb) T g1 T g S b cinng 1g 2 ),..., Inn g 1g n+1 )) ) = F tgb) T g1 S tgb) T 1,g cinn g 1g 2 ),..., Inn g 1g n+1 )) = k n Inn g, T 1,g ) n c) ) g 1,..., g n+1 ). This completes the proof that the following part of 55) is commutative: j n H n G, ξ 1 ) β,α k n H n+1 G, ξ 0 ) β,α i n+1 φ n 1, αβ j n H n G, ξ 1 ) α,β φ n+1 0, αβ k n H n+1 G, ξ 0 ) α,β i n+1 58) The commutativity of the rest of the diagram 55) can be proved analogously. The commutativity of the diagram 55) implies that the transition isomorphisms φ n i, αβ determine the long exact sequence 0 ξ G 0 i 0 ξ G j0 ξ G 1 k 0 H 1 G, ξ 0 ) i1 H 1 G, ξ) j1. 59) This answers the question posed in the beginning of this section. Namely, if k 0 ξ1 G ) = 0, then the reduced bundles ξi G = H 0 G, ξ i ) form a short exact sequence 0 ξ G 0 i 0 ξ G j0 ξ G 1 0, 60) hence in this case ξ G = ξ0 G ξ1 G. This holds when all finite-dimensional representations of the stationary group G b are completely reducible for all b B, or, in particular, when G is compact, because in these cases H 1 G, ξ 0 ) = 0 i.e., H 1 G, ξ 0 ) is a bundle with fiber a 0-dimensional vector space). This motivates the following definition. Definition 5. The dimensional reduction of the intertwining short exact sequence 47) is said to be canonical when the reduced bundles constitute the exact sequence 60). Otherwise, we call the nonzero element k 0 ξ G 1 ) H 1 G, ξ 0 ) the obstruction to canonical dimensional reduction of 47).

20 20 Petko A. Nikolov, Nikola P. Petrov We recapitulate our results from this section in the following Theorem 3. Let ξ 0, ξ and ξ 1 be reducible G-vector bundles with the same base B and the same action t of G on B, and let these bundles form an intertwining short exact sequence 0 ξ 0 i ξ j ξ 1 0. Then the reduced bundles ξ G 0, ξ G and ξ G 1 are a part of the long exact sequence 59). If the obstruction k 0 ξ G 1 ) H 1 G, ξ 0 ) to canonical dimensional reduction is zero, then the reduced bundles constitute the short exact sequence 60), hence in this case ξ G = ξ G 0 ξ G 1. Example 1. Here is an example of a non-canonical reduction of an intertwining short exact of vector bundles. Let the common base B of the vector bundles ξ 0, ξ, and ξ 1 consist of only one point. Let the typical fiber in these bundles be 2-, 3-, and 1-dimensional, respectively. Let G be the Heisenberg group, with elements g a,b,c = 1 a b 0 1 c, a, b, c R, and the actions on ξ 0, ξ, and ξ 1 be ) x T 0, ga,b,c = y ) x + ay, T y ga,b,c x x + ay + bz y = y + cz, T 1, ga,b,c z) = z). z z Let i : ξ 0 ξ be the canonical embedding and j : ξ ξ 1 be the canonical projection: x i = y) x y, j x y = z). 0 z With these definitions, the short exact sequence 47) is intertwining with respect to the actions of G. Let the maps F : ξ ξ 0 and S : ξ 1 ξ be defined by Sz) = α β z) = αz βz, F x ) y 1 0 α = x ) y x αz =, 0 1 β y βz 1 z z z where α and β are some real constants. Then the pair F, S) defines a splitting of the short exact sequence 47), i.e., the maps F and S satisfy F i = Id L0 and im S = ker F. It is easy to check that the stationary subbundles are R st ξ 0 =, st ξ = 0) R 0, st ξ 1 = ξ 1, 0

21 Dimensional Reduction I 21 and that the G-invariant sections of the three bundles have the form C ξ 0 ) G x =, C 0) ξ) G = x 0, C ξ 0 ) G = z), 0 where x and z are arbitrary real numbers. Just from counting dimensions, we see that ξ ξ 0 ξ 1, i.e., the dimensional reduction of the short exact sequence 47) is non-canonical. Let us consider the long exact sequence 59) cf. 45)), and compute the obstruction k 0 ξ1 G ) H 1 G, ξ 0 ) to canonical reduction. Clearly, the injectivity of the canonical embedding i, the fact that both st ξ 0 and st ξ are one-dimensional, and the exactness of 59) imply that jst ξ) is zero. By the exactness of 59), k 0 : st ξ 1 H 1 G, ξ 0 ) is injective, therefore the obstruction k 0 st ξ 1 ) is onedimensional. To compute the obstruction to canonical reduction more explicitly, recall that in this example H 0 G, ξ 1 ) = st ξ 1 = ξ 1, hence the equivalence class, [z] H 0 G, ξ 1 ), of an element z st ξ 1 = ξ 1, is the element z itself. Then k 0 [z]) is a map from G to ξ 0 defined modulo maps from im δ 0) ξ 0. According to 37), k 0 [z])g a,b,c ) is the equivalence class of ) ) βa + b)z F T ga,b,c Sz) = = z 1 a b 0 1 c 0 β. cz 0 1 β The space im δ 0) ξ 0 consists of maps from G to ξ 0 proportional to a Ag a,b,c ) = = 0) ) 1 a b 0 1 c 0 β Therefore, we can write [k 0 [z])] g a,b,c ) := [k 0 [z])g a,b,c )] H 1 G, ξ 0 ) as ) ) βa + b)z a z + R 0 0 [k 0 [z])] g a,b,c ) = +R = g cz 0) 0 z βz a,b,c 0 β R =, cz) 1 which makes explicit the fact that the obstruction to canonical reduction is k 0 st ξ 1 ) = R. 4. Dimensional reduction of tangent and cotangent bundles and their tensor products 4.1. Reduction of τb) and τ B).

22 22 Petko A. Nikolov, Nikola P. Petrov Vertical subbundle and reduction of τb). Let B be a manifold endowed with an action t of the Lie group G which satisfies Conditions A and B from Section 2.2. Let τb) = T B), π, B) and τ B) be respectively the tangent and the cotangent bundles to B. Let t : G T B) T B) be the tangent lift of the action t. Then t, t) is an action of G on τb) through vector bundle morphisms. This action makes τb) a reducible G-vector bundle. The base B becomes the total space of the G-bundle B, p, N) as in 11). Let τ v B) τb) be the vertical subbundle of τb), which by definition consists of the vectors tangent to the fibers of B, p, N): τ v B) b := τp 1 pb))) b b B. Let p τn) be the bundle over N induced by the projection p : B N. If we think of p τn) as the set of all pairs b, w) B T N) satisfying pb) = π N w) where π N : T N) N is the natural projection), then the action of g G is given by b, w) t g b), w). 61) Let i : τ v B) τb) be the natural embedding, and j : τb) p τn) be the natural projection. Then the short exact sequence 0 τ v B) i τb) j p τn) 0 62) is intertwining with respect to the corresponding actions of G. Hence, to perform G-dimensional reduction of 62), we can apply Theorem 3, and then the only remaining task is to find the obstruction k 0 p τn)) H 1 G, τ v B)) for canonical dimensional reduction. The obstruction for canonical reduction will be zero if for each b B the G b -invariant subspace τ v B) b has a G b -invariant complement in τb) b. For the particular case of the tangent lift t of the action t of G on B, the question of finding this obstruction is easy. Indeed, Condition A guarantees that B, π, N) see 11)) is a locally trivial G-bundle. Because of this, there exists a local section σ of B, π, N) whose graph Ũ contains b, and such that the stationary group G b of each point b Ũ is the same as the stationary group G b of b. More explicitly, if V N is an open subset of N containing pb), then W b) := { Φ 1 x, π 2 Φb)) : x V N } is a submanifold of B which is transversal to the orbits of G in B, contains b, and consists of points whose stationary group is G b we used the notation of 13)). This implies that G b acts trivially on W b), which in turn means that G b acts trivially on τw b) ), thus τw b) ) b is a G-invariant subspace of τb) b complementary to iτ v B) b ). Therefore the obstruction k 0 p τn)) is zero, and Theorem 3 yields the short exact sequence 0 st τ v B) b i st τb) b j p τn) b 0. Finally we notice that p τn) G = τn), and summarize the above in the following