Chern–Simons functional under gauge transformations on flat bundles

Accepted Manuscript Chern-Simons functional under gauge transformations on flat bundles Yanghyun Byun, Joohee Kim PII: DOI: Reference:

S0393-0440(16)30235-2 http://dx.doi.org/10.1016/j.geomphys.2016.09.012 GEOPHY 2840

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Journal of Geometry and Physics

Received date: 18 July 2015 Revised date: 16 February 2016 Accepted date: 28 September 2016 Please cite this article as: Y. Byun, J. Kim, Chern-Simons functional under gauge transformations on flat bundles, Journal of Geometry and Physics (2016), http://dx.doi.org/10.1016/j.geomphys.2016.09.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES YANGHYUN BYUN AND JOOHEE KIM Abstract. We describe the effect of a gauge transformation on the ChernSimons functional in a thorough and unifying manner. The conditions under which we work are that the structure group is compact and connected and, in particular, that the principal bundle is flat. The Chern-Simons functional we consider is the one defined by choosing a flat reference connection. The most critical step in arriving at the main result is to show both the existence and the uniqueness of a cohomology class on the adjoint bundle such that it is the class of the so-called Maurer-Cartan 3-form when restricted to each fiber.

1. Introduction The Chern-Simons functional is a function CS : A(P ) → R, where P is a principal bundle with the structure group G over an oriented connected closed 3manifold M and A(P ) is the set of all connections on P . Furthermore let ϕ : P → P be a gauge transformation. That is, assume that ϕ is a smooth G-map which maps each fiber Px , x ∈ M , onto itself. Then one expects that CS(ϕ∗ A) − CS(A) = κ deg ϕ

(1-1)

for some constant κ and for an invariant deg ϕ of ϕ. In this paper we show that this formula is valid with κ = 1 when deg ϕ is defined in a natural way under the assumptions that G is compact and connected and that the bundle is flat. For instance consider an SU (2)-bundle P over a 3-manifold M , which is necessarily isomorphic to the trivial bundle. In this case a connection A on P can be identified with an su(2)-valued 1-form on M . Assume furthermore that M is oriented, connected and closed. Then the Chern-Simons functional can be given by the equation, Z 1 1 CS(A) = − (1-2) tr(A ∧ dA + A ∧ [A ∧ A]). 2 M 3 On the other hand a gauge transformation ϕ can be identified with a smooth map ϕ : M → SU (2) ≡ S 3 . Then the degree of ϕ has an obvious meaning. In this case (1-1) above holds with κ = −4π 2 (cf. [12]). Note that the equation, hX, Y i = −tr(XY), for any X, Y ∈ su(2), provides an invariant inner product h·, ·i on su(2). By an inner product we mean a positive definite symmetric bilinear function on a real vector space. An inner product h·, ·i is invariant if it is defined on the Lie algebra G of a Lie group G and satisfies the condition that hAdg X, Adg Y i = hX, Y i for any g ∈ G and for any X, Y ∈ G. By the Lie algebra G we will mean the tangent 2010 Mathematics Subject Classification. 53C05, 58J28. Key words and phrases. Chern-Simons functional defined by a reference connection; degree of a gauge transformation; global Maurer-Cartan 3-form on the adjoint bundle of a flat bundle. 1

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space Te G at the identity e ∈ G and by Adg : G → G, the derivative at the identity e ∈ G of the adjoint homomorphism G → G defined by h → ghg −1 for any h ∈ G. More generally assume the structure group G is a compact simply connected simple Lie group. Then any principal G-bundle over a 3-manifold is isomorphic to the trivial bundle. Therefore if an invariant inner product is provided to the Lie algebra of G the discussions involving (1-2) above are valid without any essential change. Furthermore one should note the following facts: H2 (G; Z) = 0, H3 (G; Z) ∼ (1-3) =Z (see for instance the 1st paragraph of §1, [8]). Therefore the degree of a gauge transformation can be defined in an obvious way. Note that κ in (1-1) may vary depending on the choice of the invariant inner product and the sign of deg ϕ depends on the choice of the generator of H3 (G; Z). On the other hand non-trivial SO(3)-bundles exist on a 3-manifold which are studied in [5]. The authors define the degree of a gauge transformation exploiting somewhat nontrivial knowledge in algebraic topology and the equation (1-1) has been asserted to hold with κ = −8π 2 by the equation (2) in §2, [5]. Here the invariant inner product is the minus of the Killing form, or equivalently, 4-times the minus of the trace. In fact the Chern-Simons functional itself is not as simple as (1-2) above. Note that a definition similar to (1-2) is valid only when P is trivial. The authors observe that there are flat connections in the case under consideration. Therefore they may choose one of them and use it to define a Chern-Simons functional, which we also follow in this paper (see (2-2, 3) and 5.3 below). On the other hand it seems that there is no formal proof of the equation (2) not only in [5] but also even in the literature. Of course we expect that the equation (2) must be correct, considering the case of the trivial bundle and also taking into account the general properties of the Chern-Simons functional as summarized by 2.2 below. The contents of the paper are as follows. We consider a principal bundle over an oriented connected closed 3-manifold such that the structure group is compact and connected. Furthermore the bundle is assumed to be flat. Then in §2 we recall the Chern-Simons functional defined by a reference connection. At this point we do not yet require that the reference connection should be flat. In §3 we provide a detailed study of the so-called Maurer-Cartan 3-form of a compact connected Lie group. Then in §4 we construct a closed form on the adjoint bundle whose restriction to each fiber is the Maurer-Cartan 3-form. Up to this point of the section the dimension of the base manifold need not be 3. Once the ‘global’ Maurer-Cartan 3-form on the adjoint bundle is constructed for general flat bundles, we provide a definition of the degree of a gauge transformation when the dimension of the base manifold is 3 (see (4-8) below). In §5 we fix a flat connection and use it as the reference connection. Then we show that the equation (1-1) above holds with κ = 1 by 5.3 below, which is our main result. Note that the flatness condition is crucial in our work. The condition may not be as restrictive as it might appear at first sight. For instance every SO(3)-bundle on an orientable connected closed 3-manifold is flat as observed in [5]. Furthermore any critical point of a Chern-Simons functional is a flat connection as can be seen by (2-5) below. Therefore more interesting phenomena are expected under the condition. On the other hand there is another method to define a Chern-Simons functional on a general bundle over an oriented connected closed 3-manifold with a compact structure group which has been proposed in [4] (see also Appendix of [6]).

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES3

The resulting functional is R/Z-valued from the beginning. In this method one simply observes that the gauge transformation leaves the Chern-Simons functional unchanged and therefore the need to define its degree does not surface explicitly. 2. Chern-Simons functional with a reference connection Let G be a Lie group with the Lie algebra G. Consider a principal G-bundle π : P → M . Then G acts freely on P from the right so that p G = Px = π −1 {x} for any x ∈ M . A connection A on P is a G-valued 1-form on P which is equivariant in the sense that (Rg )∗ A = Adg−1 A where Rg : P → P denotes the multiplication on the right by g ∈ G. On the other hand for any p ∈ P let ιp : G → P mean the map defined by ιp (g) = pg for any g ∈ G. Then A should satisfy the condition that A(d(ιp )e X) = X

for any X ∈ G. ¯ 1 (P ; G) ⊂ Ω1 (P ; G) which consists of all horizontal equiConsider the subspace Ω variant 1-forms. Here that a ∈ Ω1 (P ; G) is horizontal means that a(X) = 0 if X is vertical, that is, if X ∈ Tp Pπ(p) ⊂ Tp P . Then the set A(P ) of all connections on P ¯ 1 (P ; G). is an affine subspace of Ω1 (P ; G) modeled after Ω On the other hand let α be any W -valued k-form on P , where W is any real vector space. Then the twisted exterior derivative dA α is defined as follows: The horizontal subbundle H of T P determined by A is such that the fiber Hp at p is given by Hp = Ker(Ap : Tp P → G) for any p ∈ P . And the vertical subbundle V of T P is defined by Vp = Tp Pπ(p) . Then we have the decomposition T P = H ⊕ V . Let πH : T P → H be the associated projection. Then we denote by β H , for any β ∈ Ωk (P ; W ), the form defined by β H (X1 , ..., Xk ) = β(πH X1 , ..., πH Xk )

for any X1 , ..., Xk ∈ Tp P and p ∈ P . Finally the twisted exterior derivative is defined by dA α = (dα)H . Now the curvature FA ∈ Ω2 (P ; G) of a connection A is defined by FA = dA A (cf. p.77, [9]). The curvature FA is equivariant. It is also horizontal, that is, we have that FA (X, Y ) = 0 if either of X, Y ∈ Tp P belongs to Vp . We say that A is flat if FA = 0 and that P is flat if it admits a flat connection. Now let us consider a multi-linear function f : G × ... × G → R which is invariant and symmetric. That f is invariant means that f (Adg X1 , ..., Adg Xk ) = f (X1 , ..., Xk ) for any g ∈ G and for any X1 , ..., Xk ∈ G. Then f (FA ) is the realvalued 2k-form on P defined by 1 X ǫσ f (F (Xσ(1) , Xσ(2) ), ..., F (Xσ(2k−1) , Xσ(2k) )) f (FA )(X1 , ..., X2k ) = (2k)! σ for any X1 , · · · , X2k ∈ Tp P and any p ∈ P , where ǫσ denotes the sign of the permutation σ. It is straightforward to see that f (FA ) is invariant in the sense that (Rg )∗ (f (FA )) = f (FA ) for any g ∈ G. Therefore f (FA ) is horizontal and invariant. This means that f (FA ) can be regarded as a form on M in the sense

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that there is a unique α ∈ Ω2k (M ; R) such that π ∗ α = f (FA ) and we may identify f (FA ) with α. It is a standard result of Chern-Weil theory that f (FA ) ∈ Ω2k (M ; R) is closed. 2k (M ) does not depend on A. To be precise, Furthermore the class of f (FA ) in HdR 1 ¯ if a is a form in Ω (P ; G), we have: Z 1  f (FA+a ) − f (FA ) = k d f (a, FA+ta , · · · , FA+ta )dt (2-1) R1

0

(cf. p.297, [10]). Note that 0 f (a, FA+ta , · · · , FA+ta )dt is also horizontal and invariant. Choose an invariant inner product h·, ·i on G in the sense of the introduction ¯ 1 (P ; G). Then we above. Fix a connection A0 on P . Write A = A0 + a where a ∈ Ω consider the 3-form 1 (2-2) cs(A) = h(FA + FA0 ) ∧ ai − ha ∧ [a ∧ a]i, 6 which is non other than the integrand of the Chern-Simons functional that appears §2 of [5] just above the equation (2). However note in (2-2) that neither the structure group is restricted to SO(3) nor A0 is required to be flat. Still cs(A) is horizontal and invariant and therefore may be regarded as a form on M . Here we would like to clarify the notational conventions in the expressions such as ha ∧ [a ∧ a]i in (2-2) above. First of all we understand a ∧ a as a G ⊗ G-valued 2-form defined by (a ∧ a)(X, Y ) = a(X) ⊗ a(Y ) − a(Y ) ⊗ a(X)

for any X, Y ∈ Tp P and any p ∈ P . Similarly we understand 1 X (FA ∧ a)(X1 , X2 , X3 ) = ǫσ FA (Xσ(1) , Xσ(2) ) ⊗ a(Xσ(3) ), 2! 1! σ∈S3

for any Xi ∈ Tp P , i = 1, 2, 3 and any p ∈ P . Furthermore we identify any bilinear function on G × G with a linear map on G ⊗ G. With these conventions in mind, one may understand such expression as the RHS of (2-2) with ease. Now assume M is an oriented connected closed 3-manifold. Then we may define a function, CS : A(P ) → R, by the equation: Z CS(A) = cs(A), (2-3) M

which we refer to as the Chern-Simons functional defined by choosing the reference connection A0 . The Chern-Simons functional given by (2-2, 3) above has the advantage that it can be applied to any principal G-bundle P over an oriented connected closed 3-manifold, even when P is not flat. This form of Chern-Simons functional has been used in [5] and also in [12]. The equality below reveals the Chern-Weil theory origin of the form cs(A): hFA ∧ FA i − hFA0 ∧ FA0 i = d(cs(A)),

(2-4)

which is in fact a special case of (2-1) above (see also [2]). Note that 12 hFA ∧FA i is the usual Chern-Weil form in the literature (cf. [5], [12]) from which one derives the Chern-Simons functional. In fact, with the factor 21 , it is the Hamiltonian in a field theory of physics. For instance the integrand in the

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES5

RHS of (1-2) above corresponds to −cs(A) with the reference connection A0 being chosen as the canonical one of the trivial bundle. However the integral is multiplied by − 21 in the end. In this paper we begin from the Chern-Weil form hFA ∧ FA i for simplicity and as the result our Chern-Simons form (2-2) in the above is 2 times the usual one. 1 ¯1 Recall that A(P ) is an affine subspace of Ω (P ; G) modeled after Ω (P ; G). d Therefore we may write dCSA (a) = dt t=0 CS(A + ta) for any A ∈ A(P ) and ¯ 1 (P ; G). Then we have the identity (see §2, [5]), for any a ∈ TA A(P ) ≡ Ω Z hFA ∧ ai. (2-5) dCSA (a) = 2 M

Thus the critical points of the Chern-Simons functional are the flat connections. Furthermore the RHS of (2-5) does not depend on the reference connection and therefore we know that CS1 − CS0 is the constant CS1 (A0 ) = −CS0 (A1 ) if CS0 and CS1 are the Chern-Simons functionals given by reference connections A0 and A1 respectively. Now we note that the group GA(P ) of all gauge transformations of P acts on A(P ). The following is also stated in [5].

Lemma 2.1. The 1-form dCS on A(P ) is invariant and horizontal. That is, we have the identities, d d d ∗ CS(ϕ (A + ta)) = CS(A + ta) and CS(ϕ∗t A) = 0, dt t=0 dt t=0 dt t=0 ¯ 1 (P ; G), ϕ ∈ GA(P ) and for any smooth 1-parameter for any A ∈ A(P ), a ∈ Ω family ϕt , −ǫ < t < ǫ, of gauge transformations such that ϕ0 = 1.

The first equality of 2.1 above implies that CS(ϕ∗ A)−CS(A) does not depend on d A. The second means in fact that dt (CS(ϕ∗t A)) = 0 for any t in the interval where the 1-parameter family ϕt is defined. Therefore it follows that CS(ϕ∗ A) − CS(A) is constant if ϕ varies within a path component of GA(P ). Since the choice of the reference connection A0 affects the Chern-Simons functional only by addition of a constant, CS(ϕ∗ A) − CS(A) does not depend on the reference connection either. Therefore we have: Proposition 2.2. The real number, CS(ϕ∗ A) − CS(A), depends only on the path component of ϕ in GA(P ). In particular it depends neither on A nor on the reference connection. Note that the main goal of the paper is to express CS(ϕ∗ A) − CS(A) in terms of an invariant of ϕ as (1-1) above implies. The proposition above shows that we are in a right direction. 3. Maurer-Cartan 3-form of a Connected Compact Lie Group Let G be a connected compact Lie group. We denote by H∗ (G) the set of all bi-invariant real valued forms on G. That a form α ∈ Ω∗ (G; R) is bi-invariant means that it satisfies (Lg )∗ α = α = (Rg )∗ α for any g ∈ G. According to Theorem 12.1 of [3], bi-invariant forms are closed and we have that ∗ H∗ (G) ≡ HdR (G).

(3-1)

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˜ → G be a surjective Lie group homomorphism with a Furthermore let p : G finite kernel. Since p is a group homomorphism, it follows that, for any bi-invariant α ∈ Ω∗ (G; R), p∗ α is also bi-invariant. Using the fact that p is in fact a local ˜ is the pull-back of a diffeomorphism it follows that any bi-invariant form of G unique bi-invariant form of G. Therefore (3-1) above implies the identity, ∗ ∗ ˜ HdR (G) ≡ HdR (G).

(3-2)

˜ can be chosen as the product, It is well-known that in fact G T k × G1 × · · · × Gl ,

of a torus group T k and simply connected simple Lie groups Gi , i = 1, · · · , l. Therefore considering (3-2) we have: ∗ ∗ ∗ ∗ HdR (G) ∼ (G1 ) ⊗ · · · ⊗ HdR (Gl ). (T k ) ⊗ HdR = HdR

Now let θ ∈ Ω1 (G; G) be the Maurer-Cartan form of G which is defined by θ(X) = d(Lg−1 )g (X) ∈ Te G ≡ G

for any X ∈ Tg G and for any g ∈ G. We observe that

(Lg )∗ θ = θ, (Rg )∗ θ = Adg−1 θ.

(3-3)

Then we consider the 3-form on G 1 (3-4) Θ = − hθ ∧ [θ ∧ θ]i 6 where h·, ·i is an invariant inner product on G. We will call Θ the Maurer-Cartan 3-form determined by the invariant inner product h·, ·i. By (3-3) above we see that Θ is bi-invariant. For further discussion we need to observe the following. Lemma 3.1. Let G be a compact simply connected simple Lie group with the Lie algebra G. Then there is a unique invariant inner product on G up to multiplication by a positive real number. 3 ∼ R (see (1-3) above). Also we note that Proof. First of all we observe that HdR (G) = 3 3 the identification H (G) ≡ HdR (G) from (3-1) above. Therefore the Maurer-Cartan 3-form Θh·,·i determined by an invariant inner product h·, ·i should be unique up to multiplication by a real number. In fact the real number should be positive: For any X, Y, Z ∈ G we have that

hX, [Y, Z]i = hZ, [X, Y ]i

(3-5)

by invariance of the inner product h·, ·i. This leads to the equality, Θh·,·i(X, Y, Z) = −hX, [Y, Z]i, for any X, Y, Z ∈ G. In particular, for X, Y ∈ G such that [X, Y ] 6= 0, we have Θh·,·i([X, Y ], X, Y ) = −h[X, Y ], [X, Y ]i < 0. Thus Θh·,·i is in fact unique up to multiplication by a positive real number. Now, since we have the equality Θh·,·i (X, Y, Z) = −hX, [Y, Z]i and [G, G] = G, we conclude the lemma.  Now assume G = G1 × · · · × Gl where Gk , k = 1, · · · , l, are simply connected simple Lie groups. Then we have the decomposition G ≡ G1 ⊕ · · · ⊕ Gl

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES7

of the Lie algebra. An invariant inner product h·, ·i on G makes this decomposition orthogonal: For instance let X, Y ∈ G1 and Z ∈ G2 ⊕ · · · ⊕ Gl . Then by applying (3-5) above we have h[X, Y ], Zi = h[Z, X], Y i = 0.

Since [X, Y ]’s generate G1 , we conclude that G1 is orthogonal to G2 ⊕ · · · ⊕ Gl . Therefore the invariant metric on G is determined by its restrictions to Gk ’s. Without loss of generality, now we may assume each Gk is equipped with an invariant inner product and G is given the direct sum inner product. Use these inner products to define the Maurer-Cartan 3-forms Θ and Θk ’s respectively of G and Gk ’s. Then we have Θ = q1∗ Θ1 + · · · + ql∗ Θl (3-6)

where qk : G → Gk , k = 1, · · · , l, are the projections. On the other hand since i HdR (Gk ) = 0 for i = 1, 2 we have that 3 3 3 HdR (G) ≡ HdR (G1 ) ⊕ · · · ⊕ HdR (Gl ).

Recall that there is a unique invariant inner product on each Gk up to multiplication by a positive real number. Therefore (3-6) implies that the set of all [Θ], each of which is determined by a choice of an invariant inner product on G, is an open cone 3 in HdR (G). Slightly more generally let G be a compact connected semi-simple Lie group. ˜ ˜ = G1 × · · · Gl for some simply connected simple Lie Then G ∼ where G = G/Z ˜ In fact the groups Gk , k = 1, · · · , l, and Z is a finite subgroup of the center of G. ˜ ˜ center of G itself is finite. Let p : G → G denote the surjective homomorphism with ∗ ∗ ˜ is an isomorphism. (G) (G) → HdR the finite kernel. Then by (3-2) above, p∗ : HdR ˜ with that of G by means of the On the other hand we identify the Lie algebra of G ˜ → Te G. Then note in particular that p∗ θ is the Maurerisomorphism dpe : Te G ˜ if θ is that of G. Now it is straightforward to see that the Cartan form of G ˜ Therefore the set of all image by p∗ of a Maurer-Cartan 3-form of G is that of G. 3 classes [Θ] is still an open cone in HdR (G). Since the image of the homomorphism 3 (G) is a full lattice, we conclude that there are H 3 (G; Z) → H 3 (G; R) ≡ HdR 3 (G) integral. invariant inner products on G which make [Θ] ∈ HdR Now let G be any compact connected Lie group and Z0 be the identity component of the center of G. Then G′ = G/Z0 is semi-simple. We have a natural decomposition of the Lie algebra, G = Z ⊕ G ′ , where Z is the center of G and G ′ = [G, G]. Note that G ′ can be identified with the Lie algebra of G′ . Let ψ : G → G′ be the quotient homomorphism. Any invariant inner product on G ′ is given, we may equip G with an invariant metric so that the restriction of dψe to G ′ is an isometry and Z is orthogonal to G ′ . In fact any invariant inner product on G makes the decomposition G = Z ⊕ G ′ orthogonal. Now let Θ and Θ′ be the Maurer-Cartan 3-forms of G and G′ respectively. Then we have: Θ = ψ ∗ Θ′ .

(3-7)

To see this equality, note first of all that ψ ∗ Θ′ is bi-invariant. Therefore it is enough to see that the equality holds when both sides of (3-7) are evaluated at any triple (X, Y, Z) ∈ G 3 . We observed that Θ(X, Y, Z) = −hX, [Y, Z]i in the proof of 3.1 above. Thus the equality (3-7) follows from the equality, hX, [Y, Z]i = hdψe X, [dψe Y, dψe Z]i, which is straightforward.

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We summarize the discussions above as follows: First of all we need to recall some of the notations used in the above. Let G be a compact connected Lie group with the Lie algebra G. Let Z0 be the identity component of the center of G and ψ : G → G/Z0 , the quotient homomorphism. Let G1 , · · · , Gl be simply connected simple Lie groups such that there is a surjective Lie group homomorphism p : G1 × · · · × Gl → ∗ ∗ G/Z0 with a finite kernel. Then p∗ : HdR (G/Z0 ) → HdR (G1 × · · · × Gl ) is an isomorphism as observed by the identity (3-2) above. Write qk : G1 × · · ·× Gl → Gk for the projection for each k = 1, · · · , l. We fix the Maurer-Cartan 3-forms of G1 , · · · , Gl and denote them respectively by Θ1 , · · · , Θl which are determined by fixing some invariant inner products on the respective Lie algebras. Then we have: Proposition 3.2. For each (t1 , · · · , tl ), tk > 0, k = 1, · · · , l, there is an invariant inner product on G so that the corresponding Maurer-Cartan 3-form is ψ ∗ p∗ −1 (t1 q1∗ Θ1 + · · · + tl ql∗ Θl ).

Conversely any invariant inner product on G gives rise to a Maurer-Cartan 3-form that can be written as in the above for some (t1 , · · · , tl ), tk > 0, k = 1, · · · , l. In particular there are invariant inner products on G each of which makes the MaurerCartan 3-form represent an integral class. 4. The Maurer-Cartan 3-form on the adjoint bundle Let π : P → M be a principal bundle with the structure group G. We regard G itself as a right G-set with the action given by the rule, x · g = g −1 xg = Adg−1 (x), for any x ∈ G and any g ∈ G. Here we are using the notation Adh , for an h ∈ G, to mean an automorphism of G even if we have used it so far to mean an automorphism of the Lie algebra. In what follows we will use the notation Adh in either sense. In fact no ambiguity will arise from this practice. Now we may regard P × G as a right G-set by considering the diagonal action. We will write Ad P to denote the orbit space. Let [p, g] denote the orbit for any (p, g) ∈ P × G. On the other hand there is the projection Ad P → M defined by [p, g] → π(p), which we will denote also by π. It is well-known that π : Ad P → M is a smooth locally trivializable bundle of Lie groups such that each fiber is isomorphic to G. The group structure on each fiber of Ad P is given by [p, g][p, h] = [p, gh] for any p ∈ P and any g, h ∈ G. The bundle Ad P is referred to as the adjoint bundle of P . It is a standard fact that the gauge transformations are in a natural 1-1 correspondence with the smooth sections of the adjoint bundle (cf. [11]). Recall that a connection A on P determines a horizontal distribution H on ¯ be P while there is the vertical distribution V independent of A. Let V¯ and H respectively the ‘vertical’ and the ‘horizontal’ distributions of Ad P which are defined as follows: Let q : P × G → (P × G)/G = Ad P denote the projection. Consider the fiber (Ad P )x = π −1 {x} for any x ∈ M . Then we define V¯ by ¯ by V¯[p,g] = T[p,g] (Ad P )π(p) for any [p, g] ∈ Ad P . On the other hand we define H ¯ [p,g] = dq(p,g) (Hp ⊕ 0g ), H

(4-1)

¯ is well-defined by the invariance of H in the sense that the equality Note that H dRg Hp = Hpg holds for any g ∈ G and any p ∈ P . Furthermore we have that ¯ [p,g] is the dimension of M for any [p, g] ∈ Ad P since dq(p,g) is injective when dim H restricted to Hp ⊕ 0g . On the other hand note that dim V¯[p,g] is the dimension of

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES9

G, where we regard G as a manifold. Furthermore we observe that ¯ [p,g] = 0 V¯[p,g] ∩ H

using the fact that V¯[p,g] ⊂ Ker dπ[p,g] while dπ[p,g] is injective when restricted to ¯ [p,g] . Therefore there is the decomposition T (Ad P ) = V¯ ⊕ H. ¯ Now we may H consider the associated projection, πA : T (Ad P ) → V¯ .

On the other hand, for each p ∈ P , there is an isomorphism κp : (Ad P )π(p) → G between the Lie groups given by the rule κp [p, g] = g for any g ∈ G. Now we recall k the identity (3-1) above which implies that any class in HdR (G) can be represented by a unique bi-invariant k-form on G. Let α be a bi-invariant k-form on G. Then we define a k-form α ˆ on Ad P by the rule: α ˆ (X1 , · · · , Xk ) = ((κp )∗ α)(πA X1 , · · · , πA Xk ) (4-2) = α(d(κp )[p,g] πA X1 , · · · , d(κp )[p,g] πA Xk ) for any X1 , · · · , Xk ∈ T[p,g] (Ad P ) and any [p, g] ∈ Ad P . Note that α ˆ does not depend on the choice of p due to the bi-invariance of α: One may use the identity, κph = Adh−1 κp , for any h ∈ G to see the well-definedness of α ˆ . Furthermore, if the connection A is flat, we have the following. Lemma 4.1. If A is flat, α ˆ is closed for any α ∈ H∗ (G).

Proof. Since A is flat, each point of M has an open neighborhood U such that there is a section s : U → PU = π −1 U , which is horizontal with respect to A. Note that here ‘horizontal’ means that dsx Tx M = Hs(x) holds for any x ∈ U , where H is the horizontal distribution given by A. Write (Ad P )U = π −1 U . Then we have the map (Ad P )U → U × G defined by [s(x), g] → (x, g) for any (x, g) ∈ U × G, which is an isomorphism between bundles of Lie groups. By applying the projection U × G → G after this isomorphism we have the map (Ad P )U → G, [s(x), g] → g, which we denote by κ. Then the lemma follows from the identity, α ˆ |(Ad P ) U = κ∗ α.

In fact this identity implies first of all that α ˆ is smooth. Considering the definition of α ˆ the identity follows from the following fact. Claim. The following equality holds for any x ∈ U and g ∈ G,

d(κs(x) )[s(x),g] πA = dκ[s(x),g] : T[s(x),g] (Ad P ) → Tg G.

Proof. Let X ∈ T[s(x),g] (Ad P ). Then there is a curve

[s(δ(t)), γ(t)] = q(s(δ(t)), γ(t)), −ǫ < t < ǫ,

whose velocity at t = 0 is X. Here δ(t) means a curve on U such that δ(0) = x and γ(t), another on G such that γ(0) = g. Then we have that ˙ X = dq(s(x),g) (dsx (δ(0)), 0g ) + dq(s(x),g) (0s(x) , γ(0)). ˙ ˙ Note that dsx (δ(0)) ∈ Hs(x) and dq(s(x),g) (0s(x) , γ(0)) ˙ ∈ V¯[s(x),g] . Therefore the expression of X in the above is consistent with the decomposition T[s(x),g] (Ad P ) = ¯ [s(x),g] ⊕ V¯[s(x),g] . Thus we conclude that H d(κs(x) )[s(x),g] πA (X) = d(κs(x) )[s(x),g] dq(s(x),g) (0s(x) , γ(0)) ˙ = γ(0). ˙

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YANGHYUN BYUN AND JOOHEE KIM

Here we used the identity, κs(x) q(s(x), γ(t)) = γ(t). On the other hand it is clear that dκ[s(x),g] (X) = γ(0), ˙ which proves the claim.  ∗ We recall again the identity H∗ (G) ≡ HdR (G). We have shown that α ˆ is a closed ∗ form on Ad P for any α ∈ H (G) if A is flat in the above. Therefore we have the map ∗ ∗ EA : HdR (G) → HdR (Ad P )

determined by EA [α] = [ˆ α] for any α ∈ H∗ (G). Now we consider the map, ιp : G → Ad P , defined by ιp (g) = [p, g] for any g ∈ G and for any p ∈ P . Then we have that ∗ ∗ (ιp )∗ EA = 1 : HdR (G) → HdR (G),

(4-3)

which is equivalent to the identity, (ιp )∗ α ˆ = α, for any bi-invariant k-form α on G. Note that κp ιp is the identity homomorphism on G. Then (4-3) follows from the fact that d(κp )[p,g] πA d(ιp )g = d(κp )[p,g] d(ιp )g is the identity on Tg G. The identity (4-3) means that the Leray-Hirsch theorem(cf. p.50, [1] or p.432, [7]) is applicable to conclude that ∗ ∗ ∗ (G) ∼ HdR (M ) ⊗ HdR (Ad P ). (4-4) = HdR ∗ In fact the isomorphism is given by a ⊗ b → (π ∗ a) ∧ (EA (b)) for any a ∈ HdR (M ) ∗ and b ∈ HdR (G). Furthermore we observe the following fact.

Lemma 4.2. Let G be a compact connected semi-simple Lie group and M , a smooth manifold. Consider a flat G-bundle P over M . Then the homomorphism EA : 3 3 HdR (G) → HdR (Ad P ) does not depend on the choice of the flat connection A.

i 0 Proof. Since HdR (G) = 0, for i = 1, 2, (see (1-3) and (3-2) above) and HdR (G) ∼ = R, by applying (4-4) above we have: 3 3 3 HdR (Ad P ) = π ∗ HdR (M ) ⊕ EA (HdR (G)).

Now consider the section s1 : M → Ad P which maps each x ∈ M to the identity element [p, e] of (Ad P )x where p is any point of Px . Note that s∗1 π ∗ is the identity ∗ ∗ ∗ (G) → HdR (M ) is the zero homomorphism: on HdR (M ). Furthermore s∗1 EA : HdR ¯ the derived Recall that we denote by H the horizontal distribution on P and by H, ¯ one on Ad P . Then it will suffice to show that ds1 Tx M = Hs1 (x) for any x ∈ M , considering the definition of EA . Since A is flat, there is an open neighborhood U of x such that there is a horizontal lifting s : U → P . Now note that the q composite U → P × G → AdP is in fact the restriction of s1 , where the first map ¯ s (x) is given by y → (s(y), e) for any y ∈ U . Now the assertion that ds1 Tx M = H 1 ¯ s (x) = dq(s(x),e) (Hs(x) ⊕ 0e ) together with the fact that follows from the equality H 1 Hs(x) = dsx Tx M . It follows that s∗

1 3 3 3 Ker(HdR (Ad P ) −→ HdR (M )) = EA (HdR (G)).

3 3 In particular, this means that EA (HdR (G)) ⊂ HdR (Ad P ) is independent of the flat connection A. 3 3 Now choose any p ∈ P and note that (ιp )∗ : EA (HdR (G)) → HdR (G) is the 3 3 3 (G) → inverse of EA : HdR (G) → EA (HdR (G)). This proves that EA : HdR 3 HdR (Ad P ) does not depend on the flat connection A. 

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES 11

As long as the main goal of the paper is concerned, the following is the only consequence of the lemma with some significance. Corollary 4.3. Let G and P be as in 4.2 above and Θ denote the Maurer-Cartan 3 (Ad P ) does not depend on the choice of the flat 3-form of G. Then EA [Θ] ∈ HdR connection A. 3 3 Considering 4.2 above we may write E : HdR (G) → HdR (Ad P ) omitting the flat connection A in the notation.

Lemma 4.4. Let G and P be as in 4.2 above. Then there is a choice of an invariant inner product on the Lie algebra of G so that, if Θ is the Maurer-Cartan 3-form, 3 E[Θ] ∈ HdR (Ad P ) is integral.

Proof. Consider the map s1 : M → Ad P as introduced in the proof of 4.2 above and the decomposition: 3 HdR (Ad P ) = Im π ∗ ⊕ Ker s∗1 .

3 (G)) = Ker s∗1 . We observed that E(HdR The decomposition above is valid also for the singular cohomology with integer coefficient up to torsion,

H 3 (Ad P ; Z) = Im π ∗ ⊕ Ker s∗1 .

3 (Ad P ) also decomThe homomorphism j : H 3 (Ad P ; Z) → H 3 (Ad P ; R) ≡ HdR poses accordingly. Therefore we conclude that the image of the map s∗

j

1 3 Ker (H 3 (Ad P ; Z) −→ H 3 (M ; Z)) −→ E(HdR (G))

3 (G)). is a full lattice in E(HdR We have observed in §3 above (see the paragraph below (3-6)) that the set of all 3 (G) determined by choosing the invariant inner products is an open cone [Θ] ∈ HdR 3 3 (G)). This in HdR (G). Therefore its image by E is also an open cone in E(HdR proves the proposition. 

Now let G be any connected compact Lie group and P → M be any principal Gbundle. Let Z0 denote the identity component of the center of G. Then G′ = G/Z0 is a semi-simple Lie group and P ′ = P/Z0 is a principal G′ -bundle. In the rest of the section we will let ψ : G → G′ , ψˆ : P → P ′ denote the respective projections. Let A be a connection on P and H be the invariant horizontal distribution of P determined by A. Then there is a horizontal distribution H ′ of P ′ defined by ′ ˆ H[p] = dψˆp Hp for any [p] = ψ(p) ∈ P ′ . Therefore there is a unique connection A′ ′ on P so that the horizontal distribution determined by A′ is H ′ . Write ψ(g) = [g] for any g ∈ G and consider the map ψ¯ : Ad P → Ad P ′ defined by ¯ g] = [[p], [g]] ψ[p, ¯ ¯ ′ denote the ‘horizontal’ distributions of Ad P for any (p, g) ∈ P × G. Let H and H ′ and Ad P respectively determined by A and A′ as in (4-1) above. Then we have that ¯ [p,g] ) = H ¯′ dψ¯[p,g] (H [[p],[g]] for any (p, g) ∈ P × G.

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YANGHYUN BYUN AND JOOHEE KIM

Let α ∈ Ω∗ (G′ ; R) be a bi-invariant form. Then ψ ∗ α ∈ Ω∗ (G; R) is also bi∗α invariant since ψ is a Lie group homomorphism. We have the forms α ˆ and ψd ′ respectively on Ad P and on Ad P . Then the following equality holds: ∗α = ψ ¯∗ α ψd ˆ.

(4-5)

In fact this equality follows from the identity,

dψg dκp πA = dκ[p] πA′ dψ¯[p,g] : T[p,g] Ad P → T[g] G′ ,

(4-6)

for any (p, g) ∈ P × G, where the maps κp , κ[p] and πA , πA′ are as introduced just before (4-2) in the above. The equality (4-6) can be seen as follows: Given any X ∈ T[p,g] Ad P , we choose a curve [γ(t), δ(t)] on Ad P such that the velocity at t = 0 is X, where γ is a horizontal curve on P such that γ(0) = p and δ is a curve on G such that δ(0) = g. Then it is straightforward to see that dψg dκp πA (X) = (ψδ) (0) = dκ[p] πA′ dψ¯[p,g] (X). Now we may restate 4.3 and 4 above in full generality as follows. Theorem 4.5. Let G be a compact connected Lie group with the Lie algebra G. Assume P → M is a flat G-bundle over a manifold M . Let Θ denote the MaurerCartan 3-form of G determined by an invariant inner product on G. Then EA [Θ] ∈ 3 (Ad P ) does not depend on the flat connection A. Furthermore, there is a HdR choice of an invariant inner product on G so that EA [Θ] is integral.

Proof. Let G′ , P ′ , ψ : G → G′ , ψˆ : P → P ′ and ψ¯ : Ad P → Ad P ′ mean the same as in the above. Flatness of A implies the same for the induced connection A′ on P ′ : If s : U → P ˆ : U → P′ is a horizontal lifting of an open set U ⊂ M with respect to A, then ψs ′ is also such with respect A . Consider the decomposition G = Z⊕G ′ where Z is the center of G and G ′ = [G, G]. Note that G ′ can be identified with the Lie algebra of G′ . We consider G ′ with the invariant inner product which is the restriction of the one on G. Let Θ′ denote the resulting Maurer-Cartan 3-form of G′ . Then we have ψ ∗ Θ′ = Θ, which is (3-7) above. Now apply (4-5) above to have EA [Θ] = ψ¯∗ EA′ [Θ′ ].

(4-7)

Since EA′ [Θ′ ] does not depend on A′ by 4.3 above, we conclude that EA [Θ] does not depend on A. Now note that any invariant inner product on G ′ can be extended to such one on G. Since there is an invariant inner product on G ′ with respect to which EA′ [Θ′ ] is integral by 4.4 above, the equation (4-7) above proves also the second assertion of the theorem.  In the current section no restriction on the smooth manifold M has been necessary so far. Now we assume that M is an oriented connected closed 3-manifold. Let ϕ : P → P be a gauge transformation. Then let u : P → G denote the map defined by ϕ(p) = pu(p) for any p ∈ P . Then it is well-known and easy to see that the map

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES 13

u ˆ : M → Ad P is well-defined by u ˆ(x) = [p, u(p)] for any x ∈ M , by choosing any p ∈ Px . Now we define deg ϕ by Z ˆ uˆ∗ Θ. (4-8) deg ϕ = M

ˆ ∈ Ω3 (Ad P ; R) As in the above Θ denotes the Maurer-Cartan 3-form of G and Θ means the form defined by (4-2) above, using any flat connection A ∈ A(P ). If G ˆ ∈ H 3 (Ad P ) does not depend on A by 4.5 is connected and compact, EA [Θ] = [Θ] dR above. Therefore deg ϕ ∈ R is well-defined. 5. The main result

Let P → M be a principal bundle with the structure group G. Let A ∈ A(P ) and ϕ : P → P be a gauge transformation. Note that ϕ determines a map u : P → G as in the previous section. Then we begin the section by observing that ϕ∗ A = A + dLu−1 dA u

(5-1)

where dA u means du πH and πH is the projection T P → H onto the horizontal distribution determined by A. Also the map dLu−1 dA u should be understood as a G-valued 1-form on P as follows: (dLu−1 dA u)(X) = d(Lu(p)−1 )u(p) (dA up (X))

for any X ∈ Tp P and any p ∈ P . Then the equality (5-1) can be seen by concentrating on the vertical vectors first and then on the horizontal ones. Now assume that the principal bundle P is flat and that A0 is a flat connection on P . Choose an invariant inner product h·, ·i on the Lie algebra G of G. Then we recall from (2-2) above the Chern-Simons form which we rewrite as follows: 1 (5-2) cs(A) = hFA ∧ (A − A0 )i − h(A − A0 ) ∧ [(A − A0 ) ∧ (A − A0 )]i 6 for any A ∈ A(P ). Note that here we are using A0 as the reference connection. Let θ and Θ respectively denote the Maurer-Cartan form and the 3-form of G. We will denote by H0 the horizontal distribution associated to A0 . Lemma 5.1. The following equality holds: cs(ϕ∗ A) − cs(A) − (u∗ Θ)H0 = dh(Adu−1 (A − A0 )) ∧ (u∗ θ)H0 i.

Proof. It is straightforward to see the equality ϕ∗ α = Adu−1 α for any horizontal ¯ ∗ (P ; G). In particular, since FA = dA+ 1 [A∧A] is horizontal equivariant form α ∈ Ω 2 and equivariant we have that Fϕ∗ A = ϕ∗ FA = Adu−1 FA . On the other hand we have that ϕ∗ A = A + dLu−1 dA u by (5-1) above. Now we have that ϕ∗ A = A0 + dLu−1 dA0 u + Adu−1 (A − A0 ). Also note that dLu−1 du = u∗ θ. Thus we have that ϕ∗ A − A0 = (u∗ θ)H0 + Adu−1 (A − A0 ). Also recall the equality, hX, [Y, Z]i = hZ, [X, Y ]i, for any X, Y, Z ∈ G from (3-5) above. Then from (5-2) above we have the following equality via a straightforward calculation: 1 cs(ϕ∗ A) − cs(A) + h(u∗ θ)H0 ∧ [(u∗ θ)H0 ∧ (u∗ θ)H0 )]i 6 1 ∗ H0 = h(u θ) ∧ (Adu−1 FA )i − h(u∗ θ)H0 ∧ [Adu−1 (A − A0 ) ∧ Adu−1 (A − A0 )]i 2 1 − hAdu−1 (A − A0 ) ∧ [(u∗ θ)H0 ∧ (u∗ θ)H0 ]i. 2 (5-3)

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YANGHYUN BYUN AND JOOHEE KIM

On the other hand the RHS of the equation of the lemma can be rewritten as follows: hdA0 (Adu−1 (A − A0 )) ∧ (u∗ θ)H0 i − h(Adu−1 (A − A0 )) ∧ dA0 ((u∗ θ)H0 )i.

(5-4)

Now assume that α ∈ Ω1 (P ; V ) where V is any real vector space. Then we have dA0 (αH0 ) = (dα)H0 ,

(5-5)

which can be seen as follows: It suffices to consider horizontal vector fields X, Y . Observe dA0 (αH0 )(X, Y ) = d(αH0 )(X, Y ) = Xα(Y ) − Y α(X) − αH0 ([X, Y ]).

Since H0 is integrable, [X, Y ] is horizontal. Thus we have dA0 (αH0 )(X, Y ) = dα(X, Y ), which proves (5-5).

Also recall that dθ = − 21 [θ ∧ θ]. Then by applying (5-5) above we have 1 dA0 ((u∗ θ)H0 ) = (u∗ dθ)H0 = − [(u∗ θ)H0 ∧ (u∗ θ)H0 ]. 2

(5-6)

We need to observe that dAdu−1 = Adu−1 ad((u−1 )∗ θ): First of all ad : G → End(G) denotes the well-known map defined by ad(Y )(Z) = [Y, Z] for any Y, Z ∈ d (Adρ(t) (Z)) where ρ is any curve on G such G = Te G. Note that ad(Y )(Z) = dt t=0 that ρ(0) = e and ρ(0) ˙ = Y . On the other hand Adu−1 is a function from P to End(G). Let X ∈ Tp P for some p ∈ P and γ(t), −ǫ < t < ǫ, be a curve on P such that γ(0) ˙ = X. Then we have that d d d(Adu−1 )p (X) = Adu−1 (γ(t)) = Adu−1 (γ(t)) dt t=0 dt t=0 d = Adu−1 (p) Adu(p)u−1 (γ(t)) dt t=0 = Adu−1 (p) ad(d(Lu(p) )u(p)−1 d(u−1 )p (X))

= Adu−1 (p) ad(((u−1 )∗ θ)(X)).

Now we have that dA0 (Adu−1 (A−A0 )) = (Adu−1 ad((u−1 )∗ θ))H0 ∧(A−A0 )+Adu−1 dA0 (A−A0 ). (5-7) The first term of the RHS of the above can be rewritten as follows,

(Adu−1 ad((u−1 )∗ θ))H0 ∧ (A − A0 ) = Adu−1 [((u−1 )∗ θ)H0 ∧ (A − A0 )].

Also note that (u−1 )∗ θ = −Adu u∗ θ. Then we may further rewrite the first term as follows, (Adu−1 ad((u−1 )∗ θ))H0 ∧ (A − A0 ) = −[(u∗ θ)H0 ∧ Adu−1 (A − A0 )].

(5-8)

On the other hand note that [α ∧ β] = [β ∧ α] for any G-valued 1-forms α and β. Then we have 1 dA0 (A − A0 ) = d(A − A0 ) + [A0 ∧ (A − A0 )] = FA − [(A − A0 ) ∧ (A − A0 )]. (5-9) 2

CHERN-SIMONS FUNCTIONAL UNDER GAUGE TRANSFORMATIONS ON FLAT BUNDLES 15

We insert (5-8, 9) to (5-7) to have: dA0 (Adu−1 (A − A0 )) = −[(u∗ θ)H0 ∧Adu−1 (A − A0 )] + Adu−1 FA 1 − [Adu−1 (A − A0 ) ∧ Adu−1 (A − A0 )]. 2 (5-10) Then we use (5-6, 10) to rewrite (5-4) and then compare it with (5-3) to prove the lemma.  For the lemma below we consider any connection A on P , which is not necessarily flat. Let H be the horizontal distribution of P given by A. Let ϕ : P → P and u : P → G be as in the above and let uˆ : M → Ad P be the section determined by ϕ. Let α ∈ Ωk (G; R) be a bi-invariant form and α ˆ ∈ Ωk (Ad P ; R) be as defined by (4-2) above. Also note the well-known fact that u(pg) = Adg−1 (u(p)) for any g ∈ G and p ∈ P . This means the equality, uRg = Adg−1 u : P → G, holds. Therefore u∗ α is an invariant form on P . It follows that (u∗ α)H can be regarded as a form on M . Lemma 5.2. The following equality holds: (u∗ α)H = u ˆ∗ α. ˆ ˜1, · · · , X ˜ k ∈ Tp P be the respective horiProof. Let X1 , · · · , Xk ∈ Tx M and let X zontal liftings with respect to H where p is any point in Px . Then we have: ˜ 1 , · · · , dup X ˜ k ). (u∗ α)H (X1 , · · · , Xk ) = α(dup X On the other hand we have that

(ˆ u∗ α)(X ˆ ˆ ux X1 , · · · , dˆ ux Xk ). 1 , · · · , Xk ) = α(dˆ

Recall the map κp : (Ad P )x → G defined by κp [p.g] = g, which is used to define α ˆ in (4-2) above. Also recall the projection πA : T (Ad P ) → V¯ given by the ¯ Then we have: decomposition T (Ad P ) = V¯ ⊕ H. α ˆ (dˆ ux X1 , · · · , dˆ ux Xk ) = α(d(κp )uˆ(x) πA dˆ ux X1 , · · · , d(κp )uˆ(x) πA dˆ ux Xk ).

Therefore the proof is complete if we observe the following:

Claim: Consider any X ∈ Tx M for an x ∈ X. Choose any p ∈ Px and let ˜ ∈ Tp P be the horizontal lifting of X with respect to A. Then we have that X ˜ d(κp )uˆ(x) πA dˆ ux X = dup X. Proof: Let X be represented by a curve γ : (−ǫ, ǫ) → M , such that γ(0) = x. Let γ˜ : (−ǫ, ǫ) → P be the horizontal lifting such that γ˜ (0) = p. Then we have that ˜ Note that uˆ(γ(t)) = [˜ γ˜˙ (0) = X. γ (t), u(˜ γ (t))] = q(˜ γ (t), u(˜ γ (t))), where q : P × G → Ad P is the projection. This means that ˜ ∈ T[p,u(p)] Ad P, ˜ dup X) dˆ ux X = dq(p,u(p)) (X,

understanding the identification, T(p,u(p)) (P × G) ≡ Tp P ⊕ Tu(p) G. By definition ˜ is ˜ and dq(p,u(p)) (0p , dup X) of πA , it follows that πA dˆ ux X = dq(p,u(p)) (0p , dup X) represented by the curve [p, u(˜ γ (t))]. Also note that [p, u(p)] = uˆ(x). Therefore d(κp )uˆ(x) πA dˆ ux X is represented by the curve u(˜ γ (t)), which proves the claim.  Now we assume that M is an oriented connected closed 3-manifold. Also we recall from (2-2, 3) above the Chern-Simons functional CS : A(P ) → R, using a

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YANGHYUN BYUN AND JOOHEE KIM

flat connection A0 as the reference connection. Then from 5.1 and 2 above we have that Z ˆ uˆ∗ Θ. (5-11) CS(ϕ∗ A) − CS(A) = M

In particular we observe that the 2-form h(Adu−1 (A − A0 )) ∧ (u∗ θ)H0 i in 5.1 above is horizontal. Furthermore note that u : P → G is a smooth map such that the equality, u(pg) = Adg−1 (u(p)), holds for any p ∈ P and g ∈ G. Then it is straightforward to see that both Adu−1 (A − A0 ) and u∗ θ are equivariant, which means that the 2-form is invariant as well. Thus all the forms in the identity of 5.1 above can be regarded as forms on M . Note that the RHS of (5-11) above is deg ϕ by the definition (4-8) above. Now we have proved the following. Theorem 5.3. Let P → M be a flat bundle over an oriented connected closed 3-manifold with a connected compact structure group. Let CS denote the ChernSimons functional defined by choosing a flat reference connection. Then for any gauge transformation ϕ : P → P and any connection A on P we have that CS(ϕ∗ A) − CS(A) = deg ϕ.

Furthermore, by 4.5 above, there is an invariant inner product on the Lie algebra so that EA [Θ], which does not depend on the flat connection A, is an integral class 3 (Ad P ). Therefore we may adopt, if necessary, such an inner product so that of HdR deg ϕ is an integer for any ϕ. References [1]

R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York, 1982. [2] S. Chern, J. Simons, Characteristic forms and geometric invariants, Ann. Math. 99 (1974) 48-69. [3] C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948) 85-124. [4] R. Dijkgraaf, E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990) 393-429. [5] S. Dostoglou, D. Salamon, Instanton homology and symplectic fixed points, Symplectic Geometry, D. Salamon, Ed., Proceedings of a Conference, LMS Lecture Notes Series 192, Cambridge University Press (1993) 57-94. [6] D.S. Freed, Remarks on Chern-Simons theory, Bull. Amer. Math. Soc. 46 (2009) 221-254. [7] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. [8] H. Kachi, Homotopy groups of compact Lie groups E6, E7 and E8, Nagoya Math. J. 32 (1968) 109-139. [9] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [10] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. [11] K. Mackenzie, Classification of principal bundles and Lie groupoids with prescribed gauge group bundle, J. Pure Appl. Algebra 58 (1989) 181-208. [12] K. Wehrheim, Energy identity for anti-self-dual instantons on C × Σ, Math. Res. Lett. 13 (2006) 161-166. Department of Mathematics, College of Natural Sciences, Hanyang University, Wangsimni-ro 222, Seongdong-gu, Seoul 133-791, Korea E-mail address: [email protected](Y. Byun) E-mail address: [email protected](J. Kim)