aa r X i v : . [ m a t h - ph ] F e b U (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS ZHI HU AND RUNHONG ZONGA
BSTRACT . In this paper, we investigate two types of U (1) -gauge field theories on G -manifolds. One is the U (1) -Yang-Millstheory which admits the classical instanton solutions, we show that G -manifolds emerge from the anti-self-dual U (1) instantons,which is an analogy of Yang’s result for Calabi-Yau manifolds. The other one is the higher-order U (1) -Chern-Simons theory as ageneralization of Kähler-Chern-Simons theory, by suitable choice of gauge and regularization technique, we calculate the partitionfunction under semiclassical approximation. C ONTENTS
1. Introduction 12. G manifolds as anti-self-dual U (1) -instantons 23. Higher-order U (1) -Chern-Simons theory 93.1. Actions 93.2. Higher order G -instanton 143.3. Partition Function 17References 221. I NTRODUCTION G manifolds appear in the compactification of M -theory or 11-dimensional supergeravity to achieve effective 4-dimensional theory with N = 1 supersymmetry [1, 2]. Mathematically, there are two equivalent approaches to define G manifolds. The first definition treats a G manifold as a 7-dimensional Riemannian manifold with holonomy group as asubgroup of G , and the other one defines a G manifold as a 7-dimensional oriented spin manifold with a torsion free G structure which is a special 3-form ϕ (called fundamental 3-form) parallel with respect to the induced Levi-Civita connection.Many examples of 7-dimensional manifold with holonomy group G have also been constructed [3, 4, 5, 6, 7]. To someextent, G manifolds can be viewed as the analog of Calabi-Yau 3-folds:Calabi-Yau 3-fold N G manifold M Kähler form ω fundamental 3-form ϕ complex structure I fundamental 4-form ∗ ϕ ϕI -holomorphic curve in N associated 3-dimensional submanifold of M special Lagrangian submanifold of N coassociated 4-dimensional submanifold of M .
It is noteworthy that the metric on a G -manifold is totally determined by the fundamental 3-form ϕ via a highly nontrivialmanner, hence the fundamental 4-form ∗ ϕ ϕ is not independent of the fundamental 3-form ϕ , where ∗ ϕ denote the Hodgestar with respect to the induced metric g ϕ . Conceptually, we should consider which results for Calabi-Yau manifolds can begeneralized to G -manifolds.In a series of papers by H.-S. Yang [8, 9, 10, 11, 12, 13], the author proposed a kind of emergent gravity , which is achievedby considering the deformation of a symplectic manifold. In this framework, Darboux theorem or the Moser lemma in sym-plectic geometry is reinterpreted as equivalence principle. From this point of view, a line bundle over a symplectic manifoldleads to a dynamical symplectic manifold described by a gauge theory of symplectic gauge fields. Then the quantizationof the dynamical symplectic manifold gives rise to a dynamical noncommutative spacetime described by a noncommutative U (1) -gauge theory. A basic idea of Yang’s emergent gravity is to realize the gauge/gravity duality using the Lie algebra ho-momorphism between noncommutative ⋆ -algebra (gauge theory side) and derivation algebra (gravity side). Then H.-S. Yangshowed that that the commutative limit of noncommutative anti-self-dual U (1) -instanton equations turns into the equationsfor spin connections of an emergent Calabi-Yau manifold [14]. In Sec. 2, we will generalize such mechanism of producingCalabi-Yau manifolds to the G -manifolds. G -instanton equations were first introduced in sd [15], which are also divided intoself-dual type and anti-self-dual type according to the irreducible representations of G on 2-forms. In the spirit of takingcommutative limit, the anti-self-dual U (1) -instanton equations give rise to equations satisfied by a collection of local orthog-onal frame fields (siebenbein), we will show that these equations force the spin connection to be valued in the Lie algebra g of G , hence determine a G -manifold.Chern-Simons theory is another important kind of gauge theory. A natural generalization of Chern-Simons theory to a G -manifold M is to consider the following action which was first introduced in dt [16] S CS = Z M CS ( A ) ∧ ∗ ϕ ϕ, (1.1)where CS ( A ) = Tr( AdA + 23 A ) (1.2)is the the standard Chern-Simons 3-form associated to the connection A on a trivial G -principal bundle over M for matrix Liegroup G . The critical point is exactly the anti-self-dual G -instanton. If one choose a special G -manifold M = CY × S ,by dimensional reduction, S CS can be reexpressed as the sum of B -model 6-brane and ¯ B -model 6-brane actions and an extraterm related to the stability of the brane bbb [17]. Similar to the work in bw [18], one may connect the partition function of S CS with the cohomology of moduli space of Hermitian-Yang-Mills connection on CY (this work will appear elsewhere). InSec. 3, we will consider higher-order Chern-Simons theory on M = M × S . This theory is viewed as a generalization ofKähler-Chern-Simons theory introduced in ns [19]. Replacing Kähler form ω with fundamental 3-form ϕ , the action is given by S = Z M CS ( A ) ∧ ϕ, (1.3)where CS ( A ) = Tr( A d A d A + 32 A d A + 35 A ) (1.4)is the the Chern-Simons 5-form associated to the connection A on a trivial G -principal bundle over M . Recently, Yamazakialso considered such type Chern-Simons theory ya [20]. The Sec. 3 is organized as follows. We first construct higer-order U (1) -Chern-Simons action, which is invariant at quantum level under the large guage transformations. If the U (1) -principal bundleis nontrivial we will use Deligne-Beilinson cohomology theory dbb [21] to discuss the problem on gauge invariance. Next weintroduce the notion of higher-order G -instanton , which will be chosen as the background of semiclassical approximation.We will consider when the corresponding linear operator is elliptic. Finally, we will calculate the semiclassical partitionfunction around higher-order U (1) -instantons for the trivial U (1) -principal bundle. To achieve the calculations, one shouldimpose torus gauge hn [22] on A as observed in bt [23], and adopt a suitable regularization technique.2. G MANIFOLDS AS ANTI - SELF - DUAL U (1) - INSTANTONS
There is a standard G -structure on R consists of a standard Euclidean metric g and a 3-form ϕ given by ϕ = e + e + e + e − e + e − e , (2.1) opi where { e , · · · , e } is an orthonormal frame that provides a coordinate { x , · · · , x } on R , and e ijk = e i ∧ e j ∧ e k for e i being the dual basis of e i . The U (1) -Yang-Mills functional on R is given by S = Z R d xF ij F ij , where F ij = ∂A j ∂x i − ∂A i ∂x j for the dual vector field A = A i ( x ) e i on R . Identifying e i with dx i , the critical point reads d ∗ g F = 0 , (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS where F = dA = F ij dx i ∧ dx j , ∗ g denotes the Hodge dual with respect to g . Obviously, if ∗ g F = cF ∧ ϕ (2.2) tr for some constant c , then the above equation holds true automatically. There is a decomposition F = F (1) + F (2) , where F (1) , F (2) satisfy ∗ g ( F (1) ∧ ϕ ) = − F (1) , ∗ g ( F (2) ∧ ϕ ) = F (2) , respectively. Therefore, only when c = − or c = 1 the equation ( tr U (1) -instanton or anti-self-dual U (1) -instanton, respectively, i.e. ( ∗ g F = − F ∧ ϕ , self-dual U (1) -instanton; ∗ g F = F ∧ ϕ , anti-self-dual U (1) -instanton, (2.3)or more explicitly ( F ij = − T ijkl F kl , self-dual U (1) -instanton; F ij = T ijkl F kl , anti-self-dual U (1) -instanton, (2.4)where T ijkl = 16 ǫ ij klpqr ( ϕ ) pqr . (2.5) We will only focus on anti-self-dual U (1) -instanton.Consider a 7-dimensional spin manifold M equipped with a Riemann metric g . Let { E i = E µi ∂∂X µ , i = 1 , · · · , } bepointwisely linearly independent local orthogonal frame fields, i.e. so-called "siebenbein", over some neighborhood withlocal coordinate { X µ , µ = 1 , · · · , } in M , namely we have E µi E νj δ ij = g µν , and let { E i = E iµ dX µ , i = 1 , · · · , } be the corresponding dual 1-forms which satisfy E iµ E jν g µν = δ ij . The spin connection 1-form { ω ij } on M is defined by dE i + ω ij ∧ E j = 0 , (2.6) lo ω ij + ω ji = 0 . (2.7)As done in [14], one does the following replacement (so-called commutative limit) D i := ∂∂x i + A i −→ E i , then the anti-self-dual U (1) -instanton equation becomes [ E i , E j ] = 16 T ijkl [ E k , E l ] , (2.8) g or equivalently, f ijs = 16 T ijkl f kls , (2.9) o where f ijk is defined by [ E i , E j ] = f ijk E k . (2.10) Writing ω ij = ω ikj E k , the equation ( lo f ijk = ω kij − ω kji =: Ω kij , ZHI HU AND RUNHONG ZONG then the equation ( o Ω sij = 16 T ijkl Ω skl , (2.11) ty which is explicitly expressed in terms of components as Ω s = Ω s − Ω s , (2.12) yu Ω s = Ω s + Ω s , (2.13) Ω s = − Ω s + Ω s , (2.14) Ω s = − Ω s − Ω s , (2.15) Ω s = Ω s + Ω s , (2.16) Ω s = Ω s − Ω s , (2.17) Ω s = Ω s − Ω s . (2.18) yui Definition 2.1.
Let M be a 7-dimensional oriented spin manifold equipped with a Riemann metric g . If there exists an opencover { U α } of M , and there is a siebenbein { E ( α ) i , i = 1 , · · · , } (compatible with the orientation) over each U α such that [ E ( α ) i , E ( α ) j ] = 16 T ijkl [ E ( α ) k , E ( α ) l ] , (2.19) g5 we call M an anti-self-dual U (1) -instanton.The covariant derivative on a spinor η over an open neighborhood U along the vector field E i is defined by ∇ E i η = E i ( η ) + 12 Ω ijk Σ jk • η, where • denotes Clifford multiplication, and { Σ ij = − Σ ji , i < j = 1 , · · · , } , satisfying the relations [Σ ij , Σ kl ] = Σ il δ jk + Σ jk δ il − Σ ik δ jl − Σ jl δ ik , (2.20) jk forms a basis of Lie algebra spin (7) . It follows from the equations ( yu yui ∇ E i η = E i ( η ) + Ω i (Σ + Σ ) • η + Ω i (Σ − Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ − Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ − Σ ) • η + Ω i (Σ − Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ − Σ ) • η + Ω i (Σ + Σ ) • η + Ω i (Σ − Σ ) • η. (2.21) ft We introduce V = Σ + Σ , W = Σ − Σ ,V = Σ + Σ , W = Σ + Σ ,V = Σ − Σ , W = Σ + Σ ,V = Σ − Σ , W = Σ − Σ ,V = Σ + Σ , W = Σ + Σ ,V = Σ + Σ , W = Σ − Σ ,V = Σ + Σ , W = Σ − Σ . (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS By the relation ( jk hh2 [24] [ V , V ] = − V , [ V , V ] = V + W , [ V , V ] = V , [ V , V ] = − W [ V , V ] = − V − W , [ V , V ] = V , [ V , W ] = W , [ V , W ] = − W , [ V , W ] = 2 V , [ V , W ] = − W , [ V , W ] = W , [ V , W ] = − W , [ V , V ] = W , [ V , V ] = 2 V , [ V , V ] = − V − W , [ V , V ] = − V , [ V , V ] = − V , [ V , W ] = W , [ V , W ] = V − W , [ V , W ] = V , [ V , W ] = − V , [ V , W ] = − W , [ V , V ] = − V − W , [ V , V ] = W , [ V , V ] = V + W , [ V , V ] = V − W , [ V , W ] = W , [ V , W ] = − W , [ V , W ] = − W , [ V , W ] = 2 W , [ V , W ] = − W , [ V , W ] = W , [ V , V ] = V , [ V , V ] = 2 V , [ V , V ] = − V − W , [ V , W ] = V − W , [ V , W ] = − V , [ V , W ] = − W , [ V , W ] = V + W , [ V , W ] = − V , [ V , W ] = W , [ V , V ] = − V , [ V , V ] = V , [ V , W ] = − W , [ V , W ] = V , [ V , W ] = − V + W , [ V , W ] = − V , [ V , W ] = V + W , [ V , W ] = − V − W , [ V , V ] = − V , [ V , W ] = − W , [ V , W ] = V , [ V , W ] = W , [ V , W ] = − V , [ V , W ] = V + W , [ V , W ] = V − W , [ V , W ] = W , [ V , W ] = − W , [ V , W ] = − V + W , [ V , W ] = − V − W , [ V , W ] = W , [ V , W ] = − W , [ W , W ] = V , [ W , W ] = 2 W , [ W , W ] = − V , [ W , W ] = − V + W , [ W , W ] = − W , [ W , W ] = V , [ W , W ] = − W , [ W , W ] = V + W , [ W , W ] = − V , [ W , W ] = V − W , [ W , W ] = V , [ W , W ] = V + W , [ W , W ] = − W , [ W , W ] = 2 W , [ W , W ] = − V , [ W , W ] = V , [ W , W ] = − V + W , [ W , W ] = − V , [ W , W ] = V , [ W , W ] = V + W , [ W , W ] = V − W , ZHI HU AND RUNHONG ZONG then we immediately find that { V , · · · , V , W , · · · , W } generate a 14-dimensional Lie subalgebra g of spin (7) .To show this subalgebra g is exactly g , we only need recover to ϕ from the invariant spinor. Dirac gamma matricessatisfying Clifford algebra in 7-dimensional Euclidean space are given by ( Γ i = Γ (6) i , i = 1 , · · · , ; Γ = √− (6)1 · · · Γ (6)6 , (2.22)where Γ (6)[ , i = 1 , · · · , , denote the Dirac gamma matrices in six dimensions. Choosing purely imaginary × matrices,one explicitly writes Dirac gamma matrices as follows Γ = √−
10 0 √− −√− √− −√− √− −√− −√− , Γ = −√− √− √− √− √− −√− −√− −√− , Γ = √− −√− √−
10 0 0 0 −√− √− √− −√− −√− , Γ = −√− −√− √− √−
10 0 −√− √− √− −√− , (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS Γ = √− −√− −√− √− √− −√− √− −√− , Γ = −√− √− −√− −√− √− √−
10 0 √− −√− , Γ = √− √− √− −√− −√− −√− √−
10 0 0 0 0 0 −√− . Then Σ ij can be realized as Σ ij = 14 [Γ i , Γ j ] , which provides explicit expressions V = 12 − − − − − − − − , W = 12 − − −
10 0 0 0 − − − − − ,V = − − , W = − − , ZHI HU AND RUNHONG ZONG V = − − , W = 12 − − − − − − − − ,V = − − , W = 12 − − − −
11 0 − − − − ,V = 12 − − − − − − − − , W = − − ,V = − − , W = 12 − − − − − − − − ,V = 12 − − − − − − − − , W = 12 − − − −
10 0 0 0 1 0 1 00 0 0 − − − − . (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS Let G be a connected subgroup of Spin (7) with Lie algebra g , then we have the unique G -invariant spinor η up to aconstant scalar determined by V i • η = W i • η = 0 (2.23)for any i = 1 , · · · , . Imposing the normalization condition, we write η = 1 √ − . (2.24) Define ψ = ψ ijk e ijk = √− η † Γ ijk η e ijk , (2.25) ui where Γ ijk = 13! X σ ( − | σ | Γ σ ( i ) Γ σ ( j ) Γ σ ( k ) with σ standing for a permutation. It is clear that ψ is G -invariant. A direct calculation shows that the nonzero coefficients ψ ijk = √− η † Γ ijk ( i < j < k ) are given by ψ = ψ = ψ = ψ = ψ = 1 ,ψ = ψ = − . If one reassigns the frame { e , e , e , e , e , e , e } to { e , e , e , e , e , e , e } , we find that ψ exactly coincides with ϕ .If M is an anti-self-dual U (1) -instanton, from the above arguments it follows that the condition ( g5 g , which implies that the holonomy group of the metric g lies in G . Conversely, if M is a G -manifold, it can be made into an anti-self-dual U (1) -instanton. As a consequence, we have the following theorem. Theorem 2.2. M is a G -manifold if and only if it is an anti-self-dual U (1) -instanton.
3. H
IGHER - ORDER U (1) -C HERN -S IMONS THEORY
Actions.
Assume M is a G -manifold with fundamental 3-form ϕ , and M = M × L for L denoting R or S . Let P bea trivial G -principal bundle over M for the matrix Lie group G , and A as a 1-form on M valued in the Lie algebra g of G bethe connection on P , then we consider higher-order Chern-Simons action, as follows S = Z M [Tr( A d A d A + 32 A d A + 35 A )] ∧ ϕ (3.1) aaa In particular, if G = U (1) , the above action is more simple as S = Z M A ∧ d A ∧ d A ∧ ϕ, which can be generalized to the U (1) -BF-type action S ABC = Z M A ∧ d B ∧ d C ∧ ϕ (3.2)for one forms A , B , C on M . Note that it would be more appropriate to call the U (1) -ABC action than U (1) -BF action. Decomposing A = A + B for A = A dt , B = A µ dX µ with coordinate t on L and coordinates { X µ , i = 1 , · · · , } on M , the action ( aaa S = − Z L dt Z M Tr[( Bd M B + d M BB + 32 B ) ˙ B ] ∧ ϕ + 3 Z L dt Z M Tr( A F B ) ∧ ϕ, (3.3) mv where d M stands for the exterior differential operator on M , ˙ B = ∂ A µ ∂t dX µ , F B = d M B + B . By Chern-Weil theory,the integral R M Tr( F B ) ∧ ϕ is a topological invariant, hence the action S has a large gauge symmetry at the quantum level A → A + f ( t ) with R L f ( t ) dt ∈ Q . Usually, the invariance at quantum level under large gauge transformations requires ϕ should be of integral period, i.e. ϕ ∈ H ( M, Z ) ֒ → H ( M , Z ) hh [25].If the G -principal bundle P is nontrivial, the story is more subtle and complicated. For example, a suitable framework tocope with the nontrivial U (1) -principal bundle is the Deligne-Beilinson cohomology theory. The original Deligne-Beilinsoncohomology is defined for algebraic varieties, and the smooth analogy of this theory is also called Cheeger-Simons cohomol-ogy dbb [21]. We recall some basis materials in this theory. For a compact manifold X , the Deligne complex of sheaves is givenby DC R ( ℓ ) : 0 → R ( ℓ ) → Λ ( X, ℓ ) d −→ Λ ( X, ℓ ) d −→ · · · d −→ Λ ℓ ( X, ℓ ) , where Λ • ( X, ℓ ) = (2 π √− ℓ Λ • ( X ) , R ( ℓ ) = (2 π √− ℓ R for a subring R of R and some integer ℓ ≥ , then the q -order Deligne-Beilinson cohomology group H q DB ( X, R ( ℓ )) is defined as the q -th hypercohomology of the Deligne complex DC R ( ℓ ) , i.e. H q DB ( X, R ( ℓ )) = H q ( DC R ( ℓ ) ) . Taking an open cover U = {U α } α ∈ I of X , i.e. X = S α ∈ I U α , we considerthe ˇCech resolution of DC R ( ℓ ) : R ( ℓ ) ι −−−−→ Λ ( X, ℓ ) d −−−−→ Λ ( X, ℓ ) d −−−−→ · · · d −−−−→ Λ ℓ ( X, ℓ ) ι y ι y ι y ι y ι y C ( U , R ( ℓ )) ι −−−−→ C ( U , Λ ( X, ℓ )) d M −−−−→ C ( U , Λ ( X, ℓ )) d −−−−→ · · · d −−−−→ C ( U , Λ ℓ ( X, ℓ )) δ y δ y δ y δ y δ y C ( U , R ( ℓ )) ι −−−−→ C ( U , Λ ( X, ℓ )) d −−−−→ C ( U , Λ ( X, ℓ )) d −−−−→ · · · d −−−−→ C ( U , Λ ℓ ( X, ℓ )) δ y δ y δ y δ y δ y ... ι −−−−→ ... d −−−−→ ... d −−−−→ ... d −−−−→ ... , where ι denotes the natural embedding, δ denotes the ˇCech operator, and C q ( U , F ) denotes the space of q -dimensional ˇCechcochains for a sheaf F over M . Therefore H q DB ( X, R ( ℓ )) = lim −→U H q (Tot U ( DC R ( ℓ ) )) , where Tot U ( DC R ( ℓ ) ) is the total complex of the ˇCech resolution of DC R ( ℓ ) associated to the cover U . In particular, if U isa simple (good) cover gt [26], we have H q DB ( M, R ( ℓ )) = H q (Tot U ( DC R ( ℓ ) )) bg [27].For our purpose, we take R = Z . From the following commutative diagram Z ( ℓ ) −−−−→ Λ ( X, ℓ ) d −−−−→ Λ ( X, ℓ ) d −−−−→ · · · d −−−−→ Λ ℓ ( X, ℓ ) y exp ℓ y π √− ℓ − y π √− ℓ − y π √− ℓ − y −−−−→ U (1) X d log −−−−→ Λ ( X, d −−−−→ · · · d −−−−→ Λ ℓ ( M, , where U (1) X denotes sheaf of U (1) -valued functions over X , and exp ℓ ( f ) = e f (2 π √− ℓ − , we find that H q DB ( X, Z ( ℓ )) ≃ H q − ( U ( ) X ( ℓ )) (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS where U ( ) X ( ℓ ) denote the complex U (1) X d log −−−→ Λ ( X, d −→ · · · d −→ Λ ℓ − ( X, . It immediately implies that Deligne-Beilinson cohomology H ( X, Z (1)) parameterize the isomorphism classes of U (1) -principal bundles with connections over M gt,bg [26, 27]. More explicitly, choosing a simple cover U = {U α } α ∈ I of X with index set I , an element in H ( X, Z (1)) isrepresented by a triple ( { A α } α ∈ I , { Γ αβ } α,β ∈ I , { Υ αβγ } α,β,γ ∈ I ) of ˇCech cochains satisfying A β − A α = d Γ αβ (3.4) Γ βγ − Γ αγ + Γ αβ = Υ αβγ (3.5) Υ βγδ − Υ αγδ + Υ αβδ − Υ αβγ = 0 , (3.6)where { A α } ∈ C ( U , Λ ( X, , { Γ αβ } ∈ C ( U , Λ ( X, , { Υ αβγ } ∈ C ( U , Z (1)) . Writing g αβ = e − Γ αβ , Γ αβ = − π √− αβ , then ( { g αβ } defines transition functions on a U (1) -principal bundle, and ( { A α } defines a connection on such U (1) -principal bundle.Now X = M = M × S , where M is a closed G -manifold whose fundamental 3-form ϕ is assumed to be an integralcohomology class. One views ϕ ∈ H ( M, Z ) as an element of Deligne-Beilinson cohomology H ( M, Z (1)) , hence an ele-ment of H ( M , Z (1)) by natural inclusion. Let A , B , C ∈ H ( M , Z (1)) describe isomorphism classes of U (1) -principalbundles with connections over M . Taking a simple cover U = {U α } α ∈ I of M , we represent A , B , C ∈ H ( M , Z (1)) as A = ( { A α } , { Γ αβ } , { Υ αβγ } ) , B = ( { B α } , { Θ αβ } , { Λ αβγ } ) , C = ( { C α } , { Ψ αβ } , { Ω αβγ } ) , and also represent ϕ ∈ H ( M , Z (1)) as ϕ = ( { χ αβ } , { τ αβγ } , { π √− θ αβγη } ) , where { π √− θ αβγη } ∈ C ( U , Z (1)) are determined via the isomorphism H ( M , Z ) ≃ H ˇCech ( M , Z ) ,ϕ
7→ { θ αβγη } , and { χ αβ } ∈ C ( U , Λ ( M , , { τ αβγ } ∈ C ( U , Λ ( M , are determined as follows χ αβ + X γ ∈ I ( dτ αβγ ) ξ γ = 0 ,τ αβγ − π √− X η ∈ I θ αβγη ξ η = 0 for a partition { ξ α } α ∈ I of unity subordinate to the simple cover U .Consider a polyhedral decomposition ( { P (8) α } α ∈ I , · · · , { P (0) α ··· α } α , ··· ,α ∈ I ) of M subordinate to U , where P ( d ) α ··· α − d is a d -dimensional submanifold of M lying in U α T · · · U α − d , then we construct the gauge invariant U (1) -ABC action. Webegin with the following action I = X α ,α ,α ,α ∈ I Z P (5) α α α α θ α α α α A α ∧ d B α ∧ d C α . Under the lacal gauge transformation A α A α + d a α , B α B α + d b α , C α C α + d c α , for a α , b α , c α ∈ C ( U , Λ ( M , , its variation is given by ∆ lol I = X α ,α ,α ,α ,α ∈ I Z P (4) α α α α α − P (4) α α α α α + P (4) α α α α α − P (4) α α α α α + P (4) α α α α α [ θ α α α α a α ∧ d B α ∧ d C α ]= X α ,α ,α ,α ,α ∈ I Z P (4) α α α α α [( θ α α α α − θ α α α α + θ α α α α − θ α α α α ) · a α ∧ d B α ∧ d C α + θ α α α α a α d B α ∧ d C α ]= X α ,α ,α ,α ,α ∈ I Z P (4) α α α α α θ α α α α ( a α − a α ) d B α ∧ d C α ] , which can be eliminated by the variation of the action I = − X α ,α ,α α ,α ∈ I Z P (4) α α α α α θ α α α α Γ α α d B α ∧ d C α under the transformation Γ αβ Γ αβ + a β − a α . However, the action I + I is not invariant under the large gaugetransformation Γ αβ Γ αβ + z αβ for z αβ ∈ C ( U , Z (1)) . Indeed, the variation of I is given by ∆ lar I = ∆ ar l I + ∆ lar I = − X α ,α ,α ,α ,α ,α ∈ I Z P (3) α α α α α α θ α α α α ( δz ) α α α B d C α − X α ,α ,α ,α ,α ,α ∈ I Z P (3) α α α α α α θ α α α α z α α ( B α d C α − B α d C α ) . ∆ lar I can be eliminated by variation of the action I = X α ,α ,α ,α ,α ,α ∈ I Z P (3) α α α α α α θ α α α α Υ α α α B α ∧ d C α under the transformation Υ αβγ Υ αβγ + ( δz ) αβγ . As well as, ∆ lar I can be calculated as ∆ lar I = − X α ,α ,α ,α ,α ,α ∈ I Z P (3) α α α α α α θ α α α α z α α d (Θ α α d C α )= X α ,α ,α ,α ,α ,α ,α ∈ I Z P (2) α α α α α α α θ α α α α ( δz ) α α α Θ α α d C α − X α ,α ,α ,α ,α ,α ,α ∈ I Z P (2) α α α α α α α θ α α α α z α α Λ α α α d C α = X α ,α ,α ,α ,α ,α ,α ∈ I Z P (2) α α α α α α α θ α α α α ( δz ) α α α Θ α α d C α − X α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (1) α α α α α α α α θ α α α α ( δz ) α α α Λ α α α C α − X α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (1) α α α α α α α α θ α α α α z α α Λ α α α d Ψ α α ≈ X α ,α ,α ,α ,α ,α ,α ∈ I Z P (2) α α α α α α α θ α α α α ( δz ) α α α Θ α α d C α − X α ,α ,α ,α ,α ,α ,β ,α ∈ I Z P (1) α α α α α α α α θ α α α α ( δz ) α α α Λ α α α C α + X α ,α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (0) α α α α α α α α α θ α α α α ( δz ) α α α Λ α α α Ψ α α , (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS where the notation ≈ means that we have omitted the term as the form of (2 π √− Z . The three terms on the right hand sideof ≈ can be eliminated by the variations of the actions I = − X α ,α ,α ,α ,α ,α ,α ∈ I Z P (2) α α α α α α α θ α α α α Υ α α α Θ α α d C α ,I = X α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (1) α α α α α α α α θ α α α α Υ α α α Λ α α α C α ,I = − X α ,α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (0) α α α α α α α α α θ α α α α Υ α α α Λ α α α Ψ α α , respectively, under the transformation Υ αβγ Υ αβγ + ( δz ) αβγ . Similar calculations exhibit I + I and I + I are invariantunder the local gauge transformations. Consequently, the gauge invariant U (1) -ABC action reads S ABC ≈ I + I + I + I + I + I = X α ,α ,α ,α ∈ I Z P (5) α α α α θ α α α α A α ∧ d B α ∧ d C α − X α ,α ,α α ,α ∈ I Z P (4) α α α α α θ α α α α Γ α α d B α ∧ d C α + X α ,α ,α ,α ,α ,α ∈ I Z P (3) α α α α α α θ α α α α Υ α α α B α ∧ d C α − X α ,α ,α ,α ,α ,α ,α ∈ I Z P (2) α α α α α α α θ α α α α Υ α α α Θ α α d C α + X α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (1) α α α α α α α α θ α α α α Υ α α α Λ α α α C α − X α ,α ,α ,α ,α ,α ,α ,α ,α ∈ I Z P (0) α α α α α α α α α θ α α α α Υ α α α Λ α α α Ψ α α . (3.7) bjk The Deligne-Beilinson cup product dbb [21] [ : H q DB ( X, Z ( ℓ )) × H t DB ( X, Z ( )) → H q + t DB ( X, Z ( ℓ + + 1)) , q = ℓ + 1 , t ≤ + 1 or t = + 1 , q ≤ ℓ + 1 ; H q + t − DB ( X, Z ( ℓ + + 1)) , q ≥ ℓ + 2 , t ≤ + 1 or t ≥ + 2 , q ≤ ℓ + 1 ; H q + t DB ( X, Z ( ℓ + + 1)) ≃ H q + t ( X, Z ) , q ≥ ℓ + 2 , t ≥ + 2 or t ≥ + 2 , q ≥ ℓ + 2 ; , other cases.defines A [ B ∈ H DB ( M , Z (3)) , A [ B [ C ∈ H DB ( M , Z (5)) ,ϕ [ A [ B [ C ∈ H DB ( M , Z (7)) ≃ R / Z . Then the action ( bjk S ABC ≈ π √− Z M ϕ [ A [ B [ C− π √− X α ,α ∈ I Z P (7) α α χ α α d A α ∧ d B α ∧ d C α + 12 π √− X α ,α ,α ∈ I Z P (6) α α α τ α α α d A α ∧ d B α ∧ d C α , (3.8) mnb where the extra two terms are obviously gauge invariant. Higher order G -instanton. In this subsection, our Lie group is chosen SU (2) or U (1) . Varying A in the action ( mv F B ∧ ϕ = 0 . (3.9) jh Let P be a G -principal bundle over M , where P is not necessary trivial, and let Ad P be the adjoint vector bundle associatedto P , then F B ∈ Λ ( M, Ad P ) . Pick an open neighbourhood U of a point p ∈ M over which P is trivial, locally F B | U lies in Λ ( U, g ) , and F B | U lies in Λ ( U ) . Indeed, over U one writes F B | U = X i =1 F i σ i for F i ∈ Λ ( U ) , i = 1 , , , where Pauli matrices σ = −√− −√− ! , σ = −
11 0 ! , σ = −√− √− ! form a basis of Lie algebra su (2) , then F B | U = − ( F ∧ F + F ∧ F + F ∧ F )Id × . It is known that the space Λ ( M ) of 4-forms on M can be decomposed into Λ ( M ) = Λ ( M ) ⊕ Λ ( M ) ⊕ Λ ( M ) , where Λ ( M ) = { f ∗ ϕ ϕ : f ∈ C ∞ ( M ) } , Λ ( M ) = { α ∧ ϕ : α ∈ Λ ( M ) } , Λ ( M ) = { η ∈ Λ ( M ) : ϕ ∧ η = ϕ ∧ ∗ ϕ η = 0 } , moreover, these decomposition are orthogonal with respect to the metric g ϕ . This inspires the following definition. Definition 3.1.
The connection B on P • is called a higher order G -instanton if F B ∈ Λ ( M ) ⊕ Λ ( M ) , • is called an anti-self-dual higher order G -instanton if F B ∈ Λ ( M ) . mklp Proposition 3.2. (1) If B is a higher order G -instanton B , then we have the identity | F B | g ϕ = | F B ∧ ϕ | g ϕ , where | • | g ϕ = Tr( • ∧ ∗ ϕ • ) .(2) If B is a higher order G -instanton B , then F B / ∈ Λ ( M ) unless F B = 0 .Proof. (1) There is also orthogonal decomposition of the the spaces Λ ( M ) of 2-forms M as Λ ( M ) = Λ ( M ) ⊕ Λ ( M ) , where Λ ( M ) = { β ∈ Λ : ∗ ϕ ( ϕ ∧ β ) = − β } , Λ ( M ) = { β ∈ Λ : ∗ ϕ ( ϕ ∧ β ) = β } . Writing F B = ( F B ) (1) + ( F B ) (2) according to the above decomposition, i.e. ( F B ) ( i ) ∈ Λ i ) ( M, Ad P ) , i = 1 , , then fromthe equation of higher order instanton it follows that Tr( F B ) ∧ ϕ = Tr[(( F B ) (1) + ( F B ) (2) ) ∧ (( F B ) (1) + ( F B ) (2) )] ∧ ϕ = − F B ) (1) ∧ ∗ ϕ ( F B ) (1) ] + Tr[( F B ) (2) ∧ ∗ ϕ ( F B ) (2) ]= 0 , (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS namely | ( F B ) (1) | g ϕ = | ( F B ) (2) | g ϕ . (3.10) pm On the other hand, we have ( F B ) (1) = 13 F B − ∗ ϕ ( ϕ ∧ F B ) , ( F B ) (2) = 23 F B + 13 ∗ ϕ ( ϕ ∧ F B ) . which together with ( pm | F B | g ϕ + 29 | F B ∧ ϕ | g ϕ = 49 | F B | g ϕ + 19 | F B ∧ ϕ | g ϕ , i.e. | F B | g ϕ = | F B ∧ ϕ | g ϕ . (3.11)(2) We only show this claim for the case of G = SU (2) . If = F B ∈ Λ ( M ) , then by definition there exists nonzero α ∈ Λ ( M ) such that F B = α ∧ ϕ. Pick a point p ∈ M such that F B | P = 0 , and let U be an open neighbourhood of p with a local coordinate system such that ϕ | p = ϕ . Then over U the above equation reduces to F ∧ F + F ∧ F + F ∧ F = − α ∧ ϕ. (3.12) fff Since G acts transitively on S , by a suitable G -action, we can assume α | p = ce for some constant c = 0 . We see that − e ∧ ϕ = − e − e + e − e (3.13) mnz cannot be expressed as the form as the left hand side of ( fff F B does not lie in Λ ( M ) . (cid:3) (cid:3) The moduli space of higher order G -instantons is defined as M P = { B is a higher order G -instanton } / ( B ∼ B ) , where B ∼ B iff B is G -gauge equivalent to B . At the point B ∈ M , the vector a ∈ Λ ( M, Ad P ) lying T B M satisfiesthe equation [ d M a + [ B ∧ a ]] ∧ F B ∧ ϕ + F B ∧ [ d M a + [ B ∧ a ]] ∧ ϕ = 0 . (3.14) mnj If B is a higher-order G -instanton, there is a complex C B,ϕ : 0 → Λ ( M, Ad P ) D B −−→ Λ ( M, Ad P ) D B,ϕ −−−→ Λ ( M ) → , for an open neighbourhood U ⊂ M , where D B = d M + [ B ∧ • ] ,D B,ϕ = ( D B • ) ∧ F B ∧ ϕ + F B ∧ ( D B • ) ∧ ϕ. Definition 3.3.
A higher order G -instanton B is called elliptic if the associated complex C B,ϕ is elliptic.At the elliptic higher order G -instanton B , the dimension of T B M P is given by dim T B M P = dim H ( C B,ϕ )= − Ind( C B,ϕ ) + dim H ( C B,ϕ ) + dim H ( C B,ϕ )= dim H ( C B,ϕ ) + dim H ( C B,ϕ ) , where the index of the complex C B,ϕ ( M ) is denoted by Ind( C B,ϕ ( M )) which automatically vanishes. m99 Proposition 3.4. (1) Assume G = U (1) , then the adjoint complex C † gϕ B,ϕ : 0 → Λ ( M ) D † gϕB,ϕ −−−→ Λ ( M ) D † gϕB −−−→ Λ ( M ) → . is exactly the Hodge dual of the complex C B,ϕ .(2) If B is a nondegenerate (namely F B is nondegenerate, in other words, the bundle map F B ∧ : Λ ( M ) → Λ ( M, g ) is injective) higher order G -instantons, then B is elliptic.Proof. (1) To show the Hodge-duality, we should check that the adjoints of D B , D B,ϕ are given by D † gϕ B = − ∗ ϕ D B ∗ ϕ : Λ ( M ) → Λ ( M ) ,D † gϕ B,ϕ = ∗ ϕ D B,ϕ ∗ ϕ : Λ ( M ) → Λ ( M ) . The latter one is seen from the following calculations up to boundary term Z M D B,ϕ α ∧ ∗ ϕ ξ = 2 Z M d M α ∧ d M B ∧ ∗ ϕ ξ ∧ ϕ = 2 Z M α ∧ d M B ∧ d M ∗ ϕ ξ ∧ ϕ = 2 Z M α ∧ ∗ ϕ ( ∗ ϕ D B,ϕ ∗ ϕ ξ ) for any α ∈ Λ ( M ) , ξ ∈ Λ ( M ) .(2) We only show this proposition for the case of G = SU (2) . Consider the sequence of bundles Σ( C B,ϕ ) : 0 → π ∗ Λ ( M, Ad P ) σ ( D B ) −−−−→ π ∗ Λ ( M, Ad P ) σ ( D B,ϕ ) −−−−−→ π ∗ Λ ( M ) → , where π : T M → M is the natural projection of the tangent bundle onto M , and σ ( • ) denotes the symbol of the differentialoperator. We first show the exactness of Σ( C B,ϕ ) at π ∗ Λ ( M, Ad P ) . Pick a point p ∈ U , and fix = ζ ∈ T M | p . Assume α ∈ Λ ( U, g ) | p satisfies ( F B | p ∧ α + α ∧ F B | p ) ∧ ζ ∧ ϕ | p = 0 (3.15) lmn Without loss of generality, it is convenient to choose ζ = e , ϕ | p = ϕ , again writing F B | U = Σ i =1 F i σ i , α = Σ i =1 α i σ i , then we have ( F | p ∧ α + F | p ∧ α + F | p ∧ α ) ∧ ( e + e − e + e ) = 0 for nondegenerate 2-forms F i | p ∈ Λ ( U ) | p , i = 1 , , . If not all α i , i = 1 , , , are equal to e up to constants, we cannotfind F i , i = 1 , , , such that the above equation holds. Hence when F B is nondegenerate, equation ( lmn α to lie in ζ ⊗ su (2) . To exhibit the exactness at π ∗ Λ ( M ) , we only need to note that β ∧ e ∧ ϕ cannot be zero when β ∈ Λ ( U ) | p isnondegenerate. (cid:3) (cid:3) One defines a 2-form Ω on M P by Ω B ( a , a ) = Z M Tr[ F B ∧ ( a ∧ a − a ∧ a )] ∧ ϕ (3.16)for a , a ∈ T B M P . Obviously, when G = SU (2) , Ω is identically zero. So we only consider the case G = U (1) . Proposition 3.5.
Assume M is closed, then Ω can be viewed as a symplectic form on the smooth locus in M P consisting ofnondegenerate higher order U (1) -instantons. (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS Proof.
Firstly, when M is closed, we have d M Ω B ( a , a , a ) = 0 for any a i ∈ T B M P , i = 1 , , , as 1-forms on M . Indeed,one can easily check that d Ω B ( a , a , a ) = 2 Z M d M Ba ∧ a ∧ a ∧ ϕ − Z M d M Ba ∧ a ∧ a ∧ ϕ + 2 Z M d M Ba ∧ a ∧ a ∧ ϕ = 2 Z M d M ( a ∧ a ∧ a ∧ ϕ ) = 0 . From the proof of Proposition m99 ∗ ϕ ( F B ∧ ϕ ) is nondegenerate if F B is nondegenerate. This meansthat for given a ∈ T B M P one can find α ∈ Λ ( M ) | p such that ( ∗ ϕ a ) | p = F B | p ∧ α ∧ ϕ | p at some point p ∈ M . Therefore, assume Ω B ( a , a ) = 0 for any a ∈ T B M P , we must have a = 0 , which implies thenondegeneracy of Ω B . (cid:3) (cid:3) Partition Function.
In this subsection, we only consider the case of P being a trivial U (1) -principal bundle over M ,which produces a trivial U (1) -principal bundle P over M . We introduce the ghost fields c , ¯ c which lie in Λ ( M ) but arefermionic, and φ ∈ Λ ( M ) which is a Lagrangian multiplier corresponding to A , then under the following transformations δ A = − ˙ c := − ∂ c ∂t , δB = − d M c ,δ ¯ c = √− φ, δφ = δ c = 0 , the action S = S + Z L dt Z M d Vol g ϕ ( A φ − √− c ˙ c ) , (3.17)where d Vol g ϕ = p det g ϕ dX ∧ · · · ∧ dX is the volume form expressed in terms of local coordinate { X i , i = 1 , · · · , } compatible with the orientation of M , is invariant up to a total derivative term − Z L dt Z M ∂∂t ( c d M B ∧ d M B ) ∧ ϕ. If one picks L = R , the gauge fixing condition A = 0 can be imposed. If L = S , although the above gauge fixing cannotbe reached, one can require ( torus gauge fixing for A : ∂ A ∂t = 0 , constraint on φ : R S dtφ = 0 . In the following, we always assume M is closed and simply-connected, and L = S , then after gauge fixing we considerthe following action S = − Z S dt Z M B ∧ d M B ∧ ˙ B ∧ ϕ + 3 Z S dt Z M A d M B ∧ d M B ∧ ϕ − √− Z S dt Z M d Vol g ϕ ¯ c ˙ c (3.18)with the residual gauge symmetries given by δB = d M f, δ A = c, δ c = δ c = 0 for f ∈ Λ ( M ) independent of t , c being a constant. The partition function is given by path integral as follows Z λ = 1Vol( G ) Z D A ∞ Y n = −∞ DB n ∞ Y n = −∞ D c n ∞ Y n = −∞ D ¯ c n e √− λ S , (3.19) dx where λ ∈ R is the coupling constant, Vol( G ) denotes the formal volume of the group G consisting of the gauge transforma-tion of the action S .By Fourier expansions B = ∞ X n = −∞ B n e √− πnt , (3.20) c = ∞ X n = −∞ c n e √− πnt (3.21)with ¯ B n = B − n due to reality of B , we have S = 4 π √− ∞ X n = −∞ ∞ X m = −∞ ( n + m ) Z M B n ∧ d M B m ∧ B − n − m ∧ ϕ + 3 ∞ X n = −∞ Z M A d M B n ∧ d M B − n ∧ ϕ + 2 π ∞ X n = −∞ n Z M d Vol g ϕ ¯ c n c n (3.22)Choose a background B b = ∞ X n = −∞ B b n e √− πnt (3.23)as the critical point of the action S , namely we have d M B b ∧ d M B b ∧ ϕ = 0 , (3.24) d M B b ∧ ˙ B b ∧ ϕ = 0 , (3.25) d M A ∧ d M B b ∧ ϕ = 0 , (3.26)or equivalently in terms of Fourier modes X m + n = k d M B b m ∧ d M B b n ∧ ϕ = 0 , (3.27) bg X m + n = k nd M B b m ∧ ˙ B b n ∧ ϕ = 0 , (3.28) d M A ∧ d M B b m ∧ ϕ = 0 , (3.29) bgg and express B = B b + λ B with B = ∞ X n = −∞ B n e √− πnt (3.30)then the action S is rewritten as S = 3 λ [4 π √− ∞ X n = −∞ ∞ X m = −∞ ( n + m ) Z M B b m ∧ d M B n ∧ B − m − n ∧ ϕ + ∞ X n = −∞ Z M A d M B n ∧ d M B − n ∧ ϕ ]+ λ [4 π √− ∞ X n = −∞ ∞ X m = −∞ ( n + m ) Z M B n ∧ d M B m ∧ B − n − m ∧ ϕ ]+ 2 π ∞ X n = −∞ n Z M d Vol g ϕ ¯ c n c n =: λ S q + λ S int + S gh . (3.31) zss (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS When λ → , Z λ can be calculated by semiclassical approximation Z sc = 1Vol( G ) Z D A ∞ Y n = −∞ DB b n ∞ Y n = −∞ D B n ∞ Y n = −∞ D c n ∞ Y n = −∞ D ¯ c n e √− S q + S gh ) . (3.32) mbv Firstly, it is clear that Z ∞ Y n = −∞ D c n ∞ Y n = −∞ D ¯ c n e √− S gh = det ′ ( ∂∂t | Λ ( M ) ⊗ Λ ( S ) ) , where the prime above det means excluding zero mode. By heat kernel regularization bt,li [23, 28], we have log det ′ ( ∂∂t | Λ ( M ) ⊗ Λ ( S ) )= Tr ′ ( e − ε ∆ (0) M log ∂∂t | Λ ( M ) ⊗ Λ ( S ) )= log det ′ ( ∂∂t | Λ ( S ) ) , where ∆ (0) M is the Laplacian on Λ ( M ) , therefore we get Z ∞ Y n = −∞ D c n ∞ Y n = −∞ D ¯ c n e √− S gh = det ′ ( ∂∂t | Λ ( S ) ) = Y n> (2 πn ) = ( √ π √ π ) = 1 . (3.33)Next we deal with the path integral G ) Z D A DB b D B e √− S q = 1Vol( G ) Z D A DB b D B e √− R S dt R M B d M ◦ (2 B b ∧ ϕ ∧ ∂∂t + d M A ∧ ϕ ∧ ) B = 1Vol( G ) Z D A DB b ∞ Y n = −∞ D B n { e √− P ∞ n = −∞ [4 π √− P ∞ m = −∞ ( n + m ) R M B b m ∧ d M B n ∧ B − m − n ∧ ϕ + A d M B n ∧ d M B − n ∧ ϕ ] } . (3.34)Define the operators D A ,B b ,ϕ = d M ◦ (2 B b ∧ ϕ ∧ ∂∂t + d M A ∧ ϕ ∧ ) , (3.35) D A ,ϕ = d M A ∧ ϕ ∧ d M . (3.36)Again by heat kernel regularization log det ′ ( ∗ ϕ D A ,B b ,ϕ | Λ ( M ) ⊗ Λ ( S ) )= Tr ′ ( e ε ∂ ∂t log ∗ ϕ D A ,B b ,ϕ | Λ ( M ) ⊗ Λ ( S ) )= log det ′ ( ∗ ϕ D A ,ϕ | Λ ( M ) ) , we find that there is only the contribution from zero mode of B left, namely we have Z DB b D B e √− S q = Z DB b D B e √− R M B ∧D A ,ϕ B . (3.37)After introducing the gauge fixing condition d † gϕ M B = 0 with Lagrange multiplier ϑ and the ghosts b , ¯ b as fermionic 1-formsassociated to the Faddeev-Popov determinant, we consider the gauge fixed action S ggq = 3( Z M B ∧ D A ,ϕ B + Z M d Vol g ϕ ϑd † gϕ M B − √− Z M d Vol g ϕ ¯ b ∆ g ϕ b ) . (3.38) with symmetry δ B = − d M b , δ ¯ b = √− ϑ, δϑ = δ b = 0 , where ∆ g ϕ = d M d † gϕ M + d † gϕ M d M : Λ • ( M ) → Λ( M ) • is the Hodge Laplacian. Then we arrive at ad [29] G ) Z D A DB b D B e √− S q = Z M b D A DB b H ) Z D B DϑD b D ¯ b e √− S ggq = 1Vol( M ) Z M b D A DB b Z D B DϑD b D ¯ b e √− S ggq , (3.39)where H stands for the isotropy subgroup of G consisting of gauge transformations which leave ( B b , A ) invariant, hencefor our case, H is exactly U (1) whose volume with respect to the induced measure on G is Vol( M ) mc [30], and the modulispace M b is defined as M b = { ( B b , A ) ∈ Λ ( M ) ⊕ Λ ( M ) : d M B b ∧ d M B b ∧ ϕ = d M A ∧ d M B b ∧ ϕ = 0 } B b ∼ B b + d M f, A ∼ A + c for f ∈ Λ ( M ) and c being a constant. Proposition 3.6.
Define the operator b D A ,ϕ := ∗ ϕ D A ,ϕ − d M ∗ ϕ d M ∗ ϕ ! : Λ ( M ) ⊕ Λ ( M ) → Λ ( M ) ⊕ Λ ( M ) , (3.40) then it is a first order self-adjoint elliptic differential operator, and b ∆ A ,ϕ | Λ ( M ) ⊕ Λ ( M ) := ( b D A ,ϕ | Λ ( M ) ⊕ Λ ( M ) ) = (∆ A ,ϕ + ∆ ′ g ϕ ) | Λ ( M ) + ∆ g ϕ | Λ ( M ) , where ∆ A ,ϕ | Λ ( M ) = D † gϕ A ,ϕ D A ,ϕ : Λ ( M ) → Λ ( M ) with D † gϕ A ,ϕ = ∗ ϕ D A ,ϕ ∗ ϕ : Λ ( M ) → Λ ( M ) , and ∆ ′ g ϕ | Λ ( M ) = d M d † gϕ M : Λ ( M ) → Λ ( M ) .Proof. The ellipticity of b D A ,ϕ can be seen from the following explicit calculations b D A ,ϕ ( α + γ ) = d † gϕ M ∗ ϕ ( χ ∧ α ) − d M ∗ ϕ α + d M ∗ ϕ γ = − p det g ϕ δ i i i i i i i ∇ i g ϕ ( χ i i i i α i ) dX i − p det g ϕ X u =2 ( − u δ i i ··· ˆ i u ··· i i i i i i ( ∇ g ϕ ) i u α i dX i ∧ · · · ∧ dX i + ( ∇ g ϕ ) i f dX i = [( ∗ ϕ χ ) i i i ( ∇ ( g ϕ ) ) i α i + ( ∇ g ϕ ) i ( ∗ ϕ χ ) i i i α i + ( ∇ g ϕ ) i f ] dX i − ( ∇ g ϕ ) i α i d Vol ϕ for α ∈ Λ ( M ) , γ = f d Vol ϕ ∈ Λ ( M ) , where χ = d M A ∧ ϕ , and ∇ g ϕ denotes the Levi-Civita connection associated tothe metric g ϕ . (1) -GAUGE FIELD THEORIES ON G -MANIFOLDS To show b D A ,ϕ = ( b D A ,ϕ | Λ ( M ) ⊕ Λ ( M ) ) † gϕ , we note the following identities Z M d Vol g ϕ h α, ∗ ϕ D A ,ϕ β i g ϕ = Z M d Vol g ϕ h∗ ϕ α, D A ,ϕ β i g ϕ = Z M d Vol g ϕ h d † gϕ M ∗ ϕ α, d M A ∧ ϕ ∧ β i g ϕ = Z M d Vol g ϕ h d M α, ∗ ϕ ( d M A ∧ ϕ ∧ β ) i g ϕ = Z M d Vol g ϕ h∗ ϕ ( d M A ∧ ϕ ∧ d M α ) , β i g ϕ , and Z M d Vol g ϕ h γ, d M ∗ ϕ α i g ϕ = Z M d Vol g ϕ h d † gϕ M γ, ∗ ϕ α i g ϕ = − Z M d Vol g ϕ h d M ∗ ϕ γ, α i g ϕ for any α, β ∈ Λ ( M ) , γ ∈ Λ ( M ) . (cid:3) (cid:3) Then formally, we have ad,sc,wit,witt,mm [29, 31, 32, 33, 34] Z D B DϑD b D ¯ b e √− S ggq = det ′ (3∆ g ϕ | Λ ( M ) ) q det ′ (3 √− b D A ,ϕ | Λ ( M ) ⊕ Λ ( M ) )= e √− π η ( b D A ,ϕ | Λ1( M ) ⊕ Λ7( M ) ) det ′ (3∆ g ϕ | Λ ( M ) )[det ′ (9(∆ A ,ϕ + ∆ ′ g ϕ ) | Λ ( M ) )] [det ′ (9∆ g ϕ | Λ ( M ) )] , (3.41)where η ( b D A ,ϕ | Λ ( M ) ⊕ Λ ( M ) ) = lim s → X = λ ∈ Spec( b D A ,ϕ | Λ1( M ) ⊕ Λ7( M ) ) sign( λ ) | λ | − s . On the other hand, it is known that mc,je [30, 35] det ′ (9∆ g ϕ | Λ ( M ) ) = 9 ζ (0) det ′ (∆ g ϕ | Λ ( M ) )= 19 det ′ (∆ g ϕ | Λ ( M ) ) , where ζ q (0) = lim s → X <λ ∈ Spec(∆ gϕ | Λ q ( M ) ) λ − s = − dim(Ker(∆ g ϕ | Λ q ( M ) )) , and similarly det ′ (9(∆ A ,ϕ + ∆ ′ g ϕ ) | Λ ( M ) ) = 19 ♭ A ,ϕ det ′ ((∆ A ,ϕ + ∆ ′ g ϕ ) | Λ ( M ) ) , where ♭ A ,ϕ = dim { α ∈ Λ ( M ) : D A ,ϕ α = d † gϕ M α = 0 } . Consequently, we arrive at Z D B DϑD b D ¯ b e √− S ggq = 3 ♭ A ,ϕ − e √− π η ( b D A ,ϕ | Λ1( M ) ⊕ Λ7( M ) ) [det ′ (∆ g ϕ | Λ ( M ) )] [det ′ ((∆ A ,ϕ + ∆ ′ g ϕ ) | Λ ( M ) )] . (3.42) So far, the partition function ( mbv Z sc = Vol( M P )Vol( M ) X κ ∈ Z ≥ κ − Z M κ D A e √− π η ( b D A ,ϕ | Λ1( M ) ⊕ Λ7( M ) ) [det ′ (∆ g ϕ | Λ ( M ) )] [det ′ ((∆ A ,ϕ + ∆ ′ g ϕ ) | Λ ( M ) )] , (3.43)where M n is defined as M κ = {A ∈ Λ ( M ) : ∃ α ∈ M P such that d M A ∧ d M α ∧ ϕ = 0 , ♭ A ,ϕ = κ }A ∼ A + c . R EFERENCES[1] P. Candelas, D.J. 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McLellan, Eta-invariants and anomalies in U (1) -Chern-Simons theory, Chern-Simons gauge theory: 20 years after, in AMS/IP Studiesin Advanced Mathematics (American Mathematical Society, Providence, RI, 2011, Vol. 50, 173-199.D EPARTMENT OF M ATHEMATICS , M
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