On the Evaluation of the Ray-Singer Torsion Path Integral

There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions. Here we provide such a calculation for the path integral representation of the Ray-Singer Torsion of a flat connection on a vector bundle on base manifolds that are themselves $S^{1}$ bundles of any dimension. The calculation relies on a suitable algebraic choice of gauge which leads to a convenient factorisation of the path integral into horizontal and vertical parts.


Introduction
There are very few explicit evaluations of path integrals for topological gauge theories in more than 3 dimensions.Here we will provide such calculations for a class of topological field theories giving a path integral realisation of the Ray-Singer Torsion.The Ray-Singer Torsion [1,2] is an invariant that can be defined on any compact closed manifold M of real dimension dim R M = m.The definition requires a bundle E over M that admits a flat connection.Schwarz [3][4][5] gave a deceptively easy topological field theory representation for the Torsion.The Schwarz action for the Ray-Singer Torsion of a flat connection A on a bundle E → M is given by with B ∈ Ω p (M, E * ), C ∈ Ω q (M, E) where q = m − p − 1.The action (1.1) has reducible symmetries as d 2 A = F A = 0. Schwarz formally quantised these reducible systems in terms of resolvents.The resolvent approach was shown to be equivalent to the more usual Faddeev-Popov gauge fixing procedure in [6][7][8].Either way the partition function Z[A, E] with action (1.1) is related to the Ray-Singer torsion τ M (A, E) by In [9] we began a study of the exact evaluation of the Schwarz path integral representation of the Ray-Singer Torsion.Particular emphasis was paid to the issue of zero modes and we advocated using the addition of a mass to take these into account.Formally, the addition of a mass is straightforward, however, computationally we focused on mapping tori as on these manifolds the mass could be understood as a part of a connection for an extended gauge group.Using path integral techniques we were able to reproduce for the Ray-Singer Torsion a result of Fried [10] that was established for the Reidemeister Torsion.That Fried's result should also hold for Ray-Singer Torsion is a consequence of the equivalence of the two Torsions conjectured by Ray and Singer and proven by Cheeger [11,12] and by Müller [13].
Here we consider the same path integrals but for the manifold M taken to be an S 1 bundle over N and where E is a flat G = SU (r) vector bundle over M .We provide an explicit evaluation of the path integrals involved for M of any dimension and E of any rank.
The natural nowhere vanishing vector field on M generating the S 1 -action allows for a decomposition of any connection A on E into a component φ along the fibre and a horizontal component A. By the same logic the fields B and C and their entire Batalin-Vilkovisky triangles of ghost and multiplier fields [14] may also be decomposed into components.This allows one to impose an algebraic gauge condition which both simplifies the field content and simultaneously allows one to perform the path integrals.
We find that quite generally (1.2) factorises as where Z [A, E] contains all the information about the zero modes associated with the flat connection.Hence in the acyclic case (and up to trivial normalisation) Z [A, E] = 1, otherwise Z [A, E] is a section of an appropriate determinant line bundle.Whether a flat connection which is acyclic exists or not depends on the SU (r) bundle E. For example if E is the fundamental representation then there are flat connections which are acyclic whereas for the adjoint bundle case no flat connection is acyclic.As how to deal with zero modes has been explained previously [7,9] we do not spend much time on this part of the factorisation formula.
The more interesting factor is Z ⊥ [φ, E] and to evaluate it we make some choices.We presume that E is the pullback of a bundle V on N .On N , A is a connection while φ is a section of the ad V bundle.In this case the flatness equations for the combined gauge field on E over M imply that either φ is zero (in which case A is a flat connection on N ) or φ is non-zero and A is a reducible connection.If φ is zero there is nothing to do, so we fix our attention on the case where φ is not zero.In this case the flatness condition implies that the connection A splits as A K ⊕ A U (1) with A U (1) lying in the same direction as φ in the Lie algebra and K × U (1) ⊂ G.With this set-up the computations are quite doable and one finds that where E ⊥ | S 1 is the restriction to the fibre of the bundle E ⊥ over M , τ S 1 is the Ray-Singer torsion on the circle and χ(N ) is the Euler characteristic of N .(The definition of E ⊥ is given in section 6; here one understands that in (1.4) it corresponds to having no zero modes.) Combining (1.2) and (1.4) in the acyclic case we arrive at which agrees with another result of Fried for Reidemeister Torsion (result (V) on page 26 of [10]) when M is an S 1 fibration.Our results for the non-acyclic case therefore represent a generalisation of that work.
While this may be of some limited interest in its own right, the ability to reproduce or generalise known mathematical results serves mainly as a check on the methods that we develop and use here to evaluate the path integral, which can then also be applied with more confidence in other cases.
In Section 2, in order to set the stage, we review the flatness equations for connections on a vector bundle over an S 1 bundle, while a more general background on the geometry of the bundles and the decomposition that we use are provided in Appendix A. In particular, we explain how to decompose forms, exterior (covariant) derivatives, connections and curvatures into horizontal and vertical components.
Section 3 is devoted to establishing that the partition function factorises as in (1.3).In the current geometric situation it is possible to impose algebraic gauge fixing conditions that simplify the ghost for ghost system of the quantisation of the theory and we carry this out in detail in Appendix B. With the gauge fixing choices that we make the factorised partition functions have the dependency on the components of the connection explicitly exhibited in (1.3).
An analysis of the factor Z [A, E] is the subject of Section 4.There it is shown that the path integrals devolve to integrals over the zero modes or, equivalently, to producing a section of a determinant line bundle of tensor powers of cohomology groups associated with the twisted de Rham operator.Such determinant lines appear in the work of Ray and Singer [2] and from the quantum gauge theory point of view [7,9] so we are brief about this here.However, we do spend some time on establishing which harmonic modes are actually zero modes and which are part of the gauge symmetry.
In Section 5 we turn our attention to the second factor and begin a first evaluation of Z ⊥ [φ, E] in generality, that is before making specific choices of the bundles and connections on them.We show that Z ⊥ [A, E] is essentially products of powers of the determinant of the covariant derivative in the fibre direction D φ acting on the spaces spanned by the forms B, C and their ghost systems.At this general level one cannot go too far but we do obtain a formal expression for the partition function that suggests that the final answer only depends on cohomology groups with values in certain vector bundles over N .
In order to proceed further with the analysis of the partition function Z ⊥ [φ, E] we take a more in-depth look at the flatness equations in Section 6.The flat bundle E over M that is considered is a pullback of a bundle V over N , so that E = π * V .In this situation A is the pullback of a connection on V that is not flat, φ is the pullback of a section of the ad V bundle while the flat connection on E is a combination of these two.We also assume that φ is not zero so that, as described above, the structure group reduces to K × U (1) ⊂ G and that the connection splits as A K ⊕ A U (1) .In such a situation V splits as the direct sum of powers of a basic line bundle L over N and the curvature 2-form of the connection A K is flat on N .
Section 7 is where we put all of the pieces together and determine Z ⊥ [φ, E] for the choices of the previous section.The determinants of D φ found in Section 5 are now over spaces of forms on N with values in tensor powers of the line bundle L. The combinations for fixed Fourier mode (on the fibre) are regularised as a Heat Kernel and the combination is the super-trace of the eigenmodes of the twisted de Rham operator, which in turn is a multiple of the Euler characteristic χ(N ) of N , which explains its appearance in (1.4).The product over the Fourier modes is regularised using a ζ function regularisation and provides the τ S 1 that appears in (1.4).
Finally, in Section 8 we briefly mention certain generalisations that can be considered, some of which straightforwardly follow from the computations that we have presented here.
There are two appendices, as mentioned above.In the first Appendix A we provide a quick summary of the geometric structure that is available on an S 1 bundle M → N .This includes the decomposition of forms and connections into horizontal and vertical components.Amongst other things we also review the relationship with basic cohomology and explain how to expand fields on M in "Fourier modes" as sections of line bundles over N .
The algebraic gauge conditions that we will make use of and that simplify our considerations in the body of the paper are introduced in Appendix B. We explain how such algebraic choices allow one to avoid the complete set of ghosts for ghosts structure that would normally be required for the reducible symmetries of the theory in covariant gauges.

Connections on Bundles over S 1 Bundles
Let M be the total space of a smooth S 1 -bundle over a smooth manifold N .We will need to make use of some aspects of the geometry of S 1 -bundles.The results that we need are quite standard, so we do not repeat them here, but for convenience we have assembled and briefly reviewed them in Appendix A.
Thus let π : M → N be a smooth S 1 -bundle over the smooth manifold N , and denote by ξ the fundamental nowhere vanishing vector field generating the S 1 -action on M .A choice of connection on this bundle is equivalent to a choice of 1-form κ on M satisfying (with L ξ = ι ξ d + dι ξ the Lie derivative), and we choose one such κ in the following.
With respect to a choice of κ, any p-form α ∈ Ω p (M ) on M can be decomposed into a "horizontal" and a "vertical" component, with α (p) = ι ξ (κ ∧ α) and α (p−1) = ι ξ α both manifestly horizontal, in the sense that The space of horizontal p-forms α, i.e. which satisfy ι ξ α = 0, is denoted by Ω p H (M ).Likewise, the exterior derivative can be decomposed as (A.8) (2.4) Now let E be a vector bundle over M .A connection A on E → M may be decomposed, as above, as so that locally A is a Lie algebra valued horizontal 1-form and φ is, locally, a Lie algebra valued 0-form.Gauge transformations decompose appropriately, and covariant derivatives split as in (2.4), i.e. one has (with the connection in the appropriate representation), and the square of the covariant derivative is The corresponding decomposition of the curvature 2-form is where for simplicity we have adopted the notation Note that F H A does not satisfy the 'horizontal' Bianchi identity d H A F H A = 0 in general as d 2 H = 0.However, if L ξ A = 0, one has and thus this condition implies the horizontal Bianchi identity, If this condition is satisfied then one can consider A to be the pullback of a connection (which we also denote by A) on a bundle on N .The pullback curvature F H A now correctly satisfies the appropriate Bianchi identity.
The equations for a flat connection are, from (2.9), The 'operator' version of these equations that follow from (2.8) are and will be useful in the next section.In particular, the second equation implies that the operators d H A and D φ commute on horizontal vector-valued forms α ∈ Ω p H (M, E),

Factorisation of the Partition Function
Let A = A + κφ be a flat connection on a vector bundle E. The Schwarz action for the corresponding Ray-Singer torsion is the action (1.1), with the fields in Ω * (M, E).Here we use •, • to denote a fixed invariant fibre scalar product on E (which in (1.1) was denoted by Tr).In case E is a complex vector bundle it is to be understood throughout that we have added the complex conjugates of the fields in (3.1) so that the action is real.Specific choices are e.g. the bundle E = F for the fundamental representation of some group G, or E = ad P for the adjoint representation (associated to the principal G-bundle P ).These are the examples that we will occasionally consider explicitly.However, we can describe both the gauge fixing and the evaluation of the path integral in a way that is largely independent of the choice of E (or representation).
The gauge fixing is described in detail in Appendix B for any E equipped with a flat connection.The gauge choices are a mixture of algebraic conditions (imposed directly on components of the fields and do not involve differential operators) and those that do involve differential operators (as in the Landau gauge in Yang-Mills theory, say).
There is a good reason to use such algebraic gauges in the present context.These gauge conditions greatly simplify the evaluation of the path integral and because of that simplification lead directly to the somewhat startling fact that the partition function factorises into two pieces, one piece only contains possible zero mode dependence and otherwise is independent of the A part of the background flat connection A = A + κφ.The second factor, one sees in a very direct fashion, depends only on the φ component of the connection.Denoting the complete partition function by Z[A, E], one thus has the factorisation (the notation will become apparent).To exhibit this simple dependence on φ one does not need to know the details of the vector bundle nor the details of the flat connection under consideration.
The key step in order to establish this factorisation (and, as it turns out, to neatly separate zero modes from non-zero modes) is to perform a decomposition of horizontal forms according to the action of D φ on them.To that end, let where Ω p is the kernel of the map and D φ is invertible on Ω p ⊥ (M, E).By virtue of (2.15) one also has that with the obvious notation.As in (2.2), the fields and the ghosts etc have the decomposition with Ψ (q) and Ψ (q−1) horizontal.As one sees from (B.1) and (B.2), or in more detail in (B.6), the field Ψ (q−1) will transform as where we have chosen the gauge parameter Λ allows one to remove the Ψ ⊥ (q−1) from Ψ (q−1) , so that only the parallel component survives, Now we can state the algebraic gauge condition: for all fields Ψ we impose the condition that In particular, for the fields in the action (3.1), the algebraic conditions are and the classical action (3.1) for the vector bundle E then becomes with the fields being sections of the appropriate bundles.Notice that 'parallel' and 'perpendicular' fields do not couple.
There are still more symmetries to gauge fix.We have only used up the Ω * ⊥ (M, E) parts of the symmetry.If Ψ satisfies (3.10) then the transformation will preserve the gauge condition.These symmetries only involve the fields in Ω * (M, E) and so gauge fixing these symmetries does not mix with the gauge fixing of the perpendicular symmetries.Consequently we conclude that for the complete partition function the total action including ghosts and other fields will split precisely into perpendicular and parallel components just as in (3.13) and this is borne out in some detail in Appendix B. This gauge fixing and ghost structure therefore guarantees that (3.2) holds.

Z [A] and the Zero Mode Measure
One can be somewhat more precise about the structure of Z [A] and at the same time come to a better understanding of the dκ component of the gauge transformations.
We gauge fix in two steps.Firstly note that by (3.14) we may covariantly gauge fix both the 'upstairs' parallel fields Ψ (p) and the 'downstairs' Ψ (p−1) ones as There is still a residual symmetry that is available even after we have used Λ Likewise there are Λ (p) harmonic modes.We will discuss the roles of the harmonic modes and their relationship with zero modes below.
All fields in Ω * (M, E) are either d H A exact or δ H A exact or harmonic with respect to ∆ H A .This follows from the fact that (d H A ) 2 = 0 on Ω * (M, E) as follows from (2.14), and that with the respect to the inner product (A.22) this Hodge decomposition is an orthogonal one.With this in mind it is convenient to set where µ is a section of an appropriate determinant line bundle of cohomology groups and Z [A] is the path integral we had before except with the harmonic mode sector projected away.µ contains all the information of the harmonic modes and is there to ensure that the overall partition function is metric independent.It is important to note that Z ⊥ [φ] on the other hand carries no zero modes as it only involves the perpendicular fields.
We now establish that Z [A] = 1.In Appendix B we show that in the gauge transformations (3.14), in the sector orthogonal to harmonic modes, the dκ terms can be ignored.We give a slightly different argument to this effect here.Start with the action (3.13), with all fields orthogonal to the harmonic modes, and send so that the last term goes over to while the others do not change.Now on taking the t → ∞ limit this last term completely vanishes.For consistency, however, we also need to scale all the fields in the symmetry transformations.The overall effect is to send the dκΛ (p−2) in (3.14) to t −1 dκΛ (p−2) so that this too vanishes in the limit.One must bear in mind that these scalings are done orthogonally to the harmonic modes, so that each mode under discussion here appears in the action.
At this point the action for the partition function Z [A] is while the symmetries have gone over to At the outset, the dκ part of the symmetries were there as (d H A ) 2 = 0.One may therefore wonder how we have managed to get away with eliminating those pieces.Essentially, this is due to the fact that on parallel fields, as we have already seen, (d H A ) 2 = 0 so that on those fields d H A is essentially flat.In this way we understand that (4.6), which only contains parallel fields, and the symmetries (4.7) (which exclude the harmonic mode symmetries) correspond to two Schwarz actions of Ray-Singer type for the 'flat' connection d H A .We thus see that the partition function Z [A] itself factorises into two parts, one for the (B (p) , C (q−1) ) system and the other for the (B (p−1) , C (q) ) system.
The path integrals in question need regularisation and one may adopt the original definition in terms of ζ-functions of Ray and Singer [1] where the Laplacian in question is Now with this definition a general (B (p) , C (q−1) ) system yields τ (A) (−1) p−1 where τ (A) is the 'Ray-Singer torsion' of the connection A. Consequently, the product of these two partition functions cancel and we are left with unity.Thus as we set out to show.In particular the definition used by Ray and Singer in [1] was to project out the harmonic modes just as we have done here.
Clearly if all of the Ω * (M, E) are trivial then (up to normalisation) one may set This fact is related to the acyclicity of the de Rham complex (d A , E).The twisted de Rham complex is said to be acyclic if all the twisted de Rham groups H * A (M, E) are trivial.The representatives of the twisted de Rham groups satisfy If Ω p−1 (M, E) = 0 then one may choose λ so that v (p−1) = 0.The condition that d A v = 0 (4.9) now includes the equation D φ v (p) = 0, but by assumption there are no solutions to this so that v = 0 and consequently H p A (M, E) = 0.More generally, when D φ has a kernel pick a v so that v ⊥ (p−1) = 0.The last equation in (4.9) now breaks into two parts of which the first, by definition, implies that v (p) ⊥ = 0. Consequently all the cohomology lives in the parallel fields so that the zero mode measure is a part of Z [A].Indeed in the gauges chosen there are two types of zero modes, namely either or v (p−1) = 0 and 12) The second of these says that any harmonic v (p−1) such that dκ ∧ v (p−1) is equal to a standard part of the gauge symmetry is a zero mode.In short, all zero modes are harmonic but the converse need not be true.In summary some of the harmonic modes are part of the gauge symmetry and are to be gauge fixed in the standard BRST fashion.Those which are not part of the gauge symmetry are zero modes and need to be dealt with in a different manner.
How to deal with zero modes in the path integral representation has been described in detail in [7,9].In particular, it has been shown in [9] how a choice of gauge matches the definition of Ray and Singer [2].We will not repeat that analysis here, but rather take the attitude that the zero modes have been dealt with according to one of the above prescriptions.Indeed we take the stronger attitude that all harmonic modes have been dealt with and in this way µ has been determined.
Before leaving this section, however, we pose a question.Could the (horizontal and parallel) "torsion" τ (A) by itself be an interesting invariant of the base space N ?From the viewpoint of the path integral, varying the metric on N amounts to a BRST exact term, just as for the standard Ray-Singer torsion, and hence formally τ (A) would be invariant under such deformations.

A Formal Expression for Z ⊥ [φ]
Apart from the zero modes, the complete partition function is just . That is the complete flat connection on E over M is not needed -rather it is just the component φ in the fibre direction that appears.In a sense φ contains almost all of the gauge invariant data of the bundle as we will see when we analyse the flatness equations in more detail.
To perform the path integral in the algebraic gauges, we can rely on the discussion in Appendix B which shows that the ghost triangle for the perpendicular fields can be disentangled from the parallel fields (discussed in the previous section).Furthermore we only need the right hand edge of the BV-triangle thanks to the algebraic conditions.
The total action for the path integral of interest (including the ghost terms and taking the algebraic conditions into account) is then where the (ω ⊥ , ω ⊥ ) are the ghosts for B ⊥ and (λ ⊥ , λ ⊥ ) are the ghosts for C ⊥ all in the appropriate representations.By making use of the Hodge star operator (A.21) we have where ⊥ .By virtue of (A.23) D φ commutes with * H in the integrals.Now all the terms in (5.1) have the same form.
At least formally, we then find that the partition function takes the form of an infinitedimensional determinant, where Ω ⊥ H (M, E) is the direct (signed) sum of infinite-dimensional vector spaces (here (−1) 2r means ⊕ while (−1) 2r+1 stands for ⊖).Note that the second equality holds by Hodge duality * H so that Ω j ⊥ (M, E) ≃ Ω n−j ⊥ (M, E) and that q = n − p while the third equality comes on setting j − n = i.Of course, in order to define the theory we still need to specify the regularisation that will be used.All in all the last equality then suggests that the sum over infinite dimensional vector spaces may reduce to a sum over some finite dimensional cohomology spaces.We cannot say more at this level of generality.However, once we have specified the flat bundles and the flat connections on them that we are interested in, the formulae we find are indeed cohomological in nature.
6 Flat Connections on a Pull Back Bundle E = π * V In previous sections we have developed the path integral for the partition function as far as possible without specifying either the vector bundle E or the allowed flat connection A on it.In this section we do not aim for complete generality but, rather, choose a natural class of bundles and connections.
Recall that the flatness equations F A = 0 by (2.13) are As a first example consider a solution with φ = 0.In this case the solutions satisfy L ξ A = 0 and the connection A can be thought of as a connection on a bundle on N .The first equation of (6.1) then tells us that the connection A is a flat connection.So the upshot is that in this case the bundle E is the pullback of a flat vector bundle on N with flat connection on E being the pullback flat connection A = π * A. Of course in the case at hand, from the results of the previous sections, the partition function is given completely by Z || [A] and only contains information about the zero modes, the flat connection A playing no other role.
A more general and more interesting situation, and the one that we concentrate on here, is where we still have a pullback bundle but with φ = 0. Let V → N be a G vector bundle over N .V is taken to be such that it admits a connection A which satisfies where φ ∈ Ω 0 (N, ad P V ) and P V is the associated principle bundle for φ = 0.By construction one admits a φ such that L ξ φ = 0 from the point of view of M .Now fix the bundle E = π * V and equip it with the connection A = A + κφ.This is not the pullback connection π * A ≃ A which is not flat.However, as one can add to any connection a Lie algebra valued 1-form and still have a connection, we have added κφ to A and, by construction, we now have A = A + κφ.This connection is a flat connection on the pullback bundle as (6.1) is equivalent to (6.2) for these fields.
We fix on G = SU (r).Solutions to the equation d A v = 0 with v ∈ Ω 0 (N, ad P V ) span the Lie algebra of the stabiliser of the connection A and a connection with non-trivial stabiliser is reducible.The first part of the flatness equation tells us that the curvature of A is non-zero only in the Abelian direction generated by φ.In this way we see that the connections of interest reduce to the form K × U (1) and are flat in the K direction.The connection can be expressed as 1) with curvature 2-form that satisfies With such a split, one has [A, φ] = 0 so that the second equation of (6.2) together with L ξ φ = 0 implies that dφ = 0 on M , i.e. that φ is constant.Note that not only the connection A is reducible, but also the flat connection A = A + κφ is, since Note that, with φ constant, the curvature formula shows that all of the Chern-Weil representatives of the characteristic classes of the pullback bundle π * V on M are trivial.However, the bundle V on N may well be non-trivial.After all, on N one has c 1 (L M ) = [dκ].This will be seen in the examples below.
In the following, in order to illustrate the evaluation of the path integral, we will not consider the most general reducible connection but restrict our attention to reducible connections with A K = 0 (though the more general case can also be dealt with), and we write A U (1) = A.Moreover, since φ is constant we may as well take it to be diagonal.From here on we do not carry around the U (1) superscript as it is clear which component of A is non-zero.
Let dκ correspond to p M times an integral class on N (for details see Appendix A).Then (6.3) makes sense for φ having any entries (as a matrix) of the form 2πib j /p M with the b j integral and r j=1 b j = 0.The non-trivial equations for A in (6.3) with G simply connected and I(G) the integral lattice of G then imply that, For example, with G = SU (2), we have By F A = −dκφ, a φ as in (6.7) implies that one has a non-trivial complex line bundle L n on N with U (1) connection A on L having integral real first Chern class (cf.(A.14)) and (A.15)) Note that L M = L p M where L M is the line bundle associated to the circle bundle which is M .The SU (2) vector bundle V → N defined by (6.7) is split according to with total Chern class Likewise for the complexified adjoint bundle Consequently both V and ad P C may well be non-trivial.For example on N = P 1 these are obviously trivial while on N = P m for m ≥ 2 they are not.
For G = SU (r), given our choices for the flat connections, the vector bundle V for a general representation is taken to split as and fields are direct sums of sections of Ω p (N, L q i ⊗ L r M ) for r ∈ Z and i = 1, . . ., k (see Fact 4 in Appendix A).Note that for ω r,q (6.13) and we see that zero modes occur precisely when L q = L −r M , and the eigenvalues are independent of the form degree of ω The determinant of the Therefore, if q is an integer multiple of p M , q = 0 mod p M , there is a zero mode eigenvalue.
In particular, if p M = ±1 then such zero modes are unavoidable in any representation.By the same logic the determinant (6.14) does not depend on q but rather on q mod p M .The zero mode in (6.14) belongs in Z and is to be deleted in the calculation for Z ⊥ .By shifting r if required, the contribution to Z ⊥ of the determinant (6.14) for q = 0 mod p M is Though strictly a contribution to Z ⊥ we will move such normalisation terms (6.15) to Z ; their value can be determined from the calculation of the following section.
With the above understood, we now decompose the vector bundle V given in (6.12) as L q i for q i = 0 mod p M (6.16) and obtain the bundles on M , with E ⊥ the bundle that will contribute to Z ⊥ [φ].Since the eigenvalues of D φ are independent of the form degree, the decomposition (3.3) is thus more explicitly given by Before turning to the actual calculation (section 7), we present some examples of the bundles under consideration.
Example 6.1.Let us consider the adjoint representation.The complexified Lie algebra splits as where the root spaces V α are copies of C and the space of positive roots is denoted by ∆ + .Just as for the SU (2) example (6.11), for a semi-simple group G the ad P C bundle over N is split into copies of line bundles corresponding to the positive and negative roots spaces Here we are taking a liberty in the notation as α(iφ) may be fractional.One should understand L ±α(iφ) M ≡ L ±p M α(iφ) .The covariant derivative (A.28) acting on sections is Example 6.2.Let M be the Lens L 2r+1 (p) that is the S 1 bundle S 1 → L 2r+1 (p) → CP r with dκ = p π * ω where ω is the Kähler 2-form dual to the integral homology 2-cycle of CP r .The only flat connection on CP r , up to gauge equivalence, is the trivial connection (as is true on all simply connected manifolds).This means that one may concentrate on the non-trivial U (1) connection.The Abelian Ray-Singer Torsion has been computed for these spaces by Ray [17] in order to corroborate the conjecture of Ray and Singer that their torsion coincides with Reidemeister Torsion.We re-derive this result in Example 7.1 below.
Example 6.3.Let (N, ω) be a Kähler manifold with dimension n = 2r and Kähler 2-form ω where dκ = π * ω and take V to be a vector bundle on N .Solutions to the flatness equations on M (6.1), which are constant along the flow L ξ A = 0, are special solutions of the Yang-Mills equations on N (here d is the exterior derivative on the base) This follows as on a Kähler manifold * ω = ω r−1 /(r − 1)! and the evolution equation together with the gauge condition on φ lead to d A φ = 0.This situation is familiar from the study of flat connections in Chern-Simons theory on contact 3-manifolds by Beasley and Witten [16] .

Evaluation of the Path Integral Representation of Z ⊥ [φ]
Summarising the previous sections we note that (6.3) tells us which line bundles one is considering and, as κφ is the vertical component of the connection, it also determines the holonomy along the fibre of the S 1 bundle.What is missing is the gauge invariant data in the possible holonomies of A but it seems that the Ray-Singer torsion does not see this data for these backgrounds.In particular the information about A is contained in Z and so enters through the zero mode measure µ in (4.3).Apart from the zero mode sector the Ray-Singer torsion is completely determined by Z ⊥ [φ] whose evaluation we now turn to.
The perpendicular fields that enter into the action (5.1) take values in E ⊥ , the pullback of V ⊥ as defined in (6.16, 6.17).The partition function of the action (5.1) is given by products of path integrals over forms in Ω p H (M, π * L q ) and its dual for q = 0 mod p M of the form which, using (6.14), can be written more explicitly as The sign in the exponents depends on the statistics of the fields being integrated over.The notation in (7.2) indicates that one gets the indicated products on the infinite-dimensional spaces Ω p (N, L q L r M ), so these clearly still require some form of regularisation.We adopt a Heat Kernel regularisation where the Laplacian ∆ A that is used is that based on the twisted de Rham operator d A on N and the Hodge operator is * H .The infinite product over the Fourier modes also requires regularisation.As both types of regularisation have been explained in detail in the 3 dimensional case [15,16,18], and more generally in higher dimensions [9] and as there are no extra subtleties in the present situation, we skip the details here.
There is one aspect of the path integral evaluation that we do need to comment on, however.The integrals are all of delta function type here (with a similar conclusion for the Grassman odd integrals) so that there is no phase.In Section 8 we give some examples where, just as in Chern-Simons theory [19], there is a phase and it needs to be determined.We do not show the absolute value symbols below, however, they are to be understood up to and including (7.7).
Up to metric-dependent factors (which also appear in Z ), which drop out of the complete path integral, the partition function is essentially By making use of the fact that for a vector bundle V on N that Ω t (N, V ) ≃ Ω n−t (N, V * ) for fixed r and q i as well as that n − q + 1 = p + 1 (and on recalling that it is the absolute value of the determinant that appears in the second term in (7.4)) one has Suitably regularised, the alternating sum of the infinite dimensional vector spaces of differential forms reduces to the alternating sum of finite dimensional spaces of harmonic forms.This comes about as follows: while the eigenvalues of the D φ operator do not depend on the form degree their multiplicity does, so by fixing on each r and q i one is calculating Here the trace is over the space of forms n s=0 Ω s (N, L (q i −rp M ) ), (−1) F is positive on forms of even degree and negative on forms of odd degree, and we have pulled out ln D φ from the trace as its eigenvalues are independent of the form degree s.One recognises the trace as the Witten index of the twisted de Rham operator d A and that trace reduces to one of (−1) F over the Hodge groups n s=0 H s (N, L (q i −rp M ) ).The index of the twisted de Rham operator (Theorem 3.4.3 in [20]) is just the Euler characteristic χ(N ) of the underlying manifold and does not depend on the line bundles L (q i −rp M ) .
Reassembling all the pieces of the path integral the partition function is Note that the infinite product is an expansion of the hyperbolic sinh function The Ray-Singer Torsion for a line bundle L over the circle is (see [17]) with φ the anti-Hermitian connection on L. Ultimately then the path integral evaluates to where E ⊥ | S 1 is the restriction to the fibre of the bundle E ⊥ over M (recall that φ does not depend on the base point of the fibre on N ).
Taking into account the contribution of Z [A] we find that (up to a suitable normalisation) the complete partition function takes the form Given the relationship between the partition function and the Ray-Singer torsion (1.2) we re-write this as where µ −1 is understood to be the appropriate section of the inverse determinant line.The formula (7.12) is our main result giving a complete evaluation of the path integral representation of Schwarz for the Ray-Singer Torsion on non-trivial S 1 bundles.
In the special case that E = E ⊥ one has The result (7.13) is a special case of Fried's theorem for the Reidemeister torsion (result (V) on page 26 of [10]) where the fibre of the bundle that he considers is taken to be S 1 while (7.12) is a generalisation of Fried's result to non-acyclic representations.
Example 7.1.The Lens space obtained by the identification (z 1 , . . ., z r+1 ) ∼ ζ(z 1 , . . ., z r+1 ) where ζ is a p-th root of unity and r+1 j=1 |z j | 2 = 1 is an S 1 bundle over CP r .Ray finds that for these Lens spaces the Abelian Ray-Singer Torsion has the explicit form (this follows from equation ( 1) in [17]) where ζ is to be interpreted as the holonomy of the Abelian connection on a line bundle L over S 1 .His result follows from (7.7) on taking N = CP r and k 1 = 1.

Some Generalisations and Observations
There are various generalisations that may be considered where the techniques of this paper can apply.We list three of them.
• We looked at the case that M is a smooth S 1 -bundle over a smooth manifold N .In the 3-dimensional case, the calculation can readily be extended to Seifert fibred 3-manifold with base an orbifold of genus g.The i'th orbifold point has order a i for i = 1, . . ., N .
In this case Z ⊥ with action S ⊥ gives a well known result, namely [15] Z It ought to be possible to perform similar calculations on smooth manifolds M which are S 1 bundles over orbifolds in higher dimensions as well.One would need, as in the 3-dimensional case, to make use of the Kawasaki index theorem [21] in order to deal with the orbifold points and to have an orbifold version of the Euler characteristic.A possible application would be to the general Lens spaces L 2r+1 (ν 1 , . . ., ν r+1 ) considered by Ray [17] where (z 1 , . . ., z r+1 ) ∼ (ζ ν 1 z 1 , . . ., ζ ν r+1 z r+1 ) for ζ a p-th root of unity and r+1 i=1 |z i | 2 = 1 which are S 1 bundles over weighted projective spaces.
• One may also consider the case of non-simply connected groups such as P SU (n) ≃ SU (n)/Z n .Some of the flat P SU (n) bundles will be flat SU (n) bundles so that the results carry over.However, not all P SU (n) bundles arise from SU (n) bundles as can be seen for the SU (2) case in (6.11)where the adjoint bundle only allows even powers of line bundles while SO(3) bundles would include odd powers (which would, in turn, correspond to having non-trivial Stiefel-Whitney classes).It is possible to envisage being able to use the results of the body of this paper by taking n in (6.11) to be half integral, though such an approach requires further investigation.
• As explained in [7] (pages 152-153) if M has real dimension m = 4k − 1 then one may refine the theory and consider an action of the form where B ∈ Ω 2k−1 (M, g).This path integral leads to the square root of the Ray-Singer Torsion up to a phase.The phase has the same source in the path integral as the familiar framing anomaly in Chern-Simons theory [19] and it ought to be possible to get explicit formulae for the phase for higher dimensional S 1 bundles.In case M has dimension m = 4k + 1 an appropriate action is where Ψ ∈ Ω 2k (M, g) and Grassman odd.

B Gauge Transformations, Gauge Fixing and BV Triangles
In this appendix we discuss the need for ghosts for ghosts in this theory and show why one can make do with just the right-hand edge of the Batalin-Vilkovisky triangle [14] when using algebraic gauges.Furthermore we explain how the algebraic gauges that we wish to impose are allowed gauge choices.We do not need to specify the representation of the fields (the vector bundle E).
The action (1.1) has the reducible symmetries These symmetries are reducible since one may vary Σ and Λ as d 2 A = F A = 0 without effecting the B and C transformations (B.1).Likewise one may transform both Ψ and Φ by covariantly exact pieces without changing (B.2) and so on.
In order to analyse these symmetries it is simplest to deal with fields that are horizontal.Consequently a p-form field Φ is always decomposed into horizontal fields as In accordance with (A.8) and (2.7), we also decompose the covariant derivative as If the p-form Φ has the usual gauge transformation then, with the (p − 1)-form Λ also decomposed as in (B.3), the gauge symmetry is Note the peculiar dκ term in the first transformation.
The symmetry (B.5, B.6) is reducible and it is convenient to exhibit the original transformation and the subsequent ones as part of a long exact sequence on spaces of horizontal forms Ω We now remind the reader of why the complete Batalin-Vilkovisky approach is required in order to take care of these reducible symmetries when using covariant gauges, as this will allow us to explain why the algebraic gauge conditions that we choose allow for a significantly reduced set of fields, compared with the complete set given by the appropriate Batalin-Vilkovisky triangles.for some metric which provides the Hodge operator * on the manifold.The need for the complete triangle can be traced to the fact that d A ω, d A ω and d A π have degeneracy given by shifting the fields by covariantly exact pieces.These symmetries arise for the anti-ghost and multiplier fields because of the gauge fixing conditions themselves (B.9), while for the ghosts they are an inherent part of the original transformations (B.8).A prudent choice of non-covariant gauge fixing conditions may mean that there will be no extra symmetries for the anti-ghosts and multiplier fields which in turn means that one may be able to ignore the plethora of ghosts for ghosts.
At this point it is convenient to regard the horizontal fields as being split according to the orthogonal decomposition (3.where Ω * (M, E) is the kernel of D φ acting on Ω * H (M, E).This split is implicitly understood with Ψ ∈ Ω p H (M, E) (there is no difference between superscripts and subscripts of ⊥ and and the positions are chosen for legibility).
The form of the symmetries suggests we start with the fields at the left of (B.7) and work our way to the right.The first transformation is δΛ (1) = d H A Λ (0) , δΛ (0) = D φ Λ (0) (B.12) Using Λ ⊥ allows us to set Λ ⊥ (0) = 0. Λ (0) is still available and only enters in the variation of the parallel components of Λ (1) (we may then use a Landau type gauge for these components).We are then left with Λ ⊥ , Λ (1) and Λ (0) .These are the available symmetries at the next step to the right of (B.7).It is important to note that thus far the gauge fixing is on Λ (0) .
The path integral should not depend on the metric that is chosen for the gauge fixing conditions, since it always enters in a BRST exact term, so that one may use this freedom to pick a metric on M adapted to the needs of the problem.The metric of choice here on M is g M as given in (A.18).In particular between horizontal fields the Hodge star operator is * H (A.21).
There are also symmetries that do not act, corresponding to zero modes that is to elements of H * A (M, E).For example, the fields Λ (q−2) and Λ (q−1) with d H A Λ (q−2) = 0 and dκΛ (q−2) = d H A Λ (q−1) correspond to zero modes of the covariant derivative and they do not appear in (B.14).The treatment of such modes using a BRST analysis can be found in [7,9] and we do not repeat that here.
The symmetries that remain unaccounted for, those that appear in (B.14), are all in Ω * (M, E) and we should employ the complete Batalin-Vilkovisky triangle for these when gauge fixing them.
Turning back to the discussion around (4.1) those harmonic Λ (p−2) for which dκ ∧ Λ (p−2) are of the form d H A σ (p) are such that one can redefine Λ (p) in (3.14) to eliminate the term proportional to dκ from the transformations.Such harmonic Λ (p−2) are zero modes of the theory.They do not appear in the action nor in the transformations.The second type of harmonic Λ (p−2) are those for which dκ ∧ Λ (p−2) is harmonic, hence cannot be expressed as d H A σ (p) and so cannot be eliminated from the gauge symmetry (3.14) transformations.Such Λ (p−2) are part of the gauge group.They are harmonic but they are not zero modes.These modes are used to gauge fix some of the harmonic Λ (p+1) modes which have not been used to impose the covariant conditions (4.1).In Example B.2 we show how this is done in practice.