Global Aspects of Abelian Duality in Dimension Three

In three dimensions, an abelian gauge field is related by duality to a free, periodic scalar field. Though usually considered on Euclidean space, this duality can be extended to a general three-manifold M, in which case topological features of M become important. Here I comment upon several of these features as related to the partition function on M. In a companion article, arXiv:1405.2483, I discuss similarly the algebra of operators on a surface of genus g.


Introduction
In classical field theory, abelian duality amounts to a simple relation between the solutions of certain linear partial differential equations on R n for n ≥ 2. The most elementary example occurs in dimension two, for which one considers a harmonic function φ ∈ C ∞ (R 2 ) satisfying the Laplace equation Here ⋆ is the Hodge star operator determined by the Euclidean metric on R 2 . According to (1.1), the one-form ⋆dφ is closed. Since R 2 has trivial cohomology in degree one, ⋆dφ is also exact. Hence one can associate to φ another function ψ ∈ C ∞ (R 2 ) via the duality relation ⋆dφ = dψ . (1. 2) The relation in (1.2) determines the function ψ up to the addition of a constant, and ψ is automatically harmonic by virtue of the identity The classical duality in (1.2) thus relates one solution of the Laplace equation on R 2 to another, distinct solution. As well-known, the dual harmonic functions φ and ψ can be combined as the real and imaginary parts of a single holomorphic function on Similarly in dimension four, if A is a U(1)-connection on R 4 which solves the source-free Maxwell equation In this paper we are concerned with abelian duality in dimension three. In that case, classical abelian duality relates a U(1)-connection A which solves the Maxwell equation on R 3 to a harmonic function φ on R 3 . By analogy to dimensions two and four, the classical duality relation in dimension three is (1.6) Unlike the preceding duality relations, the classical duality relation on R 3 involves the electric coupling e 2 , which appears in the Maxwell action , m, n = 1, 2, 3 . (1.7) The factor of 1/4π in the first line of (1.7) is simply a notational convenience, which will eliminate other factors later. In the second line of (1.7), we rewrite the Maxwell action in components with respect to an arbitrary Riemannian metric g on R 3 , and we follow the standard Einstein convention in summing over the repeated indices m and n on the curvature F A of the gauge field. The appearance of the Maxwell coupling e 2 in the duality relation (1.6) is the first of several peculiarities specific to dimension three. Most famously, the duality relations in two and four dimensions are invariant under conformal transformations, which preserve both the one-form ⋆dφ on R 2 in (1.2) and the two-form ⋆F A on R 4 in (1.5). But in dimension three, if the metric g is scaled by a constant factor g −→ Λ 2 g , Λ ∈ R + , (1.8) the dual one-form ⋆F A in (1.6) scales non-trivially as well, In components, ⋆F A is given by √ g ǫ mnp F mn dx p , where ǫ mnp is the anti-symmetric tensor on three indices, with fixed normalization ǫ 123 = +1. The scaling for ⋆F A in (1.9) follows from the combined scalings of √ g with weight Λ 3 and F mn with weight field. So on a general three-manifold, there is no hope to interpret abelian duality classically, as a one-to-one correspondence (1.6) between solutions of the Laplace and Maxwell equations on M. Instead, abelian duality on M must be interpreted as an inherently quantum phenomenon.
A basic observable in any quantum field theory is the partition function, and as an initial question, one can ask how the partition function transforms under duality. Naively, one might expect the partition function to be invariant under duality, but famously in dimensions two and four, this is not so. The most elegant statement [39] occurs for electric-magnetic duality of Maxwell theory on a four-manifold X. (See also [37] for related observations.) In this case, the Maxwell partition function Z X depends upon both the Maxwell coupling e 2 and an angular parameter θ which enters the classical Lagrangian through the topological pairing (1.11) When X is a spin-manifold, the normalization in (1.11) ensures that θ has period 2π. Otherwise, θ has period 4π. The angular parameter θ naturally complexifies the electric coupling e 2 via and electric-magnetic duality acts upon τ as a modular transformation τ → −1/τ . Moreover, as shown by direct computation in [39], the Maxwell partition function Z X (τ ) on X transforms under duality as a non-holomorphic modular form with weights 1 4 (χ − σ, χ + σ), the discussion in Section 3 of [36], where the modular anomaly in electric-magnetic duality was originally noted in the context of supersymmetric Yang-Mills theory.
Similarly on a Riemann surface Σ, the classic one-loop shift [3-5, 14, 32] in the dilaton under T-duality represents a comparable topological effect, depending again on the Euler character of Σ.
One motivation for the present work is to point out a modular property roughly analogous to (1.13) for the partition function of Maxwell theory on a closed, orientable three-manifold M.
Such modularity in three dimensions may sound surprising, because the partition function on M (as opposed to the partition function on X) can have no interesting dependence on the electric coupling e 2 . A priori, the Maxwell partition function Z M (e 2 , g) depends upon both the coupling e 2 and the Riemannian metric g on M.
However, the scale transformations in (1.8) and (1.10) together preserve Z M (e 2 , g) and can always be used to set e 2 = 1, so that any dependence on e 2 can be effectively absorbed into the dependence on the metric. In addition, the Euler character of any closed, orientable three-manifold vanishes, and there are no other local, generally covariant invariants of M that could appear in an anomaly such as (1.13).
The situation changes, though, as soon as we include additional parameters which play a role in three dimensions analogous to the role of the θ-angle in four dimensions. Very briefly, in dimension three the topological parameter ζ ∈ H 1 C (M) will be a complex harmonic one-form, which enters the classical gauge theory Lagrangian via the natural pairing (1.14) The partition function Z M then depends upon ζ as a theta-function associated to the cohomology lattice of M, and abelian duality acts as a modular transformation on that theta-function. When ζ = 0, the partition function is nonetheless invariant under duality, but in a fairly non-trivial way.
This observation appears at least implicitly in [2,29], with which the present work has some overlap, but I believe it deserves further emphasis here. I also take the opportunity to clean up a few factors in [2], which otherwise detract from a very elegant analysis. Similar questions about one-loop determinants and global aspects of duality have been addressed in the supergravity literature; see for instance [10,11,16] and references therein.

The Plan of the Paper
Very broadly, the purpose of this paper is to analyze the quantum analogue of the classical abelian duality relation in (1.6) when M is a general Riemannian three-manifold. Because the quantum field theories on both sides of the duality are free, this analysis is straightforward and can be carried out in an explicit fashion.
In fact, I will carry out the analysis two ways, working in both the Lagrangian and the Hamiltonian formalisms, since one learns different things from each. Here I focus on the Lagrangian perspective, and in a companion paper [1], I adopt the alternative Hamiltonian viewpoint. In Sections 2 and 3, I compute the respective partition functions for a periodic 1 scalar field and an abelian gauge field on the three-manifold M. Then in Section 4, I perform a direct comparison of the resulting expressions for the partition function. As mentioned above, these expressions involve a novel theta-function attached to the three-manifold M, akin to the classical theta-function on the Jacobian variety of a Riemann surface. Duality acts by a modular transformation on the theta-function.
Both to orient the reader and for sake of completeness, I conclude in Section 5 by reviewing the standard path integral explanation for abelian duality in three dimensions. A very nice exposition of the latter material appears in Lecture 8 of [40], which I largely follow. I also discuss duality for three simple classes of operators (Wilson loops, vortex loops, and monopole operators in the language of Maxwell theory) whose commutator algebra on a Riemann surface of genus g will be analyzed in [1]. See also [22] for another recent approach to abelian duality, invoking the formalism of duality walls. One coupling relevant in dimension three but with no equivalent in dimensions two and four is the Chern-Simons coupling, for which global issues feature prominently. In subsequent work, I apply ideas here and in [1] to clarify the meaning of abelian duality for Maxwell-Chern-Simons theory at level k.

Analysis of the Abelian Sigma Model
We first compute the partition function for a free, periodic scalar field on M. Throughout this paper, M is a closed, oriented three-manifold, with Riemannian metric g.
The most basic topological invariant of M is the first Betti number b 1 , which is the dimension of the vector space H 1 (M) of harmonic one-forms on M. As for the other Betti numbers, trivially b 0 = b 3 = 1, and b 2 = b 1 by Poincaré duality.
Unlike instances of topological quantum field theory, the abelian quantum field theories here will definitely depend upon the choice of the Riemannian metric g. The most elementary invariant of the metric on M is the total volume, parametrized in terms of an overall length scale ℓ, As we perform computations, we will wish to keep track of the dependence on both the length scale ℓ and the electric coupling e 2 , which enters the fundamental duality relation in (1.6). This bookkeeping is easy, for under a scale transformation Because the abelian quantum field theories under consideration are free, they can always be defined so that the transformations in (2.2) and (2.4) preserve both the classical action and the quantum partition function on M. The two parameters e 2 and ℓ are then redundant, since either e 2 or ℓ can be scaled to unity with an appropriate choice of Λ ∈ R + . Nevertheless, I leave the dependence on both e 2 and ℓ explicit, and invariance under scaling will be a small check on our later formulas.

The Classical Sigma Model
Classically, a periodic scalar field φ on M simply describes a map from M to the As indicated on the right in (2.5), we interpret φ as an angular quantity, subject to The assumption in (2.5) that φ is valued in S 1 , as opposed to R, has important global consequences. Abstractly, a given choice for φ determines a point in the space X of all maps In general, X is not connected, but rather decomposes into components labelled by the homotopy class of the map φ. By standard facts in topology (see for instance Chapter 4.3 in [18]), homotopy classes of maps from M to S 1 are in one-to-one correspondence with cohomology classes in H 1 (M; Z). Under this correspondence, the cohomology class associated to a given map φ is the pullback to M under φ of a fixed generator for H 1 (S 1 ; Z) ≃ Z. Abusing notation slightly, I write this pullback as Throughout this paper, we will treat torsion in integral cohomology with care. By the Universal Coefficient Theorem, the cohomology group H 1 (M; Z) is generated freely over Z, without torsion. Thus H 1 (M; Z) is a lattice with rank b 1 , where the notation L merely serves as a convenient shorthand. As one might guess, the lattice L will play a prominent role in what follows.
Altogether, the space X in (2.7) decomposes into connected components labelled by a winding-number ω which is valued in the cohomology lattice L, (2.10) Here X ω consists of those sigma model maps which satisfy The free sigma model action for φ takes the standard form (2.12) In the second line of (2.12), we write the sigma model action in local coordinates on M, with the Einstein summation convention applied to the index 'm'. The factor of 1/4π in the normalization of I 0 is again a numerical convenience.
The sigma model action I 0 includes a prefactor which we will eventually identify with the electric coupling e 2 under duality. As for the discussion of the Maxwell action in Section 1, the overall dependence on e 2 in (2.12) is fixed by invariance under the scale transformations in (2.2) and (2.4). Under the scaling of the metric, the field φ is necessarily invariant, since any non-trivial scaling of φ would be incompatible with the fixed angular identification in (2.6). Otherwise, due to its implicit metric dependence, the dual two-form ⋆dφ scales as ⋆dφ → Λ ⋆dφ. 2 Because the coupling e 2 scales inversely to ⋆dφ, the sigma model action I 0 is thus invariant.
As usual, the metric on M induces an inner-product on the space Ω p (M) of smooth p-forms for each p = 0, . . . , 3, In terms of the L 2 inner-product, the sigma model action can be abbreviated (2.14) More or less immediately, the critical points of the free sigma model action in (2.14) are harmonic maps from M to S 1 , Because M is compact, any R-valued harmonic function on M is constant and hence unique up to normalization. For S 1 -valued harmonic functions as in (2.15), a roughly similar statement holds. First, by classical Hodge theory, each integral cohomology class ω ∈ L admits a unique harmonic representative with integral periods on M. Abusing notation slightly, I also use ω to denote the corresponding harmonic one-form, which depends upon the Riemannian metric on M. If φ cl ∈ X ω is a circle-valued harmonic map with 2 The scaling of the two-form ⋆dφ is perhaps most easily examined in local coordinates, where ⋆dφ ≡ √ g ǫ mnp ∂ p φ dx m ∧dx n . Under the scale transformation g → Λ 2 g, the volume factor √ g transforms as √ g → Λ 3 √ g. On the other hand, ∂ p φ = g pq ∂ q φ scales as ∂ p φ → Λ −2 ∂ p φ. So ⋆dφ scales in total as ⋆dφ → Λ ⋆dφ.
winding-number ω, then necessarily φ cl is related to ω by This condition implies both that φ cl has winding-number ω and that φ cl is harmonic, since Given the integral harmonic form ω, the linear equation in (2.16) can always be solved and so determines φ cl up to the addition of a constant. As a result, the moduli space of harmonic maps with winding-number ω is a copy of S 1 .

A Topological Parameter for the Sigma Model
Given the decomposition for X in (2.10), we naturally extend the free sigma model action I 0 (φ) to include a topological term which is locally-constant on X and hence only sensitive to the winding-number ω.
The most obvious topological term for φ depends upon the choice of a de Rham cohomology class One might suppose that I β (φ) is the end of the story, since there are hardly any other topological couplings to write for the free sigma model. However, this paper is about duality, and the choice of β in (2.18) does not respect the fundamental duality on M -to wit, Poincaré duality.
The most elegant formulation of the scalar partition function on M occurs when we introduce a second cohomological parameter dual to β. The dual parameter α is a harmonic one-form on M, 20) which couples linearly to the sigma model map φ via The choice of Riemannian metric on M enters both the definition of α as a harmonic one-form and the description of the coupling I α (φ) in (2.21). However, precisely because ⋆α is closed, the value of I α (φ) does not change under variations of φ, so I α (φ) is also a locally-constant function on the space X of sigma model maps. Hence I α (φ) does not alter the harmonic equation of motion for φ either. We include the prefactor of e 2 in (2.21) to ensure that I α (φ) is invariant under the scale transformations in (2.2) and (2.4).
Including both the cohomological parameters α and β, the total sigma model action becomes Though α and β enter the total action similarly, an asymmetry exists in our description of these parameters. The parameter α is a definite harmonic one-form on M, whereas β is any closed two-form representing the given cohomology class. A loose theme running throughout this work and its companion [1] is the interpretation of abelian duality on M as a kind of quantum Poincaré duality, which here will exchange α and β. To make sense of this exchange, though, we will need to select a definite two-form on M to represent the cohomology class of β. Because α ∈ H 1 (M) is already harmonic, we also use the metric on M to determine a harmonic representative β ∈ H 2 (M) for the cohomology class in (2.18).
With this choice, the sigma model action in (2.22) only depends on the parameters α and β in the holomorphic combination where ( · , · ) is the L 2 inner-product, and · , · is the canonical intersection pairing, (2.25)

Computing the Partition Function
We now evaluate the partition function for the periodic scalar field on M using the path integral presentation Here I indicate the explicit dependence of the partition function on the holomorphic parameter γ ∈ H 2 C (M), and I leave implicit the combined dependence on the coupling e 2 and the Riemannian metric g.
Because the sigma model configuration space X decomposes into components labelled by the winding-number ω ∈ L, the sigma model path integral includes a sum over the cohomology lattice L, followed by an integral over each component X ω ⊂ X . With the free sigma model action in (2.24), those integrals are all Gaussian and hold no particular mysteries. However, we would like to assign a definite normalization to Z M (γ), and for that goal, we must be careful about the normalization of the sigma model measure Dφ itself.

More about the Sigma Model Measure
Although the configuration space X = Map(M, S 1 ) decomposes into an infinite number of components X ω labelled by the winding-number ω, each component can be identified with the distinguished component X 0 , which consists of maps with trivial winding. To make the identification X ω ≃ X 0 , we select a basepoint Φ ω ∈ X ω , corresponding to a fiducial map with winding-number ω. Given the fiducial map Φ ω , any other map φ ∈ X ω with the same winding-number can be written uniquely as a sum where ψ : M → S 1 is a sigma model map with vanishing winding. The correspondence between φ and ψ in (2.27) provides the desired identification of X ω and X 0 . As an immediate corollary, if we wish to characterize the sigma model measure Dφ on X ω , we need only characterize it on X 0 . We will characterize the measure on X 0 momentarily, but let us first make a definite choice for the fiducial map Φ ω ∈ X ω in (2.27). Our choice will depend upon the Riemannian metric g on M, as well as the data of a point p ∈ M . Using the metric, we first impose the condition that Φ ω : M → S 1 be harmonic, or equivalently The condition in (2.28) determines Φ ω up to the addition of a constant. To fix the constant, we next impose Together, the conditions in (2.28) and (2.29) determine the map Φ ω uniquely. At any given point φ ∈ X 0 , the tangent space to X 0 at φ is simply the space of real-valued functions Ω 0 (M), (2.30) Indeed globally, The metric on M immediately induces a metric on X 0 , given by The appearance of the L 2 -norm on Ω 0 (M) should come as no surprise, but the coupling-dependent prefactor in (2.32) may be one. Under a scaling of the metric g → Λ 2 g, the standard L 2 -norm on Ω 0 (M) scales according to the volume of M, Since e 2 → Λ −1 e 2 according to (2.4), the prefactor of e 6 in ||δφ|| 2 X 0 ensures invariance of the metric on X 0 under scaling. The remaining factors of 2π in (2.32) appear by convention.
Once X 0 carries a Riemannian structure, we take Dφ to be the corresponding Riemannian measure. Under the identification X ω ≃ X 0 , the sigma model measure then extends to all of the configuration space X . Finally, by construction Dφ is invariant under translations by elements in Ω 0 (M). Hence Dφ does not actually depend upon the particular choice of basepoint Φ ω ∈ X ω specified in (2.28) and (2.29).
Though the description of Dφ in terms of the metric on X 0 is purely formal, we will see later that this description, formal or no, allows us to perform a precise accounting under duality of all coupling-dependent factors in the partition function. This accounting clarifies the results in [2].

A Sum over Windings
To evaluate the partition function on M, we first employ the componentwise identification X ω ≃ X 0 to rewrite the integration variable φ in (2.34) as the sum With the substitution in (2.35), the partition function can be computed in terms of a path integral over the distinguished component X 0 alone, In general, when making changes of variables in the path integral, one must be careful about Jacobians, but according to our preceding description of Dφ, the Jacobian for the substitution in (2.36) is unity. In terms of the harmonic map Φ ω and the homotopically-trivial map ψ, the sigma model action becomes (2.37) or more explicitly, (2.38) In passing to (2.38), I recall that the fiducial map satisfies dΦ ω = 2πω. Since both ω and γ are harmonic forms on M, the cross-terms in (2.38) which involve either ω or γ together with dψ vanish, so that (2.39) From the description of the sigma model action in (2.39), the partition function on M can be rewritten more explicitly as Because the argument of the exponential in (2.40) is a sum of terms which depend separately on the variables ω ∈ L and ψ ∈ X 0 , the partition function immediately factorizes, where ∆ M is given by a Gaussian integral over the space X 0 of homotopically-trivial maps ψ : M → S 1 , and Θ M (γ) is given by a discrete sum over the cohomology lattice L = H 1 (M; Z), Due to the Gaussian damping, the lattice sum which defines Θ M (γ) in (2.43) is convergent for all γ ∈ H 2 C (M). From the perspective of duality, the more interesting term in the factorization (2.41) is Θ M (γ), which carries the dependence on the holomorphic parameter γ and arises from the quantum sum over winding-sectors in the sigma model. Clearly Θ M (γ) is a theta-function attached to the three-manifold M though the cohomology lattice L = H 1 (M; Z), and wherever a theta-function appears, the modular group lurks. For the time being, though, I postpone discussion of Θ M (γ) and its role in duality until Section 4.

The Normalization of Z M (γ)
Finally, to fix the absolute normalization of the scalar partition function, we are left to compute the factor depending on the coupling e 2 and the Riemannian metric on M. Because the path integral in (2.44) is Gaussian, the only trick will be to keep track of factors associated to the normalization of the path integral measure Dψ.
As a first step, we expand ψ in an orthonormal basis for X 0 ≃ Ω 0 (M) mod 2π with respect to the metric in (2.32), Here eachf λ is a normalized eigenfunction with eigenvalue λ for the scalar Laplacian △ on M, More important is the coefficient of the constant function also chosen to have unit-norm in Ω 0 (M). Because ψ satisfies the periodicity condition ψ ∼ ψ + 2π, the coefficient ψ 0 of the constant modef 0 must have period Though we have fixed the periodicity of the sigma model field ψ to be independent of the coupling, the effective periodicity of the zero-mode ψ 0 does depend upon e 2 and diverges as e 2 → ∞ with the volume of M held fixed. In terms of the spectral decomposition (2.45) for ψ, the Gaussian path integral in Here ψ ′ indicates the orthocomplement to the constant mode in Ω 0 (M), and at least formally, Dψ ′ is shorthand for the product measure As standard, in obtaining (2.49) we integrate by parts to produce the scalar Laplacian △ = d † d acting on ψ ′ in the argument of the exponential.
Again with care for factors of e 2 , we apply the expansion of ψ ′ in (2.50) to simplify that argument, (2.52) In passing to the second line of (2.52), we observe that the eigenfunctionf λ has L 2 -norm (f λ ,f λ ) = 4π 2 /e 6 according to (2.46). So altogether, or upon evaluating the Gaussian integrals over each spectral coefficient ψ λ , (2.54) The slightly tricky prefactor (e 2 ℓ) 3/2 arises from the integral over the zero-mode ψ 0 with the periodicity in (2.48), and det ′ indicates the determinant with kernel omitted.
Some regularization method must be chosen to define the functional determinant in (2.54) as a real number, with zeta-function regularization being one possibility. See for instance [13,27] for explicit calculations of such zeta-regularized determinants on lens spaces M = S 3 /Z k , with the round metric inherited from S 3 . The expression for ∆ M in (2.54) makes manifest the fact that ∆ M is invariant under the simultaneous scalings of the metric and coupling in (2.2) and (2.4). In particular, one can easily check that the operator △/e 4 is invariant under scaling, since △ scales with Λ as △ → Λ −2 △. However, the appearance of the coupling in the functional determinant det ′ (△/e 4 ) is slightly awkward, and if one wishes, the dependence on e 2 in ∆ M can be made more explicit by pulling e 2 out from the determinant. The same issue arises in Appendix A of [13], whose strategy of analysis we follow. In zeta-regularization, the functional determinant is defined in terms of the zetafunction for the scalar Laplacian △ on M, The sum over positive eigenvalues in (2.55) is convergent when the real part of s is sufficiently large, and ζ △ (s) is defined for other values of s by analytic continuation.
By standard manipulations, the functional determinant of △ is defined in terms of the derivative of ζ △ (s) at s = 0, Because we are interested in the determinant of the operator △/e 4 , we instead consider (2.57) Then similarly, On the other hand, directly from (2.57), from which we obtain the relation The value of ζ △ (s) at s = 0 can be interpreted as a regularized dimension for the non-zero eigenspace of the operator △. Very generally, if M is any compact manifold of odd dimension, and △ p is the de Rham Laplacian acting on the space Ω p (M) of p-forms, then the value of the associated zeta-function at s = 0 is given by (2.61) See Theorem 5.2 in [34] for a textbook proof of (2.61), which goes back to [24]. For us, the scalar Laplacian on M has a one-dimensional kernel, so ζ △ (0) = −1 in (2.60).
We thence obtain As a result, the normalization factor in (2.54) reduces to The linear dependence of ∆ M on e also follows by well-known physical arguments involving the counting of zero-modes for the Laplacian.

The Final Result
In summary, we have determined the scalar partition function on M to be The subscript serves to emphasize that △ 0 is the scalar Laplacian, acting on forms of degree zero. We will similarly meet the de Rham Laplacian △ 1 for one-forms when we consider abelian gauge theory in Section 3.

Analysis of the Abelian Gauge Theory
Just as we computed the partition function for a periodic scalar field, we now compute the partition function for an abelian gauge field on the closed three-manifold M.

The Classical Maxwell Theory
Classically, the gauge field A is a connection on a fixed principal U(1)-bundle P over M, The typical three-manifold admits many choices for the bundle P , whose topological type is characterized by the first Chern class Via (3.2), the possible types of U(1)-bundles on M are in one-to-one correspondence with elements of H 2 (M; Z).
In general, the abelian group H 2 (M; Z) contains torsion, and c 1 (P ) may be a torsion class, of finite-order in H 2 (M; Z). Throughout this work, we will be fastidious about torsion, so we recall the exact sequence Here is dual under the intersection pairing to the quotient lattice The lattice L ∨ also embeds in the vector space H 2 (M) as the set of harmonic twoforms with integral periods on M, We will often assume implicitly the embedding in (3.5).
Though the exact sequence in (3.3) can be split, the sequence does not split in any natural way, so we should not think about L ∨ as a subgroup of H 2 (M; Z). However, we can always consider the reduction of any class in H 2 (M; Z) modulo torsion, to obtain a class valued in L ∨ . Throughout this paper, we normalize the gauge field A so that the reduction of c 1 (P ) ∈ H 2 (M; Z) modulo torsion admits the de Rham As usual, F A = dA is the curvature of the connection. With the normalization in (3.6), homotopically non-trivial gauge transformations act on A by shifts Such shifts preserve the holonomy of A, as measured physically by the Wilson loop operator W n (C) attached to an oriented closed curve C ⊂ M , We have already seen that the lattice L plays an important role in characterizing the winding-number of the circle-valued map φ : M → S 1 . The dual lattice L ∨ plays a similar role for Maxwell theory, since L ∨ determines the topology of the U(1)-bundle P , at least up to torsion. Given the canonical pairing between L and L ∨ , one might wonder whether it is even necessary in the context of abelian duality to consider bundles for which c 1 (P ) is torsion. As we will see in Section 4, following the original observation in [2], a precise understanding of duality indeed requires that we consider all possibilities for c 1 (P ) in H 2 (M; Z), including torsion classes.
Once the bundle P is fixed, we introduce the Maxwell action Here e 2 is the electric coupling, and the factor of 1/4π is required to match our previous conventions for the scalar field. In the second line of (3.9), we recall the component expansion of the Maxwell action in local coordinates on M.
Critical points of I 0 (A) correspond to connections A cl on P whose curvatures Trivially dF A cl = 0, so any connection which solves the Maxwell equation on M has harmonic curvature The integrality condition in (3.6) then determines F A cl uniquely according to the topological type of P . For future reference, we set where we have been careful to divide by the large gauge equivalences in (3.7).
For each U(1)-bundle P over M, the moduli space of classical solutions to the Maxwell equation is simply a copy of the torus J M in (3.13). As mentioned in the Introduction, this moduli space bears no resemblance to the moduli space of harmonic maps φ : M → S 1 , which is instead a copy of S 1 for each winding-sector. Abelian duality on M must therefore involve a non-trivial quantum equivalence, even though the field theories involved are free. I will develop this theme further in [1], where I discuss the canonical quantization of these theories.

Adding Topological Couplings
Just as for the periodic scalar field in Section 2, we now extend the classical Maxwell action to include additional couplings which will be topological in the sense that they do not change the classical Maxwell equation in (3.10).
The more obvious topological coupling depends upon the choice of a de Rham cohomology class for which we introduce the pairing Because both α and F A are closed forms, the pairing in (3.15) depends only on the respective cohomology classes of those forms. In particular, via the identification (3.6), the value of I α (A) depends only on the image of the Chern class c 1 (P ) in the lattice L ∨ and is insensitive to torsion. The sign in (3.15) is just a convention that will make the duality formulas in Section 4 more elegant.
Dually, we also introduce a harmonic two-form β, with coupling The harmonic condition ensures that ⋆β is a closed one-form, so that the value of I β (A) also depends only on the Chern class of the bundle P . We will eventually identify α and β with the same parameters which we introduced for the periodic scalar field in Section 2. However, the defining conditions on α and β are now reversed. In Section 2, the one-form α was required to be harmonic and β was an arbitrary closed two-form, whereas here β is harmonic and α is an arbitrary closed one-form.
Including the parameters α and β, the Maxwell action on M becomes (3.18) Without loss, I select a harmonic representative for α and introduce the complex harmonic one-form appearing holomorphically in (3.18), Just as for the harmonic two-form γ in (2.23), the harmonic one-form ζ is invariant under the combined scale transformations in (2.2) and (2.4). As one can easily check, the complex harmonic forms γ and ζ are related by The total Maxwell action in (3.18) can then be written concisely in terms of the L 2 and intersection pairings, Abelian Duality at Level k As a special feature of abelian gauge theory in three dimensions, we can add to the Maxwell action (3.21) a Chern-Simons term proportional to We follow the standard practice in writing the Chern-Simons functional with respect to a local trivialization for the bundle P . However, because A can be a connection on a non-trivial U(1)-bundle P over M, the global meaning of the trivialized form (3.22) of the Chern-Simons functional may be unclear.
For an alternative presentation, one can always choose a four-manifold X such that X bounds M and the U(1)-bundle P extends from M to X. The existence of X relies upon the vanishing of H 3 (BU(1)) and is discussed thoroughly in [9]. Given X, the Chern-Simons functional on M can be rewritten in the gauge-invariant fashion Since F A /2π is an integral two-form on any closed manifold, and since the intersection pairing is also integral, the global expression for the Chern-Simons functional in (3.23) shows that the value of CS(A) is well-defined modulo 2π.
If we wish, we can then extend the Maxwell action (3.21) on M to a Maxwell-Chern-Simons action at level k ∈ Z, Just as the parameter ζ is naturally related under duality to the parameter γ for the periodic scalar field φ, one can ask about the dual interpretation for the Chern-Simons level k. One standard answer to this question would be to say that the Chern-Simons level has no simple, local description in terms of the periodic scalar field. Strictly speaking, this answer is correct, but it is unsatisfying. Another standard answer, at least when M = R 3 , would be to say that Maxwell-Chern-Simons theory at level k is equivalent [8,21] to the 'self-dual' model [35] of a massive, non-gaugeinvariant Proca vector field with Chern-Simons term. Strictly speaking, this answer is also correct, but it is not correct for a general three-manifold.
I will discuss elsewhere a better, more global answer to the question "What is the dual of the Chern-Simons level?" The answer turns out to be most clear with the Hamiltonian formalism developed in [1]. For the present, I just set k = 0 and work with only the pure Maxwell theory on M.

Computing the Partition Function
The Maxwell partition function on M can be evaluated in a manner very similar to the evaluation of the scalar partition in Section 2.2. So I will be relatively brief.
The Maxwell partition function admits the formal path integral presentation Evidently, the partition function involves both a sum over the topological type of the principal U(1)-bundle P as well as an integral over the affine space A(P ) of all connections on P . Due to the gauge invariance of the Maxwell action, we divide the path integral by the volume of the group G(P ) of gauge transformations on P . Geometrically, G(P ) can be identified with the group of maps from M to U(1), acting on P by bundle automorphisms. Again, the most delicate aspect of our computation will be to fix the normalization of the partition function, for which we must be precise about the meaning of the measure DA on A(P ).

More about the Maxwell Measure
Let P 0 be the trivial U(1)-bundle over M. For every other bundle P , the space A(P ) can be identified with A(P 0 ) as soon as we pick a basepoint in A(P ), which will correspond geometrically to a fiducial connection on P . In close analogy to the choice of the fiducial harmonic map in Section 2, we take the fiducial connection A P ∈ A(P ) to possess harmonic curvature The arbitrary connection A ∈ A(P ) can then be expressed as a sum where η is a connection on the trivial bundle. The correspondence between A and η in (3.28) provides the requisite identification A(P ) ≃ A(P 0 ) for each U(1)-bundle P . Given this identification, we need only describe the measure DA for connections on the trivial bundle over M.
The fiducial connection on the trivial bundle P 0 is flat, from which we obtain a trivialization of P 0 . We may thus regard connections on P 0 as ordinary one-forms on M. Following the same philosophy from Section 2, we characterize the measure DA on A(P 0 ) ≃ Ω 1 (M) as the Riemannian measure induced from the L 2 -norm (3.30) We have already introduced a suitable Riemannian metric on Ω 0 (M) in (2.32). We extend this metric in a translation-invariant fashion over G(P ), and we take Vol(G(P )) to be the formal Riemannian volume. This volume is independent of the bundle P .

A Sum over Fluxes
Once we substitute for A as in (3.28), the Maxwell partition function can be rewritten as a path integral over connections on the trivial bundle P 0 alone, In terms of the fiducial connection A P and the one-form η, the Maxwell action becomes 32) or more explicitly, In passing to (3.33), we recall the formula for the harmonic curvature F A P in (3.27).
Since λ and ζ are harmonic, all cross-terms which involve λ or ζ together with dη vanish, and  Like the partition function (2.41) of the periodic scalar field, the Maxwell partition function also factorizes, Here ∆ ∨ M is given by a path integral over the affine space A(P 0 ), and Θ ∨ M (ζ) is given by a sum over fluxes in the quotient lattice L ∨ = H 2 (M; Z) free , The more interesting factor in the Maxwell partition function is Θ ∨ M (ζ), which is yet another theta-function attached to the three-manifold M. Including our previous work from Section 2, we now have a dual pair of lattices L and L ∨ , as well as a pair of theta-functions Θ M and Θ ∨ M . As one might guess, and as we will demonstrate explicitly in Section 4, Θ M and Θ ∨ M are related by a modular transformation. Before we discuss modular issues though, let us finish the computation of the Maxwell partition function on M.
To fix the absolute normalization of the Maxwell partition function, which will depend upon the electric coupling e 2 and the Riemannian metric g, we are left to evaluate the Gaussian path integral (3.40) Here I abbreviate A ≡ A(P 0 ) and G ≡ G(P 0 ), since we will only consider gauge theory on the trivial U(1)-bundle P 0 for the remainder of the discussion. Also, to orient the reader, I recall that the one-form η in (3.40) is effectively identified with the gauge field A after the bundle P 0 has been trivialized. The computation of ∆ ∨ M is slightly more delicate than the analogous computation for the periodic scalar field, due to the gauge symmetry in the current problem. Because of the gauge symmetry, the argument of the exponential in (3.40) vanishes for any η ∈ Ω 1 (M) of the form η = dϕ, with ϕ ∈ Ω 0 (M). Intrinsically, ϕ can be interpreted as element in the Lie algebra of the group G, and we are simply observing that the Maxwell action is degenerate along orbits of G.
To account for the degeneracy of the integrand in (3.40), we employ the standard BRST technique to fix the gauge symmetry. We cannot possibly fix the action for the full group G of all gauge transformations, since any gauge transformation generated by a constant function ϕ 0 ∈ R acts everywhere trivially on A. Instead, we pick a point p ∈ M , and we consider the subgroup G p ⊂ G of gauge transformations which are based at p. An alternative treatment would involve the introduction of BRST ghosts-for-ghosts to deal with the constant gauge transformations, but I believe that working with the based gauge group is conceptually simpler for this example.
By definition, elements in G p are gauge transformations which are the identity at the point p, and elements in the Lie algebra of G p are functions ϕ ∈ Ω 0 (M) which vanish at p, Due to the condition in (3.41), the identity is the only constant gauge transformation in G p , and the quotient of G by G p is the group By assumption, c and c are anti-commuting, Grassmann scalar fields, and h is a commuting scalar field. If one wishes, the vanishing constraint in (3.43) amounts to the insertion of a local operator O(p), whose role is to absorb the zero-modes of (c, c, h) which would otherwise be present in the BRST path integral. To achieve the most elegant geometric formulation of the BRST procedure, I will depart somewhat from custom and introduce an extra bosonic field u, which will be an element in the based group G p . Equivalently, u is a sigma model map from M to U(1), satisfying Together, the pair (u, h) describes the cotangent bundle of the group G p , (3.45) and the anti-commuting scalars (c, c) can be interpreted as one-forms on T * G p . 3 The nilpotent BRST operator Q acts infinitesimally on the set of fields (η, h, u, c, c) according to Manifestly Q 2 = 0, and Q annihilates the Maxwell action in (3.40) by virtue of gauge invariance.
Using the BRST charge Q, we produce a gauge-fixing action I g.f. appropriate for harmonic gauge d † η = 0, Otherwise, the core of the BRST identity (3.49) is not so much the appearance of the constant '1' on the left-hand side of the identity as the independence of the righthand side on the one-form η which enters the gauge-fixing action I g.f. in (3.48). The latter property is really a property of harmonic gauge: for any one-form η, a gauge transformation by a unique u ∈ G p exists so that d † η u = 0. Given this statement, which follows from standard Hodge theory, the path integral over G p washes out all dependence on η in the integrand of (3.49).
Using the BRST identity, we enlarge the path integral which describes ∆ ∨ M in (3.40) to a path integral over the product A × T * G p , The Gaussian integral over the auxiliary scalar h can be evaluated immediately, after which the normalization factor becomes

(3.51)
To deal with the appearance of u in the integrand of (3.51), we note trivially (dη, dη) = (dη u , dη u ) , (3.52) due to gauge-invariance of the Maxwell action. Gauge-invariance for the measure on A similarly implies Dη = Dη u . As a result, η can be replaced by its gauge transform η u in (3.51), After a change-of-variables from η u back to η, the auxiliary field u ∈ G p decouples from the integrand in (3.53), so that

(3.54)
Since u appears nowhere in the integrand of (3.54), the path integral over u just contributes a factor of the group volume Vol(G p ), In passing from (3.54) to (3.55), we also integrate by parts to produce the de Rham Although both G p and G have infinite dimension, the quotient G/G p = U(1) has finite dimension, and the ratio of volumes in (3.55) is well-defined, .
· Vol(J M ) . The same factor appears in (2.54), for exactly the same reason.
To determine the volume of J M , we recall that J M is concretely the quotient Let {e 1 , · · · , e b 1 } be a basis of integral generators for L, so that J M becomes isomorphic to The extra factor of (e/2π) b 1 in the first line of (3.65) occurs due to the corresponding factor in the scale-invariant norm on Ω 1 (M) in (3.29). Finally, we extract factors of e 2 from the functional determinants in (3.58) using the zeta-function relation in (2.60). According to the general formula (2.61) for the value of the zeta-function at s = 0, We use the formulas in (3.60), (3.65), and (3.67) to simplify our result in (3.58), The overall dependence of ∆ ∨ M on the electric coupling as e 1−b 1 can also be understood more physically (and perhaps more simply) by counting modes of the gauge field A modulo gauge equivalence. The latter perspective is taken for the computations in [2] and [39].

The Final Result
In summary, we have determined the Maxwell partition function on M to be

Modularity, Duality, and All That
Having evaluated the respective scalar and Maxwell partition functions, we now compare these results. Both Z M (γ) and Z ∨ M (ζ) factorize, and we will start by comparing the respective theta-functions Θ M (γ) and Θ ∨ M (ζ) associated to the three-manifold M. See [26] or Ch. 2 in [15] for an introduction to the geometry of theta-functions, the basics of which will be useful here.

A Theta-Function for Three-Manifolds
The hallmark of any theta-function is quasi-periodic behavior under integral shifts in the argument, and both Θ M (γ) and Θ ∨ M (ζ) are easily seen to be quasi-periodic with respect to shifts in the variables γ and ζ. Recall that L is dual to the quotient lattice L ∨ = H 2 (M; Z) free . If ν ∈ L ∨ is such an integral two-form, then manifestly Similarly, if µ ∈ L is an integral one-form, then The transformation formula in (4.4) follows by a standard calculation from the lattice sum in (4.2), (4.5) In passing from the second to the third line of (4.5), I shift the summand to ω ′ = ω+µ, since µ ∈ L is also integral. Together, the transformation laws in The theta-function Θ M (γ) definitely depends on the Riemannian structure on M, but this dependence occurs only through the matrix of L 2 inner-products in (3.64), where we have selected an integral basis for L ≃ Ze 1 ⊕ · · · ⊕ Ze b 1 . In terms of the basis for L and the b 1 × b 1 matrix Q, we can write Θ M (γ) very concretely as a sum over a vector n ∈ Z b 1 of integers, In this expression, γ j ∈ C for j = 1, . . . , b 1 are the components of the complex two-form γ, expressed dually with respect to the basis for L, For instance, if M has the rational homology of S 1 × S 2 , then Θ M (γ) reduces to the classical Jacobi theta-function Θ(z; τ ) = +∞ n=−∞ exp πi n 2 τ + 2πi n z , (4.11) evaluated at the purely-imaginary complex structure Here R is a length scale naturally identified with the radius of S 1 when M = S 1 × S 2 .
More generally, if we introduce the standard multi-variable extension of (4.11), where Ω is a complex matrix with positive-definite imaginary part, then the geometric theta-function Θ M (γ) in (4.9) agrees with the classical theta-function Θ( z; Ω) under the assignments Ω = i e 2 Q , z = γ 2π . (4.14) The Modular Transform of Θ M (γ) Since Θ M (γ) agrees with the classical Jacobi theta-function Θ( z; Ω) when the period matrix Ω is imaginary, Θ M (γ) also inherits the well-known modular properties of Θ( z; Ω). A concise exposition of the latter can be found in Ch. 2.5 of [26], whose notation I follow.
Not surprising for our discussion of abelian duality, the most important modular property will be the transformation of Θ M (γ) under the analogue of the S-duality τ → −1/τ , acting here on the period matrix Ω by If Ω is purely imaginary as in (4.14), then this feature is preserved under (4.15), so that S-duality also acts on the geometric theta-function Θ M (γ) by the inversion Applied to the concrete description of Θ M (γ) in (4.9), this identity implies We evaluate the Gaussian integral over x ∈ R b 1 in the last line of (4.18) to obtain A more geometric interpretation for the right-hand side of (4.19) follows once we recognize Q −1 as the matrix whose elements encode the L 2 inner-products of the basis for L ∨ which is dual to the chosen basis for L, Here in (4.20) we introduce the Kronecker-delta, and we observe that integrality for the dual basis e * 1 , · · · , e * b 1 of L ∨ is a non-trivial consequence of Poincaré duality on M. Otherwise, the interpretation for Q −1 in (4.21) derives from the tautological relation As a result of (4.19) and (4.21), the geometric theta-function Θ M (γ) on M can be written not only in terms of a sum over the lattice L, but also in terms of a sum over the dual lattice L ∨ , where ( · , · ) now indicates the L 2 -norm on L ∨ ⊂ H 2 (M).
Of course, we also recall the description of the other theta-function Θ ∨ M (ζ) which enters the Maxwell partition function, Comparing (4.23) and (4.24), we see that (4.25) The identification ζ = i ⋆γ/e 2 in the argument of Θ ∨ M agrees with our conventions for α and β in Sections 2 and 3.

The Role of Torsion
The we compute the ratio Thus, We now reach the most important question in the present paper.
When is the ratio of partition functions in (4.29) equal to one?
If Z M (γ) is to be equal to Z ∨ M (ζ), then evidently γ = ζ = 0 in (4.29). In that case, the expression on the right-hand side of (4.29) does not depend of the Maxwell coupling e 2 , and the ratio reduces to the product

Reidemeister vs Ray-Singer Torsion
Before discussing τ M , let me briefly recall a few facts about Reidemeister torsion.
Nice expositions on Reidemeister torsion can be found in [12,23,28], and a prominent application of these ideas to gauge theory on a Riemann surface appears in [38].
Here I follow the presentation of Freed in [12], specialized to dimension three for concreteness.
The Reidemeister torsion is a combinatorial invariant of M, defined in terms of the chain complex C • associated to a given triangulation (or cellular structure) on M, We work with real coefficients throughout, and the homology of this chain complex is H * (M; R). Each C j for j = 0, . . . , 3 is a finite-dimensional vector space with a distinguished set of generators, the elementary simplices in M. Because the vector space C j has a basis, C j also has an associated metric, for which the generating simplices are orthonormal. Accompanying the metric on C j is a volume form ν j ∈ (det C j ) −1 . Here det C j = top C j indicates the top exterior power. As standard in this business, we will not worry about orientations or signs; by convention, the torsion will be positive.
I first describe the torsion assuming the complex C • to be acyclic, with trivial homology. Let k j = dim Im ∂ : C j → C j−1 , and pick an element s j ∈ k j C j so that ∂s j = 0. We now consider the following element in the alternating tensor product, The element u is independent of the choices of the s j , so we can define the torsion of the acyclic complex C • as Of course, in the geometric situation C • always has non-trivial homology, since H 0 (M) = H 3 (M) = Z for a closed, orientable three-manifold. To define τ (C • ) more generally when C • has homology, we split C • as C • = C ′ • ⊕ C ′′ • , where C ′ • is acyclic and the differential on C ′′ • vanishes, ∂ . We now apply the preceding construction to the acyclic summand C ′ • , with k j = dim Im ∂ : The torsion τ (C • ) is then defined as the element Let us be more explicit about where τ M in (4.35) is valued. In dimension three, the dual of the determinant line is given (with the obvious abbreviations) by where we apply the canonical duality between H * (M; R) and H * (M; R) in the second step. But Poincaré duality on M also implies the isomorphisms   41) and the equality between the combinatorial (4.35) and analytic (4.40) descriptions of τ M is a consequence of the Cheeger-Müller theorem [6,7,25] relating Reidemeister to Ray-Singer torsion.
As the reader may note, the Reidemeister torsion τ M and the Ray-Singer torsion T M are not precisely equal in our situation, but instead obey The correction factor det Q/ℓ 3 is discussed in Appendix B of [13] and arises due to and Here A t 1 is the transpose of A 1 . Together, the relations in (4.43), (4.44), and (4.46) produce the metric-dependent correction factor in (4.42).

Duality for the Partition Function
To summarize, the ratio of the Maxwell to scalar partition function on M is a topological invariant, Compare also to Theorem 2.39 in [28] and the surrounding discussion of the Alexander formula. 5 As a result, abelian duality for the partition function holds globally in the most naive sense, with This result strengthens the conclusions in [2] beyond the case b 1 = 0.
When M is a rational homology sphere, with b 1 = 0, a short proof of the formula (4.48) for τ M appears in [41], but the proof can be easily generalized to the arbitrary closed, orientable three-manifold. Such a manifold can always be given a cellular structure with exactly one 0-cell and one 3-cell, so that the chain complex C • in (4.31) takes the concrete form Geometrically, a chain complex of the form in (4.50) is naturally associated to any 5 I thank Maxim Braverman for pointing out this formula to me.
Heegaard decomposition of M = H 1 ∪ H 2 into handlebodies H 1 and H 2 . Because H 0 (M) = H 3 (M) = Z, the indicated maps in (4.50) vanish, and Poincaré duality otherwise implies that the chain groups satisfy C 1 , C 2 ≃ Z N for some N. Once we select integral generators {v 1 , · · · , v N } and {w 1 , · · · , w N } for C 1 and C 2 to realize the isomorphism with Z N , the non-trivial differential ∂ in (4.50) can be identified concretely with a square, N × N integer matrix. We first consider the simpler case that b 1 = 0. Necessarily, ∂ is injective and has full rank over R. . To treat C 1 and C 2 , we trivialize the determinant lines det C 1 , det C 2 ≃ R with the volume forms ν 1 = v 1 ∧ · · · ∧v N and ν 2 = w 1 ∧ · · · ∧w N . For any choice of s 2 ∈ det C 2 , the formula for τ M in (4.34) and (4.35) then reduces to the ratio (4.52) When b 1 is non-vanishing, the ideas leading to (4.48) are much the same, except for the fact that the determinant of ∂ now vanishes. By assumption, ∂ : C 2 → C 1 has a non-trivial kernel, with H 2 (M) = ker ∂. Clearly H 2 (M) ≃ Z b 1 is generated freely, and we assume without loss that the subset {w 1 , · · · , w b 1 } generates the kernel of ∂ in C 2 . Hence {w 1 , · · · , w b 1 } represent integral generators for H 2 (M). On the other hand, H 1 (M) = C 1 / Im ∂ may still contain a torsion subgroup H 1 (M) tors ≃ H 2 (M; Z) tors , with H 1 (M; R) ≃ R b 1 . Again without loss, we assume that the generating subset {v 1 , · · · , v b 1 } spans the cokernel of ∂ over R. Thus {v 1 , · · · , v b 1 } represent integral generators for H 1 (M) modulo torsion.
As before, the number of elements in H 1 (M) tors can be interpreted geometrically as the volume of the (N − b 1 )-dimensional simplex in R N spanned by the image of ∂ on the integral generators for C 2 . To compute that volume, we extend ∂ linearly to a non-degenerate integral map ∂ : C 2 → C 1 via the assignments as well as ∂ = ∂ for all other generators of C 2 . By construction, the determinant of ∂ is non-vanishing and computes the requisite volume, Here C ′ 2 is the (N − b 1 )-dimensional subspace of C 2 spanned by all generators other than {w 1 , · · · , w b 1 }, and s 2 is any non-vanishing element in the determinant line det C ′ 2 . Comparing (4.54) and (4.55), we obtain the promised reciprocal relation between the torsion invariants Tor M and τ M .

Path Integral Explanation
By somewhat laborious direct calculations, we have obtained the dual identity The relation between the scalar and the Maxwell partition functions can also be derived more economically by formal path integral manipulations of the sort in [32,39]. This approach to abelian duality in three dimensions appears already in Lecture 8 of [40], but I take the opportunity now to review it. As one application, the path integral perspective on duality neatly explains the otherwise anomalous exponential factor involving γ in (5.1), which arose from the modular transformation of Θ M (γ) in Section 4. In preparation for the Hamiltonian analysis in [1], I conclude Section 5 with a discussion of duality for three natural classes of operators on M.

Duality for the Partition Function
We start with the path integral which describes the scalar partition function on M, where the sigma model action for φ : M → S 1 is given by To find a dual reformulation for the path integral, we enlarge the space of fields to include a connection B on the trivial U(1)-bundle P 0 over M. Under a homotopicallytrivial gauge transformation, parametrized by a function f ∈ Ω 0 (M), the pair (φ, B) transforms by As a result, the combination d B φ = dφ + B is gauge-invariant. For future reference, G B will denote the group of gauge transformations acting on the pair (φ, B) ∈ X ω × A(P 0 ). Finally, the sigma model action in (5.3) can be promoted to a gauge-invariant action for the pair (φ, B) by substituting the covariant derivative d B φ everywhere for dφ, The gauged sigma model for the pair (φ, B) with the classical action in (5.5) cannot be the whole story, because we must also incorporate the Maxwell gauge field A somehow. So we let P λ be the U(1)-bundle with Chern class λ ∈ H 2 (M; Z), and we let A be a connection on P λ . To couple A to the pair (φ, B), we consider the mixed The latter property is perhaps most transparent when CS(A, B) is considered via bordism. Let X be a four-manifold bounding M, over which A and B extend. Then alternatively,

Chern-Simons interaction
Integrality of both F A /2π and F B /2π ensures that the intersection pairing in (5.8) is well-defined modulo 2π, regardless of the choice of X. We also see that the mixed Chern-Simons interaction in (5.6) occurs at level one, the minimum for gaugeinvariance in the absence of additional geometric structure (eg. a spin structure) on M.
We now couple the connection A to the pair (φ, B) through the classical action

)-bundle on which
A is a connection. We do not sum over the topological type of the bundle for the auxiliary connection B, though, for the following reason. Since φ is assumed to be defined everywhere on M, the exponential e iφ provides a global section of the U(1)-bundle P 0 on which B is a connection. This section trivializes P 0 , which must therefore be a trivial bundle. 6 The generalized partition function in (5.10) can be studied in two ways.
First, since the connection A enters the classical action in (5.9) linearly through the Chern-Simons coupling CS (A, B), the path integral over A can be performed directly. As argued carefully in [41], the result is simply a delta-function that sets B to zero modulo gauge-equivalence, DA exp i CS (A, B) . (5.11) To explain this identity, we again decompose the arbitrary connection A on the bundle P λ as a sum A = A λ + η , (5.12) where A λ is a fiducial connection with harmonic curvature 2πλ, and η (like B) is a connection on the trivial bundle P 0 . After we substitute for A in (5.12) and integrate by parts, the Chern-Simons pairing becomes Up to normalization, the path integral over η produces a formal delta-function that sets F B = 0. The remaining sum over λ ∈ H 2 (M; Z) produces a second delta-function that requires B to have trivial holonomy on M. Hence B is gauge-equivalent to zero. With a bit more work, one can verify [41] that the coefficient of the delta-function in (5.11) is precisely one, but I omit those details here. After applying the identity in (5.11) to the generalized partition function in (5.10), we find (5.14) Because B must be gauge-trivial due to the delta-function in the integrand of (5.14), we can set B = 0 by an appropriate gauge transformation. The path integral over B then contributes Vol(G B ) to cancel the prefactor in (5.14), and we obtain the simple result Z M (γ) = The path integral over the auxiliary connection B in (5.16) and (5.17) is yet another Gaussian integral, of a much simpler form than the Gaussian integrals which we analyzed in Sections 2 and 3. We immediately perform that integral to obtain a reformulation of Z M (γ) involving only the Maxwell gauge field, To determine the action for A, we substitute the classical value for B = (i/e 2 )(F A + γ) into (5.17), Comparing the classical action for A in (5.19) to the corresponding action (3.21) from Section 3, we deduce Since Z M is equal to Z M , this relation reproduces (5.1).

Duality for Operators
To conclude, let us review the dual descriptions for three natural classes of operators on the three-manifold M. For simplicity in the following, I set the cohomological parameters γ and ζ to zero.

Some Local and Non-Local Operators
Of the three operators that we consider, two are well-known: the vertex operator and the Wilson loop operator. The vertex operator is the local operator described classically in the sigma model by implies H 2 (M, C) ≃ H 1 (C) = Z. By assumption, the image ∂ * [Σ] generates H 1 (C), so [Σ] is uniquely determined once the orientation of C is fixed. Thus when M is a rational homology sphere, the choice of Seifert surface can be omitted from our notation for the fractional Wilson loop operator, and we simply write Both the vertex operator V k (p) and the Wilson loop operator W n (C) depend upon the particular choices for the point p ∈ M and the curve C ⊂ M . By contrast, the third operator L α (C) will be homological, depending only upon the class [C] ∈ H 1 (M) of the closed curve. In terms of the periodic scalar field φ, Because the periods of the one-form dφ are quantized in integral multiples of 2π, the expression on the right in (5.28) is invariant under a shift α → α + 2π. For this reason, α is best regarded as an angular parameter for the homological loop operator.

Vertex Operators and Monopoles
So far we have introduced three kinds of operators on M, labelled generally by parameters The vertex operator V k (p) and the homological loop operator L α (C) are respectively specified in (5.21) and (5.28) as classical functionals of the scalar field φ, whereas the Wilson loop operator W n (C) is a classical functional of the Maxwell gauge field A. Duality between the scalar and the Maxwell field theories on M implies not only a relation between partition functions, but also a correspondence between operators in each theory. So how do we describe the vertex operator V k (p) and the loop operator L α (C) dually in the language of Maxwell theory? And how do we describe the Wilson loop operator W n (C) in terms of the periodic scalar field?
In answer to all three questions, the duals of V k (p), W n (C), and L α (C) will be operators of disorder-type [19], which create singularities in the dual classical field.
To quickly explain both the notion of a disorder operator and its relevance for duality, let us derive the dual of the vertex operator V k (p).
At first glance, one might be tempted to consider the (unnormalized) expectation value (5.31) Unless k = 0, in which case V k (p) is the identity operator, V k (p) transforms with charge k under the global U(1) symmetry in (5.22). Hence trivially V k (p) = 0 , k = 0 , (5.32) due to cancellations in the integral over the constant mode of φ. So we cannot learn much by thinking about the expectation value of V k (p). Instead, to discuss a non-trivial expectation value, we pick distinct points p = q in M and consider vertex operators of opposite charge inserted at these points, Because the expectation value in (5.33) is invariant under the global U(1) symmetry, the expectation value need not vanish, and we can meaningfully ask for the dual description of (5.33) in terms of the Maxwell gauge field A.
Just as for the analysis in Section 5.1, the first step in dualizing the vertex operator path integral will be to promote the integrand in (5.33) to a functional of the pair (φ, B) which is invariant under the gauge transformation in (5.4). Since the vertex operators V k (p) and V −k (q) carry opposite charges, local gauge invariance can be achieved by introducing a Wilson line for the auxiliary gauge field B which runs between the vertex operators. Thus we choose an oriented curve Γ from q to p, and we consider the expectation value of the gauge-invariant composite 8  CS(A, B). Thus the path integral over A again produces a delta-function for B with support on gauge-trivial field configurations. After we integrate over B using the delta-function, the extended path integral in (5.35) reduces to the path integral over φ alone in (5.33), As a corollary, the extended path integral in (5.35) does not depend upon the choice of the curve Γ from q to p. Mimicking our previous analysis of the partition function, we alternately evaluate the path integral in (5.35) by using the local action of G B to set φ = 0, after which (5.35) reduces to a path integral involving only the gauge fields A and B, where as in (5.19), To evaluate the Gaussian integral in (5.37) further, we introduce a two-form δ Γ which has delta-function support along Γ and which represents the Poincaré dual of the curve, so that all terms in the argument of the exponential take the form of integrals over M. Because Γ is bounded by the points p and q, the two-form δ Γ is not closed but rather satisfies the distributional identity By definition, δ p and δ q are three-forms with delta-function support at the points p and q. The identity in (5.40) is most easily deduced as a consequence of Stokes' theorem for the path Γ. For if f ∈ Ω 0 (M) is any smooth function on M, then See for instance Chapter 3 of [15] for more about distributional differential forms like our δ Γ .
Via the definition in (5.39), the Gaussian integral over A and B takes precisely the same form as the generalized partition function in (5.16) with singular γ = 2πk δ Γ , (5.42) After performing the path integral over B, we obtain the desired reformulation where The interpretation of the modified Maxwell path integral in (5.43) is by now well understood. Due to the explicit delta-function in F A , the argument of the exponential diverges (and thus the integrand vanishes) unless F A itself has the appropriate singularity along Γ to cancel the delta-function in F A , where the ellipses indicate regular terms in F A . Thus, the insertion of the vertex operators V k (p) and V −k (q) in the scalar sigma model is interpreted dually as the instruction to perform the Maxwell path integral over connections with the specified singular behavior along Γ. Operators defined in this manner, as an instruction to perform the path integral over fields with given classical singularities, are said to be of disorder-type.
Because δ Γ is not closed, the Bianchi identity for F A is modified by the singularity in (5.45), where we apply the Stokes' identity in (5.40). Physically, the new source terms in the Bianchi identity for F A are interpreted as magnetic monopoles of charges ±k at the points p and q. Otherwise, so long as k is integral, the Dirac string singularity along the curve Γ is a gauge artifact. In light of (5.46), we see that the operator V k (p) itself is the monopole operator of charge k in the abelian gauge theory. By definition, the monopole operator of magnetic charge k is the local disorder operator which creates a curvature singularity in A at p of the form where r is a local radial coordinate centered at p. With this singularity, the integral of F A over any small sphere centered about p is equal to 2πk, as required by the Bianchi identity in (5.46). Also, with the given local behavior in (5.47), F A satisfies the classical source-free Maxwell equation d⋆F A = 0 on a punctured neighborhood of the point p.

Vortex Loops and Wilson Loops
The loop operators L α (C) and W n (C) can be followed through the duality in much the same fashion as the vertex operator V k (p). Very briefly, to dualize the homological loop operator L α (C) in (5.28), we consider its gauge-invariant extension in terms of the pair (φ, B), Here δ C is a two-form with delta-function support which represents the Poincaré dual of the closed curve C ⊂ M .
If we consider the expectation value of L α (C) in the extended theory of triples (φ, A, B) with total action (5.9), the path integral over A still provides a delta-function with support on gauge-trivial configurations for B. With this delta-function, B can then be gauged to zero to recover the expectation value for L α (C) in the theory of the periodic scalar field φ alone.
Alternatively, φ can be gauged to zero in the extended theory of triples (φ, A, B), after which we encounter a Gaussian integral over B taking precisely the same form as (5.42). Hence the operator L α (C) is interpreted in the dual Maxwell theory as a disorder operator which creates a curvature singularity along C, (5.49) This curvature singularity looks very much like the preceding singularity (5.45) which we interpreted in terms of monopoles. However, C is now closed, without boundary, and α is not a multiple of 2π. As a result, the physical interpretation of (5.49) is different.
In a small tubular neighborhood of C, the singularity in (5.49) implies that the gauge field A behaves as A = − α 2π dϑ + · · · , (5.50) where ϑ is an angular coordinate on the plane transverse to C, located at the origin.
Evidently, in the presence of the loop operator L α (C), the gauge field A has nontrivial monodromy Λ = exp(−i α) about any small curve which links C. Of course, the value of the monodromy only depends upon the value of α modulo 2π. Physically When α in (5.50) is an integral multiple of 2π, the singularity in A can be removed by such a gauge transformation, but not otherwise. For gauge theories in four dimensions, the codimension-two singularity in (5.50) defines the basic Gukov-Witten [17] surface operator. Hence the loop operator L α (C) in three dimensions can be interpreted as the reduction of a surface operator from four dimensions. From the purely three-dimensional perspective, L α (C) can be considered as a kind of monodromy or vortex loop.
Reversing directions, we finally discuss the interpretation for the Wilson loop operator W n (C) in terms of the periodic scalar field φ. For the time being, we do not make any assumption about the homology class of C, so the charge n ∈ Z must be an integer to maintain invariance under arbitrary gauge transformations.
In terms of the two-form δ C with delta-function support, the abelian Wilson loop operator can be rewritten as On the other hand, as also clear from (5.53), the gauge field A still enters the argument of the exponential linearly. Due to the new term involving δ C , the path integral over A now produces a delta-function for B that enforces the condition F B = 2πn δ C , (5.54) and B has trivial holonomy otherwise. At first glance, one might think that the curvature condition on B is vacuous, since we have already noted, in the discussion of the corresponding singularity for A, that the singularity in (5.54) can be removed by a gauge transformation of the local form in (5.51). However, we must remember that the group G B acts simultaneously on both B and φ via (5.4), so if we perform a gauge transformation to remove the singularity in B, we will create a singularity in φ! Specifically, once we perform the Wilson loop path integral over A in (5.53) and select a representative for the connection B satisfying (5.54), with trivial holonomies otherwise, we can rewrite the Wilson loop expectation value strictly in terms of φ, Here B is now a background, spectator field, and we have fixed the action of G B with our choice of representative connection. If we wish to eliminate B entirely, we can introduce a new periodic scalar field φ, defined so that From (5.58) we see that φ winds non-trivially around any small curve which encircles C. As a result, the Wilson loop operator W n (C) is interpreted dually as the disorder operator which creates an additive monodromy in φ of n units about the meridian of C. From the dual perspective, the integrality of the charge n is necessary to ensure that φ is single-valued as a map from the knot complement M o = M − C to S 1 . When C is trivial in H 1 (M), we have noted that the parameter n ∈ Z in the Wilson loop operator can be extended to an arbitrary real number ν ∈ R, as appears in (5.24). By our preceding discussion, the operator W ν (C) then creates a fractional monodromy in φ.
To understand the fractional monodromy better, let us think about dualizing the Wilson loop operator in the form (5.59) As before, Σ is a Seifert surface bounding C. For convenience, I assume that M is a rational homology sphere, with b 1 = 0, so that the relative homology class of Σ is unique. Associated to Σ is the Poincaré dual one-form δ Σ with delta-function support on Σ and satisfying the distributional identity dδ Σ = δ C .
By the same observations which we applied to (5.53), W ν (C) is described in terms of φ as the instruction to perform the path integral over φ in (5.55) with a background connection B which now satisfies B = 2πν δ Σ . (5.60) Equivalently, we replace φ by a new field φ so that d φ = dφ + B with the given B.
To characterize the local behavior of φ near C, we assume that M is R 3 , with coordinates (x, y, z), and that the curve C extends upwards along the z-axis. We then take Σ to be the portion of the xz-plane with x ≥ 0. Hence y is the local coordinate normal to Σ. In these local coordinates, the expression for B in (5.60) just becomes B = 2πν H(x) δ(y) dy, where H(x) is the Heaviside step-function. 9 See Figure 1 for a sketch of the situation.
To obtain a corresponding local description for φ near Σ, we integrate the relation d φ = dφ + B using our expression for B. In the region near Σ with x > 0, we find that φ is related to φ by φ = φ + 2πν H(y) , x > 0 . The Heaviside function H(y) arises from the integral of the delta-function δ(y). Thus, when C is null-homologous and the charge ν ∈ R is fractional, the role of the Wilson loop operator W ν (C) is dually to insert a discontinuity in the sigma model field φ transverse to the Seifert surface Σ. Physically, Σ can be interpreted as a kind of domain wall which is created by the fractional Wilson loop operator W ν (C).