Abelian Duality at Higher Genus

In three dimensions, a free, periodic scalar field is related by duality to an abelian gauge field. Here I explore aspects of this duality when both theories are quantized on a Riemann surface of genus g. At higher genus, duality involves an identification of winding with momentum on the Jacobian variety of the Riemann surface. I also consider duality for monopole and loop operators on the surface and exhibit the operator algebra, a refinement of the Wilson-'t Hooft algebra.


Introduction
Very broadly, dualities in quantum field theory often involve an interchange between classical and quantum data. Perhaps the simplest and best-known example occurs for the theory of a free periodic 1 scalar field φ ∼ φ + 2π on a Riemann surface Σ, with sigma model action Here R is a parameter which determines the radius of the circle for maps φ : Σ → S 1 , and ⋆ is the Hodge star associated to a given metric h on Σ. When φ is quantized on the circle, meaning that we take Σ = R × S 1 , one finds that the Hilbert space is graded by a pair of integers (p, w), The integers p and w are naturally interpreted as charges for a combined U(1) ℓ ×U(1) r action on H S 1 , under which each summand in (1.2) transforms with the specified weights. The integer p is associated to the global U(1) ℓ symmetry under which the value of φ shifts by a constant, 3) The other charge w describes the winding-number of φ as a map from the circle to itself. Hence w labels the connected components of the configuration space where Each component of the classical configuration space must be quantized separately, and those states which arise from X w span the subspace of the Hilbert space at the grade w.
Though the Hilbert space on S 1 is bigraded by (p, w) ∈ Z ⊕ Z, the individual gradings have very different physical origins. The momentum p appears only after quantization, so the grading by p is inherently quantum. Conversely, the grading by winding-number w can be understood in terms of the topology of the configuration space X , so the grading by w is classical. In a similar vein, the conserved currents on Σ associated to the U(1) ℓ × U(1) r global symmetry are respectively j ℓ = dφ , j r = ⋆dφ . (1.7) The topological current j r , whose charge is the winding-number w, trivially satisfies the conservation equation d † j r = 0 (with d † = −⋆d⋆) for arbitrary configurations of the field φ on Σ. By contrast, d † j ℓ = 0 only when φ satisfies the classical equation of motion △φ = d † dφ = 0. Thus conservation of j ℓ is a feature of the dynamics -or lack thereof -in the abelian sigma model.
Because Σ has dimension two, the conserved currents j ℓ and j r are exchanged under the action by Poincaré-Hodge duality on the space of one-forms Ω 1 Σ . As familiar, the classical action by Poincaré duality extends to an action by T-duality [6][7][8]13,22] on the quantum field φ, under which the respective quantum and classical gradings by momentum and winding are exchanged, and the parameter R in (1.1) is inverted to 1/R. See for instance Lecture 8 in [24] for further discussion of abelian duality on Σ.
The present paper is a continuation of [2], in which we examine global issues surrounding abelian duality in dimension three, on a Riemannian three-manifold M. In this case, duality now relates [5,19,21,24] the periodic scalar field φ : M → S 1 to a U(1) gauge field A on M. Both quantum field theories are free, so both can be quantized on the product M = R × Σ to produce respective Hilbert spaces H Σ and H ∨ Σ . Duality is an equivalence of quantum field theories, so we expect an isomorphism 2 Exploring how the identification in (1.8) works when Σ is a compact Riemann surface of genus g will be our main goal in this paper.
Just as T-duality acts in a non-trivial way on H S 1 by exchanging momentum and winding in (1.2), we will see that the dual identification H Σ ≃ H ∨ Σ relies upon an analogous exchange of classical and quantum data for the scalar field and the gauge field on Σ. However, the Hilbert spaces H Σ and H ∨ Σ are now more interesting due to their dependence on the geometry of the Riemann surface Σ, and unraveling the isomorphism in (1.8) turns out to be a richer story than for quantization on S 1 .
In genus zero, when Σ = CP 1 and the Hilbert space has a physical interpretation via radial quantization on R 3 , nothing that we say will be new. As usual in the world of Riemann surfaces, though, genus zero is a rather degenerate case, and several important features only emerge at genus g ≥ 1. From the perspective of the scalar field, these features are related to topological winding-modes on Σ, and from the perspective of the gauge field, they are related to the existence of a moduli space of non-trivial flat connections on Σ.
Given the current rudimentary understanding of duality, especially in dimension three, the existence of any tractable example is important. Both this work and [2] are motivated by questions about non-abelian duality for a certain topological version of the N = 8 supersymmetric Yang-Mills theory in three dimensions, considered to a certain extent in §3.3 of [25]. From the latter perspective, the abelian analysis here provides a useful toy model in which everything can be understood directly and in detail.

The Plan of the Paper
In Section 2, we construct the respective Hilbert spaces H Σ and H ∨ Σ associated to the periodic scalar field and the abelian gauge field on M = R × Σ. Because the quantum field theories are free, the quantization holds no mystery and can be carried out quite rigorously, if one wishes. Both Hilbert spaces depend on the detailed choice of the Riemannian metric on Σ. In either case, though, we identify a particularly simple, infinite-dimensional subspace of 'quasi-topological' states which depend only upon the overall volume and complex structure of Σ. These quasi-topological states are exchanged under duality, analogous to the exchange of momentum and winding states for quantization on S 1 .
In Section 3, we proceed to the consider the algebra satisfied by a natural set of operators (Wilson loops, vortex loops, and monopole operators in the language of Maxwell theory) which act on the Hilbert spaces constructed in Section 2. For a free quantum field theory, there is only one possible operator algebra that can arisenamely, the Heisenberg algebra, in a suitable geometric realization. When Σ = CP 1 there is not much to say, but in higher genus, the operator algebra has a non-trivial holomorphic dependence on Σ that seems not to have been previously noted. This algebra is a refinement of the celebrated Wilson-'t Hooft algebra [16]. See [10,11] for a somewhat related appearance of the Heisenberg algebra in four-dimensional Maxwell theory, and [3,4,9,17,20] for some recent discussions of monopoles and vortices in the setting of N = 2 supersymmetric gauge theory.
Along the way, we also consider in Section 3.1 the dual identification of operators acting on H Σ ≃ H ∨ Σ . For the convenience of the reader, a complementary review of the path integral perspective on the order-disorder correspondence for our operators can be found in Section 5.2 of [2].

Abelian Duality on a Riemann Surface
In this section, we quantize both the periodic scalar field and the abelian gauge field on a compact surface Σ of genus g, with Riemannian metric h. We then compare the results.

Some Geometric Preliminaries
Though the quantum field theories under consideration are free, they definitely depend upon the choice of the metric h on Σ. The most basic invariant of h is the total volume where ℓ is the length associated to the chosen metric. As in [2], the Hamiltonians for both the scalar field and the gauge field on Σ will depend upon a parameter e 2 , identified with the electric coupling in the Maxwell theory on M = R × Σ. All our constructions will respect the classical scaling under which the metric h transforms by along with and Hence ℓ and e 2 are redundant parameters, since either can be scaled to unity with an appropriate choice of Λ in (2.2). Nonetheless, we leave the dependence on both ℓ and e 2 explicit, so that the naive dimensional analysis holds. At least when the genus of Σ is positive, a more refined invariant of the metric h is the induced complex structure on Σ. Concretely, specifying a complex structure on Σ amounts to specifying a Hodge decomposition for complex one-forms where Ω 1,0 Σ and Ω 0,1 Σ refer to complex one-forms of given holomorphic/anti-holomorphic type. With this decomposition, one can define a Dolbeault operator ∂ by projection onto Ω 0,1 Σ , from which one obtains a notion of holomorphy on Σ. The Hodge star associated to the metric h satisfies ⋆ 2 = −1 when acting on Ω 1 Σ . The eigenspaces of the Hodge star then provide the decomposition in (2.5), where by In this manner, the metric h determines a complex structure on Σ. Finally, we will make great use of the de Rham Laplacians △ 0 and △ 1 acting on differential forms of degrees zero and one on Σ. As usual, both Laplacians are defined in terms of the L 2 -adjoint d † via △ 0,1 = d † d + dd † . With this convention, the Laplacian is a positive operator. Because Σ is smooth and compact, the spectra of △ 0 and △ 1 are discrete, and the kernels of each Laplacian are identified with the cohomology groups

The Quantum Sigma Model
As in Section 2 of [2], we consider a periodic scalar field φ on M = R × Σ, where we interpret φ as an angular quantity, subject to the identification Unlike in [2], though, for the purpose of quantization we work in Lorentz signature (− + +) on R × Σ, with the product metric Here t is interpreted as the "time" along R, and (z, z) are local holomorphic/antiholomorphic coordinates on Σ. Philosophically, quantization is more naturally carried out in Lorentz as opposed to Euclidean signature, since only in the former case does one expect to obtain a physically-sensible, unitary quantum field theory. In Lorentz signature on M = R × Σ, the free sigma model action is given by and dφ∧⋆dφ is invariant, a fact familiar in the context of two-dimensional conformal field theory. Hence I 0 (φ) will be invariant under (2.2) and (2.4) provided that the time t is also scaled by t −→ Λ t . (2.13) This scaling of the time coordinate is moreover necessary for a homogeneous scaling of the three-dimensional metric (2.10) on M.
As we observed in [2], the free sigma model action in (2.11) can be extended by topological terms Here α is a harmonic one-form on Σ, and θ is a real constant, Together, α and θ specify the components of the complex harmonic two-form γ that appears on the compact three-manifold M in [2]. The prefactor of 1/2π in (2.14) is just a convention, and the factors of e 2 and 1/ℓ 2 in the respective terms are dictated by invariance under the scaling in (2.2), (2.4), and (2.13). Note also that the two-form vol Σ /ℓ 2 which enters the second term in (2.14) is properly normalized to serve as an integral generator for H 2 (Σ; Z). We take the total sigma model action to be the sum 17) or more explicitly, Because α is closed by assumption, dα = 0, the topological terms in (2.18) do not alter the classical equation of motion a version of the usual wave equation on R × Σ. As a small check on our signs, note that △ 0 ≥ 0 is positive while ∂ 2 t ≤ 0 is negative, so the equation of motion for φ does admit non-trivial, time-dependent solutions. Clearly, α in (2.18) serves to distinguish the various topological winding-sectors associated to the circle-valued map φ. Like α, the constant θ multiplies a term in the action (2.14) which is a total derivative. Hence θ has no effect on the classical physics. However, θ does change the definition of the canonical momentum Π φ conjugate to φ, 20) in terms of which we write the classical Hamiltonian As will be clear, the quantum sigma model does depend upon θ non-trivially, and θ ∈ R/2πZ becomes an angular parameter closely analogous to the theta-angle of Yang-Mills theory in two and four dimensions.
In addition to the Hamiltonian H, another important quantity is the conserved momentum P associated to the global U(1) symmetry under which the value of φ shifts by a constant, exactly as in (1.3). Constant shifts in φ manifestly preserve the sigma model action, with conserved current j = e 2 dφ/2π and charge Because of the U(1) symmetry, the Hilbert space for φ will automatically carry an integral grading by the eigenvalue of P when we quantize.

Classical Mode Expansion
In principle, the Hilbert space for the periodic scalar field on the surface Σ is straightforward to describe, though the detailed spectrum of the Hamiltonian depends very much on the geometry of Σ.
Very briefly, just as for quantization on S 1 , the quantization on Σ involves a countable number of topological sectors, corresponding to homotopy classes of the map φ : Σ → S 1 . These homotopy classes are labelled by a winding-number ω which is valued in the cohomology lattice as discussed for instance in Section 2.1 of [2]. Abusing notation slightly, I write the cohomology class associated to the circle-valued map φ as Globally, the configuration space X = Map(Σ, S 1 ) is a union of components where As standard in quantum field theory, each component X ω ⊂ X must be quantized separately, leading to a topological grading by the cohomology lattice L on the full Hilbert space H Σ . This grading by ω ∈ L is the obvious counterpart for quantization on Σ to the grading (1.2) by winding-number w ∈ Z for quantization on S 1 .
Concretely, as the first step towards constructing the sigma model Hilbert space, we solve the classical equation of motion for φ in (2.19). For the moment, we assume φ to have trivial winding, so that the time-dependent field φ : R × Σ → S 1 can be equivalently considered as a map φ : R → X 0 to the identity component of X .
The general solution of (2.19) then takes the form Generalizing the standard solution to the wave equation on R 1,2 , this expression for φ is written in terms of an orthonormal basis {ψ λ } of eigenmodes for the scalar Laplacian △ 0 on Σ, with the convention that λ > 0 is a positive real number. For simplicity, we assume that all non-vanishing eigenvalues of the scalar Laplacian are distinct, so that each eigenfunction ψ λ (z, z) is uniquely labelled by λ. Of course, the precise spectrum for the scalar Laplacian depends sensitively on the geometry of the surface Σ, but we will not require any detailed information about the spectrum here, other than that it is discrete.
To ensure invariance of the eigenmode expansion for φ under the scaling in (2.2) and (2.4), we employ the invariant (and coupling-dependent) normalization condition A similar coupling-dependent normalization condition is used in (2.46) of [2], for the same reason. Accordingly, the constant function with unit norm on Σ is This constant function appears implicitly in (2.27) as the coefficient of the zero-mode φ 0 . Because φ is an angular quantity with period 2π, the zero-mode φ 0 must have its own periodicity Hence φ 0 effectively decompactifies in the large-volume limit ℓ → ∞ with e 2 fixed. Otherwise, p 0 , a λ , and a λ † for λ > 0 in (2.27) are constants which specify the classical solution for φ. The constant p 0 ∈ R is real and determines the classical momentum via When φ : R × Σ → S 1 has non-trivial winding, the classical mode expansion in (2.27) must be generalized only slightly. Exactly as in Section 2.2 in [2], we consider a harmonic representative for the cohomology class ω ∈ H 1 (Σ; Z). Mildly abusing notation, I re-use ω to refer to this representative in the space H 1 (Σ) of harmonic one-forms on Σ. Associated to the harmonic one-form ω with integral periods on Σ is a fiducial harmonic map Φ ω : Σ → S 1 satisfying Since d † ω = 0, we see that △ 0 Φ ω = d † dΦ ω = 0 automatically. By Hodge theory, the map Φ ω is determined by ω up to a constant. To fix that constant, we select a basepoint σ 0 ∈ Σ, which will re-occur later in Section 2. (2.36) In these terms, the coefficients φ 0 , p 0 , and (a λ , a λ † ) for all λ > 0 parametrize the classical phase space for maps φ : R → X ω .

Sigma Model Hilbert Space
To quantize, we promote both the scalar field φ and the momentum Π φ in (2.20) to operators which obey the canonical commutation relations Here δ Σ is a delta-function with support on the diagonal ∆ ⊂ Σ × Σ. We will take either of two perspectives on (2.37). From the first perspective, the commutator in (2.37) can be realized through the functional identification , (2.38) or equivalently via (2.20), .
Here D/Dφ(w) is interpreted as a covariant functional derivative incorporating the shift by θ in the canonical momentum, (2.40) Because θ is just a constant, As we will see quite explicitly, D/Dφ thus describes a flat connection with non-trivial holonomy over the configuration space X of maps from Σ to S 1 . For the alternative perspective on the commutator in (2.37), we rewrite the deltafunction δ Σ in terms of the orthonormal eigenbasis {ψ λ } for the scalar Laplacian, The first term on the right in (2.42) arises from the constant mode ψ 0 , and the factor of e 4 in the sum over the higher eigenmodes is a result of the normalization condition in (2.29).
These commutation relations hold in each winding-sector, independent of the class ω ∈ H 1 (Σ; Z), so the quantization will also be independent of ω. As the counterpart to the topological decomposition of X = Map(Σ, S 1 ) in (2.25), the total Hilbert space H Σ for the periodic scalar field on Σ decomposes into the direct sum where each subspace H ω Σ is itself a tensor product (independent of ω) Of the two factors in the tensor product, H λ is the less interesting. Up to an irrelevant choice of normalization, a λ and a λ † in (2.43) satisfy the usual commutator algebra for a harmonic oscillator with frequency λ. Hence H λ is the Fock space for that oscillator.
More interesting for us is the universal factor H 0 . This factor arises from the quantization of the zero-modes (φ 0 , p 0 ) of the scalar field and thus does not depend upon the spectral geometry of the surface Σ. Together φ 0 and p 0 simply describe the position and momentum of a free particle moving on a circle with radius e 2 ℓ. The corresponding phase space is the cotangent bundle T * S 1 with the canonical symplectic structure, and at least when θ = 0 in (2.14), the quantization is entirely standard. Directly, spanned by the Fourier wavefunctions As usual, the classical momentum p 0 becomes identified with the operator −i ∂/∂φ 0 . Via the identification P = e 2 ℓ p 0 in (2.32), each Fourier wavefunction in (2.47) is an eigenstate of the total momentum operator When the topological parameter θ is non-zero, the quantization of φ 0 and p 0 is modified. After we project (2.39) and (2.40) to the space of zero-modes, p 0 becomes identified with the θ-dependent operator Evidently, D/Dφ 0 in (2.49) is the covariant derivative for a unitary flat connection on a complex line-bundle L over the circle, with holonomy As the natural generalization of (2.46), the zero-mode Hilbert space H 0 is the space of square-integrable sections of L,   (2.54) in parallel to the bigrading by Z ⊕ Z for H S 1 in (1.2). As a convenient shorthand, I let |m; ω denote the Fourier wavefunction Ψ m (φ 0 ), considered in the topological sector with winding-number ω and satisfying the vacuum condition All other Fock states in H Σ are obtained by acting with the oscillator raising-operators a λ † on each Fock vacuum |m; ω , so the summands in H Σ above are more explicitly Philosophically, the grading by the eigenvalue m in (2.54) is a quantum grading (since we must quantize φ 0 to define it!), whereas the grading by the winding-number ω is classical, just as we saw in Section 1 for quantization on S 1 . But needless to say, because Z = L ≃ Z 2g , duality on Σ cannot exchange the two gradings, as occurs for duality on S 1 . Rather, the role of duality will be to exchange the quantum versus classical interpretations of each.
Finally, let us consider the action of the sigma model Hamiltonian H on the Hilbert space. After the identification in (2.39), the classical Hamiltonian becomes the operator Upon substituting for the momentum P in (2.52), where the ellipses indicate terms in D 2 /Dφ 2 which involve the non-zero eigenmodes of φ and thus the Fock operators (a λ , a λ † ).
The spectrum of H depends upon the corresponding spectrum of eigenvalues {λ 2 } for the scalar Laplacian △ 0 , which in turn depends upon the geometry of Σ. To simplify the situation, we consider the action of H only on the Fock vacua |m; ω in (2.55). From (2.53) and (2.58), and α, ω denotes the intersection pairing As will be important later, note that (ω, ω) in (2.60) is a conformal invariant, for which only the complex structure on Σ matters, and of course α, ω is purely topological. The energy in (2.59) also includes a constant term E 0 , independent of e 2 , m, and ω, which arises from the sum over the zero-point energies 1 2 λ of each oscillating eigenmode of φ. The factor of 1/ℓ which multiplies E 0 is fixed by the scaling in (2.33), and we have pulled out an overall factor of e 2 from H so that the quantity in brackets is scale-invariant (or dimensionless). Physically, E 0 /ℓ is a Casimir energy on the compact surface Σ, and some method of regularization must be chosen to make sense of the divergent sum E 0 ∼ λ>0 1 2 λ which naively defines E 0 , eg. by normal-ordering or use of the zeta-function. For comparison under duality, the particular method used to define E 0 will not matter, so we simply assume that E 0 has been determined in some way from the non-zero eigenvalues of the scalar Laplacian △ 0 on Σ.
Finally, let us consider the dependence of the spectrum of H on the effective coupling 1/e 2 ℓ. Though the abelian sigma model is a free quantum field theory, there remains a definite sense in which the spectrum simplifies in the weakly-coupled regime 1/e 2 ℓ ≪ 1. As apparent from (2.59), in this limit the quantum states with least energy in any given topological sector are precisely the Fock vacua |m; ω , for arbitrary values of the Fourier momentum m ∈ Z.
Conversely, when 1/e 2 ℓ is of order-one, we do not find a clean separation in energy between the Fock vacua |m; ω and oscillator states such as a λ † |0; ω for suitable λ.
Hence in the latter case, the low-lying energy spectrum of the quantum sigma model depends much more delicately on the geometry of Σ.

The Quantum Maxwell Theory
We now consider the quantization of Maxwell theory on M = R × Σ, with the same Lorentzian product metric already appearing in (2.10). Classically, the Maxwell gauge field A is a connection on a fixed principal When M = R × Σ, the restriction of P determines an associated complex line-bundle L over Σ, with Chern class Here F A = dA is the curvature, and we specify the Chern class of L by a single integer (2.64) The coincidence in notation between m in (2.47) and (2.64) is no accident. The integer m suffices to fix the topological type of both P and L. However, for purpose of quantization, we will need to endow the line-bundle L with a holomorphic structure as well. Because Σ carries a complex structure associated to its Riemannian metric h as in (2.6), L can be given a holomorphic structure uniformly for all degrees as soon as we pick a basepoint σ 0 ∈ Σ. We set By definition, holomorphic sections of L can be identified with meromorphic functions on Σ which have a pole of maximum degree m at the point σ 0 ∈ Σ. Note that the choice of basepoint is only relevant when Σ has genus g ≥ 1, since the holomorphic structure on any line-bundle of degree m over CP 1 is unique. The same remark also applies to our previous choice of basepoint for the sigma model: in genus zero, the only fiducial harmonic map Φ ω is constant, so the condition in (2.35) does not actually depend upon the choice of σ 0 . Specialized to M = R × Σ, the free Maxwell action becomes Here we stick to the assumption that ⋆ ≡ ⋆ Σ is the Hodge operator on Σ, so we have separated the curvature into the electric component E A , which transforms like a oneform on Σ, along with the magnetic component F A ≡ F A | Σ , which transforms like a two-form on Σ. Explicitly in local coordinates, Invariance under the scaling in (2.2), (2.4), and (2.13) fixes the dependence of the Maxwell action on the electric coupling e 2 , and the overall factor of 1/4π in (2.66) appears by convention. As in Section 2.1, topological terms can also be added to the Maxwell action, of the form Like α in (2.15), β is a real harmonic one-form, and θ ∈ R is a real parameter that will correspond under duality to the angle already appearing in (2.59). With some malice aforethought, the coefficient 1/e 2 ℓ 2 in (2.68) has been chosen to achieve this identification, along with invariance under the scaling in (2.2), (2.4), and (2.13). We then consider the total gauge theory action or more explicitly, Previously for the periodic scalar field, the angular parameter θ served to modify the definition (2.20) of the canonical momentum Π φ . This role is now taken by the harmonic one-form β, which appears in the canonical momentum In terms of Π A , the classical Hamiltonian is 3 The degree m of the line-bundle L is measured by the net magnetic flux through the surface, so θ in (2.73) now serves to distinguish the topological sectors labelled by m.
Because we have yet to fix a gauge, the classical Maxwell Hamiltonian H ∨ is degenerate along gauge orbits. As a remedy, we work throughout in Coulomb gauge, where A t is the time-component of the gauge field on M = R × Σ. In Coulomb gauge, the equation of motion for A t holds identically as the Gauss law constraint Harmonic gauge on Σ is particularly convenient from the geometric perspective. In this gauge, the Gauss constraint in (2.76) is automatically obeyed, and A satisfies the classical wave equation Laplacian for one-forms on Σ. We considered precisely the same equation of motion in (2.19) for the periodic scalar field φ, so quantization of A in harmonic gauge will share many features with quantization of φ, and duality will be manifest. Finally, if A is any time-independent connection on Σ, the equation of motion in (2.78) implies that the curvature is also harmonic, Thus the classical vacua of Maxwell theory on Σ correspond to harmonic connections on the line-bundle L. When Σ has genus g ≥ 1, Maxwell theory on M = R × Σ is invariant under a continuous U(1) 2g global symmetry, which does not occur at genus zero. To describe the action of the symmetry on the gauge field, we first select an integral harmonic basis {e 1 , . . . , e 2g } for the cohomology lattice The group U(1) 2g then acts on the gauge field by shifts Such shifts for any constant c j trivially preserve both the Maxwell action in (2.71) and the gauge conditions in (2.75) and (2.77). Note also that shifts by elements in the lattice 2πL are induced by homotopically non-trivial, "large" gauge transformations so these lattice elements act as the identity modulo gauge-equivalence. As a result, the parameters c j in (2.81) are circle-valued, and the global symmetry group is compact.
The group U(1) 2g acts to shift the holonomies of the gauge field, so we can think of this global symmetry group more intrinsically as the Jacobian torus of Σ, The Jacobian J Σ , in its role as the moduli space of flat U(1)-connections on Σ, will be essential in analyzing abelian duality at higher genus. The U(1) 2g global symmetry of Maxwell theory on Σ is the counterpart to the more obvious U(1) symmetry of the periodic scalar field in (1.3). Just as for the conserved momentum P in (2.22), the global symmetry of Maxwell theory leads to a set of 2g conserved charges (2.84) Conservation of W j follows from the harmonic condition de j = d † e j = 0, as well as the classical equation of motion in (2.78).
Because of the U(1) 2g global symmetry, upon quantization the Maxwell Hilbert space will automatically carry an integral grading by the eigenvalues of W j . Moreover, since ⋆E A is directly related (2.72) to the canonical momentum Π A for the gauge field, the grading by W j will again be interpreted physically as a quantum grading by total momentum.

Classical Mode Expansion
This background material out of the way, we now quantize the Maxwell gauge field on the Riemann surface Σ. As for the periodic scalar field, our main interest lies in a universal set of low-lying energy levels which are not sensitive to the detailed spectral geometry of Σ.
The quantization of A on Σ involves a countable number of topological sectors, labelled by the degree m of the line-bundle L. By analogy to the decomposition (2.25) of the configuration space for the scalar field, we write the configuration space for the Maxwell gauge field as a union of components where A m is the affine space of unitary connections on the complex line-bundle L of degree m over Σ, Each connected component A m ⊂ A of the configuration space must be quantized separately, so the Maxwell Hilbert space H ∨ Σ automatically carries an integral grading by the degree m ∈ Z.
Having broken our quantization problem into countably-many pieces, we solve the classical equation of motion (2.78) for A in harmonic gauge. As a special case, we begin by considering only time-independent classical solutions, corresponding to connections with harmonic curvature on Σ.
When L has degree m = 0 and hence is topologically trivial, a harmonic connection on L is simply a flat connection, of the form for the fixed harmonic basis {e 1 , · · · , e 2g } of H 1 (Σ; Z). The expansion coefficients ϕ 0 j for j = 1, . . . , 2g are thus angular coordinates on the Jacobian J Σ , which has already appeared in (2.83). Equivalently, (ϕ 0 1 , . . . , ϕ 0 2g ) characterize the holonomies of A around a generating set of closed one-cycles on Σ. Because Σ carries a complex structure, each harmonic one-form e j in (2.87) can be decomposed according to its holomorphic/anti-holomorphic type via (2.6), in which case J Σ itself inherits a complex structure. Intrinsically as a complex torus, where Pic 0 (Σ) denotes the group of isomorphism classes of holomorphic line-bundles of degree zero on Σ, with group multiplication given by the tensor product of linebundles.
If L has degree m = 0, then a harmonic connection on L cannot be flat. Instead, the curvature F m of any harmonic connection on L is proportional to the Riemannian volume form on Σ, Abstractly, the basepoint σ 0 ∈ Σ provides an isomorphism between distinct components of the Picard group of all holomorphic line-bundles on Σ, via the tensor product This identity is clearly compatible with the formula for the harmonic curvature F m in (2.89). For the remainder of the paper, I simplify the notation by setting A ≡ A 1 .
According to the preceding discussion, in each degree m ∈ Z, the arbitrary timeindependent solution to the classical equation of motion for A in (2.78) is given up to gauge-equivalence by a point on the Jacobian J Σ ≃ U(1) 2g . To describe the more general, time-dependent solution in harmonic gauge, we perform an expansion of A in eigenmodes of the de Rham Laplacian △ 1 , (2.93) The eigenmode expansion of A in (2.93) requires several comments.
First, p 0,j ∈ R for j = 1, . . . , 2g are the classical momenta conjugate to the angular coordinates ϕ 0 j on the Jacobian. The coefficient of e 2 which multiplies p 0,j is fixed by scaling, due to the explicit t-dependence. Also, Q is the positive-definite, symmetric matrix of L 2 inner-products 4 Coulomb gauge, the expansion of A ensures that p 0,j is equal to the conserved charge W j in (2.84), Proceeding to the second line of (2.93), (a λ , a λ † ) are a conjugate pair of complex parameters associated to the oscillating modes of A, and the coefficient √ π e 2 /λ has been chosen to simplify the commutator algebra upon quantization. We also introduce an orthonormal basis of one-forms χ λ ∈ Ω 1 Σ which satisfy the joint conditions as well as where by convention λ > 0 is positive. The first condition instantiates the harmonic gauge in (2.77), and the second condition states that χ λ is an eigenform for the de Rham Laplacian △ 1 acting on one-forms on Σ.
The eigenforms χ λ have a simple relation to the eigenfunctions ψ λ ∈ Ω 0 Σ which appear in the corresponding mode expansion for the scalar field φ. For if ψ λ is an eigenfunction of the scalar Laplacian, then we obtain a corresponding eigenform χ λ for △ 1 by setting The relative coefficient e 2 /λ in (2.99) just ensures that χ λ has unit norm, assuming that ψ λ is normalized according to (2.29). Conversely, given any eigenform χ λ satisfying (2.96) and (2.97) with λ > 0, we obtain a normalized eigenfunction ψ λ via Following the same steps in (2.100), one can verify directly that ψ λ in (2.102) is a normalized eigenfunction of the scalar Laplacian △ 0 . The minus sign in (2.102) is a nicety which ensures that the map from ψ λ to χ λ and back is the identity. Together, the relations in (2.99) and (2.102) constitute a Hodge isomorphism between the non-zero spectrum of △ 0 acting on Ω 0 Σ and the non-zero spectrum of △ 1 acting on the intersection Ω 1 Σ ∩ Ker(d † ). Consequently, the oscillator frequencies λ > 0 which appear in the harmonic expansion (2.93) of the gauge field A are precisely the same frequencies which appear in the harmonic expansion (2.27) of the periodic scalar field φ. This equality is essential for abelian duality to hold on Σ.

Canonical Commutation Relations
Naively, to quantize the Maxwell theory, we promote both A and the canonical momentum Π A to operator-valued one-forms which satisfy the canonical equal-time commutation relation In making sense of (2.103) geometrically, the Riemannian volume form on the right is to be interpreted as a section of the tensor product vol Σ ∈ Ω 1 Σ ⊗ Ω 1 Σ , and δ Σ (z, w) remains the delta-function with support along the diagonal ∆ ⊂ Σ × Σ.
Although convenient for our purposes, one feature of the canonical commutator as written in (2.103) is slightly non-standard. In local coordinates, the commutator would typically be presented as In actuality, the situation is more complicated, because the naive commutator in (2.103) is not compatible with the Coulomb gauge conditions be the Green's function for the scalar Laplacian △ 0 on Σ, such that Because Σ is compact, we are careful to subtract the contribution from the constant mode in (2.106), so that both sides of the equation for G(z, w) integrate to zero over Σ. Equivalently, G(z, w) can be expanded in terms of the orthonormal eigenbasis For the ambitious reader who enjoys keeping track of factors of e 2 , recall that e 2 appears in the normalization condition (2.29) for ψ λ and thus enters the expansion for G(z, w).
The scalar Green's function G(z, w) can be used to correct the naive commutation relation in (2.103) so that the right-hand side is actually compatible with the Coulomb gauge constraints d † A = dΠ A = 0. To wit, the corrected commutator will be (2.108) Here (d z ⊗ ⋆d w )G(z, w) is the section of Ω 1 Σ ⊗ Ω 1 Σ obtained from the action of the individual de Rham operators on G(z, w). In terms of the spectral decomposition in Most crucially, the right-hand side of the corrected commutator (2.108) does lie in the kernels of both d z † and d w . This statement can be verified either by direct computation from the defining equation for G(z, w) in (2.106) or, as will be more relevant here, by applying the spectral decomposition for (d z ⊗⋆d w )G(z, w) in (2.109). I take the latter approach.
In close analogy to (2.42), the term involving the delta-function in (2.108) can be presented as the sum Very briefly, the three terms on the right in (2.110) reflect the three terms in the Hodge decomposition The first term in (2.110), involving the harmonic forms e j for j = 1, . . . , 2g, describes the action of vol Σ · δ Σ (z, w) by wedge-product and convolution on H 1 (Σ). Otherwise, the two sets of eigenforms {χ λ } and {⋆χ λ } for λ > 0 span the respective images of d † and d, as discussed previously in relation to (2.99). The latter two terms on the right in (2.110) then account for the action of vol Σ · δ Σ (z, w) on Im(d † | Ω 2 Σ ) and Im(d| Ω 0 Σ ). The relative minus sign in (2.110) can be checked directly by working in a local frame on Σ or understood as a consequence of anti-symmetry under the exchange of the factors in Ω 1 Σ ⊗ Ω 1 Σ . Because e j is harmonic and d † χ λ = 0, the first two terms on the right of (2.110) are annihilated by d z † and d w . On the other hand, for the third term in (2.110) the relation between χ λ and ψ λ in (2.99) implies The minus sign appears since ⋆ 2 = −1 on Ω 1 Σ . Thence from (2.109) and (2.110), the right-hand side of the corrected commutation relation is given by (2.113) The expression in (2.113) is manifestly annihilated by d z † and d w , so is compatible with Coulomb gauge.

Maxwell Hilbert Space
With the corrected commutation relation in (2.108), the quantization of Maxwell theory on Σ is now straightforward. From the (slightly formal) functional perspective, the momentum Π A becomes identified with the operator .
Here the smeared term involving the Green's function G(u, w) is the necessary price of working in Coulomb gauge. I do not wish to belabor the interpretation of the non-local, smeared term, as we will be primarily interested in situations for which it does not matter. However, let me say a word about the basic geometric meaning of (2.114), which may be somewhat opaque. Because Π A ∈ Ω 1 Σ transforms as a one-form, the right-hand side of (2.114) must also transform as a one-form on Σ. Dually to A, the derivative δ/δA ∈ T Σ transforms as a vector field on Σ. The notation vol Σ · δ/δA indicates that this vector field is to be contracted with the volume form to produce a one-form on Σ, as expressed in local coordinates For the smeared term in (2.114), we take the wedge-product of (d u ⊗⋆d w )G(u, w) as a section of Ω 1 Σ ⊗ Ω 1 Σ with the one-form vol Σ · δ/δA to obtain a section of Ω 2 Σ ⊗ Ω 1 Σ . The first factor is integrated over Σ to produce yet another one-form.
The classical relation Π A = ⋆E A /2πe 2 − β/2π implies that the electric field E A acts as the covariant operator . In precise analogy to the angle θ appearing in D/Dφ(w), the harmonic one-form β in D/DA(w) will describe the holonomy of a flat connection over the Jacobian J Σ , after we reduce to zero-modes. As a more down-to-earth alternative to the functional calculus, quantization of A can be carried out in terms of the eigenmode expansion in (2.93), coupled with the spectral identity in (2.113). To realize the commutator in (2.108), the pairs (ϕ 0 j , p 0,j ) for j = 1, . . . , 2g and a λ , a λ † for all λ > 0 are promoted to operators which obey the Heisenberg algebra with all other commutators vanishing. This algebra is akin to that for the periodic scalar field φ in (2.43), but rather than quantizing a single periodic zero-mode, we quantize a set of 2g zero-modes which describe the motion of a free particle on the Jacobian J Σ of Σ. The free-field algebra holds in each topological sector labelled by m = deg(L), so the Maxwell Hilbert space H ∨ Σ is the direct sum where each summand is itself the tensor product Exactly as in (2.45), H λ is the oscillator Fock space acted upon by the pair (a λ , a λ † ).
We have already noted that an identical spectrum of non-zero frequencies λ > 0 occurs for both the periodic scalar field and the U(1) gauge field on Σ. Thus the same tensor product of Fock spaces H λ appears in both the scalar Hilbert space H Σ and the Maxwell Hilbert space H ∨ Σ . At least for the excited oscillator states in the two Hilbert spaces, abelian duality is a trivial equivalence.
The remaining factor H ∨ 0 arises from the quantization of the zero-modes for the gauge field. Classically, (ϕ 0 j , p 0,j ) for j = 1, . . . , 2g are coordinates on the cotangent bundle T * J Σ with its canonical symplectic structure, so the standard quantization yields Because J Σ ≃ U(1) 2g is just a torus, the Hilbert space H ∨ 0 is naturally spanned by the collection of Fourier wavefunctions each labelled by an element ω in the cohomology lattice L = H 1 (Σ; Z). Again, the coincidence of notation with the winding-number in Section 2.1 is no accident. Here though, integrality of ω is not due to topology per se, but rather to the requirement that the wavefunction in (2.123) be invariant under shifts ϕ 0 j → ϕ 0 j + 2π for all j = 1, . . . , 2g. Physically, ω determines the conserved momentum carried by the state Ψ ω (ϕ 0 ), where we apply the identification Directly, the Fourier wavefunction in (2.123) is a momentum eigenstate, As in (2.61), · , · is shorthand for the intersection pairing of one-forms on Σ. When the topological parameter β ∈ H 1 (Σ) is non-zero, the interpretation of the zero-mode momentum p 0,j is modified via (2.116) and (2.117) to . (2.133) In terms of the conserved momenta W j in (2.84), where E 0 /ℓ is again a Casimir energy on Σ. Because the zero-point energies 1 2 λ of the oscillating modes are the same for both the periodic scalar field and the gauge field, the constant E 0 in (2.135) will agree with the corresponding constant in (2.59) so long as we use the same regularization method to define both (as we assume).
For Maxwell theory on Σ, the spectrum of H ∨ simplifies in the regime e 2 ℓ ≪ 1 of weak electric coupling. Only then do the Fock vacua |ω; m for arbitrary Fourier momentum ω ∈ L have parametrically smaller energy than the typical oscillator state such as a λ † |0; m . Not surprisingly, this case is opposite to the strong-coupling regime 1/e 2 ℓ ≪ 1 in which the states of least-energy arise by quantizing the single zero-mode of the periodic scalar field φ.

Topological Hilbert Space
Let us summarize our results so far.
We have obtained an explicit identification between the Hilbert spaces for the U(1) gauge field A and the periodic scalar field φ on the surface Σ, The isomorphism for the oscillator Fock spaces H λ for λ > 0 follows from classical Hodge theory after we pass to Coulomb gauge for A, so it is relatively uninteresting. The non-trivial content in (2.136) is the identification between the Fock vacua |m; ω , which arise from the quantization of the harmonic modes of A and φ in each topological sector.
For the periodic scalar field, m ∈ Z is a quantum label which arises from Fourier modes on S 1 , and the lattice vector ω ∈ L is a classical label which measures the winding-number of the map φ : Σ → S 1 . Conversely for the gauge field, the integer m is the classical label, corresponding to the degree of the line-bundle L, and the lattice vector ω is the quantum label, arising from Fourier modes on the Jacobian Nonetheless, under the dual correspondence (2.138) Let us introduce the quantum partition functions for the scalar and Maxwell theories, (2.140) The same duality relation appears under a different guise in [2], where it arises from the non-trivial modular transformation of a theta-function Θ M (γ) associated to any Riemannian three-manifold M. See Section 4.1 of [2] for a complete discussion of the theta-function and Section 5.1 of the same work for a path integral derivation of the relation in (2.140). Compare especially to equation (5.1) in [2].
Finally, as we have already mentioned, the spectrum of H dramatically simplifies in either the small-volume limit e 2 ℓ ≪ 1 or the large-volume limit e 2 ℓ ≫ 1. In both cases, the quantum states of minimal energy within each topological sector are the Fock vacua |m; ω , for all pairs (m, ω) in the lattice Z ⊕ L. Hence we can sensibly restrict attention to the subspace of the full Hilbert space spanned by these states, to which we turn next. Elsewhere, I will discuss some important related notions in the context of N = 2 supersymmetric quantum field theory in three dimensions.

Operator Algebra at Higher Genus
Given the explicit construction of the Hilbert space on Σ, we now discuss the action of several natural classes of operators on that Hilbert space.
As mentioned at the end of Section 2.
The restriction to H top Σ is natural in either the regime e 2 ℓ ≪ 1 or e 2 ℓ ≫ 1, for which the Fock vacua describe states of minimal energy within each topological sector. Because the abelian theories in question are non-interacting, we do not need to worry about the effects of high-energy states, which would otherwise be integrated-out in passing from the big Hilbert space H Σ to the subspace H Σ top .
Following Section 5.2 in [2], we analyze three classes of operators on Σ. We first have the local vertex operator V k (σ) which is inserted at a point σ ∈ Σ, Periodicity of the scalar field φ ∼ φ + 2π dictates that k be an integer so that V k (σ) is single-valued.
Next we have the Wilson loop operator W n (C) associated to an oriented, smoothly embedded curve C ⊂ Σ, For generic choices of C, the charge n of the Wilson loop operator must be an integer to ensure gauge-invariance with respect to the compact gauge group U(1). Perhaps less appreciated, when C is a homologically-trivial curve which bounds a two-cycle D ⊂ Σ, the Wilson loop operator can be defined for an arbitrary real charge via This expression for W ν (C) is manifestly gauge-invariant for all values of ν, and it reduces to (3.3) by Stokes' theorem for C = ∂D. Unlike the situation for W ν (C) in three dimensions [2], where the role of D is played by a Seifert surface with some homological ambiguity, here there is no ambiguity about D. Because D is a two-cycle on Σ, the choice of D is fixed entirely by the orientations of the pair (Σ, C).
Without delay, let me emphasize that W ν (C) will act non-trivially on H top Σ even when C is trivial in homology. Likewise, W n (C) will depend upon the geometry of C ⊂ Σ, not just the homology class [C] ∈ H 1 (Σ).
By contrast, we do have a purely homological loop operator L α (C), given by functionals of the scalar field φ or the gauge field A. As well-known and reviewed for instance in Section 5.2 of [2], each of these operators admits a dual disorder description, in which the operator creates a classical singularity at the point σ or along the curve C, respectively. Very briefly, the vertex operator V k (σ) creates a local monopole singularity of magnetic charge k in the gauge field A, and the loop operator L α (C) creates a codimension-two singularity in M = R × Σ around which A has monodromy α. Note that this interpretation is consistent with the angular nature of α. For the periodic scalar field, the Wilson loop operator W n (C) dually creates an additive monodromy φ → φ + 2πn around any small path encircling C inside M. Note that integrality of n is required for the monodromy to make sense for general C. Such classical geometric descriptions of the disorder operators suffice for the path integral analysis of duality in [2]. Our goal in Section 3.1 is to provide an alternative, quantum description of these operators -in both order and disorder form -by their action on the Hilbert space H Σ top . Using these results, we then exhibit directly in Section 3.2 the combined algebra of vertex and loop operators on Σ.

Monopoles and Loops on a Riemann Surface
To discuss the action of V k (σ), W n (C), and L α (C) on the Hilbert space, we assume that each operator acts at time t = 0 on M = R × Σ. These operators do not generally preserve the topological subspace H top Σ inside the full Hilbert space H Σ . To obtain an action on H top Σ alone, we compose with the projection from H Σ onto H top Σ , which occurs naturally in either the geometric limits e 2 ℓ ≪ 1 or e 2 ℓ ≫ 1. This projection onto H top Σ will be implicit throughout. At the classical level, projection onto H top Σ amounts to the Hodge projection onto harmonic configurations for φ and A.

Monopole Operators
We begin with the action of the vertex operator V k (σ) in (3.2). Directly via the eigenmode expansion (2.36) for the periodic scalar field, Here Φ ω : Σ → S 1 is the fiducial harmonic map (2.34) with winding-number ω, and the ellipses indicate terms involving the Fock operators a λ and a λ † , whose action becomes irrelevant after the projection to H top Σ . According to the description of the Fourier wavefunction in (2.47), the Fock groundstate |m; ω can itself be written as Evidently from (3.6), V k (σ) shifts the Fourier mode number m to m + k, In higher genus, the situation is more interesting. To evaluate the phase in (3.8), we use the defining conditions for the fiducial map, namely where ω ∈ H 1 (Σ; Z) is harmonic and σ 0 ∈ Σ is the basepoint used for quantization. By Stokes' theorem, the phase factor in (3.8) can be recast in the form where Γ is any oriented path on Σ which connects the basepoint σ 0 to the point σ where the vertex operator is inserted, The homotopy class of Γ is not unique, as clear when σ = σ 0 and Γ is an arbitrary closed curve based at σ 0 . However, as usual in the business, integrality of both k and ω ensures that the phase in (3.10) is independent of the choice of the integration contour Γ. For the remainder, we suppress the appearance of Γ and simply write the vertex operator phase as Thus, even when we restrict to the low-energy subspace H top Σ ⊂ H Σ , the action of the vertex operator V k (σ) is still sensitive to the location at which the operator is inserted. How does (3.12) arise when we describe the quantum theory on Σ dually in terms of the Maxwell gauge field A? To answer this question, we recall that the Fock vacua |m; ω in H top Σ correspond to wavefunctions for A on the disjoint union of tori each isomorphic to the Jacobian of Σ. A natural guess is that the effective action of the vertex operator V k (σ) is induced from the tensor product (or Hecke modification) with the degree-k holomorphic line-bundle O Σ (k σ), as already appears in (2.91). In the gauge theory approach to geometric Langlands, this statement has been explained in §9.1 of [18], though we must make a few minor modifications to treat the non-topological (but free) theory here. According to its definition as a monopole operator, reviewed in Section 5.2 of [2], the vertex operator V k (σ) acts topologically to increase the degree of the U(1)-bundle over Σ by k units, in accord with the shift m → m + k in both (3.12) and (3.14). However, for a complete characterization of the phase in (3.12), we must also consider how the action of V k (σ) on the state |m; ω depends upon the point σ ∈ Σ and the Fourier mode ω ∈ L for the gauge field wavefunction on J Σ .
To investigate the latter dependence, let us consider the composite operator where σ is distinct from the basepoint σ 0 . Because V k (σ) and V −k (σ 0 ) carry opposite monopole charges, O k (σ, σ 0 ) does not alter the topology of the line-bundle over Σ. Nonetheless, O k (σ, σ 0 ) may still act non-trivially on the state |m; ω , at least in genus g ≥ 1.
Since we work with zero-modes, let A be an arbitrary harmonic connection on a line-bundle of degree m over Σ, of the form Here we have truncated the more general eigenform expansion for A in (2.93), and we recall that A is the fiducial harmonic connection associated to the holomorphic line-bundle O Σ (σ 0 ). As a harmonic connection, A determines a point in the Picard component Pic m (Σ) of degree m, and we may interpret the Fock groundstate |m; ω ≡ Ψ ω (A)|m (3.17) in terms of the wavefunction exactly as in (2.123). After we subtract m A in the argument of the exponential, Ψ ω (A) does not actually depend upon the degree m. On this wavefunction, the composite operator O k (σ, σ 0 ) acts via a modification of A induced from (3.14) , where In this expression, δ Γ is a one-form on Σ with delta-function support which represents the Poincaré dual of an oriented path Γ running from σ 0 to σ. Equivalently, for any smooth one-form η, the wedge product with δ Γ satisfies Because the path Γ is open, the one-form δ Γ is not closed but rather obeys where δ σ and δ σ 0 are two-forms on Σ with delta-function support at the points σ, σ 0 ∈ Σ. As a consequence of (3.22), the curvature of the modified connection A in (3.20) is singular at the locations where the vertex operators are inserted. Physically, these curvature singularities signal the creation of a monopole/anti-monopole pair of magnetic charge k on Σ. Strictly speaking, the singular connection A in (3.20) is not harmonic, and so to interpret the dual action of O k (σ, σ 0 ) on the zero-mode wavefunction, we should project the singular connection A onto the harmonic subspace of Ω 1 Σ . Thankfully, this projection is accomplished automatically for us when we evaluate (3.23) since the one-form ω is harmonic by assumption. In passing from the second to the third line of (3.23), we use the defining property of δ Γ in (3.21).
According to (3.19) and (3.23), (3.24) and again integrality of both k and ω ensures that the phase factor depends only upon the endpoints of the path Γ, where the monopoles are inserted. Clearly from its definition (3.14) via the tensor product, the monopole operator of charge k is the same as the k-th power of the unit monopole operator, (3.25) so we can rewrite the identity in (3.24) as Hence we have determined the action of the monopole operator V k (σ) for arbitrary points σ ∈ Σ in terms of the action of the monopole operator V k (σ 0 ) inserted at the basepoint σ 0 .
We are left to discuss the action of the based monopole V k (σ 0 ) on |m; ω . The first claim is that V k (σ 0 ) does not alter the quantum label ω, Otherwise, from the gauge theory perspective we have left some ambiguity in the normalization of the basis state |m; ω for fixed ω as the degree m ranges over Z. We fix this ambiguity up to an overall constant by declaring Together, (3.26) and (3.29) imply the formula in (3.12), which we deduced from the more direct description of V k (σ) as a vertex operator for the periodic scalar field φ.

Vortex Loops
Just as we consider the action by the local vertex operator V k (σ), we can also consider the action on H top Σ by the respective loop operators L α (C) and W n (C), where C is a closed curve in Σ. If C is not a spacelike curve in Σ but a timelike curve in M of the form C = R × {σ} for some point σ ∈ Σ, then L α (C) and W n (C) do not act on the Hilbert space H top Σ but lead rather to the construction of new Hilbert spaces associated to the punctured surface Σ o = Σ − {σ}. The analysis of such line operators is similar philosophically to the analysis of fibrewise Wilson loop operators in [1], so I omit the timelike case here.
Like the vertex operator V k (σ), the loop operator L α (C) admits an elementary description in terms of the periodic scalar field φ, On each state |m; ω in H top Σ , the operator L α (C) simply measures the windingnumber ω ∈ H 1 (Σ; Z), where we recall that [dφ] = 2πω for the state |m; ω . In particular, L α (C) respects the global U(1) symmetry by shifts φ → φ + c for constant c and hence preserves the mode number m of the state |m; ω . We also note that the phase in (3.31) depends only on the homology class of C in H 1 (Σ), and the integrality of ω ensures that the phase depends only on the value of the parameter α modulo 2π. Again, our main goal is to understand how the formula in (3.31) arises dually in terms of the U(1) gauge field A. As reviewed in Section 5.2 of [2], the loop operator L α (C) acts on A as a disorder operator which creates a curvature singularity along C of the form where δ C is a two-form with delta-function support that represents the Poincaré dual of C ⊂ M . Equivalently, near C the gauge field behaves as where ϑ is an angular coordinate on the plane transverse to C, located at the origin.
Globally, A has non-trivial monodromy about any small curve linking C in M, and L α (C) is the reduction to three dimensions of the basic Gukov-Witten [15] surface operator in four dimensions. Unlike the monopole singularity, the singularity in (3.32) does not change the degree m of the line-bundle L over Σ. As will be useful later, let me give an elementary argument for this statement.
We consider an arbitrary configuration for the gauge field A on M = R × Σ with the prescribed singularity in (3.32) at time t = 0, and smooth otherwise. We will measure the change in the degree m as the time t runs from −∞ to +∞ on M. By way of notation, Σ ± ⊂ M will denote the copies of Σ at the times t = ±∞. Then the change ∆m in the degree from t = −∞ to t = +∞ is computed by In the first line of (3.34) we apply Stokes' theorem, and in the second line we apply the Bianchi identity dF A = 0 on the locus where A is smooth. Finally, because C is a closed curve on Σ, the Poincaré dual current δ C is also closed, dδ C = 0. (The same computation would show that ∆m = 0 for the monopole operator, which acts as a localized source for dF A .) So as observed following (3.31), L α (C) must preserve the magnetic label m on the states |m; ω in H top Σ . On the other hand, L α (C) does change the holonomies of A on Σ, from which the phase factor in (3.31) will be induced. To setup the computation, we consider the gauge theory on M = R × Σ, and we suppose that the line operator L α (C) is inserted on Σ 0 ≡ {0} × Σ in M. We fix an initial flat connection A − on Σ − . The operator L α (C) acts as a sudden perturbation to create the singularity in (3.32) at t = 0, after which we project A back onto the subspace of harmonic (ie. flat) connections. We then let A + be the final connection on Σ + which is obtained by subsequent timeevolution. 5 We wish to compare the holonomies of A + to those of A − . See Figure 1 for a sketch of M as a cylinder over the uniformization of Σ, where for concreteness we have drawn Σ as a Riemann surface of genus two.
For any closed, oriented curve γ on Σ, we evaluate where S is oriented as the rectangle in the figure with boundary In passing from (3.36) to (3.37), we note that the vertical edges of S in Figure 1 are identified after a reversal of orientation, so they make no contribution to the boundary integral over ∂S in (3.36).
We are left to evaluate the integral of the curvature F A over S in (3.36). By assumption, F A vanishes everywhere on S except for the explicit curvature singularity (3.32) created at t = 0 along C. Consequently, where [C] ∨ ∈ H 1 (Σ; Z) is a harmonic representative for the dual of the homology class of C. The flip of sign in the second equality arises from due care with orientations. Because the curve γ is arbitrary, L α (C) must act classically by the shift We recall from (3.18) that the wavefunction Ψ ω evaluated on a harmonic connection A of degree m is given by Under the shift (3.39) induced by L α (C), we see that Ψ ω transforms by Using O k (σ, σ 0 ), we can try to make a new operator á la Verlinde [23] from the monodromy action by O k (σ, σ 0 ) on H top Σ as the point σ is moved adiabatically around a closed curve C ⊂ Σ based at σ 0 . When k is integral, the induced phase in (3.42) is trivial, and the Verlinde operator acts as the identity. However, when k = 0 mod Z is allowed to be fractional, the Verlinde operator constructed from O k (σ, σ 0 ) is nontrivial and acts by precisely the phase in (3.41), provided we set So the line operator L α (C) can be interpreted as the Verlinde operator associated to transport of a monopole/anti-monopole pair with non-integral magnetic charge. See [15] for a somewhat different explanation of the same effect in four-dimensional gauge theory. This relation can also be understood directly from the order-type expressions for V k (σ) • V −k (σ 0 ) and L α (C) in terms of the periodic scalar field φ.
Alternatively, the action by the line operator L α (C) can be understood using the Lagrangian formalism, the focus of [2]. For simplicity in the following discussion, we assume that the topological parameters θ and β from Section 2.2 are both set to zero.
In the Lagrangian formalism, the inner-product of states m; ω ′ |L α (C)|m; ω for some ω, ω ′ ∈ L is computed by the path integral Here A ± denote the boundary values for the gauge field at t = ±∞, and the path integral ranges over the affine space A m of connections on the U(1)-bundle with degree m on M = R × Σ. We also integrate over the boundary values of A with weights given by the wavefunctions Ψ ω and Ψ ω ′ as in (3.18). Finally, the term proportional to δ C in (3.45) enforces the condition that F A have the singular behavior in (3.32). For a more thorough discussion of the latter remark, see Section 5.2 in [2].
The modified action in (3.44) can be expanded in terms of F A as Here c 0 is a formally divergent constant arising from the norm-square of δ C , which we shall ignore. By comparison of (3.46) to the standard Maxwell action, L α (C) can be identified semi-classically with the operator , (3.48) so the loop operator becomes Manifestly, L α (C) acts upon any wavefunction Ψ ω (A) by the shift A → A + α [C] ∨ appearing in the first line of (3.41).
Although we began with a disorder characterization of the loop operator in gauge theory, the classical description for L α (C) in (3.47) amounts to an order expression for the same operator. A quantum operator may admit distinct classical descriptions, so there is no contradiction here. See for instance Section 4.1 of [1] for an analogous disorder presentation of the usual Wilson loop operator in Chern-Simons gauge theory.

Wilson Loops
We are left to consider the action on H top Σ of the Wilson loop operator W n (C). In terms of the gauge field, the Wilson loop operator acts simply by multiplication in the topological sector labelled by the degree m, where again [C] ∨ ∈ H 1 (Σ; Z) is the Poincaré dual of the curve C ⊂ Σ. In passing to the second line of (3.50), we recall that A is a harmonic connection with expansion arbitrary real values ν ∈ R, the orientation of C uniquely fixes the bounding two-cycle D with compatible orientation, so that the action of W ν (C) is well-defined. 7 As usual, we now wish to understand the results in (3.51)  closed curve γ ⊂ Σ as (3.54) reproducing the shift in (3.51).
The non-topological, m-dependent phase in (3.51) is slightly more subtle. For simplicity, we will reproduce this phase only in the special case that C = ∂D is homologically-trivial, as assumed in (3.52). Then As the ur-statement of abelian duality, discussed in the Introduction to [2], we have the correspondence Hence W ν (C) acts upon any wavefunction Ψ m (φ) by the shift φ → φ+2πν vol Σ (D)/ℓ 2 . According to our previous results in (2.36) and (2.47), the Fourier wavefunction Ψ m (φ) which describes the Fock state |m; ω is given explicitly by (3.59) Immediately, the action by the operator in (3.58) on this wavefunction produces the geometric phase in the second line of (3.52).
To summarize, the Wilson loop W n (C) and the vortex loop L α (C) play dual roles. When expressed in terms of the gauge field, W n (C) acts classically by multiplication on any state |m; ω . But when expressed in terms of the scalar field, W n (C) acts quantum-mechanically as the differential (or shift) operator in (3.58). Conversely, the vortex loop L α (C) acts classically by multiplication when written in terms of φ, but quantum-mechanically as the differential (or shift) operator in (3.49) when written in terms of A.

Wilson-'t Hooft Commutation Relations
Finally, let us examine the commutation relations between the operators V k (σ), L α (C), and W n (C), all acting on the topological Hilbert space H top Σ . The idea of examining these commutators goes back to 't Hooft, and we will find a holomorphic refinement of the classic results in [16].
Collecting our previous formulas in (  where (3.64) Equivalently, C · C ′ is the topological intersection number of the curves C, C ′ ⊂ Σ.
Because n and C · C ′ are integers, the phase in (3.63) only depends upon the value of α modulo 2π, consistent with its angular nature. When C ′ is trivial in homology, the charge n of the Wilson loop can be replaced by an arbitrary real parameter ν. In this special case, the phase in (3.63) remains well-defined, since C · C ′ = 0.
Of course, the commutator in (3.63) appears in direct analogy to the celebrated commutation relation for Wilson and 't Hooft operators in four-dimensional abelian gauge theory, for which the corresponding phase is proportional to the linking number of the curves C and C ′ in R 3 . See §10.2 of [24] for a review of this story in four dimensions.
Though more or less obvious, the non-trivial commutation relation in (3.63) has an interesting consequence, because it implies that the monopole operator V k (σ) and the Wilson loop operator W n (C) similarly fail to commute. As one can check directly, To make sense of (3.65), we must work with a definite, harmonic representative for the cohomology class [C] ∨ which is Poincaré dual to [C]. Otherwise, absent a definite representative, the value of the line integral from σ 0 to σ in (3.65) would be ambiguous.
One potentially unsettling feature of the commutation relation in (3.65) is that the phase on the right-hand side appears to depend upon the auxiliary choices of the basepoint σ 0 ∈ Σ and the harmonic connection A. These choices enter the definition of the states |m; ω , but they do not enter the intrinsic definitions of the operators V k (σ) and W n (C) themselves and hence should not enter the commutator. 8 Actually, the situation is slightly better than it first appears, since the fiducial connection A is itself determined by the choice of σ 0 . We recall that A is defined as the unique harmonic connection compatible with the holomorphic structure on O Σ (σ 0 ). As we now demonstrate, the explicit dependence on σ 0 in the first phase factor of (3.65) exactly cancels against the implicit dependence of A ≡ A σ 0 on σ 0 in the second phase factor.
We begin by introducing another harmonic connection A σ , associated to the degree-one holomorphic line-bundle O Σ (σ). As harmonic connections, both A σ and A σ 0 have the same curvature, proportional to the Riemannian volume form on Σ, so the difference A σ − A σ 0 is a closed one-form. For the first phase factor in (3.65), the classical Abel-Jacobi theory now provides the very beautiful reciprocity relation exp −2πi k n σ σ 0 (3.66) See Ch. 2.2 of [14] for a textbook reference on such reciprocity laws. Substituting (3.66) into (3.65), we obtain a completely intrinsic reformulation of the commutation relation between the monopole operator and the Wilson loop, (3.67) with no dependence on the arbitrary choice of the basepoint σ 0 . We emphasize that the commutation relation in (3.67) does depend on the particular curve C ⊂ Σ, not merely the homology class [C], because A σ is not flat. Moreover, the commutator depends holomorphically on the point at which the monopole operator is inserted, through the dependence of A σ on σ.
The commutation relation in (3.67) can be understood directly in terms of either the gauge field A or the periodic scalar field φ. Via the Hecke modification in (3.14), the monopole operator V k (σ) induces the shift On the other hand, the Wilson loop operator W n (C) acts multiplicatively as in (3.50), from which (3.67) follows. Alternatively in terms of φ, the vertex operator V k (σ) acts multiplicatively, and the Wilson loop operator W ν (C) for homologically-trivial C = ∂D acts as the shift operator in (3.58), from which (3.67) again follows.
To gain a final bit of additional insight into the meaning of the Wilson-'t Hooft commutation relation, let us consider the commutation relation of W n (C ′ ) with the composite operator O k (σ, σ 0 ) ≡ V k (σ) • V −k (σ 0 ) which has vanishing monopole charge. As an immediate consequence of (3.65), (3.69) But we have already identified L α (C) as the Verlinde operator derived from O k (σ, σ 0 ), describing the creation and subsequent transport of a monopole/anti-monopole pair with fractional magnetic charge α = 2πk. As σ is transported adiabatically around a curve C based at σ 0 , the commutation relation in (3.69) reproduces the topological commutation relation of loop operators in (3.63).