Table Of Content

S-duality in Abelian gauge theory revisited Ga´bor Etesi Department of Geometry, Mathematical Institute, Faculty of Science, 1 Budapest University of Technology and Economics, 1 0 Egry J. u. 1, H ép., H-1111 Budapest, Hungary ∗ 2 ´ Akos Nagy n a Budapest University of Technology and Economics † J 3 January 4, 2011 ] G D . h Abstract t a m Definition of the partition function of U(1) gauge theory is extended to a class of [ four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) 3 ones including the multi-Taub–NUT spaces. The partition function is calculated via zeta- v function regularization and heat kernel techniques with special attention to its modular 9 properties. 3 6 In the compact case, compared with the purely topological result of Witten, we find 5 a non-trivial curvature correction to the modular weights of the partition function. But . 5 S-duality can be restored by adding gravitational counter terms to the Lagrangian in the 0 usual way. 0 1 In the ALF case however we encounter non-trivial difficulties stemming from original v: non-compact ALF phenomena. Fortunately our careful definition of the partition function i makes it possible to circumnavigate them and conclude that the partition function has the X same modular properties as in the compact case. r a AMS Classification: Primary: 81T13; Secondary: 81Q30, 57M50, 11F37, 35K08 Keywords: S-duality; Partition function; L2 cohomology; Zeta-function regularization; Heat kernel 1 Introduction The long standing conjecture asserts that quantum gauge theory has a symmetry exchanging strong and weak coupling as well as electric and magnetic fields. This conjecture originated with the work of Montonen and Olive [16] from 1977 who proposed a symmetry in quantum gauge theory with the above properties and also interchanging the gauge group G with its dual group G∨. It was soon realized however that this duality is more likely to hold in an N = 4 supersymmetrized theory [17]. ∗e-mail: [email protected] †e-mail: [email protected] ´ G. Etesi, A. Nagy: Abelian S-duality revisited 2 The original Montonen–Olive conjecture proposed a Z symmetry exchanging the electric and 2 magnetic charges however in N = 4 theory it is naturally extended to an SL(2,Z) symmetry acting on the complex coupling constant θ 4π τ := + i ∈ C+ (1) 2π e2 combining the gauge coupling e and the θ parameter of the N = 4 theory. In this framework the conjecture can be formulated as follows [24]. We say that a not necessarily holomorphic function f : C+ → C on the upper half-plane is an unrestricted modular form of weight (α,β) (conventionally supposed to be integers) if with a b respect to ∈ SL(2,Z) it transforms as (cid:18)c d(cid:19) aτ +b f = (cτ +d)α(cτ +d)βf(τ). (cid:18)cτ +d(cid:19) If α 6= 0 or β 6= 0 we also say sometimes that a modular anomaly is present in f. Since SL(2,Z) 1 1 0 −1 is generated by T = and S = modular properties are sufficient to be checked (cid:18)0 1(cid:19) (cid:18)1 0 (cid:19) under the simple transformations f(τ) 7→ f(τ + 1) and f(τ) 7→ f(−1/τ). Modular forms play an important role in classical number theory [15]. The electric-magnetic duality conjecture in its simplest form asserts that over a four-manifold (M,g) the partition function of a (twisted) N = 4 supersymmetric quantum gauge theory with simply-laced gauge group G is modular in the sense that it satisfies Z(M,g,G,τ +1) = Z(M,g,G,τ) (“level 1 property”) (cid:26) Z(M,g,G,−1/τ) = Z(M,g,G∨,τ) (“S-duality property”) where G∨ is the Goddard–Nuyts–Olive or Langlands dual group to G. The first symmetry is a classical one while the second is expected to reflect the true quantum nature of gauge theories in the sense that it connects two theories on the quantum level which are classically different. In ∼ general G is not isomorphic to its dual however for example if G = U(1) then it is. Moreover if θ = 0 this duality reduces to the original Montonen–Olive conjecture. We do not attempt here to survey the long, diverse and colourful history ofthe conjecture and itsvariantsrather refer to theintroductions of[24, 14]. Wejust mention thatforinstance it ledto highly non-trivial predictions about the number of L2 harmonic forms on complete manifolds due to Sen [20] (cf. also [10, 12, 19]) and the latest chapter of the story relates the electric-magnetic duality conjecture with the geometric Langlands program of algebraic geometry due to Kapustin and Witten [14], see also [9]. In this work, motivated by papers of Witten [25, 26] we focus attention to the modular properties of the partition function Z(M,g,τ) := Z(M,g,U(1),τ) of (supersymmetric) Abelian or U(1) or Maxwell gauge theory over various four-manifolds because in this case the calculations can be carried out almost rigorously. The paper is organized as follows. In Sect. 2 we offer an extended definition of the partition function of Abelian gauge theory (cf. Eq. (11) here) such that it gives back Witten’s [25] in ´ G. Etesi, A. Nagy: Abelian S-duality revisited 3 case of compact four-manifolds moreover the definition continues to make sense for a certain class of non-compact geometries (to be called almost compact spaces, cf. Definition 2.1 here) including the multi-Taub–NUT spaces and the Riemannian Schwarzschild and Kerr solutions. The subtle point of this extended definition is imposing a natural boundary condition at infinity on finite action classical solutions of Maxwell theory in order to save the modular properties of the partition function. This boundary condition is the so-called strong holonomy condition and is used to rule out certain classical solutions having non-integer energy. It has also appeared already in the approach to SU(2) instanton moduli spaces over ALF geometries [4, 7]. Then in Sect. 3 we repeat Witten’s calculation [25] of Z(M,g,τ) in the compact case. After summing up over a discrete set leading to ϑ-functions the partition function is given by a formal infinite dimensional integral and can be calculated quite rigorously via ζ-function regularization. On the way we formulate a natural “Fubini principle” for these formal integrals which converts them into successive integrals. We come up with an expression involving the zero values of various ζ-functions and their derivatives. The zero values of these ζ-functions can be calculated by the aid of heat kernel techniques. With the help of these tools we find that Z(M,g,τ +2) = Z(M,g,τ) (2) (cid:26) Z(M,g,−1/τ) = (−i)12σ(M) τα τβ Z(M,g,τ) i.e., up to a factor it is a level 2 modular form but with non-integer weights (cf. Eq. (20) here) 1 α = (χ(M)+σ(M)+curvature corrections) 4 and 1 β = (χ(M)−σ(M)+curvature corrections) 4 yielding that the modular weights are not purely topological as claimed in [25, 26]. As a consequence S-duality fails in its simplest form but it can be saved by adding usual gravitational c-number terms to the naive Lagrangian of Maxwell theory on a curved background [25]. Before proceeding to the non-compact calculation we make a digression in Sect. 4 and clarify the role played by the strong holonomy condition. We will see that without it (i.e., simply taking the definition of the partition function from the compact case) the partition function for instance over the multi-Taub–NUT spaces would seriously fail to be modular as a consequence of the presence of classical solutions with continuous energy spectrum. Finally as a novelty in Sect. 5 we repeat the calculation over ALF spaces. S-duality over non-compact geometries is less known (cf. the case of ALE spaces in [24]). However before ob- taining some results we have toovercome another technical difficulty caused by non-compactness. Namely if one wishes to calculate the formal integrals by ζ-function regularization again then first onehas tofacethe factthat thespectra ofvariousdifferential operatorsarecontinuous hence the existence of their ζ-functions is not straightforward. Consequently we take a truncation of the original manifold and impose Dirichlet boundary condition on the boundary. This boundary condition is compatible with the finite action assumption on connections. After defining everything correctly in this framework, we let the boundary go to infinity. We will find that the modular weights converge in this limit and give back a formula very similar to the compact case above. The only difference is that the various Betti numbers which enter the modular weights are mixtures of true L2 Betti numbers and limits of Dirichlet–Betti numbers which are remnants of the boundary condition (see Eq. (26) here). ´ G. Etesi, A. Nagy: Abelian S-duality revisited 4 Consequently thepartitionfunctiontransformsakinto(2)againhoweveritsmodularanomaly cannot be cancelled by adding c-number terms. However up to a mild topological condition on the infinity of an ALF space (cf. Eq. (27) here) all these Betti numbers can be converted into L2 ones again and the cancellation of the modular anomaly goes as in the compact case hence S-duality can be saved. It is interesting that while the Riemannian Schwarzschild and Kerr not, the multi-Taub–NUT family satisfies this topological condition. We clarify at this point that throughout this paper questions related to the contributions of various determinants to the partition function will be suppressed. We close this introduction by making a comment about the way of separating rigorous mathematical steps from intuitive ones in the text. To help the reader we decided to formulate every rigorous steps in the form of a lemma (with proof) or with a clear reference to the literature while the other considerations just appear continuously in the text. Acknowledgement. The authors are grateful to Gyula Lakos and Szila´rd Szabo´ for the stim- ulating discussions. The first author was partially supported by OTKA grant No. NK81203 (Hungary). 2 Preliminary calculations In this section we formulate our problem as precisely as possible. Let (M,g) be a connected, oriented, complete Riemannian four-manifold without boundary. M can be either compact or non-compact. In the non-compact case we require (M,g) to have infinite volume. Let L be a smooth complex line bundle over M. The isomorphism classes of these bundles are classified by the elements of the group H2(M;Z) via their first Chern class c (L) ∈ H2(M;Z). 1 Putting aHermitianstructure ontoLwecanpickaU(1)-connection∇withcurvatureF . Under ∇ u(1) ∼= iR thecurvature is aniR-valued 2-formclosed by Bianchi identity hence [F∇] ∈ H2(M;R) 2πi is a de Rham cohomology class taking its values in the integer lattice. This cohomology class (the Chern–Weil class of L) is independent of the connection but it is a slightly weaker invariant of L namely it characterizes it up to a flat line bundle only. Note that flat line bundles over M satisfy c (L) ∈ Tor(H2(M;Z)). 1 With real constants e and θ the usual action of (Euclidean) electrodynamics extended with the so-called θ-term over (M,g) looks like 1 iθ S(∇,e,θ) := − F ∧∗F + F ∧F . (3) 2e2 Z ∇ ∇ 16π2 Z ∇ ∇ M M The θ-term is a characteristic class hence its variation is identically zero consequently the Euler– Lagrange equations of this theory are just the usual vacuum Maxwell-equations dF = 0, δF = 0. (4) ∇ ∇ Remark. If one considers the underlying quantum field theory then θ in (3), being a non- dynamical variable, remains well-defined at the full quantum level meanwhile the coupling constant e runs. Therefore, in order to keep its meaning at the full quantum level we extend (3), as usual, to an N = 4 supersymmetric theory [17, 24]. This extension yields additional terms to (3) however their presence do not influence our forthcoming calculations therefore we shall not mention them explicitly in this paper. ´ G. Etesi, A. Nagy: Abelian S-duality revisited 5 Rather we introduce the complex coupling constant (1) taking its values on the upper half-plane and supplementing (3) re-write the relevant part of the action as iπ 1 iπ 1 S(∇,τ) = τ  (F ∧∗F +F ∧F )+ (−τ) (F ∧∗F −F ∧F ). 2 8π2 Z ∇ ∇ ∇ ∇ 2 8π2 Z ∇ ∇ ∇ ∇  M   M  (5) For clarity we note that this expression is exactly the same as (3). The orientation and the metric on M is used to form various Sobolev spaces. Fix a line bundle L over M and a connection ∇0 on it such that iF ∈ L2(M;Λ2M). For any integer L ∇0 L l ≧ 2 then set A(∇0) := {∇ |∇ := ∇0 +a with ia ∈ L2(M;Λ1M)}. L L L L l This is the L2 Sobolev space of U(1) connections on L relative to ∇0. Notice that this is a vector l L space (not an affine space) and if L ∼= M × C is the trivial line bundle and ∇0 = d is the trivial flat connection in the trivial ga0uge on it then A(∇0 ) ∼= L2(M;Λ1M). FurthLe0rmore write L0 l U (1) for the L2 -completion of the space of gauge transformations L l+1 {γ −Id ∈ C∞(M;EndL)|kγ −Id k < +∞; γ ∈ C∞(M;AutL) a.e.}. L 0 L L2 (M) l+1 UndertheseassumptionstheSobolevmultiplicationtheoremensuresusthatiF ∈ L2(M;Λ2M) ∇L that is, the curvature 2-form is always in L2 over (M,g). The space A(∇0) is acted upon by L U (1) in the usual way; the orbit space A(∇0)/U (1) of gauge equivalence classes with its L L L quotient topology is denoted by B(∇0) as usual. L Next we record an obvious decomposition of the action. This decomposition plays a crucial role because it underlies the definition of the partition function. Suppose that ∇0 ∈ A(∇0) is a L L finite action classical solution on L i.e., it satisfies (4) and has finite action (3) or (5); moreover let ∇ ∈ A(∇0) be another connection on the same line bundle L. Then it follows from (5) that L L Imτ S(∇ ,τ) = S(∇0 +a,τ) = S(∇0,τ)+ kdak2 (6) L L L 8π L2(M) where we write kdak2 = − da∧∗da since a is pure imaginary. This decomposition is a L2(M) M straightforward calculation if MRis compact while follows from the L2 control on the curvature of ∇0 and the L2 control on the perturbation a if M is non-compact. L 1 Now we turn to clarify what shall we mean by a “partition function” in this paper. In our attempt to define it we have to be careful because we want to include the case of non-compact base spaces as well while we require that our definition should agree with Witten’s [25, 26] in the compact case. The above decomposition of the action indicates that finding the partition function i.e., the integration over all finite-action connections should be carried out in two steps: (i) summation over a certain set of classical solutions ∇0 (this will coincide with summation over L all line bundles L in the compact case) and then (ii) integration over the orbit space B(∇0 ) for L0 the trivial flat connection ∇0 in Coulomb gauge on the trivial line bundle L . L0 0 We begin with the description of the orbit space. First we have a usual gauge fixing lemma. Lemma 2.1. Let (M,g) be as above i.e., a connected, oriented complete Riemannian four- manifold which is either closed (compact without boundary) or open (non-compact without boundary) with infinite volume. Consider the trivial line bundle L ∼= M ×C on it. 0 If ∇ = d + A with A ∈ A(∇0 ) is any U(1)-connection on L then there exists an L2 L0 L0 0 l+1 gauge transformation γ : M → U(1) such that the resulting L2 connection ∇′ = d+A′ satisfies l L0 the Coulomb gauge condition δA′ = 0. ´ G. Etesi, A. Nagy: Abelian S-duality revisited 6 Proof. Take a gauge transformation γ = eif with a function f : M → R. We want to solve the equation 0 = δA′ = δ(A + idf) i.e., △ f = iδA. This equation is solvable if and only if the 0 real-valued function iδA is orthogonal in L2 to the cokernel of the scalar Laplacian △ = δd on 0 (M,g). Note that △ is formally self-adjoint hence in fact we need iδA ⊥ ker△ . However 0 L2 0 ker△ ∼= R if M is closed hence the result follows from Stokes’ theorem. While ker△ ∼= {0} 0 0 if M is non-compact but complete and has infinite volume by a theorem of Yau [28] hence the 3 result also follows. The time has come to introduce L2 cohomology groups on (M,g) in the usual way [12]: we say that a real k-form ϕ belongs to the kth reduced L2 cohomology group Hk (M) if and only if L2 kϕk < +∞ as well as dϕ = 0 and δϕ = 0. Over a complete manifold we can equivalently L2(M) say that ϕ has finite L2 norm and is harmonic i.e., △ ϕ = 0 where △ is the Laplacian on k- k k forms. If H k(M) denotes thespace of L2 harmonicforms thenby definition Hk (M) ∼= H k(M) L2 L2 L2 consequently inthis paper we will write H k(M) for these groups. In general we call a non-trivial L2 L2 harmonic form on a complete Riemannian manifold non-topological if either it is exact or not cohomologoustoacompactlysupporteddifferentialform. Roughlyspeaking theexistence ofnon- topological L2 harmonic forms are not predictable by topological means (cf. [19]). Note that in the compact case the L2 groups reduce to ordinary de Rham cohomology groups Hk(M;R) and of course there are no non-topological L2 harmonic forms. Write bk (M) := dimH k(M) for the L2 L2 corresponding L2 Betti number (if finite). Of course in the compact case bk (M) = bk(M) are L2 just the ordinary finite Betti numbers. Most of these groups admit natural interpretations in terms of our theory (3) as we will see shortly. Lemma 2.2. Let M be an arbitrary connected manifold and denote by M the space of gauge equivalence classes of L2 flat U(1)-connections on M (also called the character variety of M). c (i) There is an identification M ∼= Tb1(M) ×Tor(H (M;Z)) 1 where Tk is a k-torus. c (ii) If (M,g) is a Riemannian four-manifold as in the previous lemma then we have a more informative identification H 1(M) M ∼= L2 ×Tor(H (M;Z)) 2πΛ1 1 L2 where Λ1 is a co-compact integecr lattice in H 1(M). L2 L2 Moreover keep in mind that H 1(M) ∼= H1(M;R) is ordinary de Rham cohomology if M is L2 compact. Remark. The lemma implies that if M is a connected, oriented, non-compact, complete four- manifold with infinite volume then b1 (M) = b1(M) hence there are no non-topological L2 L2 harmonic 1-forms. Proof. The gauge equivalence classes of flat U(1)-connections are classified by the space ∼ M = Hom(π (M); U(1))/adU(1) = Hom(π (M); U(1)) 1 1 since U(1) is Abeliacn. Also commutativity implies that the commutator group [π (M),π (M)] 1 1 lies in the kernel of any such homomorphism hence π (M) can be replaced by 1 π (M)/[π (M),π (M)] ∼= H (M;Z). 1 1 1 1 ´ G. Etesi, A. Nagy: Abelian S-duality revisited 7 Write H (M;Z) = Λ × Tor(H (M;Z)) with Λ ∼= Zb1(M). Homomorphisms ρ : Z → U(1) are 1 1 1 1 classified by their trace tr(ρ(1)) ∈ S1 ⊂ C on the generator 1 ∈ Z hence the parameter space is S1. The homomorphisms of the finite Abelian group ρ : Tor(H (M;Z)) → U(1) are certainly 1 classified by its elements. This yields part (i) of the lemma. Regarding the second part, suppose that ∇0 is the trivial flat U(1)-connection on the trivial bundle L ∼= M ×C in the straightforward gauLg0e i.e., when ∇0 = d. It follows from (6) that its 0 L0 perturbation∇ := ∇0 +awitha ∈ A(∇0 )isflatifandonlyifda = 0. ImposingtheCoulomb L0 L0 L0 gauge condition δa = 0 (cf. Lemma 2.1) apparently the gauge-inequivalent perturbations are parameterized by the group H 1(M). Indeed, if two flat connections d + a and d + b on L L2 0 are in Coulomb gauge then it easily follows from the proof of Lemma 2.1 that there is no gauge transformation between them of the form γ = eif with a single-valued function M → R. However if M is not simply-connected then there may still exists a gauge transformation such that b = a+γ−1dγ with γ = eiF where F is a multi-valued real function on M (or a single-valued function on the universal cover M). More precisely, if ℓ : [0,1] → M is a smooth loop with base point x ∈ M then γ(ℓ(0)) = γ(ℓ(1)) = γ(x ) implies 0 f 0 F(x ) = F(ℓ(1)) = F(ℓ(0))+2πn = F(x )+2πn 0 ℓ 0 ℓ with some integer n . Consider a flat connection d + a with a ∈ A(∇0 ) and also pick a differentiable loop ℓ aℓs before. Associated with this loop take a gauge transLf0ormation γ = eiFℓ ℓ such that F is supported in a tubular neighbourhood of ℓ and F (ℓ(t)) = F (x ) + 2πt. Then ℓ ℓ ℓ 0 obviously γ ∈ U (1) and ℓ L0 a+γ−1dγ = a+2πidt ℓ ℓ moreover the new connection depends only on the homotopy class [ℓ]. Consider the lattice Λ′ in H 1(M) generated by the above translations when [ℓ] ∈ π (M,x ) runs over the homotopy L2 1 0 classes. It follows that the truely gauge-inequivalent flat connections are given by the quotient H 1(M)/Λ′. But comparing this quotient with part (i) of the lemma we conclude that Λ′ is L2 co-compact hence it must coincide with a full integer lattice of H 1(M). This gives a connected L2 component of M in the second picture. Note that all the flat L2 perturbations of the trivial flat connection are connections on the c same trivial bundle L . The second integer cohomology has the straightforward decomposition 0 H2(M;Z) = Λ2 ×Tor(H2(M;Z)) (7) into its free part Λ2 ∼= Zb2(M) and its torsion. We obtain that generic elements (a,α) ∈ M are flat connections of the shape ∇0 +a where ∇0 is a fixed flat connection on the non-trivial flat L L c line bundle satisfying c (L) = α ∈ Tor(H2(M;Z)) ∼= Tor(H (M;Z)) (8) 1 1 and a ∈ H 1(M)/2πΛ1 as above. This observation provides the description of M in part (ii) of L2 L2 3 the lemma. c Therefore in light of the previous lemmata and (6) elements of H 1(M) can be interpreted as L2 flat L2 perturbations of a connection ∇0 on a given line bundle L over M. L Now we can provide an explicit description of our relevant orbit space as follows. ´ G. Etesi, A. Nagy: Abelian S-duality revisited 8 Lemma 2.3. The orbit space of gauge inequivalent L2 Abelian connections (l ≧ 2) over the trivial l line bundle L with respect to the trivial flat connection ∇0 and the Coulomb gauge condition 0 L0 admits a decomposition H 1(M) B(∇0 ) ∼= L2 × (H 1(M))⊥ ∩kerδ , kerδ ⊂ L2(M;Λ1M). L0 2πΛ1 L2 l L2 (cid:0) (cid:1) That is, it is a Hilbert space bundle over a finite dimensional torus representing the gauge inequivalent flat connections on L (cf. part (ii) of Lemma 2.2). 0 We note again that if M is compact then L2 objects reduce to ordinary de Rham cohomology. Proof. Consider the trivial flat connection ∇0 on L and write any connection in the form L0 0 ∇ = ∇0 +a with a ∈ A(∇0 ) as usual. By the aid of Lemma 2.1 let us impose the Coulomb L0 L0 L0 gauge condition δa = 0 on them. An advantage of L2 cohomology is that when l ≧ 2 the usual Hodge decomposition continues to hold [12] in the form L2(M;Λ1M) ∼= imd⊕imδ ⊕H 1(M) (9) l L2 and obviously imδ⊕H 1(M) ⊆ kerδ. If γ ∈ U (1) is any gauge transformation on the trivial L2 L0 bundle then γ−1dγ is closed moreover δ(a+γ−1dγ) = 0 together with δa = 0 implies that γ−1dγ is also co-closed and of course iγ−1dγ ∈ L2(M;Λ1M). In other words iγ−1dγ ∈ H 1(M). L2 Hence Hodge decomposition shows that U (1) acts trivially on (H 1(M))⊥ ∩ kerδ. Conse- L0 L2 quently all elements here are gauge inequivalent while Lemma 2.2 shows that the finite dimensional torus H 1(M)/2πΛ1 ∼= Tb1(M) enumerates the gauge inequivalent flat connections. 3 L2 L2 Now we move on to clarify the set of classical solutions used in this paper. This will also provide us with a physical interpretation of the space H 2(M) as containing the curvatures of finite L2 action classical solutions over a 4-space (M,g). First we introduce the class of four-manifolds considered in this paper. Definition 2.1. Let (M,g) be a connected, oriented, complete Riemannian four-manifold. This space is called almost compact if it is either compact or is an ALF space with the following condition on its 2nd L2 cohomology: all non-topological L2 harmonic 2-forms are exact. For the definition of an ALF space we refer to Sect. 5. It follows from [12, Corollary 9] that an ALF space always has finite dimensional second L2 cohomology. Many important non-compact manifolds in mathematical physics are almost compact. Examples are the multi-Taub–NUT spaces, the Riemannian Schwarzschild and Kerr geometries, etc. Writing H2(M;R) ⊆ H 2(M) i.e., embedding compactly supported de Rham cohomology L2 into L2 cohomology by the unique harmonic representative in each cohomology class, we can suppose that Λ2 ⊂ H 2(M) where Λ2 is the integer lattice from (7). L2 Definition 2.2. Let (M,g) be an almost compact four-manifold and consider its 2nd L2 cohomology. We define a co-compact lattice Λ2 ⊂ H 2(M) as follows. We say that ω ∈ Λ2 if and L2 L2 L2 only if ´ G. Etesi, A. Nagy: Abelian S-duality revisited 9 (i) either ω ∈ Λ2 i.e., it is a harmonic representative of a usual compactly supported integer de Rham cohomology class; (ii) orif ω ∈ Λ2 −Λ2 then there existsan U(1)-connection∇0 on the trivial bundleL ∼= M×C L2 L0 0 with curvature F /2πi = ω and this connection satisfies the strong holonomy condition ∇0 L0 at infinity. For the definition of the strong holonomy condition we refer to [7, Definition 2.1]. Roughly speaking if a finite action U(1)-connection satisfies the strong holonomy condition then it has trivial holonomy at infinity hence in an appropriate L2 norm it approaches the trivial flat U(1)- 1 connection on the infinitely distant boundary of (M,g). This provides us that the action (3), or equivalently (5), of such a connection is an integer, cf. [7, Theorem 2.2]. Moreover it is clear that if (M,g) is compact then part (ii) of Definition 2.2 is vacuous. Chern–Weil theory says that if ω ∈ Λ2 there exists a line bundle L with c (L) ∈ Λ2 ⊆ Λ2 , L2 1 L2 the integer lattice in (7), and an U(1)-connection ∇0 on the bundle such that ω = F /2πi. L ∇0 L This connection is obviously a finite action classical solution to (4). Moreover it follows from part (ii) of Lemma 2.2 and especially from (8) that the remaining line bundles in (7) i.e., those with c (L) ∈ Tor(H2(M;Z)) also carry classical solutions ∇0 to (4) namely vacuum solutions 1 L i.e., flat connections. WeconcludethatallelementsofthelatticeΛ2 ×Tor(H2(M;Z))giverisetoclassical solutions L2 of the Abelian gauge theory (3). In terms of (5) or the projected fields F± = 1(F ±∗F ) ∇0L 2 ∇0L ∇0L all of them satisfy b± (M) 1 1 L2 F ∧∗F ±F ∧F = F± ∧∗F± = −2 n2 (10) 8π2 Z (cid:16) ∇0L ∇0L ∇0L ∇0L(cid:17) 4π2 Z ∇0L ∇0L Xk=1 k M M with n ∈ Z some integers. We emphasize again that the validity of (10) for the non-topological k sector of Λ2 follows from the boundary condition imposed on it in part (ii) of Definition 2.2. L2 Remark. In general on a given bundle these classical solutions are not unique because of two reasons. The first is if b2(M) < b2 (M) which can happen if M is not compact, cf. Sect. 4. The L2 second is if H 1(M) 6= {0} which can happen if M is not simply connected, cf. Lemma 2.2. L2 Now we are in a position to carefully define the central object of our interest here. Consider a finite action classical solution ∇0 such that either its curvature represents an element in Λ2 L L2 or it is a flat connection whose gauge class is in Tor(H2(M;Z)). In this case we write simply [∇0] ∈ Λ2 ×Tor(H2(M;Z)) and these solutions will be referred to as allowed classical solutions. L L2 Moreover let (M,g) be an almost compact four-manifold as in Definition 2.1. Our primary concern in this paper will be the calculation of the formal integral 1 Z(M,g,τ) := e−S(∇L,τ)D∇ (11) Vol(U (1) ) Z L X L [∇0L]∈(Λ2L2×Tor(H2(M;Z))) ∇L∈A(∇0L) or equivalently Z(M,g,τ) := e−S(∇L,τ)D[∇ ] L Z X [∇0L]∈(Λ2L2×Tor(H2(M;Z))) [∇L]∈B(∇0L) ´ G. Etesi, A. Nagy: Abelian S-duality revisited 10 whichgives risetotherelevant partofthe(Euclidean) partition functionofthesupersymmetrized Abelian gauge theory (3) over (M,g). We note again that in the supersymmetric setting τ is well-defined at the full quantum level hence its appearance in Z(M,g,τ) is meaningful. Here D∇ denotes the hypothetical (probably never definable) measure on the infinite dimensional L vector space A(∇0) and D[∇ ] is the induced one on the orbit space B(∇0). L L L Notice that if M is compact the summation reduces to a summation over line bundles i.e., over H2(M;Z) in accord with [25, 26]. However if M is non-compact then the summation is taken over more finite action classical solutions than bundles. Moreover we will see in Sect. 4 that these allowed classical solutions are not the whole set of finite action classical solutions due to the strong holonomy condition from Definition 2.2. Next we perform the straightforward summation in (11). Pick a bundle L and an allowed classical solution ∇0 on it. With respect to ∇0 write a connection ∇ ∈ A(∇0) on the same L L L L bundle in the form ∇0 + a. Consider the associated decomposition (6) of the action. To be L precise we regard ∇ on L ∼= L⊗L as a perturbation of ∇0 on L with a connection d+a on L 0 L L . It then follows that the integral in (11) looks like 0 e−S(∇L,τ)D[∇L] = e−S(∇0L,τ) e−I8mπτkdak2L2(M)D[a] Z Z [∇L]∈B(∇0L) [a]∈B(∇0L) where D[a] = D[∇ ] is the formal induced measure on B(∇0). L L As we have seen bundles with c (L) ∈ Tor(H2(M;Z)) are flat i.e., the action (5) vanishes 1 along them, consequently the summation in (11) over the torsion part simply gives a numerical factor in the partition function leaving us with a summation over the free lattice part as follows: Z(M,g,τ) = Tor(H2(M;Z))  e−S(∇0L,τ) e−I8mπτkdak2L2(M)D[a] Z (cid:12)(cid:12) (cid:12)(cid:12)[∇0LX]∈Λ2L2 [a]∈B(∇0L0) where ∇0 is the trivial flat connection on the trivial line bundle L . L0 0 On the way we introduce our theta function ϑ : C+ → C given by +∞ ϑ(τ) := eiπn2τ = 1+2 eiπn2τ Xn∈Z Xn=1 having the following properties (cf. e.g. [15, Theorem 7.12]): it is holomorphic on the upper half-plane moreover satisfies the functional equations ϑ(τ +2) = ϑ(τ) (“level 2 property”) (12) (cid:26) ϑ(−1/τ) = (τ/i)12 ϑ(τ) (“modularity of weight 1 property”) 2 where the square root is the principal value cut along the negative real axis. By the aid of this function we proceed as follows. The left hand side of (10) provides us with indefinite L2 scalar products on Λ2 ⊂ H 2(M) according to the splitting Λ2 = Λ+ ×Λ− into self-dual and anti- L2 L2 L2 L2 L2 self-dual parts. This splits the second L2 Betti number as b2 (M) = b+ (M)+b− (M). Note that L2 L2 L2 if M is compact and simply connected then b± (M) are just the usual signature decomposition L2 of b2(M). Therefore making use of (10) the summation in (11) over the remaining lattice Λ2 L2 gives Z(M,g,τ) = Tor(H2(M;Z)) ϑ(τ)b+L2(M)ϑ(−τ)b−L2(M) e−I8mπτkdak2L2(M) D[a]. (13) Z (cid:12) (cid:12) (cid:12) (cid:12) B(∇0 ) L0