Electric-magnetic Duality of Abelian Gauge Theory on the Four-torus, from the Fivebrane on T2 x T4, via their Partition Functions

We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T4 with flat metric using Dirac quantization. In addition to an SL(4, Z) symmetry, it possesses SL(2,Z) symmetry that is electromagnetic S-duality. We show explicitly how this SL(2, Z) S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2,0) tensor theory, by computing the partition function of a single fivebrane compactified on T2 x T4, which has SL(2,Z) x SL(4,Z) symmetry. If we identify the couplings of the abelian gauge theory \tau = {\theta\over 2\pi} + i{4\pi\over e^2} with the complex modulus of the T2 torus, \tau = \beta^2 + i {R_1\over R_2}, then in the small T2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4d gauge theory. In this way the SL(2,Z) symmetry of the 6d tensor partition function is identified with the S-duality symmetry of the 4d gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the SL(2,Z) acts suitably. For the 4d gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.


Introduction
Four-dimensional N = 4 Yang-Mills theory is conjectured to possess S-duality, which implies the theory with gauge coupling g, gauge group G, and theta parameter θ is equivalent to one with τ ≡ θ 2π + 4πi g 2 transformed by modular transformations SL(2, Z), and the group to G ∨ [1]- [3], with the weight lattice of G ∨ dual to that of G . The conjecture has been tested by the Vafa-Witten partition function on various four-manifolds [4]. More recently, a computation of the N = 4 Yang-Mills partition function on the four-sphere using the localization method for quantization, enables checking S-duality directly [5].
This duality is believed to have its origin in a certain superconformal field theory in six dimensions, the M5 brane (2, 0) theory. When the 6d, N = (2, 0) theory is compactified on T 2 , one obtains the 4d, N = 4 Yang-Mills theory, and the SL(2, Z) group of the torus should imply the S-duality of the four-dimensional gauge theory [6]- [9].
In this paper, we compare the partition function of the 6d chiral tensor boson of one fivebrane compactified on T 2 × T 4 , with that of U (1) gauge theory with a θ parameter, compactified on T 4 . We use these to show explicitly how the 6d theory is the origin of S-duality in the gauge theory. Since the 6d chiral boson has a self-dual three-form field strength and thus lacks a Lagrangian [10], we will use the Hamiltonian formulation to compute the partition functions for both theories.
In contrast, the partition function of the abelian chiral two-form on T 2 × T 4 is [12] Z 6d,chiral = tr e −2πR 6 H+i2πγ α Pα = Z 6d zero modes · Z 6d osc , where θ 1 and θ 2 are the coordinates of the two one-cycles of T 2 . The time direction θ 6 is common to both theories, the angle between θ 1 and θ 2 is β 2 , and G 5 mn is the inverse metric of G 5mn , where 1 ≤ m, n ≤ 5. The eight angles between the two-torus and the four-torus are set to zero. Section 2 is a list of our results; their derivations are presented in the succeeding sections. In section 3, the contribution of the zero modes to the partition function for the chiral theory on the manifold M = T 2 × T 4 is computed as a sum over ten integer eigenvalues using the Hamiltonian formulation. The zero mode sum for the gauge theory on the same T 4 ⊂ M is calculated with four integer eigenvalues. We find that once we identify the modulus of the T 2 contained in M , τ = β 2 + i R 1 R 2 , with the gauge couplings τ = θ 2π + i 4π e 2 , then the two theories are related by Z 6d zero modes = ǫZ 4d zero modes , where ǫ is due to the zero modes of the scalar field that arises in addition to F ij from the compactification of the 6d self-dual three-form. In section 4, the abelian gauge theory is quantized on a four-torus using Dirac constraints, and the Hamiltonian and momentum are computed in terms of oscillator modes. For small T 2 , the Kaluza-Klein modes are removed from the partition function of the chiral two-form, and in this limit it agrees with the gauge theory result, up to the scalar field contribution. In Appendix A, we show the path integral quantization gives the same result for the 4d gauge theory partition function as canonical quantization. However, the zero and oscillator mode contributions differ in the two quantizations. In Appendix B, we show how the zero and oscillator mode contributions transform under SL(2, Z) for the 6d theory, as well as for both quantizations of the 4d theory. We prove the partition functions in 4d and 6d are both SL(2, Z) invariant. In Appendix C, the vacuum energy is regularized. In Appendix D, we introduce a complete set of SL(4, Z) generators, and then prove the 4d and 6d partition functions are invariant under SL(4, Z) transformations.

Statement of the main result
We compute partition functions for a chiral boson on T 2 × T 4 and for a U (1) gauge boson on the same T 4 . The geometry of the manifold T 2 × T 4 will be described by the line element, with 0 ≤ θ I ≤ 2π, 1 ≤ I ≤ 6, and 3 ≤ α ≤ 5. R 1 , R 2 are the radii for directions I = 1, 2 on T 2 , and β 2 is the angle between them. g αβ fixes the metric for a T 3 submanifold of T 4 , R 6 is the remaining radius, and γ α is the angle between those. So, from (2.1) the metric is and the inverse metric is θ 6 is chosen to be the time direction for both theories. In the 4d expression (1.3) the indices of the field strength tensor have 3 ≤ i, j, k, l ≤ 6, whereas in (1.4), the Hamiltonian and momentum are written in terms of fields with indices 1 ≤ m, n, p, r, s ≤ 5. The 5-dimensional inverse in directions 1, 2, 3, 4, 5 is G 5 mn , g αβ is the 3d inverse of g αβ . The determinants are related by where G is the determinant for 6d metric G IJ . G 5 , g andg are the determinants for the 5d metric G mn , 4d metric G ij , and 3d metric g αβ respectively.
The zero mode partition function of the 6d chiral boson on T 2 × T 4 with the metric (2.3) is where Π α take integer values Π 3 = n 4 , Π 4 = n 5 , Π 6 = n 6 , and F 34 = n 1 , F 35 = n 2 , F 45 = n 3 , from section 3. We identify the integers whereg = g R −2 6 from (2.5), and the modulus so that as shown in section 3, we have the factorization where ǫ comes from the remaining four zero modes H αβγ and H 12α due to the additional scalar that occurs in the compactification of the 6d self-dual three-form field strength, From section 4, there is a similar relation between the oscillator partition functions (Gmp n m n p ) 3 and ǫ ′ is the oscillator contribution from the additional scalar, (2.14) Therefore, in the limit of small T 2 , we have We use this relation between the 6d and 4d partition functions to extract the S-duality of the latter from a geometric symmetry of the former. For τ = β 2 + i R 1 Z 6d zero modes and Z 6d osc are separately invariant, as are Z 4d zero modes and Z 4d osc , which we will prove in Appendix B. A path integral computation agrees with our U (1) partition function, as we review in Appendix A [13]. Nevertheless, in the path integral quantization the zero and nonzero mode contributions are rearranged, and although each is invariant under τ → τ −1, they transform differently under τ → − 1 τ , with Z P I zero modes → |τ | 3 Z P I zero modes and Z P I non−zero modes → |τ | −3 Z P I non−zero modes . For a general spin manifold, the U (1) partition function transforms as a modular form under S-duality [14], but in the case of T 4 the weight is zero.

Zero Modes
In this section, we show details for the computation of the zero mode partition functions. The N = (2, 0), 6d world volume theory of the fivebrane contains a chiral two-form B M N , which has a self-dual three-form field strength Since there is no covariant Lagrangian description for the chiral two-form, we compute its partition function from (1.4). As in [12], [15] the zero mode partition function of the 6d chiral theory is calculated in the Hamiltonian formulation similarly to string theory, where t = 2πR 6 and y l = 2π G l6 G 66 , with l = 1, ..5. However, y 1 and y 2 are zero due to the metric (2.3). Neglecting the integrations and using the metric (2.4) in (1.4), we find and the momentum components 3 ≤ α ≤ 5 are where the zero modes of the ten fields H lmp are labeled by integers n 1 , . . . n 10 [12]. Then (3.2) is given by (2.6).
If we identify the gauge couplings τ = θ 2π + i 4π e 2 with the modulus of and (3.9) becomes Z 4d zero modes = n 4 ,n 5 ,n 6 (3.11) Then the last four terms in the chiral boson zero mode sum (2.6) are equal to (2.7) since − π 2 when we identify the integers where ǫ = n 8 ,n 9 ,n 10

Oscillator modes
To compute the oscillator contribution to the partition function (1.1), we quantize the U (1) gauge theory with a theta term on the T 4 manifold using Dirac brackets. From (1.3), the equations of motion are ∂ i F ij = 0, since the theta term is a total divergence and does not contribute to them. So in Lorenz gauge, the gauge potential A i with field strength tensor The potential has a plane wave solution with momenta satisfying the on shell condition and gauge condition As in [11], [15] the Hamiltonian H 4d and momentum P 4d α are quantized with a Lorentzian signature metric that has zero angles with the time direction, γ α = 0. So we modify the metric on the four-torus (2.2), (2.3) to be Solving for k 6 from (4.3) we find where 3 ≤ α, β ≤ 5, and |k| ≡ g αβ k α k β . Employ the remaining gauge invariance This reduces the number of components of A i from 4 to 3. To satisfy (4.3), we can use the leaving just two independent polarization vectors corresponding to the physical degrees of freedom of a four-dimensional gauge theory.
From the Lorentzian Lagrangian and energy-momentum tensor we obtain the Hamiltonian and momentum operators where we have integrated by parts; and the conjugate momentum is Then we have up to terms proportional to A 6 and ∂ α Π α which vanish in Lorenz gauge. Note the term proportional to ǫ βγδ F γδ F αβ vanishes identically. (4.10) is equal to H 4d − iγ α P 4d α given in (1.2), and is used to compute the zero mode partition function in (2.7) via (3.8).
To compute the oscillator modes, the appearance of θ only in the combination Π α + θ 8π 2 ǫ αγδ F γδ in (4.10) suggests we make a canonical transformation on the oscillator fields Π α ( θ, θ 6 ), A β ( θ, θ 6 ) [17]. Consider the equal time quantum bracket, suppressing the θ 6 dependence, 11) and the canonical transformation (4.13) Therefore the exponent (4.10) contains no theta dependence when written in terms of Π α , which now reads Thus, for the computation of the oscillator partition function we will quantize with θ = 0. Note that had we done this for the zero modes, it would not be possible to pick the zero mode integer charges consistently. Since the zero and oscillator modes commute, we are free to canonically transform the latter and not the former.
In the discussion that follows we assume θ = 0 and drop the hats. We directly quantize the Maxwell theory on the four-torus with the metric (4.4) in Lorenz gauge using Dirac constraints [18,19]. The theory has a primary constraint Π 6 ( θ, θ 6 ) ≈ 0. We can express the Hamiltonian (4.7) in terms of the conjugate momentum as The primary Hamiltonian is defined by with λ 1 as a Lagrange multiplier. As in [15], we use the Dirac method of quantizing with constraints for the radiation gauge conditions A 6 ≈ 0, ∂ α A α ≈ 0, and find the equal time commutation relations: In A 6 = 0 gauge, the vector potential on the torus is expanded as The sum is on the dual lattice k = k α ∈ Z 3 = 0. Here we only consider the oscillator modes expansion of the potential and the conjugate momentum in in (4.9) with vanishing θ angle . (4.18) and the polarizations absorbed in From (4.17), the commutator in terms of the oscillators is We consider the Fourier transform (4.18) of the commutators (4.17), the commutators of the oscillator can be found: In A 6 = 0 gauge, we use the (4.18) and (4.21) to evaluate the Hamiltonian and momentum in (4.7) and (4.8) and we have used the on-shell condition G 66 L k 6 k 6 + |k| 2 = 0, and the transverse condition Inserting the polarizations as So the exponent in (1.2) is given by Then the U (1) partition function is From the usual Fock space argument we perform the trace on the oscillators, where Z 4d zero modes is given in (2.7). (4.31) and (4.37) are each manifestly SL(3, Z) invariant due to the underlying SO(3) invariance we have labeled as α = 3, 4, 5. We use the SL(3, Z) invariant regularization of the vacuum energy reviewed in Appendix C to obtain √g (g αβ n α n β ) 2 (4.32) On the other hand, one can evaluate the oscillator trace for the 6d chiral boson from (1.4) as in [12], [15]. The exponent in the trace is , and Π 6mn is the momentum conjugate to B M N . In the gauge B 6n = 0, the normal mode expansion for the free quantum fields B mn and Π mn on a torus is given in terms of oscillators B κ p and C κ † p defined in [12], with the commutation relations where 1 ≤ κ, λ ≤ 3 labels the three physical degrees of freedom of the chiral two-form, and p = (p 1 , p 2 , p α ) lies on the integer lattice Z 5 . From the on-shell condition G LM p L p M = 0, Thus the oscillator partition function of the chiral two-form on T 2 ×T 4 is obtained by tracing over the oscillators Regularizing the vacuum energy in the oscillator sum [12] yields where n ∈ Z 5 is on the dual lattice, G mp is defined in (2.2), and Z 6d zero modes is given in (2.6). Comparing the 4d and 6d oscillator traces (4.31) and (4.36), the 6d chiral boson sum has a cube rather than a square, corresponding to one additional polarization, and it contains Kaluza-Klein modes. In Appendix D, we prove that the product of the zero mode and the oscillator mode partition function for the 4d theory in (4.32) is SL(4, Z) invariant. In (D.48) we give an equivalent expression, where 5 ≤ a ≤ 6, with < H > 4d p ⊥ defined in (C.3). In Appendix D, we also prove the SL(4, Z) invariance of the 6d chiral partition function (4.37), using the equivalent form (D.65), zero modes · e πR 6 6R 3 with < H > 6d p ⊥ in (D.64). In the limit when R 1 and R 2 are small with respect to the metric parameters g αβ , R 6 of the four-torus, the contribution from each polarization in (4.38) and (4.39) is equivalent. To see this limit since we can separate the product on n ⊥ = (n 1 , n 2 , n a ) = 0 ⊥ in (4.39), into (n 1 = 0, n 2 = 0, n a = (0, 0)), (n 1 = 0, n 2 = 0 all n a ), (n 1 = 0, n 2 = 0, all n a ), (n 1 = 0, n 2 = 0, all n a )) to find, at fixed n 3 , Thus for T 2 smaller than T 4 , the last three products reduce to unity, so (4.41) The regularized vacuum energies in (C.3) and (D.64), have the same form of spherical Bessel function, but the argument differs by modes (p 1 , p 2 ). Again separating the product on n ⊥ = (n 1 , n 2 , n a ) in (4.39), into (n 1 = 0, n 2 = 0, n a = (0, 0)), (n 1 = 0, n 2 = 0 all n a ), (n 1 = 0, n 2 = 0, all n a ), (n 1 = 0, n 2 = 0, all n a )) we have In the limit R 1 , R 2 → 0, the last three products are unity. For example, the second is unity because for n 1 , n 2 = 0, Thus in the limit when T 2 is small with respect to T 4 , (4.46) We have shown the partition functions of the chiral theory on T 2 × T 4 and of gauge theory on T 4 , agree in the small T 2 limit upon neglecting less interesting contribution ǫ ′ , (4.47) Again, ǫ ′ is equivalently the contribution from one polarization, that is The relation between the 4d gauge and 6d tensor partition function is shown in the small T 2 limit,   S and T are the generators of SL(2, Z), S : τ → − 1 τ , T : τ → τ − 1, where τ = θ 2π + i 4π e 2 is also given by the modulus of the two-torus, τ = β 2 + i R 1 R 2 .

Discussion and Conclusions
We computed the partition function of the abelian gauge theory on a general four-dimensional torus T 4 and the partition function of a chiral boson compactified on T 2 ×T 4 . The coupling for the 4d gauge theory, τ = θ 2π + i 4π e 2 , is identified with the complex modulus τ = β 2 + i R 1 R 2 of T 2 . Assuming the metric of T 2 is much smaller than T 4 , the 6d partition function factorizes to a partition function for gauge theory on T 4 and a contribution from the extra scalar arising from compactification. The 6d partition function has a manifest SL(2, Z) × SL(4, Z) symmetry. Therefore the SL(2, Z) symmetry with the group action on the coupling, τ = θ 2π + i 4π e 2 , known as S-duality becomes manifest in the 4d Maxwell theory.
The 6d chiral boson has no Lagrangian, so we use the Hamiltonian approach to compute both the 4d and 6d partition functions. For gauge theory, the integration of the electric and magnetic fields as observables around one-and two-cycles respectively take integer values due to charge quantization. We sum over all possible integers to get the zero mode partition function. For the oscillator mode calculation, we quantize the gauge theory using the Dirac method with constraints. In 6d, the partition function follows from [12], [15].
We have also given the result of the 4d partition function, computed by the path integral formalism. It agrees with the partition function obtained with the Hamiltonian formulation. However, the path integral form factors into zero modes and oscillator modes differently, which leads to different SL(2, Z) transformation properties for the components. The 6d and 4d partition functions share the same SL(2, Z) × SL(4, Z) symmetry.
If we consider supersymmetry, compactification of the 6d theory on T 2 leads to N = 4 gauge theory in the limit of small T 2 . On the other hand, an N = 2 theory of class S [21], [22] arises when the 6d, (2, 0) theory is compactified on a punctured Riemann surface with genus g. Here the mapping class group of the Riemann surfaces acts as a generalized S-duality on 4d super-Yang-Mills theory [23]- [25]. In another direction, we can study the 2d conformal field theory present when 6d theory is compactified on a four-dimensional manifold. The 2d-4d relation can also be studied from a topological point of view [26], [27]. Finding explicit results, such as we have derived for T 2 × T 4 , for these more general investigations would be advantageous.

A Comparison of the 4d U(1) partition function in the Hamiltonian and path integral formulations
For convenience in comparing the 4d gauge theory and the 6d theories in sections 2 and 3, we quantized both theories using canonical quantization. Since in 4d a Lagrangian exists for the gauge theory, it is useful to check that the path integral quantization agrees with canonical quantization. We find that the two formulations distribute the zero and oscillator mode contributions differently, and thus these components transform differently under the action of SL(2, Z). We summarize the results from [13], [14], [28]. Following [14], the zero modes, So the action (1.3) is given as The zero mode partition function from path integral formalism can be expressed as a lattice sum over the integral basis of m I [14], Alternatively the zero mode sum be can written in terms of the metric using (A.3) where F ij = F ij 2π = m I are integers due to the charge quantization. To compare the zero mode partition function from the Hamiltonian and path integral formalisms, we rewrite the result (2.7) as where A αβ ≡ e 2 R 6 4π √g g αβ and x α ≡ i 4π √g e 2 R 6 g αδ F δλ γ λ − β 2 2 ǫ αγδ F γδ , we get the Hamiltonian expression as Following [13], the non-zero mode partition function is defined by a path integral as Performing the functional integration with the Fadeev-Popov approach, we get where b 1 = 4 is the dimension of the group H 1 (T 4 ). ∆ p = (d † d + dd † ) p is the kinetic energy operator acting on the p-form. G 1 is defined as follows, ω m , m = 1, .., b 1 is the harmonic one-form representative with integer coefficient on T 4 which can be chosen as dθ m 2π with the appropriate normalization Σ n ω m = δ nm and thus detG 1 = detG ij = g. Also, det(∆ 1 ) = det(∆ 0 ) 4 and thus The determinant can be computed , (A.14) where we use the identity n α n 6 + G αβ n α n β = exp − 1 2 nα n 6 ln 1 R 2 6 (n 6 + γ α n α ) 2 + g αβ n α n β .
(A. 16) Let µ(E) ≡ n 6 ln 1 R 2 6 (n 6 + γ α n α ) 2 + E 2 , where E 2 ≡ g αβ n α n β , ρ = 2πR 6 , After integration, we have where the constant ln( ) maintains the SL(4, Z) invariance of the partition function. So, Therefore, using (A.13), we have Together with (A.9), the partition functions from the two quantizations agree but they factor differently into zero and oscillator modes, B SL(2, Z) invariance of the Z 6d,chiral and Z 4d,M axwell partition functions The S-duality group SL(2, Z) group has two generators S and T which act on the parameter τ to give , the transformation S corresponds to and T corresponds to Or equivalently which for θ = 0 is the familiar electromagnetic duality transformation e 2 4π → 4π e 2 .
6d partition function The 6d chiral boson zero mode partition function (2.6), Z 6d zero modes = n 8 ,n 9 ,n 10 where (B.6) has no β 2 dependence and therefore is invariant under T . (B.7) transforms under under T to become which is equivalent to (B.7) in the sum where we shift the integer zero mode field strength H 1αβ to H 1αβ − H 2αβ . Under S, we see (B.6) as a function of R 1 R 2 is invariant, and find (B.7) transforms to − π 2 So by shifting the integer field strength tensors H 1αβ → H 2αβ and H 2αβ → −H 1αβ , the sum on (B.7) is left invariant by S. Thus we have proved SL(2, Z) invariance of the 6d zero mode partition function (2.6), and that its factors ǫ and Z 4d zeromodes in (2.9) are separately SL(2, Z) invariant.
For the oscillator modes (4.36), the only term that transforms in the sum and product is which is invariant under T by shifting the momentum p 1 → p 1 + p 2 . With the S transformation,p 2 becomes and by also exchanging the momentum p 1 → p 2 and p 2 → −p 1 , the term remains the same. So the 6d oscillator partition function (4.36) is SL(2Z) invariant, which holds also for regularized vacuum energy as given in (4.37).
4d U (1) partition function In the Hamiltonian formulation, SL(2, Z) leaves invariant the U (1) oscillator partition function (4.30). We have also checked above, starting from 6d, that the zero mode 4d partition function (2.7) is invariant. Thus the U (1) partition function from the Hamiltonian formalism is S-duality invariant.
The S-duality transformations on the corresponding quantities in the path integral quantization can be derived from (A. 8

C Regularization of the vacuum energy for 4d Maxwell theory
The sum in (4.30) is divergent. We regularize the vacuum energy following [12], [15]. For < H >= 1 2 g αβ n α n β , the regularized vacuum energy is For the discussion of SL(4, Z) invariance in Appendix D, it is also useful to write the regularized sum (C.1), as where p ⊥ = p a ∈ Z 2 , a = 4, 5, and Rewriting the 4d metric (3,4,5,6) From (2.2) the metric on the four-torus, for α, β = 3, 4, 5, is We can rewrite this metric using a, b = 4, 5, 3) The 3d inverse of g αβ is where g ab is the 2d inverse of g ab .
The line element can be written as We define The 4d inverse is Generators of GL(n, Z) The GL(n, Z) unimodular group can be generated by three matrices [29]. For GL(4, Z) these can be taken to be U 1 , U 2 and U 3 ,   11) or equivalently which leaves invariant the line element (D.5) if dθ 3 → dθ 3 − dθ 6 , dθ 6 → dθ 6 , dθ α → dθ α . U 2 is the generalization of the usual τ → τ − 1 modular transformation. The 3d inverse metric g αβ ≡ {g ab , g a3 , g 33 } does not change under U 2 . It is easily checked that U 2 is an invariance of the 4d Maxwell partition function (4.32) as well as the 6d chiral boson partition function (4.37). It leaves the zero mode and oscillator contributions invariant separately. The other generator, U 1 is related to the SL(2, Z) transformation τ → −( τ ) −1 that we discuss as follows: where M 3 is a GL(3, Z) transformation given by and U ′ is the matrix corresponding to the transformation on the metric parameters (D. 16), it will be easy to see how it transforms under the U ′ transformation. Under U ′ from (D.16), the coefficient transforms as The Euclidean action for the zero mode computation is invariant under U ′ , as we show next by first summing i = {3, a, 6}, with 4 ≤ a ≤ 5.
Letting the U ′ transformation (D.16) act on (D.25), we see the first term of (D.25) changes The second term in the exponential of (D.23) is a topological term, and under the action, it transforms trivially as If we replace the integers F 3a → F 6a and F a6 → − F a3 , the two subterms are left invariant.
(D. 28) So we showed that under the U ′ transformation (D. 16), zero modes (R 3 , R 6 , g ab , γ 3 , κ a , γ a ); (D.29) Also from (D.23), we can write (D.22) as and thus under the GL(4, Z) generator U 1 , The residual factor | τ | 2 is sometimes referred to as an SL(2, Z) anomaly of the zero mode partition function, because U ′ includes the τ → − 1 τ transformation. Finally we will show how this anomaly is canceled by the oscillator contribution.