Non-orientable surfaces and electric-magnetic duality

We consider the reduction along two compact directions of a twisted N=4 gauge theory on a 4-dimensional orientable manifold which is not a global product of two surfaces but contains a non-orientable surface. The low energy theory is a sigma-model on a 2-dimensional worldsheet with a boundary which lives on branes constructed from the Hitchin moduli space of the non-orientable surface. We modify 't Hooft's notion of discrete electric and magnetic fluxes in gauge theory due to the breaking of discrete symmetry and we match these fluxes with the homotopy classes of maps in sigma-model. We verify the mirror symmetry of branes as predicted by S-duality in gauge theory.


Introduction
In a celebrated work [1] (see [2,3,4,5,6] for subsequent developments), Kapustin and Witten showed that a twisted version of the N = 4 supersymmetric gauge theory in four dimensions compactifies to a 2-dimensional sigma-model whose target space is the Hitchin moduli space. Electric-magnetic duality or S-duality in four dimensions reduces to mirror symmetry in two dimensions that explains the geometric Langlands programme. In this paper, we consider the reduction along two compact directions of the gauge theory on a 4-dimensional orientable spacetime manifold which is not a global product of two surfaces but contains a non-orientable surface. The low energy theory is a sigma-model on a 2-dimensional worldsheet with a boundary which lives on branes constructed from the Hitchin moduli space of the non-orientable surface. We show that the discrete symmetry from the centre of the gauge group is broken by fixing the topology types of the gauge bundles and we modify 't Hooft's notion of discrete electric and magnetic fluxes accordingly. We match these rectified discrete fluxes with the homotopy classes of relative maps in the sigma-model. We verify the mirror symmetry of branes as predicted by S-duality in gauge theory.
Throughout this paper, the gauge group G is a connected compact semisimple but not necessarily simply connected Lie group. When concerned with the coupling constant or the action of the theory, we further assume for simplicity that G is simple, but this restriction is not essential as the theory always decouples to ones whose gauge groups are the simple factors of G. The rest of the paper is organised as follows.
In §2, we revisit a few global aspects of quantum gauge theories in four dimensions. We found that the usual concept of discrete electric and magnetic fluxes of 't Hooft [7,8] requires a subtle modification. As is well known, the discrete magnetic fluxes classify the topology of the gauge bundles over a time slice whereas the discrete electric fluxes are the momenta of discrete symmetries that are present when the centre of the gauge group is non-trivial. We show that a discrete symmetry may change the topology of the bundles and is therefore broken in a fixed topological sector. This leads naturally to the notion of rectified discrete fluxes. We revisit quantum gauge theory in light of this adjustment in both canonical quantisation and the path integral formalism. We establish the S-duality between the rectified discrete electric and magnetic fluxes.
In §3, we recall the reduction along a compact orientable surface C from a twisted N = 4 gauge theory in four dimensions to a 2-dimensional sigma-model whose target space is the Hitchin moduli space M H (C, G) [1], with an emphasis on the gauge group G being an arbitrary compact semisimple Lie group. We interpret the discrete fluxes and their S-duality in gauge theory from the point of view of 2-dimensional sigma-models and their mirror symmetry. Though this will be a straightforward generalisation of [1] where G is often taken as the universal covering group G or the adjoint group G ad , we introduce a few techniques in topology that will be useful in §4. Curiously, there are less topological types of gauge bundles over a 4-dimensional spacetime than the homotopy classes of maps from the worldsheet to the target. This mismatch is resolved upon summing over some flat B-fields on the target space.
In §4, we consider gauge theory on an orientable 4-manifold which is not a global product of two surfaces but contains a non-orientable surface C ′ . When the size of C ′ becomes small, the theory reduces to a sigma-model on a worldsheet with a boundary which lives on branes constructed by the Hitchin moduli space M H (C ′ , G) for C ′ . We show that the sigma-model is anomaly-free from the 2-dimensional point of view. When the worldsheet is a cylinder, the 4-dimensional spacetime splits as a product of time and a spacelike 3-manifold. We compute the sets of rectified discrete electric and magnetic fluxes in the 4-dimensional gauge theory and match them with the homotopy classes of relative maps in the 2-dimensional sigma-model. We verify the mirror symmetry of these branes as predicted by S-duality. Finally, we apply the results to the quantisation of M H (C ′ , G) via branes and mirror symmetry [9,10].
We conclude in §5 and we collect and work out various results in topology in the Appendix.
2. Global aspects of four dimensional gauge theory 2.1. Discrete electric and magnetic fluxes. If the spacetime X is an orientable 4-manifold, for any compact Lie group G, characteristic classes of a G-bundle P over X are in H 2 (X, π 1 (G)) and H 4 (X, π 3 (G)) [11]. If G is simple, then H 4 (X, π 3 (G)) ∼ = Z contains the instanton number k(P ) which is summed over in path integral, and each term is weighted by a phase determined by a theta angle in the dual group U(1) of Z [12,13,14]. More generally, k(P ) is a class in a higher rank free Abelian group H 4 (X, π 3 (G)) and the theta angle is in its dual torus. On the other hand, the class ξ(P ) in the torsion group H 2 (X, π 1 (G)) is the obstruction to lifting its structure group G to the universal cover G. It is called the discrete flux because it gives rise to the discrete electric and magnetic fluxes [7,8].
Suppose the spacetime manifold is X = T 1 × Y , where T 1 is a circle in the time direction and Y is a spacelike closed orientable 3-manifold. As in [1], we write ξ(P ) = a + m, a = ξ 1,1 (P ), m = ξ 0,2 (P ) according to the Künneth decomposition H 2 (X, π 1 (G)) ∼ = H 1 (Y, π 1 (G)) ⊕ H 2 (Y, π 1 (G)). Since m describes the topology of the G-bundle over a time slice Y , it depends only on the field configuration at a particular time. So quantisation can be performed with a fixed value of m. We call m a discrete magnetic flux until we meet the need for modification below.
In contrast, a ∈ H 1 (Y, π 1 (G)) contains information from not only the initial data but also the entire time interval. At the quantum level, fixing a is compatible with neither the cluster decomposition property nor a Hilbert space interpretation. Instead, we define discrete electric fluxes as the momenta of a discrete translation group H 1 (Y, Z(G)). Any g ∈ H 1 (Y, Z(G)) ∼ = Hom(π 1 (Y ), Z(G)) acts on a connection by modifying its holonomy along a loop γ representing [γ] ∈ π 1 (Y ) by g([γ]) ∈ Z(G). Geometrically, g ∈ H 1 (Y, Z(G)) determines a flat Z(G)-bundle Q g → Y and a thus new G-bundle P g := Q g × Z(G) P ; a connection on P and the flat connection on Q g defines a new connection on P g . If the matter fields, such as the adjoint matter, are in representations of G in which the centre Z(G) acts trivially, the action of g on them is trivial. Since the curvature and hence the action functional is invariant under H 1 (Y, Z(G)), the group acts as classical symmetries. The symmetry survives at the quantum level because it preserves the path integral measure. The quantum Hilbert space therefore decomposes according to representation types of H 1 (Y, Z(G)), each labelled by a character e ∈ H 1 (Y, Z(G)) ∨ . We tentatively call e a discrete electric flux.
In the most general situation, however, we can not fix e ∈ H 1 (Y, Z(G)) ∨ and m ∈ H 2 (Y, π 1 (G)) simultaneously because the symmetry group H 1 (Y, Z(G)) changes the topology of a bundle. In fact [15], g ∈ H 1 (Y, Z(G)) maps ξ(P ) = m to ξ(P g ) = m + δ 1 Y (g), where δ 1 Y is the connecting homomorphism in the long exact sequence of cohomology groups of Y induced by the short exact sequence of coefficient groups. In (2.1), i 1 Y is injective and j 2 Y is surjective because Y is a closed orientable 3-manifold. Actually, δ 1 Y maps only to the subgroup Ext(H 1 (Y, Z), π 1 (G)) of H 2 (Y, π 1 (G)) (see [15] for related results on flat G-bundles). So δ 1 Y is zero if H 1 (Y, Z) is torsion-free. This torsion-free assumption was made in [16,17] and is satisfied when Y is a product of S 1 and a closed orientable surface [1]. The map δ 1 Y is also zero if the orders of torsion elements in H 1 (Y, Z) are coprime to those of π 1 (G) or if the exact sequence (2.2) splits. In all cases when δ 1 Y = 0, the symmetry H 1 (Y, Z(G)) preserves the discrete magnetic flux m ∈ H 2 (Y, π 1 (G)), and the quantum Hilbert space decomposes into sectors labelled by (e, m) ∈ H 1 (Y, Z(G)) ∨ × H 2 (Y, π 1 (G)).
For a general orientable 3-manifold Y , such as the one constructed in §4 (or §A.9) from a non-orientable surface, H 1 (Y, Z) does contain torsion elements and δ 1 Y can be non-zero. A fixed m breaks the discrete symmetry to the subgroup ker(δ 1 Y ) ⊂ H 1 (Y, Z(G)), and thus the quantum Hilbert space decomposes according to the characters in e(Y, G) := ker(δ 1 Y ) ∨ . At first sight, this suggests an apparent asymmetry between the discrete electric and magnetic fluxes. But a gauge theory sector with m ∈ H 2 (Y, π 1 (G)) is isomorphic to the one with m + δ 1 Y (g) because of the symmetry g ∈ H 1 (Y, Z(G)) itself. So the actual discrete parameters labelling non-isomorphic sectors are in m(Y, G) := H 2 (Y, π 1 (G))/ im(δ 1 Y ) = coker(δ 1 Y ). The sets e(Y, G) and m(Y, G) form an exact sequence Henceforth we call elements in e(Y, G) and m(Y, G) discrete electric and magnetic fluxes, respectively, or the rectified discrete fluxes to emphasise the adjustment. We can describe these discrete fluxes using the universal cover G, which has a trivial π 1 ( G) but a maximal centre Z( G), or the adjoint group G ad := G/Z(G), which has a trivial centre but has π 1 (G ad ) ∼ = Z( G). From (2.1) we get m(Y, G) ∼ = im(i 2 Y ) = ker(j 2 Y ) ⊂ H 2 (Y, π 1 (G ad )) = m(Y, G ad ). (2.4) So a rectified discrete magnetic flux m in a gauge theory with group G can be naturally regarded as a discrete magnetic flux of a gauge theory with group G ad . The G-bundles P with ξ(P ) ∈ m, of possibly different topological types, produces the same G ad -bundle P ad := P/Z(G) with ξ(P ad ) = m, and m(Y, G) is precisely the set of discrete fluxes of G ad -bundles on Y whose structure group can be lifted from G ad to G. (2.7) We obtain the set ker(j 2 X ) ⊂ H 2 (X, π 1 (G ad )) of discrete fluxes of G ad -bundles over X that can be lifted to G-bundles. In the (relativistic invariant) path integral formulation, we sum over both the instaton number and the discrete flux in ker(j 2 X ). If X = T 1 × Y , then (2.7) reduces to two copies of (2.1). Thus in this case we have ker(δ 1 ⊂ H 2 (X, π 1 (G ad )) ∼ = H 1 (Y, π 1 (G ad )) ⊕ H 2 (Y, π 1 (G ad )). 2.2. Quantum gauge theories revisited. We revisit quantum gauge theories, first in the path integral formalism and then in canonical quantisation, taking into account the role of the rectified discrete fluxes introduced in §2.1. We pay particular attention to the relation among theories with various gauge groups of the same Lie algebra.
As before, the spacetime is a closed orientable 4-manifold X and the gauge group G is a compact semisimple Lie group. The partition function of the gauge theory is 1 vol(G(P )) DA · · · e −S(A,··· ) , (2.8) where the sum is over the topological types [P ] of principal G-bundles over X and the integral is over the gauge and matter fields. The classical action (when G is simple) is where e > 0 is the coupling constant, θ is the theta angle, "− tr" is the inner product on the Lie algebra g of G such that the long roots are of length √ 2, and the matter fields are omitted. If the matter fields are in representations of G in which the centre Z(G) acts trivially (such as the adjoint representation), the gauge group can be changed to the adjoint group G ad := G/Z(G). A G-bundle P defines a G ad -bundle P ad := P/Z(G). In the gauge theories with gauge groups G and G ad , the gauge and matter fields are the same, but the groups of gauge transformations G(P ) and G(P ad ) are related by the exact sequence (A.16) with X as the base manifold. Formally, the volumes of the two groups are related by vol(G(P ad )) = When H 1 (X, Z) has no torsion elements, we have δ 1 X = 0 and hence ker(δ 1 is the first Betti number of X. Therefore we recover from (2.10) the formula [16] vol(G(P ad )) = |Z(G)| b1(X)−1 vol(G(P )). (2.11) To relate the partition functions of the two gauge theories, we note that ξ(P ad ) = i 2 X (ξ(P )), but the map i 2 X : H 2 (X, π 1 (G)) → H 2 (X, π 1 (G ad )) is many to one with multiplicity | ker(i 2 X )| = | im(δ 1 X )|. On the other hand, k(P ad ) = k(P ) under the isomorphism π 3 (G ad ) ∼ = π 3 (G). Using these facts and (2.10), we obtain (2.13) The above trace with g ∈ ker(δ 1 Y ) contains a sum over the homotopy types of paths [A(t)] in the space B(P m ) := Here, [A] means the gauge equivalence class of a connection A on P m . If g = 0 (andĝ = id H m Y,G ), these paths in B(P m ) are loops representing elements of π 1 (B(P m )) ∼ = π 0 (G(P m )). Homotopy groups of the group G(P m ) of gauge transformations are studied in §A.2. By (A.13), an element of π 0 (G(P m )) is a pair (k, a), where a ∈ H 1 (Y, π 1 (G)) and k ∈ η −1 * (a). Here, η * : π 0 (G(P )) → H 1 (Y, π 1 (G)) maps (the homotopy class of) a gauge transformation to the primary obstruction to its topological triviality. The set η −1 * (a) is a torsor over H 3 (Y, π 3 (G)); it can be identified, though not naturally, with H 3 (Y, π 3 (G)), and we can roughly write k ∈ H 3 (Y, π 3 (G)). Using a gauge transformation on P m in the homotopy class (k, a), we can construct a Gbundle P k,a,m over X = T 1 × Y . In the path integral for the trace with g = 0, we integrate over the gauge field A ∈ A(P k,a,m ) and the matter fields that couple to it. If g = 0, the homotopy types of such paths [A(t)] in B(P m ) form a set π 1 (B(P m ), g) which is naturally identified with ζ −1 * (g) and is a torsor over π 1 (B(P m )). Here the map , and it determines a twisted G-bundle P k,a,m over X in the presence of a discrete B-field g ∈ H 1 (Y, Z(G)) ⊂ H 2 (X, Z(G)) (see §A.3). Note that P k,a,m is not determined by g alone but requires the more refined information a. To summarise, we get where Z k,a,m Y,G (β) is an integral over the gauge and matter fields associated to a possibly twisted G-bundle P k,a,m . Combining the sums over g in (2.13) and a in (2.14), we get Here we identified e ∈ e(Y, G) with an element in e(Y, G) = (H 1 (Y, π 1 (G ad ))) ∨ by the inclusion (2.5). The bundle P k,a,m is an honest G-bundle if a ∈ ker(j 1 Y ) ∼ = H 1 (Y, π 1 (G)) but is twisted if otherwise. But P k,a,m ad := P k,a,m /Z(G) is always an honest G ad -bundle over X with ξ(P k,a,m ) as before, we get a geometric formula e(g) = e(ξ 1,1 (P k,a,m ad )) (2. 16) for computing the phase e(g) in (2.13). Using (2.6) for summing over e ∈ e(Y, G), we get in which only honest G-bundles remain. The partition function of the entire gauge theory is therefore where | im(δ 1 Y )| is the size of the coset m ∈ m(Y, G). So canonical quantisation, after summing over various sectors labelled by (e, m), agrees with the manifestly relativistic invariant path integral expression (2.8).
Finally, the relation (2.12) to the G ad -theory is compatible with canonical quantisation. The G ad -bundle P m ad := P m /Z(G) over Y has ξ(P m ad ) = i 2 Y (ξ(P m )) = m. As in the derivation of (2.12), we have In agreement with (2.12), if we sum over a ∈ H 1 (Y, π 1 (G)) ∼ = ker(j 1 Y ) and m ∈ m(Y, G) ∼ = ker(j 2 Y ), we have 2.3. S-duality of the rectified discrete fluxes. Electric charges in a gauge theory with gauge group G are in one-to-one correspondence with irreducible representations of G, or characters of G. If the spacetime is 4-dimensional, magnetic charges in the theory form the set Hom(U(1), G) of homomorphisms from U(1) to G modulo conjugations by G; they are the cocharacters of G. Given a homomorphism γ : U(1) → G, we can construct, from the Hopf fibration S 3 → S 2 or from the Dirac monopole of magnetic charge 1, a principal G-bundle over S 2 or a non-Abelian monopole on R 3 \{0} with a Yang-Mills connection [18,19]. Conjugations of γ by G yield gauge equivalent bundles and connections. This classification of solutions to the Yang-Mills equation on S 2 or on R 3 \{0} by magnetic charges is more refined than the topological classification of G-bundles [20].
In [21], Goddard, Nyuts and Olive introduced a dual, magnetic group of G, which is, by an observation of Atiyah (cf. [1]), the Langlands dual group L G. The electric charges in a gauge theory with gauge group G are in one-to-one correspondence with the magnetic charges in another gauge theory with gauge group L G, and vice versa. A simple way to see this correspondence is by the natural identifications where T, L T are the respective maximal tori of G, L G and W, L W are the respective Weyl groups. A further step is the electric-magnetic duality (also called S-duality) conjecture of Montonen and Olive [22]. It states that a gauge theory with gauge group G is, at the quantum level, isomorphic to another gauge theory with gauge group L G, in the sense that we can match the states in the quantum Hilbert spaces and the operators acting on them, exchanging electricity and magnetism in the two theories. If G is simple, under the S-duality which exchanges G and L G, the complex coupling constant τ := θ 2π + 4π √ −1 e 2 (in the upper half plane) goes to L τ = − 1 ngτ , where n g = 1 if g is simply-laced and n g = 2, 3 is the ratio of the square lengths of the long and short roots if g is not [23]. The duality transformation S : τ → − 1 ngτ and the transformation T : τ → τ + 1 (periodicity in the theta angle) generate an action on τ of the modular group SL(2, Z) if n g = 1 [24] and the Hecke group G( √ n g ) if n g > 1 [25]. S-duality is more plausible in supersymmetric gauge theories [26] and is believed to be exact in the pure N = 4 gauge theory [27]. That S-duality in various twisted N = 4 gauge theories makes remarkable predictions that are compatible with highly non-trivial mathematical results [16,1] is a strong indication of its validity. We now establish the matching under S-duality of the rectified discrete electric and magnetic fluxes in §2.1. Consider a gauge theory on X = T 1 × Y , where Y is a closed orientable 3-manifold. If the gauge group G is compact and semisimple, there are isomorphisms of finite Abelian groups (2.17) and the dual of the exact sequence (2.2) is the same sequence but for L G. From (A.4), we have natural isomorphisms If δ 1 Y = 0 in (2.1), then H 1 (Y, Z(G)) ∨ and H 2 (Y, π 1 (G)) are respectively the sets of discrete electric and magnetic fluxes, and they are exchanged under S-duality by the isomorphisms (2.18). More generally, the rectified discrete electric and magnetic fluxes are elements of the sets e(Y, G) and m(Y, G) defined in §2.1. For a gauge theory with the dual gauge group L G, the rectified discrete fluxes are in e(Y, for the Langlands dual group L G. Under the isomorphisms (2.18), the sequence (2.19) is the dual of (2.1) and L δ 1 Y can be identified with (δ 1 Y ) ∨ (see §A.1). Thus, taking the dual of (2.3), we obtain two natural isomorphisms Another way to see the above isomorphisms is through the counterparts of (2.4), (2.5) for L G, which are . Now suppose the spacetime 4-manifold X is not necessarily of the product form T 1 × Y . As in (2.12), the partition function of the theory with gauge group G contains a sum over ker(j 2 X ) ⊂ H 2 (X, π 1 (G ad )), the set of discrete fluxes of G ad -bundles on X whose structure group can be lifted to G. However, ker(j 2 X ) is not respected by S-duality. Instead, when G is exchanged with L G, ker( L j 2 X ) becomes complimentary to ker(j 2 X ) in the sense that ker( L j 2 X ) can be identified with the set of homomorphisms from H 2 (X, π 1 (G ad )) to U(1) that are trivial on ker(j 2 X ). That is, So the smaller ker(j 2 X ) is in the original theory, the larger ker( L j 2 X ) becomes in the dual theory, and vice versa. For example, if G = G is simply connected, then ker(j 2 X ) = 0. But since L G = ( L G) ad and L j 2 X = 0, the set ker( L j 2 X ) = H 2 (X, π 1 (( L G) ad )) is maximally possible. On the contrary, if G = G ad , then ker(j 2 X ) = H 2 (X, π 1 (G ad )) but ker( L j 2 X ) = 0. To verify (2.21), we note that ker(j 2 X ) = im(i 2 X ) and that the Pontryagin dual of the maps i 2 X , j 2 X are L j 2 X , L i 2 X , respectively (cf. §A.1). Therefore the right hand side of (2.21) equals (coker(i 2 X )) ∨ ∼ = ker((i 2 X ) ∨ ) = ker( L j 2 X ).
3. Reduction to two dimensions along an orientable surface 3.1. Reduction from four to two dimensions. The N = 4 supersymmetric gauge theory in four dimensions admits three inequivalent twists [28,29]. In the one that is related to the geometric Langlands programme there are two independent supersymmetric transformations, δ l and δ r , that can be defined on an arbitrary curved spacetime 4-manifold X [1]. So there is a family of topological field theories parametrised by t ∈ C ∪ {∞} = CP 1 , each with a BRST operator δ t = δ l + tδ r . Among the bosonic fields are a gauge field A (connection on a G-bundle P ), a "Higgs field" φ ∈ Ω 1 (X, adP ) and a complex scalar field σ ∈ Ω 0 (X, ad P C ). Being a cohomological field theory like [30], the transformation δ t satisfies δ 2 is the curvature of the connection A + √ −1φ on the complexified bundle P C := P × G G C , and only the terms with A, φ are displayed in (3.1). The canonical parameter [1] takes values in the entire CP 1 (not just the upper half plane). It is real (i.e., in the RP 1 inside CP 1 ) if and only if |t| = 1; in particular, Ψ = ∞ if t = ± √ −1. The theory depends only on Ψ because the t-dependence of δ t can be eliminated by a redefinition of δ t , δ ′ t : . Suppose now that the spacetime manifold is X = Σ × C, where C is a closed orientable surface of small size while the surface Σ is also orientable and is either open or closed but of large size. At low energies, fields on X have to achieve (or nearly achieve) minimal energy along C but can be slowly varying along Σ. So a gauge theory on X reduces to a sigma-model on the worldsheet Σ [31,32]. For the twisted N = 4 gauge theory with the action (3.1), the equations for minimal energy configuration along C are, for all t ∈ CP 1 , Hitchin's equations [33] where A is a connection on a principal G-bundle P over C and φ ∈ Ω 1 (C, ad P ).
where U := U H (C, G) → M H (C, G) × C is the universal bundle with structure group G ad . The connection A, curvature F A and the 'Higgs' field φ in the gauge theory on X are the pull-backs via the map u × id C : X → M H (C, G) × C of the corresponding universal objects A U , F U , φ U of U (see §A.5).
We assume that C is of genus g(C) > 1. Then a generic flat G-connection or Hitchin pair on C is irreducible and its stabiliser is Z(G). So it represents a smooth point on the moduli space. We shall focus on the smooth part of M H (C, G) (which we denote by the same notation) as near singular points, there are extra light degrees of freedom that requires a more intricate study. The space M H (C, G) is hyper-Kähler [33] and is of real dimension 4(g(C) − 1) dim G. Following [1], we let I be the complex structure on M H (C, G) induced by that on C, J be the rotation from δA to δφ (both are in Ω 1 (C, adP )), and K = IJ. Let ω I , ω J , ω K be the corresponding Kähler forms. The sigma-model metric on the target space M H (C, G) is 4π e 2 = Im τ times the standard hyper-Kähler metric [1]. The theta term in the action (3.1) reduces to a B-field given by a globally defined 2-form B θ = − θ 2π ω I = −(Re τ ) ω I in the sigma-model [1]. A conceptual construction of the B-field is to use the connection A U and curvature F U of the universal bundle U. Let B θ be the 2-form on M H (C, G) obtained by integrating the 4-form θ 8π 2 tr F U ∧ F U along C. Using the (1, 1)-part of F U in (A.26), we can show that the 2-form B θ has the same expression as above. By (3.4), the theta term in gauge theory becomes justifying that B θ is the B-field to which the sigma-model couples. If G is simply connected, then 1 4π tr F U ∧ F U has integral periods and so does its integration along C. Hence the holonomy exp( In fact the resulting change in the B-field B θ → B θ + ωI 2π can be interpreted as a change of trivialisations [34] of a topologically trivial B-field on M H (C, G). If G is not simply connected, then the instanton number contribution can be fractional [16,17] and the class [ω I /2π] need not be integral [1]. So τ → τ + 1 brings a non-trivial phase in both 4-and 2-dimensional theories.
Generally, whether B is closed or not, an N = (2, 2) supersymmetry requires a pair (for the left and right movers on the worldsheet) of complex structures J + , J − on the target space. They are parallel under connections preserving an Hermitian metric g (with respect to both J ± ) but whose torsions are proportional to ±H, where H := dB [35]. These conditions are equivalent to having a pair of commuting (twisted) generalised complex structures , where ω ± = gJ ± , that form a generalised Kähler metric on the target [36]. The theory can then be twisted in two ways to make a topological sigma-model that depends only on one of J ± (cf. [37,38,39]). We choose one, say J := J + . If J + = J − , the twisted theory is a B-model in the complex structure J + , whereas if J + = −J − , it is an A-model in a symplectic form proportional to ω + . If the target space is a hyper-Kähler manifold such as M H (C, G), there is a family of complex structures [1] J w := 1−ww 1+ww I + √ −1 w−w 1+ww J + w+w 1+ww K parametrised by w ∈ C∪{∞} = CP 1 . Then for any pair (J + , J − ) of complex structures given by (w + , w − ) ∈ CP 1 ×CP 1 and for any closed B-field, the conditions for N = (2, 2) supersymmetry are automatically satisfied. The theory is a B-model if w + = w − . If w + = w − , the target space has a Kähler structure (g, ω ′ , J ′ ) given by and the theory is an A-model with the symplectic form ω ′ by a B-field transform of While the above is sufficient for the existence of a classical theory with N = (2, 2) supersymmetry, the quantum theory is anomaly-free only if c 1 (T 1,0 + ) + c 1 (T 1,0 − ) = 0, where T 1,0 ± are the holomorphic tangent bundles of the target space in the complex structures J ± , respectively [38]. Equivalently, the c 1 of the it is always satisfied [37]. Moreover, there should be a nowhere zero pure spinor of the generalised tangent bundle which is closed under the (twisted) de Rham operator [38]. When the B-field is closed, the condition reduces to the definition of generalised Calabi-Yau manifolds [40], of which the usual Calabi-Yau and symplectic manifolds are examples.
If the target is hyper-Kähler, the twisted theory is anomaly-free for all pairs (w + , w − ) ∈ CP 1 × CP 1 . If w + = w − , the target is, before the B-field transform, generalised Calabi-Yau with a pure spinor exp(B ′ + √ −1 ω ′ ), where As w ± → w ∈ CP 1 , B ′ + √ −1 ω ′ becomes proportional to a (2, 0)-form (with respect to J w ) When w + = w − = w, the B-model target space is Calabi-Yau with a holomorphic pure spinor exp( √ −1Ω ′ w ). The twisted N = 4 gauge theory on X = Σ × C with the BRST transformation δ t (t ∈ CP 1 ) reduces at low energies to the topological sigma-model on Σ with (w + , w − ) = (−t, t −1 ) [1]. The generalised complex structure is taking into account the kinetic factor Im τ from gauge theory. If t = ± √ −1, w + = w − = ∓ √ −1, the 2-dimensional theory is a B-model in the complex structures ±J. If t ∈ R ∪ {∞}, the theory is an A-model; for example, the symplectic structure is ±(Im τ ) ω K if t = ∓1 and ±(Im τ ) ω I if t = 0, ∞. For other values of t, the theory is equivalent to an A-model with a symplectic form ω t upon a B-field transform by B t , where [1] ω t := (Im τ ) As expected, these 2-dimensional theories are anomaly-free. Since ω K is exact, the complexified Kähker class is Together with the B-field B θ from the theta term, the cohomology class is showing that Ψ is the relevant parameter in the 2-dimensional theory as well [1]. These 2-dimensional topological field theories depend only on J, ω I , ω K which can be defined without choosing a complex structure on C. This reflects the metric independence of the 4-dimensional topological theories.
3.2. Two-dimensional interpretation of the discrete fluxes. Suppose the worldsheet of the sigma-model has a splitting Σ = T 1 × S 1 , where the circle T 1 is in the time direction and S 1 is in the spatial direction. Then so does the 4-dimensional spacetime X = T 1 × Y as in §2.1, where Y = S 1 × C is a closed orientable 3-manifold. Recall that H 2 (X, π 1 (G)) = H 1 (Y, π 1 (G)) ⊕ H 2 (Y, π 1 (G)). We write a = a 0 + a 1 according to H 1 (Y, π 1 (G)) = H 0 (C, π 1 (G)) ⊕ H 1 (C, π 1 (G)) and m = m 0 + m 1 ∈ H 2 (Y, π 1 (G)) according to H 2 (Y, π 1 (G)) = H 2 (C, π 1 (G)) ⊕ H 1 (C, π 1 (G)) [1]. Since H 1 (Y, Z) is torsion-free in this case, the action of a discrete symmetry group element g ∈ H 1 (Y, Z(G)) does not change the topology of G-bundles over Y . If we write g = g 0 + g 1 according to H 1 (Y, Z(G)) = H 0 (C, Z(G)) ⊕ H 1 (C, Z(G)), then g 0 and g 1 modify the holonomies of a G-bundle P → Y along S 1 and C, respectively. They both preserve the topology of the bundles or equivalently, the connected components of B(Y, G) and hence of M H (C, G). The discrete electric and magnetic fluxes are thus in e(Y, G) = H 1 (Y, Z(G)) ∨ and m(Y, G) = H 2 (Y, π 1 (G)) without the adjustment in §2.1. We write e = e 0 + e 1 according to In the sigma-model, m 0 ∈ H 2 (C, π 1 (G)) labels the connected component M := M m0 H (C, G) in which the map u takes values and m 1 ∈ H 1 (C, π 1 (G)) ∼ = Hom(π 1 (S 1 ), π 1 (M)) labels the homotopy type of u at a fixed time [1]. The symmetry group H 1 (C, Z(G)) acts as discrete translations on M. So the quantum Hilbert space further decomposes into sectors labelled by discrete momenta e 1 ∈ H 1 (C, Z(G)) ∨ . In each sector of the sigma-model labelled by e 1 , we integrate over maps u : [0, β] × S 1 → M such that u(β, ·) = g 1 · u(0, ·) for some g 1 ∈ H 1 (C, Z(G)), weighted by the phase e 1 (g 1 ) −1 ∈ U(1). If g 1 = 0, the homotopy classes of such maps along the time direction are in the fundamental group π 1 (M). In general, they form a set π 1 (M, g 1 ) which is a torsor over π 1 (M). The sums over π 1 (M, g 1 ) and g 1 ∈ H 1 (C, Z(G)) combine to a sum over a 1 ∈ H 1 (C, π 1 (G ad )). This is reminiscent of a similar pattern in gauge theory (see §2.2); in fact, such a map u yields a G ad -bundle P ad via (3.4) with ξ 1,1 (P ad ) = m 1 + a 1 and is weighted by the same of phase e 1 (a 1 ) −1 in gauge theory by (2.16). If we sum over e 1 , then a 1 is restricted to H 1 (C, π 1 (G)) by (A.3). This means g 1 = 0 and u becomes a map from the closed worldsheet Σ to M. If we also sum over m 1 , then we obtain a partition function Z a0 Σ,M which is relativistic invariant in two dimensions. Similarly, g 0 ∈ Z(G) gives rise to G-bundles that are twisted along Σ. But since it does not act on the target space M, it is not a symmetry of the sigma-model in the usual sense. Instead, each character e 0 ∈ Z(G) ∨ determines a flat B-field e 0 (ξ(U)) ∈ H 2 (M, U(1)) on M that the sigma-model couples to, producing a phase that matches the one from the gauge theory [1].
. For e 0 to be a true discrete parameter in two dimensions, we need to verify that the B-fields e 0 (ξ(U)) on M are different for different e 0 , that is, the map e 0 ∈ Z(G) ∨ → e 0 (ξ(U)) ∈ H 2 (M, U(1)) is injective. In fact, the composition of this map with ̺ 2 M,U(1) : This establishes e 0 ∈ Z(G) ∨ as faithful parameter in the 2-dimensional theory. Although the 4-dimensional gauge theory requires a 0 ∈ H 2 (Σ, π 1 (G)) ∼ = π 1 (G), we will show below that from the 2-dimensional point of view alone, a 0 can actually take values in a larger set H 2 (Σ, π 1 (G ad )) ∼ = π 1 (G ad ), and each a 0 comes with a phase e 0 (a 0 ) −1 in the path integral in the presence of the flat B-field e 0 (ξ(U)) on M. For each e 0 , is already the partition function of a relativistic invariant theory on the closed worldsheet Σ. The constraint a 0 ∈ π 1 (G) is enforced by (A.3) only after summing over e 0 ∈ Z(G) ∨ , which is not required in two dimensions but is necessary to achieve relativistic invariance in four dimensions. The resulting partition function is That a 0 ∈ π 1 (G) comes from summing over e 0 ∈ Z(G) ∨ is a relativistic invariant statement in two dimensions and is valid for worldsheets not necessarily of the product form T 1 × S 1 . Let Σ be any closed orientable worldsheet. The map u : Σ → M H (C, G) in sigma-model is related to the G-bundle P over X = Σ × C in gauge theory by (3.4). We write ξ(P ) = ξ 2,0 (P )+ξ 1,1 (P )+ξ 0,2 (P ) according to H 2 (X, π 1 (G)) = H 2 (Σ, π 1 (G))⊕H 1 (Σ, H 1 (C, π 1 (G)))⊕H 2 (C, π 1 (G)). In the special case Σ = T 1 × S 1 , we have ξ 2,0 (P ) = a 0 , ξ 1,1 (P ) = a 1 + m 1 and ξ 0,2 (P ) = m 0 . In general, let [u] 0,2 denote the element m 0 ∈ H 2 (C, π 1 (G)) such that the image of u is contained in M m0 H (C, G). Then it follows easily from (3.4) that [u] 0,2 = ξ 0,2 (P ), agreeing with the above interpretation of m 0 when Σ is a product. With m 0 fixed, we proceed to find the range of the parameter a 0 in the sigma-model defined on any closed worldsheet Σ.
With the restriction a 0 = [u] 2,0 ∈ π 1 (G), the partition function of the 2-dimensional sigma-model is A map u does not uniquely determine a G-bundle P and its connection via (3.4); there is a |H 1 (Σ, Z(G))|-fold ambiguity to reconstruct P from P ad along Σ. In addition, in the partition function of the gauge theory, we divide by the order of the centre Z(G) that acts trivially on the fields. So the relation of the partition functions of the gauge theory (in the limit of small C) and the sigma-model is Because both theories are topological, (3.13) is valid for any size of C.
For each e 0 ∈ Z(G) ∨ , the partition function of the sigma-model coupled to the flat B-field e 0 (ξ(U)) is where u * (e 0 (ξ(U))), [Σ] ∈ U(1) is the holonomy of the B-field on Σ. The sum is now over all homotopy classes in [Σ, M] as it should be from the 2-dimensional point of view. This means that we allow G-bundles in gauge theory on X = Σ × C that are twisted along Σ by a discrete B-field u * (ξ(U)) ∈ H 2 (Σ, Z(G)); such a twisted G-bundle defines an honest G ad -bundle P ad over X with ξ 2,0 (P ad ) = a 0 contributing the same phase e 0 (a 0 ) −1 . So we have for all e 0 ∈ Z(G) ∨ as a refinement of (3.13). On the sigma-model side, we have (3.9), where Z e0 Σ,M comes from all u : Σ → M with [u] 2,0 = a 0 ∈ π 1 (G ad ). Summing over e 0 and using (3.10), we obtain (3.13). In this way, the gauge theory has honest G-bundles and the sigma-model has maps with restricted homotopy classes [Σ, M] • .
We have thus completed the matching of homotopy classes of maps u in the 2-dimensional sigma-model and topological types of G-bundles P in the 4-dimensional gauge theory when X = Σ × C.
If the gauge group is G ad , consider the sigma-model whose target is M ad := M m0 H (C, G ad ). If m 0 ∈ H 2 (C, π 1 (G)), there is a regular H 1 (C, Z(G))-covering q : M → M ad of the smooth parts of the moduli spaces [41,15] (see also §A.6). A map u ad : Σ → M ad lifts to u : Σ → M if and only if (u ad ) * (π 1 (Σ)) ⊂ q * (π 1 (M)), and different lifts of a given u ad are related by the deck transformations of q. Thus the partition function Z • Σ,M can be written as Du ad · · · e −S(u ad ,... ) .
The above sum is over homotopy classes [u ad ] ∈ [Σ, M ad ] of maps u ad : ). The identity (3.15) is consistent with its gauge theoretic counterpart (2.12) via the relation (3.13) and its analogue for G ad .
3.3. S-duality and mirror symmetry. We recall the reduction along a closed orientable surface of S-duality in four dimensions to mirror symmetry in two dimensions [31,32,1]. In 4-dimensional gauge theories, the duality transformation on the complex coupling τ is S : τ → −1/n g τ . Duality also acts non-trivially on supersymmetry generators because it acts on the central charges of the extended supersymmetry algebra. The transformation on the parameter t ∈ CP 1 of topological gauge theories is S : t → − τ |τ | t [1]. Together with T : τ → τ + 1 which acts trivially on t, the transformations of the canonical parameter Ψ are S : Ψ → −1/n g Ψ and T : Ψ → Ψ + 1. For example, if Ψ = ∞, t = √ −1 and τ is purely imaginary, the dual theory has L Ψ = 0, L t = 1 and L τ is also purely imaginary. For the sigma-model with a hyper-Kähler target space and parametrised by (w + , w − ) ∈ CP 1 × CP 1 , the mirror theory has . This is compatible with the above transformation on t ∈ CP 1 when the 2-dimensional theory comes from the reduction of a 4-dimensional theory, i.e., when (w + , w − ) = (−t, t −1 ). For example, if t = √ −1 and τ is purely imaginary, the sigma-model is a B-model in the complex structure J. Its mirror, with L t = 1, is an A-model with the symplectic form (Im τ )ω K . This is the important special case in which S-duality in four dimensions gives rise to the geometric Langlands programme when reduced to two dimensions [1]. Outside the family of dimensional reductions, the sigma-model with (w + , w − ) = (0, 0) is a B-model in the complex structure I. Its mirror, with the same ( L w + , L w − ) = (0, 0), is again a B-model in the complex structure I [1].
In four dimensions, S-duality interchanges the discrete fluxes e 0 and m 0 , as well as e 1 and m 1 [1]. While they are invariant under T , the transformations under S are (e 0 , m 0 ) → (−m 0 , e 0 ) and (e 1 , m 1 ) → (−m 1 , e 1 ). This is the discrete analogue of the Hodge star operation on the curvature 2-form. For 2-dimensional sigma-models, mirror symmetry exchanges the connected components of Hitchin's moduli space with the non-trivial flat B-fields and the windings in the moduli space with the discrete momenta. Indeed, from (A.4), there are natural isomorphisms making the duality of the discrete parameters possible. The mirror symmetry between e 0 and m 0 and between e 1 and m 1 can be explained concretely from the geometry of the target space.
To see the interchange of m 0 and e 0 , recall the Hitchin fibration h : defined by evaluating the invariant polynomial of the Lie algebra g on the holomorphic part of the 'Higgs' field φ [33]. The map h is holomorphic in the complex structure and is special Lagrangian in the symplectic structures ω J , ω K but holomorphic in the complex structure I. Each is naturally identified with the space of trivialisations of the flat B-field e 0 (ξ(U)) on M H (C, G) b (see [42] for G = SU(n), PU(n), n = 2, 3 and [43] for the general case). Note that e 0 (ξ(U)) is trivial on M H (C, G) b , though not naturally trivial unless e 0 = 0. But any two trivialisations of the B-field differ by a flat line bundle [34]. The torus M e0=0 is T -duality in the sense of Strominger, Yau and Zaslow [44], extended by Hitchin [45] in the presence of non-zero B-fields. The exchange of B-fields with topology under T -duality is a general pattern that also appears in [46].
We can rephrase the above in the language of sigma-models and branes. The mirror symmetry between the two B-models in complex structure I identifies the rings of holomorphic functions on B as observables of both theories. (C, L G) represented by the trivial L G-connection on C with L φ = 0, though its internal structure at that point is not geometrical [10]. More generally, a character of π 1 (M m0 flat (C, G)) ∼ = H 1 (C, π 1 (G)) defines a flat line bundle over M m0 flat (C, G) and hence another brane of type (B, A, A) on M m0 flat (C, G). Since π 1 (M m0 flat (C, G)) ∨ ∼ = H 1 (C, Z( L G)), the character can be identified with a homomorphism from This point is the support of the mirror brane of type (B, B, B).
We study the Hitchin fibration when the gauge group changes. Since the discrete symmetry group H 1 (C, Z(G)) acts trivially on the 'Higgs' fields, it commutes with h and preserves its fibres. Each torus component M m0 . Changing the gauge group to G ad , the Hitchin fibration from bundles whose gauge group can be lifted to G, as in (A.23). On the torus component M m0=0 On the dual side, we have Hitchin maps L h : and H 1 (C, π 1 ( L G)) are dual to the kernels H 1 (C, π 1 (G)) and H 1 (C, Z(G)) ofq and q by (3.16).
The coverings q,q also determine the mappings of branes when the gauge group G becomes G ad or G. If B is a B-brane on M H (C, G) in the complex structure I, then the brane in the G ad -theory is q * B and the brane in the On the other hand,q * B is the brane whose Chan-Paton bundle isq * ℓ e on M m0 If m 0 = 0 and if a = 0 is the origin of the torus M m0 H (C, G) b , thenq −1 (a) = H 1 (C, π 1 (G)); in general it is a torsor over H 1 (C, π 1 (G)).
The above operations on branes under the changes of the gauge group are compatible with duality. Suppose the brane B on M H (C, G) is given by a flat line bundle ℓ over the torus component M m0=0 for simplicity, we take the m 0 = 0 component and assume that the B-field is zero, i.e., e 0 = 0. The dual brane is of rank 1 and is supported at the point ℓ ∨ ∈ M e0=0 H (C, L G) b on the dual torus. Changing the gauge group of the original theory to G ad , the brane B becomes q * B given by q * ℓ as above. In the dual theory, the gauge group is changed to L G and the brane ( The latter is clearly the mirror of q * B. If instead the gauge group is changed to G in the original theory, then the group of the dual theory is ( L G) ad . The braneq * B is given by the pull-back bundleq * ℓ; it is mirror to the brane L q * (B ∨ ) supported at L q(ℓ ∨ ). Suppose B is given by a flat line bundle ℓ over M m0 H (C, G) b twisted by a B-field e 0 (ξ(U)), then the mirror brane is supported at ℓ ∨ in the torus M e0 The above discussions remain valid with simple modifications. In general, by fibrewise Fourier-Mukai transform [47], the identities hold for any B-brane B in the complex structure I. This verifies the desired compatibility with duality. The interchange of winding numbers m 1 and conserved momenta e 1 under duality is standard. Here we explain a related phenomenon which will appear more pertinently in §4

Non-orientable surfaces, branes and duality
In this section, C ′ is a compact non-orientable surface with a conformal structure. Given a conformal structure on C ′ , the Hodge star * on the 1-forms on C ′ is defined up to a sign, but this is harmless to Hitchin's equations in (3.3). So we do have a moduli space M H (C ′ , G) of solutions to (3.3) on C ′ up to gauge equivalence [41] (see §A.10 for a summary and further developments). We will however avoid gauge theories on non-orientable 4-manifolds, which do not allow the full range of the theta angle or the full SL(2, Z) duality, although S-duality for Abelian gauge theories on a non-orientable spacetime was recently studied in [50]. Instead, we consider the twisted N = 4 gauge theory on a suitably constructed orientable 4-manifold so that its reduction to two dimensions is a sigma-model in which the moduli space M H (C ′ , G), where C ′ is non-orientable, plays a role. 4.1. Dimensional reduction from an orientable 4-manifold. An orientable 4-manifold that contains a nonorientable surface C ′ can not be a product space Σ × C ′ as it would be non-orientable whether the surface Σ is orientable or not. Instead, we use the orientation double cover π : C → C ′ whose non-trivial deck transformation ι is an orientation reversing involution on C. If C ′ is a connected sum of g(C ′ ) copies of RP 2 , then C is a compact surface of genus g(C) = g(C ′ ) − 1. For example, C = S 2 , g(C) = 0 if C ′ = RP 2 , g(C ′ ) = 1 and C = T 2 , g(C) = 1 if C ′ is the Klein bottle, g(C ′ ) = 2. On the other hand, letΣ be a closed orientable surface with an orientation reversing involution, also denoted by ι, and let Σ =Σ/ι. We assume that the fixed point setΣ ι of ι onΣ is non-empty and Σ is an orientable surface whose boundary is identified withΣ ι . For example,Σ = S 2 if Σ is a disc (with one boundary circle) andΣ = T 2 if Σ is a cylinder (with two boundary circles). We choose our spacetime manifold as X =Σ × ι C.
Since the diagonal action of ι onΣ × C is free and orientation preserving, the quotient space X is a closed orientable smooth 4-manifold. Globally, X is not a product of two surfaces, but there is a projection map π X : X → Σ (by forgetting C). If σ ∈ Σ 0 , the interior of Σ, then π −1 X (σ) is a copy of C. But if σ ∈ ∂Σ, then π −1 X (σ) is a copy of C ′ . We shall consider the dimensional reduction of the twisted N = 4 gauge theory on X with a compact, semisimple gauge group G. We can find a metric on X so that Σ is large but C and C ′ are small. In fact, choosing ι-invariant metrics on C and onΣ such thatΣ is large but C is small, the product metric onΣ × C descends to such a metric on X. If C ′ is the connected sum of g(C ′ ) > 2 copies of RP 2 or if g(C) > 1, a generic flat G-connection on C ′ or on C is irreducible. At low energies, for the same reason as in [31,32,1], the fields A, φ have to satisfy Hitchin's equations along the fibres C or C ′ of π X . So the gauge theory on X reduces to a sigma-model on Σ: the interior Σ 0 is mapped to M H (C, G) (we denote the map by u) while the boundary ∂Σ is mapped to M H (C ′ , G) (we denote the map by u ′ ). To stay at low energies, as a point σ ∈ Σ 0 goes to σ ′ ∈ ∂Σ, u(σ) must approach the pull-back to C of the Hitchin pair on C ′ given by u ′ (σ ′ ). Therefore u extends to a map from Σ = Σ 0 ∪ ∂Σ to M H (C, G) such that u| ∂Σ = p • u ′ , where  [2] -covering [41] (cf. §A.10) and defines a brane which wraps around its world-volume N(C, G) non-trivially. In this way, the sigma-model gains extra degrees of freedom at the boundary.
Conversely, given a pair of maps (u, u ′ ) as above, we can reconstruct the adjoint bundle P ad over X in gauge theory. Note that X = X 0 ∪ X 1 , where X 0 := π −1 X (Σ 0 ), Σ 0 is the interior of Σ as above and X 1 := π −1 X (Σ 1 ), Σ 1 is a tubular neighbourhood of ∂Σ in Σ. Then X 0 is homeomorphic to Σ 0 ×C, X 1 is homotopic to ∂Σ ×C ′ via a deformation retract from Σ 1 to ∂Σ, and X 0 ∩ X 1 is homotopic to ∂Σ × C. Recall from §A.6 the universal bundles U → M H (C, G) × C and U ′ → M H (C ′ , G) × C ′ . Consider the G ad -bundles (u| Σ0 × id C ) * U over X 0 and (u ′ × id C ′ ) * U ′ over ∂Σ × C ′ and hence over X 1 via the retract. By (A.27), we have a bundle isomorphism showing that the above two G ad -bundles over X 0 and X 1 agree on the overlap X 0 ∩ X 1 . Thus we obtain a G ad -bundle P ad over X from (u, u ′ ), generalising (3.4) when the worldsheet Σ has a boundary.
The presence of branes whose world-volume is N(C, G) is compatible with the supersymmetry δ t that is made a BRST transformation in the topological field theory. Recall that the 4-dimensional theories are parametrised by t ∈ CP 1 or by the canonical parameter Ψ in (3.2). When t = ± √ −1 or equivalently when Ψ = ∞, the reduction to two dimensions is a B-model in the complex structure ±J. For other values of t or Ψ , the reduction is an A-model (possibly after a B-field transform) with the symplectic form ω t in (3.8), which is a linear combination of ω I and ω K . Happily, N(C, G) is both a complex submanifold in J and a Lagrangian submanifold in ω I and ω K (cf. §A.10) and hence also in ω t . This can also be phrased as a statement in generalised geometry. Since the involution ι preserves J but reverses ω I , ω K , the generalised tangent bundle of N(C, G) ⊂ M H (C, G) ι , which is ker(ι * −1)⊕ker(ι * +1) ⊂ (T ⊕T * )M H (C, G), is preserved by the generalised complex structures J t in (3.7). Therefore M H (C, G) ι , and hence N(C, G), is a generalised complex submanifold with respect to J t for all t ∈ CP 1 .
At the quantum level, when the worldsheet has a non-empty boundary, the theory requires more conditions to be anomaly-free than the bulk theory (discussed in §3.1) does. For the A-model with a Lagrangian A-brane, the extra condition [51,52] is that the Lagrangian submanifold that supports the brane is gradable in the sense of [53]. (Mathematically, this is related to the orientability of the moduli space of holomorphic curves [54,55,56,57].) In particular, a special Lagrangian submanifold in a Calabi-Yau manifold is always gradable [51,58]. If the target space is hyper-Kähler such as the Hitchin moduli space M H (C, G), then for any (w + , w − ) with w + = w − , the Kähler structure in (3.6) is Calabi-Yau; it is generalised Calabi-Yau before the B-field transform. We restrict our attention to the subfamily (w + , w − ) = (−t, t −1 ) from 4-dimensional gauge theories. The submanifold N(C, G) is in fact special Langrangian for all t = ± √ −1 and is calibrated by the top exterior power of the holomorphic 2-form Consequently, the 2-dimensional theory with t = ± √ −1, which is equivalent to an A-model with an A-brane supported on N(C, G), is anomaly-free. Since being anomaly-free is a closed condition, the B-model with t = ± √ −1 and with the same brane is also anomaly-free.
The above analysis does not take into account B-fields that a sigma-model can couple to. In general, a B-field B on a target space M defines a line bundle L B over its loop space LM . If the worldsheet Σ is mapped to M by u and if, for simplicity, the boundary ∂Σ is a single circle, the holonomy exp( √ −1 Σ u * B) is a unit vector in the fibre of L B over u| ∂Σ . To obtain a phase in U(1), we often assume that u(∂Σ) is contained in the support (or world-volume) N of a brane on which there is a Chan-Paton bundle twisted by B. If the latter is a (twisted) line bundle, it trivialise the B-field on N and the line bundle L B on LN . The holonomy exp( √ −1 ∂Σ (u| ∂Σ ) * A ′ ) along ∂Σ of a connection A ′ on the Chan-Paton bundle is a unit vector in the same line over u| ∂Σ and thus the phase in the path integral is well defined [59]. When ∂Σ contains several boundary circles, exp( √ −1 Σ u * B) is an element of a tensor product of lines, each for one boundary circle. Its phase ambiguity can then be canceled by the holonomies of several (twisted) Chan-Paton bundles.
When the B-field is flat, i.e., dB = 0, we can sometimes define the holonomy exp( √ −1 Σ u * B) when ∂Σ = ∅ as a U(1) phase without twisting the Chan-Paton bundles. But this is subject to an additional anomaly, though of a classical origin. Suppose we have defined exp( √ −1 Σ u * B) for a map u : (Σ, ∂Σ) → (M, N ). We deform u smoothly until it returns to the same map. In this way, we extend u toû : I × (Σ, ∂Σ) → (M, N ) so thatû(0, ·) =û(1, ·) is the original map u. By the flatness of the B-field, we have an identity So the phase exp( √ −1 Σ u * B) can be defined if B is trivial on any torus in N . While any class in H 2 (N, Z) can be realised by a smoothly embedded closed surface [60], not all of them can be realised by tori. To circumvent this difficulty, we recall that a flat B-field on M is an element of H 2 (M, U(1)). Using the long exact sequence  N ). So if a theory on a worldsheet with boundary is already anomaly-free in the absence of B-fields, its coupling to a flat B-field B is consistent if B is trivial on N . If the original theory without a B-field is anomalous, then it is possible to cancel the anomaly by a non-trivial flat B-field on the world-volume of the brane. The topology of the normal bundle does not play a role here. This can be compared with the Freed-Witten anomaly-free condition [61] for untwisted open strings, which involves the w 2 of the normal bundle of the world-volume.
In our case, the B-field B θ on M H (C, G) induced by the theta term is a globally defined closed 2-form and it is topologically trivial. We claim that B θ = 0 on N(C, G) and hence it is also geometrically trivial on N(C, G). In fact, we have a stronger statement and because ω I satisfies the same property ι * ω I = −ω I . Alternatively, suppose as in The diagonal action of ι on M H (C, G) × C lifts to U preserving the universal connection. So the instanton density tr F U ∧F U is a ι-invariant 4-form on M H (C, G)×C. Since ι reverses the orientation of C on which the fibre integration is performed, we obtain ι * B θ = −B θ again, completing our explanation that B θ is trivial both topologically and geometrically on N(C, G).
The reduction of the theta term to two dimensions can be calculated explicitly. We note that the map u : (Σ, ∂Σ) → (M H (C, G), N(C, G)) lifts to a ι-equivariant mapũ :Σ → M H (C, G), and (ũ × id C ) * U ∼ =π * P ad , whereπ :Σ × C → X is the projection. Since ι is orientation preserving onΣ × C, we have showing explicitly that the holonomy exp(− √ −1 Σ u * B θ ) on Σ is well defined despite ∂Σ = ∅. In §4.3, we will see that the rectified discrete fluxes from four dimensions do produce non-trivial (but still untwisted) flat Chan-Paton bundles over N(C, G) through the covering map p : M H (C ′ , G) → N(C, G) in [41] or its refinement in [15] (cf. §A.10). In view of the exact sequence (4.1), these flat line bundles reflect the ambiguity in H 1 (N(C, G), U(1)) when lifting a flat B-field on M H (C, G) to a class in H 2 ( (M H (C, G), N(C, G)), U(1)). All these branes define theories which are anomaly-free from the 2-dimensional point of view.

4.2.
Discrete electric and magnetic fluxes. We consider the case when the 4-manifold X =Σ × ι C has a spacetime splitting and determine the sets of rectified discrete electric and magnetic fluxes. To achieve this, we takẽ Σ = T 1 × S 1 , and let ι be the reflection in the spatial direction S 1 with two fixed points, so that the quotient S 1 /ι = I is an interval and Σ =Σ/ι is a cylinder with two timelike boundary circles. Then X = T 1 × Y , where Y = S 1 × ι C is a smooth orientable closed 3-manifold, but X is not a global product of two surfaces.
Though H 1 (Y, Z) contains 2-torsion elements, the map δ 1 Y in (2.1) can still vanish. For example, if all elements of π 1 (G) are of odd order or if the sequence (2.2) splits, then δ 1 Y = 0. In such cases, the sets of discrete electric and magnetic fluxes remain e(Y, G) = H 1 (Y, Z(G)) ∨ and m(Y, G) = H 2 (Y, π 1 (G)), respectively. These cohomology groups of Y are calculated in §A.9. But generally, the sets of discrete fluxes need to be adjusted as in §2.1. We wish to calculate the sets m(Y, G) and e(Y, G) of rectified discrete fluxes for the above 3-manifold Y in the general case.
For subsequent applications, we start with the standard long exact sequence 0 → π 1 (G) [2] i 1 derived from the short exact sequence (2.2). It is in fact (A.5) for the classifying space BZ 2 . The map is defined abstractly but has the following explicit description. Given z ∈ Z(G), lift it toz ∈ Z( G). If 2z = 0 in Z(G), then 2z ∈ Z( G) is in the subgroup π 1 (G). With a different lift of z to Z( G), 2z changes by an element in 2π 1 (G). Thus the coset in π 1 (G)/2π 1 (G) containing 2z ∈ π 1 (G), which is precisely the image δ 1 Z2 (z), depends only on z.
We consider a similar problem for the non-orientable surface C ′ . There is a commutative diagramme containing exact columns (A.34). The first two rows are the long exact sequences (4.2) and (A.5) for C ′ . A routine diagramme chasing shows that the third row is also exact and that there is a short exact sequence ). In addition, there are isomorphisms coker(δ 1 Taking the dual of (4.4), we obtain the exact sequence There is also a commutative diagramme of homology groups whose Poincaré dual (cf. §A.8) is containing exact columns (A.38). The second row is the long exact sequence of cohomology groups from the exact sequence of local systems 0 → π 1 (G) → Z( G) → Z(G) → 0 on C ′ (cf. §A.1) whereas the last row is identical to the first in the previous diagramme. It follows that the first row is also exact; in fact, by (A.40), it is Poincaré dual to the last row of the previous diagramme. Moreover, there is an exact sequence We now return to the calculation of discrete fluxes on Y . Using (A.42), we obtain a commutative diagramme with exact rows and columns. A similar diagramme chasing yields a short exact sequence Taking the Pontryagin dual, we obtain Using instead (A.41), we have another commutative diagramme Thus we have determined the sets of rectified discrete electric and magnetic fluxes when Y = S 1 × ι C. Now consider the spacetime 4-manifold X =Σ × ι C, whereΣ is a closed orientable surface with an orientation reversing involution ι such that Σ =Σ/ι has boundary ∂Σ =Σ ι . We want to compute the set ker(j 2 X ) ⊂ H 2 (X, π 1 (G ad )) of discrete fluxes of G ad -bundles over X that can be lifted to G-bundles (cf. §2.2). This can be done by calculating the effect of j 2 X on exact sequence (A.44) with various coefficient groups in (2.2). Using (4.5) and the above exact sequence of π * H 1 (C ′ , A) with the groups A in (2.2), we get an exact sequence 0 → H 1 (∂Σ, π 1 (G) [2] ) ⊕ H 1 (Σ, ∂Σ), H 1 (C,π 1 (G)) π * H 1 (C ′ ,π 1 (G)) → ker(j 2 X ) → H 0 (∂Σ, m(Z 2 , G)) ⊕ H 1 (∂Σ, π * H 1 (C ′ , π 1 (G))) → 0 (4.11) that determines ker(j 2 X ). In particular, if Σ is a cylinder, then (4.11) reduces to (A.42) and (4.10) because . If Σ is a disc, then ker(j 2 X ) ∼ = (π 1 (G)) [2] ⊕ m(Z 2 , G). More generally, if ∂Σ is a single circle, then just as (A.45) for the cohomology of X, there is a filtration on ker(j 2 X ) whose graded components are H 0 (∂Σ, m(Z 2 , G)), H 1 (Σ, H 1 (C, π 1 (G))), H 1 (∂Σ, π 1 (G) [2] ), (4.12) whereΣ is the closed surface obtained from Σ by attaching a disc to ∂Σ.

4.3.
Two-dimensional interpretation of the discrete fluxes. We interpret the sets of rectified discrete electric and magnetic fluxes, e(Y, G) in (4.9) and m(Y, G) in (4.10) from the point of view of the 2-dimensional sigma-model. There are a number of differences between this case and the dimensional reduction in [1] along orientable surfaces when the spacetime is a product of two surfaces. First, in our case, there is an absence of m 0 ∈ H 2 (C, π 1 (G)) that labels the connected components of M H (C, G) because m 0 = 0 is the only non-empty sector. Since all G-bundles over C ′ pull back to topologically trivial bundles over C, the world-volume N(C, G) is contained solely in M m0=0 Without base points, we should count π 0 (M ′ ) twice as I has two end points. Thus we have an exact sequence which matches (A.41) when A = π 1 (G) for topological types of G-bundles over Y . From the 2-dimensional point of view, (m 2 , m ′ 2 ) ∈ (π 1 (G)/2π 1 (G)) ⊕2 labels the two components of M ′ on which an open string ends while m 1 ∈ H 1 (C, π 1 (G))/π * H 1 (C ′ , π 1 (G)) classifies the remaining relative winding of the string in M.
We must also determine the discrete symmetries in the 2-dimensional theory. The presence of branes breaks the full symmetry H 1 (C, Z(G)) on M. Although the subgroup H 1 (C, Z(G)) ι acts on the ι-invariant subset M ι , it does not always preserve the world-volume N := N(C, G) of the brane. However, the subgroup π * H 1 (C ′ , Z(G)), which is proper in H 1 (C, Z(G)) ι if χ(C ′ ) is even, does act on N and hence on the sigma-model maps (at a fixed time) u : (I, ∂I) → (M, N). By (A.34), the group H 1 (C ′ , Z(G)) is an extension of π * H 1 (C ′ , Z(G)) by Z(G) [2] . There is an action of H 1 (C ′ , Z(G)) on the pair (M, M ′ ) lifting the action of π * H 1 (C ′ , Z(G)) on M; the subgroup Z(G) [2] acts on M trivially and it acts on M ′ by deck transformations of the covering p : M ′ → N. Likewise, on the extra degrees of freedom u ′ : ∂I → M ′ at the boundary, there is an action of (Z(G) [2] ) ⊕2 (one copy for each end point of I). So the discrete symmetry group in two dimensions is an extension of π * H 1 (C ′ , Z(G)) by (Z(G) [2] ) ⊕2 , which by (A.42) is the same symmetry group H 1 (Y, Z(G)) in the 4-dimensional theory.
We then study the breaking of the discrete symmetry when the 2-dimensional theory is restricted to a topological sector. Recall from §A.10 that the Z(G) [2] -covering of M ′ over N decomposes into ker(δ 1 Z2 )-coverings of M ′m2 := M m2 H (C ′ , G) over N m2 := N m 2 (C, G), where m 2 ranges over π 1 (G)/2π 1 (G) and m 2 is its coset in m(Z 2 , G) = coker(δ 1 Z2 ). We can thus impose a more refined boundary condition on the sigma-model so that u ′ maps each end of a string to M ′m2 for some m 2 ∈ π 1 (G)/2π 1 (G). This breaks the Z(G) ⊕2 [2] part of the symmetry to ker(δ 1 Z2 ) ⊕2 . But the theory depends only on the cosets of m 2 in m(Z 2 , G) because if m 2 changes by δ 1 Z2 (z) for some z ∈ Z(G) [2] , then the covering of M ′m2 over the same N m2 is isomorphic (see §A.10) and hence it defines the same brane. Henceforth we let B m2 be the brane supported on N m 2 given by the covering q : M ′m2 → N m2 . The discrete parameters (m 2 , m ′ 2 ) of the two branes that an open string ends on and the relative winding parameter m 1 match m(Y, G) in (4.10).
The π * H 1 (C ′ , Z(G)) part of the symmetry does not change the topological types of bundles or branes. So the unbroken discrete symmetry, which combines ker(δ 1 Z2 ) ⊕2 and π * H 1 (C ′ , Z(G)), is the same discrete symmetry group ker(δ 1 Y ) in gauge theory given by (4.8). The Hilbert spaces of the two theories in two and four dimensions decompose according to the same character group e(Y, G) in (4.9), and in both theories, in the sectors labelled by e ∈ e(Y, G), the phases in path integrals associated to each discrete group element are the same. But from the 2-dimensional point of view, the two parts of ker(δ 1 Y ) play different roles. The subgroup ker(δ 1 Z2 ) ⊕2 operates on the boundary data whereas the quotient group π * H 1 (C ′ , Z(G)) is a subgroup of H 1 (C, Z(G)) which acts on the target space, and on the bulk data of the sigma-model. In (4.9), elements e 1 ∈ (π * H 1 (C ′ , Z(G))) ∨ are the discrete momenta of the translations in π * H 1 (C ′ , Z(G)). The symmetry ker(δ 1 Z2 ) ⊕2 and its characters will be analysed in more details below. Suppose one end of the open string is on the brane B m2 defined by the covering map p : M ′m2 → N m2 for some m 2 ∈ π 1 (G)/2π 1 (G). A group element g 2 ∈ ker(δ 1 Z2 ) acts as a deck transformation of p. In canonical quantisation, if g 2 = 0, then the corresponding boundary component T 1 of the cylinder Σ is mapped to a loop in M ′m2 by u ′ which lifts u : T 1 → N m 2 . If g 2 = 0 however, then the two ends of the time interval are mapped to two points on M ′m2 that differ by the deck transformation g 2 . But the projection of the curve to N m2 remains a loop, which is the restriction of the map u : Σ → M defined on the bulk of the worldsheet. In fact, by the exact sequence lifts to a loop in M ′m2 if and only if it represents an element in π 1 (N m2 ) that maps to zero in ker(δ 1 Z2 ). In the sector labelled by e 2 ∈ ker(δ 1 Z2 ) ∨ , the phase associated to g 2 is e 2 (g 2 ) −1 ∈ U(1). Summing over e 2 enforces the condition g 2 = 0 by (A.3), and thus u| T 1 can be lifted to u ′ : T 1 → M ′m2 . Thus allowing all e 2 recovers the original extra degrees of freedom that the sigma-model has on the boundary of the worldsheet.
In the e 2 sector, the phase e 2 (g 2 ) −1 is precisely the holonomy along the loop u| T 1 of the flat line bundle ℓ e2,m2 := M ′m2 × e2 C over N m2 . Therefore for each (e 2 , m 2 ) ∈ e(Z 2 , G) ⊕ m(Z 2 , G), we can construct a brane B e2,m2 supported on N m2 with the Chan-Paton bundle ℓ e2,m2 . As explained above, the extra degrees of freedom on the boundary given by the brane B m 2 turns into multiple sectors labeled by e 2 . In each sector e 2 , the theory is equivalent to having the end of the string on the brane B e2,m2 with the rank one Chan-Paton bundle ℓ e2,m2 . Similarly, the other end of the string is on another brane of the same type. If Σ has several boundary components, then each component ends on a brane of the above type. So we have m 2 ∈ H 0 (∂Σ, m(Z 2 , G)) and e 2 ∈ H 1 (∂Σ, e(Z 2 , G)). This picture is relativistic invariant on the worldsheet. But the 4-dimensional relativistic invariance requires summing over e 2 .
When Σ is closed, the discrete electric fluxes include e 0 ∈ Z(G) ∨ that defines a B-field e 0 (ξ(U)) that the sigmamodel couples to (see [1] or §3.2). When ∂Σ = ∅, the absence of e 0 in gauge theory is evident from the calculation of e(Y, G) in (4.9), but it can also be explained in a purely 2-dimensional way. For a non-zero e 0 ∈ Z(G) ∨ , not only is the B-field e 0 (ξ(U)) itself is non-zero on M (see §3.2), its restriction to N is also non-zero. Thus coupling to such the B-field is anomalous according to the anomaly-free condition at the end of §4.1. To complete the explanation, we must show that the map Z(G) ∨ → H 2 (N, U(1)) that defines the B-fields is injective. In fact, after composing it injective. This is because the latter is the dual of the surjective map j • ζ ′ * : π 2 (M ′ ) → Z(G) (cf. §A.4). Now suppose the worldsheet Σ is a general compact orientable surface with boundary ∂Σ. We compare the discrete parameters that are summed over in the path integrals of gauge theory and sigma-model. In four dimensions, they are the topological types of G ad -bundles over X that can be lifted to G-bundles and are in ker(j 2 X ) and H 4 (X, π 3 (G)). In two dimensions, they are the homotopy types of pairs of maps (u, u ′ ) : (Σ, ∂Σ) → (M, M ′ ). For simplicity, we assume that ∂Σ is a single circle. Then by (A.33), there is a short exact sequence for the homotopy types of the based maps. Here π 2 (M, M ′ ) is calculated by the exact sequence (A.48). If we consider maps without base points, there is an additional set of parameters from π 0 (M ′ ) = π 1 (G)/2π 1 (G). Combining the above information, the set [(Σ, ∂Σ), (M, M ′ )] has a filtration whose graded components are H 1 (C, π 1 (G))), π 1 (G) [2] , π 3 (G). (4.14) But the symmetry Z(G) [2] acts on the maps u ′ : ∂Σ → M ′ and on m 2 that labels the connected components of M ′ by δ 1 Z2 . So the actual parameters labelling non-isomorphic theories are in m(Z 2 , G) instead of π 1 (G)/2π 1 (G). Meanwhile, π 3 (G) ∼ = H 4 (X, π 3 (G)) contains the instanton number of the gauge bundles. If we adjust π 1 (G)/2π 1 (G) to m(Z 2 , G) and exclude the non-torsion π 3 (G) from (4.14), the graded components match those in (4.12) for ker(j 2 X ). As in §3.  .16) is equivalent to first changing the G to G ad by (2.12) and then taking the reduction to sigma-model.
On the other hand, the multiplicity of passing such possibly twisted G-bundles on X to G ad -bundles is |H 1 (X, Z(G))| but reduced by a factor |H 0 (∂Σ, ker(i 2 C ′ ))| = |H 0 (∂Σ, im(δ 1 Z2 ))| for allowing the twists. Thus (2.12) becomes and we get the same result by first changing the gauge group to G ad and then reducing to the sigma-model.

4.4.
Mirror symmetry of branes and quantisation. Mirror symmetry in 2-dimensional sigma-models is a consequence of the electric-magnetic duality, or S-duality in four dimensions [31,32,1]. For any orientable closed 3-dimensional time-slice Y , S-duality exchanges the rectified discrete electric and magnetic fluxes introduced in §2.1 by (2.20). In the present case, Y = S 1 × ι C, where C is the orientation double cover of a non-orientable surface C ′ and ι acts as orientation reversing involutions on C and S 1 . The sets e(Y, G) and m(Y, G) of discrete fluxes are given by the exact sequences (4.9) and (4.10), respectively. We recall the notations e(Z 2 , G) = ker(δ 1 Z2 ) ∨ , m(Z 2 , L G) = coker(δ 1 Z2 ), where δ 1 Z2 is defined in (4.3). When the gauge group G is exchanged with its dual L G, there are isomorphisms m(Z 2 , L G) ∼ = e(Z 2 , G), e(Z 2 , L G) ∼ = m(Z 2 , G) by (A.1) since the Pontryagin dual of δ 1 Z2 is the same map L δ 1 Z2 but for L G. Using this and (A.40), the duality (2.20) between e(Y, G) and m(Y, G) can be verified explicitly from (4.9), (4.10). In this way, S-duality exchanges m 1 with e 1 and m 2 with e 2 . For the spacetime X =Σ × ι C where Σ =Σ/ι is not necessarily a cylinder, the set ker(j 2 X ) of discrete fluxes of G ad -bundles over X that lift to G-bundles should satisfy the duality relation (2.21) when the gauge group G is exchanged with L G. This can be verified using (4.11), with the help of the isomorphisms (2.17) and (A. 40).
To see how S-duality is realised concretely from the 2-dimensional point of view, we introduce the Hitchin fibration of the moduli space M H (C ′ , G) where C ′ is a closed non-orientable surface. Let O ′ be the orientation line bundle of C ′ . Given a conformal structure on C ′ , the Hodge star * ′ on C ′ maps Ω 1 (C ′ , adP ′ ) to Ω 1 (C ′ , O ′ ⊗ adP ′ ), and vice versa. Therefore * ′ acts on Ω 1 (C ′ , adP ′ ) ⊕ Ω 1 (C ′ , O ′ ⊗ adP ′ ), satisfying ( * ′ ) 2 = −1. We can proceed with C ′ alone, but the above direct sum can be identified naturally with Ω 1 (C, adP ), where C is the orientation double cover and P = π * P ′ . Furthermore, the conformal structure on C ′ defines a complex structure on C so that the √ −1-eigenspace of * ′ matches Ω 1,0 (C, adP C ). A Hitchin pair (A ′ , φ ′ ) on C ′ pulls back to (A, φ) on C. The Hitchin fibration h ′ : Like the Hitchin map h for an orientable surface [33], h ′ is also proper and surjective. For a regular b ∈ B ι (outside the discriminant divisor), the fibre with respect to the symplectic structure ω ′ J . Therefore h ′ determines a real integrable system. Under the regular ker(δ 1 Z2 )-covering which depends only on the coset element m 2 ∈ m(Z 2 , G), and there is a disjoint union of N(C,  (4.17). In fact, it is already known that M H (C, G) ι b , if smooth, is a union of some affine real tori if ι is any anti-holomorphic involution on a compact Riemann surface C [62]. The information on the precise real tori N m2 (C, G) b appearing in the decomposition (4.17) will be necessary in verifying of the consequences of S-duality in two dimensions.
Recall that given m 2 ∈ m(Z 2 , G) and e 2 ∈ e(Z 2 , G), the flat line bundle ℓ e 2 ,m2 = M m2 H (C ′ , G) × e2 C defines a brane B e 2 ,m2 of type (A, B, A) on M H (C, G) whose world-volume is N m 2 (C, G). If b ∈ B ι is regular, then on each fibre M H (C, G) b of h, the brane B e2,m2 is supported on the real torus N m2 (C, G) b . In four dimensions, the roles of m 2 and e 2 are interchanged by S-duality. In two dimensions, the mirror of the brane B e2,m2 should be L B m2,e2 in the dual theory given by the line bundle L ℓ m2,e2 := M e2 ) labelled by the cosets e 2 ∈ m(Z 2 , L G) ∼ = e(Z 2 , G). The ι-invariant tori in M H (C, G) b and M H (C, L G) b are annihilators of one another [62]. In addition, we have (A.46), (4.17) and their duals for the world-volumes of the branes. By the fibrewise Fourier-Mukai transform [47,49], we see further that the Chan-Paton line bundle ℓ e2,m2 on N m2 (C, G) b shifts the dual torus N e2=0 (C, If the the gauge group G is changed to the adjoint group G ad , the target space of the G ad -theory is M m0=0 H (C, G ad ) and the brane B e2,m2 becomes q * B e2,m2 supported on N m2 (C, G ad ). By (4.6), (π * H 1 (C ′ , Z(G))) ∨ is a subgroup of ker(δ 1 C ′ ) ∨ and each e 2 ∈ e(Z 2 , G) can be regarded as a (π * H 1 (C ′ , Z(G))) ∨ -coset in ker(δ 1 C ′ ) ∨ . Recall from §A.10 that N m2 (C, G) is a regular π * H 1 (C, π 1 (G))-cover of N m2 (C, G ad ) while M m2 H (C ′ , G) is a regular ker(δ 1 Z2 )-cover of N m2 (C, G) and a regular ker(δ 1 If e 2 = 0, then the above simplifies to q * ℓ e2,m2 = e1∈(π * H 1 (C ′ ,Z(G))) ∨ ℓ e1,m2 ad , where ℓ e1,m2 ad := N(C, G) × e1 C. But in general, the sum is over the coset e 2 which is a torsor over (π * H 1 (C ′ , Z(G))) ∨ . Let B e,m2 ad be the brane on M m0=0 H (C, G ad ) whose world-volume is N m2 (C, G ad ) and whose Chan-Paton bundle is ℓ e,m2 ad . Then the above means Now suppose the gauge group is change to the universal cover G. A G-bundle over C ′ of topological type m 2 lifts to a G-bundle twisted by the discrete B-field m 2 , and the latter pulls back to an honest G-bundle over C. Let M m2 H (C ′ , G) be the Hitchin moduli space from such twisted G-bundles and let N m2 (C, G) ⊂ M H (C, G) consist of their pull-backs to C. Then the regular H 1 (C, π 1 (G))-coverq : M H (C, G) → M m0=0 H (C, G) restricts to a regular π * H 1 (C ′ , π 1 (G))-cover of N m2 (C, G) over N m2 (C, G) (cf. §A.10). Therefore the inverse imageq −1 (N m2 (C, G)) in M H (C, G) has H 1 (C,π1(G)) π * H 1 (C ′ ,π1(G)) connected components including N m2 (C, G). Since H 1 (C, π 1 (G)) acts on M H (C, G) as deck transformations and its subgroup π * H 1 (C ′ , π 1 (G)) acts on N m2 (C, G), the set of connected components form an H 1 (C,π1(G)) π * H 1 (C ′ ,π1(G)) -orbit, which is in general a coset in coker(δ 0 C ′ ) represented by m 2 , due to (4.7). Thus we havẽ where B e2,m is supported on the connected component N m (C, G) ⊂q −1 (N m2 (C, G)). That the set of all components N m (C, G) has an Abelian group structure will become more evident from the duality argument below. We verify that the duality (4.18) of the branes is consistent with the change of the gauge group. The mirror of the G-theory has gauge group L G and the mirror of the brane B e2,m2 is precisely L B m2,e2 by (4.18). If G is changed to G ad in the original theory, then the brane B e2,m2 becomes q * B e2,m2 in (4.19). The mirror of the G ad -theory has gauge group L G and the brane L B m2,e2 becomes Lq * ( L B m2,e2 ) = e∈e2 L B m2,e by (4.20). That (q * B e2,m2 ) ∨ = Lq * ( L B m2,e2 ), which is a special case of (3. are related by a translation of e 1 ∈ H 1 (C,π1( L G)) π * H 1 (C ′ ,π1( L G)) by the property of fibrewise Fourier-Mukai transform [47]. Moreover, there is an Abelian group structure on the set of connected components N e (C, L G) just as the line bundles ℓ e,m2 form an Abelian group in the original theory. The argument also shows that the e 1 -sector of the G-theory is mirror to the sector of the L G-theory with a relative winding e 1 . Similarly, if we change the gauge group to G in the original theory and ( L G) ad in the mirror theory, then we have ( B e 2 ,m ) ∨ = L B m,e2 ad and hence (q * B e 2 ,m2 ) ∨ = L q * ( L B m 2 ,e2 ), which verifies another part of (3.17). The relative winding m 1 in the G-theory is a character in the dual theory.
Finally, we apply the results to the quantisation of the Hitchin moduli space M H (C ′ , G) for a non-orientable surface C ′ . To quantise a symplectic manifold (M, ω), one finds a complexification M C with an anti-holomorphic involution ι such that M is a connected component of the ι-invariant part of M C . Assume that there is a holomorphic symplectic form ω C such that ω = Re(ω C ) on M . Assume also that there is a line bundle ℓ over M C with a lifted action of ι preserving a unitary connection on ℓ whose curvature is Re(ω C )/ √ −1. Clearly, ℓ restricts to a pre-quantum line bundle over M . Consider the A-model on M C with symplectic form Im(ω C ). Then ℓ defines a space-filling canonical . In our case, given any m 2 ∈ π 1 (G)/2π 1 (G) or m 2 ∈ m(Z 2 , G), the symplectic manifold (M m2 H (C ′ , G), ω ′ J ) is a finite cover of N m2 (C, G) whose complexification M m0=0 H (C, G) has the anti-holomorphic involution ι and the holomorphic symplectic form ω J + √ −1ω K (both with respect to I). Moreover, there is an Hermitian line bundle ℓ over M m0=0 H (C, G) whose curvature is ω J / √ −1 with a lifted ι-action [63]. This defines the canonical coisotropic brane B cc of type (A, B, A). It is clearly gradable in the sense of [52,64] and thus the sigma-model with this boundary condition is anomaly-free. For simplicity, we pick the trivial line bundle over M m2 H (C ′ , G) which pushes down to e2∈e(Z2,G) ℓ e2,m2 on N m2 (C, G). The quantum Hilbert space of M m2  Thus by (4.18), we can write H e2,m2 = Ext(B ∨ cc , L B m2,e2 ) in the B-model; see [63] for more details in the case δ 1 Z2 = 0.

Conclusion
In this paper, we considered the reduction of the twisted N = 4 gauge theory on an orientable spacetime 4-manifold which is not a global product of two surfaces to a 2-dimensional sigma-model on a worldsheet with boundary. The boundary conditions of the sigma-model are specified by branes constructed from the Hitchin moduli space of the non-orientable surface embedded in the 4-manifold. We showed that the resulting 2-dimensional theories are anomalyfree. We introduced the rectified discrete electric and magnetic fluxes in gauge theory, modifying the usual notion of 't Hooft, when the 3-dimensional time-slice has complicated topology. We matched these discrete fluxes, together with the instanton number, with the homotopy classes of relative maps in the low energy sigma-model, thus demonstrating the necessity of adjusting the notion of discrete fluxes in four dimensions. In addition, the agreement of topological sectors in gauge theory and in sigma-model is another non-trivial test of S-duality, or electric-magnetic duality, using the known results on the topology of Hitchin moduli spaces. Conversely, S-duality provides evidence for some of the anticipated results in §A.4 on the homotopy groups of the moduli spaces.

Appendix. Some results in topology
A.1. Pontryagin duals and Poincaré duality. If A is a finitely generated Abelian group, the Pontryagin dual of A is A ∨ := Hom(A, U (1)). If f : A → B is a homomorphism of Abelian groups, the dual map f ∨ : For example, if B = A and f is the multiplication by 2, then ker(f ) = A [2] is the 2-torsion subgroup of A while coker(f ) = A/2A is another group of 2-torsion elements, and (A.1) becomes If Y is a closed (i.e., compact and without boundary) orientable manifold of dimension n, the cap product with the fundamental class Combining the two, we obtain an isomorphism for any integer k (0 ≤ k ≤ n). Equivalently, there is a non-degenerate pairing H k (Y, A ∨ ) × H n−k (Y, A) → U(1). Given a short exact sequence 0 → A ′ i −→ A j −→ A ′′ → 0 of Abelian groups, there is a long exact sequence of cohomology groups of Y , where i k Y , j k Y are induced by i, j and δ k Y is the connecting homomorphism. From the dual exact sequence 0 → (A ′′ ) ∨ǐ −→ A ∨ǰ −→ (A ′ ) ∨ → 0, whereǐ = j ∨ andǰ = i ∨ , there is another long exact sequence with the induced mapsǐ k Y ,ǰ k Y and the connecting homomorphismδ k Y . We claim that ( under the identification (A.4). Hence the long exact sequence (A.6) is the dual of (A.5). In fact, under Poincaré duality, is the connecting homomorphism in the homology long exact sequence. On the other hand, the iso- If Y is orientable, then |H k (Y, A)| = |H n−k (Y, A ∨ )| = |H n−k (Y, A)| by Poincaré duality. So the above identity is trivial if n is odd. If n = 2l is even, it can be rewritten as Finally, let Y ′ be a closed but non-orientable manifold of dimension n. Then there is an exact sequence 1 → π 1 (Y ) → π 1 (Y ′ ) → Z 2 → 1, where Y is the orientation double cover of Y ′ . Given an Abelian group A, the action of Z 2 on A as {±id A } defines an action of π 1 (Y ′ ) on A and a local system A on Y ′ . Then Poinceré duality for Y ′ is Given an exact sequence 0 → A ′ → A → A ′′ → 0 of Abelian groups, there is a long exact sequence which is Poincaré dual to the homology long exact sequence of Y ′ .
A.2. Principal bundles and the group of gauge transformations. Let Y be a compact manifold of dimension n and let P be a principal G-bundle over Y , where G is a connected Lie group. Obstructions to the triviality of P or to constructing a global section of P are in H k+1 (Y, π k (G)), 1 ≤ k ≤ n − 1 [11]. In particular, the primary obstruction ξ(P ) ∈ H 2 (Y, π 1 (G)) is the obstruction to trivialising P on a 2-skeleton of Y or to lifting the structure group G of P to its universal cover G. For any k ≥ 1 and Abelian group A, there is a natural homomorphism For example, ̺ 2 Y,π1(G) (ξ(P )) ∈ Hom(π 2 (Y ), π 1 (G)) is the map that appears in the long exact sequence · · · → π 2 (P ) → π 2 (Y ) of homotopy groups associated to the fibration P → Y . Principal G-bundles over a compact surface C are classified by H 2 (C, π 1 (G)). If G is compact, G-bundles over a compact 3-manifold Y are also classified by H 2 (Y, π 1 (G)). Over a compact 4-manifold X, the set of topological types of G-bundles fits in the exact sequence [65] 0 If ξ(P ) = 0, then k(P ) ∈ H 4 (X, π 3 (G)) is the obstruction to extending the trivialisation of P from the 2-skeleton of X to the whole space. For a general w ∈ H 2 (X, π 1 (G)), the set ξ −1 * (w) is a torsor over H 4 (X, π 3 (G)). The group G(P ) of gauge transformations on a G-bundle P over Y is naturally the space of sections of Ad P := P × Ad G, where Ad stands for the adjoint action of G on G. Two elements in G(P ) are in the same connected component if there is a homotopy between them. Obstructions to constructing such a homotopy are in H k (Y, π k (G)), 0 ≤ k ≤ n [11]. The map η : G(P ) → H 1 (Y, π 1 (G)) to the primary obstruction is a homomorphism, and its kernel G 1 (P ) := ker(η) ⊂ G(P ) consists of sections of Ad P that can be lifted to P × Ad G. There is a commutative diagramme with exact rows and surjective vertical maps. If Y is a compact surface C, then π 0 (G(P )) ∼ = H 1 (C, π 1 (G)) and G 1 (P ) is connected. If G is compact, semisimple and Y is a closed 3-manifold, the second row of (A.12) becomes A different but related approach is based on the observation [66] that π 0 (G(P )) coincides with the set of topological types of G-bundles over X := T 1 × Y whose restrictions to a slice {t 0 } × Y are P . Given g ∈ G(P ), we can construct a bundle P g on X by gluing the two ends of the bundle I × P → I × Y using g. Obstructions to the triviality of P g are in H k+1 (X, π k (G)) ∼ = H k (Y, π k (G)) ⊕ H k+1 (Y, π k (G)). The classes in H k+1 (Y, π k (G)) are determined by the topology of P . The remaining obstructions in H k (Y, π k (G)), 1 ≤ k ≤ n, detect the homotopy class of g ∈ G(P ). We have ξ(P g ) = η(g) + ξ(P ) according to the above decomposition when k = 1. If Y is a 3-manifold and G is semisimple, then the class in H 3 (Y, π 3 (G)) is identified with k(P g ) ∈ H 4 (X, π 3 (G)).
Similarly, π 1 (G(P )) can be identified with the set of topological types of G-bundles over S 2 × Y whose restrictions to D ± × Y , where D ± are respectively the upper/lower hemispheres of S 2 , are the products D ± × P . A loop γ in G(P ) can be used to glue the two bundles D ± × P along the equator to form a bundle P γ → S 2 × Y . Topologically, G-bundles over S 2 × Y are classified by H k+1 (S 2 × Y, π k (G)) ∼ = H k−1 (Y, π k (G)) ⊕ H k+1 (Y, π k (G)). Whereas the groups H k+1 (Y, π k (G)) contain the characteristic classes of P , the obstructions to contracting a loop in G(P ) to a point are in H k−1 (Y, π k (G)). We have a map λ : LG(P ) → H 0 (Y, π 1 (G)) ∼ = π 1 (G) to the set of primary obstructions, and we denote its induced homomorphism by λ * : π 1 (G(P )) → H 0 (Y, π 1 (G)). Then ξ(P γ ) = λ * ([γ]) + ξ(P ) according to the above decomposition when k = 1. If Y = C is a closed orientable surface, there is an exact sequence 0 → H 2 (C, π 3 (G)) → π 1 (G(P )) λ * −→ H 0 (C, π 1 (G)) → 0, where the class in H 2 (C, π 3 (G)) is identified with k(P γ ) ∈ H 4 (S 2 × C, π 3 (G)). There is also an exact sequence (A.14) Now consider the space B(P ) := A(P )/G(P ) of gauge equivalence classes of connections on a G-bundle P over Y . The group G(P ) of gauge transformations acts on the space A(P ) of connections while the centre Z(G) of G, regarded as a subgroup of G(P ), acts on A(P ) trivially. We can restrict to connections whose stabiliser is precisely Z(G) and still denote the space by A(P ), which remains at least 2-connected and on which the group G(P )/Z(G) acts freely. In this way, B(P ) is smooth (though infinite dimensional) and we have π 1 (B(P )) ∼ = π 0 (G(P )/Z(G)) ∼ = π 0 (G(P )), π 2 (B(P )) ∼ = π 1 (G(P )/Z(G)). (A. 15) Concretely, for any g ∈ G(P ), a connection on the bundle P g → T 1 × Y defines a loop in B(P ) representing the class [g]. For any loop γ in G(P )/Z(G), there is a loop γ · A 0 in A(P ) by choosing a base connection A 0 . Then a disc in A(P ) bounded by γ · A 0 descends to a sphere in B(P ) representing an element of π 2 (B(P )). Finally, the action of π 0 (G(P )) on π 1 (G(P )/Z(G)) is given by the conjugation of G(P ) on the identity component of G(P )/Z(G). This action is trivial because for any g ∈ G(P ) and any loop γ in the identity component of G(P )/Z(G), the bundles on S 2 × Y defined by γ and gγg −1 can be shown to be topologically equivalent. By the isomorphisms in (A.15), the action of π 1 (B(P )) on π 2 (B(P )) is also trivial.
A.3. Gauge transformations on G-, G ad -and twisted bundles. Let G be a connected compact semisimple Lie group. A principal G-bundle P over Y defines a principal G ad -bundle P ad := P/Z(G) over the same space. There is a commutative diagramme for P ad similar to (A.12). Let q : G(P ) → G(P ad ) be the map induced by the projections G → G ad , P → P ad . Since P ad × Ad G = P × Ad G and P ad × Ad G = P × Ad G, the restriction of q to the subgroup G 1 (P ) = ker(η) is a surjective map q 1 : G 1 (P ) → G 1 (P ad ) with ker(q 1 ) ∼ = Z(G). There is a commutative diagramme with exact rows and columns, the right column being (2.1). It follows from a straightforward diagramme chasing that ker(q) ∼ = ker(q 1 ) ∼ = Z(G) and coker(q) ∼ = coker(i 1 Y ) ∼ = ker(δ 1 Y ). Therefore there is an exact sequence 16) where ζ := j 1 Y • η ad . For g ad ∈ G(P ad ) which is a section of P ad × Ad G ad , ζ(g ad ) is the obstruction to lifting it to a section of P ad × Ad G = P × Ad G or to G(P ). Taking the homotopy classes of gauge transformations, we obtain another exact sequence 1 → π 0 (G(P )) q * −→ π 0 (G(P ad )) ζ * −→ ker(δ 1 Y ) → 1. (A.17) If Y is a compact 3-manifold, then π 0 (G(P )) is given by (A.13) and thus (A.17) becomes obvious by using (2.1).
There is another interpretation of (A.17) in terms of homotopy classes of paths in B(P ) using (A.15). Since A(P ) = A(P ad ) and since Z(G) acts trivially on A(P ), we deduce from (A.16) that B(P ) is a regular ker(δ 1 Y )-cover of B(P ad ). Thus (A.17) can be regarded as the short exact sequence of fundamental groups of covering spaces A loop on B(P ad ) lifts to a loop on B(P ) if and only if its class in π 1 (B(P ad )) is in ker(ζ * ) ∼ = π 0 (G(P )). For a general g ∈ ker(δ 1 Y ), ζ −1 * (g) is the set π 1 (B(P ), g) of homotopy classes of paths [A(t)] in B(P ) such that [A(1)] = g · [A(0)]. The set ζ −1 * (g) is a torsor over π 1 (B(P )). In §2.1, we used the notion of principal bundles twisted by a discrete B-field [67,15]. Suppose {U α } is an open cover of Y . Then a principal G-bundle over Y is a collection of transition functions g αβ : U α ∩ U β → G satisfying the cocycle condition g αβ g βγ g γα = 1 on triple intersections U α ∩ U β ∩ U γ . A discrete B-field in our setting (or a Z(G)-gerbe) is an element of H 2 (Y, Z(G)) so that there is a 'holonomy' in Z(G) for each closed surface in Y . In theČech language, a cohomology class [h] ∈ H 2 (Y, Z(G)) is described by a collection of h αβγ ∈ Z(G) that satisfy a cocycle condition on quadruple intersections of the open sets. A principal G-bundle twisted by the discrete B-field can be described as a collection of transition functions g αβ : U α ∩U β → G satisfying g αβ g βγ g γα = h αβγ on all triple intersections U α ∩U β ∩U γ [67] (see [15] for a definition using bundle gerbes). If g ′ αβ is related to g αβ by g ′ αβ = f −1 α g αβ f β for some f α : U α → G, then g ′ αβ satisfy the same condition and it defines an equivalent twisted G-bundle over Y . A twisted G-bundle P induces an honest G ad -bundle P ad with ξ(P ad ) We can define gauge transformations and connections on a twisted G-bundle P . Since the failure of the cocycle condition h αβγ is in the centre Z(G), the definitions in the untwisted case work equally well here. A gauge transformation on P is a collection of maps from U α to G that are related by the conjugation of g αβ on U α ∩ U β . It can be identified as a section of AdP := P ad × Ad G which is a well-defined (though not principal) bundle over Y with fibre G. They form a group G(P ) whose Lie algebra is Ω 0 (Y, adP ), where the adjoint bundle adP := P ad × ad g (g is the Lie algebra of G) is also well defined. A connection on P is a collection of A α ∈ Ω 1 (U α , g) that are related by the gauge transformation of g αβ on U α ∩ U β . They form an affine space A(P ) modelled on Ω 1 (Y, adP ). The curvature is a collection of 2-forms F α ∈ Ω 2 (U α , g) which form adP -valued 2-form F ∈ Ω 2 (Y, adP ). We still have an action of G(P ) on A(P ) such that Z(G) as a subgroup of G(P ) acts trivially, with the quotient space B(P ) := A(P )/G(P ). The exact sequences (A.16), (A.17) and (A.18) remain valid when P is a twisted G-bundle.
A.4. Homotopy groups of the Hitchin moduli spaces. Let C be a closed orientable surface. Recall from §3.1 that the Hicthin moduli space M H (C, G) is the space of pairs (A, φ) satisfying Hitchin's equations (3.3) modulo gauge transformations. Here A is a connection on a G-bundle P over C and φ ∈ Ω 1 (C, adP ). We shall work with the smooth part of M H (C, G) for which we use the same notation.
There is a decomposition according to the topological type of G-bundles over C,

M m0
A.5. The universal bundle, its connection and curvature. Let G be a compact semisimple Lie group and let P be a principal G-bundle over a smooth manifold M . The group G(P ) of gauge transformations acts on P on the left and the action commutes with the right action of G on P . It also acts on the space A(P ) of connections on P on the right by pulling back, i.e., A → g * A by g ∈ G(P ). Therefore G(P ) acts on A(P )×P on the left by g : where g ∈ G(P ), A ∈ A(P ), p ∈ P . The Lie algebra of G(P ) can be identified with Ω 0 (M, adP ) while the tangent space of A(P ) at any A is T A A(P ) ∼ = Ω 1 (M, adP ). For v ∈ Ω 0 (M, adP ), let V v be the induced vector field on P . Then the induced vector field on A(P ) × P is (−d A v, V v (p)) at (A, p) ∈ A(P ) × P . The centre Z(G) of G is a subgroup of G(P ) as constant gauge transformations and it acts trivially on A(P ). We consider a smooth part of the orbit space B(P ) := A(P )/G(P ) that comes from the connections whose stabiliser subgroup is precisely Z(G). For these connections A, the operator d A : Ω 0 (M, adP ) → Ω 1 (M, adP ) is injective and, by picking a Riemannian metric on M , d * A d A is an invertible operator on Ω 0 (M, adP ). Then A(P ) → B(P ) is a principal bundle of structure group G(P )/Z(G). We define a connection on this bundle so that the horizontal subspaces are the orthogonal compliments of the orbits of G(P ). The connection 1-form at and the curvature at A on two horizontal vectors α, β ∈ ker(d * Here b α : Ω 0 (M, adP ) → Ω 1 (M, adP ) is the operator v → [α, v], and b * α is its adjoint operator. Consider the pull-back of the G-bundle P under the projection A(P ) × M → M . Restricting to the connections whose stabiliser is Z(G), there is a free action of G(P ) on the total space A(P ) × P covering its action on the base A(P ) × M . Taking the quotients by G(P ), we get the universal bundle U(P ) := A(P ) × G(P ) P → B(P ) × M whose structure group is G ad [82]. We define a connection of the G-bundle A(P ) × P → A(P ) × M whose horizontal subspace at (A, p) ∈ A(P ) × P consists of tangent vectors (α, S) ∈ Ω 1 (M, adP ) ⊕ T p P such that V (d * A dA) −1 d * A α (p) + S ∈ T p P is horizontal with respect to the connection A. Then for all v ∈ Ω 0 (M, adP ), the vector (−d A v, V v (p)) is horizontal at (A, p) and the connection is invariant under the action of G(P ). The curvature vanishes when contracted with the vector (−d A v, V v ). Therefore the connection descends to the universal (or tautological) connection A U on U(P ), and for α, β ∈ ker(d * A ) and tangent vectors S, T ∈ T p M , the curvature is [82] . The above construction of A U follows the general procedure of passing from connections on P (or P ad ) to one on the Borel construction A(P ) × G(P ) P = A(P ) × G(P )/Z(G) P ad of the G(P )-action on P , and the universal curvature F U can also be obtained by applying the Chern-Weil map on the G(P )-equivariant extension of F A .
In the context of Hitchin's moduli space, the universal bundle and its connection can be constructed similarly. Let P be a principal G-bundle over a closed orientable surface C (though the restriction to two dimensions is not essential) and let A H (P ) ⊂ A(P ) × Ω 1 (C, adP ) be the subset of Hitchin pairs, i.e., (A, φ) satisfying Hitchin's equations (3.3). As above, we only consider a smooth subset of the Hitchin moduli space M H (P ) := A H (P )/G(P ) from Hitching pairs with minimal stabiliser. The tangent space of A H (P ) at (A, φ) consists of vectors (α, ξ) satisfying the linearised There is a connection on the principal bundle A H (P ) → M H (P ) by choosing the horizontal spaces as the orthogonal compliments of the G(P )-orbits. A vector (α, ξ) is orthogonal to the orbits of G(P ) if and only if (α, ξ) ∈ ker(d A , b φ ) * or equivalently, d * A α + b * φ ξ = 0. The curvature on horizontal vectors (α, ξ), . The universal bundle U H (P ) := A H (P ) × G(P ) P is a principal G ad -bundle over M H (P ) × C. We define a connection on the G-bundle A H (P ) × P → A H (P ) × C by declaring that a vector (α, ξ, S) at (A, φ, p) is horizontal if φ ξ) (p) + S is so at p ∈ P with respect to A. This connection is invariant under G(P ) and for any v ∈ Ω 0 (C, adP ), the vector (−d A v, −b φ v, V v ) from the infinitesimal G(P )-action is horizontal at (A, φ, p). The curvature vanishes when contracted with the vector (−d A v, −b φ v, V v ). So the connection descends to the universal (or tautological) connection A U on U H (P ) → M H (P ) × C whose curvature we denote by F U . For (α, ξ), (β, η) ∈ ker(d A , b φ ) * and tangent vectors S, T ∈ T p C, the curvature is 26) The universal connection A U can be understood from a similar Borel construction and the universal curvature (A.26) is obtained by a Chern-Weil map on the G(P )-equivariant extension of F A .
In addition to the universal connection A U , there is a universal 'Higgs' field φ U constructed as follows. First, there is a tautological 1-form on A H (P ) × C with values in A H (P ) × adP given by (A, φ, p) → φ p . Since it is basic with respect to the action of G(P ), it descends to φ U ∈ Ω 1 (M H (P ) × C, A H (P ) × G(P ) adP ). We have where (α, ξ, S), (β, η, T ) are as above. We call (A U , φ U ) the universal Hitchin pair; it satisfies the Hitchin equations along C for each point in M H (P ). See [83] for another construction using the language of algebraic geometry.
A.6. Characteristic classes of the universal bundle. Let G be a compact semisimple Lie group. Suppose P is a G-bundle over a closed orientable surface C. Let A H (P ) be the set of Hitchin pairs and M H (P ) := A H (P )/G(P ) be the Hitchin moduli space. The quotient of A H (P ) × P by the group G(P ) of gauge transformations is the total space of the universal bundle U H (P ) → M H (P ) × C. Since G(P ) contains Z(G) which acts trivially on A H (P ), U H (P ) is a principal G ad -bundle ( §A.5). We will write M = M H (P ) and U = U H (P ) for short.

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H (C ′ , G) Z( G) [2] v v ♥ ♥ ♥ ♥ ♥ ♥ of regular coverings in which the labels on the arrows are again the groups of deck transformations.