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Generalized Seiberg-Witten equations on Riemann surface

In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra"Higgs field". Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kahler manifold and construct a pre-quantum line bundle over the moduli space of solutions.


Introduction
Dimensional reduction of gauge-theories have been instrumental in the understanding of TQFTs.
As examples, one can consider the vortex equations [Hit87], which are the dimensional reduction of 4-dimensional Yang-Mills equations, the dimensional reduction and quantization of 3-dimensional Chern-Simons gauge theory, etc. Inspired by these, a dimensional reduction of the 4-dimensional Seiberg-Witten equations -in the spirit of [Hit87] -was considered in [Dey02]. The reduction gave an additional Higgs field and "vortex-like" equations.
In this paper we extend the results in [Dey02] to a non-linear setting by considering the dimensional reduction of the 4-dimensional, non-linear Seiberg-Witten equations. These equations were introduced by Pidstrygach [Pid04] for dimension four and by Taubes for dimension three [Tau99]. The non-linearity here arises by replacing the target H in the standard Seiberg-Witten, with a hyperKähler manifold M admitting certain symmetries. As a result, our target manifold is a non-compact hyperKähler manifold. Under suitable conditions, M admits a canonically defined hyperKähler potential that plays a central role in the construction of the Quillen metric of the pre-quantum line bundle over the moduli space of solutions.
Under suitable regularity conditions, the moduli space of solutions M turns out to be a compact, finite-dimensional Kähler manifold. We then proceed to construct the Quillen determinant line bundle over M. Regarding M as the phase space, we define the Hilbert space quantization of M as the space of holomorpic sections of the Quillen determinant line-bundle as in [DM13]. In the absence of Higgs field, the equations can be thought of a symplectic vortex equations taking values in a complex submanifold of the target.
One can also define an invariant for the target M , as in the case of the symplectic vortex equations [CGiRS02]. The invariant couples the cohomology class of the Kähler symplectic form (also the first Chern class of the Quillen line-bundle) with the pull-back of the equivariant cohomology classes of the target manifold via the evaluation map and thus connects the geometry of M with that of M .

Acknowledgement
The author owes a debt of gratitude to Rukmini Dey for the immensely helpful discussions with the Quillen construction and quantization and for patiently answering many of the author's questions regarding the same.

Preliminaries and Notations
In this section, we give a quick introduction to preliminaries required to construct the non-linear Dirac operator. We will be chiefly concerned with the case where n = 4. Although most of the discussion below applies to n = 3 as well. For a detailed exposition, we refer the interested reader to [Pid04], [Sch10] and [Tha13] for the case n = 4 and to [Tau99], [Cal10] for n = 3.

Spin G ε (n) structure
Let G be a compact Lie group and ε be an element of order two in the centralizer Z(G) of G. The element (−1, ε) ∈ Spin(n) × G generates a normal subgroup of order two, denoted by (−1, ε) . For We have the following exact sequence: where Π G is the quotient map.
Let X be a n-dimensional Riemannian manifold and P SO(n) −→ X denote the principal frame bundle over X. Then a Spin G ε (n)-structure over X is a principal Spin G ε (n)-bundle, which is an equivariant double cover Q 2:1 − − → P SO(n) × X P G/{ε} , along with principal G/{ε}-bundle P G/{ε} −→ X.
The obstruction to the lift of the principal G/{ε}-bundle to a principal G-bundle is theČech cocycle that represents a cohomology class w G ∈ H 2 (X, Z 2 ). The short exact sequence (1) implies that a Spin G ε (n) structure over X will exist iff w Spin G ε (n) (P SO(n) × X P G/{ε} ) = 0. Here 1 where w 2 ∈ H 2 (X, Z 2 ) is the second Steifel-Whitney class.

HyperKähler manifolds
A HyperKähler manifold (M, g M ) manifold is a 4n-dimensional Riemannian manifold, endowed with three complex structures (I 1 , I 2 , I 3 ), satisfying quaternionic relations I 2 1 = I 2 2 = I 2 3 = I 1 I 2 I 3 = −1, which are compatible with the metric g M and are covariantly constant with respect to the Levi-Civita connection. Associated to each of these complex structures, we can define a symplectic 2-form ω i (·, ·) = g M (I i (·), ·). As a matter of convenience, we club the symplectic 2-forms into an sp(1)-valued two-form:

HyperKähler Moment map
An action of a Lie group G on a hyperKähler manifold (M, g M , I 1 , I 2 , I 3 ) is said to be hyperKähler if G acts by isometries and preserves the complex structures. If the action is such that the moment maps exist for each ω i , then, it is said to be hyperHamiltonian. We can then define moment maps µ 1 , µ 2 , µ 3 respectively. Playing the same trick as above, we can club the three moment maps into a single hyperKähler moment map µ with co-efficients in sp(1). More precisely, Definition 2.2.1. For a hyperKähler action of a compact Lie group G on M , a smooth map µ : M −→ sp(1) * ⊗ g * is said to be a hyperKähler moment map if it satisfies the following: Assume additionally, that M is endowed with an isometric action of Sp(1) such that the action permutes the complex structures I 1 , I 2 , I 3 . Such an action of Sp(1) is called as a permuting action.
Remark 2.2.1. [Pid04] If M admits a permuting Sp(1)-action that commutes with the hyperKähler action of G, then the moment map for the G-action exists and the action is thus hyperHamiltonian.

Generalized Seiberg-Witten
Assume that M admits a hyperKähler action of a compact Lie group G and a permuting action of Sp(1) which commutes with the action of G. Additionally, assume that (−1, ε) ∈ Sp(1) × G acts trivially on M . The action descends to an action of Sp(1) × (−1,ε ) G ∼ = Spin G ε (3), where we identify Sp(1) ∼ = Spin(3). Such an action of Spin G ε (3) is called permuting action. Denote by ϑ + the homomorphism We say that the action of Spin G ε (4) is permuting if the action is induced by a permuting action of Spin G ε (3) via the homomorphism ϑ + .
In what follows, we assume that (X, g X ) be a four-dimensional, oriented, compact, Riemannian manifold and M is a hyperKähler manifold admitting a permuting action of the group Spin G ε (4). Let µ : M −→ sp(1) * ⊗ g * denote the hyperKähler moment map for the hyperKähler action of G.

Generalized Spinors and Covariant Derivative
We define the space of generalized spinors to be the smooth Spin G ε (n)f our-equivariant maps Let A(Q) be the set of all connections on Q and A ⊂ A(Q) be the set of all connections which are a lift of the Levi-Civita connection ϕ. Every connection a on P G/ε uniquely determines a connection A ∈ A . We define the covariant derivative of a spinor u ∈ S , w.r.t a connection A ∈ A where, w ∈ R 4 , andw denotes the horizontal lift of π SO(4) (p)(w) ∈ T π X (p) X. Alternatively, one can write D A u as for v ∈ T p Q and the subscript "hor" denotes that D A u vanishes on vertical vector fields.

Non-linear Dirac operator
In order to define the generalized Dirac operator, we need to define the appropriate Clifford multiplication. Recall that, if W is a Cl 4 -module, restricting it to Spin(4), Clifford multiplication is a map of Spin(4)-representations: Define W + to be the Spin G ε (4)-equivariant bundle T M −→ M equipped with an action induced by ϑ + . More precisely, for any w + ∈ W + , the action is given by: [q + , q − , g] · w + = T q + T gw + and W − to be the Spin G ε (4)-equivariant bundle T M −→ M equipped with the left action (see [Sch10]): Identifying R 4 with H by mapping the standard, oriented basis (e 1 , e 2 , e 3 , e 4 ) of R 4 , to (1,ī,j,k), Since m(h) 2 = −g R 4 (h, h) · id W + ⊕W − , by universality property, m extends to a map of algebras We define the Clifford multiplication by Spin G ε (4)-equivariant map: Composing Clifford multiplication c with the covariant derivative, we get the non-linear Dirac Define the configuration space C = S × A . Then this a left G : . Then W ± is a G -equivariant vector bundle over S with the fibre at a point u ∈ S being given by the equivariant sections Γ(Q, u * W ± ) Spin G ε (4) . Hence, for A ∈ A , the Dirac operator defines a G -equivariant section of the infinite-dimensional vector bundle W − .

Generalized Seiberg-Witten equations
Denote by where γ ∈ Λ 2 + (R 4 ) ⊗ g, η l and ξ l are the basis elements of sp(1) and Λ 2 + R 4 respectively, η ∈ g and ·, · g is the Ad-invariant inner product given by the negative of the Killing form. Supressing the isomorphism, we write the generalized Seiberg-Witten equations in dimension four [Pid04] For the case G = U (1), we have the following compactness result due to [Pid04]: Let now X = R 4 . Henceforth, we shall always assume that the target hyperKähler manifold admits a permuting action of Spin G ε (4). For a connection a ∈ Ω 1 (P G , g) G , let A ∈ A be the connection on principal Spin G ε (4)-principal bundle Q over R 4 . The curvature 2-form F a ∈ Ω 2 (Q, g) G hor . In terms of trivilzation of Q over R 4 and the basic co-ordinates (x 1 , x 2 , x 3 , x 4 ), the connection A can be described as a Lie-algebra-valued 1-form Similarly, a connection a on the principal bundle P G over R 4 can be written as and the curvature of a as a Lie-algebra-valued 2-form where, We set the Levi-Civita connection ϕ = 0, which is reasonable, since we are on R 4 and corresponds to choosing the covariant derivative to be d. Then the covariant derivative of u with respect to the connection a is given by: Then, the Seiberg-Witten equations for a pair (u, a) on R 4 are equivalent to: where I 1 , I 2 , I 3 are complex structures on the target manifold and {µ 1 , µ 2 , µ 3 } are the moment maps associated to the complex structures ω 1 , ω 2 , ω 3 respectively.

Dimensional Reduction
We follow the apporach highlighted in [Dey02] for the dimensional reduction of the Seiberg-Witten equations in dimension four and modify it for the case of generalized Seiberg-Witten equations in dimension four described above.
For simplicity, we denote the principal G-bundle over R 4 simply by P . Fix a connection a ∈ Ω 1 (P, g) G and fix a trivialization of P over R 4 . We now demand that {a i } 4 i=0 be independent of (x 3 , x 4 ), then a 0 , a 1 define a connection on R 2 , which we again denote by a = a 0 dx 0 + a 1 dx 1 . The Lie-algebra-valued functions a 2 and a 3 , which we re-label as φ 1 and φ 2 , define auxillary fields(usually known as Higgs fields) on R 2 . The second equation now reads: Then, equations (5) read: The covariant derivative is an equivariant, horizontal 1-form D a u ∈ Ω 1 (P, u * T M ) G hor and therefore descends to a one form on R 2 with values in u * T M/G. The G-invariant complex structures I 1 , I 2 , I 3 define a quaternionic structure on the bundle u * T M/G −→ R 2 . We fix I 1 and regard M as a complex manifold with respect to I 1 , then∂ a u = D a u − I 1 D a u •Ĩ R 2 defines a (0, 1)-form on R 2 with values in u * T M/G with respect to the complex structure I 1 .
We rewrite the rest of the terms of the first equation in (4) as X Φ (u) : In other words, we have expressed Φ = −β 1 (Φ) + iβ 2 (Φ). The fundamental vector fields due to β i (Φ) are defined as: , v ∈ T p P.
We now write the reduction of (4) to R 2 as In what follows, we will focus our attention to the case G = U (1). In this case, Φ is just a lift of complex-valued (0, 1)-form on (R 2 , I R 2 ). With some abuse of notation, we will denote by Φ, the 1-form on Σ as well. Since [Φ, Φ * ] = 0, the above equations simplify to: This says in particular that the equations for dimensional reduction to R 2 (11) de-couples completely into symplectic vortex equations with target as the complex submanifold M c := µ −1 c {0} ⊂ M and the one for Higgs field: 2. If we restrict to the case where Φ = 0, then the case once again boils down to the symplectic vortex equations restricted to M c : Since (7) are conformally invariant, for (a, u, Φ) ∈ C, we define the dimensionally reduced Seiberg-Witten equations on Σ by: The gauge group G = C ∞ (P, U (1)) U (1) action on the configuration space, given by g · ((a, u, Φ)) −→ a + g −1 dg, g · u , Φ leaves the equations (11) invariant.
Remark 4.0.1. Although we started out with the non-linear Seiberg-Witten equations which required that the target hyperKähler manifold admit a permuting Sp(1)-action (and hence a hy-perKähler potential under vanishing of certain obstructions), we can define the equations (11) independent of this condition.

Abstract setup
For k ≥ 3, consider the infinite dimensional Banach manifold given by Here W k,2 (P, M ) U (1) ∼ = W k,2 (Σ, P × U (1) M ). The tangent to B k,2 at a point q = (a, u, Φ) is given by: The U (1)-invariant complex structure I u 1 on u * T M determines the vector bundle E k−1,2 −→ B k,2 , with fibre at a point q = (a, u, Φ) being given by The action of the gauge group G 2,2 lifts to an action on E k−1,2 . Define the equivariant section Then the solutions to (11) are just the zeroes of F.

Linearized Operator
For x ∈ M let L x : u(1) −→ T x M denote the infinitesimal action given by L x η := K M η | x . The linearization of the equations (11) at a zero q = (u, a, Φ) ∈ B k,2 of F gives the operator: Here D a,ψ,Φ ξ = (∇ a,ψ ξ) 0,1 + ∇ ψ ξ X Φ 0,1 is a real-linear Cauchy-Riemann operator, where ∇ ψ is the Levi-Civita connection due to the connector ψ : T T M −→ T M and ∇ a,ψ is the first-order differential operator that coincides with the linearization of the covariant derivative at a point (a, u), with respect to the connector and is given by where H a ⊂ T P is the horizontal sub-bundle due to connection a on P (See [Sch10] for details).
It follows from the equivariance of the equations (11) and the fact that (a, u, Φ) is a solution to (11) that the following complex is elliptic: It is said to be irreducible if there exists a point p ∈ P , such that
Therefore I 1 preserves T C ′ ⊂ T C and hence T C ′ is a complex sub-bundle of C.
Thus C ′ is a complex submanifold of the the configuration space and the L 2 -metric restricts to a Kähler metric on C ′ . The induced action of the gauge group G k+1,2 preserves the induced metric and the symplectic two-form. C ′ also admits a momentum map µ ′ I 1 which is just a restriction of the momentum mapμ I 1 on the configuration space. We denote this restriction byμ I 1 itself. The solutions to the dimensionally reduced generalized Seiberg-Witten equations now correspond to the quotient of the zero locus of the momentum mapμ I 1 by the gauge group G k+1,2 . From standard arguments in Kähler geometry, we have the following Proposition 6.0.2. [MW74] For a compact Riemann surface, the momentum map for the action of the gauge group on C ′ is given bỹ Then the L 2 -metric that is induced on the quotient M :=μ −1 I 1 {0}/G 2,k is a Kähler metric. bundle P = det(∂ a ), [Qui85], i.e. a line bundle whose the fiber over (a, u, Φ) is given by Let ρ 0 denote the hyperKähler potential on M . Following the idea in [BR02], we modify the where Φ = φdz −φdz. From the metric, one can calculate the curvature by the formula δ w δwlog||σ|| where w is the holomorphic coordinate on the configuation space and σ is the canonical section of the determinant bundle [Qui85]. The holomorphic coordinates on the configuation space is given by (a 0,1 , u, Φ) w.r.t. the complex structure I 1 . The first term in the metric, namely exp(−ζ ′ a (0)) contributes to the curvature by a term i 2π [Qui85,DM13], which is the first term in Ω 1 (X, Y ). The second term in the metric contributes to the second term in Ω 1 . This can be seen as follows: since ρ 0 is a hyperKähler potential 2 . Then γ(ζ 1 , ζ 2 ) = Σ g M u (I 1 ζ 1 , ζ 2 )ω Σ . The third term in the metric contributes to the third term in Ω 1 . This can be seen as follows: The three terms combined gives us the following proposition: Proposition 7.0.3. On the configuration space, the Quillen bundle P equipped with the modified metric mentioned above has curvature i 2π Ω 1 .
As in [DM13,Dey09], it can be shown that this line bundle descends to the moduli space as long as the descendent of Ω 1 is integral. It is holomorphic and is a prequantum bundle since its curvature is proportional to the symplectic form with the proportionality constant i 2π . One can take holomorphic square integrable sections of this bundle as the Hilbert space of quantization.
Remark 7.0.1. The Quillen metric construction can also alternatively be done in the absence of a hyperKähler potential as in [DM13] and therefore is independent of the restrictions imposed by a permuting Sp(1)-action on the target M .

Definition of Invariants
Analogous to the case to the case of symplectic vortex equations [CGMiRS02b], we define the invariants for the hyperHamiltonian U (1)-action on the target hyperKähler manifold.

Evaluation map
Let B * ⊂ B k,2 denote the space of all solutions to the equations (11) such that for every q := (a, u, Φ) ∈ B * , the linearized operator D q is surjective. Observe that action of the group G × G k+1,2 on B * × P , given by ((g, h) * (a, u, Φ), p) −→ (g −1 u, g * a, Φ, p g(p) −1 h) is free. We therefore have the The map is G k+1,2 -invariant and G-equivariant and descends to a a map ev : Fix a point p 0 ∈ P and let G 0 := {g ∈ G k+1,2 | g(p 0 ) = Id }. Consider now the space A G := A(P ) × G k+1,2 EU (1) with a free action of the group G k+1,2 given by g * (A, e) = (g * A, g(p 0 ) −1 e). We

Discussion
The prequantum line-bundle is determined by the Kähler structure on the moduli space of solutions.
A pre-condition for the existence of the prequantum line-bundle is the integrality of the symplectic form Ω 1 , i.e, Ω 1 ∈ H 2 (M, Z). This may impose restrictions on the Riemann surface and the moduli space, since the volume of the latter depends on the volume of the surface. An illustration of the same can be seen, for instance, in [MN99], where Manton and Nasir showed (in a different context) that the standard Kähler form on the vortex moduli space is in integral cohomology iff the Riemann surface is of integral volume. It might be worth exploring if the current theory proposes a similar restriction.
Modulo the adiabatic limit conjecture, the equations (8) are closely related to the "perturbed" J-holomorphic curves: Hypothesis 9.0.1. ("Adiabatic limit conjecture") There is one-one correspondence between the solutions to (11) and the equations ∂ a u − (X Φ (u)) 0,1 = 0 The idea here is to scale the metric on the Riemann surface Σ by a large number ε −2 for ε > 0 and then study the limiting case as ε → 0. The equations get modified as: For ε = 1, we get (11) and for for ε = 0, we get (24). The original idea first appeared in the work of Atiyah [Ati88]. A modification of the same has been used by Dostoglou and Salamon [DS94], to study the relation between J-holomorphic curves and self-dual instantons.
We conjecture that the invariant defined for the hyperHamiltonian action of U (1) are non-trivial Gromov-Witten-like invariants.