5d Higgs Branch Localization, Seiberg-Witten Equations and Contact Geometry

In this paper we apply the idea of Higgs branch localization to 5d supersymmetric theories of vector multiplet and hypermultiplets, obtained as the rigid limit of $\mathcal{N} = 1$ supergravity with all auxiliary fields. On supersymmetric K-contact/Sasakian background, the Higgs branch BPS equations can be interpreted as 5d generalizations of the Seiberg-Witten equations. We discuss the properties and local behavior of the solutions near closed Reeb orbits. For $U(1)$ gauge theories, we show the suppression of the deformed Coulomb branch, and the partition function is dominated by 5d Seiberg-Witten solutions at large $\zeta$-limit. For squashed $S^5$ and $Y^{pq}$ manifolds, we show the matching between poles in the perturbative Coulomb branch matrix model, and the bound on local winding numbers of the BPS solutions.


Introduction
Starting from [1], there had been numerous development in exactly computing quantities in supersymmetric field theories on curved manifolds, using the localization method. Intuitively, these developments can be loosely classified into two approaches, which should be the two sides of a whole but not fully understood story.
One side of the story mostly concerns the exactly computable quantities of theories defined on selected interesting geometries. These developments allow us to study the fine structures of these quantities.
In 3-dimension, progresses have been made to understand the structures of the supersymmetric partition functions on selected geometries. In particular, the supersymmetric partition function on squashed S 3 (smooth homological 3-spheres) is shown to be expressed in terms of double-sine functions [2,3,4,5]. Multiple-sine functions are a family of interesting functions that enjoy factorization properties. Indeed, these properties are studied in [6]; later the S 3 partition function, originally written as a matrix model, is unpacked into a product of vortexanti-vortex partition functions [7]. This result later inspired the Higgs branch localization in 3-dimension [8] [9]. See also the Higgs branch localization on S 3 × S 1 [10].
In 5-dimension, there are also similar results. Starting from the round spheres S 5 [11,12], then on the squashed S 5 [13,14,15], and later on the Sasaki-Einstein manifolds [16,17,18], the perturbative partition functions were computed, and the full non-perturbative partition functions were conjectured. Supersymmetric theories and their partition functions on other type of manifolds are also studied in detail [19,20,21]. Similar to 3-dimensional theories, the 5d perturbative results are expressed in terms of a matrix model with triple-sine functions (or their certain products) as integrand. As a member of the multiple-sine functions family, triple-sine function also has a similar factorization property: it factorizes into three pieces (two pieces when d = 3) corresponding to three closed polar Reeb orbits (two polar orbits when d = 3), which leads to the conjecture of the full non-perturbative partition function in Coulomb branch on Sasaki-Einstein manifolds [16,17].
It is therefore natural to believe that the Higgs branch localization in 3-dimension can be generalized to 5-dimension, as a manifestation of the factorization property. Namely, factorizing the matrix model and performing the contour integral should pick up residues from the poles, and these residues are the local contributions from some new BPS solutions localized to certain loci on the manifold.
There is another side of the story which concerns more about general geometric structures that support supersymmetries. Following the idea of obtaining supersymmetry on a curved manifold by taking rigid limit of suitable supergravity [22], various developments took place to understand the relation between supersymmetry and the underlying geometries [4] [23,24,25,26,27,26,28,29]. In particular, it is shown in [30] that N = 2 and N = 1 partition functions in 3d and 4d are holomorphic function of transversally holomorphic foliation moduli and complex structure moduli.
The above two approaches should somehow be consistent. For instance, it would be interesting to ask the questions like "can we start from a general supersymmetric theory on a 3-manifold as in [27] and carry out the Higgs branch localization", or "what geometric struc-tures do the ingredients of the matrix model, or the vortex/anti-vortex partition functions actually correspond to, if the whole partition function is an invariant or holomorphic function of certain geometric structures". At the moment, to the best of the author's knowledge, these kinds of questions are not fully understood.
Therefore, in this note, we try to start from general backgrounds that support 5d supersymmetry and apply the idea of Higgs branch localization in this general setting. We find that the Higgs branch localization locus can be interpreted as a 5d generalization of perturbed Seiberg-Witten equations on symplectic or Kähler 4-manifolds. If we recall that solutions to (a sequence of) perturbed Seiberg-Witten equations are related to pseudo-holomorphic curves in symplectic 4-manifolds [31], the 5d partition functions can be viewed in some sense as invariants that captures "pseudo-holomorphic" objects in contact manifolds.
The content of this note will be organized as follows: 1. In section 2, we start from 5d N = 1 supergravity and review some geometric implications of the existence of supersymmetry. In particular, we study the generalized Killing spinor equation and show how it is related to K-contact geometry.
2. In section 3, we write down the general supersymmetric theories of N = 1 vector multiplet and hypermultiplets, which can be obtained by taking rigid limit. Then we redefine the field variables, and obtain corresponding cohomological complexes. By adding Q-exact terms we obtain the new BPS equations. On a K-contact background, the Higgs branch BPS equations can be interpreted as generalization of Seiberg-Witten equations, by introducing the generalized Tanaka-Webster connection: , ∂ a α +∂ * a β = 0 L a R α = L a R β = 0 (1.1) We will show that Sasakian structures are concrete examples where solutions to the above equations have simple behavior. We also extend the discussion to more general K-contact backgrounds, and study the local behavior of solutions around closed Reeb orbits. 3. In section 4, We show that as the Higgs branch parameter ζ → +∞, one can suppress the deformed Coulomb branch if the matter content and the Chern-Simons level satisfy a certain inequality. We also show that on squashed S 5 and Y pq manifolds, the bound on local winding numbers of Higgs branch BPS solutions corresponds to poles in the Coulomb branch matrix model integrand. To do so, we interpret the shift of the form Σ i ω i /2 in the 1-loop determinant as the the R-component of the "Chern-connection" on the anti-canonical line bundle of the K-contact structure. 4. In the appendix, we summarize useful aspects of contact geometry and a review of Spin C -structures on any contact metric manifolds. The generalized Tanaka-Webster connection and its Dirac operator are also reviewed, which are closely related to the BPS equations.
2 From Supergravity to K-contact Geometry

Symplectic-Majorana Spinors, Self-duality and Chirality
In this subsection we will discuss properties of symplectic-Majorana spinors and their bilinears on 5-dimensional manifolds.

Symplectic-Majorana spinors
A symplectic-Majorana spinor ξ I with I = 1, 2 satisfies where C is the charge conjugation matrix C = C + , C αβ = −C βα . We define two products (, ) and , for any two symplectic-Majorana spinors ξ I and χ J as Note that the (, ) is anti-symmetric, while , is symmetric and positive semi-definite. We also denote the action of any differential k-form ω on any spinor ψ by

Bilinears of a symplectic-Majorana Spinor
Given any spinor ξ, one can define several bi-linears using the products defined above.
• Real vector R m ≡ − ξ, Γ m ξ 1 . The norm-squared of R is R m R m = s 2 , or equivalently ι R κ = s 2 , where we define the metric-dual 1-form κ m = g mn R n .
These bilinears satisfy various algebraic identities following from the Fierz identities, which are summarized in the Appendix [A].

5-dimensional Self-duality
Given any nowhere-vanishing spinor ξ, we construct the associated set of quantities (s, R, κ, θ IJ ). By rescaling we set s = 1. We then use them to decompose any p-forms and we call ω H (ω V respectively) is called the horizontal 2 (vertical) part of ω. We then decompose the space of p-forms Ω p (M) = Ω p V (M) ⊕ Ω p H (M), and define the projection Let * be the Hodge star operator of metric g. Then we have operator ι R * : In view of this, we can restrict ι R * onto Ω 2 H (M) and decompose horizontal 2-forms into self-dual (+) and anti-self-dual 2-forms (−), according to the eigenvalues of ι R * : So the final result is one can decompose any 2-form ω into Before moving to next subsection, we remark that following from Fierz-identities, the 2-forms Θ IJ are always self-dual: Also one can straight-forwardly extend the self-duality to the general case where s = 1.

Chirality
As reviewed in appendix [A], we define the chiral operator Γ C ≡ −R m Γ m , which satisfies chirality (following from Fierz-identities (A.10) and the assumption s = 1) Naturally, Γ C induces a decomposition of spinor bundle S = S + ⊕ S − , and we denote the projection operators (2.11)

5-dimensional N = 1 Minimal Off-shell Supergravity
In this subsection we briefly review 5-dimensional minimal off-shell supergravity discussed in [32][33] [34] (see also literatures on superspace formalism [35][36]), and then extract the generalized Killing spinor equation by taking the rigid limit, following the idea of [37]. The Weyl multiplet contains the following bosonic field content (note that there is a curly V and straight V ) Here I, J = 1, 2 are indices of SU(2) R symmetry, A m is the abelian gauge field corresponding to central charge with field strength F = dA, V is a 2-form, C is a scalar. Field t IJ and V IJ are both SU(2) R triplet, meaning that and similarly for V IJ . Define the field strength F = dA. The fermionic field content contains where ψ is the gravitino, η is the dilatino. Finally, the supergravity transformation δ Sugra has symplectic-Majorana parameter ξ I . To obtain a supersymmetric theory of some matter multiplet on some manifold M, one can first couple it to the above Weyl multiplet G, and then set all fields in G to some background values that is invariant under the supergravity transformation δ Sugra . In particular, we set the fermions (ψ, η) to zero background, and requires two spinorial differential equations (with coefficients comprised with fields {V, V, F , t IJ }) δ Sugra ψ = 0, δ Sugra η = 0, (2.15) and look for background values of {V, V, F , t IJ } that admit a solution ξ I . The result of such procedure is [22,24,23,25]: • Supersymmetry transformation Q obtained from δ Sugra by substituting in background values of {V, V, F , t IJ }.
• A Q-invariant Lagrangian from the coupled supergravity Lagrangian, where all remaining bosonic fields from G are auxiliary background fields.
• Some geometric data, including metric g, p-forms and so forth, determined by combinations of {V, V, F , t IJ }.
First of all, we focus on the equation δ Sugra ψ = 0, which we refer to as the generalized Killing spinor equation in the following discussion. Up to some numerical coefficients, the generalized Killing spinor equation reads where ∇ contains the usual Levi-Civita spin connection as well as SU(2) R gauge field V m when acting on objects with I, J indices. Equation (2.16) is studied in [38], where geometric restrictions imposed by different numbers of solutions is discussed. Subsequently, in [39] both differential equations δψ = δη = 0 are solved in a coordinates patch. It is shown that, locally, deformations of auxiliary fields that preserves (2.16) and (2.17) can be realized as Q-exact deformation or gauge transformations. This suggests that path integrals of appropriate observables may be topological or geometrical invariants. For us, it is important to note that δ Sugra η = 0 implies This will be used to ensure the closure of the rigid N = 1 supersymmetry.

Generalized Killing Spinor Equation
In this subsection we will review some basic properties the Killing spinor equations that are relevant to later discussions. Some terminology in K-contact geometry will be reviewed in the following subsection. As introduced in the previous subsection, the Killing spinor equation for symplectic-Majorana spinor ξ I is Here F is a closed 2-form, and V is a usual 2-form. The connection ∇ contains the Levi-Civita spin connection and possibly a non-zero SU(2) R background gauge field V m acting on the I-indices. All these fields are from the Weyl multiplet G and we call them auxiliary fields below. Equation (2.18) can also be written in a more convenient form

Symmetries
The Killing spinor equation enjoys several symmetries that will help simplify later discussions.
• Background SU(2) R symmetry, which acts on the I-index.
• Shifting symmetry: one can shift the auxiliary fields F and V by any anti-self-dual 3 2 (2.20) and the equation is invariant.
• Other symmetries related to the huge degrees of freedoms discussed in [39]. We will come back to this shortly.

Solving the Killing spinor equation
Let ξ I be a solution to the Killing spinor equation (2.18). Then one can construct bilinears s, R m , κ m and Θ IJ using ξ I . By directly applying equation (2.18), one obtains several differential properties of these bi-linears: (2.21) Using the above basic properties, one can partially solve Recall that the Killing spinor equation enjoys a shifting symmetry, and therefore one can always set Ω − = 0 in the above solutions; so let us do this. Then we have To further simplify later discussion, let us apply the results in [39]. The Killing spinor equation and the dilatino equation are solved locally, and it is shown that the auxiliary fields are highly unconstrained by the existence of solutions.
The freedom can be understood by looking at the Fierz identities. In some sense, solving the equations is just to properly match the "Γ-matrices structure" in (2.18). Note that one can use the Fierz-identities to alter the Γ-structures. Hence one can adjust the SU(2) R -gauge field (V m ) IJ to cancel terms with Γ-matrices in (2.18), and consequently other auxiliary fields are left unconstrained. We can use the local freedom in s and t IJ to smoothly adjust them such that s = 1 and tr(t 2 ) ≡ t I J t J I = −1/2 in a patch. Note that given a global Killing spinor solution, s and tr(t 2 ) should be patch-independent functions, and therefore, the adjustment can be made global. Therefore, let us deform the solution and auxiliary fields such that globally s ≡ 1 ⇒ ι R F = 0 and tr(t 2 ) ≡ −1/2. Furthermore, it is shown in [39] that resulting deformations in the actions are Q-exact, and therefore the above adjustment does not change the expectation values of BPS observables.

A special class of solutions
Equation (2.23) implies that it is interesting to look at a special class of solutions where the auxiliary fields F and V are such that for some constant Λ ∈ R. This implies κ is a contact 1-form, namely it satisfies (assuming

Towards a K-contact structure
Now the bi-linears from the special class of solutions satisfy various conditions: (2.27) The first row tells us that (κ, R) defines a contact structure, while the second row implies the contact structure closely resembles a K-contact structure. The only violation appears in where we defined Φ = 2 t IJ Θ IJ , instead of the standard form It is easy to bring the system to a standard K-contact structure. Let us use an adapted veilbein {e A } such that g = a e a e a + κ ⊗ κ, e 5 = κ, ι R e a=1,2,3,4 = 0, Φ(e 1 ) = e 2 , Φ(e 3 ) = e 4 .
With the new metric, the quantities (κ, R, g ′ , Φ) defines a standard K-contact structure on M: Along with the change in metric, one needs to properly deform the auxiliary fields to preserve the equation (2.18). By explicitly working out the change in spin connection ω A B , one can identify the required deformations in F and V (both are deformed by multiples of dκ), which indeed also preserve the condition (2.25), and therefore no inconsistency arises. Finally, as independent and unconstrained auxiliary fields in [39], the resulting deformations in actions are Q-exact and do not change the expectation values of BPS observables.
To summarize, any solution to (2.18) of the special class can be transformed into a standard one, such that the resulting set of geometric quantities (κ, R, g, Φ) form a K-contact structure. Later we will discuss BPS equations on K-contact and Sasakian backgrounds, where the equations are better controlled than on completely general supersymmetric backgrounds.

K-contact Geometry
In this subsection, we summarize most important aspects and formula of contact geometry that we will frequently use in later discussions. For more detail introduction, readers may refer to appendix [C].

Contact structure
A contact structure is most conveniently described in terms of a contact 1-form. A contact 1-form on a 2n + 1-manifold is a 1-form κ such that κ ∧ (dκ) n = 0. (2.32) This is analogous to the definition of a symplectic form on an even dimensional manifold. We can associate quantities (R, g, Φ) to κ called a contact metric structure, such that The vector field R is called the Reeb vector field, and Φ is like an almost complex structure in directions orthogonal to R. On a contact metric 5-manifold, we will frequently use an adapted vielbein {e A }, {e A }, such that e 5 = R, Φ (e 1 ) = e 2 , Φ (e 3 ) = e 4 , and dκ = 2 e 1 ∧ e 2 + e 3 ∧ e 4 , g = a=1,2,3,4 e a ⊗ e a + κ ⊗ κ, (2.34) Note that the first equation implies dκ is self-dual, namely ι R * dκ = dκ. We will also use the complexification of {e A }: 2. K-contact and Sasakian structure A K-contact structure is a contact structure κ and the associated (R, g, Φ), such that Note that one immediately has L R Φ = 0. For a general contact structure, the integral curves of R, or equivalently, the 1-parameter diffeomorphisms ϕ R (t) (the Reeb flow ) generated by R, can have three types of behavior. The regular or quasi-regular types are such that the flow are free or semi-free U(1) action, respectively. The irregular type is such that the flow is not U(1), and therefore the integral curves of R generally are not closed orbits.
Generic irregular Reeb flows are difficult to study, however, situation can be improved when the contact structure is K-contact. In this case, the closure of the Reeb flow (it preserves g by definition), viewed as a subgroup of the Isom(M, g), is a torus T k ⊂ Isom(M, g); k is called the rank of the K-contact structure. On a K-contact 5-manifold, 1 ≤ k ≤ 3.
Finally, a Sasakian structure is a K-contact structure with additional property Sasakian structures are the Kähler structures in the odd-dimensional world. They satisfies certain integrability condition, and all quantities discussed above, as well as some metric connections associated with g, live in great harmony. We will later see that on Sasakian structures, the Higgs branch BPS equations have very simple behavior, very much like Seiberg-Witten equations on Kähler manifolds. To end this section, we tabulate the correspondence between the structures (including some we haven't mentioned) in even and odd dimensional worlds. In this section, we begin by reviewing the 5-dimensional N = 1 vector multiplet and hypermultiplet. Then we consider deforming the theory with Q-exact terms to localize the path-integral. We discuss the deformed Coulomb branch solutions and the Higgs branch. We rewrite the Higgs branch equations and interpret them as 5-dimensional generalizations of Seiberg-Witten equations on symplectic 4-manifolds. We also discuss basic properties of solutions to the 5d Seiberg-Witten equations, including their local behavior near closed Reeb orbits.

Vector-multiplet
The Grassman odd transformation Q of vector multiplet (A m , σ, λ I , D IJ ) can be obtained directly from N = 1 supersymmetry transformation, which can be obtained by taking the rigid limit of coupled supergravity in [40]. Using a symplectic-Majorana spinor ξ I satisfying Killing spinor equation (2.19), the transformation can be written as Here the spinor ξ I is Grassman even. The transformation squares to and L is Lorentz rotation acting on spinors. The parameters are and we used the vector field R m to define self-duality Ω ± H (M), see section 2.1. Note that, as in [41], there is a violation term in δ 2 D IJ , which can be cured if there exists a function u and a vector field v m such that In the case of P mn = 0, one can show that v = 0 and the function u is proportional to the scalar curvature of the metric (g, ∇ LC ). In the presence of P, by explicitly expanding every term, one can show that We observe that the first row is just the left hand side of (2.17), and therefore using Namely, we find the function and the vector field to be Finally, we point out that function u will appear in the supersymmetric Yang-Mills Lagrangian for the vector multiplet (which is denoted as P in [39]), in the form of

Hypermultiplet
A hypermultiplet in 5-dimension consists of a set of scalars φ A I , two spinors ψ A and a set of auxiliary scalars Ξ A I ′ . Here I, I ′ = 1, 2 are two different copies of SU(2) indices (in particular, I corresponds to the SU(2) R -symmetry), while A = 1, 2 is a separate Sp(1) index. They satisfy reality conditions In the above, Ω AB is the invariant Sp(1) tensor Ω 12 = −Ω 21 = 1. The reality conditions reduces the independent components. The field φ A I can be represented by two complex scalar φ 1,2 and similarly for the field Ξ A I ′ . The field ψ A can be represented in terms of one spinor ψ One can couple the hypermultiplet to a U(N) vector multiplet by setting the independent fields to be in appropriate representation of U(N), for instance, It is well-known that one cannot write down an off-shell supersymmetry transformation for a hypermultiplet with finitely many auxiliary fields. But it is possible to write down a Grassmann odd transformation Q which squares to bosonic symmetries. As transformation parameters, we use a symplectic-Majorana spinor ξ I satisfying Killing spinor equation (2.19), and an additional SU(2) ′ -symplectic-Majorana spinorξ I ′ , satisfying One can viewξ I ′ as a orthogonal complement of ξ I in the spinor space, and therefore corresponds to anti-chiral spinors, in the sense that (3.14) The Grassman odd transformation Q is as follows: The transformation squares to the bosonic symmetries where G is the gauge transformation, R is SU(2) rotations on I, J and I ′ , J ′ indices, L is Lorentz rotation; the parameters are As in previous sections we define the function s ≡ (ξ I ξ I ), and Ω ± H (M) is defined with respect to the vector field R m ≡ −(ξ I Γξ I ).

Twisting, Q-exact Deformations and Localization Locus
In this subsection, we first review a redefinition (the twisting) of field variables in vector multiplet and hypermultiplet. Then using the redefined variables, we introduce the Q-exact deformation terms and derive the localization locus.
The twisting First introduced in [11] [42] in the context of Sasaki-Einstein backgrounds, all field variables with I or I ′ indices can be "twisted" (invertible using Fierz-identities (A.10)) using ξ I andξ I ′ . In our situation, assuming s = 1 and recalling (2.22), we define: After such redefinitions, χ and H are both horizontal self-dual two forms with respect to vector field R m , φ A + are chiral spinors while Ξ A=1,2 − are anti-chiral. In terms of these twisted field variables, the originally complicated BRST transformations can be rewritten into very simple forms: In order to derive Qψ A and QΞ A − , one needs to use the symmetry (ξ IξJ ) = (ξ JξI ) and completeness relations (3.14). Also we will use dκ · φ + ≡ 1/2(dκ) mn Γ mn φ + to simplify the notations in the following discussions.
For later convenience, we separate Qψ into chiral and anti-chiral part: 20) which implies that Note that dκ is horizontal, and therefore its Clifford multiplication does not change chirality, similar to that in 4-dimension. Also, the new spinorial variables have reality condition, for instance,

Q-exact terms
We are now ready to introduce the Q-exact terms. There are three of them: where h maps the "spinor" φ A + in the hypermultiplet to a self-dual 2-form h(φ + ). Specifically, we take it to be where ζ is a "fake" FI-parameter taking value in the U(1)-components of the Lie-algebra g of the gauge group G, A 0 is a reference gauge field taking value in the u(1)'s in g with the property ι R F A 0 /2 = 0, and α is the canonical bilinear map from chiral spinors to self-dual Up to this point, other than s = 1, we make no assumption on the background geometry.
Hence dκ does not have to be self-dual; dκ + means we extract the self-dual part from dκ. To ensure positivity, we need to analytically continue σ → −iσ, Ξ A − → iΞ A − . Now one can expand all terms, and integrate out auxiliary field H, or equivalently, impose the field equation of H: Then the bosonic Q-exact terms reads and therefore, we have the localization locus Note that using the reality condition of φ A + , the second equation on the left is equivalent to that of one component φ and similarly σ (φ + ) = 0 ⇔ σ φ A + = 0. Therefore, in the following, we will just ignore the index A, and regard φ + as in the fundamental representation of gauge group G = U(N c ).

Deformed Coulomb Branch
The deformed Coulomb branch is the class of solutions to (3.28) such that φ A + = 0. Then the equations reduces to This is a deformed version of the contact-instanton equation introduced in [42]. The undeformed version is later studied in [43,44,45,46], in the context of κ being a contact structure. So in principle, there could be a tower of instantonic solutions, very much like the deformed instantons in 4d.
To be more concrete, we consider the case when κ is a contact 1-form. Then dκ + = dκ, and one immediately has a most simple solution (assuming where σ takes constant value in the Lie-algebra g. On top of these simple solutions, one may have a lot of instantonic solutions. When (κ, R, g, Φ) give rise to a Sasakian structure, the reference A 0 can be chosen to be the restriction on K M of the Chern connection on K C(M ) , where C(M) is the Kahler cone of Sasakian manifold M. In such case, one can show that dA 0 ∝ dκ and ι R F A 0 /2 = 0.

5d Seiberg-Witten Equation
Let us consider other classes of solutions to (3.28), with non-vanishing φ + . To be concrete in many statements, we will focus on the case where (κ, R, g, Φ) form a K-contact structure, or Sasakian structures to ensure concrete existence of solutions. This will allow us to rewrite the equations in a very geometric way that resembles the 4-dimensional Seiberg-Witten equation on symplectic manifolds. We will see that Sasakian structures serve as examples where Higgs vacua always exist, and other non-trivial solutions have nice behavior. We also discuss the case of general K-contact structures.
The algebraic equation When we look for non-vanishing solution of φ + , one of the non-trivial BPS equations is (σ + m) (φ + ) = 0, where we have restored some masses for the hypermultplets by giving VEV to the scalars in the background vector multiplets that gauge the flavor symmetry. Let us consider gauge group G = U(N c ) and N f hypermultiplets, then we need to solve a matrix equation Therefore N c among the N f of φ's are selected to have non-zero values. The remaining N f − N c of φ's are fixed to be zero, and trivially satisfy all other BPS equations. These vanishing components do not have further non-trivial solutions which we will discuss shortly. The 1-loop determinants for the trivial components will be the same as that in the Coulomb branch, with the argument σ replaced by solutions (3.32). The non-zero components, on the other hand, requires extra care. First of all, given generic masses {m na = m n b if a = b}, equation d A σ = 0 implies A is also completely diagonalized. Therefore, in such favorable situations, the gauge group U(N c ) is completely broken to U(1) Nc , and for each of these non-zero components of φ, one only needs to consider a U(1)-gauge field, which we will assume from now on. These non-zero components will have to satisfy the remaining BPS equations individually, to which we will discuss the solutions shortly. To do so, we will first rewrite the remaining BPS equations in a more familiar form.

Rewriting the localization locus
In the appendix [C][D], we review in detail Spin C spinors and corresponding Dirac operators on any 5-dimensional K-contact structures. We summarize here several most relevant aspects: • The spinor bundle S has a canonical Dirac operator / ∇ TW , induced from generalized Tanaka-Webster connection on T M for any given K-contact structure [47][48] [49]. One can show that this Dirac operator can be written in terms of the Levi-Civita connection ∇ LC : which are precisely the ones appearing in Qψ ± without the gauge field A.
• There exists a canonical Any other Spin C -bundle W can be written as W = W 0 ⊗ E for some U(1)-line bundle E. It is important to note that, when the manifold is spin, namely when the genuine spinor bundle exists, then S and W 0 is related by S ⊗ K M . Therefore W can also be written as

D TW
A 0 /2 = L R ⊕ √ 2 ∂ +∂ * : Ω 0,even → Ω 0,even ⊕ Ω 0,odd (3.35) Now we can include the gauge field A onto the stage. As discussed above, we only consider G = U(1) and A is viewed as a U(1)-connection of certain line bundle L. Therefore, φ + should be really considered as a section of W + ≡ S + ⊗ L. We decompose L = K 1/2 M ⊗ E so that S ⊗ L = W 0 ⊗ E, and we also decompose the gauge field A according to Therefore, the Dirac operator / D TW A on W + = W + 0 ⊗ E can be identified as  [48], for instance) of A 0 that leads to similar identification, with the only difference that L R is replaced by L R − ia 0 (R) for some appropriate U (1) gauge field a 0 .
where L a R = L R − ia (R) ,∂ a =∂ − ia 0,1 and so forth. With such identification in mind, one can rewrite the Dirac-like equation in (3.28) In particular, we write φ + = α ⊕ β ∈ Ω 0,0 (E) ⊕ Ω 0,2 (E), and (3.28) can be written as where we have decompose F + a = F dκ a + F 2,0 a + F 0,2 a , and the bilinear map α(φ) is written more concretely as (see appendix [A, D] for choice of basis and matrix representation of Γ AB ) It is clear that the equations on the left take a similar form of ζ-perturbed Seiberg-Witten equations on a symplectic 4-manifold [31,50,51], and therefore we will call them the 5d Seiberg-Witten equations in the following discussion.
Let us pause to remark that, the operator / ∇ + 1/8dκ mn Γ mn is discussed in the context of Sasaki-Einstein manifold, and similar results were obtained in [52]. The unperturbed version of Seiberg-Witten-like equation on a contact metric manifold is also proposed in [49].
In the following we will focus on equations on the left in (3.39). They are a novel type of equations that awaits more study. Let us try to make a first step to understanding the solutions. As discussed earlier, we consider the gauge group G = U(1), and therefore σ and ζ are just real constants.
A Higgs vacuum First, we argue that the 5d Seiberg-Witten equations on Sasakian structures have one simple solution.
The remaining BPS equation is If A 0 is chosen to be induced from 6d Chern connection, this may be not true on a general K-contact background; however, if the K-contact structure is Sasakian, then (3.41) indeed holds [49] [47]. Therefore on a Sasakian structure, one always has at least one most simple solution, which we will call a Higgs vacuum.

Properties of general solutions
Let us now focus on the 5d Seiberg-Witten equations on a K-contact structure (with emphasis on Sasakian structures). First of all, the Dirac equations implȳ where N is the Nijenhuis tensor N : T 1,0 M * H → T 0,2 M * H , which vanishes for any Sasakian structure. Therefore, when (κ, R, g, Φ) is Sasakian, one has ∂ * a β =∂ a α = |α| |β| = 0. In the case of a 4-dimensional Kahler manifold, the left hand side would be replaced by the intersection number c 1 (E) · [ω], a topological number independent on ζ. Therefore, when ζ = 0, the sign of c 1 (E) · [ω] will determine whether α or β will survive; in particular, in the limit ζ ≫ +1, only the solutions with β = 0 survive. On a 5-dimensional Sasakian manifold, however, the left hand side is not a topological number, and therefore at the moment we do not have a topological criteria to determine which of the (3.44) will survive. For non Sasakian K-contact structure, one needs to take the Nijenhuis tensor into account. Combining the Weitzenbock formula, Kahler identities and triangle inequalities, we obtain several estimates (where we rescaled (α, β) → ( √ ζα, √ ζβ), z is some constant, and λ > 1 is a real constant) and In the inequalities, ∇ A 0 +a is the connection on K M ⊗ E, ρ A 0 is some function depending on A 0 but not on ζ. Again, if the integral on the left in the first estimate is bounded from above, or it scales at most of order ζ ǫ<1 (ǫ = 0 in 4-dimension, since it is topological and independent on ζ), then the above estimate tells us as ζ → +∞, almost everywhere on M |β| → 0, |α| → 1, (3.48) and |d a J α| does not grow faster than ζ. The second estimate then implies the overall derivative ∇ A 0 +a β → 0 faster than ζ ǫ−1 , and therefore ∂ * a β = ∂ a α → 0 as well. Therefore, let us make a bold conjecture that we have a similar situation as in 4-dimension. Namely for a general K-contact manifold, as ζ → +∞, β is highly suppressed, and we are left with α satisfying∂ a α = 0, which approaches α → 1 rapidly once away from any zeros α −1 (0) ∈ M. In the case of Sasakian manifold, the type of solutions with non-zero β are less and less likely to survive when ζ → +∞. With this conjecture in mind, we study the local behavior of 5d Seiberg-Witten equations with large positive ζ near any closed Reeb orbit.

The Local Model Near Closed Reeb Orbits
On a generic contact manifold, the integral curve of the Reeb vector field may have uncontrollable behavior, as we mentioned early on. However, if the structure is K-contact, then the contact flow, viewed as a subgroup of the group Isom(M, g) of isometries, has a closure of T k ⊂ Isom(M, g).
In other words, the integral curve of the Reeb vector field going through a point p ∈ M forms a torus of dimension less than or equal to k. One can think of the curves as similar to irrational flows on a torus. The integer k ≤ 3 for a K-contact five-manifold, and is called the rank of the structure. So, a rank-1 K-contact structure is a quasi-regular or regular contact structure, and k ≥ 2 are all irregular.
The isometric T k -action highly degenerates at the closed Reeb orbits, namely k − 1 of the generators do nothing to the points on closed Reeb orbits. Therefore, at a small neighborhood C × C 2 of a closed Reeb orbits C, the k − 1 generators rotates the C 2 (leaving C fixed), while the remaining 1 generator, corresponding to the Reeb field R, translates along C.
Bearing this picture in mind, one can write down an adapted coordinate (θ, z 1 , z 2 ) on a small neighborhood C × C 2 of any closed orbit C, such that T k = {t 0 , ..., t k−1 } acts on it in an intuitive way. Such a coordinate system is characterized by the numbers (λ 0 ; λ j , m 1j , m 2j ), j = 1, ..., k − 1, where λ 0 , ..., λ j are rationally independent positive real numbers, m 1j and m 2j are two lists of integers. In such a coordinate, the Reeb vector R and contact 1-form κ can be written as The isometric subgroup T k acts on the patch by Let us pick a basis for horizontal 1-forms in region C × C 2 One can also easily verify that dκ = ie z 1 ∧ ez 1 + ie z 2 ∧ ez 2 . This suggests that one can view e z i , ez i as spanning T 1,0 M * and T 0,1 M * . Under such assumption, one can show ∀α ∈ Ω 0,0 ,

Examples
Let us look at the example of squashed S 5 ⊂ C 3 One can define the Reeb vector field R and contact 1-form κ by restriction of Then it is easy to show that near the orbit C 3 ≡ θ ∈ [0, 2π] | 0, 0, e iθ ω −1 3 ∈ S 5 ω , one can rewrite R and approximate κ in the new coordinate θ = (2i) −1 log ( The natural T 3 action can be rearranged as e iϕ , e iϕ 1 , e iϕ 2 · (z 1 , z 2 , z 3 ) = e iϕ 1 e iϕ z 1 , e iϕ 2 e iϕ z 2 , e iϕ z 3 , (3.56) so that its action on the local coordinate is e iϕ e iθ , e iϕ 1 w 1 , e iϕ 2 w 3 , implying m 11 = m 22 = 1, and λ 1,2 = ω 1,2 − ω 3 . Similar steps can be done on Y pq manifolds, which has K-contact rank k = 2. Let us recall how Y pq manifolds are defined [53,54]. Y pq manifolds are Sasaki-Eintstein manifolds with topology S 2 × S 3 . They can be obtained by first looking at S 3 Then one can define a nowhere-vanishing U(1)-vector field T which rotates the phases of z i according to the charges [p + q, p − q, −p, −p]. The Y pq manifolds is then the quotient (S 3 × S 3 )/U(1) T . The Saaski-Einstein Reeb vector field is defined to be rotations of z i with irrational charges [ω 1 , ω 2 , ω 3 , ω 4 ] It is easy to show that near the closed Reeb orbit C ≡ {(z i ) ∈ Y pq |z 2 = z 4 = 0}, one has λ 0 = pω 1 + (p + q) ω 3 , λ 1 = 3, m 11 = 1, m 21 = 0. (3.59) The 5d Seiberg-Witten equation near C We study the equations near a closed orbit C. Again, we rescale (α, β) → ( √ ζα, √ ζβ) for a better looking equation: Using (3.52) and its underlying assumption, the last equation in (3.60) can be reduced to usual equation on C 2 , since L a R α = L a R β = 0, However, as we discussed early on, we conjecture that when ζ → +∞, β, ∇β → 0 and therefore the differential equations of α and β reduce to the holomorphic equation on C 2 ∂ a α = 0, ζ → +∞. (3.62) In this sense, the zero set of large-ζ 5d Seiberg-Witten solutions corresponds to pseudoholomorphic objects in K-contact manifold M. Namely near orbit C, α −1 (0) takes the form of C × Σ where Σ is "pseudo-holomorphically" mapped into M. Of course this is just a naive description and far from rigorous; more careful treatment is needed. There are known smooth solutions to the 4-dimensional Seiberg-Witten equations, which are lifts of 2-dimensional vortex solutions; however, there are more solutions that we do not yet know how to describe. Nevertheless, let us assume that α has the usual asymptotic behavior α → e in 0 θ e in 1 ϕ 1 +in 2 ϕ 2 , where n 0 ∈ Z, n 1,2 ∈ Z ≥0 is required by holomorphicity and smoothness at the origin 5 : near the origin, α ∼ e in 0 θ z n 1 1 z n 2 2 . Therefore, Note that the winding number n 0,1,2 should be bounded by ζ, similar to the situation in [9]. We demonstrate this on a Sasakian structure in the limit ζ ≫ 1. We consider the integral where Vol (κ) ≡ dκ ∧ * dκ. On the other hand, if E is a trivial line bundle and thus a can be viewed as a global 1-form, Later we will see that this bound corresponds to poles in the perturbative Coulomb branch matrix model. More general situation needs more careful treatment, and we leave it for future study.

Partition Function: Suppression and Pole Matching
Suppose one obtains a BPS solution to the localization locus (3.28), then the contribution to the partition function from this particular solution is the product where exp [−S cl ] is the exponentiated action evaluated on the BPS solution. The 1-loop determinants are where we have shifted σ → −iσ, and A cl denotes the value of A as a solution to (3.28). Let us denote for a moment H A ≡ ∇ TW R − iA(R), which we recall is part of the Dirac operator / D TW A .
In the Coulomb branch, where one does not include the deformation QV mixed , one encounters the BPS equations as a "decoupled" system of differential equations In [52], it is shown that on a Sasaki-Einstein geometry (or other geometry with a large scalar curvature), a solution A to the first line will imply the imply the second line has only trivial solution φ + = 0; namely the operator / D TW A , and in particular H A does not have zero as one of its eigenvalues. Let iλ m = 0 be an eigenvalue of H A labeled by some quantum numbers m, with the corresponding eigenstate φ m . Then This is equivalent to the statement H A+∆Am φ m = 0, where the ∆A m (R) = λ m . Namely, there exists certain new gauge field A + ∆A m with ∆A m (R) = λ m , such that H A+∆Am has zero eigenvalue. Of course, A + ∆A cannot be a solution to the original Coulomb branch BPS equations, but it could be a solution to some deformed BPS equations. In our case, they are precisely the Higgs branch BPS equations, where the QV mixed is taken into account. Therefore, solutions to the Higgs branch equations are expected to correspond to poles in the Coulomb branch matrix model, which are factors of the form (iσ − iλ m ) −1 coming from the hypermultiplet determinant. We will see this more precisely later in this section.

Suppression of the Deformed Coulomb Branch
In this subsection, we will review the supersymmetric actions for vector and hypermultiplet, and show that it is possible to achieve suppression of perturbative deformed Coulomb branch as ζ → +∞. This allows two things: 1) One can take a large ζ limit, and only focus on the contributions from 5d Seiberg-Witten solutions to the partition function.
2) One can take the Coulomb branch matrix model, close the integration contour of σ, and identify each pole of the integrand with a 5d Seiberg-Witten solution.

The supersymmetric actions
The Super-Yang-Mills and hypermultiplet action can be obtained by taking rigid limit of supergravity action. The bosonic parts read Note that we use the original field variables to write the action, and it is straight forward to use the invertible twisting to convert to new field variables. One can also add in Q-invariant Chern-Simons terms for the vector multiplet [42], and we have made the shift σ → iσ stated earlier where the pure Chern-Simons terms are The 5d Chern-Simons level k is an integer. As noted in [42], L SCS 3,2 is not invariant under rescaling of κ, while L SCS 5 is invariant.

The classical contributions
The deformed Coulomb branch equations are On a Sasakian background, ι R F A 0 /2 = 0, the perturbative solutions are To simplify the analysis further, let us assume the gauge group G = U(1), and we have only one real number ζ; more general situation where the gauge group can be completely Higgsed can be obtained straight forwardly. Evaluated on (4.11), the actions discussed above give the classical perturbative contribution to the partition function. We are interested the asymptotic behavior of these contributions as ζ → +∞.
2) There is no classical contribution from L Hyper since all fields in the hypermultiplet vanish.
3) Finally, there is classical contribution from L YM . To evaluate it, one needs to consider the field redefinition H mn = 2F + mn + 2σt IJ + D IJ (Θ IJ ) mn − 4F + mn , the equation of motion of H and BPS equation to solve D IJ in terms of σ Using some Fierz-identities, the field redefinition implies On more general background with non-vanishing F and P, the classical contribution from exp{−S YM } has the same leading behavior of ζ 2 as above, although the precise value will depend on the geometric background. The 1-loop determinant will be more complicated products of triple-sine function,

The perturbative 1-loop contributions
The perturbative 1-loop determinant from Coulomb branch was studied in [11,13,52]. It was shown that the 1-loop determinant can be expressed in terms of triple sine functions S 3 (z|ω), or their particular products.
The triple sine function S 3 (z|ω) with ω = (ω 1 , ω 2 , ω 3 ) is defined as the regularized infinite product 4.17) or in terms of generalized Γ-function Γ 3 (z|ω 1 , ω 2 , ω 3 ): What is most important to us is the asymptotic behavior of the triple-sine function: when ω i > 0, we have when z → ∞ (B 3,3 are multiple Bernoulli functions, see [55,56]) The 1-loop determinant from perturbative Coulomb branch computed in literatures are products of triple sine functions, with argument of the form where µ is a weight in the representation R that the hypermultiplet belongs, and N(ω) is a real constant determined by equivariant parameters 6 . If we consider the deformed Coulomb branch, then what we need is to compute the super-determinant of from hypermultiplet 7 , which effectively shifts σ → σ + iζ/2 + const in the Coulomb branch 1-loop determinant. In the limit of large ζ, each S 3 factor of the 1-loop determinant of hypermultiplet tends to (4.23) As an example, on the round sphere S 5 , N + and N − hypermultiplet with q = ±1 charge under gauge group G = U(1) contribute a 1-loop determinant so the overall ζ 2 classical and 1-loop contributions to the partition function is asymptotically Therefore there is a window of suppression as ζ → +∞ which comes from the competing σ and |σ| as one integrates σ from −∞ → +∞. Similar result can be obtained for squashed S 5 , where the volume Vol (κ) ∝ (ω 1 ω 2 ω 3 ) −1 , which only contributes an overall factor of the partition function as ζ → +∞. On Y pq manifolds, one needs to replace the 1-loop determinant with generalized triplesine functions, which are products of original triple-sine function. Therefore, suppression of deformed-Coulomb branch implies one needs to satisfy certain inequality of the form where ρ is some positive real number that encodes the geometry information.

Matching The Poles And The Shift
Similar to 3-dimensional Higgs branch localization [9], if one performs the integral of the Coulomb branch matrix model by closing the contour appropriately, one picks up residues from the enclosed poles. Before checking the matching between poles and 5d Seiberg-Witten equation, let us first understand the operator ∇ TW R − iι R A properly.
The operator ∇ TW R,A and L R Let φ + = ξ ⊗σ E be a section of S + ⊗E, where E is equipped with A as a U(1) connection 8 . Equivalently, noting that S M , one can choose an appropriate section σ of K 1/2 M , and rewrite φ + = (ξ ⊗σ) ⊗ (σ −1 ⊗ σ E ), where we have factored out a piece ξ ⊗σ ∈ Γ W 0 + .σ then provides the explicit connection 1-form for the abstract canonical connection "A 0 " on K M : and hence ∇ TW where we have used ∇ TW R,A 0 /2 = L R on W 0 + , a = A − A 0 /2 as a connection on In the case where A = 0, namely the perturbative Coulomb branch solution, one has ι R a = −ι R A 0 /2 and therefore the shift in eigenvalues of ∇ TW On the other hand, one of the BPS equation reads As a section of T 0,0 M * ⊕ T 0,2 M * , ξ ⊗σ contributes eigenvalues of L R of the form corresponding to modes with asymptotic behavior ∼ e in 0 z n 1 1 z n 2 2 near each closed Reeb orbit. Now the remaining puzzle is to determine the value of ι R A 0 .
Squashed S 5 and ι R A 0 As an example, let us consider matching the poles of 1-loop determinant on squashed S 5 with the local solutions to the 5d Seiberg-Witten equation. We will focus on the orbit C 3 discussed before, and recall the formula (3.55).
Note that one can define local orthonormal vielbein e A by first defining an orthonormal frame at θ = 0, then use R to translate them to almost the whole C 3 . In particular, one can define e A in such a way that it is adapted to and invariant under the K-contact structure, namely L R e A = 0. However, translating e A back to θ = 2π will in general disagree with the starting value. To obtain a vielbein well-defined on C 3 , one can rotate the original e A along the way. For instance, in terms of the complex basis Then we have In this basis, one can compute the derivative along R Let ψ + = (a, b) T ∈ S + . Using the explicit representation (A.4) the derivative ∇ LC R reduces to where we used Γ 12 ψ + = Γ 34 ψ + = iσ 3 ψ + and dκ · ψ + = 4iσ 3 ψ + .
When ω 1,2,3 = 1, one can define Killing spinor by is a solution to the above Killing spinor equation, then using the above local expression of ∇ LC , one can show that ξ behaves like ∼ exp 3i 2 θ along C 3 . Finally, if we requireσ to satisfy ξ −1/2 ⊗σ = Const ∈ Γ(T 0,0 M * ), (4.38) one deduces that along C 3 namely, along C 3 ,σ has periodic behavior exp(− 3i 2 θ) to cancel that of ξ −1/2 . This implies the shift On a general squashed S 5 ω , we continue to chooseσ such that it has exp − 3i 2 θ periodic behavior along all three closed Reeb orbits. Then near any of three orbits, we recover the shift of eigenvalues as in [16][13] Finally, the bound (3.67) on the winding numbers can now be written as i=1,2,3 where we defined n 3 = n 0 −n 1 −n 2 , which is non-negative if one consider all three closed Reeb orbits C 1,2,3 . Recall that the 1-loop determinant in deformed Coulomb branch is obtained by a shift in that of Coulomb branch Combining with the (4.42), bound saturation then means

44)
Poles of the S 5 ω perturbative 1-loop determinant Recall that the perturbative 1-loop determinant of a hypermultiplet coupled to a U(1) vector multiplet on S 5 ω is The poles are the zeros of the infinite products n 0 i=1,2,3 where we have reinstated the mass induced from a background U(1) vector multiplet. All the possible poles are The first equation above is just the equation (σ + m) φ = 0 in the Higgs branch, and the second is just the bound we obtained above, if one takes the poles with + sign. These are the poles that will be picked up when one close the contour in the upper half plane of the σ-plane. Note that this is allowed thanks to the suppression of deformed Coulomb branch as ζ ∼ Im σ → +∞.
The case of Y pq manifolds Recall (3.59) that near the orbit z 2 = z 4 = 0, the Sasaki-Einstin Reeb vector field can be written as where One can then read off again ι R A 0 = 3 by choosing the sectionσ with the same criteria as S 5 , and the bound on local winding number is also determined After redefinition n e 1 ≡ n 1 + n 0 p, n α ≡ n 0 , the bound saturation corresponds to the poles 9 Im σ = 3n e 1 + n α 3 2 (q − p) − 1 2l + 3 2 .

51)
We remark that the redefinition seems to implies n e 1 ∈ Z, but global analysis, namely, the equation (71) in [16] implies n e 1 + n α p = n e 2 ≥ 0 for the poles in the upper-half σ plane.

Summary
In this work, we apply the idea of Higgs branch localization to supersymmetric theories of N = 1 vector and hypermultiplet on general K-contact background. We show that with this generality the localization locus are described by perturbed contact instanton equations in the deformed Coulomb branch, and 5d Seiberg-Witten equations in the Higgs branch. Neither of these two types of equations is well understood. We focused on the latter, and some study basic properties of its solutions, including their local behavior near closed Reeb orbits, which is shown to reduce to 4-dimension Seiberg-Witten equations. This seems to implies that these BPS solutions corresponds to "pseudo-holomorphic" objects in K-contact manifolds, if the 4-dimensional story can some how be lifted. Finally, we study the suppression of deformed Coulomb branch as the parameter ζ → +∞, and manage to match the poles of perturbative Coulomb branch matrix model with the bound on local winding numbers. From this point on, it is straight-forward to use the factorization property of perturbative partition function on S 5 and Y pq manifolds to perform the contour integral of σ. The result should produce classical and 1-loop contributions of each local Seiberg-Witten solutions, in a form of products of contributions from each closed Reeb orbit.
However, this raises a question on whether the non-perturbative Coulomb branch partition function gives rise to more poles and therefore new Seiberg-Witten solutions, that is not described in our local study. It may well be that our assumption on β → 0 at ζ → +∞ is not always valid and therefore they correspond to more solutions that we missed. We plan on returning to the above two problems in the future.
Another question that we did not address is that whether the partition function is invariants of certain geometric structure. In [39], it is shown that the generalized Killing spinor equation (2.16) has huge degrees of local freedom, including the background metric g, κ and R, which are reflected as Q-exact deformations in the partition function. Therefore it would be interesting to explore the geometric or topological meaning of N = 1 partition functions and expectation values of BPS operators. We believe that one needs to look closely the constraint (2.17) and understand its geometric meaning. Also, one can further study the 5d Seiberg-Witten equations (3.39). For instance, it would be interesting to understand its where Λ ± n denotes restrictions on n ei n e1 + n e2 − n e3 − n e4 = n α q n e1 − n e2 = −n α p , n ei 0, n ∈ Λ + n n ei < 0, n ∈ Λ − n (4. 52) moduli spaces, which we did not take into account when matching the poles. But it is likely that on generic K-contact structures, the moduli spaces are zero-dimensional, considering the matching of perturbative poles and local solutions. Another interesting question is whether the solutions to (3.39) correspond to certain "pseudo-holomorphic" objects, similar to the 4dimensional story. If so, the partition functions will have more explicit geometrical meaning in terms of a "counting" of these objects. Finally we have the issue of A 0 . In several discussions, including obtaining the bound on winding number, we relied on the assumption that the K-contact structure is Sasakian, in order to have a simplification ι R F A 0 /2 = 0. It is not clear if this can always be achieved on general K-contact structures, or if there are other wiser choice of A 0 with the horizontal property, while simultaneously enables the identification / D TW A 0 /2 ↔ L R + ∂ +∂ * .

Acknowledgment
We thank Francesco Benini and Wolfger Peelaers for explaining their work in great detail. We thank Francesco Benini, Dario Martelli, Wolfger Peelaers, Martin Roček and Maxim Zabzine for reading the manuscript and their helpful comments. We thank Sean Fitzpatrick for discussions on related mathematics.

Appendices A Spinors and Gamma Matrices
In this appendix we review our convention on spinors and Gamma matrices, as well as useful formula.
Spinors and Gamma matrices First let us consider a 5-dimensional spin manifold M. The rank of spin bundle S is rank C S = 2 [5/2]  We use lower case Greek letters α, β, ... to denote spinor indices, and overlinez to denote usual complex conjugation of any complex number z. The complex conjugate of a spinor is defined asξ α = ξ α . We define and similarly for Γ mnk , Γ mnkl . These products of Gamma matrices satisfy One can define a chiral and anti-chiral decomposition using any unit-normed vector field. In our case, we use the Reeb vector field R and define a chiral operator Γ C ≡ −R m Γ m , and decompose S = S + ⊕ S − .
An explicit representation of Gamma matrices we will use is

Fierz identities
For arbitrary Grassmann even spinors ξ 1,2,3 , we have the basic Fierz identity It follows immediately two useful formula The Fierz-identities implies several useful formula. Let ξ I be a symplectic-Majorana spinor and (s, κ, R, Θ) be the associated quantities described in the main text. Then for any symmetric tensor t IJ and anti-self-dual (w.r.t to R m ) 2-form Ω + . In particular, if t IJ = 0 everywhere and satisfies t IJ = ǫ II ′ ǫ JJ ′ t I ′ J ′ , then the 2-form t IJ Θ IJ is nowherevanishing, since it squares to

B Conventions in Differential Geometry
In this section we review our convention in differential forms, spin connection and more tensor analysis. Differential forms For any differential p-form ω, the components ω m 1 .

Connections
Let ∇ be an arbitrary connection on T M, then for any vector X = X m ∂ m , one defines the connection coefficients Γ k mn as ∇ m X k = ∂ m X k + Γ k mn X n . The torsion tensor of such a connection is defined as T k mn ≡ Γ k mn − Γ k nm . Let {e A } be an orthonormal basis with respect to metric g. Then given any connection ∇ preserving g, one can write down Cartan structure equation and so define connection 1-form (also called spin connection) ω A It is easy to solve the spin connection for the Levi-Civita connection ∇ LC of g. Suppose One can use this to obtain ω m A B Γ AB , or one can exploit the identification to simplify computation: Given any connection ∇ that preserves metric g, maybe with torsion, one can induce a connection on the spin bundle S We will sometimes use · to denote Clifford action of any differential p-form ω on spinors: So in particular, dκ · ψ = 1 2 dκ mn Γ mn ψ. Lie derivative Let X be a smooth vector field. Then the Lie-derivative L X on a differential form ω is expressed in terms of Cartan identity When acting on another vector field Y , L X Y = [X, Y ].

C K-contact Geometry
In this appendix we review some basics aspects about contact geometry and K-contact structures. Interested readers may refer to [57] for more detail 10 . Symplectic geometry is a well-known type of geometry in even dimensions. There, a symplectic structure is defined to be a closed and non-degenerate 2-form ω. In odd dimensions, there is a similar type of structures, called contact structures, which have many similar and interesting behaviors as symplectic structures. and we call it the Reeb vector field associated to contact the 1-form κ. The Reeb vector field on a compact contact manifold generates 1-parameter family of diffeomorphisms (an effective smooth R-action on M), which is usually called the Reeb flow ϕ R (t), or the contact flow; the flow translates points along the integral curves of the R. It follows from the definition that the flow preserves the contact structure, since L R = ι R dκ + dι R and L R κ = 0, L R dκ = 0. It is important to note that the integral curves (or equivalently, the Reeb flow) of R have three types of behaviors:

Contact Structure
1) The regular type is that all the curves are closed and the Reeb flow generates free U(1)-action on M, rendering M a principal U(1)-bundle over some 2n-dimensional symplectic manifold. 10 However we point out that the convention of exterior derivative d in [57] is such that, for instance, 2) A quasi-regular type is that although the curves are all closed, the Reeb flow only generates locally-free U(1)-action.
3) The irregular type captures the generic situations, where not all the integral curves are closed. Irregular Reeb flows can have very bad behaviors, but if the Reeb vector field preserves some metric on M, then the behavior could still be tractable. In other context, irregular Reeb flows are better than the other two types, in the sense that they are nondegenerate and may provide isolated closed Reeb orbits.

Contact metric structure
Just as in symplectic geometry, one would like to have some metric and almost conplex structure into the play, so that the contact structure has more "visible" properties.
Given a contact 1-form κ, one can define a set of quantities (κ, R, g, Φ) where g is a metric and Φ is a (1, 1)-type tensor, such that where ∇ LC denotes the Levi-Civita connection of g. We call such set of quantities a contact metric structure.
There are a few useful algebraic and differential relations between quantities. First we have Φ n m R m = κ n Φ n m = 0, where Ω g is the volume form associated to metric g. From this one can show that dκ satisfies And in fact, if one takes an adapted vielbein, for instance in 5-dimension, satisfying e 5 = R, Φ (e 2i−1 ) = e 2i , κ (e 1,2,3,4 ) = 0, i = 1, 2, one has dκ = 2 e 1 ∧ e 2 + e 3 ∧ e 4 . (C.7) Moreover, using ι R dκ = 0 and κ(R) = 1, it can shown that namely R is geodesic. There are useful relations between R and Φ: for any contact metric structure, and also ∇ LC (C.10)

K-contact structure
As we have mentioned earlier, irregular Reeb flows can be more tractable if certain metric is invariant under the flow. This leads to the notion of K-contact structure, where the Reeb vector field is Killing with respect to the metric in a contact metric structure: It is called a K-contact structure, if a contact metric structure satisfies an additional condition L R g = 0. (C.11) Note that this is equivalent to, since Φ and dκ are related by metric g, it is easy to see that L R Φ = 0, and consequently, ∇ m R n = −Φ n m .

Sasakian Structure
A Sasakian structure is a K-contact structure (κ, R, g, Φ) with additional constraint Sasakian structures are Kähler structures in the odd dimensional world. Therefore, it enjoys many simple properties that allow simplification in computations.

Generalized Tanaka-Webster connection
There have been several special connections on contact metric structures introduced in various literatures. For us, the most important one is the generalized Tanaka-Webster connection. There are actually two special connections, both of which are called generalized Tanaka-Webster connection, one introduced by Tanno [58] and the other introduced in [48]. Their names comes from the property that when restricted on a integrable CR structure, the two connections reduces to the usual Tanaka-Webster connection.
On a general contact metric structure, the two connections are different. However, when the structure is K-contact, the two connections induces the same Dirac operator on the spin bundle S via the standard formula In terms of the Levi-Civita connection ∇ LC , this Dirac operator reads which is the operator that appears in the localization locus (3.28). Using the projection P ± to chiral and anti-chiral spinors, one has for chiral spinor ∀φ + ∈ Γ(S + )

D Spin C bundle and the Dolbeault-Dirac operator
In this appendix we review the Spin C bundles on a contact metric manifold and a canonical Dirac operator on any K-contact structure. Consider a contact metric structure (κ, R, g, Φ). Then on the horizontal tangent bundle T M H , Φ defines a complex structure and thus induces a (p, q)-decomposition of the complexification T M H ⊗ C = T 1,0 M ⊕ T 0,1 M, Let us focus on a 5-dimensional contact metric structure (M; κ, R, g, Φ). One can start from an adapted vielbein e A as discussed before, and consider the complexification e z 1 ≡ e 1 + ie 2 , e z 2 ≡ e 3 + ie 4 . (D.2) With this complex basis, one sees that dκ is of type-(1, 1) as expected dκ = i (e z 1 ∧ ez 1 + e z 2 ∧ ez 2 ) . (D. 3) The bundle W 0 ≡ T 0,• M * is also a Spin C bundle in the sense that T M * acts on it in a Clifford manner    ω · ψ = √ 2i ωīē¯i ∧ ψ − g¯i j ω j ι eī ψ , ω = ω i e i + ωīē¯i ∈ Γ (T M * H ) κ · ψ = e 1 · e 2 · e 3 · e 4 · ψ .
(D.4) which satisfies the Clifford algebra {ω·, µ·} = 2g (ω, µ). In particular, W 0 decomposes into chiral and anti-chiral spinor bundle according to the eigenvalue ±1 of Γ C ≡ −κ · Using the complex basis ez i , one can define an orthonormal basis of W 0 : W 0 + = span 1, If one represents φ = a 1 + a 2 2 ez 1 ∧ ez 2 + a 3 √ 2 ez 1 + a 4 √ 2 ez 2 ↔ (a 1 , a 2 , a 3 , a 4 ) T , (D.7) then the above Clifford action is represented as (A.4). On a contact metric structure, there may be other Spin C bundles. They can be obtained by tensoring an arbitrary complex line bundle E: In particular, when the manifold is spin, the spin bundle S can be obtained by where K M ≡ T 0,2 M * . More generally, any Spin C bundle W can be written as W = S ⊗ L 1/2 for some complex line bundle L 1/2 (and its square L is called the determinant line bundle of W ). For instance, W 0 = S ⊗ K This implies that given a connection on S (which can be induced from a metric connection ω A B ) and a U(1)-connection 11 A on L 1/2 , we have a connection on W = S ⊗ L 1/2 ∇ A ψ = ∇ψ − iAψ, ∀ψ ∈ Γ(W ) (D.10) The situation of W 0 is a bit special, since one can induce a canonical U(1)-connection A 0 on K M using the Chern connection ∇ C on the almost-hermitian cone C(M). Therefore, taking A 0 as a reference connection, any connection A on a Spin C bundle W can be written in terms of a U(1)-connection a on E as A = 1 2 A 0 + a. The above construction is good for any contact metric structure. Now let us focus on a Kcontact structure, and use the generalized Tanaka-Webster connection to induce a connection ∇ TW on S. Combining with a U(1)-connection A on L 1/2 , one can define a Dirac operator / D In [48], it is shown that when E is trivial and a = 0, namely A = 1/2A 0 , / D TW A 0 /2 (α + β) = L R (α + β) +∂α +∂ * β, α + β ∈ Ω 0,0 ⊕ Ω 0,2 = Γ(W 0 + ) (D. 12) where the Dolbeault operator ∂ and∂ are defined in the usual way 12 ∂ ≡ π p+1,q • d : T p,q M * → T p+1,q M * ,∂ ≡ π p,q+1 • d : T p,q M * → T p,q+1 M * (D.14) Note that the two operators do not square to zero in general; define N (ω p,q ) ≡ π p−1,q+2 (dω p,q ) andN (ω p,q ) ≡ π p+2,q−1 (dω p,q ), then one has ∂ 2 α p,q = −N (∂α p,q ) − ∂N (α p,q ) , ∂ 2 α p,q = −N ∂ α p,q −∂N (α p,q ) , (D.15) 11 A local basis σ on L 1/2 is assumed, such that ∇ A (f σ) = df ⊗ σ − iA ⊗ (f σ) 12 On a K-contact structure, on has in general (recall that L R preserves Φ and therefore the (p, q)-

Weitzenböck Formula
We review several useful formula for studying 5d Seiberg-Witten equations, which are direct generalization from those on symplectic 4-manifolds.