N=2 gauge theories on the hemisphere $HS^4$

Using localization techniques, we compute the path integral of $N=2$ SUSY gauge theory coupled to matter on the hemisphere $HS^4$, with either Dirichlet or Neumann supersymmetric boundary conditions. The resulting quantities are wave-functions of the theory depending on the boundary data. The one-loop determinant are computed using $SO(4)$ harmonics basis. We solve kernel and co-kernel equations for the relevant differential operators arising from gauge and matter localizing actions. The second method utilizes full $SO(5)$ harmonics to reduce the computation to evaluating $Q_{SUSY}^2$ eigenvalues and its multiplicities. In the Dirichlet case, we show how to glue two wave-functions to get back the partition function of round $S^4$. We will also describe how to obtain the same results using $SO(5)$ harmonics basis.


Introduction
Since the seminal work [14] on supersymmetric localization on round S 4 there has been an intense activity in the field. This computation was soon generalized to more general curved manifolds in various dimensions [7,11,8] and to various number of supersymmetries [13]. Manifolds with boundaries in two and three dimensions were also considered and various quantities computed [16,10,15,9]. For manifolds without boundaries Atiyah-Singer Index Theory, together with a clever cohomological reorganization of fields introduced in [14], provides a shortcut to the computation of the one-loop determinants factor and avoids going through the diagonalization of the relevant differential operators arising from the quadratic part of the localizing action expanded around the saddle points. In particular, in [14], it has been shown that the one-loop factor is given by the ratio of determinants of the operator Q 2 on the kernel and cokernel of certain differential operator determined from the localizing action QV , Q being the supercharge entering in the localization procedure and V being some fermionic field. To compute the latter determinants one then uses Atiyah-Bott fixed point theorem which localizes the computation near the fixed points of Q 2 on the base manifold. In this note we will work out the partition function of N = 2 SUSY G gauge theory coupled to matter in some G-representation R, on the hemisphere HS 4 , both for Dirichlet and Neumann boundary conditions at the equator of the round S 4 . Since applying the above mathematical machinery in the case of manifolds with boundary may be subtle, we will rather solve the partial differential equations which determine the kernel and cokernel of the relevant differential operators , together with the corresponding Q 2 quantum numbers and multiplicities. With this data we write down the one-loop determinant for both the vector multiplet and the hypermultiplets, for the corresponding boundary conditions. This direct approach turns out to be feasible because of the symmetry of the problem: we choose the S 4 metric with a manifest SO(4) symmetry which is preserved by the localizing action. This allows to reduce the system of (coupled) partial differential equations depending on the four coordinates of S 4 , to a system of ordinary differential equations depending on a radial coordinate r. We will then write the expressions for the wave functions on HS 4 with Dirichlet and Neumann boundary conditions respectively. We finally show that the partition function of N = 2 theory on the round S 4 can be obtained by gluing two Dirichlet wave functions corresponding to the two hemispheres HS 4 s.
Next, for the purpose of completion, we briefly describe how one can obtain the same results using spherical harmonics of the full isometry group SO (5). The logic is essentially the same as that for SO (4).
It is interesting to note that the vector multiplet one loop determinant for Dirichlet BCs is not 'half' of that for the full round S 4 [14] as one would naively expect. As one solves the zero mode equations for kernel and co-kernel of vector multiplet ,to find the multiplicity of the unpaired bosonic and fermionic modes, it turns out that for the Dirichlet BCs the net multiplicity is one unit less and for the Neumann BCs one unit more that that required for the one loop determinant to be exactly 'half'. This deficiency or access of one unit of multiplicity for bulk theory has interesting interpretation in terms of whether one chooses to freeze some degrees of freedom or to add some extra degrees of freedom at the boundary. The contents in this note are arranged as follows: After a lightening review of N = 2 SUSY gauge theory coupled to matter in section 2 we move on to compute one-loop determinants of vectormultiplet and hypermultiplet in sections 3 and 4. Everything in the path integral computation is put together in section 5 to get the sought after wave functions of N = 2 system on hemisphere with Dirichlet and Neumann boundary conditions. Factorization of round sphere S 4 N = 2 partition function is discussed in section 6. Discussion of SO(5) harmonics and subsequent computation of Z vec 1−loop is given in sections 7 and 8. A brief set of conclusions are given in section 9.

N = 2 SUSY Field theory
We will consider 4d N = 2 extended supersymmetric field theories which are defined by eight supercharges in the flat space limit. These supercharges correspond to choosing a pair of chiral, anti-chiral Killing spinors denoted in four component notation by ξ, which satisfies following set of equations.
where ξ p ≡ γ µ D µ ξ and T µν , M are background non-dynamical fields belonging to parent supergravity theory [7]. Moreover the covariant derivative D µ also contains a background SU (2) R R-symmetry gauge field V A µB . The values of these auxiliary fields are found, if they exist, for which one is able to solve the Killing spinor equations. This Killing spinor ξ defines the supercharge Q with respect to which we localize the path integral.

Physical actions
The physical actions for vector multiplet and the matter multiplet of N = 2 SUSY on curved background are written in [7]. For SYM action we have: When restricted to the round sphere S 4 (appendix [B]), the other supergravity background fields reduce to T mn = 0,T mn = 0, V m=0 , M = −4.

(2.2)
There are also reality conditions for the fields, chosen to ensure a well defined path integral. In particular, φ = φ 2 + iφ 1 andφ = −φ 2 + iφ 1 . The gauginos λ A andλ A of opposite SO(4) chirality carry and SU (2) R doublet index. D AB is an SU (2) R triplet of auxiliary fields. For non-trivial topological sectors, characterized by instanton number k, one has to add θ-term to the full action: The action for for the matter hypermultiplets is: where again the supergravity background satisfies eq.( 2.2) and a with proper reality conditions for the fields is understood. The scalars q A carry an SU (2) R index A and in addition an index I = 1, ..., 2q of a symplectic representation of the gauge symmetry group G ⊂ Sp(q), therefore it is possible to impose a reality condition on them in the usual way. The spinors ψ carry index I and F A are auxiliary fields.

Localizing actions
The localization technique proceeds first by identifying a superchargeQ and then aQ-exact localizing action, with positive definite bosonic part, by which one perturbs the physical action in such a way that the path integral is independent of the perturbation. One then shows that the path integral localizes at the supersymmetric saddle points of the localizing action, in the sense that the one-loop approximation around them is exact.
The localizing superchargeQ depends on a choice of Killing spinors, ξ Aα andξ Aα , which we arrange in a four-by-two matrix using four component SO(4) spinors which are also SU (2) R doublets. Killing spinor ξ is taken as Grassmann-even andQ is Grassmann-odd. With the background metric: which can be written in terms of SU (2) left-invariant one-forms, a solution of the N = 2 Killing spinor equations is given by: with the index structure ξ ≡ (ξ αA ,ξα A ) where α,α are Lorentz indices and A is R symmetry index 1 . And it is normalized to ξ A ξ A +ξ Aξ A = 1. These Killing spinors give rise to a Killing vector v = 2 ∂ ∂ψ . The corresponding superchargeQ squares on all fields to bosonic symmetry generators: where L v is the Lie derivative along v, R and Gauge Λ are, respectively, R-symmetry and field dependent gauge transformation parameter Λ: The localizing action, S loc =QV 2 , is determined by the fermionic field V , for which, a convenient expression in terms of original fields is:  to raise and and lower the α,α indices 2Q should be defined by including the BRST component and correspondingly V should include the ghost part, as it will be shown in the following.

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where † means complex conjugation. One can show that the localization locus is given by where ω AB = −8(ξ A ξ pB +ξ BξpA ) and ξ p = γ µ D µ ξ. Therefore φ 2 = 0 and φ 1 = a 0 , a 0 being a constant element of the Lie algebra of the gauge group G. Note that at the saddle point, choosing A µ = 0, Λ reduces to a constant gauge parameter, Λ = −2ia 0 , given by the v.e.v. of the scalar φ 1 .
Here we stress that although A µ is a pure gauge, due to non-trivial π 3 (SU (2)) ≃ π 3 (S 3 ) ≃ Z of the boundary there are equivalence classes of gauge connections characterized by integer k. And to compute perturbative part the quadratic fluctuations will be in general around vacuum labelled by k.
As mentioned in the introduction, it is convenient to introduce new fermionic fields which are linear combinations of the gauginos and carry integer spins, which are part of the cohomological fields, The main point about introducing these fields is that, after gauge fixing and introducing the ghost-antighost system c,c, and extendingQ to include the BRST generator, the bosonic fields X 0 = (A µ , φ 2 ) and the fermionic ones, X 1 = ( χ, c,c), are lowest components ofQ multiplets 3 . One can rewrite 2.9 in terms of cohomological variables [7], however after some linear algebra, one can show that the relevant super determinant corresponding to the quadratic part ofQV can be expressed in terms of the super determinant of theQ 2 operator on the spaces X 0 and X 1 .
Furthermore, since these spaces are related by a differential operator D 10 which commutes witĥ QV and can be read off from V , at the end the super determinant, upto an overall sign ambiguity, reduces to that is, it is enough to compute the spectrum ofQ 2 on the kernel and cokernel of D 10 , where the latter equals the kernel D † 10 . The differential operator D 10 is identified from the terms in V which are bilinear in X 0 and X 1 , after expanding around the saddle point field configuration. The terms relevant to our analysis are the following Here e = √ g.
Similarly, for the matter part the localizing action is obtained from: (2.14) and a trivial localization locus As for the vector case, it is convenient to change fermionic variables from ψ α to Σ A =ξ A ψ, wherê ξ is a spinor orthogonal to ξ. The cohomological fields are X 0 = q A and X 1 = Σ A and the same linear algebra argument as before shows that the matter one-loop contribution is given by the superdeterminant ofQ 2 on the kernel and cokernel of the operator D m 10 mapping X 0 to X 1 , which can be read off from V mat by keeping the terms bilinear in q and Σ.

Main content of computation
The main content of this work is the computation of the partition function (actually, wavefunction ) of N = 2 supersymmetric gauge theory on a hemisphere HS 4 , with supersymmetric boundary conditions. The relevant one-loop determinants for gauge and matter multiplets are computed by direct analysis of partial differential equations defining the kernel and cokernel of D 10 differential operators. In other words , we take the differential operator D 10 and its adjoint counterpart (D 10 ) † from the fermionic functional V and then solve the zero mode partial differential equations for them. We explicitly find the solutions and their multiplicities by diagonalizing the differential equations by expanding fields in SO(4) ∼ SU (2) L × SU (2) R harmonics. The kernel equations for X 0 fields are obtained by varying V loc with respect to X 1 fields and vice versa of the cokernel equations. We write them in the appendix. The differential equations can be expressed in terms of SU (2) L generators and r-derivatives, so that SU (2) R is spectator and provides the multiplicities, once we solve the ordinary differential equations in the r coordinate. We work with the round S 4 and then we adapt it to the case of hemisphere HS 4 , where we impose appropriate boundary conditions to the zero modes of D 10 and (D 10 ) † at the equator of S 4 , taken to be at r = π/2. As it turns out the solution set for the zero modes of D 10 is empty and those of (D 10 ) † is non-empty for the vector multiplet, whereas the converse is true for the matter multiplet. TheQ 2 eigenvalues and their multiplicities will then give us, by using eq.(2.12), the expression for the determinants. The analysis for kernel and cokernel equations respectively in sections 3 and 4 is done for U (1) gauge group. The generalization for a non-abelian gauge group G is straightforward. For vector multiplet we have to multiply the index by the character α∈Roots e iα.a of adjoint representation of G and for matter multiplet by the character ρ∈R e iρ.a in the representation R of G. Some comments on BRST analysis are in order. The standard covariant way to fix the gauge redundancy of the action is BRST formalism. In an abelian gauge theory like U (1) the BRST charge Q B which parametrized the gauge freedom is nilpotent. To generalize it to non-abelian gauge group the corresponding BRST charge Q B squares to a constant gauge transformation a that is ultimately identified with the zero mode of the scalar field as a solution to localization equations. This constant gauge parameter a is integrated over to get the partition function on S 4 .

Vector-multiplet contribution
Perturbative part of partition function corresponds to computing one loop determinant of the quadratic fluctuations around classical field configuration given by the saddle point solutions (2.10)(2.15). The differential operator D 10 from which we get kernel and cokernel equations, contains only the fluctuating part of the quantum fields. In the vector multiplet the fluctuating part of only φ 2 ∈ X 0 is relevant, whereas φ 1 and D AB contributes only classically. Hence in our discussion of kernel equations we will set φ 1 = 0, D AB = 0. As detailed in the appendix, it is convenient to work with tangent space basis for the gauge fields A µ : for the S 3 directions µ = 1, 2, 3, the flat basis is A a = l aµ A µ where a = 1, 2, 3 or a = +, −, 3 where Similarly for the fermionic fields in X 1 we choose a complex basis χ + ≡ χ 1 + iχ 2 , χ − ≡ χ 1 − iχ 2 . The first thing to observe is that, since D 10 commutes withQ 2 , it will close in X 0 on fields of one can show that the kernel equations can be written in the following form: x sin(r)∂ r A + (θ, r) , + tan(r)(A 3 (θ, r)(3 + cos(2r)) + sin(2r)∂ r A 3 (θ, r))) , + sin(r)∂ r A 4 (θ, r))) , Where, in the last equation we have traded φ 2 with Λ , for fluctuating part of φ 1 = 0, in order to simplify the equation. Note that E 5 = 0 can be written as Laplacian acting of Λ This equation has no smooth solution on S 4 apart from the constant and since we will be considering here in j L > 0, we set it to 0. Whereas on hemisphere HS 4 with boundary there is constant function as solution for the special value j L = 0. But if we impose supersymmetric boundary conditions this constant solution must be set to zero. This is discussed briefly in appendix. Therefore E 5 drops from our further discussions. The coefficient tan(r) or sec(r) in kernel equations (3.4) may appear problematic because it blows up at r = π 2 . However it is only an artifact of the redefinition of φ 1 (θ, r) in favor of Λ(θ, r) to simplify the form of differential equations. On the other hand if redefine A 3 (θ, r) instead of φ 1 (θ, r) we will get kernel equations which will be well defined at r = π 2 . The only change in the kernel equations will be that A 3 (θ, r) is replaced by φ 1 (θ, r) but the rest of the analysis will remain the same resulting in the same eigenvalue spectrum and degeneracies. A similar argument holds for the cokernel equations. The following modes carry the same value ofQ 2 , taking into account the shifts in (3.1): (r).
where the Y functions are the θ dependent part of the scalar spherical harmonics with the indicated quantum numbers. Now we analyze the kernel equations separately for all the possible values of q L for which we may get a non-trivial solution.
The kernel equations evaluate to Solving the differential equation E 4 = 0 we get the solution This is clearly a singular solution at the two poles of round S 4 . So we have to set A 0 + = 0. If there is a physical boundary at r = π 2 the result does not change. For q L = −j L − 1: Here E 3 can be solved to give which is again a singular solution and A 0 + = 0. ii)q L = +j L , −j L First for q L = j L solving E 1 = 0 gives the solution ))). (3.12) for The last equation is singular at the North or South Poles r = 0, π which implies that A 0 (r). (3.14) Next E 3 = 0 and E 4 can be solved to give a 0 4 comes with cos(r) while b 0 4 is multiplied by a polynomial in cos(r) 2 , this means that the absence of singularity at r = 0 or r = π implies that a 0 4 = 0, b 0 4 = 0. Similarly for q L = −j L we get a singular solutions. So far there is no Kernel, now we go to |q L | < j L case. iii) |q L | < j L For the isolated case of q L = 0 , solving the kernel equations we get the result that the solution set is empty. The details are given in appendix H. Hence the conclusion is that Kernel of D vec 10 is empty.

Cokernel equations
The Fourier series expansion of fields contributing to the cokernel is given in eq. (B.19), except the follwoing field redefinitionc Next the Cokernel equations can be written in terms of these generators as + sin(r)∂ r χ − (θ, r))), + sin(r)∂ r χ + (θ, r))), We can express the fields in terms of scalar harmonics(i.e.j R = j L ) whose ψ and φ coordinates dependence has already been extracted above. Similar to the case of kernel equations, using the inventory of various identities given in appendix G. Let's begin the analysis. α) q L = j L + 1 In this case CE 1 = 0, CE 2 = 0, CE 4 = 0, CE 5 = 0 give empty solution set and CE 3 = 0 can be solved to give the follwoing solution for χ , the multiplicity of this solution is 2j L + 1 = n − 1. Siimilarly for q L = −j L − 1 only χ + survives and is given as (r) as n = −2j L − 2, the multiplicity of this solution is 2j L + 1 = |n| − 1. β) q L = j L For this value of q L , CE 1 = 0 can be solved to give Using this value of χ By considering the regularity of the solution at r = 0 and r = π , solvingCE 3 = 0 yields for c (r) is a new function to be determined. Plugging this solution into CE 3 , it is converted to a differential equation for c We multiply this equation by c (r) and integrate over r from 0 to π, if there is smooth solution the result must be zero. On the other hand by partial integrating the term containing Note that 2j L ((2j L − 3) cos 2 (r) + 5) is always positive for all j L and all r, and sin(r) is positive, so for this integral to be zero, c (r) must be zero.
CE 4 is just the complex conjugate of CE 3 . so the same analysis goes through. The CE 5 is the conjugate equation to the kernel equation for φ 2 or φ p and is given for q L = j L by If the homogeneous piece is zero then this is just the Laplacian, which has not smooth solution on S 4 . So it is sufficient to construct one smooth solution of the inhomogeneous equation and that will be the unique solution. It is given by Summarizing the solution set for q L = j L is For q L = −j L We get only one new solution from this In this case the Q 2 eigenvalue= n = 2j L and the multiplicity is |n| + 1.
Combining the results for q L = ±(j L + 1) and q L = ±(j L ) the total multiplicity forQ 2 is (|n| − 1) + (|n| + 1) = 2|n|. γ) |q L | < j L For q L = 0 there is no solution of the cokernel equations. Next we consider the case when q L = 0 by following the same argument as given in H one can show that solution set is empty for this range of q L . For round S 4 solution set of only cokernel differential equations is non-empty. Eq.(3.30) shows that for qL = ±j L the solution set for χ 3 , c,c depends on a single constant parameter and we will count it only once. For our choice of normalization the eigenvalue of Q 2 = n = ±2j L , with multiplicity 2j L + 1 = |n| + 1. Therefore it will contribute a factor (n + iα.a) n+1 . Similarly from eqs.(3.20),(3.21), for q L = ±(j L + 1) we have eigenvalue Q 2 = n = ±2(j L + 1), with multiplicity 2j L − 1 = |n| − 1 and the corresponding contribution (n + iα.a) n−1 . Using this data the one loop determinant for round S 4 can be immediately written down where ∆ is the set of roots of G, which matches with Pestun's result [14].

Hypermultiplet contribution
For matter multiplet the fields in the kernel and cokernel of D hyper 10 in cohomological form are Notice here too the shift due to the R-charge. The relevant kernel and cokernel equations are obtained by varying the localizing fermionic field of eq. ( 2.15) with respect to Σ's and q's respectively 4 .

Analysis of kernel and cokernel equations
Since the set of fields with flavor index I = 2 form a copy of that of I = 1 we will discuss the kernel and co-kernel equations for I = 1 only. After Fourier transforming in coordinates φ and ψ as follows: the kernel equations become: and the cokernel are: As in the vector multiplet case, the isometry group SO(4) ≃ SU (2) L × S(U (2) R of foliated S 3 s plays an important role in solving these equations. It turns out that kernel and cokernel equations can be written in terms of generators of SU (2) L , whereas the SU (2) R remains a spectator. For this reason the degeneracy of the solutions to these equations is determined by q R quantum number.
To convert these partial differential equations into ordinary ones in the variable r we further expand the fields in terms of spherical harmonics: and get the solutions which we summarize below. For kernel equations, solutions, which are regular at the North or South poles of S 4 , exists only with eigenvalue for theQ 2 action equal to (2j L + 1).
with eigenvalue for theQ 2 action equal to −(2j L + 1) for constants C 1 , C 2 , C 3 , C 4 . Analysis of kernel and cokernel equations with physical boundary at r = π 2 , with regularity at one of the poles, and following the logic of appendix H agains shows that only the solution set of kernel is non-empty. For instance for q L = (j L + 1 2 ): which are not regular at r = 0 or π. For q L = (j L − 1 2 ): we get identical set of ordinary differential equations for two fields with the solution (4.14) with the solution where P and Q are Legendre functions. These solutions are not regular at r = 0 or π for the case of round S 4 or at one pole and the equator r = π 2 in case of half-S 4 . Therefore the solution set is empty for cokernel equations.

Wave function on hemisphere HS 4
As is clear from the above analysis that for the round S 4 solution set of kernel equations is empty and that for the cokernel equations is nonempty. In order for the boundary to preserve supersymmetry, the component of the supercurrent normal to the boundary must vanish. Also the boundary conditions must be consistent with the localization locus given in eqs.(2.10), (2.15). If we consider hemisphere HS 4 with supersymmetric BCs at r = π 2 , the analysis of kernel and cokernel equations remains identical except that one has to take account of possible boundary contributions. Also spectrum remains same with the result that kernel of D vec 10 is empty and cokernel solutions set is non-trivial with the same eigenvalues and multiplicities if one imposes supersymmetric boundary conditions. Thus practically the only change is to take the range of coordinate r to be 0 ≤ r ≤ π 2 .

Supersymmetric boundary conditions Vector multiplet
First we recall that for manifolds with boundary e.g. HS 4 the supersymmetric variation of physical actionQS vanishes upto total derivative terms. These total derivative terms break supersymmetry at the boundary unless on adds extra terms to Lagrangian such that theQ variation of the modified action vanishes. The other way to get rid of the boundary terms is to impose supersymmetric boundary conditions on all the fields. For the vector multiplet the boundary contribution iŝ where Λ is the gauge parameter which takes the scalar zero mode as its value at the localization locus. For supersymmetry consistent BCs the boundary contribution vanishes. Dirichlet type boundary conditions correspond to choosing while keeping χ + , χ − arbitrary. Whereas for Neumann type boundary conditions: keeping χ 3 , c,c arbitrary at the boundary. This can be understood in the following way: at the boundary r = π 2 the Killing spinor satisfies which motivates choosing the following BC's on the gaugino The above conditions on χ follow from its definition in terms of λ. Moreover for the consistency of supersymmetry it follows that for the lower sign. On the other hand, for the upper sign choice, we get: So, for lower sign choice we get Dirichlet and for the upper sign we get Neumann.
If we act once more withQ they are closed and trivially satisfied. For example acting on eq.(5.6), The second line holds upto a constant gauge transformation a. we see that it is trivially satisfied for Dirichlet BCs. Similar is the case for Neumann BCs. Therefore these boundary conditions are closed under the action of supersymmetry and hence consistent with it. However there is one subtle point about the SUSY closure of BCs. Since we are working in Euclidean signature and the fields entering the Lagrangian are analytically continued for Lorentzian signature, the BCs imposed on fields are closed under SUSY only if we take the fields as complex valued functions. If one tries to impose BCs on real and imaginary parts of various fields separately, it turns out that they are not closed under SUSY and will generate infinite number of differential constraints on gauge field and gaugino at the boundary. In other words the BC in eq. (5.6) is written in covariant form and due to this reason it is closed under SUSY trivially as shown in eq.(5.10). On the other hand if we do not work covariantly and instead consider the action of Q on ∂ r φ 2 (θ, π 2 ) = 0 it is easy to see that Note the important inversion of sign from ± to ∓. Now this BC should itself be closed under SUSY. But it is easy to convince oneself that due to the inversion of sign ∓ at each step if we act once more with Q it will generate BC other than already imposed and infact one has to impose infinite number of boundary conditions. However it turns out that our solution set of kernel and co-kernel equations satisfy BCs irrespective of whether we impose them covariantly or and separately in terms of real and imaginary parts of individual fields. Second constraint that the BC conditions have to satisfy is the action principle. Taking arbitrary variations δ of the fields in the the action, for round metric, to get the equations of motion, we obtain boundary contributions coming from integration by parts. Keeping in mind that the boundary conditions have to be consistent with saddle point solutions and that the later break the gauge symmetry G to its maximal torus, some non abelian terms drop out and we get the following With the set of BCs (5.4),(5.5),(5.6), the boundary term from the action principle vanishes. δ is an arbitrary variation in the sense that it may represent supersymmetry variation Q too. SUSY variation of theQV produces total derivative in the ψ, θ, or φ direction and not in r direction.
Hence there is no non-trivial contribution from here.

Hyper multiplet
In the case of hyper multiplet for the boundary conditions to preserve N = 2 SUSY in 3 − d at r = π 2 , one has to impose complementary BC's on scalars with different R-charges .i.e. Dirichlet BC's on the scalars q 11 , q 21 and Neumann BC's on q 21 , q 22 or vice versa. Only this choice of BC's satisfy the constraints coming from the vanishing of supercurrent normal to the boundary [6,5]. In either case we get ((2n+1)+(2n−1)) 2 = 2n multiplicity of theQ 2 eigenstates with eigenvalue n+ia.ρ, with ρ the wight vector of the complex conjugate representation R of the Gauge group G. The one loop factor for hypermultiplet on hemisphere with Dirichlet or Neumann boundary conditions at the equator can immediately be written down (5.14) When one reads off the cokernel equations fromQV matter , an integration by parts is done, which in the case of a manifold with boundary produces following boundary termŝ There are two choices: If we act with supersymmetry Q on this BC, it will be closed if we choose Dirichlet BCs on the following fields q 12 (θ, π 2 ) = 0, q 21 (θ, π 2 ) = 0, Σ 11 (θ, π 2 ) = 0 Σ 22 (θ, π 2 ) = 0, and Neumann BCs on the following Acting once more with a supersymmetry operator Q we get However note that using the BCs in case (1). It is clear that this will go on to produce infinite number of boundary conditions, not closed within themselves. The resolution is that in working on HS 4 with Euclidean signature, the real and imaginary parts of all fields on the Lorentzian signature get mixed when they are analytically continued to Euclidean signature. So when we check the SUSY closure of BCs on the individual fields thinking of them the same way as on Lorentzian space-time, the SUSY fails to close. On the other hand the full covariant expression for BCs ψ αI | r= π 2 = +iψ αJ τ 3J I | r= π 2 is closed under SUSY by construction. So the conclusion is that the BCs can closed under supersymmetry when written in covariant form in the Euclidean signature.
If we act with supersymmetry Q on this BC, it will be closed if we choose Dirichlet BCs on the following fields q 11 (θ, π 2 ) = 0, q 22 (θ, π 2 ) = 0, Σ 12 (θ, π 2 ) = 0 Σ 21 (θ, π 2 ) = 0, and Neumann BCs on the following Like that previous case (1) this choice of BCs is closed under supersymmetry except for the properly writing the BCs in a covariant way with respect to Euclidean signature. However fortunately the solution set of kernel and co-kernel equations that we have found satisfy BCs irrespective of whether we impose them covariantly or and separately in terms of real and imaginary parts. Applying variational principle to S hyper , to get equations of motion ,we get boundary terms Choosing one of the above BCs, the action principle will be satisfied as well as these BCs are consistent with supersymmetry. Knowing that the eigenvalue ofQ 2 for χ + (ψ, θ, φ, r) is n + ia.α, with corresponding multiplicity |n| − 1 with n ∈ Z, a the zero mode of the imaginary part of the scalar of the vector multiplet and α the roots of the gauge group G. The expression for the one loop determinant can be written ∆ representing the root system. The regularized form of this ill defined [14] product is Using the identity a.α sinh(πa.α) (5.29) getting following hemisphere wave function and Z k inst is the contribution of k − th sector of the Nekrasov instanton partition function. Recall that the instanton configurations contributing to the path integral are point-like instantons localised at the pole of the hemisphere and in particular are pure (large) gauge at the boundary S 3 . Given the large gauge transformation T , which maps the boundary S 3 to the SU (N ) Lie group, the corresponding winding number is given as: Neumann BCs by definition imply that the components of fields tangential to the boundary r = π 2 are kept arbitrary and consequently in performing the path integral one has to integrate over all field configurations. However to be able to apply localization, the field configurations satisfying some BCs must be solution of the saddle point equations. As is evident from solution of saddle equations (2.10),(2.15), the infinite dimensional field space is reduced to a single scalar zero mode a. For Dirichlet BCs this zero mode is fixed but for Neumann BCs a takes arbitrary values at the boundary and so one has to integrate over it to get the wave function. In general the Neumann wave function depends on the variables canonically conjugate to those fixed by the Dirichlet BCs. If at the boundary we see 4d vectormultiplet as composed of one 3d vectormultiplet plus a 3d chiral multiplet, then the Neumann BCs data corresponds to the fixed value of 3d chiral multiplet at the boundary. The dynamical fields of 3d chiral multiplet are given in terms of 4d fields as with i = ψ, θ and φ. Therefore ∆ and g representing the root system and Lie algebra respectively, of SU (N ). The regularized form of this infinite product is [14] Using the identity For Neumann BC's, instanton configurations contributing to the path integral are again localised at the pole and are pure (large gauge) at the equator S 3 , at r = π 2 , with winding number k as before.

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Therefore the full partition function is: and where Z inst is the full holomorphic part of Nekrasov partition function. Note that here we have written the Neumann wave function after summing over all instanton sectors and thus getting Z int (a, τ ). However in principle Neumann wave function is computed for each instanton sector labelled by an integer k and hence depends on the discrete parameter k. After summing over all values of k one gets the last expression.

Large Radius limit R → ∞
In our computation we have set the radius R S 4 ≡ R = 1 ǫ = 1. For illustration let's take G = SU (2) in this subsection. Then restoring it one gets the following expressions for the bulk one-loop part of HS 4 with Dirichlet and Neumann BCs.
Now using the following identities it is easy to see that to leading order in R → ∞ limit we get the following simplified expressions where the positive sign in the exponential is accounted by taking a to be anti-hermitian. So we reach an interesting conclusion that at leading order in R → ∞ limit Dirichlet and Neumann BCs lead to same perturbative result. This exponential contribution of one loop in the large radius limit can be interpreted as producing an RG flow that renormalizes the classical gauge coupling constant g Y M .
6. Gluing back two hemisphere HS 4 wave functions

Wave functions with Dirichlet BCs
If we see N = 2, d = 4 vector multiplet as a combination of an N = 2, d = 3 vector and a chiral multiplet, then imposing Dirichlet BC's at the equator r = π 2 of S 4 amounts to freezing the 3d vector multiplet to fixed value and consequently decoupling the gauge theory dynamics from two sides of the equator. Gluing the two wave functions then naturally implies that one has to put back N = 2, d = 3 gauge multiplet at the equator. See also the discussion in [3]. It has being argued that the two wave functions for Dirichlet BC's are glued along the equator r = π 2 by gauging the global symmetry under which the boundary values of the dynamical fields transform [4,2]. In other words one has to put an N = 2, 3d vector multiplet at the equator and include the corresponding partition function. In fact, our results confirm this general argument. As for the matter multiplet, the wave functions from two hemispheres are joined together by turning on a super potential coupling at the equator [5]. However being a Q-exact term, the superpotential does not contribute to the localization computation.
where q = e 2πiτ , The last identity can also be interpreted as factorization of round sphere S 4 partition function, though more precisely it is a convolution of two Dirichlet wave functions of two hemispheres with non-trivial integral kernel, the latter being due to a 3D vector multiplet at the equator. Perhaps the instanton contribution requires a comment: one would have naively thought that in glueing two Dirichlet wave functions one should have matched the k-th instanton sector on one side with the the −k-th anti-instanton sector on the other side. This would have produced a function of |q|, i.e. no Θ dependence. This is not the S 4 answer however, which is not diagonal in the instanton number: to get the S 4 result one has actually to sum over all (anti-)instanton sectors on each hemisphere before glueing. Put it differently, the identification of fields at the boundary is up to large gauge transformations.

Wave functions with Neumann BCs
We will only be sketchy here to describe the gluing of two Neumann wave functions. Roughly speaking Neumann BCs are canonically conjugate to Dirichlet BCs. In the semiclassical approximation Neumann wave function is related to Dirichlet wave function through Legendre transformation. From section 5.3 we know that the 4d vector multiplet when restricted to 3d boundary, can be decomposed into a 3d vector plus a 3d chiral multiplet. In the same vein Dirichlet wave function depends on the value of scalar of the 3d gauge multiplet at the boundary, whereas Neumann depends on the value of 3d chiral multiplet at the boundary. These boundary conditions are constrained by localizing equations and for this reason 3d chiral vev at the boundary is taken to be zero in our case. Compared to the Dirichlet case, where we needed to include a 3d vector multiplet partition function in the glueing procedure, in the Neumann case we are facing an over counting problem, i.e. we count twice the contribution of the boundary 3D vector multiplet, since in this case the corresponding boundary degrees of freedom from each side are not frozen, as it is clear from eq.(5.36). Therefore in the glueing procedure we insert a factor: α α(a) sinh(π(α(a))) (6.2) to remove the redundant degrees of freedom. This glueing procedure gives rise to the round S 4 partition function. However it would be very nice to have a better understanding of how this measure arises from the full path integral. See e.g. [12].

Wave functions with Dirichlet-Neumann BCs
If we impose Dirichlet BCs on the vector multiplet fields on one hemisphere with the resulting one-loop part and on the complementary hemisphere we impose Neumann BCs with the following one-loop part it is obvious that if we naively glue these two hemispheres we get This shows that no extra measure is needed to glue Z vecDir.

HS 4
with Z vecN em.

HS 4
to get Z vec S 4 . Intuitively the over counting of modes from the hemisphere with Neumann BCs is compensated by the removal of boundary modes by imposing Dirichlet BCs on the other hemisphere. However a more satisfactory explanation in terms of path integral will be illuminating.

One loop determinant and SO(5) harmonics
One loop determinants can also be computed more directly using the full SO(5) harmonics. Like the case of SO(4) harmonics as given in the first part of work, we are only interested in the spectrum ofQ 2 on the kernel(D 10 ) and cokernel(D 10 ). The purpose of this and the next section is to compute the net multiplicity ofQ 2 on kernel and cokernel of D vec 10 for round S 4 , hemisphere HS 4 with Dirichlet BCs , Neumann BCs at the equator. We show that it matches with the results obtained using SO(4). The task can be simplified by observing how vector and scalar harmonics of SO(4) irreps. are embedded in SO(5) irreps. Here it is helpful to recall some useful results from Lie group Representation theory [1]. Irreducible representations of SO(2k + 1) determined by their highest weights (n 1 , n 2 , ..., n k ) with integer or half-integer entries, when restricted to the subgroup SO(2k), contains all irreps. of the later with highest weights (p 1 , p 2 , ..., p k ) with integer or half-integer entries, satisfying the following constraints If n i are integers ( half integers) so are p i . Quadratic Casimir is an important operator for a Lie algebra whose eigenvalues for different irreps. are used to regularize infinite sums using heat kernel technique. For irreps. of orthogonal group it is given by C 2 (n 1 , n 2 , ..., n k+1 ) = n.n + 2w.m (7.2) with Euclidean dot product assumed and where the Weyl vector w given by Assume that the weights are given in the basis of Cartan generators (j 3 L , j 3 R ) for SU (2) L ×SU (2) R ∼ SO(4) ⊂ SO(5), with SO(4) being the isometry group of S 3 at constant value of coordinate r. Then the irreps. of scalar and vector SO(5) harmonics can be constructed by starting with the simple roots in the above basis. Here we describe only the final results of the construction. First of all it is easy to check that SO(5) Lie algebra is generated by the following generators (− cot(r)), 2i sin θ 2 cot(r), sec θ 2 (− cot(r)), −i cos θ 2 ).(7.5) Keeping in mind the fact that the above generators act on the fields as a differential and hence the fourth entry corresponds to derivative w.r.t. r, we conclude that first six generators are even under Z 2 action r → π − r, whereas the last four generators are odd.

Harmonics
The logic for contstructing SO(5) harmonics is simple. One repeatedly applies negative roots (− 1 2 , − 1 2 ), ( 1 2 , − 1 2 ) to the highest weight state of an SO(5) irrep. to get a state which is a linear combination SO(4) highest weight and SO(4) descendants. One then removes the descendants part to get the irreps. of SO(4) given by its highest weight. This construction has following properties • The highest weights of SO(5) both for scalars and vectors are Z 2 even.
• Since we already know from the branching rule given in 7.1 which SO(4) irreps. appear in a given SO(5) irrep. one can easily see that by counting how many times one needs to apply these two negative roots to reach an allowed SO(4) highest weight state starting from a given SO(5) highest weight state • If the count is even (odd) the corresponding SO(4) irrep. is even(odd).

Scalar harmonics
So for SO(5) case we will only describe the final results. For scalars the SO(5) highest weight state appears only for j R = j L and is given by (j L , j L ) cos 2j L θ 2 sin 2j L (r)e ij L (ψ+φ) (7.6) and is clearly even under Z 2 action. So if we apply j − 6 or j − 5 on it the result will be odd. As a result decomposing this SO(5) representation in terms of irreps. of SO(4) we get the following For n an even integer this irrep. is even under Z 2 and for n odd the irrep. is Z 2 odd.
8. Z vec 1−loop via SO(5) harmonics For SO(4) irreps. (j L − n 2 , j L − n 2 ) f or n = 0, 1, .., 2j L contained in (j l , j L ) irrep. of SO(5), various dynamical scalar fields will contribute the following to the net multiplicity of the one-loop determinant. Scalar contribution for Z 2 even irreps. is denoted as S e and for odd irreps. as S o .
Similarly for vector harmonics of SO(5) one gets the following individual contribution to the net multiplicity of one loop determinant. For even irreps. of SO(4) ], (8.2) and for odd irreps. of SO(4) ].

Regularizing the Infinite sums
As an example we will describe in detail the regularization of S e . For the other contributions we will only give the final results. Consider the following expression in the heat kernel regularization where 2j 2 + 3j is the regularization factor for j L = j R ≡ j representation of SO (5). Taking the Mellin transform of S e w.r.t. tS Now applying the Binomial expansion and using Γ function analytic continuation Substituting this in the expression forS Since the above summation is absolutely convergent, one can exchange the order of summation and at the same time shift j to j + m 2 and perform the j summation in terms of Hurwitz Zeta function to get ) k (8.10) In the last step we take the inverse Mellin transform S e (t, m) = 1 2πi C t −sS e (s, m)ds (8.11) and perform complex integration along a contour C which encloses all the poles of integrand. Interestingly when we evaluate the last integral for various poles of s, we fine that the series terminates for finite values of k. We thus obtain Following the same procedure we find and similarly for vector harmonics and In the application of localization to supersymmetric theory, fermions are written in cohomological form. Different components of fermion transform as scalars and vector of SO(4). Therefore the above results will suffice in determining their contribution.
Net multiplicity N It is easy to see that for round S 4 the net multiplicity can be found as In the next step keeping in mind the Dirichlet and Neumann BCs on the hemisphere given in section 5.1 for Neumann BCs. Next since we know the eigenvalues ofQ 2 , it is trivial to write down the expressions for one-loop determinant. However it is important to note that there is some arbitrariness in the regularization scheme used here. For instance if one multiplies a factor of e tC for constant C ∈ R , the net multiplicity N Dir HS 4 and N N eum.

HS 4
is modified to m + p and m − p respectively, for some positive integer p, in such a way that N N eum.

Conclusions
Despite extensive activity in Supersymmetric Localization computations on curved manifolds in various dimensions, there were no first principle computations available on hemisphere in four dimensions, although some educated guesses were given in [2]. We have done detailed computation of wave functions on hemisphere HS 4 with supersymmetric BCs of Dirichlet type, and also discussed briefly the Neumann BCs. In the first part of this work, various one-loop determinants are computed using SO(4) harmonics as the complete set of basis functions. The results obtained in the first part are re-checked in the second part where we do the same analysis in the framework of full SO(5) harmonics. We have also briefly discussed how the N = 2 SUSY partition function on round S 4à la Pestun [14], can be seen as composed of two Dirichlet type wave functions on southern and northern hemispheres properly glued together. The last observation can also be interpreted as kind of factorization of Z S 4 wave function in terms of two hemisphere wave functions. Though this factorization should be seen as a convolution of two wave functions with non-trivial kernel, the later being the one-loop determinant of N = 2, 3d gauge multiplet.

Acknowledgements
We would like to thank Maszumi Honda for pointing out an important typo and Alejandro Cabo Bizet for insightful comments. NM would like to thank Bruno Le Floch for pointing out important issues in Instanton contributions discussed in section 6 and for suggesting many improvements in the draft.

A. Notation
We use the same notation as in [7]. In the flat basis of the tangent space on S 4 we use the following set of Dirac matrices γ 1 , γ 2 , γ 3 , γ 4 Pauli matrices are defined as usual In Weyl basis, with the decomposition SO(4) ≈ SU (2) R × SU (2) L in chiral and anti-chiral basis, the sigma matrices (σ a ) αβ , (σ a )α β are related to Pauli matrices as follows The R-symmetry indices A, B.. and chiral and anti-chiral indices α,α are raised and lowered with antisymmetric matrices ǫ αβ , ǫ αβ , ǫαβ, ǫαβ, ǫ AB , ǫ AB with the following matrix elements

Background geometry
The isometry group of round S 4 is SO(5) and the most general way to compute the one-loop determinant is to use SO(5) harmonics. However since we are interested in applying localization on a hemisphere, the boundary at r = π 2 breaks translational symmetry in r coordinate, only SO(4) ⊂ SO(5) is left intact and the best we can do is to use SO(4) spherical harmonics. We take the following metric on S 4 ds 2 = g µν dx µ dx ν = dr 2 + sin(r) 2 4 dθ 2 + sin θ 2 dφ 2 + (dψ + cos θdφ) 2 (B.1) with coordinates µ = (ψ, θ, φ, r), such that the coordinates for Hopf fibration of unit S 3 part are: for 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π and 0 ≤ ψ ≤ 4π. As for the radial coordinate r, 0 ≤ r ≤ π. The metric above describes S 4 as an S 3 fibration over the r interval(0, π). S 3 has radius sin(r) which vanishes at 0 and π. The vielbeins for SU (2) L -frame and SU (2) R -frame are and The SO(4) = SU (2) R × SU (2) L Killing vectors of SU (2) L , J a L and SU (2) R , J a R , a = 1, 2, 3 are given by J a L = l aµ ∂ µ and J a R = r aµ ∂ µ , where: They obey the algebra: Notice that and showing that e i(q L ψ+q R φ) is an eigenfunction of J 3 R and J 3 L .

Scalar harmonics
Highest weight states with respect to SU (2) L and SU (2) R for the scalar functions e i(j L ψ+j R φ) f (θ) are constructed by forming raising (lowering) operators J ± L = J 1 L ± iJ 2 L and J ± R = J 1 R ± iJ 2 R . Highest weight states are annihilated by J + L,R : one can easily prove that this implies that j L = j R and f (θ) = cos θ 2 2j L , up to a constant: Applying lowering operators on Φ we can get all the other harmonics. In particular by applying J − R s times we get states and one can check that when s = 2j L this is annihilated by J − R , i.e. it is a lowest weight state.

Vector harmonics
Let us move on to the vector harmonics: now we have to consider Lie derivatives along the Killing vectors of SO(4) acting on one-forms. The Lie derivative with respect to a vector field K on a 1-form ω is defined as One can verify that Lie derivatives satisfy the Lie algebra relations: [L a , L b ] = L [a,b] . A basis of eigenstates of of the Cartan generators of both SU (2) L,R is given by: Now for q L = j L , q R = j R , this will be a highest weight state if it is annihilated by the raising operators of SU (2) R × SU (2) L Lie algebra Solving these differential equations gives the following general solution: along with the following three choices of constant parameters a)j R = j L , α 1 = 0, In all these cases the solutions are regular at the North and South Poles, r = 0 and r = π respectively.

(B.19)
To write the kernel equations in a more suggestive form, we redefine the φ 1 (θ, r) field as where Λ(θ, r)) is field dependent gauge transformation.

Regularity at North and South poles
At the two poles of S 4 the space locally looks like R 4 and to check the regularity of A µ one has to expand its components as polynomials in z 1 , z 2 ,z 1 ,z 2 and find the leading behavior in the limit of z 1 → 0, z 2 → 0,z 1 → 0,z 2 → 0 or in terms of polar variable r → 0. It is easy to find that the highest weight state with q L = j L and q R = j R and because of expansion in scalar harmonics whereas for A i with i = ψ, θ, ψ one gets the following leading order behavior However in tangent space basis all the components of gauge field A a = e µ a A µ with a = 1, 2, 3, 4 have identical leading behavior The consequences of these regularity properties are analyzed in detail in section 3 and in appendix H. Since for computational purposes fermions are written in terms of cohomological variables, their regularity behavior can be easily deduced from that of the gauge field A µ given above.

C. N = 2 off-shell SUSY transformations
For completeness we reproduce here the supersymmetric transformation rules of vector and matter multiplet for general background auxiliary fields.

G.1. For kernel
For simplicity we take the basis for the harmonics as This basis is not normalized but it is irrelevant for the present analysis. It is trivial to see the following identities hold To evalute the Kernel equations for different SU (2) R charges we need the following identities.
and for E 4
(H. 13) and observe that for q L = 3 2 , X N < 0 and X D > 0. Now for all q L > 1 i.e. for q L = p + 1 for p increasing in increments of 1 2 , let' s perform a Taylor series expansion of X N around n = 0 (H.14) For q L > 1 this is negative. Furthermore the denominator X D = 4(1 − q 2 L )(5 − 4q 2 L ) is positive for q L > 1 as then q L ≥ 3 2 . Next we are going to show that |X N | = −X N > X D , or in other words |X N | − X D is positive. To show this again perform Taylor series expansion of |X N | − X D around n = 0 f or q L = p + 1 |X N | − X D = 16n 2 2p 2 + 4p + 1 + 16n 4p 3 + 18p 2 + 22p + 5 + 96p 3 + 396p 2 + 456p + 93 + O(n) 5 .

(H.17)
Each term is positive, so stationary point at r = π 2 is minimum for q L ≥ 1. In the final step we evaluate the coefficient of F (r) 2 at the minimum and show that it is non-negative in all the cases. First consider q L = 1 2 and evaluating the coefficient at the minimum r = 0 4(417 + 928n + 744n 2 + 256n 3 + 32n 4 ) > 0 (H.18) Next consider q L ≥ 1, in this case the minimum is at r = π 2 and the series expansion of the coefficient around n = 0 and q L = p + 1 evaluates to 12(305 + 328p + 88p 2 ) + 32(215 + 206p + 48p 2 )n + 32(143 + 104p + 16p 2 )n 2 +256(5 + 2p)n 3 + 128n 4 + O(n) 7 (H.19) showing that each coefficient in this series expansion is positive, so the coefficient of F (r) 2 is already positive at the minimum value as a function of r. This proves that S bulk is a sum of non-negative terms, therefore each term must vanish, which implies that there is no non-singular solution for F (r). Hence the conclusion is that Kernel of D 10 is empty.