BPS Equations in Omega-deformed N=4 Super Yang-Mills Theory

We study supersymmetry of N=4 super Yang-Mills theory in four dimensions deformed in the Omega-background. We take the Nekrasov-Shatashvili limit of the background so that two-dimensional super Poincare symmetry is recovered. We compute the deformed central charge of the superalgebra and study the 1/2 and 1/4 BPS states. We obtain the Omega-deformed 1/2 and 1/4 BPS dyon equations from the deformed supersymmetry transformation and the Bogomol'nyi completion of the energy.


Introduction
The Ω-deformation of supersymmetric gauge theories has been studied extensively since it gives a very useful regularization to calculate the non-perturbative instanton corrections to the partition functions [1,2]. This deformation has been developed for theories with N = 2 supersymmetry in four dimensions [3,4,5]. It has been generalized to the N = 4 super Yang-Mills theory in four dimensions [6,7,8] and also theories in various dimensions [9,10,11,12].
The Ω-background is realized as a spacetime fibration over a torus with the actions of an abelian group. This is also obtained by the dimensional reduction from higher dimensional curved spacetime compactified on torus. For this reduction it is necessary to introduce the R-symmetry Wilson line gauge fields along the torus in order to recover a part of supersymmetry. The R-symmetry Wilson lines are identified with the contorsion along the torus [7]. This deformation however breaks Poincaré symmetry in general and the deformed supersymmetry algebra takes the form of nilpotent type, which means that the BPS spectrum is not well-defined.
The Nekrasov-Shatashvili (NS) limit of the Ω-background is the limit such that a part of the torus action is removed [13]. In this limit we recover the translational symmetry in the subspace and the supersymmetry algebra is enhanced to the super Poincaré algebra.
Then one can study the BPS spectrum based on the deformed supersymmetry. Moreover in the NS limit the effective two-dimensional theory has the non-trivial vacuum structure related to the integrable models [13,14,15]. It will be an interesting problem how such integrable structure appears from the UV point of view. The BPS solitons would play an important role to study this integrable structure.
For N = 2 supersymmetric Yang-Mills theory, the deformed BPS equations were studied in [16]. In particular the deformed monopole equations can be solved by the Bäcklund transformations as the undeformed one [17]. Moreover in [15] (see also [18]), they proposed the non-trivial vortex solution induced by the Ω-background. The purpose of this paper is to study deformed BPS states for Ω-deformed N = 4 super Yang-Mills theory. This is also interesting because these BPS states play an important role for understanding S-duality of the Ω-deformed gauge theories.
In this paper we focus on the 1/2 and 1/4 BPS dyon equations in N = 4 supersymmetric Yang-Mills theory in the NS limit of the Ω-background. In a previous paper [19] three of the present authors studied the deformed supersymmetry algebra in the Ω-background as well as its NS limit. There are three types of Ω-deformations which are associated with three types of topological twist of N = 4 supersymmetry [20]. These are the half (or Donaldson-Witten [21]), the Vafa-Witten [22] and the Marcus (or generalized Langlands or Kapustin-Witten) [23,24] twists. In the NS limit, the Ω-deformations for the Vafa-Witten and the Marcus twists are shown to be equivalent by some identification. It turns out that only the two types of deformations are independent. One of the new and interesting properties in N = 4 theory is the 1/4 BPS states [25,26,27,28,29,30,31].
In this paper we will find the deformed 1/4 BPS equations for the Vafa-Witten twist.
The BPS equations are similar to the vortex type equations and might have vortex type central charge [15]. We will calculate the central charge carefully from the deformed supersymmetry algebra and show that there is no vortex type central charge in N = 4 theory.
This paper is organized as follows: in section 2, we review the Ω-deformation of N = 4 super Yang-Mills theory. In section 3, we study deformed supersymmetry algebra in the NS limit and calculate the central charges. In section 4, we argue the BPS equations from the superalgebra and also the energy bound. Section 5 is devoted to conclusions and discussion. In appendix A, we summarize the notation of the Dirac matrices in four and six dimensions. In appendix B, we investigate the vacuum conditions based on the Ω-deformed Lagrangian.

Setup
In this section, we introduce four-dimensional N = 4 super Yang-Mills theory with gauge group G in Ω-background. The theory is obtained by the dimensional reduction of the ten-dimensional N = 1 super Yang-Mills theory in the Ω-background metric with torsion [7]. We start from the N = 1 super Yang-Mills theory in general curved background with Euclidean signature 1 . We denote x M (M = 1, . . . , 10) as spacetime coordinates.
Here κ is the normalization factor for the generators of the gauge group and  (10) local Lorentz generator and ω M,N P is the spin connection including the torsion.
We will consider the invariance of the action under the following supersymmetry transformation: where the supersymmetry parameter ζ satisfies the parallel spinor condition If the action is invariant under the transformation (2.2), we can derive the supersymmetry algebra from the supersymmetry transformation of the supercurrent [32] (see also [33]).
From the Noether procedure we have the supercurrent with the parameter ζ as Its supersymmetry transformation with the parameter ξ is calculated as where we have used the equation of motion for Ψ. Γ MN P and Γ MN PQR are the totally antisymmetrized products of the Dirac matrices with the weights 1 3! and 1 5! , respectively. The supercharge Q is defined byζQ = d 9 x j ♯ ζ , where j ♯ ζ is the temporal component of the supercurrent, and d 9 x is the spatial volume element. Since the left hand side of (2.5) can be rewritten as [iξQ, j M ζ ], the supersymmetry algebra is obtained by the spatial integration of the temporal component of (2.5) as where P M = d 9 x e N M T ♯ N is the ten-dimensional momentum. T ♯ N is the temporal component of the energy momentum tensor T M N , which is given by The central charge Z M N P QR is defined by where ε M 1 ···M 10 is the totally antisymmetric tensor in ten dimensions and F M N = e M M e N N F MN . Now we introduce the Ω-background. The Ω-background metric in ten dimensions is defined by the R 4 fibration over the torus T 6 : where x m (m = 1, 2, 3, 4) and x a+4 (a = 1, . . . , 6) are the spacetime and the internal space coordinates. The antisymmetric matrices Ω mna are parameterized as where ǫ 1 a , ǫ 2 a are real constant parameters. The matrices satisfy the following commutation relation, is the gauge covariant derivative. g = g 10 V The torsion is identified with the constant SU(4) I R-symmetry Wilson line gauge field (A a ) A B by the following relation [7]: (2.14) The four-dimensional supersymmetry transformation is obtained by the dimensional re-duction of (2.2): where Σ ab ,Σ ab are the six-dimensional Lorentz generators defined in appendix A. The constant supersymmetry parameters ζ = (ζ α A ,ζα A ) satisfy the parallel spinor conditions in four dimensions: In (2.16), the first terms represent the four-dimensional rotation of the spinor by the parameter Ω mna . The second terms represent the six-dimensional rotation by the parameter It is convenient to introduce the topological twist to see the cancellation between the four-and six-dimensional rotations. The topological twists in N = 4 super Yang-Mills theory are classified into three types [20]. They are called the half twist [21], the Vafa-Witten twist [22] and the Marcus twist [23,24]. We note that supersymmetry requires the additional conditions for Ω mna and (A a ) A B [7], where we find solutions of Ω mna and (A a ) A B for which ζ α A ,ζα A satisfy (2.16) and a part of supercharges is conserved.
Although a part of supersymmetry is preserved in each twisted theory, the fourdimensional translational symmetry is broken by the Ω-background (2.10). In [7], we have shown that the supersymmetries are enhanced in the Nekrasov-Shatashvili (NS) limit ǫ 1 a → 0 (or ǫ 2 a → 0) [13], where the translational symmetry in a two-dimensional subspace is recovered. In the following subsections, we summarize the solutions to the conditions and the enhanced supersymmetries in the three types of the topological twists.

Topological twist and supersymmetry in the NS limit
The topological twist is defined by an embedding of the Lorentz group SO(4) ≃ SU(2) L × SU(2) R into the subgroups of the SU(4) I R-symmetry group. We take the SU(2) L ′ × SU(2) R ′ subgroup of SU(4) I , where the SU(4) I index A = 1, 2, 3, 4 is decomposed into A ′ = 1, 2 andÂ = 3, 4. Here A ′ andÂ are indices for the two-dimensional representations of SU(2) R ′ and SU(2) L ′ , respectively.
Half twist In the half twist, SU(2) R subgroup of the Lorentz group SO(4) is replaced by the diagonal subgroup of SU(2) R ′ × SU(2) R . Then the new Lorentz group becomes where the subscript "diag" stands for the diagonal subgroup. The spinor indexα is identified with the R-symmetry index A ′ in the half twist. We define the vector, the scalar and the anti-self-dual tensor supercharges Q m ,Q, where Q αA ,Qα A are the supercharges associated with the supersymmetry transformation (2.15). The background Ω mna and (A a ) A B such thatQ andQ 12 are conserved [7] is , where m a (a = 1, 2) are real parameters. These parameters are identified with the mass of the adjoint hypermultiplet in the N = 2 * theory [4,6]. The theory has N = (0, 2) supersymmetry for the background (2.18). Here the notation N = (m, n) means that the theory has m chiral, n anti-chiral supercharges.
In the NS limit 2 , two components of the vector supercharges Q 3 , Q 4 are conserved in addition to the scalar and the tensor superchargesQ,Q 12 . In the ordinary basis, the conserved supercharges in the NS limit correspond to (Q 11 , Q 22 ) in Q αA and (Q˙1 1 ,Q˙2 2 ) in Qα A . Therefore the supersymmetry is enhanced to N = (2, 2).

Vafa-Witten twist In the Vafa-Witten twist, the new Lorentz group is
The spinor indexα is identified with the R-symmetry indices A ′ andÂ. We define the two scalarsQ,Q, the two vectors Q m ,Q m and the two anti-self-dual tensor superchargesQ mn ,Q mn by The background Ω mna and (A a ) A B such thatQ,Q 12 ,Q andQ 12 are conserved is , The theory has N = (0, 4) supersymmetry for the background (2.20). In the NS limit, four vector supercharges Q 3 , Q 4 ,Q 3 ,Q 4 are conserved in addition toQ,Q,Q 12 ,Q 12 . In the ordinary basis, these conserved supercharges correspond to (Q 11 , Q 22 , Q 13 , Q 24 ) in Q αA and Therefore the supersymmetry is enhanced to N = (4, 4).

Marcus twist In the Marcus twist, the new Lorentz group is [SU
where the spinor indices α andα are identified with the Rsymmetry indicesÂ and A ′ , respectively. We define the two scalars Q,Q, the two vectors Q m ,Q m and the two tensor supercharges Q mn ,Q mn by Table 1: Conserved supercharges in the NS limit for topological twists. The signs (+) and (−) in the Marcus twist correspond to the choice of the sign of (A a )ÂB in (2.22).
The background Ω mna and (A a ) A B such that the theory has N = (2, 2) supersymmetry is , For the background with the minus sign in the lower-right block in (A a ) A B , the conserved supercharges are Q, Q 12 ,Q andQ 12 [7]. If we choose the plus sign, Q 13 , Q 14 ,Q andQ 12 are conserved. But it can be shown that the theory defined in the background (2.22) with the plus sign is the same as the one with the minus sign by suitable field redefinition.
In the NS limit, for the minus sign in the lower-right block in (A a ) A B , four vector supercharges Q 1 , Q 2 ,Q 1 ,Q 2 are conserved in addition to Q,Q, Q 12 ,Q 12 . In the ordinary basis, these supercharges correspond to (Q 11 , The supersymmetry is enhanced to N = (4, 4). If we choose the plus sign, the conserved supercharges in the ordinary basis are (Q 11 , Q 22 , Q 13 , Q 24 ) in Q αA and (Q˙1 1 ,Q˙2 2 ,Q˙1 3 ,Q˙2 4 ) inQα A , which are the same as the case of the Vafa-Witten twist.
Moreover in the NS limit, the background is obtained from (2.20) by setting ǫ 1 5 = ǫ 1 6 = 0. Therefore the Marcus twist is regarded as the special case of the Vafa-Witten twist in the NS limit.
The conserved supercharges in the NS limit are summarized in Table 1.

Central charges and BPS bounds in the NS limit
We now study the supersymmetry algebra for Ω-deformed N = 4 super Yang-Mills theory in the NS limit ǫ 2 a → 0 such that the translational symmetry in the (x 3 , x 4 )-plane is recovered. We also perform the inverse Wick rotation x 4 = ix 0 and consider the theory in the spacetime with the Minkowski signature, which implies that the energy and the third component of the momentum are well-defined and conserved. In this section we will calculate the central charges of the algebras and the BPS bounds for the mass.
Now we calculate the central charges of the supersymmetry algebra (3.2). The sixdimensional momentum P a+4 is obtained by the spatial integral of the energy-momentum tensor (2.7). After the dimensional reduction, we have Here the fermionic part is omitted since we are interested in the BPS configuration, in which the fermions are set to be zero. This momentum is interpreted as the electric charge [32]. Then we define the electric charge q The other charges Z i,abcd , Z ij,abc and Z ijk,ab correspond to the charges of the BPS vortices, the BPS domain-walls and the space-filling BPS objects. In the next subsections, we examine the Ω-deformation of the central charges in (3.2) for each twist.

Half twist
In the case of the half twist with the NS limit ǫ 2 a → 0, we have the conserved supercharges Q 11 , Q 22 ,Q˙1 1 andQ˙2 2 . The supersymmetry generator ζ αA Q αA +Qα Aζα A becomes Let us examine the right hand side in the supersymmetry algebra (3.2). For the energy and the momentum, only P 0 and P 3 contribute to the algebra. This is consistent with the recovery of the translational invariance for x 0 -and x 3 -directions under the NS limit.
where T m n is the bosonic part of the four-dimensional energy momentum tensor: The first term in the right hand side of (3.6) is the electric charge density of the unde-  .7) is not deformed.
For the vortex charge Z i,abcd , it is convenient to consider its Hodge dual, which is given From the spinorial structure of the conserved supercharges, the components V 1212 , V 1234 and V 1256 can contribute to the algebra (3.2). These are computed as Here the torsion T ab c is written in terms of the deformation parameters and is given in appendix B. In the case of the half twist, ϕ 1 and ϕ 2 can be nonzero, which commute with each other, while ϕ 3 , ϕ 4 , ϕ 5 and ϕ 6 are zero. Using this vacuum configuration we find that the surface term from the first line in (3.10) vanishes. The second and the third lines vanish also in the NS limit. Therefore V ijab does not contribute to the algebra (3.2). In a similar way, we can show that neither Z ij,abc nor Z ijk,ab contributes to the algebra. This implies that under the vacuum boundary condition the BPS vortices, the BPS domain walls and the space-filling BPS objects do not exist in this theory. This is similar to the case of the undeformed N = 2 * and N = 4 theories. For Ω-deformed N = 2 super Yang-Mills theory, see [15]. We finally obtain the supersymmetry algebra 12) and the other (anti-)commutators vanish. Here the central charge Z is given by (3.14) The supercharges preserved by the BPS states are given by Here θ is the argument of Z and satisfies (3.16) Since two supercharges out of four are preserved, these states are the 1/2 BPS states.

From (3.18) the BPS bound is given by
where |q (e) | and |q (m) | are the lengths of the vectors q (e) = (q 6 ), respectively. α is the angle between the vectors q (e) and q (m) . The bound (3.22) has the same form as the undeformed theory [32,34] except that the charges with the indices a = 3, 4 do not contribute. This is because those charges vanish due to the vacuum conditions. The number of supercharges preserved by the BPS states depends on whether sin α is zero or nonzero.
In the case of sin α = 0, the supercharges preserved by the BPS states are and their complex conjugates from the algebras (3.18). Here the coefficients v i (i = 1, . . . , 4) are defined by Since four supercharges out of eight are preserved, these states are the 1/2 BPS states.
For nonzero sin α, the supercharges preserved by the BPS states are and its complex conjugate. Here the coefficients w i (i = 1, . . . , 4) are defined by  Since two supercharges out of eight are preserved, these states are the 1/4 BPS states.

Ω-deformed BPS equations
In this section we study the BPS equations deformed in the Ω-background in the NS limit.

1/2 BPS equations for the half twist
(4.8) The 1/2 BPS dyon equations become [ϕ a , D 0 ϕ a ] = 0, and their complex conjugates. As in the case of the half twist, we have the 1/2 BPS dyon equations. But in order to obtain the BPS equations which have the nontrivial electric and magnetic fields, we need to impose the following conditions for v i 's: These four cases are equivalent up to the field redefinition by the R-symmetry transformation. We consider the case v 1 = v 4 and v 2 = v 3 = 0, where q 2 = q 5 = q 6 = 0. Since we have |v 1 | = 1 from (3.24), we can introduce the angle θ by (4.14) In this case, the preserved supercharges are given by and their complex conjugates. These are included in the supercharges preserved by the undeformed 1/2 BPS equations. The deformed 1/2 BPS dyon equations in the Vafa- where the function f (a) is given by (4.10). Comparing the deformed 1/2 BPS equations

1/4 BPS equations for the Vafa-Witten and the Marcus twists
and their complex conjugates.
In order to obtain the BPS equations which have the nontrivial electric and magnetic fields, we must take the limit w 1 , w 3 → 0, w 1 , w 4 → 0, w 2 , w 3 → 0, or w 2 , w 4 → 0. These four limits are shown to be equivalent to each other by the field redefinition. Let us consider the limit w 2 , w 4 → 0. More precisely we need to take the limit w 2 , w 4 → 0 such that are satisfied. Here the angle parameter θ is given by This can be done by the limit q 5 , q 6 → 0 with fixed q 5 /q 6 , q 1 and q 2 . The preserved supercharges now have the same form as (3.15). The deformed 1/4 BPS dyon equations where φ a , ω m a and ε 1 a (a = 1, 2) are given by (4.8), and the functions f (a) and g(a) are given by (4.10). The equations (4.20) are the Ω-deformation of the undeformed 1/4 BPS equation (4.5). They become the same as in the half twist (4.9) when ǫ 1 5 = ǫ 1 6 = 0 and µ a = ε 1 a /2 (a = 1, 2) [19]. We have to impose the Gauss' law constraint in the NS limit: [ϕ a , D 0 ϕ a ] = 0, (4.21) for the consistency with the equations of motion.

BPS equations from Bogomol'nyi completion
We can derive the deformed BPS equations (4.9), (4.16) and (4.20) from the Bogomol'nyi completion of the energy E = P 0 , which is given by Here the fermionic part is omitted. We will study the completion of the energy for each BPS equation.
1/2 BPS equations in the half twist We first perform the Bogomol'nyi completion of the energy in the half twist and will derive the 1/2 BPS equations (4.9). We find that the energy (4.22) is rewritten as the sum of the complete squared forms, the magnetic charges and the deformed electric charges: Here we define Then the energy is bounded from below as In order that this inequality holds for any θ, the energy must satisfy the following inequality: The inequality is saturated when each squared form in (4.23) vanishes and θ is given by (3.16). When the equality holds, we obtain the 1/2 BPS equations (4.9). The energy bound (4.26) is given by the absolute value of the central charge (3.14).  where θ is a phase factor and ε abc (a, b, c = 2, 5, 6) is the totally antisymmetric tensor with ε 256 = 1. f (a) is given by (4.10). Maximizing the energy bound with respect to θ, we obtain the inequality The inequality is saturated when each squared form in (4.27) vanishes and θ satisfies 1 , which is the same as (4.14). The condition that each squared form vanishes gives the 1/2 BPS equations (4.16). The energy bound (4.28) is given by the BPS bound of (3.22) for q (m) a = q (e) a = 0 (a = 2, 5, 6), which corresponds to the condition such that the supercharges (4.15) are preserved.

1/4 BPS equations in the Vafa-Witten and the Marcus twists
We next discuss the 1/4 BPS equations (4.20) from the Bogomol'nyi completion. We find that the energy (4.22) is rewritten as Then the energy is bounded from below as The inequality is saturated when θ is given by (4.19)

Conclusion
In this paper, we studied the central charge of the supersymmetry algebra and the de- Since the present BPS equations reduce to the N = 2 ones by the projection, we expect that for the N = 4 theory the equations have similar type of solutions. It would be interesting to study the Nahm construction of the monopoles [36] for the construction of the solutions with higher charges and their moduli space. It is also interesting to study S-duality [8,37,38,39] of the Ω-deformed N = 4 theory as well as the relation to the integrable systems [13,14,15].

A Four-and six-dimensional Dirac matrices
In this appendix, we provide the decomposition of the ten-dimensional Dirac matrices by the dimensional reduction to four dimensions and introduce the four and the six- The Lorentz generators (Σ ab ) A B and (Σ ab ) A B are defined by We also define the antisymmetrized products of the six-dimensional Dirac matrices (Σ abc ) AB , where the square bracket denotes the antisymmetrization defined in subsection 3.1.

B Vacuum conditions
In this appendix, we study the vacuum structure of the N = 4 super Yang-Mills theory in the Ω-background. This is necessary to determine the surface terms in the integral of belong to the Cartan subalgebra of the Lie algebra of the gauge group and ϕ a = 0 (a = 3, . . . , 6). The vacuum solution does not change in the NS limit.