Two-dimensional superconformal field theories from Riemann surfaces with boundary
OOU-HET 846IPMU14-0365
Two-dimensional superconformal field theoriesfrom Riemann surfaces with a boundary
Koichi Nagasaki and Satoshi Yamaguchi Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan [email protected] Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan [email protected]
Abstract
We consider a 2-dimensional conformal field theory (CFT) obtained from twisted com-pactification of the 4-dimensional N = 4 super Yang-Mills theory on a Riemann surfacewith boundary. We find the boundary conditions for preserving some of the supersymmetry.In particular an N = (2 ,
2) superconformal field theory is obtained from supersymmetrybreaking due to the boundary from N = (4 , We often find an interesting relationship between a geometry and a supersymmetric quantum fieldtheory by compactifying a higher dimensional conformal field theory. The class S theories [1] arefamous examples which are obtained by compactification of 6-dimensional (2,0) superconformalfield theories (SCFTs) by Riemann surfaces. Alday-Gaiotto-Tachikawa correspondence [2, 3] isa relation between a class S theory and a 2-dimensional CFT on the Riemann surface. SCFTsobtained from a d -dimensional theory compactified on various manifolds are studied, for example,[4, 5, 6, 7, 8, 9, 10].It is also interesting to consider a Riemann surface with boundary. However for the class S theories it seems difficult to introduce a boundary of the Riemann surface since an M5-branecannot have a supersymmetric boundary.In this paper we construct 2-dimensional CFTs obtained from compactification of 4-dimensionalgauge theories on Riemann surfaces with boundary. To realize a boundary theory we considertype IIB superstring in this paper. Our gauge theory is a 4-dimensional N = 4 super Yang-Millstheory (SYM) realized on the world-volume of D3-branes. These D3-branes can end on D5-branesor NS5-branes, and thus can have a boundary.The 2-dimensional CFTs obtained from compactification on closed Riemann surfaces [4] arestudied by using c -extremization [11, 12, 13, 14]. This method is an analogue to a -maximizationin 4-dimensions [15, 16] and F -maximization in 3-dimensions [17]. For a -maximization its gravitydual is studied in [18, 19, 20, 21, 22]. 1 a r X i v : . [ h e p - t h ] M a r n this paper we study the 4-dimensional N = 4 SYM on R , × Σ o where Σ o is a Riemannsurface with a boundary. In the low energy limit this theory is expected to become a 2-dimensionalCFT. We find a class of boundary conditions at the boundary of Σ o which preserve some ofthe supersymmetry, following the strategy of [23]. The boundary is a geodesic and preservesthe N = (0 , , (1 , , (2 ,
2) supersymmetry out of the N = (0 , , (2 , , (4 ,
4) original bulksupersymmetry, respectively. It is an interesting future work to study more general boundaryconditions as in [23, 24, 25, 26, 27] and S-duality. In this paper we also show some attempt tofind a different class of boundary conditions.Among these theories we calculate the central charge for the N = (2 ,
2) case because in thiscase the central charge is related to the ’t Hooft anomaly coefficients which are invariant underthe renormalization group flow [28]. We obtain a positive central charge only when the Eulernumber χ o of Σ o is negative. In this case the central charge is written as c = 3 d G | χ o | , (1.1)where d G is the dimension of the gauge group. This theory has the N = (2 ,
2) superconformalsymmetry with c = 3 × (integer). Therefore this theory seems to be a sigma model with a Calabi-Yau target space. Further study of this theory, in particular the relationship with the theory of[4], is also an interesting problem. This result coincides with the case for the central charge oftheories compactified on the closed Riemann surfaces [11, 12, 29]. Studying the reason of thecoincidence between out result (1.1) and previous works [11, 12, 29] is an interesting future work.Another interesting future work is to investigate the realization in the string theory andAdS/CFT correspondence [30]. Our setup is realized by D3-branes wrapping on a holomorphiccycle in a local Calabi-Yau manifold and ending on a 5-brane system [31, 32, 23, 24, 25, 26, 27].The construction of this paper is as follows: In section 2 we introduce a twisted compactifi-cation of 4-dimensional gauge theories following [33, 34, 29]. In section 3 we find a condition forpreserving supersymmetry and calculate the central charge. N = 4 SYM
We first review a 4-dimensional N = 4 SYM on a curved spacetime following [11, 12]. Insubsection 2.1, first we obtain the action on the flat spacetime. In subsections 2.2 and 2.3 weintroduce a closed Riemann surface with constant curvature and twist the theory. We also showhow many supersymmetries are preserved by compactification on closed Riemann surfaces. N = 4 SYM on the flat spacetime
The 4-dimensional N = 4 SYM action on the flat spacetime is obtained by the trivial dimensionalreduction from the 10-dimensional SYM. It contains a 10-dimensional vector field A M , M =0 , , . . . , Ψ = Ψ. Bothof them are in the adjoint representation of the gauge group G . The vector field is decomposed intoa 4-dimensional vector A µ , µ = 0 , , , , and 6 scalars Φ A = A A , A = 4 , . . . , S = 1 g YM2 (cid:90) d x Tr (cid:48) (cid:26) − F MN F MN + i M D M Ψ (cid:27) , (2.1)where g YM is the 4-dimensional gauge coupling. F MN M, N = 0 , , . . . , F µν = ∂ µ A ν − ∂ ν A µ + i [ A µ , A ν ] , µ, ν = 0 , , , , (2.2) F µA = − F Aµ = ∂ µ Φ A + i [ A µ , Φ A ] =: D µ Φ A , (2.3) F AB = i [Φ A , Φ B ] . (2.4)The covariant derivative for Ψ is defined as D µ Ψ = ∂ µ Ψ + i [ A µ , Ψ] , D A Ψ = i [Φ A , Ψ] . (2.5)Tr (cid:48) is a trace normalized as Tr (cid:48) = h ∨ Tr adjoint where h ∨ is the dual Coxeter number. For example,Tr (cid:48) = 2Tr fundamental for SU( N ). The action is rewritten as S = 1 g YM2 (cid:90) d x Tr (cid:48) (cid:26) − F µν F µν − D µ Φ A D µ Φ A + 14 [Φ A , Φ B ][Φ A , Φ B ]+ i µ D µ Ψ −
12 ΨΓ A [Φ A , Ψ] (cid:27) . (2.6)This action is invariant under the supersymmetry transformation: δA M = i ¯ (cid:15) Γ M Ψ , δ Ψ = 12 Γ MN F MN (cid:15). (2.7)The parameters (cid:15) are Majorana-Weyl fermions satisfyingΓ (cid:15) = (cid:15). (2.8)Then the supersymmetry current is obtained as J µ = i (cid:48) { F µN Γ N − F KL Γ KLµ } Ψ = i (cid:48) { F KL Γ KL Γ µ Ψ } . (2.9) We will consider this 4-dimensional N = 4 SYM theory compactified on a compact Riemannsurface Σ. In this paper we concentrate on a Riemann surface with constant curvature R = 2 κ ,where κ = +1 ( g = 0)0 ( g = 1) − g > , (2.10)for a genus g closed Riemann surface. We denote the coordinates of this Riemann surface by( x , x ), the vielbein by E a , a = 2 ,
3, and the spin connection by Ω . The curvature 2-form iswritten as R = d Ω , and thus the Gauss-Bonnet theorem reads (cid:90) Σ d Ω = 12 (cid:90) Σ √ gR = 4 π (1 − g ) . (2.11)3or g (cid:54) = 1 the volume of the Riemann surface isvol Σ = 4 π | − g | , (2.12)and the volume form is d vol Σ = κd Ω . (2.13) Now we consider the 4-dimensional N = 4 SYM theory on a curved spacetime with the metric g µν and a background SO(6) gauge field A µ = A ABµ M AB , where M AB , A, B = 4 , · · · , S = 1 g YM2 (cid:90) d x √ g Tr (cid:48) (cid:26) − F µν F µν − D (cid:48) µ Φ A D (cid:48) µ Φ A + 14 [Φ A , Φ B ][Φ A , Φ B ]+ i µ D (cid:48) µ Ψ −
12 ΨΓ A [Φ A , Ψ] (cid:27) , (2.14)where the covariant derivative D (cid:48) µ includes the spin connection and the SO(6) gauge field D (cid:48) µ Φ A := ∂ µ Φ A + i [ A µ , Φ A ] + (cid:88) B A ABµ Φ B , (2.15) D (cid:48) µ Ψ := ∂ µ Ψ + i [ A µ , Ψ] + 14 Ω abµ Γ ab − i A µ Ψ . (2.16)Here A µ Ψ := i A ABµ Γ AB Ψ. In order to preserve the supersymmetry, a parameter of the supersym-metry transformation (2.7) should satisfy the Killing spinor equation. The twisted Killing spinorequation is D (cid:48) µ (cid:15) := (cid:18) ∂ µ + 14 Ω abµ Γ ab − i A µ (cid:19) (cid:15) = 0 . (2.17)We choose the external gauge field A µ in SO(2) ⊂ SO(6), such that the field strength, F = d A , A = A µ dx µ , (2.18)satisfies F = − T d vol Σ ( g (cid:54) = 1) − T π vol Σ d vol Σ ( g = 1) . (2.19)Here T is an SO(2) generator T = a T + a T + a T , (2.20)where a i are parameters of twisting and T i , i = 1 , , T = i , T = i , T = i . (2.21)4he condition for existing covariantly constant spinors is, from eq. (2.17), D (cid:48) µ (cid:15) = 0 ⇒ [ D (cid:48) , D (cid:48) ] (cid:15) = 0 ⇒ (cid:18) d Ω Γ − id A (cid:19) (cid:15) = 0 . (2.22)Using the relations (2.13) and (2.19), (cid:18) κ d vol Σ · Γ + id vol Σ · T (cid:19) (cid:15) = 0 . (2.23)Finally, substituting eq.(2.20), the supersymmetry condition is (cid:0) − κi Γ + a i Γ + a i Γ + a i Γ (cid:1) (cid:15) = 0 . (2.24)The amount of the supersymmetry depends on the number of the non-zero parameters among a i ,i = 1 , ,
3. Let us classify them here:1. All a i are non-zero: ( − κ Γ + a Γ + a Γ + a Γ ) (cid:15) = 0.In this case the number of the supersymmetries is N = (0 , a i is a + a + a = κ. (2.25)2. Two of a i are non-zero: ( − κ Γ + a Γ + a Γ ) (cid:15) = 0.In this case the number of the supersymmetries is N = (2 , a i is a + a = κ. (2.26)3. One of a i is non-zero: ( − κ Γ + a Γ ) (cid:15) = 0.In this case the number of the supersymmetries is N = (4 , a is a = κ. (2.27)4. No background field: ( − κ Γ ) (cid:15) = 0.In this case the number of the supersymmetries is N = (8 , κ = 0, i.e. g = 1.These results are summarized in Table 1. a i (cid:54) = 0 N g ,
2) all2 (2 ,
2) all1 (4 ,
4) all0 (8 ,
8) 1Table 1: Remaining supersymmetries for closed Riemann surfaces.5
Supersymmetric boundary condition and central charge
In this section we introduce a boundary on the Riemann surface. We assume that the boundaryis a geodesic. First we explain this assumption is appropriate and simplifies our argument. Afterthat we study the boundary condition for preserving some supersymmetries. We obtain thecentral charge when the N = (2 ,
2) supersymmetry is preserved. In this calculation we assumethat the two-dimensional theory at low energies is conformal. However, if the calculation gives thenegative central charge then this indicates the assumption is violated. We also show an attemptto find other class of boundary conditions.
In this paper we focus on Riemann surfaces with one boundary. We also assume that these surfaceshave constant curvature. In this paper we only consider a geodesic boundary for simplicity. Therecould be a non-geodesic boundary which preserves some supersymmetry, although we do not findan example. The analysis is rather simple for the geodesic boundary for the following reasons.Let ( x , x ) the coordinates of the Riemann surface and the geodesic boundary x = 0. Thenwe can choose a gauge such that locally A = A = 0 on the boundary since A is proportionalto Ω and we can choose the gauge Ω = 0 on a geodesic. Then terms including the externalgauge field A µ in the covariant derivative (2.15) can be omitted and D (cid:48) = D is satisfied at leastlocally. However we cannot ignore the holonomy along the boundary. The boundary conditionmust be consistent with this holonomy. Another reason for choosing the geodesic boundary isthat we want to use the doubling trick later. If the boundary is a geodesic, one can join togetherthe Riemann surface and a copy of it with the opposite orientation to construct a closed surfacewith constant curvature.Let us see the holonomy of this external gauge field along the boundary. First for simplicitywe consider an S with a boundary at the equator — a northern (or southern) hemisphere S ( S − ). This holonomy is given by (cid:73) ∂S A = (cid:90) S d A = (cid:90) S F = Magnetic flux . (3.1)Here we use Stokes’ theorem to express it as an integral of the gauge field strength. This integralgives a magnetic flux through the surface S . Due to the Dirac quantization condition, theintegral of magnetic flux on the S is an integral multiplication of 2 π . Now this gauge fieldis distributed isotropically. Then the integral only over the northern hemisphere (3.1) gives aninteger or a half integer times 2 π . We can use the same strategy for a general Riemann surfaceΣ o with one geodesic boundary. Let Σ be the closed Riemann surface made by gluing Σ o and acopy of it with the opposite orientation Σ o along their boundaries. Notice that the genus g of Σis an even number and thus it is not 1. The holonomy along this boundary can be written byusing eqs. (2.19), (2.20), (2.12) as H := exp (cid:18) i (cid:73) ∂ Σ o A (cid:19) = exp (cid:18) i (cid:90) Σ o F (cid:19) = exp (cid:18) i (cid:90) Σ F (cid:19) = (cid:89) i =1 , , exp( − iπn i T i ) , (3.2)6here n i := 2 | − g | a i are integers [12]. Later we use the fact H = exp (cid:18) i (cid:90) Σ F (cid:19) = 1 (3.3)following from the Dirac quantization condition. The boundary condition considered in this paperlater (3.11) is consistent with this holonomy (3.2). Let us here consider the boundary conditions which preserve some part of the supersymmetry.For preserving the supersymmetry the current component normal to the boundary must be zeroat the boundary ( x = 0). From eq. (2.9) this condition is expressed as (cid:15)J = 0 ⇔ Tr (cid:48) (cid:0) (cid:15)F KL Γ KL Γ Ψ (cid:1) = 0 . (3.4)In this condition we can replace D (cid:48) by D since we can choose the gauge where A = 0 at theboundary. Thus we can employ the same strategy as [23] (see also [26]). Define the followingmatrices: B = Γ , (3.5) B = Γ , (3.6) B = Γ , (3.7)and redefine the scalar fields ( X , X , X ) := (Φ , Φ , Φ ) , (3.8)( Y , Y , Y ) := (Φ , Φ , Φ ) . (3.9)The boundary condition (3.4) is decomposed into the following equations as done in [23]:Tr (cid:48) (cid:15) (Γ µν F µν + 2Γ µ F µ )Γ Ψ = 0 , Tr (cid:48) (cid:15) (2Γ a D X a + Γ ab [ X a , X b ])Γ Ψ = 0 , Tr (cid:48) (cid:15) (2Γ m D Y m + Γ mn [ Y m , Y n ])Γ Ψ = 0 , Tr (cid:48) (cid:15) Γ µa D µ X a Γ Ψ = 0 , Tr (cid:48) (cid:15) Γ µm D µ Y m Γ Ψ = 0 , Tr (cid:48) (cid:15) Γ am [ X a , Y m ]Γ Ψ = 0 , (3.10)where µ, ν = 0 , , , a, b = 4 , , , and m, n = 5 , ,
9. An example of the boundary condition isthe NS5-brane like boundary condition D X a = 0 , Y m = 0 , F µ = 0 , (3.11)for the bosonic fields. For the fermionic fields we impose B Γ Ψ = − Γ Ψ (3.12)7t the boundary. Actually the NS5-brane like boundary conditions (3.11) and (3.12) preserve thesupersymmetry if the parameter (cid:15) satisfies B (cid:15) = (cid:15). (3.13)The conditions (3.10) is verified. The condition (3.13) for (cid:15) kills half of the supersymmetry asfollows. If an i Γ eigenvector (cid:15) satisfies (2.24), B (cid:15) also satisfies (2.24) and they are independent.Therefore among the linear combinations of these two independent parameters, one combination (cid:15) = (1 + B ) (cid:15) satisfies the condition (3.13). Since (cid:15) and B (cid:15) have the same chirality (Γ eigenvalue), the preserved supersymmetry is as follows.1. N = (0 ,
2) bulk ⇒ N = (0 , N = (2 ,
2) bulk ⇒ N = (1 , N = (4 ,
4) bulk ⇒ N = (2 , H Φ) A = ± Φ A , so the conditions for the bosons (3.11) areconsistent. The consistency of the condition for the fermions (3.13) is verified by (3.3) H = 1and B Γ H Ψ = H − B Γ Ψ. N = (4 , case and the central charge The case where the bulk N = (4 ,
4) supersymmetry is broken to N = (2 ,
2) by the boundaryis interesting because of the R-symmetry of the N = 2 superconformal symmetry. In this case a = a = 0 and a = κ , and T in eq. (2.20) becomes T = κ i . (3.14)The preserved supersymmetry parameters satisfy eq. (2.24), which is rewritten asΓ (cid:15) = − (cid:15). (3.15)Then the exact central charge is obtained from the ’t Hooft anomaly coefficient as in [12]. Howeverin our case the situation is much simpler since there is only one candidate U(1) symmetry Q R forthe R-symmetry Q R = i + i . (3.16)This is determined such that for the right moving supersymmetry parameters (cid:15) (Γ (cid:15) = + (cid:15) )satisfy Q R (cid:15) = ± (cid:15) and the left moving ones satisfy Q R (cid:15) = 0. The right moving central charge isexpressed as c = 3Tr Weyl fermion (Γ ( Q R ) ) . (3.17)In the above expression Tr Weyl fermion means counting the number of the 2-dimensional Weylfermions.The number of the chiral fermions can be counted by the index theorem as in [11]. In thispaper we use the doubling trick to map the problem to the index theorem in the closed Riemann8urface. We take the Riemann surface Σ o and a copy with the opposite orientation Σ o , and jointhem together, Σ o (cid:83) Σ o =: Σ, so that their boundaries are the same (See Figure 1). Originally Ψincludes four 4-dimensional Weyl spinors. Half of them satisfying Γ Ψ = − Ψ have charge ± Q R and the others are neutral. Let us denote these two charged 4-dimensional Weyl spinorsΨ ± which satisfy i Γ Ψ ± = ± Ψ ± and B Ψ ± = Ψ ∓ . These two fermions on R , × Σ o are treatedas a fermion Ψ c on R , × Σ. Ψ c is defined asΨ c = Ψ − ( z ) , (Im( z ) ≥ + ( z ∗ ) , (Im( z ) ≤ . (3.18)Here we use the complex coordinate z = x + ix of Σ such that Σ o is parametrized by Im z ≥ o is parametrized by Im z ≤ z → z ∗ is the symmetry which exchanges Σ o and Σ o . Actuallythis Ψ c is continuous at the boundary due to the boundary condition (3.12). Furthermore, wedefine extended spin connections and gauge fieldsΩ z ( z ) := Ω z ( z ) (Im( z ) ≥ − Ω z ( z ∗ ) (Im( z ) ≤ , A z ( z ) := A z ( z ) (Im( z ) ≥ −A z ( z ∗ ) (Im( z ) ≤ . (3.19)Then according to the above definitions, the Dirac equations for Ψ ± on R , × Σ o are equivalentto the one for Ψ c on R , × Σ Γ µ D (cid:48) µ Ψ c ( z ) = 0 . (3.20)We denote the number of 2-dimensional right(left)-moving massless fermions by n R ( L ) for the4-dimensional Weyl fermion Ψ c . The index theorem gives the difference of these numbers and itis rewritten using eqs. (2.19), (2.12) : n R − n L = − π (cid:90) Σ Tr Ψ c F = t | g − | , (3.21)where Tr Ψ c is taken in the representation of Ψ c , and t is the eigenvalue of T for the fermion Ψ c which is given by t = − κ/ c = 3 d G ( n R − n L )= − d G κ | g − | , (3.22) ⌃ o ⌃ boundary of ⌃ o ⌃ o Figure 1: (Doubling trick) We construct a closed surface Σ by taking Σ o and one with the oppositeorientation Σ o . 9here d G is the dimension of the gauge group. This expression gives the positive c only when κ = − , ( g > c = 3 d G | χ o | . (3.23)In the final expression we use the Euler number of the original Riemann surface with the boundary.We are now considering a case where the Riemann surface has only one boundary, b = 1. Thenthe Euler number of the original surface, Σ o , is χ o = 2 − g / − b = 1 − g . In this subsection, we examine boundary conditions different from the NS5-like shown in theprevious subsections. We show some cases where the original bulk supersymmetries are N =(0 , ,
2) and (4 , (cid:15) I , I =1 , · · · ,
8. We diagonalize Γ , i Γ M,M +1 , M = 2 , , , (cid:40) Γ (cid:15) I = λ I (cid:15) I i Γ M,M +1 (cid:15) I = λ MI (cid:15) I ⇒ (cid:40) (cid:15) I Γ = − λ I (cid:15) I (cid:15) I ( i Γ M,M +1 ) = − λ MI (cid:15) I , (3.24)where eigenvalues λ I , λ MI take values +1 or − λ I λ I λ I λ I λ I I − + + + +2 − − − − − − − − − +5 − + + − − − − − + +7 + + + − +8 + − − + − Table 2: Eigenvalues of (cid:15) I . N = (0 , case The supersymmetry parameters preserved in the bulk are (cid:15) and (cid:15) .The current condition (3.4) for these generators isTr (cid:48) (cid:15) (cid:0) F Γ + F Γ + F Γ + F Γ + F Γ (cid:1) Γ Ψ = 0 , (3.25)Tr (cid:48) (cid:15) (cid:0) F M,N Γ M,N + F M,N +1 Γ M,N +1 + F M +1 ,N Γ M +1 ,N + F M +1 ,N +1 Γ M +1 ,N +1 (cid:1) Γ Ψ = 0 , (3.26)( M, N ) = (0 , , (0 , , (0 , , (0 , , (2 , , (2 , , (2 , , (4 , , (4 , , (6 , . We impose the boundary condition for the fermion field: − i Γ Ψ = Ψ . (3.27)10rom the first equation (3.25),Tr (cid:48) (cid:15) I (cid:0) F Γ + F Γ + F Γ + F Γ + F Γ (cid:1) Γ ( − i Γ Ψ) = 0 ↔ Tr (cid:48) (cid:15) I ( i Γ ) (cid:0) F Γ + F Γ + F Γ + F Γ + F Γ (cid:1) Γ Ψ = 0 , ( I = 1 , . (3.28)The lefthand side is trivially satisfied for (cid:15) which satisfies (cid:15) ( i Γ M,M +1 ) = − (cid:15) . For (cid:15) this equationgives the condition Tr (cid:48) (cid:15) (cid:16) F − i ( F + F + F + F ) (cid:17) Γ Ψ = 0 . (3.29)Then, F = 0 , F + F + F + F = 0 . (3.30)The second equation (3.26) of ( M, N )=(0 , , ,
6) and (2 ,
8) are trivially satisfied for thecase of (cid:15) in the same way and in the cases ( M, N )=(0 , , (0 , , (4 ,
8) and (6 ,
8) thisequation becomes trivial for (cid:15) .The condition for the supersymmetry generated by (cid:15) I to be preserved is summarized as follows: (i) Supersymmetry generated by (cid:15) F ,M + F ,M = 0 ( M = 2 , ,F ,M − F ,M +1 = F ,M +1 + F ,M = 0 ( M = 4 , , . (3.31) (ii) Supersymmetry generated by (cid:15) F , = 0 , F + F + F + F = 0 F ,M + F ,M = 0 ( M = 4 , , , , , ,F M,N − F M +1 ,N +1 = F M,N +1 + F M +1 ,N = 0 (( M, N ) = (4 , , (4 , , (6 , . (3.32)Let us define complex fields Z := Φ + i Φ , Z := Φ + i Φ , Z := Φ + i Φ . (3.33)We define coordinates on the 2d CFT and the Riemann surface and redefine gauge field on them.On ( x , x ) : x ± x =: x ± , A x ± := 12 ( A ± A ) , (3.34)On ( x , x ) : x ± ix =: w ± , A w ± := 12 ( A ∓ iA ) . (3.35)Then, the following new derivatives can be defined:12 (cid:16) D ± D (cid:17) = (cid:18) ∂∂x ± + [ A x ± , ∗ ] (cid:19) =: D x ± , (3.36)12 (cid:16) D ∓ iD (cid:17) = (cid:18) ∂∂w ± + [ A w ± , ∗ ] (cid:19) =: D w ± . (3.37)Using these notations the supersymmetry conditions (3.31), (3.32) are respectively rewritten asfollows. 11. Supersymmetry generated by (cid:15) :(3.31) ⇒ F M + F M = 0 ( M = 2 , ,D w − Z A = 0 . (3.38)2. Supersymmetry generated by (cid:15) :(3.32) ⇒ F = 0 , F = − i (cid:80) i [ Z i , Z i ] ,D x + Z i = 0 , [ Z i , Z j ] = 0 . (3.39)In the second case we find that this equation looks like a Hitchin system [35]. For more detailsof these types of equations, see [36]. N = (2 , case The supersymmetry parameters preserved in the bulk are (cid:15) , . . . , (cid:15) in Table 2. In this case wecan use the same method to the previous N = (0 ,
2) case. The normal component of the currentsatisfies:Tr (cid:48) (cid:15) (cid:0) F Γ + F Γ + F Γ + F Γ + F Γ (cid:1) Γ Ψ = 0 , (3.40)Tr (cid:48) (cid:15) (cid:0) F M,N Γ M,N + F M,N +1 Γ M,N +1 + F M +1 ,N Γ M +1 ,N + F M +1 ,N +1 Γ M +1 ,N +1 (cid:1) Γ Ψ = 0 . (3.41)The first equation (3.40) becomes trivial for (cid:15) I having eigenvalue λ I = +1 in the same way to N = (0 ,
2) case and for (cid:15) I having eigenvalue λ I = − F = 0 , (cid:88) i =1 λ iI F i, i +1 = 0 . (3.42)The second equation (3.41) splits into two groups( M, N ) = (0 , , (2 , , (2 , , (2 , , ( M, N ) = (0 , , (0 , , (0 , , (4 , , (4 , , (6 , . The former becomes trivial for λ I = − λ I = +1. The nontrivialconditions are for λ I = +1 F − λ I F = F − λ I F = 0 , (3.43) F M,N − λ MI λ NI F M +1 ,N +1 = + λ MI F M +1 ,N + λ NI F M,N +1 = 0( M, N ) = (2 , , , , (3.44)and for λ I = − F ,M − λ I F ,M = 0 , M = 4 , , , , , , (3.45) F M,N − λ MI λ NI F M +1 ,N +1 = λ MI F M +1 ,N + λ NI F M,N +1 = 0( M, N ) = (4 , , , . (3.46)Summarizing the above, the supersymmetries generated by (cid:15) I are respectively as follows:12. (cid:15) I ( λ I = +1) F ,M − λ I F ,M = 0 ( M = 2 , F M,N − λ MI λ NI F M +1 ,N +1 = λ MI F M +1 ,N + λ NI F M,N +1 = 0 ( M, N ) = (2 , , , , (3.47)2. (cid:15) I ( λ I = − F = 0 , (cid:80) i =1 λ iI F i, i +1 = 0 F ,M − λ I F ,M = 0 ( M = 4 , , , , , F M,N − λ MI λ NI F M +1 ,N +1 = λ MI F M +1 ,N + λ NI F M,N +1 = 0 ( M, N ) = (4 , , , . (3.48)The case we studied before in the subsection 3.4.1 corresponds to the case of λ I = − N = (4 , case The supersymmetry parameters preserved in the bulk are (cid:15) , . . . , (cid:15) in Table 2. The conditionsfor the bosonic fields are F , = 0 , (cid:88) i =1 λ iI F i, i +1 = 0 , (3.49)where I = 2 , , , F M − λ I F M = 0 , (3.50)where M = 2 , I = 2 , , , M = 4 , , , , , I = 1 , , ,
7, and F M,N − λ MI λ NI F M +1 ,N +1 = λ NI F M,N − λ MI F M +1 ,N +1 = 0 , (3.51)where ( M, N ) = (2 , , (2 ,
6) and (2 ,
8) for I = even, while ( M, N ) = (4 , , (4 ,
8) and (6 ,
8) for I = odd. The case of N = (2 ,
2) with the boundary is an interesting case and the central chargeis obtained only from the calculation of the ’t Hooft anomaly, as shown in subsection 3.3.
Acknowledgement
We are pleased to thank Dongmin Gang, Kentaro Hori, Masahito Yamazaki, and Yutaka Yoshidafor useful discussion. K.N. would like to thank the organizers of “the 7th Taiwan String Workshop”at National Taiwan University for giving him an opportunity to talk about this work. K.N. wassupported in part by JSPS Research Fellowship for Young Scientists and JSPS KAKENHI GrantNumber 13J02068. This work was supported in part by World Premier International ResearchCenter Initiative (WPI), MEXT, Japan. 13 eferences [1] D. Gaiotto, “N=2 dualities,”
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