OU-HET 846
IPMU14-0xxx

Two-dimensional superconfomal field theories
from Riemann surfaces with boundary
Koichi Nagasaki1 and Satoshi Yamaguchi2
1 Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan

kohichi.nagasaki@ipmu.jp
2 Department of Physics, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan

yamaguch@het.phys.sci.osaka-u.ac.jp
Abstract
We consider a 2-dimensional conformal field theory (CFT) obtained from twisted compactification of the 4-dimensional N = 4 super Yang-Mills theory on a Riemann surface
with boundary. We find the boundary conditions to preserve some of the supersymmetry.
In particular an N = (2, 2) superconformal field theory is obtained from supersymmetry
breaking due to the boundary from N = (4, 4). In this case we calculate the central chargeof the CFT and show its dependence on the topology of the Riemann surface.

1

Introduction and summary

We often find an interesting relationship between a geometry and a supersymmetric quantum field
theory 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) superconformal
field theories (SCFTs) by Riemann surfaces. AGT correspondence [2, 3] is a relation betweena class S theory and a 2-dimensional CFT on the Riemann surface. SCFTs obtained from ad-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-dimensional
gauge theories on Riemann surfaces with boundary. To realize a boundary theory we consider
type IIB superstring in this paper. Our gauge theory is a 4-dimensional N = 4 super Yang-Mills
theory (SYM) realized on the world-volume of D3-branes. These D3-branes can end on D5-branes
or 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]. This method is an analogue to a-maximization in
4-dimensions [13, 14] and F -maximization in 3-dimensions [15]. For a-maximization its gravity
dual is studied in [16, 17, 18, 19, 20].
1

In this paper we study the 4-dimensional N = 4 SYM on R1,1 × Σo where Σo is a Riemannsurface with a boundary. In the low energy limit this theory is expected to become a 2-dimensional
CFT. We find a class of boundary conditions at the boundary of Σo which preserve some ofthe supersymmetry, following the strategy of [21]. The boundary is a geodesic and preserves
the N = (0, 1), (1, 1), (2, 2) supersymmetry out of the N = (0, 2), (2, 2), (4, 4) original bulk
supersymmetry, respectively. It is an interesting future work to study more general boundary
conditions as in [21, 22, 23, 24, 25] 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 under
the RG flow [26]. We obtain a positive central charge only when the Euler number χo of Σo isnegative. In this case the central charge is written asc = 3dG |χo |,(1.1)

where dG is the dimension of the gauge group. This theory has the N = (2, 2) superconformal
symmetry with c = 3×(integer). Therefore this theory seems to be a sigma model with a CalabiYau target space. Further study of this theory, in particular the relationship with the theory of[4], is also an interesting problem.
Another interesting future work is to investigate the realization in the string theory and
AdS/CFT correspondence [27]. Our setup is realized by D3-branes wrapping on a holomorphic
cycle in a local Calabi-Yau manifold and ending on a 5-brane system [28, 29, 21, 22, 23, 24, 25].
The construction of this paper is as follows: In section 2 we introduce a twisted compactification of 4-dimensional gauge theories following [30, 31, 32]. In section 3 we find a condition forpreserving supersymmetry and calculate the central charge.

2

Twisted compactification of N = 4 SYM

We first review a 4-dimensional N = 4 SYM on a curved spacetime and twisting following [11, 12].
In subsection 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 show
how many supersymmetries are preserved by compactification on closed Riemann surfaces.

2.1

N = 4 SYM on the flat spacetime

The 4-dimensional N = 4 SYM action on the flat spacetime is obtained by the trivial dimensional
reduction from the 10-dimensional SYM. It contains a 10-dimensional vector field AM , M =0, 1, . . . , 9 and a 10-dimensional Majorana-Weyl spinor Ψ, which satisfies Γ0123456789 Ψ = Ψ. Bothof them are in the adjoint representation of the gauge group G. The vector field is decomposed into
a 4-dimensional vector Aµ , µ = 0, 1, 2, 3, and 6 scalars ΦA = AA , A = 4, . . . , 9 in 4-dimensions.
The action is written as


Z
1
1
i
0
4
MN
M
S=
d xTr − FM N F+ ΨΓ DM Ψ ,
(2.1)
gYM 2
4
2
2

where gYM is the 4-dimensional gauge coupling. FM N M, N = 0, 1, . . . , 9 is defined as
Fµν = ∂µ Aν − ∂ν Aµ + i[Aµ , Aν ],µ, ν = 0, 1, 2, 3,(2.2)

FµA = −FAµ = ∂µ ΦA + i[Aµ , ΦA ] =: Dµ ΦA ,(2.3)

FAB = i[ΦA , ΦB ].

(2.4)

The covariant derivative for Ψ is defined asDµ Ψ = ∂µ Ψ + i[Aµ , Ψ],DA Ψ = i[ΦA , Ψ].

(2.5)

Tr0 is a trace normalized as Tr0 = h1∨ Tradjoint where h∨ is the dual Coxeter number. For example,
Tr0 = 2Trfundamental for SU(N ). The action is rewritten as

Z
1
1
1
1
0
4
S=
d xTr − Fµν F µν − Dµ ΦA Dµ ΦA + [ΦA , ΦB ][ΦA , ΦB ]
2
gYM
4
2
4

i
1 A
µ
+ ΨΓ Dµ Ψ − ΨΓ [ΦA , Ψ] .
(2.6)
2
2
This action is invariant under the supersymmetry transformation:1
δAM = i¯ΓM Ψ, δΨ = ΓM N FM N .
2
The parameters  are Majorana-Weyl fermions satisfying
Γ0123456789  = .
Then the supersymmetry current is obtained asii
J µ = Tr0 {2F µN ΓN − FKL ΓKLµ }Ψ = Tr0 {FKL ΓKL Γµ Ψ}.
2
2

2.2

(2.7)

(2.8)

(2.9)

Riemann surfaces

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 = 2κ,
where



(g = 0)

 +1
κ=

0



 −1

(g = 1)(2.10)(g > 1),for a genus g closed Riemann surface. We denote the coordinates of this Riemann surface by(x2 , x3 ), the vielbein by E a , a = 2, 3, and the spin connection by Ω23 . The curvature 2-form is
written as R23 = dΩ23 , and thus the Gauss-Bonnet theorem reads
Z
Z
1
√
23
dΩ =
gR = 4π(1 − g).
(2.11)2 Σ
ΣFor g 6= 1 the volume of the Riemann surface isvolΣ = 4π|1 − g|,(2.12)

dvolΣ = κdΩ23 .

(2.13)and the volume form is

3

2.3

Twisted gauge theory on the curved spacetime

Now we consider the 4-dimensional N = 4 SYM theory on a curved spacetime with the metricgµν and a background SO(6) gauge field Aµ = 12 AABµ MAB , where MAB , A, B = 4, · · · , 9 are theSO(6) generators. The action becomes

Z
1
1
1
1
0
4 √
S=
d x gTr − Fµν F µν − Dµ0 ΦA D0µ ΦA + [ΦA , ΦB ][ΦA , ΦB ]
2
gYM
4
2
4

i
1 A
µ 0
+ ΨΓ Dµ Ψ − ΨΓ [ΦA , Ψ] ,(2.14)
22
where the covariant derivative Dµ0 includes the spin connection and the SO(6) gauge fieldX
Dµ0 ΦA := ∂µ ΦA + i[Aµ , ΦA ] +AAB(2.15)µ ΦB ,B

1
Dµ0 Ψ := ∂µ Ψ + i[Aµ , Ψ] + ΩabΓab − iAµ Ψ.
4 µ(2.16)

Here Aµ Ψ := 4i AABµ ΓAB Ψ. In order to preserve the supersymmetry, a parameter of the SUSY
transformation (2.7) should satisfy the Killing spinor equation. The twisted Killing spinor equation is


1 ab
0
Dµ  := ∂µ + Ωµ Γab − iAµ  = 0.
(2.17)
4
We choose the external gauge field Aµ in SO(2)3 ⊂ SO(6), such that the field strength,
F = dA, A = Aµ dxµ ,(2.18)

satisfiesF = −T dvol(g 6= 1)

 −T 2π dvolΣ(g = 1).

Σ

volΣ(2.19)

Here T is an SO(2)3 generator
T = a1 T1 + a2 T2 + a3 T3 ,(2.20)

where ai are parameters of twisting and Ti , i = 1, 2, 3 are generators expressed in the spinor
representationiii
T1 = Γ45 , T2 = Γ67 , T3 = Γ89 .
2
2
2
The condition for existing covariantly constant spinors is, from eq. (2.17),


1
0
0
0
23
Dµ  = 0 ⇒ [D2 , D3 ] = 0 ⇒dΩ23 Γ − idA  = 0.
2
Using the relations (2.13) and (2.19),

123
κ dvolΣ · Γ + idvolΣ · T  = 0.
2
Finally, substituting eq.(2.20), the supersymmetry condition is−κiΓ23 + a1 iΓ45 + a2 iΓ67 + a3 iΓ89  = 0.

(2.21)(2.22)(2.23)(2.24)

The amount of the supersymmetry depends on the number of the non-zero parameters among ai ,i = 1, 2, 3. Let us classify them here:41. All ai are non-zero: (−κΓ23 + a1 Γ45 + a2 Γ67 + a3 Γ89 )  = 0.
In this case the number of the supersymmetries is N = (0, 2). The constraint for theparameters ai is
a1 + a2 + a3 = κ.
(2.25)
2. Two of ai are non-zero: (−κΓ23 + a1 Γ45 + a2 Γ67 )  = 0.
In this case the number of the supersymmetries is N = (2, 2). The constraint for theparameters ai is
a1 + a2 = κ.
(2.26)
3. One of ai is non-zero: (−κΓ23 + a1 Γ45 )  = 0.
In this case the number of the supersymmetries is N = (4, 4). The constraint for thenon-zero parameter a1 is
a1 = κ.
(2.27)
4. No background field: (−κΓ23 )  = 0.
In this case the number of the supersymmetries is N = (8, 8). This situation is realized
only for the zero curvature case κ = 0, i.e. g = 1.
These results are summarized in Table 1.
# of ai 6= 0
3
2
1
0

N
(0, 2)(2, 2)(4, 4)(8, 8)

g
all
all
all
1

Table 1: Remaining supersymmetries for closed Riemann surfaces.

3

SUSY boundary condition and central charge

In this section we introduce a boundary on the Riemann surface. We assume that the boundary
is 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. We also show an attempt tofind other class of boundary conditions.

3.1

Shape of the Boundary

In this paper we focus on Riemann surfaces with one boundary. We also assume that these surfaces
have constant curvature. In this paper we only consider a geodesic boundary for simplicity. There
could 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 (x2 , x3 ) the coordinates of the Riemann surface and the geodesic boundary x3 = 0. Then5

we can choose a gauge such that locally A2 = A3 = 0 on the boundary since A is proportional
to Ω23 and we can choose the gauge Ω23 = 0 on a geodesic. Then terms including the externalgauge field Aµ in the covariant derivative (2.15) can be omitted and D0 = D is satisfied at least
locally. However we cannot ignore the holonomy along the boundary. The boundary condition
must 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 together
the Riemann surface and a copy of it with the opposite orientation to construct a closed surface
with constant curvature.
Let us see the holonomy of this external gauge field along the boundary. First for simplicity
we consider an S 2 with a boundary at the equator — a northern (or southern) hemisphere S+2(S−2 ). This holonomy is given by
I
Z
Z
A=
dA =
F = Magnetic flux.
(3.1)
2
∂S+

2
S+

2
S+

Here we use Stokes’ theorem to express it as an integral of the gauge field strength. This integral
gives a magnetic flux through the surface S+2 . Due to the Dirac quantization condition, theintegral of magnetic flux on the S 2 is an integral multiplication of 2π. Now this gauge field
is distributed isotropically. Then the integral only over the northern hemisphere (3.1) gives aninteger or a half integer times 2π. We can use the same strategy for a general Riemann surface
Σo with one geodesic boundary. Let Σ be the closed Riemann surface made by gluing Σo and a
copy of it with the opposite orientation Σo 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
 I

 Z

 Z 
Y
i
H := exp i
A = exp i
F = exp
F =exp(−iπni Ti ),
(3.2)
2 Σ
∂Σo
Σo
i=1,2,3
where ni := 2|1 − g|ai are integers [12]. Later we use the fact Z 
2
H = exp i F = 1

(3.3)

Σ

following from the Dirac quantization condition. The boundary condition considered in this paper
later (3.11) is consistent with this holonomy (3.2).

3.2

Boundary condition

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 (x3 = 0). From eq. (2.9) this condition is expressed asJ 3 = 0 ⇔ Tr0 FKL ΓKL Γ3 Ψ = 0.

(3.4)

In this condition we can replace D0 by D since we can choose the gauge where A = 0 at theboundary. Thus we can employ the same strategy as [21] (see also [24]). Define the following
6matrices:
B0 = Γ468579 ,(3.5)

B1 = Γ3468 ,(3.6)

B2 = Γ3579 ,(3.7)(X4 , X6 , X8 ) := (Φ4 , Φ6 , Φ8 ),(3.8)(Y5 , Y7 , Y9 ) := (Φ5 , Φ7 , Φ9 ).

(3.9)

and redefine the scalar fields

The boundary condition (3.4) is decomposed into the following equations as done in [21]:Tr0 (Γµν Fµν + 2Γ3µ F3µ )Γ3 Ψ = 0,
Tr0 (2Γ3a D3 Xa + Γab [Xa , Xb ])Γ3 Ψ = 0,
Tr0 (2Γ3m D3 Ym + Γmn [Ym , Yn ])Γ3 Ψ = 0,
Tr0 Γµa Dµ Xa Γ3 Ψ = 0,
Tr0 Γµm Dµ Ym Γ3 Ψ = 0,
Tr0 Γam [Xa , Ym ]Γ3 Ψ = 0,

(3.10)

where µ, ν = 0, 1, 2, a, b = 4, 6, 8, and m, n = 5, 7, 9. An example of the boundary condition is
the NS5-brane like boundary condition
D3 Xa = 0, Ym = 0, Fµ3 = 0,

(3.11)

for the bosonic fields. For the fermionic fields we impose
B2 Γ3 Ψ = −Γ3 Ψ(3.12)

at the boundary. Actually the NS5-brane like boundary conditions (3.11) and (3.12) preserve thesupersymmetry if the parameter  satisfies
B2  = .

(3.13)

The conditions (3.10) is verified. The condition (3.13) for  kills half of the supersymmetry as
follows. If an iΓ23 eigenvector 1 satisfies (2.24), B2 1 also satisfies (2.24) and they are independent.
Therefore among the linear combinations of these two independent parameters, one combination
 = (1 + B2 )1 satisfies the condition (3.13). Since 1 and B2 1 have the same chirality (Γ01eigenvalue), the preserved supersymmetry is as follows.
1. N = (0, 2) bulk ⇒ N = (0, 1).
2. N = (2, 2) bulk ⇒ N = (1, 1).
3. N = (4, 4) bulk ⇒ N = (2, 2).
Let us verify the boundary conditions (3.11) and (3.12) are consistent with the holonomy(3.2). For the vector representation (HΦ)A = ±ΦA , 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 2 = 1
and B2 Γ3 HΨ = H −1 B2 Γ3 Ψ.
7

3.3

N = (4, 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 casea2 = a3 = 0 and a1 = κ, and T in eq. (2.20) becomesi
T = κ Γ45 .
2(3.14)The preserved supersymmetry parameters satisfy eq. (2.24), which is rewritten asΓ2345  = −.

(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 QR for
the R-symmetryii
QR = Γ68 + Γ79 .
(3.16)
2
2
This is determined such that for the right moving SUSY parameters  (Γ01  = +) satisfy QR  =± and the left moving ones satisfy QR  = 0. The right moving central charge is expressed as
c = 3TrWeyl fermion (Γ01 (QR )2 ).

(3.17)In the above expression TrWeyl fermion means counting the number of the 2-dimensional Weyl
fermions.
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 Riemannsurface. We take the Riemann surface Σo and its orientation flipped one Σo , and join themS
together, Σo Σo =: Σ, so that the boundary is corresponding (See Figure 1). Originally Ψ
includes four 4-dimensional Weyl spinors. Half of them satisfying Γ6789 Ψ = −Ψ have charge ±1
of QR and the others are neutral. Let us denote these two charged 4-dimensional Weyl spinors
Ψ± which satisfy iΓ45 Ψ± = ±Ψ± and B2 Ψ± = Ψ∓ . These two fermions on R1,1 × Σo are treatedas a fermion Ψc on R1,1 × Σ. Ψc is defined as Ψ (z), (Im(z) ≥ 0)−
Ψc =(3.18) Ψ+ (z ∗ ), (Im(z) ≤ 0).

⌃

boundary of ⌃o⌃o⌃oFigure 1: (Doubling trick) We construct a closed surface Σ by taking Σo and its orientation
flipped copy Σo .
8Here we use the complex coordinate z = x2 + ix3 of Σ such that Σo is parametrized by Imz ≥ 0,
Σo is parametrized by Imz ≤ 0 and z → z ∗ is the symmetry which exchanges Σo and Σo . Actually
this Ψc is continuous at the boundary due to the boundary condition (3.12). Furthermore, we
define extended spin connections and gauge fields
Ω23 (z)A45 (z)(Im(z)≥
0)(Im(z) ≥ 0)
z̄
z̄
23
45
Ωz̄ (z) :=Az̄ (z) :=(3.19)
−Ω23 (z ∗ ) (Im(z) ≤ 0),
−A45 (z ∗ ) (Im(z) ≤ 0).
zzThen according to the above definitions, the Dirac equations for Ψ± on R1,1 × Σo are equivalent
to the one for Ψc on R1,1 × ΣΓµ Dµ0 Ψc (z) = 0.

(3.20)We denote the number of 2-dimensional right(left)-moving massless fermions by nR(L) for the
4-dimensional Weyl fermion Ψc . The index theorem gives the difference of these numbers and it
is rewritten using eqs. (2.19), (2.12) :Z
1nR − nL = −TrΨc F = t2|g − 1|,(3.21)
2π Σ
where TrΨc is taken in the representation to Ψc , and t is the eigenvalue of T for the fermion Ψcwhich is given by t = −κ/2 using eqs. (3.18), (3.14). Taking the multiplicity of the Lie algebra
into account we obtain the resultc = 3dG (nR − nL )= −3dG κ|g − 1|,(3.22)

where dG is the dimension of the gauge group. This expression gives the positive c only whenκ = −1, (g > 1). In this casec = 3dG |χo |.

(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, Σo , is χo = 2 − 2g/2 − b = 1 − g.

3.4

Candidate for other types of boundary condition

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), (2, 2) and (4, 4). We study how these supersymmetries are broken when introducing theboundary.
In this subsection we use the following notation for the SUSY parameters I , I = 1, · · · , 8.
We diagonalize Γ01 , iΓM,M +1 , M = 2, 4, 6, 8 and denote the eigenvalues as follows:((Γ01 I = λ0I I
I Γ01 = −λ0I I⇒,(3.24)iΓM,M +1 I = λM
I (iΓM,M +1 ) = −λM
I I
I I
where eigenvalues λ0I , λM
I take values +1 or −1 and are summarized in Table 2.
9

I

1
2
3
4
5
6
7
8

λ0I
−
−
+
+
−
−
+
+

λ2I
+
−
+
−
+
−
+
−

λ4I
+
−
+
−
+
−
+
−

λ6I
+
−
+
−
−
+
−
+

λ8I
+
−
−
+
−
+
+
−

Table 2: Eigenvalues of I .
3.4.1

N = (0, 2) case

The SUSY parameters preserved in the bulk are 1 and 2 .
The current condition (3.4) for these generators is
Tr0  F01 Γ01 + F23 Γ23 + F45 Γ45 + F67 Γ67 + F89 Γ89 Γ3 Ψ = 0,
Tr0  FM,N ΓM,N + FM,N +1 ΓM,N +1 + FM +1,N ΓM +1,N + FM +1,N +1 Γ

(3.25)

M +1,N +1

Γ3 Ψ = 0,(3.26)(M, N ) = (0, 2), (0, 4), (0, 6), (0, 8), (2, 4), (2, 6), (2, 8), (4, 6), (4, 8), (6, 8).
We impose the boundary condition for the fermion field:−iΓ23 Ψ = Ψ.

(3.27)

From the first equation (3.25),
Tr0 I F01 Γ01 + F23 Γ23 + F45 Γ45 + F67 Γ67 + F89 Γ89 Γ3 (−iΓ23 Ψ) = 0
↔ Tr0 I (iΓ23 ) F01 Γ01 + F23 Γ23 + F45 Γ45 + F67 Γ67 + F89 Γ89 Γ3 Ψ = 0,

(I = 1, 2).

(3.28)The lefthand side is trivially zero for 1 which satisfies 1 (iΓM,M +1 ) = −1 . For 2 this equation
gives the condition
Tr0 2 F01 − i (F23 + F45 + F67 + F89 ) Γ3 Ψ = 0.
(3.29)
Then,
F01 = 0,F23 + F45 + F67 + F89 = 0.

(3.30)For the second equation (3.26) of (M, N )=(0, 2), (2, 4), (2, 6) and (2, 8) are trivially zero for thecase of 2 in the same way and in the cases (M, N )=(0, 4),(0,6), (0, 8),(4,6), (4, 8) and (6, 8) thisequation becomes trivial for 1 .
The condition for the supersymmetry generated by I to be preserved is summarized as follows:(i) Supersymmetry generated by 1F(M = 2, 3),0,M + F1,M = 0
F2,M − F3,M +1 = F2,M +1 + F3,M = 0 (M = 4, 6, 8).
10(3.31)

(ii) Supersymmetry generated by 2F0,1 = 0, F23 + F45 + F67 + F89 = 0

F0,M + F1,M = 0(M = 4, 5, 6, 7, 8, 9),FM,N − FM +1,N +1 = FM,N +1 + FM +1,N = 0 ((M, N ) = (4, 6), (4, 8), (6, 8)).

(3.32)Let us define complex fields
Z1 := Φ1 + iΦ2 , Z2 := Φ3 + iΦ4 , Z3 := Φ5 + iΦ6 .

(3.33)We define coordinates on the 2d CFT and the Riemann surface and redefine gauge field on them.
1(A0 ± A1 ),2
1
On (x2 , x3 ) : x2 ± ix3 =: w± , Aw± := (A2 ∓ iA3 ).
2
Then, the following new derivatives can be defined:

  ∂
1
D0 ± D1 =+ [Ax± , ∗] =: Dx± ,
2
∂x±



1
∂
D2 ∓ iD3 =+ [Aw± , ∗] =: Dw± .
2∂w±
On (x0 , x1 ) : x0 ± x1 =: x± , Ax± :=(3.34)(3.35)(3.36)(3.37)

Using these notations the supersymmetry conditions (3.31), (3.32) are respectively rewritten as
follows.
1. Supersymmetry generated by 1 :(3.31) ⇒F

0M + F1M = 0 (M = 2, 3),(3.38)

Dw− ZA = 0.
2. Supersymmetry generated by 2 :(3.32) ⇒


P

i

F
01 = 0, F23 = − 2
i [Zi , Z i ],

Dx+ Zi = 0,
[Zi , Zj ] = 0.

(3.39)In the second case we find that this equation looks like a Hitchin system [33]. For more detailsof these types of equations, see [34].
3.4.2

N = (2, 2) case

The supersymmetry parameters preserved in the bulk are 1 , . . . , 4 in Table 2. In this case we
can use the same method to the previous N = (0, 2) case. The normal component of the current
satisfies:
Tr0  F01 Γ01 + F23 Γ23 + F45 Γ45 + F67 Γ67 + F89 Γ89 Γ3 Ψ = 0,(3.40)

Tr0  FM,N ΓM,N + FM,N +1 ΓM,N +1 + FM +1,N ΓM +1,N + FM +1,N +1 ΓM +1,N +1 Γ3 Ψ = 0.
(3.41)
11

The first equation (3.40) becomes trivial for I having eigenvalue λ2I = +1 in the same way to
N = (0, 2) case and for I having eigenvalue λ2I = −1 this equation becomes
F01 = 0,4X

2i,2i+1
λ2i
= 0.
I F

(3.42)

i=1

The second equation (3.41) splits into two groups(M, N ) = (0, 2), (2, 4), (2, 6), (2, 8),(M, N ) = (0, 4), (0, 6), (0, 8), (4, 6), (4, 8), (6, 8).
The former becomes trivial for λ2I = −1 and the latter becomes trivial for λ2I = +1. The nontrivial
conditions are for λ2I = +1
F02 − λ0I F12 = F03 − λ0I F13 = 0,(3.43)N
M
N
FM,N − λM
I λI FM +1,N +1 = +λI FM +1,N + λI FM,N +1 = 0(M, N ) = (2, 4)(2, 6)(2, 8),(3.44)

and for λ2I = −1
F0,M − λ0I F1,M = 0,

M = 4, 5, 6, 7, 8, 9,(3.45)N
M
N
FM,N − λM
I λI FM +1,N +1 = λI FM +1,N + λI FM,N +1 = 0(M, N ) = (4, 6)(4, 8)(6, 8).

(3.46)

Summarizing the above, the supersymmetries generated by I are respectively as follows:
1. I (λ2I = +1)F0(M = 2, 3)
0,M − λI F1,M = 0
FM,N − λM λN FM +1,N +1 = λM FM +1,N + λN FM,N +1 = 0 (M, N ) = (2, 4)(2, 6)(2, 8),
I
I
I
I(3.47)
2. I (λ2I = −1)F01 = 0,P42i 2i,2i+1=0
i=1 λI F

F0,M − λ0I F1,M = 0(M = 4, 5, 6, 7, 8, 9)FM,N − λM λN FM +1,N +1 = λM FM +1,N + λN FM,N +1 = 0 (M, N ) = (4, 6)(4, 8)(6, 8).
I
I
I
I(3.48)

The case we studied before in the subsection 3.4.1 corresponds to the case of λ0I = −1
(eqs.(3.47)) in the current case.

12

3.4.3N = (4, 4) case

The supersymmetry parameters preserved in the bulk are 1 , . . . , 8 in Table 2. The conditions
for the bosonic fields are
F0,1 = 0,4X

λ2i
I F2i,2i+1 = 0,

(3.49)

i=1

where I = 2, 4, 6, 8 and
F0M − λ0I F1M = 0,(3.50)

where M = 2, 3 for I = 2, 4, 6, 8 while M = 4, 5, 6, 7, 8, 9 for I = 1, 3, 5, 7, andN
N
M
FM,N − λM
I λI FM +1,N +1 = λI FM,N − λI FM +1,N +1 = 0,(3.51)

where (M, N ) = (2, 4), (2, 6) and (2, 8) for I = even, while (M, N ) = (4, 6), (4, 8) and (6, 8) for
I = odd. In the case of N = (2, 2) after the breaking is an interesting case and the central charge
is 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. was
supported in part by JSPS Research Fellowship for Young Scientists and JSPS KAKENHI Grant
Number 13J02068. This work was supported in part by World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.

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