SciPost Phys. 11, 077 (2021)

A domain wall in twisted M-theory
Jihwan Oh1* and Yehao Zhou2
1 Mathematical Institute, University of Oxford,
Woodstock Road, Oxford, OX2 6GG, United Kingdom
2 Perimeter Institute for Theoretical Physics,
31 Caroline St. N., Waterloo, ON N2L 2Y5, Canada? jihwan.oh@maths.ox.ac.uk

Abstract
We study a four-dimensional domain wall in twisted M-theory. The domain wall is engineered by intersecting D6 branes in the type IIA frame. We identify the classical algebraof operators on the domain wall in terms of a higher vertex operator algebra, which
describes the holomorphic subsector of a 4d N = 1 supersymmetric field theory, andcompute the associated mode algebra. We conjecture that the quantum deformationof the classical algebra is isomorphic to the bulk algebra of operators from which weestablish twisted holography of the domain wall.
Copyright J. Oh and Y. Zhou.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 21-05-2021
Accepted 22-09-2021
Check forupdatesPublished 14-10-2021
doi:10.21468/SciPostPhys.11.4.077

Contents
1 Introduction

2

2 Twisted M-theory
2.1 Background
2.2 Compatibility of M2 and M5 branes in twisted M-theory
2.3 KK monopole, intersecting D6 branes and a domain wall

4
4
6
6

3 4d VOA in the 5d Chern-Simons theory
3.1 Holomorphic twist of 4d N = 1 theory
3.2 Higher vertex operator algebra
3.3 The boundary condition of the 5d gauge field
3.4 Physical observables on the domain wall
3.5 5d/4d system and anomaly cancellation

9
9
12
14
14
16

4 Algebra of operators on the domain wall
4.1 The commutation relation
4.2 The large N limit
4.3 Conjecture

17
17
19
19

5 Conclusion and open questions

20
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SciPost Phys. 11, 077 (2021)

A 4d N = 1 SUSY transformation of a chiral multiplet

22

B Mode algebra computation

23

References

26

1

Introduction

Topological twists [1, 2] have been a standard tool to study a supersymmetric quantum field
theory along with Ω−deformation [3, 4]. Since the twists make the translation generator Qexact with respect to the scalar supercharge Q of the twisted theory, the Q-cohomology class ofa Q-closed local operator is independent of its position. One can slightly generalize the notionof the topological twist and discuss the holomorphic version. In this case, the Q-cohomology
class only retains the holomorphic dependence. By construction, the resulting holomorphic
field theory needs to be even dimensional. Outstanding examples include the 2d (0, 2) theory[5–7] and 4d N = 1 theory [8–10]. The protected subsector of the 2d example naturally
forms a familiar vertex operator algebra. On the other hand, the holomorphic sector of 4d
N = 1 theory reorganizes itself into a higher vertex operator algebra (higher VOA) [11–14],
which will be the main player of this paper. Higher VOA naturally appears in twisted Mtheory [15, 16], where there are 4 holomorphic directions and 7 topological directions.
Twisted M-theory [16,17] is an eleven-dimensional supergravity background specified by a
product geometry of a hyperKähler 4-manifold and a 7-manifold with G2 -holonomy, equippedwith the above mentioned twist and Ω−deformation. The hyperKähler 4-manifold is equippedwith the holomorphic twist and the G2 -manifold is equipped with the topological twist, alongwith the Ω−deformation. Due to the Ω-deformation, the M-theory degree of freedom localizes to the 5d Chern-Simons theory, which is topological in one direction and holomorphic infour directions; it comes from Taub-NUT geometry inside the G2 −manifold, or a KK-monopole.
Once we introduce M2 and M5 branes, the worldvolume theories of M2 and M5 branes are
twisted and deformed(and localized to the subspace of the spacetime). To preserve supersymmetry and to be compatible with the twisted background, the M2 branes should wrap athree-cycle in the G2 manifold and the M5 branes should wrap a four-cycle in the G2 manifold.
The supersymmetric coupling between the membranes and the supergravity localizes to thecoupling between defects and the 5d Chern-Simons theory; the M2 branes and M5 branes giverise to line and surface defects.
Twisted holography in the context of M-theory is an exact isomorphism between the algebra of operators of the 5d Chern-Simons theory and the algebra of operators on membranes.
The relevant algebras are affine g l(1) Yangian [18, 19] and its shifted and truncated version [20–24]. The isomorphism is induced by Koszul duality,1 which is a common name fora duality between a certain pair of algebras. Conjecturally, twisted holography describes theprotected subsectors of well-established M2 and M5 brane holography [27,28]. We will discuss
the relevant background for twisted M-theory in §2.1, §2.2.
One can also consider a network of M2 and M5 branes [24, 29] and identify the relevant
algebra [30]. Following the idea of [17], where the author beautifully derived the M2 brane
algebra from the perturbative computation in the 5d Chern-Simons theory, one can reproduce
the algebra associated to the network of M2 and M5 branes [31].
1

For another nice set of physical examples of Koszul duality not in the context of twisted holography, see [25,26].

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SciPost Phys. 11, 077 (2021)

Twisting supergravity is not necessarily restricted to the eleven-dimensional example. Indeed, the most complete example for twisted holography is [32] the twisted type IIB theory
and a protected subsector of D3 brane holography. See also [33] as an example of a D2-D4
system and [34] as an example of a D1-D5 system. Literatures that employ direct localizationin both supergravity and field theory include [35, 36].
In this paper, we study the seemingly last available BPS defect in the twisted M-theory thathas not been not studied: an extra KK-monopole, which is different from the KK monopolethat descends to the 5d Chern-Simons theory. The new KK-monopole creates a domain walllike defect wrapping the 4 holomorphic directions in the 5d Chern-Simons theory. Different
from the line and surface defects engineered by membranes, the domain wall-like defect isengineered by changing the G2 -holonomy 7-manifold [37, 38]. It passes tight constraints forthe allowed BPS objects in twisted M-theory. (1) It preserves some amount of supersymmetry.
(2) It is compatible with the topological holomorphic background. (3) It preserves the G2 holonomy structure. (4) It preserves the U(1) isometry of the G2 manifold.
For a practical computation and convenient visualization, it is better to work in the type
IIA frame by reducing along a circle in the G2 manifold. In type IIA, we obtain two sets ofintersecting D6 branes [39], one set from the original Taub-NUTK geometry and the other set
from the change we made. The original K D6 brane worldvolume theory gives the 5d G = U(K)
Chern-Simons theory and the new N D6’ brane intersects with the 5d Chern-Simons theory
as a domain wall. On the wall, D6-D6’ strings are localized and realize the 4d N = 1 chiralmultiplet [39]. Due to the holomorphic twist on the wall, we get a higher VOA. We will give
more details about the intersecting D6 brane configuration in §2.3.
The major difference between the higher VOA coming from the holomorphic twist and thefamiliar 2d VOA is that in the former case the descent operators play the key role to generatemeromorphicity, which is the essential ingredient for the algebraic structure under the vertex
operator algebra. Like the usual cohomological field theory, the set of Q-cohomology classes
in the higher VOA includes a certain integrated version of the descent operators and in ourcase they are holomorphic descents of local Q-cohomology classes. The holomorphic descentsthemselves are not Q-closed, but combining with the holomorphic top form of the 4d spacetime
manifold C2 and integrating over S 3 ⊂ C2 , it becomes a Q-cohomology class. We will explainin more detail about this set of operators and how it generates the higher VOA in §3.1, §3.2.
Since the domain wall forms a boundary of the 5d Chern-Simons theory, we need to account forthe boundary condition of the 5d Chern-Simons gauge field, which we did in §3.3. Collecting
these ingredients, we propose the physical observables living at the domain wall in §3.4. The
5d/4d system that we have discussed is anomalous due to the chiral fermion at the domain
wall. It can be canceled by tuning a parameter of the 5d Chern-Simons theory on one side of
the wall, as we will discuss in §3.5.
In §4, we will compute the commutation relation of the physical observables that we haveconstructed in the previous section. We prove that the resulting algebra is the universal enveloping algebra of Diffε2 C⊗ glK . We conjecture that the quantum deformation of our classicalcomputation is isomorphic to the Koszul dual algebra of operators in the 5d Chern-Simons theory. A priori, this conjecture must be true due to the powerful theorem by Costello [17]: there
is a unique deformation of U(Diffε2 C ⊗ glK ) and it is the algebra of operators in the 5d ChernSimons theory. In this sense, the proof for the conjecture is already given. However, we preferto call it the conjecture since we do not know how to systematically incorporate the quantum
corrections only using our machinery but not relying on the Costello’s theorem.
We conclude in §5 by summarizing the discussion and provide open questions that we wantto answer in the future. In Appendix, we provide details of some computations that we haveabbreviated in the main text. We collect relevant 4d N = 1 supersymmetry transformation ofa chiral multiplet in Appendix A. Lastly, we provide details of the key commutation relation

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SciPost Phys. 11, 077 (2021)

computation in Appendix B.

2

Twisted M-theory

In §2.1, we will review how to define a twisted supergravity and its deformation under theΩ−background. We will focus on the 11-dimensional example, which we call the twisted
M-theory. Due to the Omega background, the bulk dynamics drastically simplifies to a noncommutative version of the 5d Chern-Simons theory, which is topological in 1 direction andholomorphic in 4 directions. We will briefly explain the algebra of operators in the 5d ChernSimons theory. In §2.2, we will explore various BPS objects in the twisted M-theory and study
their compatibility with the twisted background. Finally, in §2.3, we identify the BPS object in
M-theory that realizes the domain wall in the 5d Chern-Simons theory.

2.1

Background

Twisted supergravity [15] is defined as the supergravity in a background where the bosonicghost Ψ of the local supersymmetry takes a nonzero value. To satisfy the supergravity equationof motion, Ψ needs to square to zero, i.e., it is nilpotent. In the presence of U(1) isometry inthe background geometry, we may turn on the Ω-background for the twisted supergravity
background X and deform it to X ε . X ε is equipped with Ψε , which squares not to zero but toεV , where V is the vector field generating the S 1 action.
In [16] Costello defines the twisted and Ω−deformed 11-dimensional supergravity andproves that it satisfies the equation of motion of the 11-dimensional supergravity. We will callthis background twisted M-theory. The twisted M-theory background2 is specified by the triple(Ψε , g, C), which are a bosonic ghost, a metric, and an M-theory 3-form. The 11-dimensional
manifold is a product of a 7-manifold M7T with G2 holonomy and a hyperKähler 4-manifold
M4H parametrized by z and w. The background induced by Ψε makes the dependence on M7T
topological and the dependence on M4H holomorphic.
The background geometry is given by(Cz × Cw )H × (R t × C × TNK )T .

(1)

TNK is the Taub-NUTK manifold, which can be thought of as a circle(S T1 N ) fibration over thebase R3 . There is an Sε1 × Sε1 action on C × TNK . The first Sε1 action, parametrized by ε1 ,1
2
1
acts on C and the base of the Taub-NUT manifold simultaneously. We will therefore put thesubscript ε1 on the C and denote it as Cε1 . Also, we will denote the Taub-NUT base R3 as
R3T N B . The second Sε1 action, parametrized by ε2 , acts on the Taub-NUT circle. Finally, the2

S 1 × S 1 action preserves the holomorphic volume form of Cε1 × TNK .
For a practical computation, it is helpful to reduce along S T1 N to go to type IIA theory. Notethat we do not lose essential information by the circle reduction, since it is a Q-exact operation, as proven in [16]. Under the reduction, the M-theory geometry produces K D6 branes.
Furthermore, due to the localization effect of the Omega background on Cε1 , the 7d maximal
SYM of D6 brane worldvolume reduces to the 5d Chern-Simons theory on R t × Cz × Cw :
Z


1
2
dz ∧ d w ∧ AdA + A ∧ε2 A ∧ε2 A ,(2)ε1 R ×C ×C
3
t

z

w

Appendix A of [16] contains the exact expression of the background and the proof for it to be a consistentsupergravity background.
2

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SciPost Phys. 11, 077 (2021)

where ∧ε2 is a combination of a wedge product and a Moyal product ?ε2 . The Moyal productbetween two holomorphic functions is defined as1∂∂
f ?ε2 g = f g + ε2 εi jf
g + ...
2 ∂zi ∂z j

(3)

Due to the holomorphic top form in the action, the 5d gauge field has only three components:
A = A t d t + Az̄ dz̄ + Aw̄ d w̄.

(4)

The origin of the non-commutativity that induces the Moyal product is the B-field thatdescends from the M-theory 3-form C-fieldC = dz ∧ d w ∧ V [ ,(5)where V is a vector field that generates the rotation on S T1 N and V [ is the corresponding dual
TN νTN
1-form, whose component is given by Vµ[ = gµν
V , where gµν
is the Taub-NUT metric. We

reduce along S T1 N when we pass to the type IIA frame.
The reason that we may entirely focus on the open string part(the D6 brane worldvolume
theory) ignoring the closed strings from the geometry is that the B-field combines with the
B-model background(holomorphic) on Cz × Cw and effectively produces the A-model background(topological). As we already have A-model background in the other 6 directions,
R t × Cε1 × R3T N B , all 10 directions in type IIA are topological. In other words, physical closed
string observables in the relevant Q-cohomology have a trivial dependence on spacetime coordinates. Hence, they can be ignored [16].
Let us discuss the algebra of operators in the 5d Chern-Simons theory: Obs5d
ε ,ε . We will
1

2

first describe the case with ε1 = 0, Obs5d0,ε2 and turn on the deformation parameter ε1 later.
Since the equation of motion of the 5d Chern-Simons theory is F = 0, all operators have apositive ghost number. The ghosts have a trivial dependence on t and holomorphic dependence on z and w. Due to the non-commutative background on Cz × Cw , [z, w] = ε2 , theoperators form a non-commutative algebra of holomorphic functions on Cz × Cw . Together
with the BRST differential δ, the algebra of operators of the 5d Chern-Simons theory forms agraded associative algebra isomorphic to∧∗ C[z, w]ε2 ⊗ glK ∼= ∧∗ Diffε2 C ⊗ glK ,(6)

where Diffε2 C is the algebra of differential operators on Cz with [z, ∂z ] = ε2 . In other words,∗
the classical algebra Obs5d0,ε2 is the Chevalley-Eilenberg algebra C (g) of cochains on the Liealgebra g = Diffε2 C⊗glK . A Koszul dual algebra of the Lie algebra cochain C ∗ (g) is the universal
enveloping algebra U(g).
3When ε1 6= 0, Obs5d0,ε receives quantum corrections and deforms into an A∞ algebra
2Obs5d
ε1 ,ε2 and we denote the Koszul dual of it as Uε1 (g). Costello showed that U(g) has a nontrivial(section 9 of [17]) and unique(section 15 of [17]) deformation Uε1 (g) and identified itwith the algebra of operators on N M2 branes in the large N limit.
As long as we do not break the supersymmetry and respect the topological holomorphic
twist of the twisted supergravity, we can introduce M2 and M5 branes that inherit all twists inthe background. They span the following directions in the twisted M-theory:
M2: R t × Cε1 ,M5: {0} × TNK × Cz .
3

For a physical description of A∞ algebra, see [29].

5(7)SciPost Phys. 11, 077 (2021)

Each of the membranes supports an interesting quantum field theory [16, 17], which is a
twisted subsector of 3d N = 4 ADHM gauge theory and 6d N = (2, 0) SCFT, respectively. Dueto the Omega background applied to each of the worldvolume, the field theories that encode
the protected subsector are localized to R t and Cz :M2: Topological quantum mechanics on R t ,M5: Free fermion vertex operator algebra on Cz .

(8)

In the limit of a large number of membranes, the algebra of operators on the M2 branes is 1shifted affine g l(1) Yangian with one central element eliminated and the algebra of operatorson the M5 branes is W∞ algebra, or affine g l(1) Yangian.
Note that in some literature, Cε1 ×TNK part of the twisted M-theory background is replacedwithCε2 × Cε3(9)
Cε1 ×, where ε1 + ε2 + ε3 = 0.
ZKThis presentation is useful to explain a triality automorphism of the algebra associated with
the M2 and M5 branes(see [29], for instance). However, we will follow the original presentation [16], which did not show ε2 , ε3 subscripts explicitly, since the triality does not play animportant role in our discussion.

2.2

Compatibility of M2 and M5 branes in twisted M-theory

Let us comment on the compatibility of the M2 and M5 branes and the 11d topological holomorphic background and explain why these two BPS defects cannot engineer the domain walllike defect, which is the main object of this paper.
M2 branes are entirely embedded in M7T . It preserves 3d N = 4 supersymmetry in thepresence of the Taub-NUT geometry. There are two types of topological twists available in 3d:
Rozansky-Witten twist [41] and the mirror version of it. Both twists are compatible with thetwisted supergravity background and it is useful to work in either of the two twists to give acomplete picture of the algebra of operators on the M2 branes. Clearly, the three-dimensional
M2 branes cannot engineer the four-dimensional domain wall.
Next, out of the 6 directions of M5 brane worldvolume, 4 directions lie in M7T and 2
directions are in M4H . There are two possible twists available to the 6d (2, 0) theory.4 First, a
topological holomorphic twist that gives 4 topological directions and 2 holomorphic directions.
Second, a holomorphic twist that gives all 6 directions holomorphic. Since we only have 4
holomorphic directions in the twisted M-theory background, only the first option is feasible
and this background exactly applies to the M5 branes that appear in (8). In other words,
since the 6d (2,0) theory does not admit a twist generating 2 topological and 4 holomorphic
directions, we are not allowed to use M5 branes to engineer the domain wall-like defects inthe 5d Chern-Simons theory, which require the M5 branes to wrap M4H .
There are remaining BPS objects in M-theory: the M9 brane and the KK monopole. The
KK monopole wraps different directions to those of the already existing KK monopole, whichis equivalent to the Taub-NUT geometry.

2.3

KK monopole, intersecting D6 branes and a domain wall

We would like to engineer a codimension-1 defect in the 5d Chern-Simons theory from theremaining BPS objects. However, the M9 brane only admits a holomorphic twist (on the entire
4

We are grateful to Kevin Costello for the discussion on this point.

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SciPost Phys. 11, 077 (2021)

10 directions) on its worldvolume, so we exclude the M9 brane since it is not compatible withthe 7 topological directions. Next, let us consider
New KK monopole: Cz × Cw × R3 ,(10)

where R3 ⊂ M7T (we will specify the R3 more precisely later in the IIA frame.) Since the 7d
SYM on the worldvolume of the KK monopole admits a topological holomorphic twist thatinduces 3 topological and 4 holomorphic directions, like the original KK monopole associatedwith T NK , the new KK monopole is compatible with the twisted background.
Let us now argue that the new KK monopole is compatible with the other necessary conditions for twisted M-theory: 1. G2 -holonomy, 2. Sε1 × Sε1 action.
1
2
First, the multi(original plus new) KK monopole configuration corresponds to a singularG2 -holonomy manifold [37, 38], if we tune R3 to intersect with R t × Cε1 at one point. If
there are N new KK monopoles, the singularity of the G2 -holonomy manifold represents theunfolding of an AN +K singularity into AN and AK singularities [42]. More precisely, it is a coneÝT .
on a weighted projective space WCP3N ,N ,K,K [37, 38]. Let us call this new geometry M
7
ÝT admits S 1 ×S 1 isometry. The original Taub-NUT circle S 1
Second, we will argue that M
7

ε1

ε2

TN

ÝT has an S 1 isometry that rotates S 1 .
is intact by the introduction of new KK monopoles. M
7
ε
TN
2

Indeed, when we go to the type IIA frame, we take this circle as the M-theory circle to reduce.5
Next, it is helpful to go to the type IIA frame to see the presence of Sε1 action. Reducing along
1

S T1 N , we map two types of KK monopoles in the M-theory to K D6 and N D6’ branes in thetype IIA frame that wrap the 4 holomorphic directions and 3 topological directions:
D6: Cz × Cw × R t × Cε1 ,D6’: Cz × Cw × R3 .

(11)

In general, such a D6-D6’ configuration does not preserve supersymmetry. However, if we alignthem to make certain angles, we can preserve 4d N = 1 supersymmetry on Cz ×Cw [39]. MoreÝT /S 1 ) into three 2-planes P t , P1 , P2 .
precisely, we may split the 6 topological directions(M7TN
Then, we embed each Ri of R t × Cε1 = R t × R1 × R2 of D6 branes in Pi , respectively. We
parametrize Ri with x i and the orthogonal axis in Pi with yi . Next, we orient each R0i of
R3 = R0t × R01 × R02 of the D6’ branes in Pi to make an angle ϕi with Ri such that [39]ϕ t ± ϕ1 ± ϕ2 = 0 mod 2π(12)for some choice of signs.

Figure 1: P1 and P2 . Each of them is parametrized by (x 1 , y1 ) and (x 2 , y2 ) respectively.
5
Historically, the brane configuration, which we get after the circle reduction in type IIA, was first discovered[39] and M-theory lift was introduced later [37, 38].

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SciPost Phys. 11, 077 (2021)

ÝT /S 1 ∼ R t × Cε × R3 .
Given the supersymmetry, let us check the Sε1 isometry in M
7
TNB
1
1

Recall that we took T NK ≈ S T1 N × R3T N B . In the present case, it suffices to show that the metricon both D6 and D6’ branes is invariant under Sε1 action that simultaneously rotates Cε1 ×R2t nb ,1

where R2t nb is a 2-plane embedded in R3T N B .
Let us first consider two independent rotations, parametrized by θ1 , θ2 , respectively. Theyact on the coordinates (x 1 , x 2 ) of Cε1 and ( y1 , y2 ) of R2t nb as follows 
cos θ1 x 1 − sin θ1 x 2
x1,→
sin θ1 x 1 + cos θ1 x 2
x2




 
cos θ2 y1 − sin θ2 y2
y1.
→
sin θ2 y1 + cos θ2 y2
y2

(13)

The relevant part of the metric on the original D6 branes is preserved trivially under the rotation:
ds2 = d x 12 + d x 22 → d x 12 + d x 22 .
(14)
If we denote a = tan ϕ1 , b = tan ϕ2 , we may write down the relevant part of the metric on
the new D6’ branes as
ds2 = d(x 1 + a y1 )d(x 1 + a y1 ) + d(x 2 + b y2 )d(x 2 + b y2 )= d x 12 + d x 22 + a2 d y12 + b2 d y22 + a(d x 1 d y1 + d y1 d x 1 ) + b(d x 2 d y2 + d y2 d x 2 ).

(15)

For (15) to be invariant under (13), a, b, θ1 , θ2 need to satisfy the following set of equations:a2 sin2 θ2 + b2 cos2 θ2 = b2 ,a2 cos2 θ2 + b2 sin2 θ2 = a2 ,(a2 − b2 ) cos θ2 sin θ2 = 0,a cos θ1 cos θ2 + b sin θ1 sin θ2 = a,(16)

a sin θ1 sin θ2 + b cos θ1 cos θ2 = b,a cos θ1 sin θ2 − b sin θ1 cos θ2 = 0,a sin θ1 cos θ2 − b cos θ1 sin θ2 = 0.
There is an obvious solution:a = b and θ1 = θ2 .

(17)

In principle, a = b can be anything, but to be consistent with Figure 4, let us choose them
to be 1. As this condition fixes ϕ1 = ϕ2 = π/4, by (12), we can also fix ϕ t . Since we wantto preserve exactly N = 1 supersymmetry on the domain wall, ϕ t should not be 0,6 but any
value, which satisfies (12). The existence of the solution (17) guarantees the U(1)ε1 isometry,
which simultaneously(θ1 = θ2 ) rotates Cε1 × R2t nb .
Now that we have shown the compatibility of the new KK monopoles with the twisted Mtheory, let us discuss the degree of freedom that they introduce. It is helpful to stay in the type
IIA frame to do so. There are D6-D6’ strings localized at the 4d intersection Cz × Cw . The low
energy mode of the quantization of D6-D6’ strings corresponds to the 4d N = 1 massless chiral multiplet [39], charged under both Chan-Paton symmetries associated with the D6 and D6’
branes. We can also understand this matter content as off-diagonal components in the decomposition of AN +K singularity into AN and AK singularities in the M-theory [37, 38, 42]. Since
the worldvolume theory of D-branes probing the twisted supergravity background inherits thetwist, the D6-D6’ string or the 4d N = 1 chiral multiplet on M4H = Cz × Cw is holomorphicallytwisted.
6
In this case, the D6 and D6’ branes intersect in 5 directions, which is not of our interest. In fact, configuration
with ϕ t = 0 and its M-theory uplift was discussed in [37, 38], but another set of KK monopoles other than the two
sets of KK monopoles should be considered in that case.

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SciPost Phys. 11, 077 (2021)

Finally, as we have seen in §2.1, because of the Omega background on the 2-plane R2t nb
inside the N D6’ branes, the theory on the N D6’ branes localizes on another copy of the
5d U(N ) CS theory. However, this theory will not play an important role in the following
discussion, as we will focus on the 4d intersection.

34d VOA in the 5d Chern-Simons theory

To construct a set of physical operators on the domain wall, we need to collect all fields, whichlive on the domain wall. In §3.1, we will describe the first ingredient, the holomorphicallytwisted 4d N = 1 chiral multiplet coming from the D6-D6’ strings. We can recast the 4d holomorphic field theory in a first-order formalism, called a twisted formalism in [44], and findthat the 4d Lagrangian resembles that of the 2d βγ system. The authors of [14] derived the 4dβγ system. In §3.1, §3.2, we will translate and explain this mathematical notion in the morefamiliar language to physicists using holomorphic descent, which is an analog of the topological descent defined in Witten’s classical paper [1]. This is in the line of [43–45], where theauthors defined the topological holomorphic descent and the product between a local operator
and a descent operator, which is called a secondary product. In particular, see the beautiful
exposition in the paper [45] by Beem, Ben-Zvi, Bullimore, Dimofte, and Neitzke. In §3.3, we
identify the second ingredient: the boundary condition of the 5d Chern-Simons gauge field on
the domain wall. Combining these ingredients, we construct the physical operators living on
the domain wall in §3.4. The 4d βγ system on the domain wall, however, is anomalous. The
anomaly cancelation condition is a shift of the non-commutativity parameter ε2 on one sideof the domain wall, compared to that on the other side, as we show in §3.5.

3.1

Holomorphic twist of 4d N = 1 theory

Let us study the D6-D6’ strings that are localized on the D6, D6’ intersection Cz × Cw .
The D6-D6’ strings give the 4d N = 1 chiral multiplet Φ = (φ, ψα ) and the D6’-D6strings give the complex conjugate Φ̄ = (φ̄, ψ̄α̇ ), where α and α̇ are spinor indices ofSU(2)l × SU(2) r = SO(4) Lor entz . Since branes moving in the twisted supergravity backgroundinherit the twist of the background, we get the holomorphically twisted 4d N = 1 chiral multiplet theory. We will define and study the holomorphic twist of 4d N = 1 theory in thissubsection.
4d N = 1 supersymmetry algebra is generated by four supercharges Q α and Q̄ α̇ :Q α , Q β = [Q α̇ , Q β̇ ] = 0,(18)µ[Q α , Q β̇ ] = σ Pµ ,αβ̇where µ is the 4d spacetime index, σ0 = −Id, and σ1,2,3 are Pauli matrices. The graded
commutator7 [ , ] is defined by [a, b] = a b − (−1) F (a)F (b) ba, where F (a) is a fermion numberof a.
Under the holomorphic twist, the Lorentz symmetry algebra is redefined in such a way thatthe supersymmetry generators are reorganized into a scalar Q and a 1-form Q under the new
Lorentz symmetry algebra. They satisfy the following relations:
Q2 = 0,[Q, Q] = i Pz̄ dz̄ + i Pw̄ d w̄.

(19)7
We unify the convention for the commutator to distinguish it from the secondary product { , }, which we willsoon introduce.

9

SciPost Phys. 11, 077 (2021)

We may solve (19) utilizing (18) and find the scalar supercharge
Q = Q−,(20)

which defines the Q-cohomology and the 1-form supercharge Q, which anti-commutes with Q
to produce anti-holomorphic translation generators Pz̄ , Pw̄ :Q = Q̄ +̇ dz̄ − Q̄ −̇ d w̄.

(21)

Given a local operator O, one may act with Q n-times to obtain an n-form operator O(n)Qn O = O(n) .

(22)

There are two kinds of operators in the Q-cohomology. The first type is a local operator
O(z). Examples of the first type operator in 4d N = 1 chiral multiplet theory are
φ̄,

ψ+ ,

(23)

including all of their holomorphic derivatives with respect to ∂z and ∂w . See Appendix A for thederivation. Even though ψ+ now transforms as (2, 0) form under the new Lorentz symmetryof the twisted theory, we will keep the notation and not explicitly show the differential formin the later discussion for a concise presentation. We will also do so for the other componentsof the spinor ψ− and ψ̄α̇ , which are a scalar and (0, 1)-form under the new Lorentz symmetry.
The product of two first-type operators

O1,2 (z1 , z2 ) = O1 (z1 )O2 (z2 )(24)

is again in the Q-cohomology. By Hartog’s theorem, in d ≥ 2-complex dimension, every function on U ⊂ Cd that is holomorphic away from singularities can be uniquely extended to aholomorphic function on the entire U. Therefore, there cannot be a singular OPE between
two local operators. This is in contrast with a complex one-dimension holomorphic theory,
which is the usual vertex operator algebra, where one obtains interesting mode algebras ofvarious holomorphic currents due to the singular OPE’s between the currents.
The second type of operator in the Q-cohomology is more interesting. We can construct
it by first applying the 1-form supercharge to the first-type operator. The resulting O(1) is not
Q-closed, but we can check(25)
QO(1) = ∂¯ O.
If we then wedge O with the holomorphic top form Ω of C2 , we get
Q(ΩO(1) ) = d(ΩO).

(26)

By integrating both sides along S 3 , which does not have a boundary, we get a Q-closed operator
at x
Z

Ω ∧ O(1) = 0,

Q

(27)

S 3x

where S 3x is a 3-sphere centered at x. It may look nonlocal, as it involves the integral overS 3 , but it is a local operator at x after the integral. We can avoid a potential intersection withother second-type operators centered at different points by shrinking S 3 ’s. For two secondtype operators sharing the same center, there may be a nontrivial algebraic structure. We will
describe more the algebra in the next subsection.

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SciPost Phys. 11, 077 (2021)

Examples of the second-type operator in 4d N = 1 chiral multiplet theory areZZ
dzd w φ̄ (1) =dzd w (ψ̄+̇ dz̄ − ψ̄−̇ d w̄),

S3

S3

Z
S3

(1)
dzd w ψ+ =Z

(28)
dzd w (∂w φdz̄ − ∂z φd w̄).

S3

We have collected the supersymmetry transformations of the components of the 4d N = 1
chiral multiplet under Q and Q in Appendix A.
Furthermore, we can define a secondary product between the first-type operator and thesecond-type operator by wrapping the first-type operator with a 3-sphere that is used to define
the second-type operatorZ(1){O1 , O2 }(z0 , w0 ) =Sz3 ,w (z,w)
0

O1 (z0 , w0 ) O2 (z, w)dzd w,

(29)

0

where Sz3 ,w (z, w) is a 3-sphere centered at (z0 , w0 ) and parametrized by two complex coordi0 0nates z, w. Here O1 and O2 are cochain representatives and the secondary product does notdepend on the choice of the representatives, since Q-exact deformation of Oi will not change
the result.

Figure 2: It illustrates a product between the first and second-type operators. We(1)place O1 at the origin and wrap it with ΩO2 , which is supported on the 3-sphereS 3 , parametrized by z, w.
Again, {O1 , O2 }(z0 , w0 ) is a local operator in the Q-cohomology, supported at (z0 , w0 ). To(1)
compute the secondary product, we need to make a contraction between O1 and O2 . Hence,
the result of the secondary product is in general a first-type operator. It is the remaining
operator, which is not contracted. More details of the secondary product along with the studyof the holomorphic twist of a general 4d N = 1 gauge theory will be discussed in [51].
It is helpful to write down the Lagrangian for the 4d N = 1 chiral multiplet to extract the
propagators used in the contraction:Z∂ φ ∂¯ φ̄ + ψ+ ∂z̄ ψ̄−̇ + ψ+ ∂w̄ ψ̄+̇ + . . .
(30)
Cz ×Cw

From the kinetic terms, we read off the nonvanishing propagators that involve the first-type
operators in the Q-cohomology:〈φ̄∂ φ〉 ∼ 〈ψ+ ψ̄〉 ∼
11

ωBM
,
dzd w

(31)

SciPost Phys. 11, 077 (2021)

where ωBM is Bochner-Martinelli kernel
ωBM = dzd w

z̄d w̄ − w̄dz̄
.
|z|2 + |w|2

(32)

The Bochner-Martinelli kernel is a four-dimensional analog of the S 1 residue integral measuredz
z , used in 2d VOA mode algebra computation. It has the following nice property that willplay an important role in the next subsection:ZωBM f (z, w) = f (z0 , w0 ).

Sz3 ,w
0

3.2

(33)

0

Higher vertex operator algebra

Before we discuss the 4d VOA, let us recall the familiar example of the OPE and residue pairingin a simple 2d VOA; we will find the analogous properties in the 4d soon. Consider 2d βγ
system, which has the following OPE:
β(z) · γ(0) ∼1+ ...,
z(34)

where . . . are regular terms. Then, we may compute the following integralZZdz
dz(β(z) · γ(0)) =
= 1,
S1 z
S1

(35)

0

0

by using the familiar residue formula.
Similar to 2d, in 4d VOA we form an “OPE" between the first-type and second-type operators(1)(36)
O1 (z0 , w0 ) · O2 (z, w).
Note that a more general OPE in holomorphically twisted field theory was discussed in [11].
Next, we can compute the S 3 residue integral, using the nontrivial propagators defined in
(31):
Z
(1)

Sz3 ,w
0 0

dzd w O1 (z0 , w0 ) · O2 (z, w).

(37)This is precisely the secondary product {O1 , O2 }, which we defined in (29).
Specializing to the 4d N = 1 theory of a chiral multiplet, we can compute a simple secondary product:ZZ{φ̄, ψ+ } =

(1)

S03

dz ∧ d w ∧ φ̄ · ψ+ =

ωBM = 1.

(38)

S03

We will soon analyze a product involving two second-type operators as well.
We can make the analogy more concrete by rewriting the Lagrangian in the descent field
utilizing the BV Lagrangian [14]. The physical Lagrangian (30) admits a compact representation in terms of the following twisted superfields
γ = (φ̄, φ̄ (1) , φ̄ (2) ),
(1)

(2)

β = (ψ+ , ψ+ , ψ+ ).

12

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SciPost Phys. 11, 077 (2021)

By multiplying β, γ so that the form degree of the integrand sums up to (2, 2), we can rewrite(30) as a part of the bigger BV Lagrangian that follows
Z
dzd wβ ∂¯ γ.
(40)
Cz ×Cw

This free field Lagrangian resembles 2d VOA βγ system. As our main intention to present thisexpression is to show the resemblance of the higher VOA to the usual 2d VOA, we will not try
to give details on the BV formalism. The reader who is interested in this may refer to [14, 44].
Let us now define a higher current that generates a higher VOA, which is an analog of
the affine current operator in the 2d VOA. Rather than giving an abstract definition, we will
explain using the 4d N = 1 chiral multiplet example, as it will be our main interest. We definea precurrent J
J = φ̄ψ+(41)
and consider its first descendant J (1) :J (1) = (−ψ̄+̇ d w̄ + ψ̄−̇ dz̄)ψ+ + φ̄(∂w φdz̄ − ∂z φd w̄).

(42)This is not in Q-cohomology by the general discussion above, but we can obtain a nontrivial
Q-cohomology class by combining it with the holomorphic top form on C2 and integrating thewedged product over S 3 :Z
dz ∧ d w ∧ J (1) .

(43)

S3

We call dz ∧ d w∧ J (1) a higher current. Notice that J (1) captures some components of the flavor
current operator in the 4d N = 1 theory of a free chiral multiplet. In this sense, the expression(43) is a charge Q of a conserved global symmetry current j, which is defined as
Z
?j

Q=

(44)

S 2n−1in the 2n−dimensional spacetime, where ? is the Hodge star of the spacetime geometry. The
higher current can be compared with ? j in (43). Indeed, it satisfies the conservation law up
to a Q-exact term,8 which we can treat as zero:d(dzd wJ (1) ) = Q(dzd wJ (1) ) ≡ 0.

(45)

In principle, one can think about the multiplication operation by a monomial z m wn in
the integrand of (43) and obtain a double graded associative algebra of modes,9 which is a
generalization of the single-graded algebra of modes of the 2d VOA. Therefore, we can think(43) as a zero mode(m = n = 0) of the higher current. This observation will be useful when
we construct the algebra of operators on the domain wall by combining all ingredients.
We have studied the holomorphically twisted 4d N = 1 chiral multiplet, which comes fromthe D6-D6’ string. Since the domain wall separates the 5d Chern-Simons theory, we also needto take care of the boundary condition of the 5d gauge field on the domain wall to construct
the algebra of operators on the domain wall. It will be the second ingredient.
See also the relevant discussion in Section 3.5 of [43]. We thank Junya Yagi for useful discussion on this point.
One should distinguish a higher VOA from a toroidal Lie algebra [46], since the former is related to (C2 )× ∼ S 3
rather than (C× )2 ∼ T 2 by construction (43). Indeed, the original reference of the higher VOA [12] called it thesphere algebra.
8

9

13

SciPost Phys. 11, 077 (2021)

3.3

The boundary condition of the 5d gauge field

Before considering the boundary condition of the 5d gauge field A5d , we first notice that we
can interpret the boundary condition of the 5d gauge field as the background gauge field forthe flavor symmetry U(K) of the 4d βγ system. The induced non-dynamical coupling modifies
the free kinetic term of the 4d βγ system in the following way,Z

dzd w ∧ β ∂¯ + A5d |∂ γ.
(46)
Cz ×Cw

Here, the anti-holomorphic derivative is replaced by the gauge covariant derivative with respect to the 5d gauge field A5d at the boundary t = 0, A5d |∂ . The form of this minimal interaction indicates that the 5d gauge field A5d = A t d t + Az̄ dz̄ + Aw̄ d w̄ loses its t-component A twhen it approaches to the boundary.
Next, we will follow the standard approach used to study the boundary condition of the 3d
Chern-Simons theory. We will turn off the non-commutativity parameter temporarily as it does
not affect the following analysis. We need to make sure of the locality of the 5d Chern-Simons
theory near the boundary t = 0. To show that, let us vary the action, obtainingZdz ∧ d w ∧ Tr (δA ∧ dA + d(δA ∧ A)) .

δS5d C S =(47)

R t ×Cz ×Cw

Using Stokes theorem, we may rewrite the second term asZ

dz ∧ d w ∧ δA ∧ A.

(48)

Cz ×Cw

From this, we see two types of equivalently allowed boundary conditions
Az̄ = 0or

Aw̄ = 0,(49)

which makes the boundary variations (48) zero. This resembles the familiar holomorphic
Dirichlet boundary condition of the 3d Chern-Simons theory. We will choose to work with thesecond one Aw̄ = 0.
By the equation of motion of the 5d Chern-Simons theory,(d t + ∂¯Cz ×Cw )A5d = 0,(50)

and the boundary condition that we have just fixed, we conclude the boundary condition of
the 5d gauge field at the domain wall is a holomorphic function in z and w, valued in the Liealgebra glK :A5d |∂ = Az̄ (z, m) ∈ C[z, w] ⊗ glK .
(51)

3.4

Physical observables on the domain wall

So far, we have collected the ingredients that live at the domain wall in the 5d Chern-Simons
theory. Those are the holomorphic twist of 4d N = 1 chiral multiplet and the boundary
condition of the 5d Chern-Simons theory. We will combine those degrees of freedom andconstruct the gauge invariant physical operators living on the domain wall. The relevant gauge
symmetry is U(N ), where N is the number of D6’ branes. On the other hand, we treat U(K)as a flavor symmetry,10 where K is the number of D6 branes.
Since both D6 and D6’ branes are non-compact, one may think both U(N ) and U(K) groups to be nondynamical. However, D6 is related to the bulk dynamics and D6’ is related to the boundary dynamics. Since
we are now studying the boundary dynamics on D6’, we may treat D6 heavy and only freeze the associated gauge
field. [16, 40] also discussed such an assignment of the flavor and gauge symmetry in an analogous intersecting
D4-D6 brane system.
10

14

SciPost Phys. 11, 077 (2021)

We have already seen a candidate for the gauge invariant algebra of operators in §3.2
Zdz ∧ d w ∧ J (1) .

C [1] =

(52)

S03 (z,w)

We will shortly clarify the notation used in the LHS of (52). Restoring all indices of the precurrent J, we have
J ji (z, w) = φ̄ai ψ+ aj ,(53)
where a is the U(N ) gauge symmetry and i, j are the U(K) flavor symmetry index. Note that
we treat J ji (z, w) as a composite operator, which is a holomorphic function; there is no UVdivergence in forming such an operator out of φ̄ and ψ+ , since there is no non-trivial OPE
between a boson and a fermion. As we have remarked in §3.2, we can multiply a holomorphic
function to the integrand of (52), the result is again well-defined. Now, recall from §3.3 thatthe boundary condition of the 5d gauge field is precisely a holomorphic function in z and w,
valued in the Lie algebra glK . Therefore, we can combine (52) andA5d |∂ ∈ C[z, w] ⊗ glK(54)

to form a generating set of the algebra of operators on the domain wall.
Consequently, we propose that the generators of the algebra of operators on the domain
wall areZ
dzd w z m wn T A J (1) .

C [z m wn T A] =(55)

S03 (z,w)

m, n are non-negative integers and T A ∈ glK , where A ∈ {1, . . . , K 2 }. A few remarks are inorder.
First, if there is a 4d gauge field localized at the domain wall, we need to introduce b,
c ghosts that would modify the precurrent J and the higher current J (1) , as well. However,
the gauge field is a 7d field that lives in the 7d worldvolume of the N D6’ branes. Unless wecompactify the three transverse directions normal to the domain wall, this cannot be treatedas the 4d gauge field. Hence, we do not have to introduce the ghosts for the 7d gauge field.
Moreover, we do not introduce ghosts for constant gauge transformations. Therefore, all weneed to do is to take a trace in glN , which we have already done in (53).11 Second, noticethat (55) is not a secondary product, defined in (29), between C [1] and A5d |∂ . Even though
the secondary product involves a second-type operator, the result of the product is a first-type
operator, which is not expected to generate a nontrivial algebra. We will see why such an
operator does not form a nontrivial algebra in §4.
This generator is a four-dimensional version of the two-dimensional observable consideredin Proposition 15.3.1 of [16], which discussed the twisted holography of N M5 branes,
Z


m n A
C2d A5d |Cz = z ∂z T =dz z m ∂zn T A J.
(56)
S01 (z)In this case, the stack of M5 branes forms a surface defect supported on Cz in the 5d CS theory.
A5d |Cz is a z̄ component of the 5d gauge field on Cz . J is a fermion bilinear ψψ̄ and it is a local
operator, which does not involve any descent. The pair of fermions is a quantization of D6-D4
strings. The D4 branes, which are avatars of the M5 branes in type IIA frame, intersect with
the D6 branes along Cz . The author of [16] proved the isomorphism between the algebra ofoperators on N D4 branes, generated by (56), and the WK+∞ algebra in the large N limit. Notethat we may also write w in (55) as ∂z , due to the non-commutative background on Cz × Cw ,which induces a non-trivial commutator between z and w:[z, w] = ε2
11or[z, ∂z ] = ε2 .

We are thankful to Kevin Costello, who patiently explained the relevant part of his paper [16].

15

(57)

SciPost Phys. 11, 077 (2021)

3.5

5d/4d system and anomaly cancellation

Before we close the section, we would like to point out a subtle modification of the 5d ChernSimons theory on one side of the domain wall compared to another side of the wall, due to theanomaly coming from the domain wall. After the modification, the 5d/4d system is anomalyfree. Although we have treated the U(K) gauge field as a background field when we studied
the boundary dynamics, we still need to cancel the U(K) gauge anomaly for the bulk 5d U(K)gauge theory to make sense.
The Lagrangian of the 5d-4d coupled system is
Z
Z


1
2 3
¯
β ∂Aγ +
dzd wTr AdA + A + 2ε2 A{A, A} + . . .
(58)
ε1 R ×C ×C
3{0}×C ×C
z

w

t

z

w

Recall that in the presence of the gauge field A, the partition function of the βγ system isa local section of a determinant line bundle Ldet on the moduli space of connections of thebase manifold (here is C2zw ). In order to quantize A, we need to have a well-defined partition
function in the first place and the local anomaly is the curvature of the determinant line bundle
Fdet , which is determined by the following formula [47, Theorem 10.35]:
Z
Z
Tr(FA3 ),Fdet = n(59)

Cz ×Cw ×D

D

where D is an arbitrary 2 dimensional region in the moduli of connections, n is some constant
independent of D, and FA = dA + A ∧ A is the curvature form of A. To trivialize Ldet , we startfrom the partition function at A = 0, find a path L connecting 0 and A, and define the partition
function at A to be the parallel transport of ZA=0 along L. If we perturb RL to L 0 connecting 0
and A, then the difference between parallel transports along L and L 0 is D Fdet , where D is adisk bounded by L and L 0 . Using Stokes formula, we can write it asZ
Z
CS5d (A),CS5d (A) − n

n

(60)

L 0 ×Cz ×Cw

L×Cz ×Cw

where CS5d (A) is the ordinary 5d Chern-Simons form33
CS5d (A) = Tr A ∧ dA ∧ dA + A3 ∧ dA + A5 .
2
5

(61)

Since A only has anti-holomorphic components along Cz × Cw , we can rewrite A ∧ dA ∧ dA as
A ∧ ∂ A ∧ ∂ A, and all other parts of CS5d (A) vanish, and we see that the action functionalZZ¯
β ∂Aγ − n
Tr (A ∧ ∂ A ∧ ∂ A)
(62){0}×Cz ×Cw

R t≥0 ×Cz ×Cw

is free of gauge anomaly, where we put the boundary condition that A → 0 when t → ∞.
On the other hand, separating out the 5d holomorphic Chern-Simons action into two domains, we get
Z
Z


1
2 3
¯
β∂ γ +
dzd wTr AdA + A + 2ε2 A{A, A}
ε1 R− ×C ×C
3{0}×Cz ×Cw
z
w
t
Z


1
2
+
dzd wTr AdA + A3 + 2ε02 A{A, A} ,(63)
ε1 R+ ×C ×C
3
t

z

w

16

SciPost Phys. 11, 077 (2021)

which can be rewritten as
Z
 Z

1
2 3
β ∂¯Aγdzd wTr AdA + A + 2ε2 A{A, A} +ε1 R ×C ×C
3{0}×Cz ×Cw
t
z
w
Z
0
ε − ε2− 2
Tr (A ∧ ∂ A ∧ ∂ A) .
ε1
R+ ×C ×C
t

z

(64)

w

Therefore, we can cancel the anomaly of the domain wall by imposing the non-commutativity
parameter ε2 on the one side of the wall jumps to a different non-commutativity parameter ε02
on the other side of the wall:
ε02 = ε2 + nε1 .
(65)Note that n ∼ N up to a proportionality constant since the source of the anomaly, which wehave just canceled, is the fermions in the N 4d N = 1 chiral multiplets.

4

Algebra of operators on the domain wall

In this section, we will compute the commutation relation of two generic elements of thealgebra of physical operators on the domain wall. In §4.1, we prove that the algebra is adeformation of the universal enveloping algebra of Diffε2 C ⊗ glK . For a concise presentation,
we simply summarize the idea here. The reader who is interested in the detail of the calculation
may refer to Appendix B. The computation is only valid in a careful large N limit and this is
implicitly assumed in §4.1. We explain how we took the large N limit in some detail in §4.2.
Finally, we conjecture the universal enveloping algebra that we discovered is the classical partof its unique deformation Uε1 (Diffε2 C ⊗ glK ), which coincides with the tree-level part of thebulk algebra in §4.3,

4.1

The commutation relation

Now, we can compute the commutator of the physical observables. We will prove that themode algebra is the universal enveloping algebra of U(Diffε2 C ⊗ glK ). ConsiderC (z m wn T A),C (z r ws T B ) =
Z
Z


(66)
(1)
(1)
r s B
m n A
dz2 d w2 J2 z2 w2 T.
dz1 d w1 J1 z1 w1 T ,S03 (z2 ,w2 )S03 (z1 ,w1 )We can compute the commutator12 by contracting pairs of free fields that give nonzero values.
The nonzero propagators that appear in the computation are〈∂ φ φ̄〉,〈ψ+ ψ̄+̇ 〉,〈ψ+ ψ̄−̇ 〉.

(67)

There are two sorts of terms in the RHS: terms from a single contraction and terms from double
contractions. As shown in detail in the first part of Appendix B, the single contracted terms
reorganize intoC ([z m wn TA, z r ws TB ]) ,(68)
where we used the S 3 residue integral in the computationZωBM (z1 − z0 , w1 − w0 ) f (z1 , w1 ) = f (z0 , w0 ).

Sz3 ,w (z1 ,w1 )
0

(69)

0

12

Note that the product used in the above commutator is not a secondary product, since the one we are discussing
is the product between two descent operators.

17

SciPost Phys. 11, 077 (2021)

A few remarks follow. One may wonder why there is no gauge coupling g dependence13 in(68), since there can be gauge boson loop corrections to the propagators (67) that we use in
the computation. The reason that we do not see such corrections is that the Yang-Mills terms
that provide the gauge coupling are Q-exact14 in our set-up. Therefore, all g−dependence is
Q-exact.
To compute the double contraction, it is useful to rewrite the integral contour, appealing
to the usual 2d CFT S 1 contour rearrangement used in the computation of mode algebra in
the affine Kac-Moody VOA. We perform the following change of S 3 contours
Sz3 (z1 ) [Sz3 (z2 )] − [Sz3 (z2 )][Sz3 (z3 )] = [Sz3 (z2 )][Sz3 (z10 )].
(70)
0

0

0

0

0

2

Figure 3: The S 3 contour rearrangement. S 3 (zi ) is a 3-sphere embedded in Cz × Cw
and parametrized by zi and w i .
Using this contour and performing the double contractions in (66), we end up withZ
Z
S03 (z2 ,w2 )=Z
S03 (z2 )Sz3 ,w (z1 ,w1 )2ωBM (z1 − z2 , w1 − w2 )(w̄12 dz̄2 − z̄12 d w̄2 )z1m wn1 z2r ws2 tr T A T B

2

(71)

(0)z2m+r wn+s
tr T A T B
2= 0.
We wrote down the details of the double contractions in the second part of Appendix B. Therefore, we getC (z m wn T A), C (z r ws T B ) = C ([z m wn T A, z r ws T B ]),(72)
where the commutator in the argument of C (•) in the RHS is that of glK ⊗ Diffε2 C. From this,we see that the domain wall algebra of operators satisfies the commutation relations of thealgebra U(glK ⊗ Diffε2 C).
Before closing this subsection, we would like to point out that Johansen derived the 2d
W −algebra, using second-type operators in 4d N = 1 holomorphic field theory on
M4H = Σz × C×w in [9]. Here Σz is a Riemann surface parametrized by z and C×w is a punctured complex plane parametrized by w. The second-type operator that Johansen used is“2-dimensional”, different from our “3-dimensional” second-type operator, as he integrated
13

We are thankful to Christopher Beem and Kevin Costello for discussion on this point.
Here, the relevant Q is the supercharge that defines the topological holomorphic sector of the 7d SYM on theworldvolume of the N D6’ branes.
14

18

SciPost Phys. 11, 077 (2021)

the first descent operator over the Riemann surface Σz by wedging it with a holomorphic top
form dz of Σz :Z
dz ∧ QO.

(73)Σz ×{w}×
Since this operator is a point on C×
w and “wraps" Σz , when two such operators approach in Cw ,we can expect the meromorphic structure in the OPE. By effectively15 reducing the spacetime
dimension to 2d via the second-type operator, Johansen avoided Hartog’s theorem in 4d.
We can see a hint of the 2d WK+∞ algebra from a bare form of our universal enveloping
algebra. Taking M4H = Σ × C× = C × C× , and inserting it into our universal enveloping
algebra U(glK ⊗ Oε2 (M4T )), we obtain U(glK ⊗ Cε2 [z, w, w−1 ]). Setting ε2 = 1, we almost get
the universal enveloping algebra, which is one of the definition of the WK+∞ algebra [16].
However, importantly, we do not reproduce the central extension of WK+∞ in this way, since
our commutation relation lacks the central extension due to the vanishing double contraction.
It would be interesting to clarify the relation between Johansen’s algebra and our algebra.

4.2

The large N limit

Trying to avoid potential confusion, we have not discussed the large N limit yet. Let us describe
how to take the large N limit and explain the implicit assumption that we make in the previous
subsection. We will mostly follow [16], where the author explained how to take the large N
limit of the algebra of operators living on N M5 branes. The difference is that we do not needsupergroups to proceed.
Let us denote by O bsε4d (N ) the algebra of gauge invariant operators on the domain wall2
coming from a stack of N D6’ branes. For N 0 ≥ N , there is a natural block diagonal embedding
glN ,→ glN 0 , which gives rise to a map between algebras0ρNN : O bsε4d (N 0 ) → O bsε4d (N ).

(74)

2

2

0

It maps the generator C (z r ws T A) to its restriction to glN fields. Note that ρNN is a homomorphism of algebras since the computation of commutation relations in the last section holds for0
all N . ρNN is also surjective as generators are mapped to generators. Therefore, we see that the
map defined in the last section U(Diffε2 C ⊗ glK ) → O bsε4d (N ) is compatible with the transition
20

map ρNN . The only relations are trace relations and by sending N → ∞ all trace relations aregone. Hence, the map U(Diffε2 C ⊗ glK ) → O bsε4d (N ) becomes an isomorphism in the large N
2limit.

4.3

Conjecture

Let us discuss the implication of the result that we obtained. We proved that the algebra ofoperators living on the domain wall is the universal enveloping algebra of Diffε2 C × glK . It
is important to notice that this algebra has no ε1 parameter. Classically, since both the 4d
VOA and the boundary condition of the 5d gauge field that were used to construct the set ofgenerators of the enveloping algebra do not depend on ε1 , the resulting algebra constructed
out of these ingredients should not depend on ε1 . Still, there can be potential quantum corrections that depend on ε1 at the level of both generators and relations, similar to the M5
brane example [16]. The quantum correction is reflected in the change of the original structure constants of the algebra U(Diffε2 C ⊗ glK )[[ε1 ]] to a new one, i.e. this is a flat deformation
15
See also [48–50] for another example, where the meromorphic structure naturally arises in 4d N = 2 superconformal field theory without relying on the second-type operators.

19

SciPost Phys. 11, 077 (2021)of U(Diffε2 C ⊗ glK ) over the base ring C[[ε1 ]]. Let us denote this deformed algebra as AεK K .
1
Then, what is this algebra?
It is in general a hard question to characterize a deformation of the universal enveloping
algebra of a certain Lie algebra, but this particular case of interest was explicitly worked out in
the N M2 brane twisted holography [17]. One important result of [17] is the uniqueness of thedeformation of U(Diffε2 C⊗ glK ).16 Now, recall from §2.1 that O bsε5d,εC S the algebra of operators
1 2
in the 5d Chern-Simons theory is Koszul dual of Uε1 (Diffε2 C ⊗ glK ). Since Uε1 (Diffε2 C ⊗ glK )is the unique deformation of U(Diffε2 C ⊗ glK ), AεK K must be isomorphic to Uε1 (Diffε2 C ⊗ glK ).
1

This leads us to conjecture17 that the quantum deformation of our classical result is Koszul dual
to O bsε5d,εC S up to a coordinate change ε1 → ε1 + f2 (κ)ε21 + f3 (κ)ε31 + · · · , where f i (κ) ∈ C[κ]1 2
are polynomials of κ, establishing the conjectural twisted holography for the domain wall
example.
Let us give more details about the unique deformation of the enveloping algebra. Even
though the deformation of the algebra is uniquely fixed, we cannot immediately deduce thealgebraic relations of the deformed algebra. This algebra actually appeared in another exampleof twisted M-theory. In [17], the author considered N M2 branes in the twisted M-theory
background. Similar to our case, the twisted M-theory background localizes to the 5d ChernSimons theory, whose algebra of operators is again O bsε5d,εC S . The worldvolume theory of the
1 2
N M2 branes is the 3d N = 4 G = U(N ) gauge theory with K fundamental hypermultiplets
and an adjoint hypermultiplet with Rozansky-Witten twist and Ω-deformation applied to two
directions out of the 3d worldvolume. The author computed Aε1 ,ε2 the algebra of operatorson the N M2 branes and drew an isomorphism between Aε1 ,ε2 and the Koszul dual algebra ofO bsε5d,εC S . Hence, we can equally understand Uε1 (Diffε2 ⊗ glK ) as Aε1 ,ε2 . The generators and1 2
relations of this algebra can be found in [23, 24].
The conjectural isomorphism between the quantum deformation of the algebra of operators on the domain wall Uε1 (Diffε2 C ⊗ glK ) and the Koszul dual of O bsε5d,εC S leads to the
1 2statement of the twisted holography of the domain wall in twisted M-theory. It is rather surprising that the domain wall algebra is related to the M2 brane algebra Aε1 ,ε2 , since there is no
obvious structural similarity between the two quantum field theories on the M2 brane and the
KK monopole intersection. It would be interesting to figure out a more clear physical reason.

5

Conclusion and open questions

In this paper, we study the last available BPS defect in the twisted M-theory, a domain wall-like
defect, building on the already established line and surface-like defects in the literature. We
identified the algebra of physical operators on the domain wall and establish the isomorphism
between the boundary algebra and the bulk operator algebra, which is the algebra of operatorsof the 5d Chern-Simons theory. The higher vertex operator algebra, which is the algebraic
structure under the holomorphically twisted 4d N = 1 theory, plays a key role in the derivation.
We engineered a domain wall-like defect in the 5d Chern-Simons theory by introducing
a new KK monopole in the twisted M-theory. The new KK monopole along with the alreadyexisting Taub-NUT geometry forms a new G2 -holonomy 7-manifold, which is topologicallytwisted. In the type IIA frame, the singular G2 -holonomy manifold is mapped to an intersecting
D6-D6’ brane configuration. The D6-D6’ strings are localized at the 4d intersection, where theMore precisely, the deformation space of U(Diffε2 C ⊗ glK+R|R ) for all R in a compatible way is isomorphic toC[κ], where κ is the central element 1 ⊗ IdK+R|R .
17
As we have said in the introduction, we believe this conjecture is true a priori. However, since we do not know
how to compute the quantum corrections using the defining properties of the higher VOA, not relying on Costello’s
theorem, we modestly call it as a conjecture.
16

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SciPost Phys. 11, 077 (2021)

holomorphic twist is applied. The intrinsic degree of freedom living on the wall is the 4d
N = 1 bi-fundamental chiral multiplet. After passing to the Q-cohomology, specified by theholomorphic twist, one gets a higher version of the vertex operator algebra, which has a concise
presentation as the 4d βγ system. The algebra of operators in the 4d βγ system is generated
by the integrated higher current. Combining it with the boundary condition of the 5d gauge
field on the domain wall, we form a set of generators of the physical operators living on thedomain wall. We prove that this algebra is the universal enveloping algebra of Diffε2 C ⊗ glK
and conjecture an isomorphism with the Koszul dual of the bulk algebra using the uniquenessof deformation of U(Diffε2 C ⊗ glK ).
We can summarize the operator algebras on M2, M5, and the new KK monopole in thetwisted M-theory on R t × Cz × Cw × Cε1 × TNK in the following table.
Table 1: Summary of BPS defects, associated field theory, twists, protected subsectorsof the field theory, and algebras. ADHM means the G = U(N ) gauge theory with
1 adjoint hypermultiplet and K fundamental hypermultiplets. T and H stand for
topological and holomorphic twists. Field theories I and II are the original theoriesof the relevant branes and their protected subsectors. TQM is an abbreviation of
topological quantum mechanics.
BPS defect
M2
M5
KKM

Field theory I
3d N = 4 ADHM
6d N = (2, 0)
4d N = 1 chiral

Twist
3T
4T, 2H3T, 4H

Field theory II
1d TQM
2d fermions
4d βγ system

AlgebraUε1 (Diffε2 C ⊗ glK )Uc=N (Oε2 (C × C× ) ⊗ glK )Uε1 (Diffε2 C ⊗ glK )There is a notable difference between the last example and the first two examples. Although the first two examples have an algebra of operators constructed solely from the firsttype operators, in other words, local operators in the Q-cohomology, the algebra of operators of
the last example involves the second-type operators, in other words, Q−descendant operators.
On the other hand, there is a similarity shared by all three examples: the effective codimension of the elements of algebras is one. Let us explain more about what we mean by theeffective codimension. First, in 1d TQM we easily see that the codimension of a local operator
is 1. Second, in 2d VOA, when one computes the mode algebra, one extracts a mode froma current operator by using S 1 residue integral. The codimension of the associated S 1 in 2dspacetime is 1. Lastly, for higher VOA, when we derive the mode algebra on the domain wall,
we used S 3 contour to extract elements of the algebra. The non-commutative structure comesfrom the codimension-1 property of S 3 in 4d spacetime.

Figure 4: From left to right, 1d TQM, 2d VOA, and 4d VOA operators, depicted by
the circles. The thick straight lines are the transverse directions of the operators,
showing the codimension-1 nature of the algebra of operators in each case.

21

SciPost Phys. 11, 077 (2021)

Let us now list some open questions that we hope to answer in the future:• What would be the canonical Koszul-dual pair interaction between the 5d gauge theory
and 4d holomorphic field theory at the level of modes? Can we derive a deformed
version of the commutation relation (72) by imposing the BRST invariance of a certain
set of Feynman diagrams, like [17]? It is not clear to us how to define a local interaction
between a ghost and a second-type operator. Once it is identified, the computation will
be as straightforward as that of [17, 24].
• Can we identify a supersymmetric or at least stable configuration of the combined systemof domain wall, line, and surface defects? [30, 31] studied the network of M2 and M5
branes and identified the relevant fusion algebra using free field realization of the M2
and M5 brane algebras and Feynman diagram computation in 5d Chern-Simons theory.
Incorporating the domain wall defect in this line of analysis seems to be a very natural
direction to pursue.
• In [52], Li and Yamazaki gave a complete answer for algebras acting on BPS partition
functions that count D6-D2-D0 bound states on toric Calabi-Yau 3-folds. We wonder
if there is a connection between the algebra associated to our 5d/4d/5d system and apotential BPS partition function that counts M2 branes wrapping associative 3-cycles in
the G2 manifold.

Acknowledgements
We thank Christopher Beem, Kevin Costello, and Junya Yagi for illuminating discussions. We
especially thank Kevin Costello for his insightful comments on our draft. Finally, we are grateful to two anonymous SciPost referees who gave us valuable remarks on various parts of thefirst version of this paper.
Funding information Research of JO was supported by ERC Grants 682608 and 864828.
Research at the Perimeter Institute is supported by the Government of Canada through Industry
Canada and by the Province of Ontario through the Ministry of Economic Development &Innovation.

A

4d N = 1 SUSY transformation of a chiral multiplet

In this appendix, we summarize the supersymmetry transformation of the 4d N = 1 chiralmultiplet Φ = (φ, ψα , F ) and its conjugate Φ̄ = (φ̄, ψ̄α̇ , F̄ ). In particular, we only record
the transformations, generated by the scalar Q, which we used to define the Q-cohomology,
and the 1-form supercharge Q, which we used to define the second-type operators in the Q-

22

SciPost Phys. 11, 077 (2021)

cohomology:
Qφ = ψ− ,Qψ+ = −F,Qψ− = 0,
QF = 0,(75)Qφ̄ = 0,Qψ̄+̇ = −i∂z̄ φ̄,Qψ̄−̇ = i∂w̄ φ̄,
Q F̄ = i∂w̄ ψ̄+̇ + i∂z̄ ψ̄−̇ .
Up to on-shell condition, there are three Q-closed operators:
ψ+ ,

ψ− ,

φ̄ .

(76)

Since ψ− is Q-exact by the first equation, the operators in the Q-cohomology are
φ̄,

ψ+

(77)

and their holomorphic derivatives with respect to ∂z , ∂w .
Next, we record the action of 1-form supercharge on the on-shell chiral multiplet fields:
Qφ = 0,Qψ+ = dz̄∂w φ − d w̄∂z φ,
Qψ− = dz̄∂z̄ φ + d w̄∂w̄ φ,
Qφ̄ = idz̄ ψ̄+̇ − id w̄ψ̄−̇ ,(78)

Qψ̄+̇ = −id w̄ F̄ ,Qψ̄−̇ = −idz̄ F̄ .
Since we only use the action of 1-form supercharge on (77), only second and fourth lines aremeaningful in the main text of this paper.

B

Mode algebra computation

In this appendix, we will provide the details on the commutator computation summarized in
§4.1.




C (z m wn T A), C (z r ws T B ) = C z m wn T A, z r ws T B + 0,
(79)
where
m

n

A

C (z w T ) =Z(1)

S03 (z1 ,w1 )rsBC (z w T ) =Z
S03 (z2 ,w2 )(1)

dz1 d w1 J1 (z1m wn1 T A),
(80)

(1)
dz2 d w2 J2 (z2r ws2 T B ).
(1)

We will first focus on expanding J1 and J2 and identify the nonzero propagators that
we will contract later:(1) (1)J1 J2 = (ψ̄−̇ dz̄1 − ψ̄+̇ d w̄1 )ψ+ + φ̄(∂w1 φdz̄1 − ∂z1 φd w̄1 ) 1(81)· (ψ̄−̇ dz̄2 − ψ̄+̇ d w̄2 )ψ+ + φ̄(∂w2 φdz̄2 − ∂z2 φd w̄2 ) 2 .
23

SciPost Phys. 11, 077 (2021)

Out of 16 terms in the expansion, 8 terms can host nonzero contractions. Those aredz̄1 dz̄2 (ψ̄−̇ ψ+ )1 (ψ̄−̇ ψ+ )2 + (φ̄∂w1 φ)1 (φ̄∂w2 φ)2+ d w̄1 d w̄2 (−ψ̄+̇ ψ+ )1 (−ψ̄+̇ ψ+ )2 + (−φ̄∂z1 φ)1 (−φ̄∂z2 φ)2+ dz̄1 d w̄2 (ψ̄−̇ ψ+ )1 (−ψ̄+̇ ψ+ )2 + (φ̄∂w1 φ)1 (−φ̄∂z2 φ)2+ d w̄1 dz̄2 (−ψ̄+̇ ψ+ )1 (ψ̄−̇ ψ+ )2 + (−φ̄∂z1 φ)1 (φ̄∂w2 φ)2 .

(82)

Now, we will use〈∂zi φi (zi , w i )φ̄ j (z j , w j )〉 =〈ψ̄+̇,i (zi , w i )ψ+, j (z j , w j )〉 =

z̄i j

,

〈∂w i φi (zi , w i )φ̄ j (z j , w j )〉 =,2〈ψ̄−̇,i (zi , w i )ψ+, j (z j , w j )〉 =

di2j
z̄i j

di j

w̄ i jdi2j,w̄ i j

(83)

,
2

di j

where
di2j = |zi j |2 + |w i j |2 ,(84)(AB)1 (AB)2 ,(85)

in the following.

Single contraction
For each term in (82) that looks
we may contract either 〈A1 B2 〉 or 〈B1 A2 〉. Summing up all possible single contractions, we get1dz̄1 dz̄2 w̄12 (ψ+,1 ψ̄−̇,2 + ψ̄−̇,1 ψ+,2 + ∂w1 φ1 φ̄2 + φ̄1 ∂w2 φ2 )2d12
+ d w̄1 d w̄2 z̄12 (ψ+,1 ψ̄+̇,2 + ψ̄+̇,1 ψ+,2 + ∂z1 φ1 φ̄2 + φ̄1 ∂z2 φ2 )− dz̄1 d w̄2 (w̄12 ψ+,1 ψ̄+̇,2 + z̄12 ψ̄−̇,1 ψ+,2 + z̄12 ∂w1 φ1 φ̄2 + w̄12 φ̄1 ∂z2 φ2 )− d w̄1 dz̄2 (z̄12 ψ+,1 ψ̄−̇,2 + w̄12 ψ̄+̇,1 ψ+,2 + w̄12 ∂z1 φ1 φ̄2 + z̄12 φ̄1 ∂w2 φ2 ) .

(86)

Next, rearrange the above so that we can find the following S 3 integral kernelωBM (z, w) =w̄dz̄ − z̄d w̄
(|z|2 + |w|2 )2(87)

manifestlydz̄1+ dz̄1w̄12 dz̄2 − z̄12 d w̄22
d12w̄12 dz̄2 − z̄12 d w̄2ψ̄−̇,1 ψ+,2 + d w̄12
d12
z̄12 d w̄2 − w̄12 dz̄2ψ̄+̇,1 ψ+,2

∂z1 φ1 φ̄2
2d12
z̄12 d w̄1 − w̄12 dz̄1
+ dz̄2
ψ+,1 ψ̄−̇,2 + d w̄2
ψ+,1 ψ̄+̇,2
2
2
d12
d12
z̄12 d w̄1 − w̄12 dz̄1w̄12 dz̄1 − z̄12 d w̄1+ dz̄2φ̄1 ∂w2 φ2 + d w̄2
φ̄1 ∂z2 φ2 .
2
2
d12
d12
2
d12
w̄12 dz̄1 − z̄12 d w̄1∂w1 φ1 φ̄2 + d w̄1z̄12 d w̄2 − w̄12 dz̄2

(88)Then, make the following change of variables(z2 , w2 ) → (z̃2 , w̃2 ) = (z2 − z1 , w2 − w1 )
24

(89)

SciPost Phys. 11, 077 (2021)

in the first two lines of (88) to get ωBM (z̃2 , w̃2 ) and make the following change of variables(z1 , w1 ) → (z̃1 , w̃1 ) = (z1 − z2 , w1 − w2 )

(90)
(1)

(1)

in the last two lines of (88) to get ωBM (z̃1 , w̃1 ). We have expanded J1 · J2 . Let us plug thisback inC (z m wn T A) · C (z r ws T B ).
(91)
We will omit the elements of C[z, w] ⊗ glK and restore those at the end.
Then, we can rearrange (91) intoZZ
ωBM (z̃2 , w̃2 )

S3

+

Z0

S3

ωBM (z̃1 , w̃1 )

S03

Z0

S03


dz1 d w1 (ψ̄−̇,1 dz̄1 − ψ̄+̇,1 d w̄1 )ψ+,2 + φ̄2 (∂w1 φ1 dz̄1 − ∂z1 φ1 d w̄1 )dz2 d w2 (ψ̄−̇,2 dz̄2 − ψ̄+̇,2 d w̄2 )ψ+,1 + φ̄1 (∂w2 φ2 dz̄2 − ∂z2 φ2 d w̄2 ) ,(92)
where S03 is a 3-sphere centered at the origin of Cz ×Cw . Note that ψ+,2 and φ̄2 in the first lineof (92) are functions of z̃2 + z1 and w̃2 + w1 due to (89). Similarly, ψ+,1 and φ̄1 are functionsof z̃1 + z2 and w̃1 + w2 due to (90). Therefore, so far, the integrands are bi-local functions on
two sets of variables. Evaluating the first S 3 residue integrals in each line of (92), we getZdz1 d w1 (ψ̄−̇,1 dz̄1 − ψ̄+̇,1 d w̄1 )ψ+,1 + φ̄1 (∂w1 φ1 dz̄1 − ∂z1 φ1 d w̄1 )
S3

+

Z0

S03

(93)

dz2 d w2 (ψ̄−̇,2 dz̄2 − ψ̄+̇,2 d w̄2 )ψ+,2 + φ̄2 (∂w2 φ2 dz̄2 − ∂z2 φ2 d w̄2 ) .

Due to the residue integrals, all fields with subscript i now depend on a single set of variables(zi , w i ), so there is no bilocal behavior anymore. Therefore, we observe that each integrand(1)
corresponds to Ji and rewrite (93) as
Z
S3

(1)
dz1 d w1 J1 +

Z
S3

(1)

dz2 d w2 J2 .

(94)

Since the subscripts are now dummy variables, we can unify (94) as one term. After restoring
the omitted elements of C[z, w] ⊗ glK , we getZ
dzd wJ (1) z m wn T A z r ws T B .
(95)
S3This is the result of the single contractions of a product of two algebra generators (91). To get
the commutator, we permute two elements of C[z, w] ⊗ glK in (95) and subtract the resulting
term from (95). We get
Z




dzd wJ (1) z m wn T A, z r ws T B =: C z m wn T A, z r ws T B .
(96)
S3

It reproduces the first term in the RHS of (79).

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SciPost Phys. 11, 077 (2021)

Double contractions
Let us start from the single-contracted expression (88) and perform another contraction:
1
dz̄1 (w̄12 dz̄2 − z̄12 d w̄2 )w̄12 + d w̄1 (z̄12 d w̄12 − w̄12 dz̄12 )z̄12
4d12+dz̄1 (w̄2 dz̄12 − z̄12 d w̄2 )w̄12 + d w̄1 (z̄12 d w̄2 − w̄12 dz̄2 )z̄12+dz̄2 (w̄12 dz̄1 − z̄12 d w̄1 )w̄12 + d w̄2 (z̄12 d w̄1 − w̄12 dz̄1 )z̄12(97)+dz̄2 (w̄12 dz̄1 − z̄12 d w̄1 )(−w̄12 ) + d w̄12 (z̄12 d w̄1 − w̄12 dz̄1 )(−z̄12 ) ,where in the third and fourth line we used
〈ψ+,1 ψ̄−̇,2 〉 = −〈ψ̄−̇,2 ψ+,1 〉 = 〈ψ̄−̇,1 ψ+,2 〉,〈ψ+,1 ψ̄+̇,2 〉 = −〈ψ̄+̇,2 ψ+,1 〉 = 〈ψ̄+̇,1 ψ+,2 〉,〈φ̄1 ∂w2 φ2 〉 = 〈∂w2 φ2 φ̄1 〉 = −〈∂w1 φ1 φ̄2 〉,〈φ̄1 ∂z2 φ2 〉 = 〈∂z2 φ2 φ̄1 〉 = −〈∂z1 φ1 φ̄2 〉.

(98)

We can simplify (97) into2(w̄12 dz̄2 − z̄12 d w̄2 )w̄12 dz̄1 − z̄12 d w̄1
4
d12

.

(99)

By integrating (99) along the modified contour depicted in Figure 3, we can finish the computation for the double contracted term of (79):Z
Zw̄12 dz̄1 − z̄12 d w̄1
2(w̄12 dz̄2 − z̄12 d w̄2 )(z1m wn1 z2r ws2 tr T A T B )43d3
S0 (z2 ,w2 ) Sz ,w (z1 ,w1 )
12
Z 2 2
Z
= 2

S03 (z2 ,w2 )= 2ZS03 (z1 ,w1 )Sz3 ,w (z1 ,w1 )2 2

ωBM (z1 − z2 , w1 − w2 )(w̄12 dz̄2 − z̄12 d w̄2 )(z1m wn1 z2r ws2 tr T A T B )(0)(z2m wn2 z2r ws2 tr T A T B )

= 0,
(100)
where Sz3 ,w (z j , w j ) is a 3-sphere centered at (zi , w i ) and parametrized by (z j , w j ). In the firstij42equality, we used the fact that d12= d12
on Sz3 ,w (z2 , w2 ), assuming the radius of the 3-sphere1 1
is 1. The result is independent on the radius, as it should. We conclude that there is no central
element in the commutation relation.
Therefore, combining with (96), we recover (79)C (z m wn T A), C (z r ws T B ) = C z m wn T A, z r ws T B .
(101)

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