SciPost Phys. 9, 017 (2020)

Topological holography: The example of the D2-D4 brane system
Nafiz Ishtiaque1,2? , Seyed Faroogh Moosavian1,2 and Yehao Zhou1,2
1 Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
2 University of Waterloo, Waterloo, ON, Canada? nishtiaque@perimeterinstitute.ca

Abstract
We propose a toy model for holographic duality. The model is constructed by embedding a stack of N D2-branes and K D4-branes (with one dimensional intersection) in a 6d
topological string theory. The world-volume theory on the D2-branes (resp. D4-branes)
is 2d BF theory (resp. 4D Chern-Simons theory) with GL N (resp. GL K ) gauge group. We
propose that in the large N limit the BF theory on R2 is dual to the closed string theory
on R2 × R+ × S3 with the Chern-Simons defect on R × R+ × S2 . As a check for the duality we compute the operator algebra in the BF theory, along the D2-D4 intersection –the algebra is the Yangian of gl K . We then compute the same algebra, in the guise ofa scattering algebra, using Witten diagrams in the Chern-Simons theory. Our computations of the algebras are exact (valid at all loops). Finally, we propose a physical string
theory construction of this duality using a D3-D5 brane configuration in type IIB – using
supersymmetric twist and Ω-deformation.
Copyright N. Ishtiaque et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 07-04-2020
Accepted 14-07-2020
Check forupdatesPublished 05-08-2020
doi:10.21468/SciPostPhys.9.2.017

Contents
1 Introduction and Summary

12 Isomorphic algebras from holography

3

3 The dual theories
3.1 Brane construction
3.2 The closed string theory
3.3 BF: The theory on D2-branes
3.4 4d Chern-Simons: The theory on D4-branes

5
5
5
8
10

4 AOp (Tbd ) from BF ⊗ QM theory
4.1 Free theory limit, O(ħh0 )4.2 Loop corrections from BF theory
4.3 Large N limit: The Yangian

13
15
16
22

5 ASc (Tbk ) from 4d Chern-Simons Theory
5.1 Relation to anomaly of Wilson line

22
25
1

SciPost Phys. 9, 017 (2020)

5.2
5.3
5.4

Classical algebra, O(ħh0 )Loop corrections
Large N limit: The Yangian

27
29
35

6 Physical String Theory Construction of The Duality
6.1 Brane Configuration
6.2 Twisting Supercharge
6.3 Omega Deformation
6.4 Takeaway from the Brane Construction

36
37
37
41
42

7 Concluding Remarks and Future Works

42

A Integrating the BF interaction vertex

44

B Yangian from 1-loop Computations
B.1 Tannaka formalism

44
45

C Technicalities of Witten Diagrams
C.1 Vanishing lemmas
C.2 Comments on integration by parts

54
54
56

D Proof of Lemma 2

57

References60

1

Introduction and Summary

Holography is a duality between two theories, referred to as a bulk theory and a boundary
theory, in two different space-time dimensions that differ by one [1–3]. A familiar manifestation of the duality is an equality of the partition function of the two theories - the boundary
partition function as a function of sources, and the bulk partition function as a function ofboundary values of fields. This in turns implies that correlation functions of operators in theboundary theory can also be computed in the bulk theory by varying boundary values of its
fields [2,3]. This dictionary has been extended to include expectation values of non-local operators as well [4–7]. This is a strong-weak duality, relating a strongly coupled boundary theory
to a weakly coupled bulk theory. As is usual in strong-weak dualities, exact computations on
both sides of the duality are hard. Topological theories have provided interesting examples ofholographic dualities where exact computations are possible [8–13].
Recently, it has been shown that some instances of holography can be described as analgebraic relation, known as Koszul duality, between the operator algebras of the two dual
theories [14, 15]. It was previously known that the algebra of operators restricted to a line in
the holomorphic twist of 4d N = 1 gauge theory with the gauge group GLK is the Koszul dualof the Yangian of glK [16]. In light of the connection between Koszul duality and holography,
this result suggests that if there is a theory whose local operator algebra is the Yangian of glKthen that theory could be a holographic dual to the twisted 4d theory. Since the inception ofholography, brane constructions played a crucial role in finding dual theories. It turns out that2

SciPost Phys. 9, 017 (2020)

the particular twisted 4d theory is the world-volume theory of K D4-branes1 embedded in aparticular 6d topological string theory [18]. Since the operators whose algebra is the Koszul
dual of the Yangian lives on a line, it is a reasonable guess that we need to include some otherbranes that intersect this stack of D4-branes along a line. Beginning from such motivations weeventually find (and demonstrate in this paper) that the correct choice is to embed a stack of N
D2-branes in the 6d topological string theory so that they intersect the D4-branes along a line.
The world-volume theory of the D2-branes is 2d BF theory with GLN gauge group coupled toa fermionic quantum mechanics along the D2-D4 intersection. The algebra of gauge invariant
local operators along this D2-D4 intersection is precisely the Yangian of glK .
This connected the D2 world-volume theory and the D4 world-volume theory via holography in the sense of Koszul duality. The connection between these two theories via holography
in the sense of [2, 3] was still unclear. In this paper we begin to establish this connection.
We take the D2-brane world-volume theory to be our boundary theory. This implies that theclosed string theory in some background, including the D4-brane theory should give us thedual bulk theory. In the boundary theory, we consider the OPE (operator product expansion)
algebra of gauge invariant local operators, we argue that this algebra can be computed in thebulk theory by computing a certain algebra of scatterings from the asymptotic boundary in thelimit N ! ∞. Our computation of the boundary local operator algebra using the bulk theory
follows closely the computation of boundary correlation functions using Witten diagrams [3].
The Feynman diagrams and Witten diagrams we compute in this paper have at most twoloops, however, we would like to emphasize that the identification we make between the operator algebras and the Yangian is true at all loop orders. In the boundary theory (D2-brane
theory) this will follow from the simple fact that, for the operator product that we shall compute, there will be no non-vanishing diagrams beyond two loops. In the bulk theory thisfollows from a certain classification of anomalies in the D4-brane theory [19] and independently from the very rigid nature of the deformation theory of the Yangian. We explain someof these mathematical aspects underlying our results in appendix B – we begin the appendix
with motivations for and a light summary of the purely mathematical results to follow.
We note that there is a long history of studying links between quantum integrable systems
soluble by Yangians and quantum affine algebras on one hand and supersymmetric gauge
theory dynamics in the vacuum sector on the other hand – including early pioneering work [20]and subsequent developments [21–25]. In this paper we study particular examples of (quasi)
topological gauge theories with similar connection to Yangins, the novelty is that we propose
a certain holographic duality linking the theories.
A particular motivation for studying these topological/holomorphic theories and their duality is that these theories can be constructed from certain brane setup in a physical 10d string
theory. In particular, we can identify these theories as certain supersymmetric subsectors ofsome theories on D-branes in type IIB string theory by applying supersymmetric twists andΩ-deformation.
The organization of the paper is as follows. In §2 we describe, in general terms, howholographic duality in the sense of [2, 3] leads to the construction of two isomorphic algebras
from the two dual theories. In §3 we start from a brane setup involving N D2-branes and K
D4-branes in a 6d topological string theory and describe the two theories that we claim to beholographic dual to each other. In §4 we compute the local operator algebra in the D2-brane
theory, this algebra will be the Yangian Y (glK ) in the limit N ! ∞. In §5 we show thatthe same algebra can be computed using Witten diagrams in the D4-brane theory. In the lastsection, §6, we propose a physical string theory realization of the duality.
We are following the convention of [17], according to which, by a Dp-brane in topological string theory wemean a brane with a p-dimensional world-volume. In §6 when we discuss branes in type IIB string theory, we ofcourse use the standard convention that a Dp-brane refers to a brane with (p + 1)-dimensional world-volume.
1

3

SciPost Phys. 9, 017 (2020)

Relevant New Literature. Since the preprint of this paper appeared online, there has
been a series of interesting developments exploring the idea of topological holography – also
referred to as twisted holography – we mention the ones we found conceptually related to thispaper. In [26] the authors studied the holographic duality between a gauged βγ system and a
Kodaira-Spencer theory on the SL(2, C) manifold (the deformed conifold) – emphasizing therole of global symmetry matching. Authors of [27] studied a twisted holography closely related
to AdS3 /CFT2 duality, they highlight in particular the link to Koszul duality and contains a rare
(in physics literature) introduction to Koszul duality. [28] computes certain operator algebraof a topological quantum mechanics living at the intersection of M2 and M5 branes in an Ωdeformed M-theory using Feynman diagram techniques similar to the ones we shall use in our
computations. In this setup the M2 and M5 branes play roles analogous to certain D3 and D5
branes that we shall study in §6. The M5 brane world-volume theory in the Ω-backgroundis the 5d Chern-Simons theory, a close cousin of the 4d Chern-Simons theory we are goingto study. This M2-M5 brane setup in Ω-deformed M-theory is also studied in [29, 30] in thecontext of twisted holography.

2

Isomorphic algebras from holography

In [2,3], two theories, Tbd and Tbk were considered on two manifolds M1 and M2 respectively,
with the property that M1 was conformally equivalent to the boundary of M2 . The theory
Tbd was considered with background sources, schematically represented by φ. The theory Tbk
was such that the values of its fields at the boundary ∂ M2 can be coupled to the fields of Tbd ,then Tbk was quantized with the fields φ as the fixed profile of its fields at the boundary ∂ M2 .
These two theories were considered to be holographic dual when their partition functions wereequal:
Zbd (φ) = Zbk (φ) .
(1)This equality leads to an isomorphism of two algebras constructed from the two theories,
as follows. Consider local operators Oi in Tbd with corresponding sources φ i . The partition
function Zbd (φ) with these sources has the form:


Z
X
1
Zbd (φ) = D X exp − Sbd +(2)
Oi φ i ,ħhiwhere X schematically represents all the dynamical fields in Tbd . Correlation functions of theoperators Oi can be computed from the partition function by taking derivatives with respect
to the sources:〈O1 (p1 ) · · · On (pn )〉 =

1
δ
δ
·
·
·
Zbd (φ).
Zbd (φ) δφ 1 (p1 )δφ n (pn )
φ=φ0

(3)

We can consider the algebra generated by the operators Oi using operator product expansion(OPE). However, this algebra is generally of singular nature, due to its dependence on thelocation of the operators and the possibility of bringing two operators too close to each other.
In specific cases, often involving supersymmetry, we can consider sub-sectors of the operator
spectrum that can generate algebras free from such contact singularity, so that a position independent algebra can be defined.2 Suppose the set {Oi } represents such a restricted set withan algebra:
Oi O j = Cikj Ok .
(4)
2

Various chiral rings, for example.

4

SciPost Phys. 9, 017 (2020)

Let us call this algebra AOp (Tbd ). In terms of the partition function and the sources the relation(4) becomes:δ δδ= CikjZbd (φ)Zbd (φ)
.
(5)
j
i
δφ δφ
δφ k
φ=0
φ=φ0
The statement of duality (1) then tells us that the above equation must hold if we replace Zbd
by Zbk :δδ δ= CikjZbk (φ)Zbk (φ)
.
(6)
j
i
δφ δφ
δφ k
φ=0
φ=φ0
This gives us a realization of the operator algebra AOp (Tbd ) in the dual theory Tbk .
This suggests a check for holographic duality. The input must be two theories, say Tbd and
Tbk , with some compatibility:• Tbd can be put on a manifold M1 and Tbk can be put on a manifold M2 such that∂ M2 ∼= M1 , where equivalence between ∂ M2 and M1 must be equivalence of whatevergeometric/topological structure is required to define Tbd .3• Quantum numbers of fields of the two theories are such that the boundary values of the
fields in Tbk can be coupled to the fields in Tbd .4
Suppose Tbd has a sub-sector of its operator spectrum that generates a suitable algebra5
AOp (Tbd ). We denote the operators in this algebra by {Oi } with corresponding sources φ i .
According to the first compatibility condition these sources can be thought of as boundary values for the fields in Tbk , so that we can quantize Tbk by fixing the values of the fields at theboundary to be φ. Then, we can define another algebra by taking functional derivatives ofthe partition function of Tbk with respect to φ, as in (6). Let’s call this algebra the scattering
algebra, ASc (Tbk ). Now a check of holographic duality is the following isomorphism:

AOp (Tbd ) ∼= ASc (Tbk ) .

(7)This is the general idea that we employ in this paper to check holographic duality. The operator
algebra AOp (Tbd ) can be computed in perturbation theory using Feynman diagrams and we
can use Witten diagrams, introduced in [3], to compute the scattering algebra ASc (Tbk ). We
will do this concretely in the rest of this paper.

3

The dual theories

3.1

Brane construction

The quickest way to introduce the theories we claim to be holographic dual to each other is touse branes to construct them. Our starting point is a 6d topological string theory, in particular,
the product of the A-twisted string theory on R4 and the B-twisted string theory on C [18].
The brane setup is the following:
Rv
D2 0
D4 0

Rw R x
×
×
0
×

Ry
0
×

3

Cz
0
×

No. of branes
N
K

(8)

In case of AdS/CFT, it is conformal equivalence, perfect for defining the CFT. In this paper we shall only beconcerned with topology.
4
To clarify, this is merely a compatibility condition for the duality, the two dual theories are not supposed to becoupled, they are supposed to be alternative descriptions of the same dynamics.
5
Ideally we should consider the OPE algebra of all the operators, but if that is too hard, we can restrict to smaller
sub-sectors which may still provide a non-trivial check.

5

SciPost Phys. 9, 017 (2020)

The subscripts denote the coordinates we use to parameterize the corresponding directions,and it is implied that the complex direction is parameterized by the complex variable z, alongwith its conjugate variable z.
Our first theory, denoted by Tbd , is the theory of open strings on the stack of D2-branes. Thisis a 2d topological gauge theory with the complexified gauge group GLN [18]. The intersectionof the D2-branes with the D4-branes introduces a line operator in this theory. We describe thistheory in §3.3.
Next, we consider the product of two theories, open string theory on the stack of D4branes, and closed string theory on the 6d background sourced by the stack of D2-branes. The
theory on the stack of D4-branes is a 4d analogue of Chern-Simons (CS) gauge theory with thecomplexified gauge group GLK [18]. As it does in the theory on the D2-branes, the intersection
between the D2 and the D4-branes introduces a line operator in this theory as well. This line
sources a flux supported on the 3-sphere linking the line. Our bulk theory is the Kaluza-Klein
compactification of the total 6d theory – 6d closed string theory coupled to 4d CS theory – on
the 3-sphere. We describe the 4d CS theory in §3.4. Let us describe the closed sting theory in
the next section.

3.2

The closed string theory

The closed string theory, denoted by Tcl , is a product of Kodaira-Spencer (also known as BCOV)
theory [31, 32] on C and Kähler gravity [33] on R4 , along with a 3-form flux sourced by thestack of D2-branes.6 Fields in this theory, including ghosts and anti-fields, are given by:
Set of fields,

F := Ω• (R4 ) ⊗ Ω•,• (C) ,(9)i.e., the fields are differential forms on R4 and (p, q)-forms on C.7 The linearized BRST differential acting on these fields is a sum of the de Rham differential on R4 and the Dolbeault
differential on C, leading to the following equation of motion:
dR4 + ∂ C α = 0 ,
α∈F.
(10)
The background field sourced by the D2-branes, let it be denoted by F3 ∈ F , measures theflux through a topological S 3 surrounding the D2-branes, it can be normalized as:
Z
F3 = N .

(11)

S3

Note that the S 3 is only topological, i.e., continuous deformation of the S 3 should not affect theabove equation. This is equivalent to saying that, the 3-form must be closed on the complementof the support of the D2-branes:
dR4 ×C F3 (p) = 0 ,p 6∈ D2 .

(12)

Here the differential is the de Rham differential for the entire space, i.e., dR4 ×C = dR4 +∂ C +∂C .
Moreover, as a dynamically determined background it is also constrained by the equation of
motion (10). In addition to satisfying these equations, F3 must also be translation invariant
along the directions parallel to the D2-branes. The solution is:
F3 =iN
2π(v 2 + y 2 + zz)2

(v d y ∧ dz ∧ dz − y dv ∧ dz ∧ dz − 2z dv ∧ d y ∧ dz) .

(13)

6This flux is analogous to the 5-form flux sourced by the stack of D3-branes in Maldacena’s setup of AdS/CFT
duality between N = 4 super Yang-Mills and supergravity on AdS5 × S 5 [1].
7
We are not being careful about the degree (ghost number) of the fields since this will not be used in this paper.

6

SciPost Phys. 9, 017 (2020)

In general, a closed string background like this might deform the theory on a brane, however,the pullback of the form (13) to the D4-branes vanishes:ι ∗ F3 = 0 ,(14)where ι : R2x, y × Cz ,! R4v,w,x, y × Cz is the embedding of the D4-branes into the entire space.
So the closed string background leaves the D4-brane world-volume theory unaffected.8 Sucha lack of backreaction is a rather drastic simplification of the holographic setup which canoccur in a topological setting such as ours (see also [15]) but this is not a general feature. For
examples of topological holography with nontrivial backreaction see [26, 27].
The flux (13) signals a change in the topology of the closed string background:
R4v,w,x, y × Cz ! R2w,x × R+ × S 3 ,(15)pwhere the R+ is parameterized by r := v 2 + y 2 + zz. This change follows from requiringtranslation symmetry in the directions parallel to the D2-branes and the existence of an S 3
supporting the flux F3 . This S 3 is analogous to the S 5 in the D3-brane geometry supportingthe 5-form flux sourced by the said D3-branes in Maldacena’s AdS/CFT [1]. The coordinater measures distance from the location of the D2-branes. In the absence of a metric we canonly distinguish between the two extreme limits r ! 0 and r ! ∞. The r ! 0 region would
be analogous to Maldacena’s near horizon geometry. In our topological setting there is no
distinction between near and distant, and we treat the entire R2w,x × R+ × S 3 as analogous to
the near horizon geometry. This makes R2w,x × R+ analogous to the AdS geometry. We recall
that, in the AdS/CFT correspondence the location of the black branes and the boundary of AdS
correspond to two opposite limits of the non-compact coordinate transverse to the branes. In
our case r = 0 corresponds to the location of the D2-branes, and we treat the plane at r = ∞,namely:(16)
R2w,x × {∞} ,as analogous to the asymptotic boundary of AdS.
The D4-branes in (8) appear as a defect in the closed string theory, they are analogous
to the D5-branes that were considered in [34] in Maldacena’s setup of AdS/CFT, where they
were presented as holographic duals of Wilson loops in 4d N = 4 super Yang-Mills. For theworld-volume of these branes, the transformation (15) corresponds to:R2x, y × Cz ! R x × R+ × S 2 ,(17)pwhere the R+ direction is parameterized by r 0 := y 2 + zz. The intersection of the boundary
plane (16) and this world-volume is then the line:
R x × {∞},

(18)

at infinity of r 0 . We draw a cartoon representing some aspects of the brane setup in figure 1.
We can now talk about two theories:
1. The 2d world-volume theory of the D2-branes. This is our analogue of the CFT (with aline operator) in AdS/CFT.
8

The flux (13) is the only background turned on in the closed string theory. This can be argued as follows: The
D2-branes introduce a 4-form source (the Poincaré dual to the support of the branes) in the closed string theory.
This form can appear on the right hand side of the equation of motion (10) only for a 3-form field α, which canthen have a non-trivial solution, as in (13). Furthermore, since the equation of motion (10) is free, the non-trivial
solution for the 3-form field does not affect any other field.

7

SciPost Phys. 9, 017 (2020)

2. The effective9 3d theory on world-volume R2w,x ×R+ with a defect supported on R x ×R+ .
This is our analogue of the gravitational theory in AdS background (with defect) in
AdS/CFT.
To draw parallels once more with the traditional dictionary of AdS/CFT [1–3], we should
establish a duality between the operators in the D2-brane world-volume theory and variations of boundary values of fields in the “gravitational" theory on R2w,x × R+ (the boundary is
Rw,x × {∞}). Both of these surfaces have a line operator/defect and this leads to two types ofoperators, ones that are restricted to the line, and others that can be placed anywhere. Local
operators in a 2d surface are commuting, unless they are restricted to a line. Therefore, in bothof our theories, we have non-commutative associative algebras whose centers consist of operators that can be placed anywhere in the 2d surface. For this paper we are mostly concernedwith the non-commuting operators:
1. Operators in the world-volume theory of the D2-branes that are restricted to the D2-D4
intersection.
2. Variations of boundary values of fields in the effective theory along the intersection (18)of the boundary R2w,x × {∞} and the defect on R x × R+ .
In physical string theory, the analogue of the D4-branes would be coupled to the closed stringmodes. In an appropriate large N low energy limit such gravitational couplings can be ignored,
leading to the notion of rigid holography [35]. Since we are working with topological theory
at large N , we are assuming such a decoupling.
The computations in the “gravitational" side will be governed by the effective dynamics onthe defect on R x × R+ . This is the Kaluza-Klein compactification of the world-volume theoryof the D4-branes (with a line operator due to D2-D4 intersection). This 4d theory (which
we describe in §3.4) is familiar from previous works such as [19]. Therefore we use the 4d
dynamics, instead of the effective 2d one for our computations. In terms of Witten diagrams(which we compute in §5) this means that while we have a 1d boundary, the propagators arefrom the 4d theory and the bulk points are integrated over the 4d world-volume R2 × C. We
take the boundary line to be at y = ∞ with some fixed coordinate z in the complex direction.
In future we shall refer to this line as `∞ (z):`∞ (z) := R x × { y = ∞} × {z} .

(19)

A cartoon of our setup
Let us make a diagrammatic summary of our brane setup in Fig 1. In the figure we draw the
non-compact part, namely R2w,x × R+ , of the closed string background (the right hand side of(15)). We identify the location of the 2d black brane and the defect D4-branes, the asymptotic
boundary R2w,x × {∞}, and the intersection between the boundary and the defect. At the topof the picture, parallel to the asymptotic boundary, we also draw the D2-branes. We draw theD2-branes independently of the rest of the diagram because the D2-branes do not exist in thebackreacted bulk, they become the black brane. However, traditionally, parallels are drawn
between the asymptotic boundary and the brane sourcing the bulk (the D2-brane in this case).
The dots on the asymptotic boundary represent local variations of boundary values of fields inthe bulk theory Tbk . The corresponding dots on the D2-brane represent the local operators inthe boundary theory Tbd that are dual to the aforementioned variations. By the duality map in
the figure we schematically represent boundary excitations in the bulk theory corresponding
to some local operators in the dual description of the same dynamics in terms of the boundary
theory.
9

Effective, in the sense that this is the Kaluza-Klein reduction of a 6d theory with three compact directions,though we don’t want to lose any dynamics, i.e., we don’t throw away massive modes.

8

SciPost Phys. 9, 017 (2020)

Duality map

D2-brane•Belongs to thecenter.
R2w,x × {∞}•`∞ (z)

r

D4-branex
2d black branew

Figure 1: D2-brane, and the non-compact part of the backreacted bulk.

3.3

BF: The theory on D2-branes

This is a 2d topological gauge theory on the stack of N D2-branes (see (8)), supported on R2w,x ,with complexified gauge group GLN . The field content of this theory is:
Field

B
A

Valued inΩ0 (R2 ) × glN .
Ω1 (R2 ) × glN(20)

A is a Lie algebra valued connection and B is a Lie algebra valued scalar, both complex. The
curvature of the connection is denoted as F = dA + A ∧ A. The action is given by:Z

trN (BF) ,SBF :=

(21)

R2w,x

where the trace is taken in the fundamental representation of glN .
We consider this theory in the presence of a line operator supported on R x × {0}, causedby the intersection of the D2 and D4-branes. The line operator is defined by a fermionic
quantum mechanical system living on it.10 The fields in the quantum mechanics (QM) are K
fundamental (of glN ) fermions and their complex conjugates:
Field Valued inψiΩ0 (R x ) × N ,ψiΩ0 (R x ) × N

i ∈ {1, · · · , K} ,(22)

where N refers to the fundamental representation of glN and N to the anti-fundamental. The
fermionic system has a global symmetry GLN × GLK . These fermions couple naturally to theglN connection A of the BF theory. The action for the QM is given by:Zj
SQM :=
ψi dψi + ψi Aψi + ψ j Ai ψi ,(23)
Rx

where we have introduced a background glK -valued gauge field A ∈ Ω1 (R x )× glK . Note that theterms in the above action are made glN invariant by pairing up elements of N with elementsof the dual space N.
This closely resembles the D3-D5 system in type IIB string theory considered in [34], there too a fermionic
quantum mechanics lived on the intersection, giving rise to Wilson lines upon integrating out the fermions. Notethat we could have considered bosons, instead of fermions, living on the line, without any significant change to
our following computations. This would be similar to the D3-D3 system considered in [34, 36].
10

9

SciPost Phys. 9, 017 (2020)

Our first theory is this BF theory with the line operator, schematically:

Tbd := BFN ⊗N QM N ×K ,(24)

where the subscripts on BF and QM refer to the symmetries (GLN and GLN × GLK respectively)of the respective theories and the subscript on ⊗ implies that the GLN is gauged. There are
two types of gauge (glN ) invariant operators in the theory:11
for n ∈ N≥0 ,operators restricted to R x : O ji [n] := ħ1h ψ j Bn ψi ,operators not restricted to R x : O[n] := ħ1h trN Bn .

(25)

Unrestricted local operators in two topological dimensions can be moved around freely, implying that for any n ≥ 0, the operator O[n] commutes with all of the operators defined above.12
The operator algebra of the 2d BF theory consists of all theses operators but in this paper we
focus on the non-commuting ones, in other words we, focus on the quotient of the full operator
algebra of the boundary theory by its center.13 We shall compute their Lie bracket in §4, which
will establish an isomorphism with the Yangian. Had we included the commuting operators
as well we would have found a central extension of the Yangian. In sum, the operator algebra
we construct from the theory Tbd is:(26)
AOp (Tbd ) := O ji [n], O[n] /(O[n]) .
By the notation (x, y, · · · ) we mean the algebra generated by the set of operators {x, y, · · · }over C.
Remark 1 (A speculative link). Note that it is possible to lift our D2 and D4 branes to type IIB
string theory while maintaining a one dimensional intersection. This results in a D3-D5 setup(studied in particular in [34]) where on the D3 brane we find the N = 4 Yang-Mills theory witha Wilson line.14 In [38–40], the authors considered local operators in the N = 4 Yang-Mills
that are restricted to certain Wilson lines. With the proper choice of Wilson lines, Localization
reduces this setup to 2d Yang-Mills theory with Wilson lines – local operator insertions along
the Wilson lines in 4d reduce to local operator insertions along the Wilson lines in 2d [41]. 2d
BF theory is the zero coupling limit of 2d Yang-Mills theory. We therefore expect the algebra
constructed in this section to be related to the algebra constructed in the aforementioned
references, at least in some limit.15 The algebra in [40] would correspond to the K = 1
instance of our algebra, it may be an interesting check to compute the analogue of the algebrain [40] for higher K.
4

3.4

4d Chern-Simons: The theory on D4-branes

This is a 4d gauge theory on the stack of K D4-branes, supported on R2x, y × Cz with the line
L := R x × (0, 0, 0) removed and with the (complexified) gauge group GLK . The notation of
distinguishing directions by R and C is meant to highlight the fact that observables in thistheory depend only on the topology of the real directions and depend holomorphically on
the complex direction. In particular, they are independent of the coordinates x and y thatparameterize the R2 , and depend holomorphically on z which parameterizes the C. Due tothe removed line, we can represent the topology of the support of this theory as (c.f. (17)):
M := R × R+ × S 2 .

(27)


The ħ
h−1 appears in these definitions because the action (23) will appear in path integrals as exp −ħh−1 SQM ,which means functional derivatives with respect to Aij inserts operators that carry ħ
h−1 .
12These operators are represented by the red dot on the D2-brane in figure 1.
13
We shall similarly quotient out the center in the bulk theory as well.
14
It is also interesting to note that the D5 brane in an Omega background reproduces the 4d CS theory [37].
15
We thank Shota Komatsu for pointing out this interesting possibility.
11

10

SciPost Phys. 9, 017 (2020)

The field of this theory is just a connection:
Field

Valued inAΩ1 (R2 ×C\L)
⊗ glK
(dz)

.

(28)

The above notation simply means that A is a glK -valued 1-form without a dz component. The
theory is defined by the action:16
Z
i
SCS :=dz ∧ CS(A) ,
(29)
2π M
where CS(A) refers to the standard Chern-Simons Lagrangian:
2
CS(A) = trK A ∧ dA + A ∧ A ∧ A ,3(30)

where the trace is taken over the fundamental representation of glK . This theory is a 4d
analogue of the, perhaps more familiar, 3d Chern-Simons theory. We shall therefore refer to it
as the 4d Chern-Simons theory and sometimes denote it by CS4K or just CS.
The removal of the line L from R2 × C is caused by the D2-D4 brane intersection. Notethat from the perspective of the CS theory, the D2-D4 intersection looks like a Wilson line.
This means that we should be quantizing the CS theory on M with a background electric flux
supported on the S 2 inside M . Alternatively, we can quantize the CS theory on R2 × C with a
Wilson line inserted along L.17 The choice of representation for the Wilson line is determinedby the number, N , of D2-branes – let us denote this representation as % : glK ! End(V ). With
this choice, the Wilson line is defined as the following operator:
Z%(A) ,

W% (L) := P exp

(31)

L

where P exp implies path ordered exponentiation, made necessary by the fact that the exponent is matrix valued. The above operator is valued in End(V ). This in general means that thefollowing expectation value:R
DAW% (L) exp − ħ1h SCSRW% (L) =,(32)
DA exp − ħ1h SCS
is valued in Hom(H−∞ ⊗ V, H+∞ ⊗ V ), where H±∞ are the Hilbert spaces of the CS4K theory
on the Cauchy surfaces perpendicular to L at x = ±∞, in the absence of the Wilson line.
However, for the particular CS theory, these Hilbert spaces are trivial and we end up with amap that transports vectors in V from x = −∞ to x = +∞:W% (L) : V−∞ ! V+∞ .

(33)

In picture this operator may be represented as:

W% (L) :W% (L)

Vx = −∞

V.

(34)

x = +∞

This theory was proposed in [42] to explain the representation theory of quantum affine algebras and more
recently studied in [16, 19, 43, 44] as a way of producing integrable lattice models using Wilson lines.
17
Recall that in case of the BF theory the line operator at the D2-D4 intersection was described by a fermionic
QM. We could do the same in this case. However, in this case it proves more convenient to integrate out thefermion, leaving a Wilson line in its place. The mechanism is the same that appeared for intersection of D3 andD5-branes in physical string theory [34].
16

11

SciPost Phys. 9, 017 (2020)

The CS theory is quantized with some fixed boundary profile of the connection along theboundary R x ×{∞}×S 2 .18 To express the dependence of expectation values on this boundary
value we put a subscript, such as W% (L) A. Since we are essentially interested in the KaluzaKlein reduced theory on R x × R+ we mostly care about the value of the connection along theboundary line (defined in (19)) `∞ (z) ⊂ R x × {∞} × S 2 .
To define our second theory, we start with the product of the closed string theory and the
CS theory, Tcl ⊗ CS4K , supported on R2w,x × R+ × S 3 and compactify on S 3 , our notation for thistheory is the following:3
Tbk := πS∗ Tcl ⊗ CS4K .
(35)
We can put the theory Tbd (24) on the plane R2w,x at infinity of R2w,x × R+ . This plane has
a distinguished line R x × {∞} (18) where the D4-brane world volume intersects.19 Alongthis line we have the glK gauge field which couples to the fermions of the QM in Tbd (thiscoupling corresponds to the last term in (23)). Boundary excitations from arbitrary points on
Rw,x ×{∞} will correspond to operators in the BF theory that are commuting, since these local
excitations on a plane are not ordered. The non-commutative algebra we are interested in the
BF theory is the algebra of gauge invariant operators restricted to a particular line. Similarly,
in the “gravitational" side of the setup, we are interested in boundary excitations restricted to
the line `∞ (z). Let us look a bit more closely at the coupling between the connection A andthe fermions:
Z
i j
1
Iz :=
ψ Ai ψ j ,`∞ (z) = R x × { y = ∞} × {z} .
(36)ħh ` (z)∞A small variation of z leads to coupling between the fermions and z-derivatives of the connection:Z∞X(δz)n i n j
1
Iz+δz =(37)
ψ ∂z Ai ψ j .
ħ
h ` (z) n!
n=0∞

In the BF theory, the field B corresponds to the fluctuation of the D2-branes in the transverse C
direction [18]. Therefore, we can interpret the above varied coupling term as saying that thejoperator in the boundary theory Tbd that couples to the derivative ∂zn Ai is precisely the operatori

O ji [n] = ħ
h−1 ψ Bn ψ j (c.f. (25), (26)). This motivates us to look at functional derivatives ofj

W% (L) A with respect to ∂zn Ai at fixed points along `∞ (z), such as:δ
n jδ∂z 1 Ai1 (p1 )
1

···

δ
n
j
δ∂z m Aim (pm )m

W% (L) A ,p1 , · · · , pm ∈ `∞ (z) .

(38)

Just as the expectation value W% (L) A is End(V )-valued, these functional derivatives are
End(V )-valued as well. The action is given by applying the functional derivative on W% (L) A (ψ)
for any ψ ∈ V . Let us denote this operator as
T ji [n] : `∞ (z) × V ! V ,p ∈ `∞ (z) ,18
19

T ji [n](p) : ψ 7!

δj
δ∂zn Ai (p)W% (L) A (ψ) ,The boundary was chosen to respect the symmetry of the Wilson line along L.
After aligning the v-coordinates of the plane and the D4-branes.

12

(39)

SciPost Phys. 9, 017 (2020)

which can be pictorially represented by slight modifications of (34):
W% (L)

y = 0, ψT ji [n](p)(ψ).

δ
jδ∂zn Aiy =∞x = −∞x=p

x = +∞

ii(40)

Composition of these operators, such as T j 1 (p1 ) · · · T j m (pm ), is defined by the expression (38).
1m
A more precise and computable characterization of these operators and their composition interms of Witten diagrams [3] will be given in §5 (see (122)). Due to topological invariancealong the x-direction, the operator T ji [n](p) must be independent of the position p. However,since these operators are positioned along a line, their product should be expected to dependon the ordering, leading to a non-commutative associative algebra. We can now define thesecond algebra to appear in our example of holography:
ASc (Tbk ) := T ji [n] ,(41)i.e., the complex algebra generated by the set {T ji [n]}.
Remark 2 (Center of the algebra). In the BF theory we mentioned gauge invariant operatorsthat belong to the center of the algebra. Clearly, the holographic dual of those operators do
not come from the CS theory, rather they come from the closed string theory. A 2-form field
φ = φwx dw ∧ dx + · · · from the closed string theory deforms the BF theory as:Z
dw ∧ dx ∂zn φwx trN (Bn ) .
SBF ! SBF +
(42)
R2w,x

Functional derivatives with respect to the fields ∂zn φw,x placed at arbitrary locations on theasymptotic boundary R2w,x × {∞} correspond to inserting the operators trN Bn in the BF theory.20 As we did in the BF theory, we are going to ignore these operators now as well.
4
After all this setup, we can present the main result of this paper:
Theorem 1. In the limit N ! ∞, both the algebra of local operators (26) along the line operator
in the theory Tbd = BFN ⊗N QM N ×K , and the algebra of scatterings from a line in the boundary3(41) of the theory Tbk = πS∗ Tcl ⊗ CS4K are isomorphic to the Yangian of glK , i.e.:N !∞AOp (Tbd ) ∼=Yħh (glK )

N !∞

∼
=

ASc (Tbk ) .

(43)

The rest of the paper is devoted to the explicit computations of these algebras.

AOp (Tbd ) from BF ⊗ QM theory

4

In this section we prove the first half of our main result (Theorem 1):
Proposition 1. The algebra AOp (Tbd ), defined in the context of 2d BF theory with the gauge
group GLN coupled to a 1d fermionic quantum mechanics with global symmetry GLN × GLK , isisomorphic to the Yangian of glK in the limit N ! ∞:N !∞AOp (Tbd ) ∼=20

Yħh (glK ) .

These functional derivatives are represented by the red dot on the asymptotic boundary in figure 1.

13

(44)

SciPost Phys. 9, 017 (2020)

The BF theory coupled to a fermionic quantum mechanics was defined in §3.3, let us repeat
the actions here:
STbd = SBF + SQM ,(45)
where:SBF =Z

trN (BdA + B[A, A])(46)
ψi dψi + ψi Aψi .

(47)

R2w,x

and SQM =

Z



Rx

We no longer need the source term, i.e., the coupling to the background glK connection (c.f.
(23)). Let us determine the propagators now.
The BF propagator is defined as the 2-point correlation function:Pαβ (p, q) := Bα (p)Aβ (q) .

(48)

We choose a basis {τα } of glN which is orthonormal with respect to the trace trN :trN (τα τβ ) = δαβ .

(49)Then the two point correlation function becomes diagonal in the color indices:Pαβ (p, q) ≡ δαβ P(p, q) .

(50)

We shall often refer to just P as the propagator, it is determined by the following equation:21
1
dP(0, p) = δ2 (p)dw ∧ dx .
ħ
h

(51)

Once we impose the following gauge fixing condition, analogous to the Lorentz gauge:d ? P(0, p) = 0 ,(52)

the solution is (using translation invariance to replace the 0 with an arbitrary point):P(p, q) =ħhdφ(p, q) ,
2π

(53)

where φ(p, q) is the angle (measured counter-clockwise) between the line joining p-q andany other reference line passing through p. In Feynman diagrams we shall represent thispropagator as:q .
P(p, q) = p(54)
Similarly, the propagator in the QM is defined by:with the solution:

¬ a
¶
1
j
j
∂ x 2 ψi (x 1 )ψ b (x 2 ) = δ ab δi δ1 (x 1 − x 2 ) ,
ħ
h

(55)

¶
a
j
j
ψi (x 1 )ψ b (x 2 ) = δ ab δi ħh ϑ(x 2 − x 1 ) ,

(56)

¬

where ϑ(x 2 − x 1 ) is a unit step function. Anti-symmetry of the fermion fields dictates:
¬
¶
¬ a
¶
a
j
j
j
ψ b (x 1 )ψi (x 2 ) = − ψi (x 2 )ψ b (x 1 ) = −δ ab δi ħh ϑ(x 1 − x 2 ) .

(57)

A minor technicality: P(p, q) is a 1-form on R2p ×R2q and in (51), by P(0, p) we mean the pull-back of P ∈ Ω2 (R4 )by the diagonal embedding R2 ,! R2 × R2 .
21

14

SciPost Phys. 9, 017 (2020)

We take the step function to be:

 1/2
1
0
ϑ(x) = sgn(x) =2
 −1/2for x > 0
for x = 0 .
for x < 0(58)

Then we can write:
¬

¶ ¬
¶
a
a
h
j
j
jħ
ψi (x 1 )ψ b (x 2 ) = ψ b (x 1 )ψi (x 2 ) = δ ab δi sgn(x 2 − x 1 ) .
2(59)This propagator does not distinguish between ψ and ψ and it depends only on the order ofthe fields, not their specific positions. In Feynman diagrams we shall represent this propagatoras:ħhsgn(x 2 − x 1 ) = x
(60)
x2 ,
1
2
where the curved line refers to the propagator itself and the horizontal line refers to the supportof the QM, i.e., the line w = 0. We now move on to computing operator products that will giveus the algebra AOp (Tbd ).
Remark 3 (Fermion vs. Boson - Propagator). We might as well have considered a bosonic
QM instead of a fermionic QM. At present, this is an arbitrary choice, however, if one starts
from some brane setup in physical string theory and reduce it to the topological setup we areconsidering by twists and Ω-deformations,22 then depending on the starting setup one mightend up with either statistics. Let us make a few comments about the bosonic case. In the firstorder formulation of bosonic QM the action looks exactly as in the fermionic action 47 except
the fields would be commuting – let us denote the bosonic counterpart of ψ and ψ by φ and φrespectively. Then, instead of the propagator (59), we would have the following propagator:23
¶ ¬
¬ a
¶
a
h
j
j
jħ
− φ i (x 1 )φ b (x 2 ) = φ b (x 1 )φ i (x 2 ) = δ ab δi sgn(x 2 − x 1 ) .
(61)
2Note that the extra sign in the first term (compared to (59)) is consistent with the commutativity of the bosonic fields:
¬ a
¶ ¬
¶
a
j
j
φ i (x 1 )φ b (x 2 ) = φ b (x 2 )φ i (x 1 ) .
(62)
The bosonic propagator (61) distinguishes between φ and φ, in that, the propagator is positiveif φ(x 1 ) is placed before φ(x 2 ), i.e., x 1 < x 2 , and negative otherwise.
4

4.1

Free theory limit, O(ħh0 )Interaction in the quantum mechanics is generated via coupling to the glN gauge field (see(47)). Without this coupling, the quantum mechanics is free. In this section we compute theoperator product between O ji [m] and Olk [n] in this free theory, which will give us the classicalalgebra.
Let us denote the operator product by ?, as in:
O ji [m] ? Olk [n] .
22

(63)

We describe one such specific procedure in §6.
We have chosen the overall sign of the propagator to make comparision between Feynman diagrams involving
bosonic operators and fermionic operators as simple as possible. However, the overall sign is not important for thedetermination of the algebra. The parameter ħh enters the algebra as the formal variable deforming the universal
enveloping algebra U(glK [z]) to its Yangian, and the sign of ħ
h is irrelevant for this purpose.
23

15

SciPost Phys. 9, 017 (2020)

The classical limit of this product has an expansion in Feynman diagrams where we ignore all
diagrams with BF propagators. Before evaluating this product let us illustrate the computationsof the relevant diagrams by computing one exemplary diagram in detail.
Consider the following diagram:24
G ik
jl [Í· Î](x 1 , x 2 ) :=(64)
x1
O ij [m]x2
Olk [n]a

We are representing the operator O ji [m] = ħ1h ψ j (Bm )ab ψib by the symbol

where the threeadots represent the three fields ψ j , (Bm )ab , and ψib respectively. The coordinate below an op-

erator in (64) represents the position of that operator and the lines connecting different dots
are propagators. Depending on which dots are being connected a propagator is either the BF
propagator (53) or the QM propagator (59). The value of the diagram is then given by:
1 a
1 c i1ψ (x 1 )(B(x 1 )m )ab ħhδ δ (B(x 2 )n )dc ψkd (x 2 ) ,ħ
h j
2 b lħh1 i=δ ψ (x 1 )B(x 1 )m B(x 2 )n ψk (x 2 ) .
2ħ
h l j

G ik
jl [Í· Î](x 1 , x 2 ) =(65)

In the second line we have hidden away the contracted glN indices. In computing the operator
product (63) only the following limit of the diagram is relevant:lim G ikjl [Í· Î](x 1 , x 2 ) =x2 ! x1

1 i
1
δ ψ Bm+n ψk = δli O jk [m + n] .
2ħ
h l j
2

(66)

We have ignored the positions of the operators, because the algebra we are computing mustbe translation invariant. Reference to position only matters when we have different operators
located at different positions.
We can now give a diagramatic expansion of the operator product (63) in the free theory:
O ji [m] ? Olk [n]x 2 ! x1

=

+
x1

x1

x2

x2
(67)

+

x1

x2

+

.
x1

x2

We have omitted the labels for the operators in the diagrams. It is understood that the first
operator is O ji [m] and the second one is Olk [n]. Summing these four diagrams we find:1
1
1
O ji [m] ? Olk [n] = O ji [m]Olk [n] + δli O jk [m + n] − δ kj Oli [m + n] + δli δ kj trN Bm+n .
2
2
4

(68)

The product in the first term on the right hand side of the above equation is a c-number product,hence commuting. The sign of the third term comes from the first diagram in the second linein (67). In short, this comes about by commuting two fermions, as follows:
lim G ikjl [Î ·Í](x 1 , x 2 ) =x2 ! x1

1
1
1 k i m+n
δj ψ Bψl = − δ kj ψl Bm+n ψi = − δ kj Oli [m + n] .
2ħ
h
2ħ
h
2

(69)

24
The reader can ignore the elaborate symbols (triangles and as such) that we use to refer to a diagram. They
are meant to systematically identify a diagram, but for practical purposes the entire expression can be thought ofas an unfortunately long unique symbol assigned to a diagram, just to refer to it later on.

16

SciPost Phys. 9, 017 (2020)

Using (68) we can compute the Lie bracket of the algebra AOp (Tbd ) in the classical limit:
(70)
O ji [m], Olk [n] = δli O jk [m + n] − δ kj Oli [m + n] .
?

This is the Lie bracket in the loop algebra glK [z].25
Remark 4 (Fermion vs. Boson - Classical Algebra). How would the bracket (70) be affected if
we had a bosonic QM? It would not. The first and the fourth diagrams from (67) would stillcancel with their counterparts when we take the commutator. The value of the second diagram,(66), remains unchanged. In computing the value of the third diagram (see (69)) we get an
extra sign compared to the fermionic case because we don’t pick up any sign by commuting
bosonos, however, we pick up yet another sign from the propagator relative to the fermionicpropagator (see Remark 3 – compare the bosonic (61) and fermionic (59) propagators).

4.2

Loop corrections from BF theory

Interaction in the BF theory comes from the following term in the BF action (46):Z

Bα Aβ ∧ Aγ ,

fαβγ

(72)

R2

where the structure constant fαβγ comes from the trace in our orthonormal basis (49):fαβγ = trN (τα [τβ , τγ ]) .

(73)

In Feynman diagrams this interaction will be represented by a trivalent vertex with exactly 1
outgoing and 2 incoming edges. Including the propagators for the edges, such a vertex will
look like:
q2 , β

ħ
h2
=
f αβγ
(2π)3

p
q3 , γq1 , αZp∈R2dq1 φ(p, q1 ) ∧ dq2 φ(p, q2 ) ∧ dq3 φ(p, q3 ) ,(74)=: V αβγ (q1 , q2 , q3 ) .

We have given the name V αβγ to this vertex function.
Possibilities of Feynman diagrams are rather limited in the BF theory. In particular, there
are no cycles.26 This means that there is only one possible BF diagram that will appear in our
computations, which is the following:
.
j

(75)

j

The isomorphism is given by: O ji [m] 7! z m ei , where ei are the elementary matrices of dimension K × K
satisfying the relation:jjj[ei , elk ] = δil ek − δk eli .
(71)
25

26

By cycle we mean loop in the sense of graph theory. In this paper when we write loop without any explanation,
we mean the exponent of ħh, as is customary in physics. This exponent is related but not always equal to the numberof loops (graph theory). Therefore, we reserve the word loop for the exponent of ħh, and the word cycle for what
would be loop in graph theory.
Let us illustrate why there are no cycles in BF Feynman diagrams. Consider the cycle. The three propa-

gators in the cycle contribute the 3-form dφ1 ∧ dφ2 ∧ dφ3 to a diagram containing the cycle, where the φ’s are theangles between two successive vertices. However, due to the constraint φ1 + φ2 + φ3 = 2π, only two out of thethree propagators are linearly independent. Therefore, their product vanishes.

17

SciPost Phys. 9, 017 (2020)

The middle operator looks slightly different because this operator involves the connection A
and an integration, as opposed to just the B field, to be specific,Z
1=
ψ Aψ i .
(76)
ħ
h R i
This term is the result of the insertion of the term coupling the fermions to the glN connection in
the QM action (47). In doing the above integrationover R we shall take ψ and ψ to be constant.
In other words, we are taking derivatives of the fermions to be zero. The reason is that, theequations of motion for the fermions (derived from the action (47)), namely dψi = −Aψi anddψi = Aψi , tell us that derivatives of the fermions are not gauge-invariant quantities – andwe want to expand the operator product of gauge invariant operators in terms of other gauge
invariant operators only.27
In the following we shall consider the diagram (75) with all possible fermionic propagators
added to it.
4.2.1 0 fermionic propagators
We are mostly going to compute products of level 1 operators, i.e., O ji [1], this is because
together with the level 0 operators, they generate the entire algebra. Without any fermionicpropagators, we just have the diagram (75):G ikjl [··](x 1 , x 2 ) :=.
x1
O ij [1]

(77)

x2
R x
1
ψAψ O k [1]ħhl

In future, we shall omit the labels below the operators to reduce clutter. In terms of the BF
vertex function (74), the above diagram can be expressed as:
Z
1
ik
i
k
G jl [··](x 1 , x 2 ) = 3 ψ j τα ψ ψτβ ψψl τγ ψV αβγ (x 1 , x, x 2 ) .
(78)
ħ
h
Rx
We have used the expansions of B = Bα τα and A = Aβ τβ in the orthonormal glN basis{τα }. As defined in (74), the vertex function V αβγ is a 2d integral of a 3-form, therefore, theintegration of the vertex function on a line gives us a number. It will be convenient to divideup the integral of the vertex function into three integrals depending on the location of thepoint x relative to x 1 and x 2 :
Z
αβγ

αβγ

αβγ

V αβγ (x 1 , x, x 2 ) = V·|| (x 1 , x 2 ) + V|·| (x 1 , x 2 ) + V||· (x 1 , x 2 ) ,

(79)

Rx

where,
αβγ
V·|| (x 1 , x 2 ) :=αβγ
V|·| (x 1 , x 2 ) :=αβγ
V||· (x 1 , x 2 ) :=Z

V αβγ (x 1 , x, x 2 ) =
x<x 1

Z

ħ
h2 αβγf,24

V αβγ (x 1 , x, x 2 ) =x 1 <x<x 2ZV αβγ (x 1 , x, x 2 ) =x 2 <xħ
h2 αβγf,24

ħh2 αβγ
f
.
24

(80a)
(80b)
(80c)

27
An alternative, and perhaps more streamlined, way to say this would be to formulate all the theories in the
BV/BRST formalism, where operators are defined, a priori, to be in the cohomology of the BRST operator, which
would exclude derivatives of the fermions to begin with.

18

SciPost Phys. 9, 017 (2020)

We evaluate these integrals in Appendix §A. Adding them up and substituting in (78) we getfrom the diagram (77):G ikjl [··](x 1 , x 2 )x1! x2=1ψ τα ψi ψτβ ψψl τγ ψk f αβγ .
8ħ
h j

(81)

Since the glN indices are all contracted, we can choose a particular basis to get an expression
independent of any reference to glN . Choosing the elementary matrices as the basis we get
the following expression:
π2
bd fψ ea ψi ψecd ψψl eef ψk f ace
.
2ħ
h j b
c
the elementary matrices
eab d

G ik
jl [··] =definition of
=

d
a
c
ψ j eab ψi = ψ j eab d ψic = ψ j ψib and in this basis the structure constant is:
Using

the

(82)
δda δ cb

we

bd f
f ace
= δad δcf δeb − δcb δed δaf .

get

(83)

Using these expressions in (82) we get:
1 
ψ j ψm ψm ψk ψl ψi − ψl ψm ψm ψi ψ j ψk ,8ħh

1 2 mik[0]O jk [0] .
[0]Oli [0] − Olm [0]Omh O j [0]Om
= ħ
8

G ik
jl [··] =(84)

The above expression is anti-symmetric under the exchange (i, j) ↔ (k, l), therefore, the contribution of this diagram to the Lie bracket (70) is twice the value of the diagram.
4.2.2

1 fermionic propagator

We have the following six diagrams:

G ik
jl [·Í· Î] =, G ikjl [· Î ·Í] =,G ik
jl [Í· Î ·] =, G ikjl [Î ·Í·] =,G ik
jl [Í · · Î] =, G ikjl [Î · · Í] =

(85)

.

In all the above diagrams, the left and the right most operators are O ji [1] and Olk [1] respectively, and all the graphs are functions of x 1 and x 2 , where these two operators are located.
Let us explain the evaluation of the top left diagram in detail. Written explicitly, this diagram
is:
Z
¶
b ¬
a
c
1
ik
G jl [·Í· Î](x 1 , x 2 ) = 3
ψ j (x 1 )τα ψi (x 1 )ψm (x) τβ a ψmb (x)ψl (x 2 )
ħ
h Rx
d
× τγ c ψkd (x 2 ) V αβγ (x 1 , x, x 2 ) ,(86)
19

SciPost Phys. 9, 017 (2020)

where the two point correlation function is the QM propagator (59). The integrand above
depends on the position only to the extend that they depend on the ordering of the positions,since we are only quantizing the constant modes of the fermions.28 The propagator between
the two fermions gives a propagator which depends on the sign of x 2 − x (see (59), (60)),
since we are integrating over x, this propagator will change sign depending on whether x isto the left or to the right of x 2 .29 Therefore, we can write this graph as:
1



αβγ
αβγ
αβγ
i
k
ψ
τ
ψ
ψ
τ
τ
ψ
V
+
V
−
V
,
α
β
γ
j
l
·||
|·|
||·
ħ
h2
1
1
ψ τα ψi ψl τβ τγ ψk f αβγ =ψ τα ψi ψl τδ ψk fβγ δ f αβγ .
=
24 j
24 j

G ik
jl [·Í· Î] =(87)

Due to the symmetry fβγ δ f αβγ = fβγ α f δβγ , the above expression is symmetric under theexchange (i, j) ↔ (k, l), therefore this diagram does not contribute to the Lie bracket (70).
The diagrams G ik[· Î ·Í], G ik
[Í· Î ·], and G ik[Î ·Í·] do not contribute to the Lie bracket forjljljl
exactly the same reason. The remaining two diagrams evaluate to the following expressions:
1 αβγ ifδl ψ j τα τγ ψk ψτβ ψ ,8ħh1 αβγ kG ikf
δ j ψl τγ τα ψi ψτβ ψ .
jl [Î · · Í] = −
8ħ
h

G ik
jl [Í · · Î] =

(88a)
(88b)

Their sum is symmetric under the exchange (i, j) ↔ (k, l),30 and therefore these diagrams do
not contribute to the Lie bracket either.
None of the diagrams with one fermionic propagator contributes to the Lie bracket.
4.2.3

2 fermionic propagators

There are nine ways to join two pairs of fermions with propagators:

G ik[Í·ÎÏ·È]jl

G ik[ÈÍ·Î·Ï]jl

G ik[Î·Ï·ÈÍ]
jl

(89)
G ik
[Î·ÈÍ·Ï]jl

G ik[ÈÍ·ÎÏ·]jl

G ik[ÎÏ·Í·È]jl

G ik[ÈÍ··ÎÏ]
jl

28

G ik
[Í·È·ÎÏ]jl

G ik[·ÈÍ·ÎÏ]jl

Derivatives of the fermions are not gauge invariant.
This is the reason why we computed the integrals (80) separately depending on the position of x.
30
The opposite ordering of τα and τγ cancels the sign, using the anti-symmetry of the indices on the structure
constant.
29

20

SciPost Phys. 9, 017 (2020)

The left and the right most operators in all of the above diagrams are O ji [1] and Olk [1] respectively.
All three of the diagrams in the bottom line vanish. This is because joining all the fermionsin two operators with propagators introduces a trace trN (τα τβ ) of glN generators when thesame color indices, α and β in this case, are contracted with the structure constant comingfrom the BF interaction vertex, as in trN (τα τβ ) f αβγ . Since the trace is symmetric and thestructure constant is anti-symmetric, these three diagrams vanish.
Computation also reveals the following relations among the four diagrams at the top right2 × 2 corner of (89):ik
G ik
jl [È Í· ÎÏ] = G jl [Î · Ï · È Í] ,ik
G ik
jl [ÎÏ · Í· È] = G jl [Í· È · ÎÏ] ,(90)

together with the fact that G ik[È Í· Î · Ï] + G ik
[ÎÏ ·Í· È] is symmetric under the exchangejljl(i, j) ↔ (k, l). The above relations and symmetry implies that when anti-symmetrized withrespect to (i, j) ↔ (k, l), the sum of the four diagrams appearing in the above relations vanish.
In a similar vein, the sum G ik
[Í· ÎÏ·È]+G ik[Î·È Í· Ï] also turns out to be symmetric underjljl(i, j) ↔ (k, l) and therefore these two diagrams do not contribute to the Lie bracket either.
None of the diagrams with two fermionic propagators contributes to the Lie bracket.
4.2.4 3 fermionic propagators
There are two ways to join all the fermions with propagators:

,

(91)

As before, the left and the right most operators are O ji [1] and Olk [1] respectively. Both ofthese diagrams are proportional to δli δ kj , in particular, they are symmetric under the exchange(i, j) ↔ (k, l), and therefore do not contribute to the Lie bracket.
4.2.5

Lie bracket

Since only the diagram with zero fermionic propagator (84) survives the anti-symmetrization,
the Lie bracket (70) up to O(ħh2 ) corrections becomes:ki
O ji [1], Olk [1] = δli O jk [2] − δ kj Oli [2] + G ik
jl [··] − Gl j [··] ,?

= δli O jk [2] − δ kj Oli [2] +ħh2  mki
O j [0]Om[0]Oli [0] − Olm [0]Om[0]O jk [0] . (92)4Though we have only computed up to 2-loops diagrams, this result is exact, because there are
no more non-vanishing Feynman diagrams that can be drawn.
Since (92) is not among the standard relations of the Yangian that are readily available in
the literature, we shall now make a change of basis to get to a standard relation. First note
that, the product of operators in the right hand side of the above equation is not the operator
product, this product is commutative (anti-commutative for fermions) and therefore we canwrite it in an explicitly symmetric form, such as:¦©
kk
O jm [0]Om[0]Oli [0] = O jm [0], Om[0], Oli [0] ,
(93)

21

SciPost Phys. 9, 017 (2020)

where the bracket means complete symmetriazation, i.e., for any three symbols O1 , O2 and O3
with a product we have:{O1 , O2 , O3 } =1 X
Os(1) Os(2) Os(3) ,3! s∈S(94)

3

where S3 is the symmetric group of order 3!. With this symmetric bracket, let us now define:¦©
iun vpq r t k
Q ik
Onm [0], Opo [0], Ots [0] ,(95)
jl := f j vm f uor f qsl
i jk

where f l mn are the glK structure constants in the basis of elementary matrices. Using the formof the gl structure constant in the basis of elementary matrices (c.f. (83)) we can write:
¦
©
¦
©
¦
©
 m n i
i
m
k
k
m
i
k
i
m
n
k
Q ik
(96)
jl = 3 Ol , O j , Om − 3 O j , Ol , Om + δ j Ol , Om , On − δl O j , Om , On .
We have ignored to write the [0] for each of the operators. Using the above expression we canre-write (92) as:


ħ
h2 ik
e i [2] +e k [2] − δ k O
Q ,(97)
O ji [1], Olk [1] = δli O
j l
j
?
12 jl
with the redefinition:ħh2 ¦ m n k ©
e k [2] := O k [2] −(98)
O j , Om , On .
O
j
j
12
¦
©
n
Note that, O jm , Om
, Onk does indeed transform as an element of glK , since it only has a pair offundamental-anti-fundamental glK indices free. This makes the redefinition of O jk [2] possible.
The Lie bracket (97) is how the Yangian was presented in [19].
Remark 5 (Fermion vs. Boson - Quantum Algebra). In Remark 4 we pointed out that theclassical part of the algebra (97) remains unchanged if we replace the fermionic QM on thedefect with a bosonic QM. This remains true at the quantum level – though a bit tedious, it canbe readily verified by using the bosonic propagator (61) and keeping track of signs throughthe computations of this section without any other modifications.
4

4.3

Large N limit: The Yangian

In (97) we have already found a defining relation for the Yangian, and this relation holds at
finite N . However, for finite N there can be extra relations among the operators O ji [m]. For
example, for finite N , B is a finite dimensional (namely N × N ) matrix and therefore BN canbe written as a linear combination of Bm with m < N . This can lead to relations among theoperators O ji [m]. To find all the relations precisely one must quantize the BF theory taking thedirection of the 1d defect to be time and establish the relations among the operators O ji [m]on the resulting Hilbert space. Any such relation reduces the operator algebra to a quotientof the Yangian. This will be done in a future work, for now we note that we can avoid theseincidental relations by taking the large N limit where all matrices are infinite dimensional.
In this limit we therefore have the standard Yangian, as opposed to some quotient of it. Thisconcludes our proof for Proposition 1.

5 ASc (Tbk ) from 4d Chern-Simons Theory
In this section we prove the second half of our main result (Theorem 1):

22

SciPost Phys. 9, 017 (2020)

Proposition 2. The algebra ASc (Tbk ), defined in (41) in the context of 4d Chern-Simons theory,is isomorphic to the Yangian Yħh (glK ):N !∞ASc (Tbk ) ∼=Yħh (glK ) .

(99)

The 4d Chern-Simons theory with gauge group GLK , also denoted by CS4K , is defined by
the action (29), which we repeat here for convenience:
Z


i
2
SCS :=(100)dz ∧ trK A ∧ dA + A ∧ A ∧ A .
2π R2 ×C
3zx, y

The trace in the fundamental representation defines a positive-definite metric on glK , moreover, we choose a basis of glK , denoted by {t µ }, in which the metric becomes diagonal:
trK (t µ t ν ) ∝ δµν .

(101)

We consider this theory in the presence of a Wilson line in some representation % : glK ! End(V ),supported along the line L defined by y = z = 0:Z%(A) .

W% (L) = P exp

(102)

L

Consideration of fusion of Wilson lines to give rise to Wilson lines in tensor product representation shows that it is not only the connection A that couples to a Wilson line but also its
derivatives ∂zn A [19]. Furthermore, gauge invariance at the classical level requires that ∂zn A
couples to the Wilson line via a representation of the loop algebra glK [z]. So the line operator
that we consider is the following:ZX∂zn Aµ

%µ,n

,

(103)

where the matrices %µ,n ∈ End(V ) satisfy:%µ,m , %ν,n = fµν ξ %ξ,m+n .

(104)

P exp

L

n≥0

The structure constant fµν ξ is that of glK . In particular, we have %µ,0 = %(t µ ).
In (28), A was defined to not have a dz component. The reason is that, due to the appearance of dz in the above action (100), the dz component of the connection A never appears in
the action anyway.31
Though the theory is topological, in order to do concrete computations, such as imposing
gauge fixing conditions, computing propagator, and evaluating Witten diagrams etc. we needto make a choice of metric on R2x, y × Cz , we choose:32
ds2 = dx 2 + d y 2 + dzdz .

(107)

31
Had we defined the space of connections to be Ω1 (R2x, y × Cz ) ⊗ glK , then, in addition to the usual GLK gauge
symmetry, we would have to consider the following additional gauge transformation:

A ! A + f dz ,(105)

for arbitrary function f ∈ Ω (R × C). We could fix this gauge by imposing:02

Az = 0 .
This would get us back to the space Ω1 (R2x, y × Cz )/(dz) ⊗ glK .
32
For this theory we follow the choices of [19] whenever possible.


23

(106)

SciPost Phys. 9, 017 (2020)

For the GLK gauge symmetry we use the following gauge fixing condition:∂ x A x + ∂ y A y + 4∂z Az = 0 .

(108)

The propagator is defined as the two-point correlation function:
P µν (v1 , v2 ) := 〈Aµ (v1 )Aν (v2 )〉 .

(109)

Since in the basis of our choice the Lie algebra metric is diagonal (101), this propagator is
proportional to a Kronecker delta in the Lie algebra indices:
P µν (v1 , v2 ) = δµν P(v1 , v2 ) ,(110)

where P is a 2-form on R4v × R4v . We can fix one of the coordinates to be the origin, this12amounts to taking the projection:$ : R4v × R4v ! R4v ,1

1$ : (v1 , v2 ) 7! v1 − v2 =: v .

(111)

Due to translation invariance, P can be written as a pullback of some 2-form on R4 by $, i.e.,
P = $∗ P for some P ∈ Ω2 (R4 ). The propagator P can be characterized as the Green’s functionifor the differential operator 2πħ
h dz ∧ d that appears in the kinetic term of the action SCS . For P
this results in the following equation:idz ∧ dP(v) = δ4 (v)dx ∧ d y ∧ dz ∧ dz ,
2πħ
h

(112)

The propagator P, and in turns P, must also satisfy the gauge fixing condition (108):∂ x P x + ∂ y P y + 4∂z P z = 0 .

(113)

The solution to (112) and (113) is given by:P(x, y, z, z) =ħh x d y ∧ dz + y dz ∧ dx + 2z dx ∧ d y.
2π(x 2 + y 2 + zz)2

(114)

The propagator P(v1 , v2 ) will be referred to as the bulk-to-bulk propagator, since the points
v1 and v2 can be anywhere in the world-volume R2x, y × Cz of CS theory. To compute Witten diagrams we also need a boundary-to-bulk propagator. We will denote it as Kµ (v, x) ≡ K(v, x)t µ ,where v ∈ R2x, y × Cz and x ∈ `∞ (z) is restricted to the boundary line. The boundary-to-bulk
propagator is a 1-form defined as a solution to the classical equation of motion:
dz v ∧ d v K(v, x) = 0 ,(115)

and by the condition that when pulled back to the boundary, in this case `∞ (z), it must becomea delta function supported at x:" ∗ K(x 0 , x) = δ1 (x 0 − x)dx 0 ,x 0 ∈ `∞ (z) ,(116)

where " : `∞ (z) ,! R2 × C is the embedding of the line in the larger 4d world-volume. As ourboundary-to-bulk propagator we choose the following:K(v, x) = d v θ (x v − x) = δ1 (x v − x)dx v ,(117)

where x v refers to the x-coordinate of the bulk point v. The function θ is the following step
function:for x > 0
 1
1/2 for x = 0 .
θ (x) =
(118)
 0
for x < 024

SciPost Phys. 9, 017 (2020)

Note that we have functional derivatives with respect to ∂zn A for n ∈ N≥0 . The propagator(117) corresponds to the functional derivative with n = 0. Let us denote the propagatorcorresponding to δ∂δn A , more generally, as Kn , and for n ≥ 0, we modify the condition (116)z
by imposing:
lim0 " ∗ ∂zn K(v, x) = δ1 (x 0 − x)dx 0 ,x 0 ∈ `∞ (z) .
(119)
v! x

This leads us to the following generalization of (117):

Kn (v, x) = z vn δ1 (x v − x)dx v .

(120)Apart from the two propagators, we shall need the coupling constant of the theory tocompute Witten diagrams. The coupling constant of this theory can be read off from theinteraction term in the action SCS, it is:if ξ dz .
2πħ
h µν

(121)

Now we can give a diagrammatic definition of the operators in the algebra ASc (Tbk ),namely the ones defined in (39), and their products:%ν1 , j1q1

%νl , jl
ql

···

jl

j1

Tµ1 [n1 ](p1 ) · · · Tµm [nm ](pm ) =

···

∞ X
X

.

l=1 ji ≥0

(122)

···
p1
µ1 , n 1· · · µmp,mnm

Let us clarify some points about the picture. We have replaced the pair of fundamental-antifundamental indices on T with a single adjoint index. The bottom horizontal line represents
the boundary line `∞ (z), and the top horizontal line represents the Wilson line in representation % : glK ! V at y = 0. The sum is over the number of propagators attached to the Wilsonline and all possible derivative couplings. The orders of the derivatives are mentioned in theboxes. The points q1 ≤ · · · ≤ ql on the Wilson line are all integrated along the line withoutchanging their order. The gray blob represents a sum over all possible graphs consistent withthe external lines. We use different types of lines to represent different entities:
Bulk-to-bulk propagator, P(v1 , v2 ) = v1v2 ,Boundary-to-bulk propagator, K(v, x) = v

x ,The boundary line `∞ (z) :,Wilson line :

.

(123)

The labels µi , ni below the points along the boundary line implies that the corresponding
boundary-to-bulk propagator is Kni = z ni K and that it carries a glK -index µi . Finally, thejth derivative of Aν couples to the Wilson line via the matrix %ν, j . Such a diagram with mboundary-to-bulk propagators and l bulk-to-bulk propagators attached to the Wilson lines will
be evaluated to an element of End(V ) which will schematically look like:···µl(Γm!l )µν 1···ν
%µ1 , j1 · · · %µl , jl ,
1

m

(124)

···µm
where (Γm!l )µν 1···ν
is a number that will be found by evaluating the Witten diagram. Since1lthe bulk-to-bulk propagator (114) is proportional to ħh and the interaction vertex (121) is

25

SciPost Phys. 9, 017 (2020)

proportional to ħ
h−1 , each diagram will come with a factor of ħh that will be related to the Euler
33
character of the graph. In the following we start computing diagrams starting from O(ħh0 )2
and up to O(ħh ), by the end of which we shall have proven the main result (Proposition 2) ofthis section.
Remark 6 (Diagrams as m ! l maps, and deformation). Each m ! l Witten diagram thatappears in sums such as (122) can be interpreted as a map whose image is the value of the
diagram:mlOOni
Γ m! l :z glK !
z ji glK ! End(V ) ,Γ m! l :

i=1
m
O

i=1

z

ni

i=1

(125)

···µl
t µi 7! (Γm!l )µν 1···ν
%µ1 , j1 · · · %µl , jl .
1m

As we shall see explicitly in our computations, diagrams in (122) without loops (diagrams ofO(ħh0 )) define an associative product that leads to classical algebras such as U(glK [z]). However, there are generally more diagrams in (122) involving loops (diagrams of O(ħh) and higher
order) that change the classical product to something else. Since loops in Witten or Feynman
diagrams are the essence of the quantum interactions, classical algebras deformed by suchloop diagrams are aptly called quantum groups (of course, why they are called groups is adifferent story entirely [45].)4

5.1

Relation to anomaly of Wilson line

As we shall compute relevant Witten diagrams of the 4d Chern-Simons theory in detail in later
sections, we shall find that the computations are essentially similar to the computations ofgauge anomaly of the Wilson line [19] in this theory. This of course is not a coincidence. To
see this, let us consider the variation of the expectation value of the Wilson line, W% (L) A, as
we vary the connection A along the boundary line `∞ (z):
δ W% (L) A =

∞ Z
X
n=0

p∈`∞

δ
W% (L) A δ∂zn Aµ (p) .
n Aµ (p)
δ∂
z
(z)

(126)

Let us make the following variation:δ∂z Aµ (x) = δ1 (x − p)ηµ = d x θ (x − p)ηµ ,(127)

for some fixed Lie algebra element ηµ t µ ∈ glK . Then we find:δ W% (L) A =δ
W% (L) A ηµ .
δ∂z Aµ (p)(128)

An exact variation of the boundary value of the connection is like a gauge transformation that
does not vanish at the boundary. In [19] it was proved that such a variation of the connection
leads to a variation of the Wilson line which is a local functional supported on the line:Zδ W% (L) A = %µ,1 , %ν,1 + Θµ,1,ν,1∂z Aµ ∂z cν ,
(129)
L

33

In a Feynman diagram all propagators are proportional to ħh and the power of ħh of a diagram relates simply
to the number of faces of the diagram, which is why ħ
h is called the loop counting parameter. In a Witten diagram
the boundary-to-bulk propagators do not carry any ħh and therefore the power of ħh depends also on the number ofboundary-to-bulk propagators. However, we are going to ignore this point and simply refer to the power of ħh in adiagram as the loop order of that diagram.

26

SciPost Phys. 9, 017 (2020)

where c was the generator of the gauge transformation:∂z dcµ = δ∂z Aµ ,(130)%µ,1 ∈ End(V ) is part of the representation of glK [z] that couples ∂z Aµ to the Wilson line (see(103)), and Θµ,1,ν,1 , which is anti-symmetric in µ and ν, is a matrix that acts on V . Variations
such as the above measure gauge anomaly associated to the line, though in our case it is
not an anomaly since we are varying the connection at the boundary, and such “large gauge"transformations are not actually part of the gauge symmetry of the theory. The matrix Θµ,1,ν,1
which signals the presence of anomaly is not an arbitrary matrix and in [19], all constraints on
this matrix were worked out, we shall not need them at the moment. Comparing with (127)
we see that for us ∂z c µ (x) = θ (x − p)ηµ , which leads to:Zξ∂z Aµ ην ,(131)
δ W% (L) A = fµν %ξ,2 + Θµ,1,ν,1x>pwhere we have used the fact that the matrices %µ,1 satisfy the loop algebra (104). The integral
above is along L. The connection A above is a background connection satisfying the equationof motion, i.e., it is flat. Since the D4 world-volume, even in the presence of a Wilson line, has
no non-contractible loop, all flat connections are exact. Symmetry of world-volume dictatesin particular that the connection must also be translation invariant along the direction of the
Wilson line L. By considering the integral of A along the following rectangle:
x=p

x =∞
y =0

L

(132)

dA = 0`∞ (z)y =∞and using translation invariance in the x-direction along with Stoke’s theorem, we can change
the support of the integral in (131) from L to `∞ (z), to get:Zξ
δ W% (L) A = fµν %ξ,2 + Θµ,1,ν,1∂z Aµ ην .
(133)`∞ (z)3x>pComparing with (128) we find:


δ
ξ
W
(L)
=
f
%
+
Θ
%
ξ,2
µ,1,ν,1µνAδ∂z Aν (p)Z∂z Aµ ,

(134)

x>p

where the integral is now along the boundary line `∞ (z). This leads to the following relation
between our algebra and anomaly:Tµ [1], Tν [1]


δ
δ
δ
δ
−
= lim
W% (L) A
ι !0 δ∂z Aµ (p + ι) δ∂z Aν (p)δ∂z Aν (p) δ∂z Aµ (p + ι)
= fµν ξ %ξ,2 + Θµ,1,ν,1 .

(135)

The first term with the structure constant gives us the loop algebra glK [z], which is theclassical result. The anomaly term is the result of 2-loop dynamics [19], i.e., it is proportional
to ħ
h2 . This term gives the quantum deformation of the classical loop algebra. This also explains
27

SciPost Phys. 9, 017 (2020)

why our two loop computation of the algebra is similar to the two loop computation of anomaly
from [19].
At this point, we note that we can actually just use the result of [19] to find out what Θµ,1,ν,1
is and we would find that the deformed algebra of the operators T µ [n] is indeed the YangianYħh (glK ). However, we think it is illustrative to derive this result from a direct computation ofWitten diagrams.

5.2

Classical algebra, O(ħ
h0 )

5.2.1

Lie bracket

We denote a diagram by Γnd!m when there are n boundary-to-bulk propagators, m propagators
attached to the Wilson line, and the diagram is of order ħ
hd . If there are more than one suchdiagrams we denote them as Γnd!m,i with i = 1, · · · .
Our aim in this section is to compute the product Tµ [m](p1 )Tν [n](p2 ) and eventually thecommutatorTµ [m], Tν [n] := lim Tµ [m](p1 )Tν [n](p2 ) − Tν [n](p1 )Tµ [m](p2 ) ,(136)
p2 ! p1

at 0-loop.34
We have the following two 2 ! 2 diagrams:
q1

q2

m

n


p1 p2
; ν,n =
Γ20!2,1 µ,m

,
p1
µ,m

q2

q1

n

m


p1 p2
; ν,n =
Γ20!2,2 µ,m

p2
ν,n

,
p1
µ,m

(137)

p2
ν,n

where a label m in a box on the Wilson line refers to the coupling between the Wilson line andthe mth derivative of the connection. The first diagram evaluates to (note that p1 < p2 andq1 < q2 ):
Z

p1 p2
0
µ ν
%n
Γ2!2,1 µ,m ; ν,n =dq1 dq2 δ1 (q1 − p1 )δ1 (q2 − p2 )%mq1 <q2= %µ,m %ν,n ,(138)

and the second one (with p1 < p2 and q1 > q2 ):
Z

p1 p2
0
Γ2!2,2 µ,m ; ν,n =dq2 dq1 δ1 (q1 − p1 )δ1 (q2 − p2 )%ν,n %µ,m
q1 >q2

= 0.

(139)

Therefore their contribution to the commutator is:





p1 p2
p1 p2
Tµ [m], Tν [n] = lim Γ20!2,1 µ,m; ν,n − Γ20!2,1 ν,n; µ,mp2 ! p1= %µ,m , %ν,n = fµν ξ %ξ,m+n = fµν ξ Tξ [m + n] ,Tµ [m](p1 ), Tν [n](p1 ) may be a more accurate notation but this algebra must be position invariant and therefore we shall ignore the position. Reference to the position only matters when different operators are positioned
at different locations.
34



(140)

28

SciPost Phys. 9, 017 (2020)

where the last equality is established by evaluating the diagram:
m+n

.

(141)

p
ξ,m+n

The bracket (140) is precisely the Lie bracket in the loop algebra glK [z]. Note in passingthat had we considered the same diagrams as the ones in (137) except with different derivative
couplings at the Wilson line then the diagrams would have vanished, either because there
would be more z-derivatives than z, or there would be less, in which case there would be z’s
floating around which vanish along the Wilson line located at y = z = 0.
There is one 2 ! 1 diagram as well:
m+n

,
p1
µ,m

(142)

p2
ν,n

however, since the two boundary-to-bulk propagators are two parallel delta functions, i.e.,their support are restricted to x = p1 and x = p2 respectively with p1 6= p2 , they never meetin the bulk and therefore the diagram vanishes. There are no more classical diagrams, so the
Lie bracket in the classical algebra is just the bracket in (140).
5.2.2

Coproduct

Apart from the Lie algebra structure, the algebra ASc (Tbk ) also has a coproduct structure. Thiscan be seen by considering the Wilson line in a tensor product representation, say U ⊗ V . Sucha Wilson line can be produced by considering two Wilson lines in representations U and V
respectively and bringing them together, and asking how Tµ [n] acts on U ⊗ V . Since there
are going to be multiple vector spaces in this section, let us distinguish the actions of Tµ [n]on them by a superscript, such as, TµU [n], TµV [n], etc. At the classical level the answer to thequestion we are asking is simply given by computing the following diagrams:
V
U

V
U

m

m

+
p
µ,m

.

(143)

p
µ,m

Evaluation of these diagrams is very similar to that of the diagrams in (137) and the result is:
TµU⊗V [m] = TµU [m] ⊗ idV + idU ⊗ TµV [m] .

(144)

This is the same coproduct structure as that of the universal enveloping algebra U(glK [z]).
Combining the results of this section and the previous one we find that, at the classical level
we have an associative algebra with generators Tµ [n] with a Lie bracket and coproduct givenby the Lie bracket of the loop algebra glK [z] and the coproduct of its universal enveloping
algebra. This identifies ASc (Tbk ), clasically, as the universal enveloping algebra itself:
Lemma 1. The large N limit of the algebra ASc (Tbk ) at the classical level is the universal enveloping algebra U(glK [z]):N !∞ASc (Tbk )/ħh ∼=U(glK [z]) ∼h.
= Yħh (glK )/ħ
29

(145)

SciPost Phys. 9, 017 (2020)

The reason why we need to take the large N limit is that, the operators Tµ [m] acts on avector space which is finite dimensional for finite N . This leads to some extra relations in thealgebra, which we can get rid of in the large N limit. A similar argument was presented forthe operator algebra coming from the BF theory in §4.3 and the argument in the context of
the CS theory will be explained in more detail in §5.4.

5.3

Loop corrections

5.3.1 1-loop, O(ħh)
Now we want to compute 1-loop deformation to both the Lie algebra structure and the coproduct structure of ASc (Tbk ).
Lie bracket. The 2 ! 1 diagrams at this loop order are:35

,

,

,

.

(146)

All of these vanish due to Lemma 6 of §C.1.
The 2 ! 2 diagrams at this loop order are:

+

.

(147)

Note that, since the bulk points are being integrated over, crossing the boundary-to-bulk propagators does not produce any new diagram, it just exchanges the two diagrams that we havedrawn:
crossing

−−−−!

=

.

(148)

For this reason, in future we shall only draw diagrams up to crossing of the boundary-to-bulk
propagators that are connected to bulk interaction vertices.
Now let us comment on the evaluation of the diagrams in (147). We start by doing integration by parts with respect the differential corresponding to either one of the two boundaryto-bulk propagators. As mentioned in §C.2, this gives two kinds of contributions, one comingfrom collapsing a bulk-to-bulk propagator, the other coming from boundary terms. Collapsing
any of the bulk-to-bulk propagators leads to a configuration which will vanish due to Lemma
7 (§C.1). Therefore, doing integration by parts will only result in a boundary term. Recall
from the general discussion in §C.2 that only the boundary component of the integrals along
the Wilson line can possibly contribute. Since there are two points on the Wilson line, let us
call them p1 and p2 , the domain of integration is:∆2 = {(p1 , p2 ) ∈ R2 | p1 < p2 } .

(149)

35
Sometimes we ignore to specify the derivative couplings at the Wilson line, when the diagrams we draw arevanishing regardless.

30

SciPost Phys. 9, 017 (2020)

The boundary of this domain is:∂ ∆2 = {(p1 , p2 ) ∈ R2 | p1 = p2 } .

(150)

Once restricted to this boundary, both of the diagrams in (147) will involve a configurationsuch as the following:,(151)
which vanishes due to Lemma 6.36 The diagrams (147) thus vanish.
There are four other 2 ! 2 diagrams at 1-loop, they can be generated by starting with:
,

(152)

and then
1. Permuting the two points on the Wilson line.
2. Permuting the two points on the boundary.
3. Simultaneously permuting the two points on the Wilson line and the two points on theboundary.
All of these diagrams vanish due to Lemma 6.
There are also six 2 ! 3 diagrams. All of these can be generated from the following:
,

(153)

by permuting the points along the Wilson line and the boundary. However, due to Lemma 7,
all of these diagrams vanish.
There are no more 2 ! m diagrams at 1-loop. Thus, we conclude that there is no 1-loop
contribution to the Lie bracket in ASc (Tbk ).
Coproduct. We use the same superscript notation we used in §5.2.2 to distinguish between
the actions of Tµ [m] on different vector spaces. The 1-loop diagram that deforms the classicalcoproduct is the following:VUp Γ11!2 µ,1 =

(154)
p
µ,1

Happily for us, precisely this diagram was computed in eq. 5.6 of [19] to answer the question “how does a background connection couple to the product Wilson line?". The result of
36

These diagrams actually require a UV regularization due to logarithmic divergence coming from the two points
on the Wilson line being coincident. To regularize, the domain of integration needs to be restricted from ∆2 toe 2 := {(p1 , p2 ) ∈ R2 | p1 ≤ p2 −ε} for some small positive number ε, which leads to the modified boundary equation∆
p1 = p2 − ε, however, this does not affect the arguments presented in the proof of Lemma 6 (essentially because
ε is a constant and dε = 0, resulting in no new forms other than the ones considered in the proof), and therefore
we are not going to describe the regularization of these diagrams in detail.

31

SciPost Phys. 9, 017 (2020)

that paper involved an arbitrary background connection where we have our boundary-to-bulk
propagator, so we just need to replace that with K1 (v, p) = z v δ1 (x v − p) and we find:ħhp Γ11!2 µ,1 = − fµ νξ TνU [0] ⊗ TξV [0] .
2

(155)

This deforms the classical coproduct (144) as follows:ħh νξ UfTν [0] ⊗ TξV [0] .
(156)
2 µ
p 
The exact same computation with K0 instead of K1 shows that Γ11!2 µ,0 = 0, i.e., the classicalalgebra of the 0th level operators remain entirely undeformed at this loop order.37
Thus we see that at 1-loop, the Lie algebra structure in ASc (Tbk ) remains undeformed,
but there is a non-trivial deformation of the coalgebra structure. At this point, there is amathematical shortcut to proving that the algebra ASc (Tbk ), including all loop corrections, isthe Yangian. The proof relies on a uniqueness theorem (Theorem 12.1.1 of [45]) concerning
the deformation of U(glK [z]). Being able to use the theorem requires satisfying some technicalconditions, we discuss this proof in Appendix B. This proof is independent of the rest of thepaper, where we compute two loop corrections to the commutator (140) which will directly
show that the algebra is the Yangian.
TµU⊗V [1] = TµU [1] ⊗ idV + idU ⊗ TµV [1] −5.3.2 2-loops, O(ħh2 )The number of 2-loop diagrams is too large to list them all, and most of them are zero. Insteadof drawing all these diagrams let us mention how we can quickly identify a large portion of
the diagrams that end up being zero.
Consider the following transformations that can be performed on a propagator or a vertex
in any diagram:

!

!

,

,
(157)

!

,

!

,

!

.

All these transformations increase the order of ħh by one, however, all the diagrams constructedusing these modifications are zero due to Lemma 6. We will therefore ignore such diagrams.
Let us now identify potentially non-zero 2 ! m diagrams at 2-loops.
All 2-loop 2 ! 1 diagrams are created from lower order diagrams using modifications suchas (157). All of them vanish.
For 2 ! 2 diagrams, ignoring those that are results of modifications such as (157) orthat are product of lower order vanishing diagrams, we are left with the sum of the following
37

Note that the 0th level operators form a closed algebra which is nothing but the Lie algebra glK . Reductive Liealgebras belong to discrete isomorphism classes and therefore they are robust against continuous deformations. So
the algebra of Tµ [0] will in fact remain undeformed at all loop orders. We will not make more than a few remarks
about them in the future.

32

SciPost Phys. 9, 017 (2020)

diagrams:
Γ22!2,1 =Γ22!2,2 =

,

,

(158)
Γ22!2,3 =Γ22!2,4 =,.

Let us first consider the first two diagrams Γ22!2,1 and Γ22!2,2 . Collapsing any of the bulk-to-bulk
propagators will result in a configuration where either Lemma 6 or 7 is applicable. Therefore,
when we do integration by parts with respect to the differential in one of the two boundaryto-bulk propagators we only get a boundary term. The boundary corresponds to the boundaryof ∆2 (defined in (149)), and when restricted to this boundary, the integrand vanishes due toLemma 7, in the same way as for the diagrams in (147).38
The diagrams Γ22!2,3 and Γ22!2,4 are symmetric under the exchange of the color labels
associated to the boundary-to-bulk propagators, for a proof see the discussion following (239).
So these diagrams don’t contribute to the anti-symmetric commutator we are computing.
Now we come to the most involved part of our computations, 2 ! 3 diagrams at 2-loops.
We have the sum of the following diagrams:

Γ22!3,1 =,Γ22!3,2 =,Γ22!3,3 =,(159)
Γ22!3,4 =,Γ22!3,5 =,Γ22!3,6 =.

All of these diagrams are in fact non-zero. We proceed with the evaluation of the diagramΓ22!3,1 :q1 q2 q3
p1 p2 
Γ22!3,1 µ,1; ν,1 = v1

v2

p1
µ,m

v3

(160)

p2
ν,n

The glK factor of the diagram is easily evaluated to be:fµ ξo fξπρ

fνπ σ %(t o )%(t ρ )%(t σ ) .

(161)

The numerical factor takes a bit more care. Each of the bulk points vi = (x i , yi , zi , z i ) is
integrated over M = R2 × C and the points qi on the Wilson line take value in the simplex
38

These diagrams are linearly divergent when the two points on the Wilson line are coincident and they require
similar UV regularization as their 1-loop counterparts.

33

SciPost Phys. 9, 017 (2020)∆3 = {(q1 , q2 , q3 ) ∈ R3 | q1 < q2 < q3 }. For the sake of integration we can partially compactify
the bulk to M = R × S 3 . So the domain of integration for this diagram is:
M 3 × ∆3 .

(162)

However, this domain needs regularization due to UV divergences coming from the points qi
all coming together. As in [19], we use a point splitting regulator, by restricting integration to
the domain:e 3 := {(q1 , q2 , q3 ) ∈ ∆3 | q1 < q3 − ε} ,∆(163)
for some small positive number ε. We are not going to discuss the regulator here, as it would
be identical to the discussion in [19]. We shall now do integration by parts with respect to thedifferential in the propagator connecting p1 and v1 . Note that collapsing any of the bulk-tobulk propagators leads to a configuration where the vanishing Lemma 7 applies. Therefore,e 3 . The boundary of thecontribution to the integral only comes from the boundary M 3 × ∂ ∆simplex has three components, respectively defined by the constraints q1 = q2 , q2 = q3 , and
q1 = q3 − ε. However, when q1 = q2 or q2 = q3 , we can use the vanishing Lemma 6 andthe integral vanishes. Therefore the contribution to the diagram comes only from integrationover:e 3 | q1 = q3 − ε} .
M 3 × {(q1 , q2 , q3 ) ∈ ∆(164)
Further simplification can be made using the fact that the propagator connecting p2 and v3 iszδ1 (x 3 − p2 ). This restricts the integration over v3 to {p2 } × S 3 . However, using translation
symmetry in the x-direction we can fix the position of q1 at (0, 0, 0, 0) and allow the integrationof v3 over all of M . However, due to the presence of the delta function δ1 (x 3 − p2 ) in theboundary-to-bulk propagator, x 3 and p1 = p2 − δ are rigidly tied to each other. This way, we
end up with the following integration for the numerical factor:39

3 Z
1
i
dq2 d4 v1 d4 v2 d4 v3 θ (x 1 − x 3− )z1 z3 P(v2 , v1 )0<q2 <ε
2 2πħ
h
(165)
v1 ,v2 ,v3× P(v3 , v2 )P(q1 , v1 )P(q2 , v2 )P(q3 , v3 ) ,where q1 = (0, 0, 0, 0), q2 = (p2 , 0, 0, 0), q3 = (ε, 0, 0, 0), and x 3− := x 3 − δ, and since all theforms that appear are even we have ignored the wedge product symbols to be economic.
Before evaluating the above integral, we note that the diagram Γ22!3,4 evaluates to thesame color factor and almost same numerical factor, except for a different step function:

3 Z
1
i
dq2 d4 v1 d4 v2 d4 v3 θ (x 3 − x 1− )z1 z3 P(v2 , v1 )0<q2 <ε
2 2πħ
h
(166)
v1 ,v2 ,v3× P(v3 , v2 )P(q1 , v1 )P(q2 , v2 )P(q3 , v3 ) .
Since we have to sum over all the diagrams, we use the fact that:lim θ (x 1 − x 3− ) + θ (x 3 − x 1− ) = 1 ,
δ!0

(167)

to write:p1 p2 p1 p2 
Γ22!3,1 µ,1; ν,1 + Γ22!3,4 µ,1; ν,1
p2 ! p1
3 Z

1
i
πρ
ξo
σ
dq2 d4 v1 d4 v2 d4 v3= fµ fξ fνπ %(t o )%(t ρ )%(t σ )2πħ
h 2 0<q2 <εlimv1 ,v2 ,v3

× z1 z3 P(v2 , v1 )P(v3 , v2 )P(q1 , v1 )P(q2 , v2 )P(q3 , v3 ) .
39

The factor of 1/2 comes from diagram automorphisms.

34

(168)

SciPost Phys. 9, 017 (2020)

Let us refer to the above integral by ħh2 I1 , so that we can write the right hand side of the above
equation as:πρħ
h2 fµ ξo fξ fνπ σ %(t o )%(t ρ )%(t σ ) I1 .
(169)
Similar considerations for the rest of the diagrams in (159) lead to similar expressions:
p1 p2 p1 p2 πρlim Γ22!3,2 µ,1; ν,1 + Γ22!3,5 µ,1; ν,1 = ħh2 fµ ξo fξ fνπ σ %(t ρ )%(t o )%(t σ ) I2 , (170a)
p2 ! p1

p1 p2 p1 p2 πρlim Γ22!3,2 µ,1; ν,1 + Γ22!3,5 µ,1; ν,1 = ħh2 fµ ξo fξ fνπ σ %(t o )%(t σ )%(t ρ ) I3 , (170b)
p2 ! p1

for two integrals I2 and I3 thatdifferent from I1 .40 To get the 2-loop contri are only slightlybutions to the commutator Tµ [1], Tν , [1] we need only to anti-symmetrize the expressions(169), (170). Putting them together with the classical result (140) we get the Lie bracket up
to 2-loops:


πρ
ξo
Tµ [1], Tν , [1] = fµν ξ Tξ [2] + 2ħ
h2 f[µ fξ fν]π σ To [0]Tρ [0]Tσ [0] I1(171)
+ Tρ [0]To [0]Tσ [0] I2 + To [0]Tσ [0]Tρ [0] I3 ,where we have replaced matrix products such as %(t ρ )%(t o )%(t σ ) with Tρ [0]To [0]Tσ [0]which is accurate up to the loop order shown. Thus we see that quantum corrections deformthe classical Lie algebra of glK [z].

5.4

Large N limit: The Yangian

The deformed Lie bracket (171) may not look quite like the standard relations of the Yangian
found in the literature, but we can choose a different basis to get to the standard relations,which we shall do momentarily.41 However, for finite N , our algebra has more relations. Recall
that the generators Tµ [1] act on the space V where classically V is a representation space,% : glK [z] ! End(V ), of the loop algebra glK [z] and the representation % was determinedby the number N . The representation % depends on N because % is the representation that
couples the glK connection A to the Wilson line generated by integrating out N × K fermions.
The representation is found by integrating out the fermions that define the line defect (23).42
The important point for us is that, for finite N , % is finite dimensional. This implies thatthe generators Tµ [1] satisfy degree d polynomial equations where d = dim(V ). In the limit
N ! ∞ these relations disappear and we have our isomorphism with the Yangian Y (glK ).43
1
12 − π32 , I1 = 36
1 + π32 though we
These integrals can be performed and their values are I2 = I3 = 72
postpone computing them until we no longer need to compute them.
41
We can also appeal to the uniqueness theorem 12.1.1 of [45], in conjunction with the result of Appendix B, toconclude that the deformed algebra must be the Yangian Yħh (glK ).
42
By integrating out the fermions from the action (23) we get the holonomy of the connection (A, A) ∈ glN ⊕ glK
in the following representation [34]:M
YNT ⊗ Y K ,
(172)
40

Y

where the sum is over all possible Young tableaux. Y T is the tableau we get by transposing the tableau Y (i.e.,turning rows into columns). YNT is the representation of GLN denoted by the tableau Y T , and Y K is the representation of GLK dual to the representation (of GLK ) denoted by Y . Had we started with a bosonic quantum mechanics
instead, integrating out the bosons would result in a holonomy in the following representation [34]:M
YN ⊗ Y K .
(173)
Y

An important difference between (172) and (173) is that while the former is finite dimensional for finite N and K,
the latter is always infinite dimensional.
43
In the case of bosonic quantum mechanics the representation % is actually infinite dimensional, however, forfinite N our Witten diagram computations can not be trusted, as the decoupling between closed string modes anddefect (4d Chern-Simons) modes that we have assumed relies on the large N limit [35].

35SciPost Phys. 9, 017 (2020)

The Yangian in a more standard basis
To get to a standard defining bracket for the Yangian, we change basis as follows. There is an
ambiguity in Tξ [2]. In (140) it was equal to %ξ,2 at the classical level, but it can be shifted at
2-loops (i.e., by a term proportional to ħh2 ) by the image ϑ(t ξ ) for an arbitrary glK -equivariant
map ϑ : glK ! End(V ). This shift simply corresponds to a different choice for the countertermthat couples ∂z2 Aξ to the Wilson line. Using this freedom we want to replace products such as%(t o )%(t ρ )%(t σ ) with the totally symmetric product %(t o ), %(t ρ ), %(t σ ) (defined in (94)).
To this end, Consider the difference:


πρ
ξo
(174)
∆µν := 2ħ
h2 f[µ fξ fν]π σ %(t o )%(t ρ )%(t σ ) − %(t o ), %(t ρ ), %(t σ ) .
The square brackets around µ and ν in the above equation implies anti-symmetrization withrespect to µ and ν. The difference ∆µν can be viewed as the image of the following glK equivariant map:
∆ : ∧2 glK ! End(V ) ,∆ : t µ ∧ t ν 7! ∆µν .
(175)
We now propose the following lemma:
Lemma 2. The map ∆ factors through glK , i.e., ∆ : ∧2 glK ! glK ! End(V ).
The proof of this lemma involves some algebraic technicalities which we relegate to the
Appendix §D. The utility of this lemma is that, it establishes the difference (174) as the imageof an element of glK which, according to our previous argument, can be absorbed into a redefinition of %ξ,2 (equivalently Tξ [2]). Therefore, with a new Tξnew [2] we can rewrite (171)as:h2 (I1 + I2 + I3 )Q µν ,(176)
Tµ [1], Tν , [1] = fµν ξ Tξnew [2] + ħ
where we have also defined:Q µν := 2 f[µ

ξo

fξ

πρ


fν]π σ To [0], Tρ [0], Tσ [0] .

(177)

The reason why we have postponed presenting the evaluations of the individual integrals I1 ,I2 , and I3 is that we don’t need their individual values, only the sum, and precisely this sum
was evaluated in eq. (E.23) of [19] with the result:1.
12
We can therefore write (ignoring the “new" label on Tξ [2]):I1 + I2 + I3 =

(178)


ħ
h2
Q µν .
(179)
Tµ [1], Tν [1] = fµν ξ Tξ [2] +12This is the relation for the Yangian that was presented in §8.6 of [19] and how to relate thisto other standard relations of the Yangian was also discussed there. This is also the exact
relation we found in the boundary theory (c.f. (97)). Note furthermore that, if we had used
the relation between our algebra and anomaly (135) to derive the algebra Lie bracket, we
would have arrived at precisely the same conclusion, as the second term in (179) is indeed theanomaly of a Wilson line (c.f. eq. (8.35) of [32]).
Thus we see that the algebra ASc (Tbk ), defined in (41), at 2-loops, is the Yangian Yħh (glK ):

ASc (Tbk )/ħ
h3

N !∞

∼
=

Yħh (glK )/ħ
h3 .

(180)

The two loop result in the BF theory was exact. The above two loop result is exact as well.
Though we do not prove this by computing Witten diagrams, we can argue using the formof the algebra in terms of anomaly (135). In [19] it was shown that there are no anomalies
beyond two loops. This concludes our second proof of Proposition 2.44
44

The first one, which is significantly more abstract, being in Appendix B.

36

SciPost Phys. 9, 017 (2020)

6

Physical String Theory Construction of The Duality

The topological theories we have considered so far can be constructed from a certain brane
setup in type IIB string theory and then applying a twist and an Ω-deformation. This brane
construction will show that the algebras we have constructed are in fact certain supersymmetric subsectors of the well studied N = 4 SYM theory with defect and its holographic dual. We
note that the idea of embedding Chern-Simons type theories inside 1 and 2 higher dimensional
supersymmetric gauge theories as (quasi)-topological subsectors can be traced back to [46].
A Caveat. The most transparent way of probing (quasi) topological subsectors of therelevant physical (defect) AdS/CFT correspondence would be to apply twist and deformation
to 4d N = 4 SYM with a domain wall on one hand and to AdS5 × S 5 supergravity with D5brane probes on the other hand. Localization in the AdS background is yet to be developed
and this is not what we do in this section. We apply twist and deform the gauge theories thatappear in a certain D3-D5 brane configuration in flat background and argue that we end up
with the topological brane setup of section 3.1. Thus in making the claim that our topologicalholography embeds into physical holography we are relying on the assumption that the processof twist and deformation commutes with taking the decoupling limit.
We describe our construction below.

6.1

Brane Configuration

Our starting brane configuration involves a stack of N D3 branes and K D5 branes in type IIB
string theory on a 10d target space of the form R8 × C where C is a complex curve which wetake to be just the complex plane C. The D5 branes wrap R4 × C and the D3 branes wrap an
R4 which has a 3d intersection with the D5 branes. Let us express the brane configuraiton by
the following table:
0 1 2 3 4 5 6 7 8 9
R4
C
R4
(181)
D5 × × × × × ×
D3 ×
× ×
×
The world-volume theory on the D5 branes is the 6d N = (1, 1) SYM theory coupled to a 3d
defect preserving half of the supersymmetry. Similarly, the world-volume theory on the D3
branes is the 4d SYM theory coupled to a 3d defect preserving half of the supersymmetry. To
this setup we apply a particular twist, i.e., we choose a nilpotent supercharge and consider its
cohomology.

6.2

Twisting Supercharge

6.2.1

From the 6d Perspective

We use Γi with i ∈ {0, · · · , 9} for 10d Euclidean gamma matrices. We also use the notation:
Γi1 ···in := Γi1 · · · Γin .

(182)

Type IIB has 32 supercharges, arranged into two Weyl spinors of the same 10 dimensional
chirality – let us denote them as Q l and Q r . A general linear combination of them is written as
ε L Q l + εRQ r where ε L and εR are chiral spinors parameterizing the supercharge. The chirality
constraints on them are:iΓ0···9 ε L = ε L ,iΓ0···9 εR = εR .
(183)
We shall discuss constraints on the supercharge by describing them as constraints on the parameterizing spinors.
37

SciPost Phys. 9, 017 (2020)

The supercharges preserved by the D5 branes are constrained by:
εR = iΓ012345 ε L .

(184)

This reduces the number of supercharges to 16. The D3 branes imposes the further constraint:
εR = iΓ0237 ε L .

(185)This reduces the number of supercharges by half once more. Therefore the defect preserves
just 8 supercharges. Since εR is completely determined given ε L , in what follows we refer toour choice of supercharge simply by referring to ε L .
We want to perform a twist that makes the D5 world-volume theory topological along R4
and holomorphic along C. This twist was described in [37]. Let us give names to the two
factors of R4 in the 10d space-time:
M 0 := R46789 .

M := R40123 ,(186)

The spinors in the 6d theory transform as representations of Spin(6) under space-time rotations. N = (1, 1) algebra has two left handed spinors and two right handed spinors transforming as 4l and 4 r respectively. There are two of each chirality because the R-symmetry is
Sp(1) × Sp(1) = Spin(4) M 0 such that the two left handed spinors transform as a doublet ofone Sp(1) and the two right handed spinors transform as a doublet of the other Sp(1). The
subgroup of Spin(6) preserving the product structure R4 × C is Spin(4) M × U(1). Under thissubgroup 4l and 4 r transform as (2, 1)−1 ⊕ (1, 2)+1 and (2, 1)+1 ⊕ (1, 2)−1 respectively, where
the subscripts denote the U(1) charges. Rotations along M 0 act as R-symmetry on the spinors– the spinors transform as representations of Spin(4) M 0 such that 4+ transforms as (2, 1) and4− transforms as (1, 2). In total, under the symmetry group Spin(4) M × U(1) × Spin(4) M 0 the
16 supercharges of the 6d theory transform as:((2, 1)−1 ⊕ (1, 2)+1 ) ⊗ (2, 1) ⊕ ((2, 1)+1 ⊕ (1, 2)−1 ) ⊗ (1, 2) .

(187)

The twist we seek is performed by redefining the the space-time isometry:
Spin(4) M

Spin(4)new
M ⊆ Spin(4) M × Spin(4) M 0 ,(188)

where the subgroup Spin(4)new
M of Spin(4) M × Spin(4) M 0 consists of elements (x, θ (x)) which∼
is defined by the isomorphism θ : Spin(4) M −! Spin(4) M 0 . More, explicitly, the isomorphism
acts as:θ (Γµν ) = Γµ+6,ν+6 ,µ, ν ∈ {0, 1, 2, 3} .
(189)
The generators of the new Spin(4)new
M are then:Γµν + Γµ+6,ν+6 .

(190)

After this redefinition, the symmetry Spin(4) M × U(1) × Spin(4) M 0 of the 6d theory reduces toSpin(4)new
M ×U(1) and under this group the representation (187) of the supercharges becomes:2(1, 1)−1 ⊕ (3, 1)−1 ⊕ (1, 3)−1 ⊕ 2(2, 2)+1 .

(191)

We thus have two supercharges that are scalars along M , both of them have charge −1 under
the U(1) rotation along C. We take the generator of this rotation to be −iΓ45 , then if ε is oneof the scalar (on M ) supercharges that means:iΓ45 ε = ε .
38

(192)

SciPost Phys. 9, 017 (2020)

We identify the supercharge ε by imposing invariance under the new rotation generators on
M , namely (190):(Γµν + Γµ+6,ν+6 )ε = 0 .
(193)
The constraints (184) and (185) put by the D-branes and the U(1)-charge on C (192) together
are equivalent to the following four independent constraints:iΓµ,µ+6 ε = ε ,µ{0, 1, 2, 3} .

(194)

Together with the chirality constraint (183) in 10d we therefore have 5 equations, each reducing the number degrees of freedom by half. Since a Dirac spinor in 10d has 32 degrees of
freedom, we are left with 32×2−5 = 1 degree of freedom, i.e., we have a unique supercharge,45
which we call Q. It was shown in [37] that the supercharge Q is nilpotent:
Q2 = 0 ,(195)

and the 6d theory twisted by this Q is topological along M – which is simply a consequenceof (193) – and it is holomorphic along C. The latter claim follows from the fact that there isanother supercharge in the 2d space of scalar (on M ) supercharges in the 6d theory, let’s callit Q0 , which has the following commutator with Q:{Q, Q0 } = ∂z ,(196)

where z = 12 (x 4 − i x 5 ) is the holomorphic coordinate on C. This shows that z-dependence istrivial (Q-exact) in the Q-cohomology.
6.2.2

From the 4d Perspective

What is new in our setup compared to the setup considered in [37] is the stack of D3 branes.
We can figure out what happens to the world-volume theory of the D3 branes – we get the
Kapustin-Witten (KW) twist [47], as we now show. The equations (194) can be used to to get
the following six (three of which are independent) equations:(Γ02 + Γ68 )ε = 0 ,(Γ03 + Γ69 )ε = 0 ,(Γ23 + Γ89 )ε = 0 ,(Γ07 + Γ16 )ε = 0 ,(Γ27 + Γ18 )ε = 0 ,(Γ37 + Γ19 )ε = 0 .

(197)

These are in fact the equations that defines a scalar supercharge in the KW twist of N = 4
theory on R40237 for a particular homomorphism from space-time ismoetry to the R-symmetry.46
Space-time isometry of the theory on R40237 acts on the spinors as Spin(4)iso , generated by thesix generators:Γµν ,µ, ν ∈ {0, 2, 3, 7} and µ 6= ν .
(198)
Rotations along the transverse directions act as R-symmetry, which is Spin(6), though thesubgroup of the R-symmetry preserving the product structure C × R41689 is U(1) × Spin(4)R .
The KW twist is constructed by redefining space-time isometry to be a Spin(4) subgroup of∼Spin(4)iso × Spin(4)R consisting of elements (x, ϑ(x)) where ϑ : Spin(4)iso −! Spin(4)R is an
isomorphism. The particular isomorphism that leads to the equations (197) is:Γ02 7! Γ68 ,Γ03 7! Γ69 ,Γ23 7! Γ89 ,Γ07 7! Γ16 ,Γ27 7! Γ18 ,Γ37 7! Γ19 .

45

(199)

Note that without using the constraint put by the D3 branes we would get two supercharges that are scalars
on M , i.e., there are two superhcarges in the 6d theory (by itself) that are scalars on M .
46
Note that we are using subscripts simply to refer to particular directions.

39

SciPost Phys. 9, 017 (2020)

Remark 7 (A member of a CP1 family of twists). In [47] it was shown that there is a family of
KW twists parameterized by CP1 . The unique twist (by the supercharge Q) we have found isa specific member of this family. Let us identify which member that is.
The CP1 family comes from the fact that there is a 2d space of scalar (on M ) supercharges(in (191)) in the twisted theory.47 Also note from the original representation of the spinors(187) that the two scalar supercharges come from spinors transforming as (1, 2) and (2, 1)
under the original isometry Spin(4)old .48 Let us choose two Spin(4)new scalar spinors with
opposite Spin(4)old chiralities and call them εl and ε r . The Spin(4)old chirality operator isΓ old := Γ0237 . Let us choose εl and ε r in such a way that they are related by the following
equation:
1(200)
ε r = N εl where N = (Γ06 + Γ28 + Γ39 + Γ17 ) .
4This relation is consistent with the spinors being Spin(4)new invariant because N anti-commuteswith Spin(4)new (thus invariant spinors are still invariant after being operated with N ), as wellas with Γ old (changing Spin(4)old chirality). An arbitrary scalar supercharge in the twisted theory is a complex linear combination of εl and ε r , such as αεl + βε r , however, since the overall
normalization of the spinor is irrelevant, the true parameter identifying a spinor is the ratio
t := β/α ∈ CP1 . Furthermore, due to the equations (197), N 2 acts as −1 on any Spin(4)newscalar, leading to:εl = −N ε r .
(201)
To see the value of the twisting parameter t for the supercharge identified by the equations(194) (in addition to the 10d chirality (183)), we first pick a linear combination ε := εl + tε rwith t ∈ CP1 . Then using (201) and (194) we get:−iε = N ε = ε r − tεl ,(202)

where the first equality follows from (194) and the second from (201). Equating the two sides
we find the twisting parameter:t = i.
(203)
4
6.2.3

From the 3d Perspective

Finally, at the 3 dimensional D3-D5 intersection lives a 3d N = 4 theory consisting of bifundamental hypermultiplets coupled to background gauge fields which are restrictions of the gauge
fields from the D3 and the D5 branes [48]. Considering Q-cohomology for the 3d theory reduces it to a topological theory as well. To identify the topological 3d theory we note that for
the twisting parameter t = i, the 4d theory is an analogue of a 2d B-model49 [47] and this canbe coupled to a 3d analogue of the 2d B-model50 – a B-type topological twist of 3d N = 4 iscalled a Rozansky-Witten (RW) twist [49]. The flavor symmetry of the theory is U(N ) × U(K)which acts on the hypers and is gauged by the background connections.
We can reach the same conclusion by analyzing the constraints on the twisting supercharge viewed from the 3d point of view. The bosonic symmetry of the 3d theory includes
Though we began the discussion with a view to identifying topological-holomorphic twist of 6d N = (1, 1)
theory, what we found in the process in particular are supercharges that are scalars on M . If we forget that we had
a 6d theory on M × C and just consider a theory on M with rotations on C being part of the R-symmetry then, firstof all, we find a N = 4 SYM theory on M and the twist we described is precisely the KW twist.
48
We are writing Spin(4)old instead of Spin(4) M since the support of the 4d theory is not M ≡ R40123 but R40237 .
49
In particular, the 4d Theory on R2 × T 2 can be compactified on the two-torus T 2 to get a B-model on R2 .
50
We want to be able to take the 3d theory on R2 × S 1 and compactify it on S 1 to get a B-model on R2 . If wehave a 4d theory on R2 × T 2 coupled to a 3d theory on R2 × S 1 , compactifying on T 2 should not make the twosystems incompatible.
47

40

SciPost Phys. 9, 017 (2020)

SU(2)iso × SU(2)H × SU(2)C where SU(2)iso is the isometry of the space-time R3023 , SU(2)C are
rotations in R3689 , and SU(2)H are rotations in R3145 . The hypers in the 3d theory come fromstrings with one end attached to the D5 branes and another end attached to the D3 branes.
Rotations in R3145 – the R-symmetry SU(2)H – therefore act on the hypers. This means that
SU(2)H acts on the Higgs branch of the 3d theory. This leaves the other R-symmetry group
SU(2)C which would act on the coulomb branch of the theory if the theory had some dynamical 3d vector multiplets. We now note that the topological twist, from the 3d perspective,
involves twisting the isometry SU(2)iso with the R-symmetry group SU(2)C , as evidenced explicitly by the three equations in the first line of (197). This particular topological twist (asopposed to the topological twist using the other R-symmetry SU(2)H ) of 3d N = 4 is indeed
the RW twist [50].
To summerize, taking cohomology with respect to the supercharge Q leaves us with the
KW twist (twisting parameter t = i) of N = 4 SYM theory on R4 with gauge group U(N )and a topological-holomorphic twist of N = (1, 1) theory on R4 × C with gauge group U(K),and these two theories are coupled via a 3d RW theory of bifundamental hypers with flavor
symmetry U(N )×U(K) gauged by background connections.51 Note that we have not describedthe effect of the twist on the closed string theory. This is because we are assuming a decoupling
between the closed string modes and the D5-defect modes in the large N limit (referred to as
rigid holography in [35]) and therefore, the operator algebra that we will concern ourselveswith will be insensitive to the closed string modes.52 We will ignore the closed string modes
moving forward as well.

6.3

Omega Deformation

We start by noting that the dimensional reduction of the topological-holomorphic 6d theory
from R4 × C to R4 reduces it to the KW twist of N = 4 SYM on the R4 .53 This observation
allows us to readily tailor the results obtained in [37] about Ω-deformation of the 6d theory
to the case of Ω-deformation of 4d KW theory.
The fundamental bosonic field in the 10d N = 1 SYM theory is the connection A I whereI ∈ {0, · · · , 9}. When dimensionally reduced to 6d, this becomes a 6d connection A M withM ∈ {0, · · · , 5} and four scalar fields φ0 , φ1 , φ2 , and φ3 which are just the remaining four components of the 10d connection. The Spin(4) M space-time isometry acts on the first four components of the connection, namely A0 , A1 , A2 , and A3 via the vector representation. The four
scalars – φ0 , φ1 , φ2 , and φ3 – transform under the vector representation of the R-symmetry
Spin(4) M 0 .
Once twisted according to (188), only the diagonal subgroupnew
Spin(4) M of Spin(4) M × Spin(4) M 0 acts on the fields, under which the first four componentsof the connection and the four scalars transform in the same way – apart from the inhomogeneous transformation of the connection – and therefore we can package them together into
one complex valued gauge field:

Aµ := Aµ + iφµ ,µ ∈ {0, 1, 2, 3} .

(204)

We also write the remaining components of the connection in complex coordinates on C:
Az := A4 + iA5

and Az := A4 − iA5 .

(205)

It was shown in [37] that this topological-holomorphic 6d theory can be viewed as a 2dgauged B-model on R223 where the fields are valued in maps Map(R201 × C, glK ). This is a51

Though it is customary to decouple the central U(1) subgroup from the gauge groups as it doesn’t interact with
the non-abelian part, our computations look somewhat simpler if we keep the U(1).
52This is the same argument we used in §3.2.
53
Both the 6d N = (1, 1) SYM and the 4d N = 4 SYM are dimensional reductions of the 10d N = 1 SYM anddimensional reduction commutes with the twisting procedure.

41

SciPost Phys. 9, 017 (2020)

vector space which plays the role of the Lie algebra of the 2d gauge theory. From the 2d
point of view A2 and A3 are part of a connection on R223 and there are four chiral multipletswith the bottom components A0 , A1 , Az , and Az . The 2d theory consists of a superpotential
which is a holomorphic function of these chiral multiplets – the superpotential can be written
conveniently in terms of a one form Ae := A0 dx 0 + A1 dx 1 + Az dz + Az dz on R201 × C consistingof these chiral fields:54
Z
2 e e eee
W (A0 , A1 , Az , Az ) =dz ∧ tr A ∧ dA + A ∧ A ∧ A .
(206)
3
R2 ×C
01

The superpotential is the action functional of a 4d CS theory on R201 × C for the connection Ae.
One of the results of [37] is the following: Ω-deformation55 applied to this topologicalholomorphic 6d theory with respect to rotation on R223 reduces the theory to a 4d CS theory
on R201 × C with complexified gauge group GLK .
The twisted 4d theory (the D3 world-volume theory) wraps the plane R223 as well andtherefore is affected by the Ω-deformation. By noting that the 4d theory is a dimensional
redcution of the 6d theory from R4 × C to R4 and assuming that Ω-deformation commuteswith dimensional reduction,56 we can deduce what the Ω-deformed version of the twisted 4dtheory is. This will be a 2d gauge theory with complexified gauge group GLN and the action
will be the dimensional reduction of the 4d CS action (206) from R2 × C to R2 – this is the 2d
BF theory where Az plays the role of the B field:
Z
Z


1
Reduce on C
trAz dAR2 + AR2 ∧ AR2
dz ∧ CS(AR2 ×C ) −−−−−−!
2
R2
R2 ×C
Z
(207)
=

trAz F (AR2 ) ,R2where, as before, z is the anti-holomorphic coordinate on C.
Finally, it was shown in [52] that the RW twist of a 3d N = 4 theory on R2Ω × R with only
hypers reduces, upon Ω-deformation with respect to rotation in the plane R2Ω , to a free quantum mechanics on R. A slight modification of this result, involving background connections
gauging the flavor symmetry of the hypers leads to the result that the omega deformed theory
is a gauged quantum mechanics, the kind of theory we have considered on the defect in the
2d BF theory.57

6.4

Takeaway from the Brane Construction

Via supersymmetric twists and Ω-deformation, we have made contact with precisely the setup
we have considered in this paper. We have a 4d CS theory with gauge group GLK and a 2d BF
theory with gauge group GLN and they intersect along a topological line supporting a gauged
quantum mechanics with GLK ×GLN symmetry. We thus claim that the topological holographic
duality that we have established in this paper is indeed a topological subsector of the standard
holographic duality involving defect N = 4 SYM.
54

Up to some overall numerical factors.
Introduced for the first time in [51] in the context of 4d N = 2 gauge theories on R2 × R2 . The relevant
space-time rotation in that case was a U(1) × U(1) action rotating the two planes – which ultimately localized
the 4d theory to a 0d matrix model. Analogously, Ω-deformation with respect to rotation on a plane localizes our
6d/4d/3d theory to a 4d/2d/1d theory.
56
Alternatively, one can redo the localization computations of [37] for the 4d case, confirming that Ω-deformation
does indeed commute with dimensional reduction.
57
The bosonic version, which leads to the same Yangian with minor modifications to the computations as remarked in 3, 4, and 5.
55

42

SciPost Phys. 9, 017 (2020)

7Concluding Remarks and Future Works

In the previous sections we have been able to exactly (at all loops) match a subsector of the operator algebra in the 2d BF theory with a line defect, with a subsector of the scattering algebra
in a 3d closed string theory with a surface defect. The subsectors of operators we focused on
are restricted to the defects on both sides of the duality. This matching provides a non-trivial
check of the proposed holographic duality. Furthermore, we have shown that this holographic
duality between topological/holomorphic theories is in fact a supersymmetric subsector of themore familiar AdS5 /CFT4 duality. From the considerations of this paper several immediate
questions and new directions arise that we have not yet addressed. Let us comment on a few
such issues that we think are interesting topics to pursue for future research.
Central extensions on two sides of the duality: To ease computation we restricted ourattention to the quotients of the full operator algebra and scattering algebra by their centers.
The inclusion of the central operators will change the associative structure of the algebras. A
stronger statement of duality will be to compare the centrally extended Yangians coming fromthe boundary and the bulk theory.
Brane probes: Using branes in the bulk to probe local operators in the boundary theory has
been a useful tool [53, 54]. In our setup, a brane must be Lagrangian in the A-twisted R4
directions. Looking at the brane setup (8) (which we reproduce in (209) for convenience)
we see that the real directions of the D2 and D4 branes are Lagrangian with respect to thefollowing symplectic form:
dv ∧ dx + dw ∧ d y .
(208)This leaves the possibility of two more different embeddings for D2-branes:

D2
D4
D20
D200

Rv
0
0
×
×

Rw R x
×
×
0
×
0
0
×
0

Ry
0
×
×
0

Cz
0
×
z
z

(209)

The D20 -branes are Wilson lines in the CS theory on the D4-branes perpendicular to the original
Wilson line at thte D2-D4 intersection. Such crossing Wilson lines were studied in [19, 44]with the result that this corssing (of two Wilson lines carrying representations U and V of glK
respectively) inserts an operator TV U (z) : U ⊗ V ! V ⊗ U in the CS theory which solves theYang-Baxter equation, which is described more easily with diagrams:
z1

z2

U
z0 V

W

TU V (z10 )V

TW V (z20 )U

W
TW U (z21 )

W

z1

z2

U

W

V

TW U (z21 )

=
z0

U

V

TW V (z20 )W, (210)U

WV

TU V (z10 )VUwhere z1 , z2 , and z3 are the spectral parameters (location in the complex plane) of the lines
carrying representations V , U, and W respectively, and z21 := z2 − z1 and so on. Solutionsof the above equation are closely tied to Quantum Groups. The operators TU V (z), which arecommonly referred to as R-matrices, can be explicitly constructed using Feynman diagrams[19]. When the complex directions of the theory are parameterized by C (as in our case),these R-matrices are rational functions of z. If we choose U and W to be the fundamental
43

SciPost Phys. 9, 017 (2020)

representation of glK , then by providing an incoming and an outgoing fundamental state, we
can view 〈 j|TKV (z)|i〉 as a map Tij (z) : V ! V which has an expansion is z −1 :Tij (z) = idV δ ij − ħh

X

−z −1

n+1

T ji [n] ,

(211)

n≥0

where the T ji [n] are precisely the operators that generate the scattering algebra ASc (Tbk ) (see(39) and (41)). This suggests that in the dual picture we should be able to interpret the D20
branes as a generating function for the operators O ji [n].
The interpretations of the D200 branes are missing on both sides of the duality.
Finite N duality: We considered the large N limit to decouple the closed string modes fromthe defect (CS) mode in the bulk side of the duality and to eliminate any relations among ouroperators that would arise from having finite dimensional matrices (see §4.3 and §5.4). It
would of course be a stronger check if we could match the algebras at finite N , when they canbe quotients of the Yangian by some extra relations.
Duality for other quantum groups: In [19, 44] it was shown that by replacing our complex
direction C with the punctured plane C× or an elliptic curve, we can get, instead of the Yangian,the trigonometric or elliptic solutions to the Yang-Baxter equation (210). It will be interesting
to have an analogous analysis of holographic duality for the corresponding quantum groups
as well.

Acknowledgements
We are grateful to Kevin Costello, Davide Gaiotto, Jaume Gomis, Shota Komatsu, Natalie Paquette, and Masahito Yamazaki for valuable discussions and feedbacks on the manuscript. We
specially thank Kevin Costello, whose works and ideas have directly motivated and guided thisproject.
Funding information All authors are supported by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada through Industry
Canada and by the Province of Ontario through the Ministry of Research and Innovation.

A

Integrating the BF interaction vertex

In this appendix we evaluate the integrals in (80).
φ2

φ1

φ1

φ

,

.

(212)

φ2

φ

We split up each integral into two, based on whether the bulk point is above or below theline operator. We use angular coordinates defined as in the above diagrams. One subtlety isthat, from the definition of the propagators in the Cartesian coordinate we can see that the
integrand (including the measure) is even under reflection with respect to the line. So, we justhave to make sure that when we divide up the integral in the aforementioned way, even when
written in angular coordinates, the integrand does not change sign under reflection. With this44

SciPost Phys. 9, 017 (2020)in mind, the integrals we have to evaluate are:ħh2αβγ
V·|| (x 1 , x 2 ) =
f αβγ
(2π)3
ħ
h2
αβγ
f αβγ
V|·| (x 1 , x 2 ) =
(2π)3
ħ
h2
αβγ
V||· (x 1 , x 2 ) =
f αβγ
(2π)3

Zπ

Z 2π
dφ1

Z φ +π
1

dφ2
φ1
Zπ

0
Z 2π

dφ1

π
Z φ +π
2

dφ2
φ1
Zπ

0
Z 2π

dφ1

φ1 +π
Z 2π

dφ2
φ1

0

dφ +
dφ +

dφ +

φ2 +π

Z φ1 −πdφ

π
Z φ2 −π

,


dφ

φ1 −π
Z0

dφ

,


.

φ2 −π

2

ħ
h
f αβγ .
All three terms are equal to 24

B

Yangian from 1-loop Computations

At the end of §5.3.1, by computing 1-loop diagrams, we concluded that quantum corrections
deform the coalgebra structure of the classical Hopf algebra U(glK [z]). Since ASc (Tbk ) is analgebra to begin with, we conclude that at one loop, we have a deformation of the classicalalgebra as a Hopf algebra. We are using the term “deformation" (alternatively, “quantization")in the sense of Definition 6.1.1 of [45], which essentially means that:• ASc (Tbk ) becomes the classical algebra U(glK [z]) in the classical limit ħ
h ! 0.
• ASc (Tbk ) is isomorphic to U(glK [z])Jħ
hK as a CJħ
hK-module.

• ASc (Tbk ) is a topological Hopf algebra (with respect to ħ
h-adic topology).

The reason that we adhere to these conditions is that, there is a well known uniqueness theorem (Theorem 12.1.1 of [45]) which says that the Yangian is the unique deformation ofU(glK [z]) in the above sense. Therefore, if we can show that our algebra ASc (Tbk ) satisfies allthese conditions and it is a nontrivial deformation of U(glK ) then we can conclude that it is the
Yangian. From 1-loop computations we already know that it is a non-trivial deformation. Thatthe first condition in the list above is satisfied is the content of Lemma 1. The second conditionis satisfied because ħh acts on the generators of our algebra by simply multiplying the external
propagators by ħh in the relevant Witten diagrams, this action does not distinguish between
classical diagrams and higher loop diagrams. Satisfying the last condition is less trivial. While
it seems known to people working in the field, we were unable to find a reference to cite,
therefore, for the sake of completion, we provide a proof in this appendix, that the algebra
ASc (Tbk ) is indeed an (ħh-adic) topological Hopf algebra.
We shall prove this by reconstructing the algebra ASc (Tbk ) from its representations. As
mentioned in §3.4, representations of this algebra are carried by Wilson lines, which form an
abelian monoidal category. A morphism between two representations V and U in this category
is constructed by computing the expectation value of two Wilson lines in representations U
and V ∨ and providing a state at one end of each of the lines. For example, if % and % 0 are
two homomorphisms from glK to End(U) and End(V ∨ ) respectively, then for two lines L and
L 0 in the topological plane of the CS theory and any ψ ⊗ χ ∨ ∈ U ⊗ V ∨ , the expectation value
W% (L)W%0 (L 0 ) is valued in End(U) ⊗ End(V ∨ ) and plugging in states we find a morphism
W% (L)W%0 (L 0 ) (ψ ⊗ χ ∨ ) : V ! U.
Classically, these same Wilson lines carry representations of the classical algebra U(glK [z]).
When viewed as representations of the deformed (alternatively, quantized) algebra ASc (Tbk ),we shall call the category of Wilson lines the quantized category and viewed as representations
45

SciPost Phys. 9, 017 (2020)of U(glK [z]) we shall refer to the category as the classical category. Given any two Wilson lines
U and V , any non-trivial morphism between them in the quantized category is a quantizationof a non-trivial morphism in the classical category. As we mentioned, a morphism between two
Wilson lines is the expectation value of the lines provided with states at one end. A classicalmorphism is computed with classical diagrams and its quantization amounts to adding loop
diagrams. A zero morphism is constructed by providing zero states, this is independent of
quantization, i.e., a quantized morphism is zero, if the provided states are zero, but thenso is the original classical morphism. There is in fact a one-to-one correspondence between
morphisms between two lines in the classical category and the morphisms between the samelines in the quantized category.
For the sake of proof, let us abstract the information we have. We start with a C-linear rigid
abelian monoidal category C = RepC (H) which is the representation category of a Hopf algebra
H. We then find a CJħ
hK-linear abelian monoidal category Cħh , whose objects are representationsof some, yet unknown, Hopf algebra Hħh , with the following properties:• ob(Cħh ) = ob(C ) ,• HomCħh (U, V ) ∼hK as CJħhK-modules .
= HomC (U, V )Jħ
Given this information we shall now prove that Hħh is unique and that it is topological withrespect to ħ
h-adic topology. Then specializing to the case H = U(glK [z]) completes the proof ofSc
A (Tbk ) being topological.

B.1

Tannaka formalism

The aim of this formalism is to realize certain abelian rigid monoidal categories as the representation (or corepresentation) categories of Hopf algebras (possibly with extra structures).
To avoid running into some subtlety in the beginning (we shall explain the subtlety later inthis section), we first consider the reconstruction from the category of corepresentations.
Reconstruction from corepresentation. Let k be a field, C an abelian (resp. abelianmonoidal and End(1) = k) category such that morphisms are k-bilinear, and let R be a commutative algebra over k – if there is an exact faithful (resp. monoidal) functor ω from C to
Mod f (R)58 such that the image of ω is inside the full subcategory Proj f (R)59 , then we shallsay that C has a fiber functor ω to Mod f (R).
Theorem 2 (Tannaka Reconstruction for Coalgebra and Bialgebra). With the notation above,
if moreover R is a local ring or a PID60 , then there exists a unique flat R-coalgebra (resp. Rbialgebra) A, up to unique isomorphism, such that A represents the endomorphism of ω in thesense that ∀M ∈ IndProj f (R) 61
HomR (A, M ) ∼= Nat(ω, ω ⊗ M ) .
Moreover, there is a functor φ : C ! CorepR (A) which makes the following diagram commutativeCφω

CorepR (A)f or getMod f (R)58

finitely generated modules of R
finitely generated projective modules of R
60
PID=Principal Ideal Domain
61
IndProj f (R) means category of inductive limit of finite projective R-modules, which is equivalent to categoryof flat R-modules.
59

46

SciPost Phys. 9, 017 (2020)and φ is an equivalence if R = k.
Our strategy in proving this theorem basically follows [55]. First of all, we need the following
Lemma 3. C is both Noetherian and Artinian.
Proof. Take X ∈ ob(C ), and an ascending chain X i of subobjects of X , apply the functor
ω to this chain, so that ω(X i ) is an ascending chain of finitely generated projective submodules of finitely generated projective module ω(X ), thus there is an index j such thatrank(ω(X j )) = rank(ω(X )). Now the quotient of ω(X ) by ω(X j ) is ω(X /X j ), which is again
finitely generated projective, so it has zero rank, hence trivial. Faithfulness of ω implies that
X /X j is zero, i.e. X = X j , so C is Noetherian. It follows similarly that C is Artinian as well.
Next, we define a functor⊗ : Proj f (R) × C ! C
by sending (Rn , X ) to X n , recall that every finitely generated projective module over a local
ring or a PID is free, thus isomorphic to Rn for some n. Define Hom(M , X ) to be M ∨ ⊗ X . For
V ⊂ M and Y ⊂ X , we define the transporter of V to Y to be(Y : V ) := Ker(Hom(M , X ) ! Hom(V, X /Y )) .
We now have the following:
Lemma 4. Take the full abelian subcategory CX of C generated by subquotients of X n , consider
the largest subobject PX of Hom(ω(X ), X ) whose image in Hom(ω(X )n , X n ) under diagonal embedding is contained in (Y : ω(Y )) for all subobjects Y of X n and all n. Then the Theorem (2) istrue for CX with coalgebra defined by AX := ω(PX )∨ .
Proof. PX exists because C is Artinian. Notice that ω takes Hom(M , X ) to HomR (M , X ) and(Y : V ) to (ω(Y ) : V ), so it takes PX , which is defined by\Hom(ω(X ), X ) ∩ (Y : ω(Y ))to\(EndR (ω(X )) ∩ (ω(Y ) : ω(Y ))) .

Hence ω(PX ) is the largest subring of EndR (ω(X )) stabilizing ω(Y ) for all Y ⊂ X n and all n. It’sa finitely generated projective R module by construction, and so is AX . Note that only finitely
many intersection occurs because Hom(ω(X ), X ) is Artinian.
Next, take any flat R module M ,62 since CX is generated by subquotients of X , an element
λ ∈ Nat(ω, ω ⊗ M ) is completely determined by it is value on X , so λ ∈ EndR (ω(X )) ⊗ M .
Since − ⊗R M is an exact functor, we have:\(HomR (ω(X ), ω(X ) ⊗R M ) ∩ (ω(Y ) ⊗R M : ω(Y )))\
=(EndR (ω(X )) ∩ (ω(Y ) : ω(Y ))) ⊗R M .
This follows because there are only finitely many intersections and finite limit commutes withtensoring with flat module. Therefore,
λ ∈ ω(PX ) ⊗ M .
R

62

Recall that a R module is flat if and only if it is a filtered colimit of finitely generated projective modules.

47

SciPost Phys. 9, 017 (2020)

Conversely, every element in ω(PX )⊗R M gives rise to a natural transform in the way describedabove. Hence we establish the isomorphism
Nat(ω, ω ⊗ M ) ∼= ω(PX ) ⊗R M ∼= HomR (AX , M ) .
AX is unique up to unique isomorphism (as a flat R module) because it represents the functor
M 7! Nat(ω, ω ⊗ M ).
Next, we shall define a co-action of AX on ω, a counit and a coproduct on AX which makes
AX an R-coalgebra and ω a corepresentation:ρ ∈ Nat(ω, ω ⊗ AX ) ∼= EndR (AX )corresponds to the identity map of AX , andε ∈ HomR (AX , R) ∼= Nat(ω, ω)corresponds to Idω . The co-action ρ tensored with IdAX gives a natural transform between
ω⊗AX and ω⊗AX ⊗AX , whose composition with ρ gives the following commutative diagram:ωρω ⊗ AXψρ⊗IdAX.

ω ⊗ AX ⊗ AX
Take ∆ to be the image of ψ in HomR (AX , AX ⊗R AX ). It follows from definition that AX iscounital and ρ : ω ! ω⊗AX is a corepresentation. It remains to check that ∆ is coassociative.
Observe that the essential image of ω ⊗ AX is a subcategory of the essential image of ω,hence every functor that shows up here can be restricted to ω ⊗ AX , in particular, ρ, whose
restriction to ω ⊗ AX is obviously ρ ⊗ IdAX . It follows from the definition that(ρ ⊗ IdAX ) ◦ ρ = (Idω ⊗ ∆) ◦ ρ ∈ Nat(ω, ω ⊗ AX ⊗ AX ) .
Restrict this equation to ω ⊗ AX and we get(ρ ⊗ IdAX ⊗ IdAX ) ◦ (ρ ⊗ IdAX ) = (Idω ⊗ IdAX ⊗ ∆) ◦ (ρ ⊗ IdAX ) .
Composing with ρ, the LHS corresponds to (∆⊗IdAX )◦∆ and the RHS corresponds to (IdAX ⊗∆)◦∆
whose equality is exactly the coassociativity of AX .
It follows that ∀Z ∈ CX ,ρ(Z) : ω(Z) ! ω(Z) ⊗R AX

gives ω(Z) a AX corepresentation structure and this is functorial in Z, thus ω factors througha φ : CX ! CorepR (AX ).
Back to the uniqueness of AX . It has been shown that it is unique up to unique isomorphism as a flat R module. Additionally, if φ : AX ! A0X is an isomorphism such that it induces
identity transformation on the functor M 7! Nat(ω, ω ⊗ M ) then, φ automatically maps thetriple (∆, ε, ρ) to (∆0 , ε0 , ρ 0 ), so φ is a coalgebra isomorphism.
Finally, it remains to show that when R = k, φ is essentially surjective63 and full:63

In fact, φ is essentially surjective even without the assumption that R = k.

48

SciPost Phys. 9, 017 (2020)• Essentially Surjective: If M ∈ Corepk (AX ), then define
e := Coker(M ⊗ ω(PX ) ⊗ PX ⇒ M ⊗ PX ) ,M
where two arrows are ω(PX ) representation structure of M and PX respectively, thene ) = M ⊗ ω(PX ) = M .
ω( Mω(PX )• Full: If f : M ! N is a AX -corepresentation morphism, then by the k-linearlity of CX , f
lifts to morphismsf ⊗ Id PX : M ⊗ PX ! N ⊗ PX ,andf ⊗ Idω(PX ) ⊗ Id PX : M ⊗ ω(PX ) ⊗ PX ! N ⊗ ω(PX ) ⊗ PX .
e !Ne which is mapped to f by ω.
Thus, passing to cokernel gives rise to fe : M

Next we move on to recover the category C by its subcategories CX . Define an index category I such that its objects are isomorphism classes of objects in C , denoted by X i for eachindex i, and a unique arrow from i to j if X i is a subobject of X j . I is directed because for any
two objects Z and W , they are subobjects of Z ⊕ W . Observe that if X is a subobject of Y , then
CX is a full subcategory of CY , so a functorial restriction
HomR (AY , M ) ∼= Nat(ωY , ωY ⊗ M ) ! Nat(ωX , ωX ⊗ M ) ∼= HomR (AY , M ) ,gives rise to a coalgebra homomorphism AX ! AY . Futhermore, this homomorphism is injective because ω(PY ) ! ω(PX ) is surjective, otherwise Coker(ω(PY ) ! ω(PX )) will be mapped
to the zero object in CorepR (AY ), which contradicts with ω being faithful.
Lemma 5. Define the coalgbra
A := lim AX i ,−!
i∈Ithen it is the desired coalgebra in Theorem 2.
Proof. A is flat because it is an inductive limit of flat R modules. Moreover
HomR (A, M ) = lim HomR (AX i , M ) ∼Nat(ωX i , ωX i ⊗ M ) = Nat(ω, ω ⊗ M ) ,
= lim
−
−
i∈I

i∈I

which gives the desired functorial property and this implies that A is unique up to unique isomorphism. Finally, when R = k, the functor φ is defined and it is fully faithful because it is fully
faithful on each subcategory CX i . It’s also essentially surjective because every corepresentation
V of A comes from a corepresentation of a finite dimensional sub-coalgebra of A,64 and A is afiltered union of sub-coalgebras AX i , so V comes from a corepresentation of some AX i .
Proof of Theorem 2. It remains to prove the theorem when C is monoidal. This amounts toincluding m : C  C ! C and e : 1 ! C with associativity and unitarity constrains, where 1 isTake a basis {ei } for V , the co-action ρ takes ei todimensional sub-coalgebra of A.
64Pj e j ⊗ a ji , then it is easy to see that span{a ji } is a finite

49

SciPost Phys. 9, 017 (2020)

the trivial tensor category with objects {0, 1} and only nontrivial morphisms are End(1) = k.
Using the isomorphism:
HomR (A ⊗R A, A ⊗R A) ∼= Nat(ω  ω, ω  ω ⊗ A ⊗R A) ,we get a homomorphismτ : HomR (A ⊗R A, M ) ! Nat(ω  ω, ω  ω ⊗ M ) .
It is an isomorphism because for each pair of subcategories (CX , CY )HomR (AX ⊗R AY , M ) ∼= HomR (AX , R) ⊗R HomR (AY , M )∼= Nat(ω , ω ) ⊗ Nat(ω , ω ⊗ M )
X

X

R

Y

Y

∼
= Nat(ωX  ωY , ωX  ωY ⊗ M )and it is compatible with the homomorphism given above, so after taking limit, τ is an isomorphism. We also have a homomorphism:
Nat(ω, ω ⊗ M ) ! Nat(ω  ω, ω  ω ⊗ M ) ,by taking any α ∈ Nat(ω, ω⊗M ), and composing with the isomorphism ωω(X Y ) ∼= ω(X ⊗Y ).
This homomorphism in turn becomes a homomorphismµ : A ⊗R A ! A.
And the obvious isomorphism
HomR (R, M ) = M ! Nat(ω1 , ω1 ⊗ M ) ,together with the unit functor e : 1 ! C give a homomorphismι : R ! A.
All of the homomorphisms are functorial with respect to M so µ and ι are homomorphisms
between coalgebras. Now the associativity and unitarity of monoidal category C translates
into associativity and unitarity of µ and ι, which are exactly conditions for A to be a bialgebra.
This concludes the proof of Theorem 2.
Remark 8. In the statement of Theorem 2, it is assumed that R is a local ring or a PID, for thefollowing technical reason: we want to introduce the functor⊗ : Proj f (R) × C ! C ,which is defined by sending (Rn , X ) to X n . This is feasible only if every finite projective module
is free, which is not always true for an arbitary ring. Nevertheless, this is true when R is local
or a PID. It is tempting to eliminate this assumption when C is rigid, since we only use theHom(ω(X ), X ) to define the crucial object PX , and there is no need to define a Hom when the
category is rigid. In fact, there is no loss of information if we define PX by\
Hom(X , X ) ∩ (Y : Y ) ,then the fiber functor ω takes PX to\(EndR (ω(X )) ∩ (ω(Y ) : ω(Y ))) ,since ω is monoidal by definition and a monoidal functor between rigid monoidal categories
preserves duality and thus preserves inner Hom.
450

SciPost Phys. 9, 017 (2020)

Following the above remark, we drop the assumption on ring R and state the following
version of Tannaka reconstruction for Hopf algebras:
Theorem 3 (Tannaka Reconstruction for Hopf Algebra). Let R be a commutative k-algebra, Ca k-linear abelian rigid monoidal category (resp. abelian rigid braided monoidal) with a fiber
functor ω to Mod f (R), then there exists a unique flat R-Hopf algebra A (resp. R-coquasitriangular
Hopf algebra), up to unique isomorphism, such that A represents the endomorphism of ω in thesense that ∀M ∈ IndProj f (R)HomR (A, M ) ∼= Nat(ω, ω ⊗ M ) .
Moreover, there is a functor φ : C ! CorepR (A) which makes the following diagram commutative:Cφω

CorepR (A)f or getMod f (R)and φ is an equivalence if R = k.
Sketch of proof. The idea of proof basically follows [56]. Accoring to Remark 8 and Theorem
2, there exists a bialgebra A which satisfies all conditions in the theorem, so it remains to provethat there are compatible structures on A when C has extra structures.
(a) C is rigid. This means that there is an equivalence between k-linear abelian monoidal
categories
σ : C ! C op ,by taking the right dual of each object, so it turns into an isomophism between R modules
σ : Nat(ω, ω ⊗ M ) ! Nat(ωop , ωop ⊗ M ) .
According to the functoriality of the construction of the bialgebra A, there is a bialgebra
isomorphism:

S : A ! Aop ,put it in another way, a bialgebra anti-automorphism of A. To prove that it satisfies therequired compatibility:µ ◦ (S ⊗ Id) ◦ ∆ = ι ◦ ε = µ ◦ (Id ⊗ S ) ◦ ∆ ,we observe that ι ◦ ε gives the natural transformation
Id ⊗ ρω(1) : ω(X ) = ω(X ) ⊗ ω(1) 7! ω(X ) ⊗ ρ(ω(1)) ,but 1 is the trivial corepresentation of A, so ρ(ω(1)) is canonically identified with ω(1),so ι ◦ ε is just the identity morphism on ω(X ). On the other hand, µ ◦ (S ⊗ Id) ◦ ∆corresponds to the homomorphism
ω(X ) ! ω(X ) ⊗ ω(X )∨ ⊗ ω(X ) ! ω(X ) ⊗ ω(X ∨ ⊗ X ) ! ω(X ) ⊗ ω(1) = ω(X ) ,which is identity by the rigidity of C , hence µ ◦ (S ⊗ Id) ◦ ∆ = ι ◦ ε. The other equation
is similiar.
51

SciPost Phys. 9, 017 (2020)(b) C is rigid braided. This means that there is a natural transformation:r : ω  ω ! ω  ω,
which gives the braiding. This corresponds to a homomorphism of R-modules

R : A⊗ A ! R,
let’s define it to be the universal R-matrix. The fact that r is a natural transformation isequivalent to the diagram below being commutative
ω(U) ⊗ ω(V )ρ⊗ρ

ω(U) ⊗ ω(V ) ⊗ A ⊗ A

Id⊗Id⊗µω(U) ⊗ ω(V ) ⊗ Ar

ω(V ) ⊗ ω(U)

r⊗Id
ρ⊗ρ

ω(V ) ⊗ ω(U) ⊗ A ⊗ A

Id⊗Id⊗µ,ω(V ) ⊗ ω(U) ⊗ A

which in turn translates to the following equation of R:

R12 ◦ µ24 ◦ (∆ ⊗ ∆) = R23 ◦ µ13 ◦ τ13 ◦ (∆ ⊗ ∆) ,where τ : A ⊗ A ! A ⊗ A sends x ⊗ y to y ⊗ x. The compactibility of r with the identity
ω(X )ω(X ) ⊗ ω(1)
,

r

Id

ω(X )

ω(1) ⊗ ω(X )translates to R ◦ (IdA ⊗ 1) = ε. And symmetrically R ◦ (1 ⊗ IdA) = ε.
Finally, the hexagon axiom of braiding:(ω(X ) ⊗ ω(Y )) ⊗ ω(Z)r⊗1

(ω(Y ) ⊗ ω(X )) ⊗ ω(Z)ω(X ) ⊗ (ω(Y ) ⊗ ω(Z)),

r

ω(Y ) ⊗ (ω(X ) ⊗ ω(Z))(ω(Y ) ⊗ ω(Z)) ⊗ ω(X )1⊗r

ω(Y ) ⊗ (ω(Z) ⊗ ω(X ))translates to the commutativity of the diagramA⊗ A⊗ A

Id⊗Id⊗∆µ⊗Id

A⊗ A⊗ A⊗ A
R13 ·R24

R

A⊗ A

,

R

and the same hexagon but with r −1 instead of r gives another one:

52

SciPost Phys. 9, 017 (2020)

A⊗ A⊗ A∆⊗Id⊗IdA⊗ A⊗ A⊗ A
R14 ·R23

Id⊗µ

R

A⊗ A

.

R

So we end up confirming all the properties that universal R-matrix should satisfy, andwe conclude that A is indeed a coquasitriangular Hopf algebra.

Reconstruction from representation It is tempting to dualize everything above to formalize the Tannaka reconstruction for the category of representations. In other words, we cantake the dual of A instead of A itself, and a corepresentation becomes the representaion, andwhen the category has extra structures, those structures will be dualized, for example, when C
is a k-linear abelian rigid braided monoidal category, it should come from the representation
category of a flat R-quasitriangular Hopf algebra, since the dual of those diagrams involved in
the proof of Theorem 3 are exactly properties of universal R-matrix of a quasitriangular Hopf
algebra.
This is naive because the statement:
HomR (U, V ⊗ A) ∼= HomR (U ⊗ A∗ , V ) ,is not true in general, since A can be infinite dimensional, thus the naive dualizing procedure is
not feasible. To resolve this subtlety, we observe that A is constructed from a filtered colimit offinite projective R-modules, each is an R-coalgebra, and any finitely generated corepresentationof A comes from a corepresentation of a finite coalgebra, so it is natural to define the actionof A∗ on those modules by factoring through some finite quotient A∗X for some X ∈ ob(C ).
Similiarly, the multiplication structure on A∗ can be defined by first projecting down to somefinite quotient and taking multiplicationA∗ ⊗ A∗ = lim AX i ⊗ lim AX i ! AX i ⊗ AX i ! AX i
−
−
i∈I

i∈I

which is compatible with transition map AX j ! AX i then taking the inverse limit gives themultiplication of A∗ . For antipode S , its dual is a map A∗ ! A∗ .
On the other hand, the comultiplication on A∗ , is still subtle. If we dualize the multiplication of A, cut-off at some finite submodule
AX i ⊗ AX j ! A,
we only get an inverse system of morphisms from A∗ to A∗X ⊗A∗X and the latter’s inverse limit isijb A∗ , instead of A∗ ⊗ A∗ . So we actually get a topological Hopf algebra with topological basisA∗ ⊗Ni := ker(A∗ ! A∗X ) ,iso that the comultiplication is continuous. Similiarly the counit, multiplication, and anipodeb A∗ ,are continuous as well. Finally when C is braided, there exists an invertible element R ∈ A∗ ⊗and the dual of the structure homomorphism in A is exactly the condition that R is the universal R-matrix of a topological quasitriangular Hopf algebra.
So we can restate Theorem 3 in terms of representations of topological Hopf algebras:
53

SciPost Phys. 9, 017 (2020)

Theorem 4. Let R be a commutative k-algebra, C a k-linear abelian rigid monoidal category(resp. abelian rigid braided monoidal) with a fiber functor ω to Mod f (R), then there exists a
unique topological R-Hopf algebra H (resp. R-quasitriangular Hopf algebra) which is an inverse
limit of finite projective R-modules endowed with discrete topology, up to unique isomorphism,
such that H represents the endomorphism of ω in the sense thatH∼
= Nat(ω, ω) .
Moreover, there is a functor φ : C ! RepR (H) which sends an object in C to a continuous representation of H and makes the following diagram commutative:Cφω

RepR (H)f or get,Mod f (R)and φ is an equivalence if R = k.
Application to Quantization We now consider the case that we have a category Cħh , which
is a quantization of the category of representations of some Hopf algebra H over C. The
quantization, namely Cħh , of RepC (H) is a C-linear abelian monoidal category which has thesame set of generators as RepC (H), together with a fiber functor ωħh : Cħh ! Mod f (CJħhK) whichacts on generators of RepC (H) by tensoring with CJħ
hK, andHomCħh (X , Y ) ∼h = HomRepC (H) (X , Y )= HomCħh (X , Y )/ħ

for any pair of generators X and Y . For example, the classical algebra of local observables
in 4d Chern-Simons theory is U(g[z]), the universal enveloping algebra of Lie algebra g[z],which has the category of representations generated by classical Wilson lines. Quantized Wilson lines naturally generated a C-linear abelian monoidal category.
Applying Theorem 4, (Cħh , ωħh ) gives us a (topological) CJħ
hK-Hopf algebra Hħh . Since Cħh
and C shares the same set of generators, and the construction of those Hopf algebras as CJħ
hKmodules only involves generators of corresponding categories, so Hħh is isomorphic to the completion of H ⊗ CJħ
hK in the ħh-adic topology:
Hħh := lim H X i ⊗ CJħhK ∼lim H ⊗ C[ħ
h]/(ħ
hn )
= lim
−
− − Xi
i∈I

i∈I

n

∼
lim H ⊗ C[ħ
h]/(ħ
hn )
= lim
− − Xini∈I∼
H ⊗ C[ħh]/(ħhn ) .
= lim−n

For the same reason, tensor product of two copies of Hħh and completed in the inverse limit
topology is isomorphic to the completion of Hħh ⊗CJħhK Hħh in the ħh-adic topology:b Hħh ∼
Hħh ⊗
H ⊗
H /(ħ
hn ) .
= lim− ħh CJħhK ħhnFrom the construction of those Hopf algebras and the condition that a morphism in Cħh modulo
ħh is a morphism in RepC (H), it is easy to see that modulo ħh respects all structure homomorphisms, thus Hħh modulo ħh and H are isomorphic as Hopf algebras. Finally, structure homomorphisms of Hħh are continuous in the ħh-adic topology because they are ħ
h-linear. Thus weconlude that:54

SciPost Phys. 9, 017 (2020)

Theorem 5. Hħh is a quantization of H in the sense of Definition 6.1.1 of [45], i.e. it is a topological Hopf algebra over CJħ
hK with ħ
h-adic topology, such that(i) Hħh is isomorphic to H Jħ
hK as a CJħ
hK-module;(ii) Hħh modulo ħ
h is isomorphic to H as Hopf algebras.
In our case, H = U(g[z]) for g = glK [z], so Hħh is a quantization of U(glK [z]), and according
to Theorem 12.1.1 of [45], this is unique up to isomorphisms. This proves Proposition (2).

CTechnicalities of Witten Diagrams

C.1

Vanishing lemmas

We introduce some lemmas to allow us to readily declare several Witten diagrams in the 4d
Chern-Simons theory to be zero.
Lemma 6. The product of two or three bulk-to-bulk propagators vanish when attached cyclically,
diagrammatically this means:
v1
v1

v0 =

v0 = 0 .

(213)

v2
Proof. Two propagators: We can choose one of the two bulk points, say v0 , to be at the origin
and denote v1 simply as v. This amounts to taking the projection (111), namely:
R4v × R4v 3 (v0 , v1 ) 7! v1 − v0 =: v ∈ R4 . Then the product of the two propagators become:0

1

P(v0 , v1 ) ∧ P(v1 , v0 ) 7! P(v) ∧ P(−v) = −P(v) ∧ P(v) .

(214)This is a four form at v, however, P does not have any dz component, therefore the four form
P(v) ∧ P(v) necessarily contains repetition of a one form and thus vanishes.
Three propagators: By choosing v0 to be the origin of our coordinate system we can turnthe product to the following:P(v1 ) ∧ P(v2 ) ∧ P(v1 , v2 ) .

(215)

We now need to look closely at the propagators (see (111) and (114)):ħh x i d yi ∧ dz i + yi dz i ∧ dx i + 2z i dx i ∧ d yi,
2πd(vi , 0)4
ħ
h x 12 d y12 ∧ dz 12 + y12 dz 12 ∧ dx 12 + 2z 12 dx 12 ∧ d y12
P(v1 , v2 ) =,2πd(v1 , v2 )4P(vi ) =(216a)(216b)

where vi := (x i , yi , zi , z i ), x i j := x i − x j , yi j := yi − y j , · · · , and d(vi , v j )2 := (x i2j + yi2j + zi j z i j ).
Since theVpropagators don’t have any dz component the product (215) must be proportionalto ω := i∈{1,2} dx i ∧ d yi ∧ dz i . In the product there are six terms that are proportional to ω.
For example, we can pick dx 1 ∧ d y1 from P(v1 ), dz 2 ∧ dx 2 from P(v2 ) and d y12 ∧ dz 12 fromP(v1 , v2 ), this term is proportional to:dx 1 ∧ d y1 ∧ dz 2 ∧ dx 2 ∧ d y12 ∧ dz 12 = −dx 1 ∧ d y1 ∧ dz 2 ∧ dx 2 ∧ d y2 ∧ dz 1 = +ω .
55

(217)

SciPost Phys. 9, 017 (2020)

The other five such terms are:d y1 ∧ dz 1 ∧ dz 2 ∧ dx 2 ∧ dx 12 ∧ d y12 = − ω ,d y1 ∧ dz 1 ∧ dx 2 ∧ d y2 ∧ dz 12 ∧ dx 12 = + ω ,dz 1 ∧ dx 1 ∧ d y2 ∧ dz 2 ∧ dx 12 ∧ d y12 = + ω ,(218)

dz 1 ∧ dx 1 ∧ dx 2 ∧ d y2 ∧ d y12 ∧ dz 12 = − ω ,dx 1 ∧ d y1 ∧ d y2 ∧ dz 2 ∧ dz 12 ∧ dx 12 = − ω .
These signs can be determined from a determinant, stated differently, we have the following
equation:
d y1 ∧ dz 1
dz 1 ∧ dx 1
dx 1 ∧ d y1
dz 2 ∧ dx 2
dx 2 ∧ d y2  = −6ω ,det  d y2 ∧ dz 2(219)
d y12 ∧ dz 12 dz 12 ∧ dx 12 dx 12 ∧ d y12
where the product used in taking determinant is the wedge product. The above equation
implies that in the product (215) the coefficient of −ω is given by the same determinant if we
replace the two forms with their respective coefficients as they appear in (216). Therefore,
the coefficient is:x1
y1 z 11y2 z 2  = 0 .
det  x 2
(220)
8π3 d(v1 , 0)4 d(v2 , 0)4 d(v1 , v2 )4x 12 y12 z 12
The determinant vanishes because the three rows of the matrix are linearly dependent. Thus
we conclude that the product (215) vanishes.
Lemma 7. The product of two bulk-to-bulk propagators joined at a bulk vertex where the othertwo endpoints are restricted to the Wilson line, vanishes, i.e., in any Witten diagram:
p1

p2
= 0.

(221)

v
Proof. This simply follows from the explicit form of the bulk-to-bulk propagator. Computation
verifies that:(222)ι∂ x ∧∂ x (P(v, p1 ) ∧ P(v, p2 )) = 0 ,12where x 1 and x 2 are the x-coordinates of the points p1 and p2 respectively.
The world-volume on which the CS theory is defined is R2x, y × Cz , which in the presenceof the Wilson line at y = z = 0 we view as R x × R+ × S 2 . When performing integration overthis space we approximate the non-compact direction by a finite interval and then taking thelength of the interval to infinity. In doing so we introduce boundaries of the world-volume,
namely the two components B±D := {±D} × R+ × S 2 at the two ends of the interval [−D, D].
Our next lemma concerns some integrals over these boundaries.
Lemma 8. The integral over a bulk point vanishes when restricted to the spheres at infinity, indiagram:
Z
v1
.
= 0.
(223)
lim
v0
..
D!∞
v0 ∈B±Dvn

56

SciPost Phys. 9, 017 (2020)

Proof. Symbolically, the integration can be written as:
Z
lim

D!∞

B±D

dvolB±D ι∂ y ∧∂z (P(v0 , v1 ) ∧ · · · ∧ P(v0 , vn )) ,(224)

where y and z are coordinates of v0 . Note that the dz required for the volume form on B±D
comes from the structure constant at the interaction vertex, not from the propagators. In theabove integration the x-component of v0 is fixed at ±D, which introduces D dependence in
the integrand. The bulk-to-bulk propagator has the following asymptotic scaling behavior:65
D !∞P((D, y, z, z), v j ) ∼D−2 + O(D−3 ) .

(225)

The integration measure on B±D is independent of D, therefore the integral behaves as D−2n
for large D, and consequently vanishes in the limit D ! ∞.

C.2

Comments on integration by parts

Finally, let us make a few general remarks about the integrals involved in computing Witten
diagrams. Since the boundary-to-bulk propagators are exact and the bulk-to-bulk propagators
behave nicely when acted upon by differential (see (112)), we want to use Stoke’s theorem tosimplify any given Witten diagram. Suppose we have a Witten diagram with m propagatorsconnected to the boundary, n propagators connected to the Wilson line, and l bulk points.
Let us denote the bulk points by vi for i = 1, · · · , l, the points on the Wilson line by p j for
j = 1, · · · , n, and the points on the boundary as x k for k = 1, · · · , m. The domain of integration
for the diagram is then M l × ∆n , where M = R × R+ × S 2 and ∆n is an n-simplex defined as:∆n := {(p1 , · · · , pn ) ∈ Rn | p1 ≤ p2 ≤ · · · ≤ pn } .

(226)This domain may need to be modified in some Witten diagrams due to the integral over thisdomain having UV divergences. UV divergences can occur when some points along the Wilsonline collide with each other. To avoid such divergences we shall use a point splitting regulator,i.e., we shall cut some corners from the simplex ∆n . Let us denote the regularized simplex
e n . The exact description of ∆e n will vary from diagram to diagram, and we shall describe
as ∆them as we encounter them.
When we do integration by parts with respect to the differential in a boundary-to-bulk
propagator, we get the following three types of terms:
1. A boundary term. Boundaries of our integration domain comes from boundaries of M
e n . For M we get:and ∆∂ M = B+∞ t B−∞ .
(227)
Due to Lemma 8, integrations over ∂ M will vanish. Therefore, nonzero contribution
to the boundary integration, when we do integration by parts, will only come from thee n . Schematically, the appearance ofboundary of the regularized simplex, namely ∂ ∆such a boundary integral will look like:ZZdθ ∧ (· · · ) =en
M l ×∆θ ∧ (· · · ) + · · · .

(228)

en
M l ×∂ ∆2. The differential acts on a bulk-to-bulk propagator. Due to (112), this identifies the two
end points of the propagator, schematically:ZZdθ ∧ P ∧ (· · · ) =b ∈ {0, 1} ,en
M l ×∂ b ∆65

θ ∧ (· · · ) + · · · .

en
M l−1 ×∂ b ∆Keep in mind that ħ
h has a (length) scaling dimension 1.

57

(229)

SciPost Phys. 9, 017 (2020)

3. The differential acts on a step function left by a previous integration by parts. This does
not change the domain of integration.
The third option does not to lead a simplification of the domain of integration. Therefore, at thepresent abstract level, our strategy to simplify an integration is: first go to the boundary of thesimplex, and then keep collapsing bulk-to-bulk propagators until we have no more differential
left or when no more bulk-to-bulk propagator can be collapsed without the diagram vanishing
due the vanishing lemmas from §C.1.

D

Proof of Lemma 2

All the diagrams that we draw in this section only exist to represent color factors, their numerical values are irrelevant. Which is why we also ignore the color coding we used in the
diagrams in the main body of the paper.
We start with yet another lemma:
Lemma 9. The color factor of any Witten diagram with two boundary-to-bulk propagators connected by a single bulk-to-bulk propagator, that is any Witten diagrams with the following configuration:
..
..
.
.
(230)
µ

ν

upon anti-symmetrizing the color labels of the boundary-to-bulk propagators, involves the following factor:(231)
fµν ξ X ξ ,for some matrix X ξ that transforms under the adjoint representation of glK . In particular, thiscolor factor is the image in End(V ) of some element of glK where V is the representation of somedistant Wilson line.
Proof. The two bulk vertices in the diagram results in the following product of structure constants: fµo π fνρ o where the indices π and ρ are contracted with the rest of the diagram.
Anti-symmetrizing the indices µ and ν we get fµo π fνρ o − fνo π fµρ o , which using the Jacobi
identity becomes − fµν o fρo π . Once π and ρ are contracted with the rest of the diagram we
get an expression of the general form (231). Furthermore, any expression of the form (231) is
an image in End(V ) of some element in glK , since the structure constant fµν ξ can be viewed
as a map:f : ∧2 glK ! glK ,f : t µ ∧ t ν 7! fµν ξ t ξ .
(232)
Now composing the above map with a representation of glK on V gives the aforementioned
image.
Let us now look at the color factor (161) of the diagram (160), both of which we repeat
here:
,
µ

fµ ξo fξ

πρ

ν

58

fνπ σ %(t o )%(t ρ )%(t σ ) .

(233)

SciPost Phys. 9, 017 (2020)

By commuting %(t o ) and %(t ρ ) in the color factor we create a difference which is the color
factor of the following diagram:
.
µ

(234)

ν

The key feature of the above diagram is the loop with three propagators attached to it. Sucha loop produces a color factor which is a glK -invariant inside (glK )⊗3 , explicitly we can write
a loop and its associated color factor respectively as:
ν
µ

and

ρ

fµo π fνρ o fξπ .

(235)

ξ
The color factor is glK -invariant since the structure constant itself is such an invariant. To findthe invariants in (glK )⊗3 we start by writing glK as:

glK = slK ⊕ C ,(236)

where by slK we mean the complexified algebra sl(K, C). This gives us the decomposition(glK )⊗3 = (slK )⊗3 ⊕ · · · ,(237)

where the “· · · " contains summands that necessarily include at leas one factor of the center
C. However, none of the three indices that appear in the diagram in (235) can correspondto the center, because each of these indices belong to an instance of the structure constant,which vanishes whenever one of its indices correspond to the center.66 This means that the
glK invariant we are looking for must lie in (slK )⊗3 . For K > 2, there are exactly two suchinvariants [57], one of them is the structure constant itself, which is totally anti-symmetric.
The other invariant is totally symmetric. However the structure constant is even (invariant)under the Z2 outer automorphism of slK whereas the symmetric invariant is odd. Since ourtheory has this Z2 as a symmetry, only the structure constant can appear as the invariant ina diagram.67 This means, as far as the color factor is concerned, we can collapse a loop suchas the one in (235) to an interaction vertex. As soon as we do this operation to the diagram(234), Lemma 9 tells us that the color factor of the diagram is an image in End(V ) of anelement in glK . This shows that we can swap the positions of any of the two pairs of theadjacent matrices in the color factor in (233) and the difference we shall create is an image ofa map glK ! End(V ). To achieve all permutations of the three matrices wee need to be able
to keep swaping positions, let us therefore keep looking forward.
Suppose we commute %(t o ) and %(t ρ ) in (233), then we end up with the color factor of the
diagram (159). Now if we commute %(t o ) and %(t σ ), we create a difference that corresponds
the color factor of the following diagram:

.
µ

(238)

ν

66

In other words, the central abelian photon in glK interacts with neither itself nor the non-abelian gluons andtherefore can not contribute to the diagrams we are considering.
67
This is also apparent from the way this invariant is written in (235), since the structure constant is invariant
under this Z2 , certainly a product of them is invariant as well.

59

SciPost Phys. 9, 017 (2020)

The key feature of this diagram is a loop with four propagators attached to it. The loop andits associated color factor can be written as:
ξ

µ
,

ν

fµπ τ f oτ σ fνσ ρ fξρ π .

(239)

o

As before, the color factor is a glK -invariant in (glK )⊗4 . This time, it will be more convenient
to write the color factor as a trace. Noting that the structure constants are the adjoint representations of the generators of the algebra we can write the above color factor as:
trad (t µ t o t ν t ξ ) .

(240)

The adjoint representation of glK factors through slK , and the adjoint representation of slK has
a non-degenerate metric with which we can raise and lower adjoint indices. Suitably changing
positions of some of the indices in the color factor we can conclude:
trad (t µ t o t ν t ξ ) = trad (t µ t ξ t ν t o ) .

(241)

Using the cyclic symmetry of the trace we then find that the color factor is symmetric underthe exchange of µ and ν, therefore when we anti-symmetrize the diagram with respect to µand ν it vanishes.
In summary, starting from the color factor in (233), we can keep swapping any two adjacentmatrices and the difference can always be written as an image of some map glK ! End(V ).
The same argument applies to the color factors of all the diagrams in (159). This proves thelemma.

References[1] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int.
J. Theor. Phys. 38, 1113 (1999), doi:10.1023/A:1026654312961.
[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from non-criticalstring theory, Phys. Lett. B 428, 105 (1998), doi:10.1016/S0370-2693(98)00377-3.
[3] E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys 2, 253 (1998),
doi:10.4310/ATMP.1998.v2.n2.a2.
[4] J. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80, 4859 (1998),
doi:10.1103/PhysRevLett.80.4859.
[5] S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks: Large-N gauge theory andanti-de Sitter supergravity, Eur. Phys. J. C 22, 379 (2001), doi:10.1007/s100520100799.
[6] S. Yamaguch, Bubbling geometries for half BPS Wilson lines, Int. J. Mod. Phys. A 22, 1353
(2007), doi:10.1142/S0217751X07035070.
[7] J. Gomis, S. Matsuura, T. Okuda and D. Trancanelli, Wilson loop correlators at strong
coupling: from matrices to bubbling geometries, J. High Energ. Phys. 08, 068 (2008),
doi:10.1088/1126-6708/2008/08/068.
[8] R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor.
Math. Phys 3, 1415 (1999), doi:10.4310/ATMP.1999.v3.n5.a5.
60

SciPost Phys. 9, 017 (2020)[9] H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577, 419(2000), doi:10.1016/S0550-3213(00)00118-8.
[10] H. Ooguri and C. Vafa, Worldsheet derivation of a large N duality, Nucl. Phys. B 641, 3(2002), doi:10.1016/S0550-3213(02)00620-X.
[11] J. Gomis and T. Okuda, Wilson loops, geometric transitions and bubbling Calabi-Yau’s, J.
High Energ. Phys. 02, 083 (2007), doi:10.1088/1126-6708/2007/02/083.
[12] J. Gomis and T. Okuda, D-branes as a bubbling Calabi-Yau, J. High Energ. Phys. 07, 005(2007), doi:10.1088/1126-6708/2007/07/005.
[13] K. Costello, M-theory in the Omega-background and 5-dimensional non-commutative gauge
theory (2016), arXiv:1610.04144.
[14] K. Costello and S. Li, Twisted supergravity and its quantization (2016), arXiv:1606.00365.
[15] K. Costello, Holography and Koszul duality: the example of the M 2 brane (2017),
arXiv:1705.02500.
[16] K. Costello, Supersymmetric gauge theory and the Yangian (2013), arXiv:1303.2632.
[17] P. S. Aspinwal, D-branes on Calabi-Yau manifolds, Prog. string theory, Proc., 1 (2005),
doi:10.1142/9789812775108_0001.
[18] K. Costello, String theory for mathematicians, http://pirsa.org/C17014, Accessed Aug
31, 2018 (2017).
[19] K. Costello, E. Witten and M. Yamazaki, Gauge theory and integrability, I, Not. Int. Congr.
Chin. Math. 6, 46 (2018), doi:10.4310/ICCM.2018.v6.n1.a6.
[20] G. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs Branches, Commun.
Math. Phys. 209, 97 (2000), doi:10.1007/PL00005525.
[21] A. A. Gerasimov and S. L. Shatashvili, Higgs bundles, gauge theories and quantum groups,
Commun. Math. Phys. 277, 323 (2007), doi:10.1007/s00220-007-0369-1.
[22] A. A. Gerasimov and S. L. Shatashvili, Two-dimensional gauge theories and quantum integrable systems, arXiv:0711.1472.
[23] N. A. Nekrasov and S. L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nuc. Phys.
B - Proc. Supp. 192-193, 91 (2009), doi:10.1016/j.nuclphysbps.2009.07.047.
[24] N. Nekrasov and S. Shatashvili, Quantum integrability and supersymmetric vacua, Prog.
Theor. Phys. Suppl. 177, 105 (2009), doi:10.1143/PTPS.177.105.
[25] N. A. Nekrasov and S. L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, Proc. 16th Int. Congr. Math. Phys. ICMP09, 265 (2010),
doi:10.1142/9789814304634_0015.
[26] K. Costello and D. Gaiotto, Twisted holography (2018), arXiv:1812.09257.
[27] K. Costello and N. M. Paquette, Twisted supergravity and Koszul duality: A case study in
AdS3 (2020), arXiv:2001.02177.
[28] J. Oh and Y. Zhou,
arXiv:2002.07343.

Feynman diagrams and Ω-deformed M-theory (2020),

61

SciPost Phys. 9, 017 (2020)[29] D. Gaiotto and J. Oh, Aspects of Ω-deformed M-theory (2019), arXiv:1907.06495.
[30] D. Gaiotto and J. Abajian, Twisted M2 brane holography and sphere correlation functions(2020), arXiv:2004.13810.
[31] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity andexact results for quantum string amplitudes, Commun. Math. Phys. 165, 311 (1994),
doi:10.1007/BF02099774.
[32] K. Costello and S. Li,
arXiv:1505.06703.

Quantization of open-closed BCOV theory,

I (2015),[33] M. Bershadsky and V. Sadov, Theory of Kähler gravity, Int. J. Mod. Phys. A 11, 4689
(1996), doi:10.1142/S0217751X96002157.
[34] J. Gomis and F. Passerini, Holographic Wilson loops, J. High Energ. Phys. 08, 074 (2006),
doi:10.1088/1126-6708/2006/08/074.
[35] O. Aharony, M. Berkooz and S.-J. Rey, Rigid holography and six-dimensional
N = (2, 0) theories on AdS5 × S1 , J. High Energ. Phys. 03, 121 (2015),
doi:10.1007/JHEP03(2015)121.
[36] J. Gomis and F. Passerini, Wilson loops as D3-branes, J. High Energ. Phys. 01, 097 (2007),
doi:10.1088/1126-6708/2007/01/097.
[37] K. Costello and J. Yagi, Unification of integrability in supersymmetric gauge theories(2018), arXiv:1810.01970.
[38] N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Wilson loops: From 4D supersymmetric Yang-Mills theory to 2D Yang-Mills theory, Phys. Rev. D 77, 047901 (2008),
doi:10.1103/PhysRevD.77.047901.
[39] S. Giombi, R. Roiban and A. A. Tseytlin, Half-BPS Wilson loop and AdS2 /CFT1 , Nucl. Phys.
B 922, 499 (2017), doi:10.1016/j.nuclphysb.2017.07.004.
[40] S. Giombi and S. Komatsu, Exact correlators on the Wilson loop in N = 4 SYM:
Localization, defect CFT, and integrability, J. High Energ. Phys. 05, 109 (2018),
doi:10.1007/JHEP05(2018)109.
[41] S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops
on S 2 from 2d YM and matrix models, J. High Energ. Phys. 10, 033 (2010),
doi:10.1007/JHEP10(2010)033.
[42] N. Nekrasov, Four dimensional
doi:10.13140/RG.2.1.4116.2481.

holomorphic

theories,

Ph.D.

thesis(1996),[43] K. Costello, Integrable lattice models from four-dimensional field theories, Proc. Symp. Pure
Math. 88, 3 (2014), doi:10.1090/pspum/088/01483.
[44] K. Costello, E. Witten and M. Yamazaki, Gauge theory and integrability, II, Not. Int. Congr.
Chin. Math. 6, 120 (2018), doi:10.4310/ICCM.2018.v6.n1.a7.
[45] V. Chari and A. Pressley, A guide to quantum groups (1994).
[46] L. Baulieu, A. Losev and N. Nekrasov, Chern-Simons and twisted supersymmetry in various
dimensions, Nucl. Phys. B 522, 82 (1998), doi:10.1016/S0550-3213(98)00096-0.

62

SciPost Phys. 9, 017 (2020)[47] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program(2006), arXiv:hep-th/0604151.
[48] A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71, 983(1999), doi:10.1103/RevModPhys.71.983.
[49] L. Rozansky and E. Witten, Hyper-Kähler geometry and invariants of three-manifolds, Sel.
math., New ser. 3, 401 (1997), doi:10.1007/s000290050016.
[50] K. Costello and D. Gaiotto, Vertex operator algebras and 3d N = 4 gauge theories, J. HighEnerg. Phys. 05, 018 (2019), doi:10.1007/JHEP05(2019)018.
[51] N. A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math.
Phys. 7, 831 (2003), doi:10.4310/ATMP.2003.v7.n5.a4.
[52] J. Yagi, Ω-deformation and quantization, J. High Energ. Phys. 08, 112 (2014),
doi:10.1007/JHEP08(2014)112.
[53] J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from anti-de Sitter space, J. High Energ. Phys. 06, 008 (2000), doi:10.1088/1126-6708/2000/06/008.
[54] V. Balasubramanian, M. Berkooz, A. Naqvi and M. J. Strassler, Giant gravitons inconformal field theory, J. High Energ. Phys. 04, 034 (2002), doi:10.1088/11266708/2002/04/034.
[55] P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Springer, 111 (2007).
[56] A. Joyal and R. Street, An introduction to tannaka duality and quantum groups, Category
theory, Springer, 413 (1991).
[57] W. McKay, J. Patera and D. Rand, Tables of representations of simple Lie algebras: Exceptional simple lie algebras, Tables Represent, Simple Lie Algebras (1990).

63