Topological Holography: The Example of The D2-D4 Brane System
TTopological Holography: The Example of The D2-D4 BraneSystem
N. Ishtiaque , S. F. Moosavian , Y. Zhou Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada University of Waterloo, Waterloo, ON, Canada* [email protected]
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. Theworld-volume theory on the D2-branes (resp. D4-branes) is 2d BF theory (resp. 4D Chern-Simonstheory) with GL N (resp. GL K ) gauge group. We propose that in the large N limit the BF theory on R is dual to the closed string theory on R × R + × S with the Chern-Simons defect on R × R + × S . Asa 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 of a scatteringalgebra, using Witten diagrams in the Chern-Simons theory. Our computations of the algebras areexact (valid at all loops). Finally, we propose a physical string theory construction of this dualityusing a D3-D5 brane configuration in type IIB – using supersymmetric twist and Ω-deformation. Contents A Op ( T bd ) from BF ⊗ QM theory 13 O ( (cid:126) ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Loop corrections from BF theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Large N limit: The Yangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A Sc ( T bk ) from 4d Chern-Simons Theory 21 O ( (cid:126) ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 Large N limit: The Yangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 a r X i v : . [ h e p - t h ] J u l Physical String Theory Construction of The Duality 35
B.1 Tannaka formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
C Technicalities of Witten Diagrams 53
C.1 Vanishing lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53C.2 Comments on integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
D Proof of Lemma 2 55
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, 2, 3]. A familiar manifestation of theduality is an equality of the partition function of the two theories - the boundary partition functionas a function of sources, and the bulk partition function as a function of boundary values of fields.This in turns implies that correlation functions of operators in the boundary theory can also becomputed in the bulk theory by varying boundary values of its fields [2, 3]. This dictionary has beenextended to include expectation values of non-local operators as well [4, 5, 6, 7]. This is a strong-weakduality, relating a strongly coupled boundary theory to a weakly coupled bulk theory. As is usual instrong-weak dualities, exact computations on both sides of the duality are hard. Topological theorieshave provided interesting examples of holographic dualities where exact computations are possible[8, 9, 10, 11, 12, 13].Recently, it has been shown that some instances of holography can be described as an algebraicrelation, known as Koszul duality, between the operator algebras of the two dual theories [14, 15]. Itwas 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 GL K is the Koszul dual of the Yangian of gl K [16]. In lightof the connection between Koszul duality and holography, this result suggests that if there is a theorywhose local operator algebra is the Yangian of gl K then that theory could be a holographic dual tothe twisted 4d theory. Since the inception of holography, brane constructions played a crucial role infinding dual theories. It turns out that the particular twisted 4d theory is the world-volume theory of K D4-branes embedded in a particular 6d topological string theory [18]. Since the operators whosealgebra is the Koszul dual of the Yangian lives on a line, it is a reasonable guess that we need toinclude some other branes that intersect this stack of D4-branes along a line. Beginning from suchmotivations we eventually find (and demonstrate in this paper) that the correct choice is to embed astack of N D2-branes in the 6d topological string theory so that they intersect the D4-branes alonga line. The world-volume theory of the D2-branes is 2d BF theory with GL N gauge group coupled We are following the convention of [17], according to which, by a D p -brane in topological string theory we mean abrane with a p -dimensional world-volume. In § p -brane refers to a brane with ( p + 1)-dimensional world-volume.
2o a fermionic quantum mechanics along the D2-D4 intersection. The algebra of gauge invariant localoperators along this D2-D4 intersection is precisely the Yangian of gl K .This connected the D2 world-volume theory and the D4 world-volume theory via holography inthe sense of Koszul duality. The connection between these two theories via holography in the senseof [2, 3] was still unclear. In this paper we begin to establish this connection. We take the D2-braneworld-volume theory to be our boundary theory. This implies that the closed string theory in somebackground, including the D4-brane theory should give us the dual bulk theory. In the boundarytheory, we consider the OPE (operator product expansion) algebra of gauge invariant local operators,we argue that this algebra can be computed in the bulk theory by computing a certain algebra ofscatterings from the asymptotic boundary in the limit N ! ∞ . Our computation of the boundarylocal operator algebra using the bulk theory follows closely the computation of boundary correlationfunctions using Witten diagrams [3].The Feynman diagrams and Witten diagrams we compute in this paper have at most two loops,however, we would like to emphasize that the identification we make between the operator algebras andthe Yangian is true at all loop orders. In the boundary theory (D2-brane theory) this will follow fromthe simple fact that, for the operator product that we shall compute, there will be no non-vanishingdiagrams beyond two loops. In the bulk theory this follows from a certain classification of anomaliesin the D4-brane theory [19] and independently from the very rigid nature of the deformation theoryof the Yangian. We explain some of these mathematical aspects underlying our results in appendix B– we begin the appendix with motivations for and a light summary of the purely mathematical resultsto follow.We note that there is a long history of studying links between quantum integrable systems soluble byYangians and quantum affine algebras on one hand and supersymmetric gauge theory dynamics in thevacuum sector on the other hand – including early pioneering work [20] and subsequent developments[21, 22, 23, 24, 25]. In this paper we study particular examples of (quasi) topological gauge theorieswith similar connection to Yangins, the novelty is that we propose a certain holographic duality linkingthe theories.A particular motivation for studying these topological/holomorphic theories and their duality isthat these theories can be constructed from certain brane setup in a physical 10d string theory. Inparticular, we can identify these theories as certain supersymmetric subsectors of some theories onD-branes in type IIB string theory by applying supersymmetric twists and Ω-deformation.The organization of the paper is as follows. In § § N D2-branes and K D4-branes in a 6d topologicalstring theory and describe the two theories that we claim to be holographic dual to each other. In § Y ( gl K )in the limit N ! ∞ . In § §
6, we propose a physical string theory realization of theduality.
Relevant New Literature.
Since the preprint of this paper appeared online, there has been aseries of interesting developments exploring the idea of topological holography – also referred to as twisted holography – we mention the ones we found conceptually related to this paper. In [26] theauthors studied the holographic duality between a gauged βγ system and a Kodaira-Spencer theoryon the SL(2 , C ) manifold (the deformed conifold) – emphasizing the role of global symmetry matching.Authors of [27] studied a twisted holography closely related to AdS /CFT duality, they highlightin particular the link to Koszul duality and contains a rare (in physics literature) introduction toKoszul duality. [28] computes certain operator algebra of a topological quantum mechanics living atthe intersection of M2 and M5 branes in an Ω-deformed M-theory using Feynman diagram techniquessimilar to the ones we shall use in our computations. In this setup the M2 and M5 branes play rolesanalogous to certain D3 and D5 branes that we shall study in §
6. The M5 brane world-volume theoryin the Ω-background is the 5d Chern-Simons theory, a close cousin of the 4d Chern-Simons theory weare going to study. This M2-M5 brane setup in Ω-deformed M-theory is also studied in [29, 30] in the3ontext of twisted holography.
In [2, 3], two theories, T bd and T bk were considered on two manifolds M and M respectively, with theproperty that M was conformally equivalent to the boundary of M . The theory T bd was consideredwith background sources, schematically represented by φ . The theory T bk was such that the values ofits fields at the boundary ∂M can be coupled to the fields of T bd , then T bk was quantized with thefields φ as the fixed profile of its fields at the boundary ∂M . These two theories were considered tobe holographic dual when their partition functions were equal: Z bd ( φ ) = Z bk ( φ ) . (1)This equality leads to an isomorphism of two algebras constructed from the two theories, as follows.Consider local operators O i in T bd with corresponding sources φ i . The partition function Z bd ( φ ) withthese sources has the form: Z bd ( φ ) = (cid:90) D X exp (cid:32) − (cid:126) S bd + (cid:88) i O i φ i (cid:33) , (2)where X schematically represents all the dynamical fields in T bd . Correlation functions of the operators O i can be computed from the partition function by taking derivatives with respect to the sources: (cid:104) O ( p ) · · · O n ( p n ) (cid:105) = 1 Z bd ( φ ) δδφ ( p ) · · · δδφ n ( p n ) Z bd ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ . (3)We can consider the algebra generated by the operators O i using operator product expansion (OPE).However, this algebra is generally of singular nature, due to its dependence on the location of theoperators and the possibility of bringing two operators too close to each other. In specific cases, ofteninvolving supersymmetry, we can consider sub-sectors of the operator spectrum that can generatealgebras free from such contact singularity, so that a position independent algebra can be defined. Suppose the set { O i } represents such a restricted set with an algebra: O i O j = C kij O k . (4)Let us call this algebra A Op ( T bd ). In terms of the partition function and the sources the relation (4)becomes: δδφ i δδφ j Z bd ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ =0 = C kij δδφ k Z bd ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ . (5)The statement of duality (1) then tells us that the above equation must hold if we replace Z bd by Z bk : δδφ i δδφ j Z bk ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ =0 = C kij δδφ k Z bk ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ . (6)This gives us a realization of the operator algebra A Op ( T bd ) in the dual theory T bk .This suggests a check for holographic duality. The input must be two theories, say T bd and T bk ,with some compatibility: • T bd can be put on a manifold M and T bk can be put on a manifold M such that ∂M ∼ = M ,where equivalence between ∂M and M must be equivalence of whatever geometric/topologicalstructure is required to define T bd . Various chiral rings , for example. In case of AdS/CFT, it is conformal equivalence, perfect for defining the CFT. In this paper we shall only beconcerned with topology. Quantum numbers of fields of the two theories are such that the boundary values of the fields in T bk can be coupled to the fields in T bd . Suppose T bd has a sub-sector of its operator spectrum that generates a suitable algebra A Op ( T bd ).We denote the operators in this algebra by { O i } with corresponding sources φ i . According to the firstcompatibility condition these sources can be thought of as boundary values for the fields in T bk , sothat we can quantize T bk by fixing the values of the fields at the boundary to be φ . Then, we candefine another algebra by taking functional derivatives of the partition function of T bk with respectto φ , as in (6). Let’s call this algebra the scattering algebra , A Sc ( T bk ). Now a check of holographicduality is the following isomorphism: A Op ( T bd ) ∼ = A Sc ( T bk ) . (7)This is the general idea that we employ in this paper to check holographic duality. The operator algebra A Op ( T bd ) can be computed in perturbation theory using Feynman diagrams and we can use Wittendiagrams, introduced in [3], to compute the scattering algebra A Sc ( T bk ). We will do this concretely inthe rest of this paper. The quickest way to introduce the theories we claim to be holographic dual to each other is to usebranes to construct them. Our starting point is a 6d topological string theory, in particular, theproduct of the A-twisted string theory on R and the B-twisted string theory on C [18]. The branesetup is the following: R v R w R x R y C z No. of branesD2 0 × × N D4 0 0 × × × K (8)The subscripts denote the coordinates we use to parameterize the corresponding directions, and it isimplied that the complex direction is parameterized by the complex variable z , along with its conjugatevariable z .Our first theory, denoted by T bd , is the theory of open strings on the stack of D2-branes. This isa 2d topological gauge theory with the complexified gauge group GL N [18]. The intersection of theD2-branes with the D4-branes introduces a line operator in this theory. We describe this theory in § K [18]. As it does in the theory on the D2-branes, the intersection between the D2 and theD4-branes introduces a line operator in this theory as well. This line sources a flux supported on the3-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 § The closed string theory, denoted by T cl , is a product of Kodaira-Spencer (also known as BCOV)theory [31, 32] on C and K¨ahler gravity [33] on R , along with a 3-form flux sourced by the stack of 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. Ideally we should consider the OPE algebra of all the operators, but if that is too hard, we can restrict to smallersub-sectors which may still provide a non-trivial check. Fields in this theory, including ghosts and anti-fields, are given by:Set of fields, F := Ω • ( R ) ⊗ Ω • , • ( C ) , (9)i.e., the fields are differential forms on R and ( p, q )-forms on C . The linearized BRST differentialacting on these fields is a sum of the de Rham differential on R and the Dolbeault differential on C ,leading to the following equation of motion: (cid:0) d R + ∂ C (cid:1) α = 0 , α ∈ F . (10)The background field sourced by the D2-branes, let it be denoted by F ∈ F , measures the fluxthrough a topological S surrounding the D2-branes, it can be normalized as: (cid:90) S F = N . (11)Note that the S is only topological, i.e., continuous deformation of the S should not affect the aboveequation. This is equivalent to saying that, the 3-form must be closed on the complement of thesupport of the D2-branes: d R × C F ( p ) = 0 , p (cid:54)∈ D2 . (12)Here the differential is the de Rham differential for the entire space, i.e., d R × C = d R + ∂ C + ∂ C .Moreover, as a dynamically determined background it is also constrained by the equation of motion(10). In addition to satisfying these equations, F must also be translation invariant along the directionsparallel to the D2-branes. The solution is: F = iN π ( v + y + zz ) ( v d y ∧ d z ∧ d z − y d v ∧ d z ∧ d z − z d v ∧ d y ∧ d z ) . (13)In general, a closed string background like this might deform the theory on a brane, however, thepullback of the form (13) to the D4-branes vanishes: ι ∗ F = 0 , (14)where ι : R x,y × C z (cid:44) ! R v,w,x,y × C z is the embedding of the D4-branes into the entire space. Sothe closed string background leaves the D4-brane world-volume theory unaffected. Such a lack ofbackreaction is a rather drastic simplification of the holographic setup which can occur in a topologicalsetting such as ours (see also [15]) but this is not a general feature. For examples of topologicalholography with nontrivial backreaction see [26, 27].The flux (13) signals a change in the topology of the closed string background: R v,w,x,y × C z ! R w,x × R + × S , (15)where the R + is parameterized by r := (cid:112) v + y + zz . This change follows from requiring translationsymmetry in the directions parallel to the D2-branes and the existence of an S supporting the flux F .This S is analogous to the S in the D3-brane geometry supporting the 5-form flux sourced by thesaid D3-branes in Maldacena’s AdS/CFT [1]. The coordinate r measures distance from the locationof the D2-branes. In the absence of a metric we can only distinguish between the two extreme limits This flux is analogous to the 5-form flux sourced by the stack of D3-branes in Maldacena’s setup of AdS/CFT dualitybetween N = 4 super Yang-Mills and supergravity on AdS × S [1]. We are not being careful about the degree (ghost number) of the fields since this will not be used in this paper. The flux (13) is the only background turned on in the closed string theory. This can be argued as follows: TheD2-branes introduce a 4-form source (the Poincar´e dual to the support of the branes) in the closed string theory. Thisform can appear on the right hand side of the equation of motion (10) only for a 3-form field α , which can then have anon-trivial solution, as in (13). Furthermore, since the equation of motion (10) is free, the non-trivial solution for the3-form field does not affect any other field. ! r ! ∞ . The r ! R w,x × R + × S as analogous to the near horizon geometry. This makes R w,x × R + analogous to theAdS geometry. We recall that, in the AdS/CFT correspondence the location of the black branes andthe boundary of AdS correspond to two opposite limits of the non-compact coordinate transverse tothe branes. In our case r = 0 corresponds to the location of the D2-branes, and we treat the plane at r = ∞ , namely: R w,x × {∞} , (16)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 theD5-branes that were considered in [34] in Maldacena’s setup of AdS/CFT, where they were presentedas holographic duals of Wilson loops in 4d N = 4 super Yang-Mills. For the world-volume of thesebranes, the transformation (15) corresponds to: R x,y × C z ! R x × R + × S , (17)where the R + direction is parameterized by r (cid:48) := (cid:112) y + zz . The intersection of the boundary plane(16) and this world-volume is then the line: R x × {∞} , (18)at infinity of r (cid:48) . 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 a lineoperator) in AdS/CFT.2. The effective
3d theory on world-volume R w,x × R + with a defect supported on R x × R + . Thisis 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, 2, 3], we shouldestablish a duality between the operators in the D2-brane world-volume theory and variations ofboundary values of fields in the “gravitational” theory on R w,x × R + (the boundary is R w,x × {∞} ).Both of these surfaces have a line operator/defect and this leads to two types of operators, ones thatare restricted to the line, and others that can be placed anywhere. Local operators in a 2d surfaceare commuting, unless they are restricted to a line. Therefore, in both of our theories, we have non-commutative associative algebras whose centers consist of operators that can be placed anywhere inthe 2d surface. For this paper we are mostly concerned with the non-commuting operators:1. Operators in the world-volume theory of the D2-branes that are restricted to the D2-D4 inter-section.2. Variations of boundary values of fields in the effective theory along the intersection (18) of theboundary R w,x × {∞} and the defect on R x × R + .In physical string theory, the analogue of the D4-branes would be coupled to the closed string modes.In an appropriate large N low energy limit such gravitational couplings can be ignored, leading tothe notion of rigid holography [35]. Since we are working with topological theory at large N , we areassuming such a decoupling.The computations in the “gravitational” side will be governed by the effective dynamics on thedefect on R x × R + . This is the Kaluza-Klein compactification of the world-volume theory of the D4-branes (with a line operator due to D2-D4 intersection). This 4d theory (which we describe in § Effective, in the sense that this is the Kaluza-Klein reduction of a 6d theory with three compact directions, thoughwe don’t want to lose any dynamics, i.e., we don’t throw away massive modes. §
5) this meansthat while we have a 1d boundary, the propagators are from the 4d theory and the bulk points areintegrated over the 4d world-volume R × C . We take the boundary line to be at y = ∞ with somefixed coordinate z in the complex direction. In future we shall refer to this line as (cid:96) ∞ ( z ): (cid:96) ∞ ( 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-
2d black brane R w,x × {∞} D2-braneD4-brane (cid:96) ∞ ( z ) Belongs to thecenter.Duality map wxr •• Figure 1: D2-brane, and the non-compact part of the backreacted bulk.compact part, namely R w,x × R + , of the closed string background (the right hand side of (15)). Weidentify the location of the 2d black brane and the defect D4-branes, the asymptotic boundary R w,x ×{∞} , and the intersection between the boundary and the defect. At the top of the picture, parallelto the asymptotic boundary, we also draw the D2-branes. We draw the D2-branes independently ofthe rest of the diagram because the D2-branes do not exist in the backreacted bulk, they become theblack brane. However, traditionally, parallels are drawn between the asymptotic boundary and thebrane sourcing the bulk (the D2-brane in this case). The dots on the asymptotic boundary representlocal variations of boundary values of fields in the bulk theory T bk . The corresponding dots on theD2-brane represent the local operators in the boundary theory T bd that are dual to the aforementionedvariations. By the duality map in the figure we schematically represent boundary excitations in thebulk theory corresponding to some local operators in the dual description of the same dynamics interms of the boundary theory. This is a 2d topological gauge theory on the stack of N D2-branes (see (8)), supported on R w,x , withcomplexified gauge group GL N . The field content of this theory is:Field Valued in B Ω ( R ) × gl N A Ω ( R ) × gl N . (20) A is a Lie algebra valued connection and B is a Lie algebra valued scalar, both complex. The curvatureof the connection is denoted as F = d A + A ∧ A . The action is given by: S BF := (cid:90) R w,x tr N ( BF ) , (21)8here the trace is taken in the fundamental representation of gl N .We consider this theory in the presence of a line operator supported on R x × { } , caused by theintersection of the D2 and D4-branes. The line operator is defined by a fermionic quantum mechanicalsystem living on it. The fields in the quantum mechanics (QM) are K fundamental (of gl N ) fermionsand their complex conjugates:Field Valued in ψ i Ω ( R x ) × N ψ i Ω ( R x ) × N , i ∈ { , · · · , K } , (22)where N refers to the fundamental representation of gl N and N to the anti-fundamental. The fermionicsystem has a global symmetry GL N × GL K . These fermions couple naturally to the gl N connection A of the BF theory. The action for the QM is given by: S QM := (cid:90) R x (cid:16) ψ i d ψ i + ψ i A ψ i + ψ j A ji ψ i (cid:17) , (23)where we have introduced a background gl K -valued gauge field A ∈ Ω ( R x ) × gl K . Note that the termsin the above action are made gl N invariant by pairing up elements of N with elements of the dualspace N .Our first theory is this BF theory with the line operator, schematically: T bd := BF N ⊗ N QM N × K , (24)where the subscripts on BF and QM refer to the symmetries (GL N and GL N × GL K respectively) ofthe respective theories and the subscript on ⊗ implies that the GL N is gauged. There are two typesof gauge ( gl N ) invariant operators in the theory: for n ∈ N ≥ , operators restricted to R x : O ij [ n ] := (cid:126) ψ j B n ψ i , operators not restricted to R x : O [ n ] := (cid:126) tr N B n . (25)Unrestricted local operators in two topological dimensions can be moved around freely, implying thatfor any n ≥
0, the operator O [ n ] commutes with all of the operators defined above. The operatoralgebra 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 boundarytheory by its center. We shall compute their Lie bracket in §
4, which will establish an isomorphismwith the Yangian. Had we included the commuting operators as well we would have found a centralextension of the Yangian. In sum, the operator algebra we construct from the theory T bd is: A Op ( T bd ) := (cid:0) O ij [ n ] , O [ n ] (cid:1) / ( O [ n ]) . (26)By the notation ( x, y, · · · ) we mean the algebra generated by the set of operators { x, y, · · · } over C . Remark . Note that it is possible to lift our D2 and D4 branes to type IIB stringtheory while maintaining a one dimensional intersection. This results in a D3-D5 setup (studied inparticular in [34]) where on the D3 brane we find the N = 4 Yang-Mills theory with a Wilson line. In [38, 39, 40], the authors considered local operators in the N = 4 Yang-Mills that are restricted to This closely resembles the D3-D5 system in type IIB string theory considered in [34], there too a fermionic quantummechanics lived on the intersection, giving rise to Wilson lines upon integrating out the fermions. Note that we could haveconsidered 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]. The (cid:126) − appears in these definitions because the action (23) will appear in path integrals as exp (cid:0) − (cid:126) − S QM (cid:1) , whichmeans functional derivatives with respect to A ij inserts operators that carry (cid:126) − . These operators are represented by the red dot on the D2-brane in figure 1. We shall similarly quotient out the center in the bulk theory as well. It is also interesting to note that the D5 brane in an Omega background reproduces the 4d CS theory [37]. The algebra in [40]would correspond to the K = 1 instance of our algebra, it may be an interesting check to compute theanalogue of the algebra in [40] for higher K . (cid:52) This is a 4d gauge theory on the stack of K D4-branes, supported on R x,y × C z with the line L := R x × (0 , ,
0) removed and with the (complexified) gauge group GL K . The notation of distinguishingdirections by R and C is meant to highlight the fact that observables in this theory depend only on thetopology of the real directions and depend holomorphically on the complex direction. In particular,they are independent of the coordinates x and y that parameterize the R , and depend holomorphicallyon z which parameterizes the C . Due to the removed line, we can represent the topology of the supportof this theory as (c.f. (17)): M := R × R + × S . (27)The field of this theory is just a connection:Field Valued in A Ω ( R × C \ L )(d z ) ⊗ gl K . (28)The above notation simply means that A is a gl K -valued 1-form without a d z component. The theoryis defined by the action: S CS := i π (cid:90) M d z ∧ CS( A ) , (29)where CS( A ) refers to the standard Chern-Simons Lagrangian:CS( A ) = tr K (cid:18) A ∧ d A + 23 A ∧ A ∧ A (cid:19) , (30)where the trace is taken over the fundamental representation of gl K . 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-Simonstheory and sometimes denote it by CS K or just CS.The removal of the line L from R × C is caused by the D2-D4 brane intersection. Note that fromthe perspective of the CS theory, the D2-D4 intersection looks like a Wilson line. This means that weshould be quantizing the CS theory on M with a background electric flux supported on the S inside M . Alternatively, we can quantize the CS theory on R × C with a Wilson line inserted along L . The choice of representation for the Wilson line is determined by the number, N , of D2-branes – letus denote this representation as (cid:37) : gl K ! End( V ). With this choice, the Wilson line is defined as thefollowing operator: W (cid:37) ( L ) := P exp (cid:18)(cid:90) L (cid:37) ( A ) (cid:19) , (31) We thank Shota Komatsu for pointing out this interesting possibility. This theory was proposed in [42] to explain the representation theory of quantum affine algebras and more recentlystudied in [43, 16, 19, 44] as a way of producing integrable lattice models using Wilson lines. 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 the fermion, leavinga Wilson line in its place. The mechanism is the same that appeared for intersection of D3 and D5-branes in physicalstring theory [34]. P exp implies path ordered exponentiation, made necessary by the fact that the exponent ismatrix valued. The above operator is valued in End( V ). This in general means that the followingexpectation value: (cid:104) W (cid:37) ( L ) (cid:105) = (cid:82) D A W (cid:37) ( L ) exp (cid:0) − (cid:126) S CS (cid:1)(cid:82) D A exp (cid:0) − (cid:126) S CS (cid:1) , (32)is valued in Hom( H −∞ ⊗ V, H + ∞ ⊗ V ), where H ±∞ are the Hilbert spaces of the CS K theory on theCauchy surfaces perpendicular to L at x = ±∞ , in the absence of the Wilson line. However, for theparticular CS theory, these Hilbert spaces are trivial and we end up with a map that transports vectorsin V from x = −∞ to x = + ∞ : (cid:104) W (cid:37) ( L ) (cid:105) : V −∞ ! V + ∞ . (33)In picture this operator may be represented as: (cid:104) W (cid:37) ( L ) (cid:105) : W (cid:37) ( L ) V Vx = −∞ x = + ∞ . (34)The CS theory is quantized with some fixed boundary profile of the connection along the boundary R x × {∞} × S . To express the dependence of expectation values on this boundary value we put asubscript, such as (cid:104) W (cid:37) ( L ) (cid:105) A . Since we are essentially interested in the Kaluza-Klein reduced theoryon R x × R + we mostly care about the value of the connection along the boundary line (defined in (19)) (cid:96) ∞ ( z ) ⊂ R x × {∞} × S .To define our second theory, we start with the product of the closed string theory and the CStheory, T cl ⊗ CS K , supported on R w,x × R + × S and compactify on S , our notation for this theoryis the following: T bk := π S ∗ (cid:0) T cl ⊗ CS K (cid:1) . (35)We can put the theory T bd (24) on the plane R w,x at infinity of R w,x × R + . This plane has adistinguished line R x × {∞} (18) where the D4-brane world volume intersects. Along this line wehave the gl K gauge field which couples to the fermions of the QM in T bd (this coupling correspondsto the last term in (23)). Boundary excitations from arbitrary points on R w,x × {∞} will correspondto operators in the BF theory that are commuting, since these local excitations on a plane are notordered. The non-commutative algebra we are interested in the BF theory is the algebra of gaugeinvariant operators restricted to a particular line. Similarly, in the “gravitational” side of the setup,we are interested in boundary excitations restricted to the line (cid:96) ∞ ( z ). Let us look a bit more closelyat the coupling between the connection A and the fermions: I z := 1 (cid:126) (cid:90) (cid:96) ∞ ( z ) ψ i A ji ψ j , (cid:96) ∞ ( z ) = R x × { y = ∞} × { z } . (36)A small variation of z leads to coupling between the fermions and z -derivatives of the connection: I z + δz = ∞ (cid:88) n =0 (cid:126) (cid:90) (cid:96) ∞ ( z ) ( δz ) n n ! ψ i ∂ nz A ji ψ j . (37)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 the operatorin the boundary theory T bd that couples to the derivative ∂ nz A ji is precisely the operator O ij [ n ] = 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. − ψ i B n ψ j (c.f. (25), (26)). This motivates us to look at functional derivatives of (cid:104) W (cid:37) ( L ) (cid:105) A withrespect to ∂ nz A ji at fixed points along (cid:96) ∞ ( z ), such as: δδ∂ n z A j i ( p ) · · · δδ∂ n m z A j m i m ( p m ) (cid:104) W (cid:37) ( L ) (cid:105) A , p , · · · , p m ∈ (cid:96) ∞ ( z ) . (38)Just as the expectation value (cid:104) W (cid:37) ( L ) (cid:105) A is End( V )-valued, these functional derivatives are End( V )-valued as well. The action is given by applying the functional derivative on (cid:104) W (cid:37) ( L ) (cid:105) A ( ψ ) for any ψ ∈ V . Let us denote this operator as T ij [ n ] : (cid:96) ∞ ( z ) × V ! V ,p ∈ (cid:96) ∞ ( z ) , T ij [ n ]( p ) : ψ δδ∂ nz A ji ( p ) (cid:104) W (cid:37) ( L ) (cid:105) A ( ψ ) . (39)which can be pictorially represented by slight modifications of (34): W (cid:37) ( L ) δδ∂ nz A ji x = py = 0 , ψ T ij [ n ]( p )( ψ ) x = −∞ x = + ∞ y = ∞ (40)Composition of these operators, such as T i j ( p ) · · · T i m j m ( p m ), is defined by the expression (38). A moreprecise and computable characterization of these operators and their composition in terms of Wittendiagrams [3] will be given in § x -direction, theoperator T ij [ n ]( p ) must be independent of the position p . However, since these operators are positionedalong a line, their product should be expected to depend on the ordering, leading to a non-commutativeassociative algebra. We can now define the second algebra to appear in our example of holography: A Sc ( T bk ) := (cid:0) T ij [ n ] (cid:1) , (41)i.e., the complex algebra generated by the set { T ij [ n ] } . Remark . In the BF theory we mentioned gauge invariant operators thatbelong to the center of the algebra. Clearly, the holographic dual of those operators do not come fromthe CS theory, rather they come from the closed string theory. A 2-form field φ = φ wx d w ∧ d x + · · · from the closed string theory deforms the BF theory as: S BF ! S BF + (cid:90) R w,x d w ∧ d x ( ∂ nz φ wx ) tr N ( B n ) . (42)Functional derivatives with respect to the fields ∂ nz φ w,x placed at arbitrary locations on the asymptoticboundary R w,x × {∞} correspond to inserting the operators tr N B n in the BF theory. As we did inthe BF theory, we are going to ignore these operators now as well. (cid:52)
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 inthe theory T bd = BF N ⊗ N QM N × K , and the algebra of scatterings from a line in the boundary (41) ofthe theory T bk = π S ∗ (cid:0) T cl ⊗ CS K (cid:1) are isomorphic to the Yangian of gl K , i.e.: A Op ( T bd ) N ! ∞ ∼ = Y (cid:126) ( gl K ) N ! ∞ ∼ = A Sc ( T bk ) . (43)The rest of the paper is devoted to the explicit computations of these algebras. These functional derivatives are represented by the red dot on the asymptotic boundary in figure 1. A Op ( T bd ) from BF ⊗ QM theory In this section we prove the first half of our main result (Theorem 1):
Proposition 1.
The algebra A Op ( T bd ) , defined in the context of 2d BF theory with the gauge group GL N coupled to a 1d fermionic quantum mechanics with global symmetry GL N × GL K , is isomorphicto the Yangian of gl K in the limit N ! ∞ : A Op ( T bd ) N ! ∞ ∼ = Y (cid:126) ( gl K ) . (44)The BF theory coupled to a fermionic quantum mechanics was defined in § S T bd = S BF + S QM , (45)where: S BF = (cid:90) R w,x tr N ( B d A + B [ A , A ]) (46)and S QM = (cid:90) R x (cid:0) ψ i d ψ i + ψ i A ψ i (cid:1) . (47)We no longer need the source term, i.e., the coupling to the background gl K connection (c.f. (23)).Let us determine the propagators now.The BF propagator is defined as the 2-point correlation function: P αβ ( p, q ) := (cid:10) B α ( p ) A β ( q ) (cid:11) . (48)We choose a basis { τ α } of gl N which is orthonormal with respect to the trace tr N :tr N ( τ α τ β ) = δ αβ . (49)Then the two point correlation function becomes diagonal in the color indices: P αβ ( p, q ) ≡ δ αβ P ( p, q ) . (50)We shall often refer to just P as the propagator, it is determined by the following equation: (cid:126) d P (0 , p ) = δ ( p )d w ∧ d x . (51)Once we impose the following gauge fixing condition, analogous to the Lorentz gauge:d (cid:63) P (0 , p ) = 0 , (52)the solution is (using translation invariance to replace the 0 with an arbitrary point): P ( p, q ) = (cid:126) π d φ ( p, q ) , (53)where φ ( p, q ) is the angle (measured counter-clockwise) between the line joining p - q and any otherreference line passing through p . In Feynman diagrams we shall represent this propagator as: P ( p, q ) = p q . (54) A minor technicality: P ( p, q ) is a 1-form on R p × R q and in (51), by P (0 , p ) we mean the pull-back of P ∈ Ω ( R )by the diagonal embedding R (cid:44) ! R × R . (cid:126) ∂ x (cid:68) ψ ai ( x ) ψ jb ( x ) (cid:69) = δ ab δ ji δ ( x − x ) , (55)with the solution: (cid:68) ψ ai ( x ) ψ jb ( x ) (cid:69) = δ ab δ ji (cid:126) ϑ ( x − x ) , (56)where ϑ ( x − x ) is a unit step function. Anti-symmetry of the fermion fields dictates: (cid:68) ψ jb ( x ) ψ ai ( x ) (cid:69) = − (cid:68) ψ ai ( x ) ψ jb ( x ) (cid:69) = − δ ab δ ji (cid:126) ϑ ( x − x ) . (57)We take the step function to be: ϑ ( x ) = 12 sgn( x ) = / x >
00 for x = 0 − / x < . (58)Then we can write: (cid:68) ψ ai ( x ) ψ jb ( x ) (cid:69) = (cid:68) ψ jb ( x ) ψ ai ( x ) (cid:69) = δ ab δ ji (cid:126) x − x ) . (59)This propagator does not distinguish between ψ and ψ and it depends only on the order of the fields,not their specific positions. In Feynman diagrams we shall represent this propagator as: (cid:126) x − x ) = x x , (60)where the curved line refers to the propagator itself and the horizontal line refers to the support ofthe QM, i.e., the line w = 0. We now move on to computing operator products that will give us thealgebra A Op ( T bd ). Remark . We might as well have considered a bosonic QM insteadof a fermionic QM. At present, this is an arbitrary choice, however, if one starts from some branesetup in physical string theory and reduce it to the topological setup we are considering by twists andΩ-deformations, then depending on the starting setup one might end up with either statistics. Letus make a few comments about the bosonic case. In the first order formulation of bosonic QM theaction looks exactly as in the fermionic action 47 except the fields would be commuting – let us denotethe bosonic counterpart of ψ and ψ by φ and φ respectively. Then, instead of the propagator (59), wewould have the following propagator: − (cid:68) φ ai ( x ) φ jb ( x ) (cid:69) = (cid:68) φ jb ( x ) φ ai ( x ) (cid:69) = δ ab δ ji (cid:126) x − x ) . (61)Note that the extra sign in the first term (compared to (59)) is consistent with the commutativity ofthe bosonic fields: (cid:68) φ ai ( x ) φ jb ( x ) (cid:69) = (cid:68) φ jb ( x ) φ ai ( x ) (cid:69) . (62)The bosonic propagator (61) distinguishes between φ and φ , in that, the propagator is positive if φ ( x )is placed before φ ( x ), i.e., x < x , and negative otherwise. (cid:52) We describe one such specific procedure in § We have chosen the overall sign of the propagator to make comparision between Feynman diagrams involving bosonicoperators and fermionic operators as simple as possible. However, the overall sign is not important for the determinationof the algebra. The parameter (cid:126) enters the algebra as the formal variable deforming the universal enveloping algebra U ( gl K [ z ]) to its Yangian, and the sign of (cid:126) is irrelevant for this purpose. .1 Free theory limit, O ( (cid:126) ) Interaction in the quantum mechanics is generated via coupling to the gl N gauge field (see (47)).Without this coupling, the quantum mechanics is free. In this section we compute the operatorproduct between O ij [ m ] and O kl [ n ] in this free theory, which will give us the classical algebra.Let us denote the operator product by (cid:63) , as in: O ij [ m ] (cid:63) O kl [ n ] . (63)The classical limit of this product has an expansion in Feynman diagrams where we ignore all diagramswith BF propagators. Before evaluating this product let us illustrate the computations of the relevantdiagrams by computing one exemplary diagram in detail.Consider the following diagram: G ikjl [ (cid:77) · (cid:78) ]( x , x ) := x O ij [ m ] x O kl [ n ] (64)We are representing the operator O ij [ m ] = (cid:126) ψ aj ( B m ) ba ψ ib by the symbol where the three dotsrepresent the three fields ψ aj , ( B m ) ba , and ψ ib respectively. The coordinate below an operator in (64)represents the position of that operator and the lines connecting different dots are propagators. De-pending on which dots are being connected a propagator is either the BF propagator (53) or the QMpropagator (59). The value of the diagram is then given by: G ikjl [ (cid:77) · (cid:78) ]( x , x ) = 1 (cid:126) ψ aj ( x )( B ( x ) m ) ba (cid:126) δ cb δ il (cid:126) ( B ( x ) n ) dc ψ kd ( x ) , = 12 (cid:126) δ il ψ j ( x ) B ( x ) m B ( x ) n ψ k ( x ) . (65)In the second line we have hidden away the contracted gl N indices. In computing the operator product(63) only the following limit of the diagram is relevant:lim x ! x G ikjl [ (cid:77) · (cid:78) ]( x , x ) = 12 (cid:126) δ il ψ j B m + n ψ k = 12 δ il O kj [ m + n ] . (66)We have ignored the positions of the operators, because the algebra we are computing must be transla-tion invariant. Reference to position only matters when we have different operators located at differentpositions.We can now give a diagramatic expansion of the operator product (63) in the free theory: O ij [ m ] (cid:63) O kl [ n ] x ! x = x x + x x + x x + x x . (67)We have omitted the labels for the operators in the diagrams. It is understood that the first operatoris O ij [ m ] and the second one is O kl [ n ]. Summing these four diagrams we find: O ij [ m ] (cid:63) O kl [ n ] = O ij [ m ] O kl [ n ] + 12 δ il O kj [ m + n ] − δ kj O il [ m + n ] + 14 δ il δ kj tr N B m + n . (68) The reader can ignore the elaborate symbols (triangles and as such) that we use to refer to a diagram. They aremeant to systematically identify a diagram, but for practical purposes the entire expression can be thought of as anunfortunately long unique symbol assigned to a diagram, just to refer to it later on. x ! x G ikjl [ (cid:78) · (cid:77) ]( x , x ) = 12 (cid:126) δ kj ψ i B m + n ψ l = − (cid:126) δ kj ψ l B m + n ψ i = − δ kj O il [ m + n ] . (69)Using (68) we can compute the Lie bracket of the algebra A Op ( T bd ) in the classical limit: (cid:2) O ij [ m ] , O kl [ n ] (cid:3) (cid:63) = δ il O kj [ m + n ] − δ kj O il [ m + n ] . (70)This is the Lie bracket in the loop algebra gl K [ z ]. Remark . How would the bracket (70) be affected if wehad a bosonic QM? It would not. The first and the fourth diagrams from (67) would still cancel withtheir counterparts when we take the commutator. The value of the second diagram, (66), remainsunchanged. In computing the value of the third diagram (see (69)) we get an extra sign compared tothe fermionic case because we don’t pick up any sign by commuting bosonos, however, we pick up yetanother sign from the propagator relative to the fermionic propagator (see Remark 3 – compare thebosonic (61) and fermionic (59) propagators).
Interaction in the BF theory comes from the following term in the BF action (46): f αβγ (cid:90) R B α A β ∧ A γ , (72)where the structure constant f αβγ comes from the trace in our orthonormal basis (49): f αβγ = tr N ( τ α [ τ β , τ γ ]) . (73)In Feynman diagrams this interaction will be represented by a trivalent vertex with exactly 1 outgoingand 2 incoming edges. Including the propagators for the edges, such a vertex will look like: q , βq , γ q , αp = (cid:126) (2 π ) f αβγ (cid:90) p ∈ R d q φ ( p, q ) ∧ d q φ ( p, q ) ∧ d q φ ( p, q ) , =: V αβγ ( q , q , q ) . (74)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 nocycles. This means that there is only one possible BF diagram that will appear in our computations, The isomorphism is given by: O ij [ m ] z m e ji , where e ji are the elementary matrices of dimension K × K satisfyingthe relation: [e ji , e lk ] = δ li e jk − δ jk e li . (71) By cycle we mean loop in the sense of graph theory. In this paper when we write loop without any explanation, wemean the exponent of (cid:126) , as is customary in physics. This exponent is related but not always equal to the number ofloops (graph theory). Therefore, we reserve the word loop for the exponent of (cid:126) , and the word cycle for what would beloop in graph theory.Let us illustrate why there are no cycles in BF Feynman diagrams. Consider the cycle . The three propagatorsin the cycle contribute the 3-form d φ ∧ d φ ∧ d φ to a diagram containing the cycle, where the φ ’s are the angles betweentwo successive vertices. However, due to the constraint φ + φ + φ = 2 π , only two out of the three propagators arelinearly independent. Therefore, their product vanishes. . (75)The middle operator looks slightly different because this operator involves the connection A and anintegration, as opposed to just the B field, to be specific,= 1 (cid:126) (cid:90) R ψ i A ψ i . (76)This term is the result of the insertion of the term coupling the fermions to the gl N connection inthe QM action (47). In doing the above integrationover R we shall take ψ and ψ to be constant. Inother words, we are taking derivatives of the fermions to be zero. The reason is that, the equationsof motion for the fermions (derived from the action (47)), namely d ψ i = − A ψ i and d ψ i = A ψ i , tellus that derivatives of the fermions are not gauge-invariant quantities – and we want to expand theoperator product of gauge invariant operators in terms of other gauge invariant operators only. In the following we shall consider the diagram (75) with all possible fermionic propagators addedto it. fermionic propagators We are mostly going to compute products of level 1 operators, i.e., O ij [1], this is because together withthe level 0 operators, they generate the entire algebra. Without any fermionic propagators, we justhave the diagram (75): G ikjl [ ·· ]( x , x ) := x O ij [1] x (cid:126) (cid:82) ψ A ψ x O kl [1] . (77)In future, we shall omit the labels below the operators to reduce clutter. In terms of the BF vertexfunction (74), the above diagram can be expressed as: G ikjl [ ·· ]( x , x ) = 1 (cid:126) ψ j τ α ψ i ψτ β ψψ l τ γ ψ k (cid:90) R x V αβγ ( x , x, x ) . (78)We have used the expansions of B = B α τ α and A = A β τ β in the orthonormal gl N basis { τ α } . Asdefined in (74), the vertex function V αβγ is a 2d integral of a 3-form, therefore, the integration of thevertex function on a line gives us a number. It will be convenient to divide up the integral of the vertexfunction into three integrals depending on the location of the point x relative to x and x : (cid:90) R x V αβγ ( x , x, x ) = V αβγ ·|| ( x , x ) + V αβγ |·| ( x , x ) + V αβγ ||· ( x , x ) , (79)where, V αβγ ·|| ( x , x ) := (cid:90) x Proposition 2. The algebra A Sc ( T bk ) , defined in (41) in the context of 4d Chern-Simons theory, isisomorphic to the Yangian Y (cid:126) ( gl K ) : A Sc ( T bk ) N ! ∞ ∼ = Y (cid:126) ( gl K ) . (99)The 4d Chern-Simons theory with gauge group GL K , also denoted by CS K , is defined by the action(29), which we repeat here for convenience: S CS := i π (cid:90) R x,y × C z d z ∧ tr K (cid:18) A ∧ d A + 23 A ∧ A ∧ A (cid:19) . (100)21he trace in the fundamental representation defines a positive-definite metric on gl K , moreover, wechoose a basis of gl K , denoted by { t µ } , in which the metric becomes diagonal:tr K ( t µ t ν ) ∝ δ µν . (101)We consider this theory in the presence of a Wilson line in some representation (cid:37) : gl K ! End( V ),supported along the line L defined by y = z = 0: W (cid:37) ( L ) = P exp (cid:18)(cid:90) L (cid:37) ( A ) (cid:19) . (102)Consideration of fusion of Wilson lines to give rise to Wilson lines in tensor product representationshows that it is not only the connection A that couples to a Wilson line but also its derivatives ∂ nz A [19]. Furthermore, gauge invariance at the classical level requires that ∂ nz A couples to the Wilson linevia a representation of the loop algebra gl K [ z ]. So the line operator that we consider is the following: P exp (cid:88) n ≥ (cid:37) µ,n (cid:90) L ∂ nz A µ , (103)where the matrices (cid:37) µ,n ∈ End( V ) satisfy:[ (cid:37) µ,m , (cid:37) ν,n ] = f ξµν (cid:37) ξ,m + n . (104)The structure constant f ξµν is that of gl K . In particular, we have (cid:37) µ, = (cid:37) ( t µ ).In (28), A was defined to not have a d z component. The reason is that, due to the appearanceof d z in the above action (100), the d z component of the connection A never appears in the actionanyway. Though the theory is topological, in order to do concrete computations, such as imposing gaugefixing conditions, computing propagator, and evaluating Witten diagrams etc. we need to make achoice of metric on R x,y × C z , we choose: d s = d x + d y + d z d z . (107)For the GL K gauge symmetry we use the following gauge fixing condition: ∂ x A x + ∂ y A y + 4 ∂ z A z = 0 . (108)The propagator is defined as the two-point correlation function: P µν ( v , v ) := (cid:104) A µ ( v ) A ν ( v ) (cid:105) . (109)Since in the basis of our choice the Lie algebra metric is diagonal (101), this propagator is proportionalto a Kronecker delta in the Lie algebra indices: P µν ( v , v ) = δ µν P ( v , v ) , (110) Had we defined the space of connections to be Ω ( R x,y × C z ) ⊗ gl K , then, in addition to the usual GL K gaugesymmetry, we would have to consider the following additional gauge transformation: A ! A + f d z , (105)for arbitrary function f ∈ Ω ( R × C ). We could fix this gauge by imposing: A z = 0 . (106)This would get us back to the space (cid:0) Ω ( R x,y × C z ) / (d z ) (cid:1) ⊗ gl K . For this theory we follow the choices of [19] whenever possible. P is a 2-form on R v × R v . We can fix one of the coordinates to be the origin, this amounts totaking the projection: (cid:36) : R v × R v ! R v , (cid:36) : ( v , v ) v − v =: v . (111)Due to translation invariance, P can be written as a pullback of some 2-form on R by (cid:36) , i.e., P = (cid:36) ∗ P for some P ∈ Ω ( R ). The propagator P can be characterized as the Green’s function forthe differential operator i π (cid:126) d z ∧ d that appears in the kinetic term of the action S CS . For P thisresults in the following equation: i π (cid:126) d z ∧ d P ( v ) = δ ( v )d x ∧ d y ∧ d z ∧ d z , (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 ) = (cid:126) π x d y ∧ d z + y d z ∧ d x + 2 z d x ∧ d y ( x + y + zz ) . (114)The propagator P ( v , v ) will be referred to as the bulk-to-bulk propagator, since the points v and v can be anywhere in the world-volume R x,y × C z of CS theory. To compute Witten diagrams we alsoneed a boundary-to-bulk propagator. We will denote it as K µ ( v, x ) ≡ K ( v, x ) t µ , where v ∈ R x,y × C z and x ∈ (cid:96) ∞ ( z ) is restricted to the boundary line. The boundary-to-bulk propagator is a 1-form definedas a solution to the classical equation of motion:d z v ∧ d v K ( v, x ) = 0 , (115)and by the condition that when pulled back to the boundary, in this case (cid:96) ∞ ( z ), it must become adelta function supported at x : ε ∗ K ( x (cid:48) , x ) = δ ( x (cid:48) − x )d x (cid:48) , x (cid:48) ∈ (cid:96) ∞ ( z ) (116)where ε : (cid:96) ∞ ( z ) (cid:44) ! R × 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 ) = δ ( x v − x )d x v , (117)where x v refers to the x -coordinate of the bulk point v . The function θ is the following step function: θ ( x ) = x > / x = 00 for x < . (118)Note that we have functional derivatives with respect to ∂ nz A for n ∈ N ≥ . The propagator (117)corresponds to the functional derivative with n = 0. Let us denote the propagator corresponding to δδ∂ nz A , more generally, as K n , and for n ≥ 0, we modify the condition (116) by imposing:lim v ! x (cid:48) ε ∗ ∂ nz K ( v, x ) = δ ( x (cid:48) − x )d x (cid:48) , x (cid:48) ∈ (cid:96) ∞ ( z ) . (119)This leads us to the following generalization of (117): K n ( v, x ) = z nv δ ( x v − x )d x v . (120)23part from the two propagators, we shall need the coupling constant of the theory to computeWitten diagrams. The coupling constant of this theory can be read off from the interaction term inthe action S CS , it is: i π (cid:126) f ξµν d z . (121)Now we can give a diagrammatic definition of the operators in the algebra A Sc ( T bk ), namely theones defined in (39), and their products: T µ [ n ]( p ) · · · T µ m [ n m ]( p m ) = ∞ (cid:88) l =1 (cid:88) j i ≥ · · ·· · · (cid:37) ν ,j q (cid:37) νl,jl q l j j l p µ , n p m µ m , n m · · ·· · · . (122)Let us clarify some points about the picture. We have replaced the pair of fundamental-anti-fundamentalindices on T with a single adjoint index. The bottom horizontal line represents the boundary line (cid:96) ∞ ( z ),and the top horizontal line represents the Wilson line in representation (cid:37) : gl K ! V at y = 0. The sumis over the number of propagators attached to the Wilson line and all possible derivative couplings.The orders of the derivatives are mentioned in the boxes. The points q ≤ · · · ≤ q l on the Wilson lineare all integrated along the line without changing their order. The gray blob represents a sum over allpossible graphs consistent with the external lines. We use different types of lines to represent differententities: Bulk-to-bulk propagator, P ( v , v ) = v v , Boundary-to-bulk propagator, K ( v, x ) = v x , The boundary line (cid:96) ∞ ( z ) : , Wilson line : . (123)The labels µ i , n i below the points along the boundary line implies that the corresponding boundary-to-bulk propagator is K n i = z n i K and that it carries a gl K -index µ i . Finally, the j th derivative of A ν couples to the Wilson line via the matrix (cid:37) ν,j . Such a diagram with m boundary-to-bulk propagatorsand l bulk-to-bulk propagators attached to the Wilson lines will be evaluated to an element of End( V )which will schematically look like: (Γ m ! l ) µ ··· µ l ν ··· ν m (cid:37) µ ,j · · · (cid:37) µ l ,j l , (124)where (Γ m ! l ) µ ··· µ m ν ··· ν l is a number that will be found by evaluating the Witten diagram. Since thebulk-to-bulk propagator (114) is proportional to (cid:126) and the interaction vertex (121) is proportionalto (cid:126) − , each diagram will come with a factor of (cid:126) that will be related to the Euler character of thegraph. In the following we start computing diagrams starting from O ( (cid:126) ) and up to O ( (cid:126) ), by theend of which we shall have proven the main result (Proposition 2) of this section. Remark m ! l maps, and deformation) . Each m ! l Witten diagram that appears In a Feynman diagram all propagators are proportional to (cid:126) and the power of (cid:126) of a diagram relates simply tothe number of faces of the diagram, which is why (cid:126) is called the loop counting parameter. In a Witten diagram theboundary-to-bulk propagators do not carry any (cid:126) and therefore the power of (cid:126) depends also on the number of boundary-to-bulk propagators. However, we are going to ignore this point and simply refer to the power of (cid:126) in a diagram as theloop order of that diagram. 24n sums such as (122) can be interpreted as a map whose image is the value of the diagram:Γ m ! l : m (cid:79) i =1 z n i gl K ! l (cid:79) i =1 z j i gl K ! End( V ) , Γ m ! l : m (cid:79) i =1 z n i t µ i (Γ m ! l ) µ ··· µ l ν ··· ν m (cid:37) µ ,j · · · (cid:37) µ l ,j l . (125)As we shall see explicitly in our computations, diagrams in (122) without loops (diagrams of O ( (cid:126) ))define an associative product that leads to classical algebras such as U( gl K [ z ]). However, there aregenerally more diagrams in (122) involving loops (diagrams of O ( (cid:126) ) and higher order) that change theclassical product to something else. Since loops in Witten or Feynman diagrams are the essence ofthe quantum interactions, classical algebras deformed by such loop diagrams are aptly called quantumgroups (of course, why they are called groups is a different story entirely [45].) (cid:52) 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 of gauge anomaly ofthe Wilson line [19] in this theory. This of course is not a coincidence. To see this, let us consider thevariation of the expectation value of the Wilson line, (cid:104) W (cid:37) ( L ) (cid:105) A , as we vary the connection A alongthe boundary line (cid:96) ∞ ( z ): δ (cid:104) W (cid:37) ( L ) (cid:105) A = ∞ (cid:88) n =0 (cid:90) p ∈ (cid:96) ∞ ( z ) δδ∂ nz A µ ( p ) (cid:104) W (cid:37) ( L ) (cid:105) A δ∂ nz A µ ( p ) . (126)Let us make the following variation: δ∂ z A µ ( x ) = δ ( x − p ) η µ = d x θ ( x − p ) η µ , (127)for some fixed Lie algebra element η µ t µ ∈ gl K . Then we find: δ (cid:104) W (cid:37) ( L ) (cid:105) A = δδ∂ z A µ ( p ) (cid:104) W (cid:37) ( L ) (cid:105) A η µ . (128)An exact variation of the boundary value of the connection is like a gauge transformation that doesnot vanish at the boundary. In [19] it was proved that such a variation of the connection leads to avariation of the Wilson line which is a local functional supported on the line: δ (cid:104) W (cid:37) ( L ) (cid:105) A = ([ (cid:37) µ, , (cid:37) ν, ] + Θ µ, ,ν, ) (cid:90) L ∂ z A µ ∂ z c ν , (129)where c was the generator of the gauge transformation: ∂ z d c µ = δ∂ z A µ , (130) (cid:37) µ, ∈ End( V ) is part of the representation of gl K [ z ] that couples ∂ z A µ to the Wilson line (see (103)),and Θ µ, ,ν, , which is anti-symmetric in µ and ν , is a matrix that acts on V . Variations such as theabove measure gauge anomaly associated to the line, though in our case it is not an anomaly since weare varying the connection at the boundary, and such “large gauge” transformations are not actuallypart of the gauge symmetry of the theory. The matrix Θ µ, ,ν, which signals the presence of anomalyis not an arbitrary matrix and in [19], all constraints on this matrix were worked out, we shall not needthem at the moment. Comparing with (127) we see that for us ∂ z c µ ( x ) = θ ( x − p ) η µ , which leads to: δ (cid:104) W (cid:37) ( L ) (cid:105) A = (cid:0) f ξµν (cid:37) ξ, + Θ µ, ,ν, (cid:1) (cid:90) x>p ∂ z A µ η ν , (131)25here we have used the fact that the matrices (cid:37) µ, satisfy the loop algebra (104). The integral above isalong L . The connection A above is a background connection satisfying the equation of motion, i.e., itis 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 dictates in particular that the connectionmust also be translation invariant along the direction of the Wilson line L . By considering the integralof A along the following rectangle: d A = 0 y = 0 y = ∞ x = ∞ x = p (cid:96) ∞ ( z ) L (132)and using translation invariance in the x -direction along with Stoke’s theorem, we can change thesupport of the integral in (131) from L to (cid:96) ∞ ( z ), to get: δ (cid:104) W (cid:37) ( L ) (cid:105) A = (cid:0) f ξµν (cid:37) ξ, + Θ µ, ,ν, (cid:1) (cid:90) (cid:96) ∞ ( z ) (cid:51) x>p ∂ z A µ η ν . (133)Comparing with (128) we find: δδ∂ z A ν ( p ) (cid:104) W (cid:37) ( L ) (cid:105) A = (cid:0) f ξµν (cid:37) ξ, + Θ µ, ,ν, (cid:1) (cid:90) x>p ∂ z A µ , (134)where the integral is now along the boundary line (cid:96) ∞ ( z ). This leads to the following relation betweenour algebra and anomaly:[ T µ [1] , T ν [1]]= lim ι ! (cid:20) δδ∂ z A µ ( p + ι ) δδ∂ z A ν ( p ) − δδ∂ z A ν ( p ) δδ∂ z A µ ( p + ι ) (cid:21) (cid:104) W (cid:37) ( L ) (cid:105) A = f ξµν (cid:37) ξ, + Θ µ, ,ν, . (135)The first term with the structure constant gives us the loop algebra gl K [ z ], which is the classicalresult. The anomaly term is the result of 2-loop dynamics [19], i.e., it is proportional to (cid:126) . Thisterm gives the quantum deformation of the classical loop algebra. This also explains why our two loopcomputation 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 Θ µ, ,ν, isand we would find that the deformed algebra of the operators T µ [ n ] is indeed the Yangian Y (cid:126) ( gl K ).However, we think it is illustrative to derive this result from a direct computation of Witten diagrams. O ( (cid:126) ) We denote a diagram by Γ dn ! m when there are n boundary-to-bulk propagators, m propagators at-tached to the Wilson line, and the diagram is of order (cid:126) d . If there are more than one such diagramswe denote them as Γ dn ! m,i with i = 1 , · · · .Our aim in this section is to compute the product T µ [ m ]( p ) T ν [ n ]( p ) and eventually the commu-tator [ T µ [ m ] , T ν [ n ]] := lim p ! p ( T µ [ m ]( p ) T ν [ n ]( p ) − T ν [ n ]( p ) T µ [ m ]( p )) , (136)26t 0-loop. We have the following two 2 ! ! , ( p µ,m ; p ν,n ) = p µ,m p ν,n q q m n , Γ ! , ( p µ,m ; p ν,n ) = p µ,m p ν,n q q n m , (137)where a label m in a box on the Wilson line refers to the coupling between the Wilson line and the m th derivative of the connection. The first diagram evaluates to (note that p < p and q < q ):Γ ! , ( p µ,m ; p ν,n ) = (cid:90) q The large N limit of the algebra A Sc ( T bk ) at the classical level is the universal envelopingalgebra U ( gl K [ z ]) : A Sc ( T bk ) / (cid:126) N ! ∞ ∼ = U ( gl K [ z ]) ∼ = Y (cid:126) ( gl K ) / (cid:126) . (145)The reason why we need to take the large N limit is that, the operators T µ [ m ] acts on a vectorspace which is finite dimensional for finite N . This leads to some extra relations in the algebra, whichwe can get rid of in the large N limit. A similar argument was presented for the operator algebracoming from the BF theory in § § -loop, O ( (cid:126) )Now we want to compute 1-loop deformation to both the Lie algebra structure and the coproductstructure of A Sc ( T bk ). Lie bracket. The 2 ! , , , . (146)All of these vanish due to Lemma 6 of § C.1. Sometimes we ignore to specify the derivative couplings at the Wilson line, when the diagrams we draw are vanishingregardless. ! . (147)Note that, since the bulk points are being integrated over, crossing the boundary-to-bulk propagatorsdoes not produce any new diagram, it just exchanges the two diagrams that we have drawn: crossing −−−−−! = . (148)For this reason, in future we shall only draw diagrams up to crossing of the boundary-to-bulk propa-gators that are connected to bulk interaction vertices.Now let us comment on the evaluation of the diagrams in (147). We start by doing integration byparts with respect the differential corresponding to either one of the two boundary-to-bulk propagators.As mentioned in § C.2, this gives two kinds of contributions, one coming from collapsing a bulk-to-bulkpropagator, the other coming from boundary terms. Collapsing any of the bulk-to-bulk propagatorsleads to a configuration which will vanish due to Lemma 7 ( § C.1). Therefore, doing integration byparts will only result in a boundary term. Recall from the general discussion in § C.2 that only theboundary component of the integrals along the Wilson line can possibly contribute. Since there aretwo points on the Wilson line, let us call them p and p , the domain of integration is:∆ = { ( p , p ) ∈ R | p < p } . (149)The boundary of this domain is: ∂ ∆ = { ( p , p ) ∈ R | p = p } . (150)Once restricted to this boundary, both of the diagrams in (147) will involve a configuration such asthe following: , (151)which vanishes due to Lemma 6. The diagrams (147) thus vanish.There are four other 2 ! , (152)and then1. Permuting the two points on the Wilson line. These diagrams actually require a UV regularization due to logarithmic divergence coming from the two pointson the Wilson line being coincident. To regularize, the domain of integration needs to be restricted from ∆ to (cid:101) ∆ := { ( p , p ) ∈ R | p ≤ p − (cid:15) } for some small positive number (cid:15) , which leads to the modified boundary equation p = p − (cid:15) ,however, this does not affect the arguments presented in the proof of Lemma 6 (essentially because (cid:15) is a constant andd (cid:15) = 0, resulting in no new forms other than the ones considered in the proof), and therefore we are not going to describethe regularization of these diagrams in detail. 29. Permuting the two points on the boundary.3. Simultaneously permuting the two points on the Wilson line and the two points on the boundary.All of these diagrams vanish due to Lemma 6.There are also six 2 ! , (153)by permuting the points along the Wilson line and the boundary. However, due to Lemma 7, all ofthese diagrams vanish.There are no more 2 ! m diagrams at 1-loop. Thus, we conclude that there is no 1-loop contribu-tion to the Lie bracket in A Sc ( T bk ). Coproduct. We use the same superscript notation we used in § T µ [ m ] on different vector spaces. The 1-loop diagram that deforms the classical coproductis the following: Γ ! (cid:0) pµ, (cid:1) = pµ, UV (154)Happily for us, precisely this diagram was computed in eq. 5.6 of [19] to answer the question “howdoes a background connection couple to the product Wilson line?”. The result of that paper involvedan arbitrary background connection where we have our boundary-to-bulk propagator, so we just needto replace that with K ( v, p ) = z v δ ( x v − p ) and we find:Γ ! (cid:0) pµ, (cid:1) = − (cid:126) f νξµ T Uν [0] ⊗ T Vξ [0] . (155)This deforms the classical coproduct (144) as follows: T U ⊗ Vµ [1] = T Uµ [1] ⊗ id V + id U ⊗ T Vµ [1] − (cid:126) f νξµ T Uν [0] ⊗ T Vξ [0] . (156)The exact same computation with K instead of K shows that Γ ! (cid:0) pµ, (cid:1) = 0, i.e., the classical algebraof the 0th level operators remain entirely undeformed at this loop order. Thus we see that at 1-loop, the Lie algebra structure in A Sc ( T bk ) remains undeformed, but there isa non-trivial deformation of the coalgebra structure. At this point, there is a mathematical shortcutto proving that the algebra A Sc ( T bk ), including all loop corrections, is the Yangian. The proof relieson a uniqueness theorem (Theorem 12.1.1 of [45]) concerning the deformation of U( gl K [ z ]). Being ableto use the theorem requires satisfying some technical conditions, we discuss this proof in AppendixB. This proof is independent of the rest of the paper, where we compute two loop corrections to thecommutator (140) which will directly show that the algebra is the Yangian. Note that the 0th level operators form a closed algebra which is nothing but the Lie algebra gl K . Reductive Liealgebras belong to discrete isomorphism classes and therefore they are robust against continuous deformations. So thealgebra of T µ [0] will in fact remain undeformed at all loop orders. We will not make more than a few remarks aboutthem in the future. .3.2 -loops, O ( (cid:126) )The number of 2-loop diagrams is too large to list them all, and most of them are zero. Instead ofdrawing all these diagrams let us mention how we can quickly identify a large portion of the diagramsthat end up being zero.Consider the following transformations that can be performed on a propagator or a vertex in anydiagram: ! , ! , ! , ! , ! . (157)All these transformations increase the order of (cid:126) by one, however, all the diagrams constructed usingthese modifications are zero due to Lemma 6. We will therefore ignore such diagrams. Let us nowidentify potentially non-zero 2 ! m diagrams at 2-loops.All 2-loop 2 ! ! ! , = , Γ ! , = , Γ ! , = , Γ ! , = . (158)Let us first consider the first two diagrams Γ ! , and Γ ! , . Collapsing any of the bulk-to-bulkpropagators will result in a configuration where either Lemma 6 or 7 is applicable. Therefore, when wedo integration by parts with respect to the differential in one of the two boundary-to-bulk propagatorswe only get a boundary term. The boundary corresponds to the boundary of ∆ (defined in (149)),and when restricted to this boundary, the integrand vanishes due to Lemma 7, in the same way as forthe diagrams in (147). The diagrams Γ ! , and Γ ! , are symmetric under the exchange of the color labels associatedto the boundary-to-bulk propagators, for a proof see the discussion following (239). So these diagramsdon’t contribute to the anti-symmetric commutator we are computing.Now we come to the most involved part of our computations, 2 ! These diagrams are linearly divergent when the two points on the Wilson line are coincident and they require similarUV regularization as their 1-loop counterparts. ! , = , Γ ! , = , Γ ! , = , Γ ! , = , Γ ! , = , Γ ! , = . (159)All of these diagrams are in fact non-zero. We proceed with the evaluation of the diagram Γ ! , :Γ ! , (cid:0) p µ, ; p ν, (cid:1) = p µ,m p ν,n v v v q q q (160)The gl K factor of the diagram is easily evaluated to be: f ξoµ f πρξ f σνπ (cid:37) ( t o ) (cid:37) ( t ρ ) (cid:37) ( t σ ) . (161)The numerical factor takes a bit more care. Each of the bulk points v i = ( x i , y i , z i , z i ) is integratedover M = R × C and the points q i on the Wilson line take value in the simplex ∆ = { ( q , q , q ) ∈ R | q < q < q } . For the sake of integration we can partially compactify the bulk to M = R × S .So the domain of integration for this diagram is: M × ∆ . (162)However, this domain needs regularization due to UV divergences coming from the points q i all comingtogether. As in [19], we use a point splitting regulator, by restricting integration to the domain: (cid:101) ∆ := { ( q , q , q ) ∈ ∆ | q < q − (cid:15) } , (163)for some small positive number (cid:15) . We are not going to discuss the regulator here, as it would beidentical to the discussion in [19]. We shall now do integration by parts with respect to the differentialin the propagator connecting p and v . Note that collapsing any of the bulk-to-bulk propagators leadsto a configuration where the vanishing Lemma 7 applies. Therefore, contribution to the integral onlycomes from the boundary M × ∂ (cid:101) ∆ . The boundary of the simplex has three components, respectivelydefined by the constraints q = q , q = q , and q = q − (cid:15) . However, when q = q or q = q , wecan use the vanishing Lemma 6 and the integral vanishes. Therefore the contribution to the diagramcomes only from integration over: M × { ( q , q , q ) ∈ (cid:101) ∆ | q = q − (cid:15) } . (164)Further simplification can be made using the fact that the propagator connecting p and v is zδ ( x − p ). This restricts the integration over v to { p } × S . However, using translation symmetry in the x -direction we can fix the position of q at (0 , , , 0) and allow the integration of v over all of M .However, due to the presence of the delta function δ ( x − p ) in the boundary-to-bulk propagator, x p = p − δ are rigidly tied to each other. This way, we end up with the following integration forthe numerical factor: (cid:18) i π (cid:126) (cid:19) (cid:90) To get to a standard defining bracket for the Yangian, we change basis as follows. There is an ambiguityin T ξ [2]. In (140) it was equal to (cid:37) ξ, at the classical level, but it can be shifted at 2-loops (i.e., by aterm proportional to (cid:126) ) by the image ϑ ( t ξ ) for an arbitrary gl K -equivariant map ϑ : gl K ! End( V ).This shift simply corresponds to a different choice for the counterterm that couples ∂ z A ξ to the Wilsonline. Using this freedom we want to replace products such as (cid:37) ( t o ) (cid:37) ( t ρ ) (cid:37) ( t σ ) with the totally symmetricproduct { (cid:37) ( t o ) , (cid:37) ( t ρ ) , (cid:37) ( t σ ) } (defined in (94)). To this end, Consider the difference:∆ µν := 2 (cid:126) f ξo [ µ f πρξ f σν ] π ( (cid:37) ( t o ) (cid:37) ( t ρ ) (cid:37) ( t σ ) − { (cid:37) ( t o ) , (cid:37) ( t ρ ) , (cid:37) ( t σ ) } ) . (174)The square brackets around µ and ν in the above equation implies anti-symmetrization with respectto µ and ν . The difference ∆ µν can be viewed as the image of the following gl K -equivariant map:∆ : ∧ gl K ! End( V ) , ∆ : t µ ∧ t ν ∆ µν . (175)We now propose the following lemma: Lemma 2. The map ∆ factors through gl K , i.e., ∆ : ∧ gl K ! gl K ! 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 image of an element of gl K which, according to our previous argument, can be absorbed into a redefinition of (cid:37) ξ, (equivalently T ξ [2]). Therefore, with a new T new ξ [2] we can rewrite (171) as:[ T µ [1] , T ν , [1]] = f ξµν T new ξ [2] + (cid:126) ( I + I + I ) Q µν , (176) 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 Y (cid:126) ( gl K ). By integrating out the fermions from the action (23) we get the holonomy of the connection ( A , A ) ∈ gl N ⊕ gl K inthe following representation [34]: (cid:77) Y Y TN ⊗ Y K , (172)where the sum is over all possible Young tableaux. Y T is the tableau we get by transposing the tableau Y (i.e., turningrows into columns). Y TN is the representation of GL N denoted by the tableau Y T , and Y K is the representation ofGL K dual to the representation (of GL K ) 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]: (cid:77) Y Y N ⊗ Y K . (173)An important difference between (172) and (173) is that while the former is finite dimensional for finite N and K , thelatter is always infinite dimensional. In the case of bosonic quantum mechanics the representation (cid:37) is actually infinite dimensional, however, for finite N our Witten diagram computations can not be trusted, as the decoupling between closed string modes and defect (4dChern-Simons) modes that we have assumed relies on the large N limit [35]. Q µν := 2 f ξo [ µ f πρξ f σν ] π { T o [0] , T ρ [0] , T σ [0] } . (177)The reason why we have postponed presenting the evaluations of the individual integrals I , I , and I is that we don’t need their individual values, only the sum, and precisely this sum was evaluated ineq. (E.23) of [19] with the result: I + I + I = 112 . (178)We can therefore write (ignoring the “new” label on T ξ [2]):[ T µ [1] , T ν [1]] = f ξµν T ξ [2] + (cid:126) Q µν . (179)This is the relation for the Yangian that was presented in § A Sc ( T bk ), defined in (41), at 2-loops, is the Yangian Y (cid:126) ( gl K ): A Sc ( T bk ) / (cid:126) N ! ∞ ∼ = Y (cid:126) ( gl K ) / (cid:126) . (180)The two loop result in the BF theory was exact. The above two loop result is exact as well. Thoughwe do not prove this by computing Witten diagrams, we can argue using the form of the algebra interms of anomaly (135). In [19] it was shown that there are no anomalies beyond two loops. Thisconcludes our second proof of Proposition 2. The topological theories we have considered so far can be constructed from a certain brane setup intype IIB string theory and then applying a twist and an Ω-deformation. This brane constructionwill show that the algebras we have constructed are in fact certain supersymmetric subsectors ofthe well studied N = 4 SYM theory with defect and its holographic dual. We note that the ideaof embedding Chern-Simons type theories inside 1 and 2 higher dimensional supersymmetric gaugetheories as (quasi)-topological subsectors can be traced back to [46]. A Caveat. The most transparent way of probing (quasi) topological subsectors of the relevantphysical (defect) AdS/CFT correspondence would be to apply twist and deformation to 4d N = 4SYM with a domain wall on one hand and to AdS × S supergravity with D5-brane probes on theother hand. Localization in the AdS background is yet to be developed and this is not what we doin this section. We apply twist and deform the gauge theories that appear in a certain D3-D5 braneconfiguration in flat background and argue that we end up with the topological brane setup of section3.1. Thus in making the claim that our topological holography embeds into physical holography weare relying on the assumption that the process of twist and deformation commutes with taking thedecoupling limit.We describe our construction below. The first one, which is significantly more abstract, being in Appendix B. .1 Brane Configuration Our starting brane configuration involves a stack of N D3 branes and K D5 branes in type IIB stringtheory on a 10d target space of the form R × C where C is a complex curve which we take to be justthe complex plane C . The D5 branes wrap R × C and the D3 branes wrap an R which has a 3dintersection with the D5 branes. Let us express the brane configuraiton by the following table:0 1 2 3 4 5 6 7 8 9 R C R D × × × × × × D × × × × (181)The world-volume theory on the D5 branes is the 6d N = (1 , 1) SYM theory coupled to a 3d defectpreserving half of the supersymmetry. Similarly, the world-volume theory on the D3 branes is the 4dSYM theory coupled to a 3d defect preserving half of the supersymmetry. To this setup we apply aparticular twist, i.e., we choose a nilpotent supercharge and consider its cohomology. We use Γ i with i ∈ { , · · · , } for 10d Euclidean gamma matrices. We also use the notation:Γ i ··· i n := Γ i · · · Γ i n . (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 (cid:15) L Q l + (cid:15) R Q r where (cid:15) L and (cid:15) R are chiral spinors parameterizing the supercharge. The chirality constraints on themare: i Γ ··· (cid:15) L = (cid:15) L , i Γ ··· (cid:15) R = (cid:15) R . (183)We shall discuss constraints on the supercharge by describing them as constraints on the parameterizingspinors.The supercharges preserved by the D5 branes are constrained by: (cid:15) R = i Γ (cid:15) L . (184)This reduces the number of supercharges to 16. The D3 branes imposes the further constraint: (cid:15) R = i Γ (cid:15) L . (185)This reduces the number of supercharges by half once more. Therefore the defect preserves just 8supercharges. Since (cid:15) R is completely determined given (cid:15) L , in what follows we refer to our choice ofsupercharge simply by referring to (cid:15) L .We want to perform a twist that makes the D5 world-volume theory topological along R andholomorphic along C . This twist was described in [37]. Let us give names to the two factors of R inthe 10d space-time: M := R , M (cid:48) := R . (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 l and r respectively. There are two of each chirality because the R-symmetry is Sp(1) × Sp(1) =Spin(4) M (cid:48) such that the two left handed spinors transform as a doublet of one Sp(1) and the tworight handed spinors transform as a doublet of the other Sp(1). The subgroup of Spin(6) preservingthe product structure R × C is Spin(4) M × U(1). Under this subgroup l and r transform as( , ) − ⊕ ( , ) +1 and ( , ) +1 ⊕ ( , ) − respectively, where the subscripts denote the U(1) charges.36otations along M (cid:48) act as R-symmetry on the spinors – the spinors transform as representations ofSpin(4) M (cid:48) such that + transforms as ( , ) and − transforms as ( , ). In total, under the symmetrygroup Spin(4) M × U(1) × Spin(4) M (cid:48) the 16 supercharges of the 6d theory transform as:(( , ) − ⊕ ( , ) +1 ) ⊗ ( , ) ⊕ (( , ) +1 ⊕ ( , ) − ) ⊗ ( , ) . (187)The twist we seek is performed by redefining the the space-time isometry:Spin(4) M (cid:32) Spin(4) new M ⊆ Spin(4) M × Spin(4) M (cid:48) , (188)where the subgroup Spin(4) new M of Spin(4) M × Spin(4) M (cid:48) consists of elements ( x, θ ( x )) which is definedby the isomorphism θ : Spin(4) M ∼ −! Spin(4) M (cid:48) . More, explicitly, the isomorphism acts as: θ (Γ µν ) = Γ µ +6 ,ν +6 , µ, ν ∈ { , , , } . (189)The generators of the new Spin(4) new M are then:Γ µν + Γ µ +6 ,ν +6 . (190)After this redefinition, the symmetry Spin(4) M × U(1) × Spin(4) M (cid:48) of the 6d theory reduces toSpin(4) new M × U(1) and under this group the representation (187) of the supercharges becomes:2( , ) − ⊕ ( , ) − ⊕ ( , ) − ⊕ , ) +1 . (191)We thus have two supercharges that are scalars along M , both of them have charge − C . We take the generator of this rotation to be − i Γ , then if (cid:15) is one of the scalar (on M ) supercharges that means: i Γ (cid:15) = (cid:15) . (192)We identify the supercharge (cid:15) by imposing invariance under the new rotation generators on M , namely(190): (Γ µν + Γ µ +6 ,ν +6 ) (cid:15) = 0 . (193)The constraints (184) and (185) put by the D-branes and the U(1)-charge on C (192) together areequivalent to the following four independent constraints: i Γ µ,µ +6 (cid:15) = (cid:15) , µ { , , , } . (194)Together with the chirality constraint (183) in 10d we therefore have 5 equations, each reducing thenumber degrees of freedom by half. Since a Dirac spinor in 10d has 32 degrees of freedom, we are leftwith 32 × − = 1 degree of freedom, i.e., we have a unique supercharge, which we call Q . It wasshown in [37] that the supercharge Q is nilpotent: Q = 0 , (195)and the 6d theory twisted by this Q is topological along M – which is simply a consequence of (193) –and it is holomorphic along C . The latter claim follows from the fact that there is another superchargein the 2d space of scalar (on M ) supercharges in the 6d theory, let’s call it Q (cid:48) , which has the followingcommutator with Q : { Q, Q (cid:48) } = ∂ z , (196)where z = ( x − ix ) is the holomorphic coordinate on C . This shows that z -dependence is trivial( Q -exact) in the Q -cohomology. 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 . .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 canfigure 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 (threeof which are independent) equations:(Γ + Γ ) (cid:15) = 0 , (Γ + Γ ) (cid:15) = 0 , (Γ + Γ ) (cid:15) = 0 , (Γ + Γ ) (cid:15) = 0 , (Γ + Γ ) (cid:15) = 0 , (Γ + Γ ) (cid:15) = 0 . (197)These are in fact the equations that defines a scalar supercharge in the KW twist of N = 4 theoryon R for a particular homomorphism from space-time ismoetry to the R-symmetry. Space-timeisometry of the theory on R acts on the spinors as Spin(4) iso , generated by the six generators:Γ µν , µ, ν ∈ { , , , } and µ (cid:54) = ν . (198)Rotations along the transverse directions act as R-symmetry, which is Spin(6), though the subgroupof the R-symmetry preserving the product structure C × R is U(1) × Spin(4) R . The KW twistis 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 particularisomorphism that leads to the equations (197) is:Γ Γ , Γ Γ , Γ Γ , Γ Γ , Γ Γ , Γ Γ . (199) Remark CP family of twists) . In [47] it was shown that there is a family of KWtwists parameterized by CP . The unique twist (by the supercharge Q ) we have found is a specificmember of this family. Let us identify which member that is.The CP family comes from the fact that there is a 2d space of scalar (on M ) supercharges (in(191)) in the twisted theory. Also note from the original representation of the spinors (187) thatthe two scalar supercharges come from spinors transforming as ( , ) and ( , ) under the originalisometry Spin(4) old . Let us choose two Spin(4) new scalar spinors with opposite Spin(4) old chiralitiesand call them (cid:15) l and (cid:15) r . The Spin(4) old chirality operator is Γ old := Γ . Let us choose (cid:15) l and (cid:15) r insuch a way that they are related by the following equation: (cid:15) r = N (cid:15) l where N = 14 (Γ + Γ + Γ + Γ ) . (200)This relation is consistent with the spinors being Spin(4) new invariant because N anti-commutes withSpin(4) new (thus invariant spinors are still invariant after being operated with N ), as well as with Γ old (changing Spin(4) old chirality). An arbitrary scalar supercharge in the twisted theory is a complexlinear combination of (cid:15) l and (cid:15) r , such as α(cid:15) l + β(cid:15) r , however, since the overall normalization of the spinoris irrelevant, the true parameter identifying a spinor is the ratio t := β/α ∈ CP . Furthermore, due tothe equations (197), N acts as − new scalar, leading to: (cid:15) l = − N (cid:15) 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 (cid:15) := (cid:15) l + t(cid:15) r with t ∈ CP .Then using (201) and (194) we get: − i(cid:15) = N (cid:15) = (cid:15) r − t(cid:15) l , (202) Note that we are using subscripts simply to refer to particular directions. 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 theoryon M × C and just consider a theory on M with rotations on C being part of the R-symmetry then, first of all, we finda N = 4 SYM theory on M and the twist we described is precisely the KW twist. We are writing Spin(4) old instead of Spin(4) M since the support of the 4d theory is not M ≡ R but R . t = i . (203) (cid:52) Finally, at the 3 dimensional D3-D5 intersection lives a 3d N = 4 theory consisting of bifundamentalhypermultiplets coupled to background gauge fields which are restrictions of the gauge fields from theD3 and the D5 branes [48]. Considering Q -cohomology for the 3d theory reduces it to a topologicaltheory 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-model [47] and this can be coupled to a 3d analogue of the2d B-model – a B-type topological twist of 3d N = 4 is called 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 thebackground connections.We can reach the same conclusion by analyzing the constraints on the twisting supercharge viewedfrom the 3d point of view. The bosonic symmetry of the 3d theory includes SU(2) iso × SU(2) H × SU(2) C where SU(2) iso is the isometry of the space-time R , SU(2) C are rotations in R , and SU(2) H arerotations in R . The hypers in the 3d theory come from strings with one end attached to the D5branes and another end attached to the D3 branes. Rotations in R – 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. Thisleaves the other R-symmetry group SU(2) C which would act on the coulomb branch of the theory ifthe theory had some dynamical 3d vector multiplets. We now note that the topological twist, fromthe 3d perspective, involves twisting the isometry SU(2) iso with the R-symmetry group SU(2) C , asevidenced explicitly by the three equations in the first line of (197). This particular topological twist(as opposed to the topological twist using the other R-symmetry SU(2) H ) of 3d N = 4 is indeed theRW 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 R with gauge group U( N ) and a topological-holomorphic twist of N = (1 , 1) theory on R × C with gauge group U( K ), and these two theoriesare coupled via a 3d RW theory of bifundamental hypers with flavor symmetry U( N ) × U( K ) gaugedby background connections. Note that we have not described the effect of the twist on the closedstring theory. This is because we are assuming a decoupling between the closed string modes andthe D5-defect modes in the large N limit (referred to as rigid holography in [35]) and therefore, theoperator algebra that we will concern ourselves with will be insensitive to the closed string modes. We will ignore the closed string modes moving forward as well. We start by noting that the dimensional reduction of the topological-holomorphic 6d theory from R × C to R reduces it to the KW twist of N = 4 SYM on the R . This observation allowsus to readily tailor the results obtained in [37] about Ω-deformation of the 6d theory to the case ofΩ-deformation of 4d KW theory. In particular, the 4d Theory on R × T can be compactified on the two-torus T to get a B-model on R . We want to be able to take the 3d theory on R × S and compactify it on S to get a B-model on R . If we havea 4d theory on R × T coupled to a 3d theory on R × S , compactifying on T should not make the two systemsincompatible. Though it is customary to decouple the central U(1) subgroup from the gauge groups as it doesn’t interact with thenon-abelian part, our computations look somewhat simpler if we keep the U(1). This is the same argument we used in § 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. N = 1 SYM theory is the connection A I where I ∈{ , · · · , } . When dimensionally reduced to 6d, this becomes a 6d connection A M with M ∈ { , · · · , } and four scalar fields φ , φ , φ , and φ which are just the remaining four components of the 10dconnection. The Spin(4) M space-time isometry acts on the first four components of the connection,namely A , A , A , and A via the vector representation. The four scalars – φ , φ , φ , and φ –transform under the vector representation of the R-symmetry Spin(4) M (cid:48) . Once twisted according to(188), only the diagonal subgroup Spin(4) new M of Spin(4) M × Spin(4) M (cid:48) acts on the fields, under whichthe first four components of the connection and the four scalars transform in the same way – apart fromthe inhomogeneous transformation of the connection – and therefore we can package them togetherinto one complex valued gauge field: A µ := A µ + iφ µ , µ ∈ { , , , } . (204)We also write the remaining components of the connection in complex coordinates on C : A z := A + iA and A z := A − iA . (205)It was shown in [37] that this topological-holomorphic 6d theory can be viewed as a 2d gaugedB-model on R where the fields are valued in maps Map( R × C , gl K ). This is a vector space whichplays the role of the Lie algebra of the 2d gauge theory. From the 2d point of view A and A are part of a connection on R and there are four chiral multiplets with the bottom components A , A , A z , and A z . The 2d theory consists of a superpotential which is a holomorphic functionof these chiral multiplets – the superpotential can be written conveniently in terms of a one form (cid:101) A := A d x + A d x + A z d z + A z d z on R × C consisting of these chiral fields: W ( A , A , A z , A z ) = (cid:90) R × C d z ∧ tr (cid:18) (cid:101) A ∧ d (cid:101) A + 23 (cid:101) A ∧ (cid:101) A ∧ (cid:101) A (cid:19) . (206)The superpotential is the action functional of a 4d CS theory on R × C for the connection (cid:101) A .One of the results of [37] is the following: Ω-deformation applied to this topological-holomorphic6d theory with respect to rotation on R reduces the theory to a 4d CS theory on R × C with complexified gauge group GL K .The twisted 4d theory (the D3 world-volume theory) wraps the plane R as well and thereforeis affected by the Ω-deformation. By noting that the 4d theory is a dimensional redcution of the 6dtheory from R × C to R and assuming that Ω-deformation commutes with dimensional reduction, we can deduce what the Ω-deformed version of the twisted 4d theory is. This will be a 2d gauge theorywith complexified gauge group GL N and the action will be the dimensional reduction of the 4d CSaction (206) from R × C to R – this is the 2d BF theory where A z plays the role of the B field: (cid:90) R × C d z ∧ CS( A R × C ) Reduce on C −−−−−−−−! (cid:90) R tr A z (cid:18) d A R + 12 A R ∧ A R (cid:19) = (cid:90) R tr A z F ( A R ) , (207)where, 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 R × R with only hypersreduces, upon Ω-deformation with respect to rotation in the plane R , to a free quantum mechanics on Up to some overall numerical factors. Introduced for the first time in [51] in the context of 4d N = 2 gauge theories on R × R . The relevant space-timerotation in that case was a U (1) × U (1) action rotating the two planes – which ultimately localized the 4d theory to a0d matrix model. Analogously, Ω-deformation with respect to rotation on a plane localizes our 6d/4d/3d theory to a4d/2d/1d theory. Alternatively, one can redo the localization computations of [37] for the 4d case, confirming that Ω-deformation doesindeed commute with dimensional reduction. . A slight modification of this result, involving background connections gauging the flavor symmetryof the hypers leads to the result that the omega deformed theory is a gauged quantum mechanics, thekind of theory we have considered on the defect in the 2d BF theory. Via supersymmetric twists and Ω-deformation, we have made contact with precisely the setup we haveconsidered in this paper. We have a 4d CS theory with gauge group GL K and a 2d BF theory withgauge group GL N and they intersect along a topological line supporting a gauged quantum mechanicswith GL K × GL N symmetry. We thus claim that the topological holographic duality that we haveestablished in this paper is indeed a topological subsector of the standard holographic duality involvingdefect N = 4 SYM. In the previous sections we have been able to exactly (at all loops) match a subsector of the oper-ator algebra in the 2d BF theory with a line defect, with a subsector of the scattering algebra ina 3d closed string theory with a surface defect. The subsectors of operators we focused on are re-stricted to the defects on both sides of the duality. This matching provides a non-trivial check of theproposed holographic duality. Furthermore, we have shown that this holographic duality between topo-logical/holomorphic theories is in fact a supersymmetric subsector of the more familiar AdS /CFT duality. From the considerations of this paper several immediate questions and new directions arisethat we have not yet addressed. Let us comment on a few such issues that we think are interestingtopics to pursue for future research. Central extensions on two sides of the duality: To ease computation we restricted our attentionto the quotients of the full operator algebra and scattering algebra by their centers. The inclusion ofthe central operators will change the associative structure of the algebras. A stronger statement ofduality will be to compare the centrally extended Yangians coming from the boundary and the bulktheory. Brane probes: Using branes in the bulk to probe local operators in the boundary theory has been auseful tool [53, 54]. In our setup, a brane must be Lagrangian in the A-twisted R directions. Lookingat the brane setup (8) (which we reproduce in (209) for convenience) we see that the real directions ofthe D2 and D4 branes are Lagrangian with respect to the following symplectic form:d v ∧ d x + d w ∧ d y . (208)This leaves the possibility of two more different embeddings for D2-branes: R v R w R x R y C z D2 0 × × × × × D2 (cid:48) × × z D2 (cid:48)(cid:48) × × z (209)The D2 (cid:48) -branes are Wilson lines in the CS theory on the D4-branes perpendicular to the originalWilson line at thte D2-D4 intersection. Such crossing Wilson lines were studied in [19, 44] with theresult that this corssing (of two Wilson lines carrying representations U and V of gl K respectively)inserts an operator T V U ( z ) : U ⊗ V ! V ⊗ U in the CS theory which solves the Yang-Baxter equation, The bosonic version, which leads to the same Yangian with minor modifications to the computations as remarkedin 3, 4, and 5. V V VU U U WWWT UV ( z ) T W V ( z ) T W U ( z ) z z z = V V VW W W UUUT W V ( z ) T UV ( z ) T W U ( z ) z z z , (210)where z , z , and z are the spectral parameters (location in the complex plane) of the lines carryingrepresentations V , U , and W respectively, and z := z − z and so on. Solutions of the above equationare closely tied to Quantum Groups. The operators T UV ( z ), which are commonly referred to as R -matrices, can be explicitly constructed using Feynman diagrams [19]. When the complex directionsof 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 representation of gl K , then by providing an incomingand an outgoing fundamental state, we can view (cid:104) j | T K V ( z ) | i (cid:105) as a map T ij ( z ) : V ! V which has anexpansion is z − : T ij ( z ) = id V δ ij − (cid:126) (cid:88) n ≥ (cid:0) − z − (cid:1) n +1 T ij [ n ] , (211)where the T ij [ n ] are precisely the operators that generate the scattering algebra A Sc ( T bk ) (see (39)and (41)). This suggests that in the dual picture we should be able to interpret the D2 (cid:48) branes as agenerating function for the operators O ij [ n ].The interpretations of the D2 (cid:48)(cid:48) branes are missing on both sides of the duality. Finite N duality: We considered the large N limit to decouple the closed string modes from thedefect (CS) mode in the bulk side of the duality and to eliminate any relations among our operatorsthat would arise from having finite dimensional matrices (see § § N , when they can be quotients of the Yangianby some extra relations. Duality for other quantum groups: In [19, 44] it was shown that by replacing our complexdirection C with the punctured plane C × or an elliptic curve, we can get, instead of the Yangian, thetrigonometric or elliptic solutions to the Yang-Baxter equation (210). It will be interesting to have ananalogous 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 thankKevin Costello, whose works and ideas have directly motivated and guided this project. 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 Canadaand 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).42 φφ , φ φφ . (212)We split up each integral into two, based on whether the bulk point is above or below the line operator.We use angular coordinates defined as in the above diagrams. One subtlety is that, from the definitionof 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 just have to make sure that when we divide upthe integral in the aforementioned way, even when written in angular coordinates, the integrand doesnot change sign under reflection. With this in mind, the integrals we have to evaluate are: V αβγ ·|| ( x , x ) = (cid:126) (2 π ) f αβγ (cid:90) π d φ (cid:90) πφ d φ (cid:32)(cid:90) φ + ππ d φ + (cid:90) φ − ππ d φ (cid:33) , V αβγ |·| ( x , x ) = (cid:126) (2 π ) f αβγ (cid:90) π d φ (cid:90) πφ d φ (cid:32)(cid:90) φ + πφ + π d φ + (cid:90) φ − πφ − π d φ (cid:33) , V αβγ ||· ( x , x ) = (cid:126) (2 π ) f αβγ (cid:90) π d φ (cid:90) πφ d φ (cid:18)(cid:90) πφ + π d φ + (cid:90) φ − π d φ (cid:19) . All three terms are equal to (cid:126) f αβγ . B Yangian from 1-loop Computations At the end of § U ( gl K [ z ]). Since A Sc ( T bk ) is an algebra to beginwith, we conclude that at one loop, we have a deformation of the classical algebra 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: • A Sc ( T bk ) becomes the classical algebra U ( gl K [ z ]) in the classical limit (cid:126) ! • A Sc ( T bk ) is isomorphic to U ( gl K [ z ]) (cid:74) (cid:126) (cid:75) as a C (cid:74) (cid:126) (cid:75) -module. • A Sc ( T bk ) is a topological Hopf algebra (with respect to (cid:126) -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 of U ( gl K [ z ]) in theabove sense. Therefore, if we can show that our algebra A Sc ( T bk ) satisfies all these conditions andit is a nontrivial deformation of U ( gl K ) then we can conclude that it is the Yangian. From 1-loopcomputations we already know that it is a non-trivial deformation. That the first condition in thelist above is satisfied is the content of Lemma 1. The second condition is satisfied because (cid:126) actson the generators of our algebra by simply multiplying the external propagators by (cid:126) in the relevantWitten 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, wewere unable to find a reference to cite, therefore, for the sake of completion, we provide a proof in thisappendix, that the algebra A Sc ( T bk ) is indeed an ( (cid:126) -adic) topological Hopf algebra.We shall prove this by reconstructing the algebra A Sc ( T bk ) from its representations. As mentionedin § V and U in this category is constructed by com-puting the expectation value of two Wilson lines in representations U and V ∨ and providing a state43t one end of each of the lines. For example, if (cid:37) and (cid:37) (cid:48) are two homomorphisms from gl K to End( U )and End( V ∨ ) respectively, then for two lines L and L (cid:48) in the topological plane of the CS theory andany ψ ⊗ χ ∨ ∈ U ⊗ V ∨ , the expectation value (cid:104) W (cid:37) ( L ) W (cid:37) (cid:48) ( L (cid:48) ) (cid:105) is valued in End( U ) ⊗ End( V ∨ ) andplugging in states we find a morphism (cid:104) W (cid:37) ( L ) W (cid:37) (cid:48) ( L (cid:48) ) (cid:105) ( ψ ⊗ χ ∨ ) : V ! U .Classically, these same Wilson lines carry representations of the classical algebra U ( gl K [ z ]). Whenviewed as representations of the deformed (alternatively, quantized) algebra A Sc ( T bk ), we shall call thecategory of Wilson lines the quantized category and viewed as representations of U ( gl K [ z ]) we shallrefer to the category as the classical category . Given any two Wilson lines U and V , any non-trivialmorphism between them in the quantized category is a quantization of a non-trivial morphism in theclassical category. As we mentioned, a morphism between two Wilson lines is the expectation value ofthe lines provided with states at one end. A classical morphism is computed with classical diagramsand its quantization amounts to adding loop diagrams. A zero morphism is constructed by providingzero states, this is independent of quantization, i.e., a quantized morphism is zero, if the provided statesare zero, but then so is the original classical morphism. There is in fact a one-to-one correspondencebetween 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 rigidabelian monoidal category C = Rep C ( H ) which is the representation category of a Hopf algebra H .We then find a C (cid:74) (cid:126) (cid:75) -linear abelian monoidal category C (cid:126) , whose objects are representations of some,yet unknown, Hopf algebra H (cid:126) , with the following properties: • ob( C (cid:126) ) = ob( C ) , • Hom C (cid:126) ( U, V ) ∼ = Hom C ( U, V ) (cid:74) (cid:126) (cid:75) as C (cid:74) (cid:126) (cid:75) -modules .Given this information we shall now prove that H (cid:126) is unique and that it is topological with respect to (cid:126) -adic topology. Then specializing to the case H = U ( gl K [ z ]) completes the proof of A Sc ( T bk ) beingtopological. 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 runninginto some subtlety in the beginning (we shall explain the subtlety later in this section), we first considerthe reconstruction from the category of corepresentations. Reconstruction from corepresentation. Let k be a field, C an abelian (resp. abelian monoidaland End(1) = k ) category such that morphisms are k -bilinear, and let R be a commutative algebraover k – if there is an exact faithful (resp. monoidal) functor ω from C to Mod f ( R ) such that theimage of ω is inside the full subcategory Proj f ( R ) , then we shall say that C has a fiber functor ω toMod f ( R ). Theorem 2 (Tannaka Reconstruction for Coalgebra and Bialgebra) . With the notation above, ifmoreover R is a local ring or a PID , then there exists a unique flat R -coalgebra (resp. R -bialgebra) A , up to unique isomorphism, such that A represents the endomorphism of ω in the sense that ∀ M ∈ IndProj f ( R ) Hom R ( A, M ) ∼ = Nat ( ω, ω ⊗ M ) . Moreover, there is a functor φ : C ! Corep R ( A ) which makes the following diagram commutative finitely generated modules of R finitely generated projective modules of R PID=Principal Ideal Domain IndProj f ( R ) means category of inductive limit of finite projective R -modules, which is equivalent to category of flat R -modules. Corep R ( A ) Mod f ( R ) ωφ forget 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 thischain, so that ω ( X i ) is an ascending chain of finitely generated projective submodules of finitelygenerated projective module ω ( X ), thus there is an index j such that rank( 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 haszero 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 ( R n , X ) to X n , recall that every finitely generated projective module over a local ring ora PID is free, thus isomorphic to R n 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 C X of C generated by subquotients of X n , consider thelargest subobject P X of Hom ( ω ( X ) , X ) whose image in Hom ( ω ( X ) n , X n ) under diagonal embedding iscontained in ( Y : ω ( Y )) for all subobjects Y of X n and all n . Then the Theorem (2) is true for C X with coalgebra defined by A X := ω ( P X ) ∨ .Proof. P X exists because C is Artinian. Notice that ω takes Hom( M, X ) to Hom R ( M, X ) and ( Y : V )to ( ω ( Y ) : V ), so it takes P X , which is defined by (cid:92) (Hom( ω ( X ) , X ) ∩ ( Y : ω ( Y )))to (cid:92) (End R ( ω ( X )) ∩ ( ω ( Y ) : ω ( Y ))) . Hence ω ( P X ) is the largest subring of End R ( ω ( X )) stabilizing ω ( Y ) for all Y ⊂ X n and all n . It’s afinitely generated projective R module by construction, and so is A X . Note that only finitely manyintersection occurs because Hom( ω ( X ) , X ) is Artinian.Next, take any flat R module M , since C X is generated by subquotients of X , an element λ ∈ Nat( ω, ω ⊗ M ) is completely determined by it is value on X , so λ ∈ End R ( ω ( X )) ⊗ M . Since − ⊗ R M is an exact functor, we have: (cid:92) (Hom R ( ω ( X ) , ω ( X ) ⊗ R M ) ∩ ( ω ( Y ) ⊗ R M : ω ( Y ))) Recall that a R module is flat if and only if it is a filtered colimit of finitely generated projective modules. (cid:16)(cid:92) (End R ( ω ( X )) ∩ ( ω ( Y ) : ω ( Y ))) (cid:17) ⊗ R M . This follows because there are only finitely many intersections and finite limit commutes with tensoringwith flat module. Therefore, λ ∈ ω ( P X ) ⊗ R M . Conversely, every element in ω ( P X ) ⊗ R M gives rise to a natural transform in the way described above.Hence we establish the isomorphismNat( ω, ω ⊗ M ) ∼ = ω ( P X ) ⊗ R M ∼ = Hom R ( A X , M ) .A X is unique up to unique isomorphism (as a flat R module) because it represents the functor M Nat( ω, ω ⊗ M ).Next, we shall define a co-action of A X on ω , a counit and a coproduct on A X which makes A X an R -coalgebra and ω a corepresentation: ρ ∈ Nat( ω, ω ⊗ A X ) ∼ = End R ( A X )corresponds to the identity map of A X , and (cid:15) ∈ Hom R ( A X , R ) ∼ = Nat( ω, ω )corresponds to Id ω . The co-action ρ tensored with Id A X gives a natural transform between ω ⊗ A X and ω ⊗ A X ⊗ A X , whose composition with ρ gives the following commutative diagram: ω ω ⊗ A X ω ⊗ A X ⊗ A Xψρ ρ ⊗ Id AX .Take ∆ to be the image of ψ in Hom R ( A X , A X ⊗ R A X ). It follows from definition that A X is counitaland ρ : ω ! ω ⊗ A X is a corepresentation. It remains to check that ∆ is coassociative.Observe that the essential image of ω ⊗ A X is a subcategory of the essential image of ω , henceevery functor that shows up here can be restricted to ω ⊗ A X , in particular, ρ , whose restriction to ω ⊗ A X is obviously ρ ⊗ Id A X . It follows from the definition that( ρ ⊗ Id A X ) ◦ ρ = (Id ω ⊗ ∆) ◦ ρ ∈ Nat( ω, ω ⊗ A X ⊗ A X ) . Restrict this equation to ω ⊗ A X and we get( ρ ⊗ Id A X ⊗ Id A X ) ◦ ( ρ ⊗ Id A X ) = (Id ω ⊗ Id A X ⊗ ∆) ◦ ( ρ ⊗ Id A X ) . Composing with ρ , the LHS corresponds to (∆ ⊗ Id A X ) ◦ ∆ and the RHS corresponds to (Id A X ⊗ ∆) ◦ ∆whose equality is exactly the coassociativity of A X .It follows that ∀ Z ∈ C X , ρ ( Z ) : ω ( Z ) ! ω ( Z ) ⊗ R A X gives ω ( Z ) a A X corepresentation structure and this is functorial in Z , thus ω factors through a φ : C X ! Corep R ( A X ).Back to the uniqueness of A X . It has been shown that it is unique up to unique isomorphismas a flat R module. Additionally, if φ : A X ! A (cid:48) X is an isomorphism such that it induces identitytransformation on the functor M Nat( ω, ω ⊗ M ) then, φ automatically maps the triple (∆ , (cid:15), ρ ) to46∆ (cid:48) , (cid:15) (cid:48) , ρ (cid:48) ), so φ is a coalgebra isomorphism.Finally, it remains to show that when R = k , φ is essentially surjective and full: • Essentially Surjective: If M ∈ Corep k ( A X ), then define (cid:102) M := Coker( M ⊗ ω ( P X ) ⊗ P X ⇒ M ⊗ P X ) , where two arrows are ω ( P X ) representation structure of M and P X respectively, then ω ( (cid:102) M ) = M ⊗ ω ( P X ) ω ( P X ) = M . • Full: If f : M ! N is a A X -corepresentation morphism, then by the k -linearlity of C X , f lifts tomorphisms f ⊗ Id P X : M ⊗ P X ! N ⊗ P X , and f ⊗ Id ω ( P X ) ⊗ Id P X : M ⊗ ω ( P X ) ⊗ P X ! N ⊗ ω ( P X ) ⊗ P X . Thus, passing to cokernel gives rise to (cid:101) f : (cid:102) M ! (cid:101) N which is mapped to f by ω .Next we move on to recover the category C by its subcategories C X . Define an index category I such that its objects are isomorphism classes of objects in C , denoted by X i for each index i , and aunique 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 C X is a full subcategoryof C Y , so a functorial restrictionHom R ( A Y , M ) ∼ = Nat( ω Y , ω Y ⊗ M ) ! Nat( ω X , ω X ⊗ M ) ∼ = Hom R ( A Y , M ) , gives rise to a coalgebra homomorphism A X ! A Y . Futhermore, this homomorphism is injectivebecause ω ( P Y ) ! ω ( P X ) is surjective, otherwise Coker( ω ( P Y ) ! ω ( P X )) will be mapped to the zeroobject in Corep R ( A Y ), which contradicts with ω being faithful. Lemma 5. Define the coalgbra A := lim −! i ∈ I A X i , then it is the desired coalgebra in Theorem 2.Proof. A is flat because it is an inductive limit of flat R modules. MoreoverHom R ( A, M ) = lim − i ∈ I Hom R ( A X i , M ) ∼ = lim − i ∈ I Nat( ω X i , ω X i ⊗ M ) = Nat( ω, ω ⊗ M ) , 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 eachsubcategory C X i . It’s also essentially surjective because every corepresentation V of A comes from acorepresentation of a finite dimensional sub-coalgebra of A , and A is a filtered union of sub-coalgebras A X i , so V comes from a corepresentation of some A X i . In fact, φ is essentially surjective even without the assumption that R = k . Take a basis { e i } for V , the co-action ρ takes e i to (cid:80) j e j ⊗ a ji , then it is easy to see that span { a ji } is a finitedimensional sub-coalgebra of A . roof of Theorem 2. It remains to prove the theorem when C is monoidal. This amounts to including m : C (cid:2) C ! C and e : ! C with associativity and unitarity constrains, where is the trivial tensorcategory with objects { , } and only nontrivial morphisms are End(1) = k . Using the isomorphism:Hom R ( A ⊗ R A, A ⊗ R A ) ∼ = Nat( ω (cid:2) ω, ω (cid:2) ω ⊗ A ⊗ R A ) , we get a homomorphism τ : Hom R ( A ⊗ R A, M ) ! Nat( ω (cid:2) ω, ω (cid:2) ω ⊗ M ) . It is an isomorphism because for each pair of subcategories ( C X , C Y )Hom R ( A X ⊗ R A Y , M ) ∼ = Hom R ( A X , R ) ⊗ R Hom R ( A Y , M ) ∼ = Nat( ω X , ω X ) ⊗ R Nat( ω Y , ω Y ⊗ M ) ∼ = Nat( ω X (cid:2) ω Y , ω X (cid:2) ω Y ⊗ 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 (cid:2) ω, ω (cid:2) ω ⊗ M ) , by taking any α ∈ Nat( ω, ω ⊗ M ), and composing with the isomorphism ω (cid:2) ω ( X (cid:2) Y ) ∼ = ω ( X ⊗ Y ).This homomorphism in turn becomes a homomorphism µ : A ⊗ R A ! A . And the obvious isomorphism Hom R ( R, M ) = M ! Nat( ω , ω ⊗ M ) , together with the unit functor e : ! C give a homomorphism ι : R ! A . All of the homomorphisms are functorial with respect to M so µ and ι are homomorphisms betweencoalgebras. Now the associativity and unitarity of monoidal category C translates into associativityand unitarity of µ and ι , which are exactly conditions for A to be a bialgebra. This concludes theproof of Theorem 2. Remark . 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 ( R n , X ) to X n . This is feasible only if every finite projective module isfree, 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 the Hom( ω ( X ) , X ) todefine the crucial object P X , 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 P X by (cid:92) (Hom( X, X ) ∩ ( Y : Y )) , then the fiber functor ω takes P X to (cid:92) (End R ( ω ( X )) ∩ ( ω ( Y ) : ω ( Y ))) , since ω is monoidal by definition and a monoidal functor between rigid monoidal categories preservesduality and thus preserves inner Hom. (cid:52) R and state the following version ofTannaka reconstruction for Hopf algebras: Theorem 3 (Tannaka Reconstruction for Hopf Algebra) . 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 flat R -Hopf algebra A (resp. R -coquasitriangular Hopfalgebra), up to unique isomorphism, such that A represents the endomorphism of ω in the sense that ∀ M ∈ IndProj f ( R ) Hom R ( A, M ) ∼ = Nat ( ω, ω ⊗ M ) . Moreover, there is a functor φ : C ! Corep R ( A ) which makes the following diagram commutative: C Corep R ( A ) Mod f ( R ) ωφ forget 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, thereexists a bialgebra A which satisfies all conditions in the theorem, so it remains to prove that there arecompatible 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( ω op , ω op ⊗ M ) . According to the functoriality of the construction of the bialgebra A , there is a bialgebra isomor-phism: S : A ! A op , put it in another way, a bialgebra anti-automorphism of A . To prove that it satisfies the requiredcompatibility: µ ◦ ( S ⊗ Id) ◦ ∆ = ι ◦ (cid:15) = µ ◦ (Id ⊗ S ) ◦ ∆ , we observe that ι ◦ (cid:15) gives the natural transformationId ⊗ ρ ω (1) : ω ( X ) = ω ( X ) ⊗ ω (1) 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 ◦ (cid:15) is just the identity morphism on ω ( X ). On the other hand, µ ◦ ( S ⊗ Id) ◦ ∆ corresponds to thehomomorphism 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 ◦ (cid:15) . The other equation is similiar.49b) C is rigid braided. This means that there is a natural transformation: r : ω (cid:2) ω ! ω (cid:2) 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 ) ω ( U ) ⊗ ω ( V ) ⊗ A ⊗ A ω ( U ) ⊗ ω ( V ) ⊗ Aω ( V ) ⊗ ω ( U ) ω ( V ) ⊗ ω ( U ) ⊗ A ⊗ A ω ( V ) ⊗ ω ( U ) ⊗ A ρ ⊗ ρr Id ⊗ Id ⊗ µ r ⊗ Id ρ ⊗ ρ Id ⊗ Id ⊗ µ which in turn translates to the following equation of R : R ◦ µ ◦ (∆ ⊗ ∆) = R ◦ µ ◦ τ ◦ (∆ ⊗ ∆) , 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) ω ( X ) ω (1) ⊗ ω ( X ) Id r ,translates to R ◦ (Id A ⊗ 1) = (cid:15) . And symmetrically R ◦ (1 ⊗ Id A ) = (cid:15) .Finally, the hexagon axiom of braiding:( ω ( X ) ⊗ ω ( Y )) ⊗ ω ( Z )( ω ( Y ) ⊗ ω ( X )) ⊗ ω ( Z ) ω ( X ) ⊗ ( ω ( Y ) ⊗ ω ( Z )) ω ( Y ) ⊗ ( ω ( X ) ⊗ ω ( Z )) ( ω ( Y ) ⊗ ω ( Z )) ⊗ ω ( X ) ω ( Y ) ⊗ ( ω ( Z ) ⊗ ω ( X )) r ⊗ r ⊗ r ,translates to the commutativity of the diagram A ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ AA ⊗ A R Id ⊗ Id ⊗ ∆ µ ⊗ Id R ·R R ,and the same hexagon but with r − instead of r gives another one:50 ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ AA ⊗ A R ∆ ⊗ Id ⊗ IdId ⊗ µ R ·R R .So we end up confirming all the properties that universal R-matrix should satisfy, and we concludethat A is indeed a coquasitriangular Hopf algebra. Reconstruction from representation It is tempting to dualize everything above to formalize theTannaka reconstruction for the category of representations. In other words, we can take the dual of A instead of A itself, and a corepresentation becomes the representaion, and when the category has extrastructures, those structures will be dualized, for example, when C is a k -linear abelian rigid braidedmonoidal category, it should come from the representation category of a flat R -quasitriangular Hopfalgebra, since the dual of those diagrams involved in the proof of Theorem 3 are exactly properties ofuniversal R-matrix of a quasitriangular Hopf algebra.This is naive because the statement:Hom R ( U, V ⊗ A ) ∼ = Hom R ( U ⊗ A ∗ , V ) , is not true in general, since A can be infinite dimensional, thus the naive dualizing procedure isnot 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 corepresentation of A comes from a corepresentation of a finite coalgebra, so it is natural to define the action of A ∗ onthose modules by factoring through some finite quotient A ∗ X for some X ∈ ob( C ). Similiarly, themultiplication structure on A ∗ can be defined by first projecting down to some finite quotient andtaking multiplication A ∗ ⊗ A ∗ = lim − i ∈ I A X i ⊗ lim − i ∈ I A X i ! A X i ⊗ A X i ! A X i which is compatible with transition map A X j ! A X i then taking the inverse limit gives the multipli-cation 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 A X i ⊗ A X j ! A , we only get an inverse system of morphisms from A ∗ to A ∗ X i ⊗ A ∗ X j and the latter’s inverse limit is A ∗ (cid:98) ⊗ A ∗ , instead of A ∗ ⊗ A ∗ . So we actually get a topological Hopf algebra with topological basis N i := ker( A ∗ ! A ∗ X i ) , so that the comultiplication is continuous. Similiarly the counit, multiplication, and anipode are con-tinuous as well. Finally when C is braided, there exists an invertible element R ∈ A ∗ (cid:98) ⊗ A ∗ , and thedual of the structure homomorphism in A is exactly the condition that R is the universal R-matrix ofa topological quasitriangular Hopf algebra.So we can restate Theorem 3 in terms of representations of topological Hopf algebras:51 heorem 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 theendomorphism of ω in the sense that H ∼ = Nat ( ω, ω ) . Moreover, there is a functor φ : C ! Rep R ( H ) which sends an object in C to a continuous representationof H and makes the following diagram commutative: C Rep R ( H ) Mod f ( R ) ωφ forget ,and φ is an equivalence if R = k . Application to Quantization We now consider the case that we have a category C (cid:126) , which is a quantization of the category of representations of some Hopf algebra H over C . The quantization,namely C (cid:126) , of Rep C ( H ) is a C -linear abelian monoidal category which has the same set of generatorsas Rep C ( H ), together with a fiber functor ω (cid:126) : C (cid:126) ! Mod f ( C (cid:74) (cid:126) (cid:75) ) which acts on generators of Rep C ( H )by tensoring with C (cid:74) (cid:126) (cid:75) , andHom C (cid:126) ( X, Y ) ∼ = Hom C (cid:126) ( X, Y ) / (cid:126) = Hom Rep C ( H ) ( X, Y )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 categoryof representations generated by classical Wilson lines. Quantized Wilson lines naturally generated a C -linear abelian monoidal category.Applying Theorem 4, ( C (cid:126) , ω (cid:126) ) gives us a (topological) C (cid:74) (cid:126) (cid:75) -Hopf algebra H (cid:126) . Since C (cid:126) and C sharesthe same set of generators, and the construction of those Hopf algebras as C (cid:74) (cid:126) (cid:75) -modules only involvesgenerators of corresponding categories, so H (cid:126) is isomorphic to the completion of H ⊗ C (cid:74) (cid:126) (cid:75) in the (cid:126) -adictopology: H (cid:126) := lim − i ∈ I H X i ⊗ C (cid:74) (cid:126) (cid:75) ∼ = lim − i ∈ I lim − n H X i ⊗ C [ (cid:126) ] / ( (cid:126) n ) ∼ = lim − n lim − i ∈ I H X i ⊗ C [ (cid:126) ] / ( (cid:126) n ) ∼ = lim − n H ⊗ C [ (cid:126) ] / ( (cid:126) n ) . For the same reason, tensor product of two copies of H (cid:126) and completed in the inverse limit topologyis isomorphic to the completion of H (cid:126) ⊗ C (cid:74) (cid:126) (cid:75) H (cid:126) in the (cid:126) -adic topology: H (cid:126) (cid:98) ⊗ H (cid:126) ∼ = lim − n H (cid:126) ⊗ C (cid:74) (cid:126) (cid:75) H (cid:126) / ( (cid:126) n )From the construction of those Hopf algebras and the condition that a morphism in C (cid:126) modulo (cid:126) isa morphism in Rep C ( H ), it is easy to see that modulo (cid:126) respects all structure homomorphisms, thus H (cid:126) modulo (cid:126) and H are isomorphic as Hopf algebras. Finally, structure homomorphisms of H (cid:126) arecontinuous in the (cid:126) -adic topology because they are (cid:126) -linear. Thus we conlude that: Theorem 5. H (cid:126) is a quantization of H in the sense of Definition 6.1.1 of [45], i.e. it is a topologicalHopf algebra over C (cid:74) (cid:126) (cid:75) with (cid:126) -adic topology, such that i) H (cid:126) is isomorphic to H (cid:74) (cid:126) (cid:75) as a C (cid:74) (cid:126) (cid:75) -module;(ii) H (cid:126) modulo (cid:126) is isomorphic to H as Hopf algebras. In our case, H = U ( g [ z ]) for g = gl K [ z ], so H (cid:126) is a quantization of U ( gl K [ z ]), and according toTheorem 12.1.1 of [45], this is unique up to isomorphisms. This proves Proposition (2). C Technicalities 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: v v = v v v = 0 . (213) Proof. Two propagators: We can choose one of the two bulk points, say v , to be at the origin anddenote v simply as v . This amounts to taking the projection (111), namely: R v × R v (cid:51) ( v , v ) v − v =: v ∈ R . Then the product of the two propagators become: P ( v , v ) ∧ P ( v , v ) P ( v ) ∧ P ( − v ) = − P ( v ) ∧ P ( v ) . (214)This is a four form at v , however, P does not have any d z component, therefore the four form P ( v ) ∧ P ( v )necessarily contains repetition of a one form and thus vanishes.Three propagators: By choosing v to be the origin of our coordinate system we can turn theproduct to the following: P ( v ) ∧ P ( v ) ∧ P ( v , v ) . (215)We now need to look closely at the propagators (see (111) and (114)): P ( v i ) = (cid:126) π x i d y i ∧ d z i + y i d z i ∧ d x i + 2 z i d x i ∧ d y i d ( v i , , (216a) P ( v , v ) = (cid:126) π x d y ∧ d z + y d z ∧ d x + 2 z d x ∧ d y d ( v , v ) , (216b)where v i := ( x i , y i , z i , z i ), x ij := x i − x j , y ij := y i − y j , · · · , and d ( v i , v j ) := ( x ij + y ij + z ij z ij ).Since the propagators don’t have any d z component the product (215) must be proportional to ω := (cid:86) i ∈{ , } d x i ∧ d y i ∧ d z i . In the product there are six terms that are proportional to ω . For example,we can pick d x ∧ d y from P ( v ), d z ∧ d x from P ( v ) and d y ∧ d z from P ( v , v ), this term isproportional to:d x ∧ d y ∧ d z ∧ d x ∧ d y ∧ d z = − d x ∧ d y ∧ d z ∧ d x ∧ d y ∧ d z = + ω . (217)The other five such terms are:d y ∧ d z ∧ d z ∧ d x ∧ d x ∧ d y = − ω , d y ∧ d z ∧ d x ∧ d y ∧ d z ∧ d x = + ω , d z ∧ d x ∧ d y ∧ d z ∧ d x ∧ d y = + ω , d z ∧ d x ∧ d x ∧ d y ∧ d y ∧ d z = − ω , d x ∧ d y ∧ d y ∧ d z ∧ d z ∧ d x = − ω . (218)53hese signs can be determined from a determinant, stated differently, we have the following equation:det d y ∧ d z d z ∧ d x d x ∧ d y d y ∧ d z d z ∧ d x d x ∧ d y d y ∧ d z d z ∧ d x d x ∧ d y = − ω , (219)where the product used in taking determinant is the wedge product. The above equation implies thatin the product (215) the coefficient of − ω is given by the same determinant if we replace the two formswith their respective coefficients as they appear in (216). Therefore, the coefficient is:18 π d ( v , d ( v , d ( v , v ) det x y z x y z x y z = 0 . (220)The determinant vanishes because the three rows of the matrix are linearly dependent. Thus weconclude that the product (215) vanishes. Lemma 7. The product of two bulk-to-bulk propagators joined at a bulk vertex where the other twoendpoints are restricted to the Wilson line, vanishes, i.e., in any Witten diagram: vp p = 0 . (221) Proof. This simply follows from the explicit form of the bulk-to-bulk propagator. Computation verifiesthat: ι ∂ x ∧ ∂ x ( P ( v, p ) ∧ P ( v, p )) = 0 , (222)where x and x are the x -coordinates of the points p and p respectively.The world-volume on which the CS theory is defined is R x,y × C z , which in the presence of theWilson line at y = z = 0 we view as R x × R + × S . When performing integration over this space weapproximate the non-compact direction by a finite interval and then taking the length of the intervalto infinity. In doing so we introduce boundaries of the world-volume, namely the two components B ± D := {± D } × R + × S at the two ends of the interval [ − D, D ]. Our next lemma concerns someintegrals over these boundaries. Lemma 8. The integral over a bulk point vanishes when restricted to the spheres at infinity, in diagram: lim D ! ∞ (cid:90) v ∈ B ± D v v n ... v = 0 . (223) Proof. Symbolically, the integration can be written as:lim D ! ∞ (cid:90) B ± D dvol B ± D ι ∂ y ∧ ∂ z ( P ( v , v ) ∧ · · · ∧ P ( v , v n )) , (224)where y and z are coordinates of v . Note that the d z required for the volume form on B ± D comes fromthe structure constant at the interaction vertex, not from the propagators. In the above integration the x -component of v is fixed at ± D , which introduces D dependence in the integrand. The bulk-to-bulkpropagator has the following asymptotic scaling behavior: P (( D, y, z, z ) , v j ) D ! ∞ ∼ D − + O ( D − ) . (225)The integration measure on B ± D is independent of D , therefore the integral behaves as D − n for large D , and consequently vanishes in the limit D ! ∞ . Keep in mind that (cid:126) has a (length) scaling dimension 1. .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 nicelywhen acted upon by differential (see (112)), we want to use Stoke’s theorem to simplify any givenWitten diagram. Suppose we have a Witten diagram with m propagators connected to the boundary, n propagators connected to the Wilson line, and l bulk points. Let us denote the bulk points by v i 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 and ∆ n is an n -simplex defined as:∆ n := { ( p , · · · , p n ) ∈ R n | p ≤ p ≤ · · · ≤ p n } . (226)This domain may need to be modified in some Witten diagrams due to the integral over this domainhaving UV divergences. UV divergences can occur when some points along the Wilson line collide witheach other. To avoid such divergences we shall use a point splitting regulator, i.e., we shall cut somecorners from the simplex ∆ n . Let us denote the regularized simplex as (cid:101) ∆ n . The exact description of (cid:101) ∆ n will vary from diagram to diagram, and we shall describe 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 and (cid:101) ∆ n .For M we get: ∂M = B + ∞ (cid:116) B −∞ . (227)Due to Lemma 8, integrations over ∂M will vanish. Therefore, nonzero contribution to theboundary integration, when we do integration by parts, will only come from the boundary of theregularized simplex, namely ∂ (cid:101) ∆ n . Schematically, the appearance of such a boundary integralwill look like: (cid:90) M l × (cid:101) ∆ n d θ ∧ ( · · · ) = (cid:90) M l × ∂ (cid:101) ∆ n θ ∧ ( · · · ) + · · · . (228)2. The differential acts on a bulk-to-bulk propagator. Due to (112), this identifies the two endpoints of the propagator, schematically: b ∈ { , } , (cid:90) M l × ∂ b (cid:101) ∆ n d θ ∧ P ∧ ( · · · ) = (cid:90) M l − × ∂ b (cid:101) ∆ n θ ∧ ( · · · ) + · · · . (229)3. The differential acts on a step function left by a previous integration by parts. This does notchange the domain of integration.The third option does not to lead a simplification of the domain of integration. Therefore, at the presentabstract level, our strategy to simplify an integration is: first go to the boundary of the simplex, andthen keep collapsing bulk-to-bulk propagators until we have no more differential left or when no morebulk-to-bulk propagator can be collapsed without the diagram vanishing due the vanishing lemmasfrom § C.1. D Proof of Lemma 2 All the diagrams that we draw in this section only exist to represent color factors, their numericalvalues are irrelevant. Which is why we also ignore the color coding we used in the diagrams in themain body of the paper.We start with yet another lemma: 55 emma 9. The color factor of any Witten diagram with two boundary-to-bulk propagators connectedby a single bulk-to-bulk propagator, that is any Witten diagrams with the following configuration:... ... µ ν (230) upon anti-symmetrizing the color labels of the boundary-to-bulk propagators, involves the followingfactor: f ξµν X ξ , (231) for some matrix X ξ that transforms under the adjoint representation of gl K . In particular, this colorfactor is the image in End( V ) of some element of gl K where V is the representation of some distantWilson line.Proof. The two bulk vertices in the diagram results in the following product of structure constants: f πµo f oνρ where the indices π and ρ are contracted with the rest of the diagram. Anti-symmetrizingthe indices µ and ν we get f πµo f oνρ − f πνo f oµρ , which using the Jacobi identity becomes − f oµν f πρo .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 gl K ,since the structure constant f ξµν can be viewed as a map: f : ∧ gl K ! gl K , f : t µ ∧ t ν f ξµν t ξ . (232)Now composing the above map with a representation of gl K 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 ν , f ξoµ f πρξ f σνπ (cid:37) ( t o ) (cid:37) ( t ρ ) (cid:37) ( t σ ) . (233)By commuting (cid:37) ( t o ) and (cid:37) ( t ρ ) in the color factor we create a difference which is the color factor of thefollowing diagram: µ ν . (234)The key feature of the above diagram is the loop with three propagators attached to it. Such a loopproduces a color factor which is a gl K -invariant inside ( gl K ) ⊗ , explicitly we can write a loop and itsassociated color factor respectively as: µνξ and f πµo f oνρ f ρξπ . (235)56he color factor is gl K -invariant since the structure constant itself is such an invariant. To find theinvariants in ( gl K ) ⊗ we start by writing gl K as: gl K = sl K ⊕ C , (236)where by sl K we mean the complexified algebra sl ( K, C ). This gives us the decomposition( gl K ) ⊗ = ( sl K ) ⊗ ⊕ · · · , (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 correspond to the center, becauseeach of these indices belong to an instance of the structure constant, which vanishes whenever oneof its indices correspond to the center. This means that the gl K invariant we are looking for mustlie in ( sl K ) ⊗ . For K > 2, there are exactly two such invariants [57], one of them is the structureconstant itself, which is totally anti-symmetric. The other invariant is totally symmetric. However thestructure constant is even (invariant) under the Z outer automorphism of sl K whereas the symmetricinvariant is odd. Since our theory has this Z as a symmetry, only the structure constant can appearas the invariant in a diagram. This means, as far as the color factor is concerned, we can collapse aloop such as 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 an element in gl K . This shows that we can swap the positions of any of the two pairs of the adjacent matrices inthe color factor in (233) and the difference we shall create is an image of a map gl K ! End( V ). Toachieve all permutations of the three matrices wee need to be able to keep swaping positions, let ustherefore keep looking forward.Suppose we commute (cid:37) ( t o ) and (cid:37) ( t ρ ) in (233), then we end up with the color factor of the diagram(159). Now if we commute (cid:37) ( t o ) and (cid:37) ( t σ ), we create a difference that corresponds the color factor ofthe following diagram: µ ν . (238)The key feature of this diagram is a loop with four propagators attached to it. The loop and itsassociated color factor can be written as: µξν o , f τµπ f σoτ f ρνσ f πξρ . (239)As before, the color factor is a gl K -invariant in ( gl K ) ⊗ . This time, it will be more convenient to writethe color factor as a trace. Noting that the structure constants are the adjoint representations of thegenerators of the algebra we can write the above color factor as:tr ad ( t µ t o t ν t ξ ) . (240)The adjoint representation of gl K factors through sl K , and the adjoint representation of sl K has anon-degenerate metric with which we can raise and lower adjoint indices. Suitably changing positionsof some of the indices in the color factor we can conclude:tr ad ( t µ t o t ν t ξ ) = tr ad ( t µ t ξ t ν t o ) . 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Springer(2007).[56] A. Joyal and R. Street, An introduction to tannaka duality and quantum groups , In Categorytheory , pp. 413–492. Springer (1991). 6057] W. McKay, J. Patera and D. Rand, q ):Γ ! , ( p µ,m ; p ν,n ) = (cid:90) q >q d q d q δ ( q − p ) δ ( q − p ) (cid:37) ν,n (cid:37) µ,m , = 0 . (139)Therefore their contribution to the commutator is:[ T µ [ m ] , T ν [ n ]] = lim p ! p (cid:0) Γ ! , ( p µ,m ; p ν,n ) − Γ ! , ( p ν,n ; p µ,m ) (cid:1) , = [ (cid:37) µ,m , (cid:37) ν,n ] = f ξµν (cid:37) ξ,m + n = f ξµν T ξ [ m + n ] , (140)where the last equality is established by evaluating the diagram: m + npξ,m + n . (141)The bracket (140) is precisely the Lie bracket in the loop algebra gl K [ z ]. Note in passing that hadwe considered the same diagrams as the ones in (137) except with different derivative couplings at theWilson line then the diagrams would have vanished, either because there would be more z -derivativesthan z , or there would be less, in which case there would be z ’s floating around which vanish alongthe Wilson line located at y = z = 0.There is one 2 ! p µ,m p ν,nm + n , (142)however, since the two boundary-to-bulk propagators are two parallel delta functions, i.e., their supportare restricted to x = p and x = p respectively with p (cid:54) = p , they never meet in the bulk and thereforethe diagram vanishes. There are no more classical diagrams, so the Lie bracket in the classical algebrais just the bracket in (140). [ T µ [ m ]( p ) , T ν [ n ]( p )] may be a more accurate notation but this algebra must be position invariant and thereforewe shall ignore the position. Reference to the position only matters when different operators are positioned at differentlocations. .2.2 Coproduct Apart from the Lie algebra structure, the algebra A Sc ( T bk ) also has a coproduct structure. This canbe seen by considering the Wilson line in a tensor product representation, say U ⊗ V . Such a Wilsonline can be produced by considering two Wilson lines in representations U and V respectively andbringing them together, and asking how T µ [ n ] acts on U ⊗ V . Since there are going to be multiplevector spaces in this section, let us distinguish the actions of T µ [ n ] on them by a superscript, such as, T Uµ [ n ], T Vµ [ n ], etc. At the classical level the answer to the question we are asking is simply given bycomputing the following diagrams: UV pµ,mm + UV pµ,mm . (143)Evaluation of these diagrams is very similar to that of the diagrams in (137) and the result is: T U ⊗ Vµ [ m ] = T Uµ [ m ] ⊗ id V + id U ⊗ T Vµ [ m ] . (144)This is the same coproduct structure as that of the universal enveloping algebra U( gl K [ z ]).Combining the results of this section and the previous one we find that, at the classical level wehave an associative algebra with generators T µ [ n ] with a Lie bracket and coproduct given by the Liebracket of the loop algebra gl K [ z ] and the coproduct of its universal enveloping algebra. This identifies A Sc ( T bk ), clasically, as the universal enveloping algebra itself: Lemma 1.