DDefect a -Theorem and a -Maximization Yifan Wang , Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
Conformal defects describe the universal behaviors of a conformal field theory (CFT)in the presence of a boundary or more general impurities. The coupled critical system ischaracterized by new conformal anomalies which are analogous to, and generalize those ofstandalone CFTs. Here we study the conformal a - and c -anomalies of four dimensionaldefects in CFTs of general spacetime dimensions greater than four. We prove that underunitary defect renormalization group (RG) flows, the defect a -anomaly must decrease, thusestablishing the defect a -theorem. For conformal defects preserving minimal supersymmetry,the full defect symmetry contains a distinguished U (1) R subgroup. We derive the anomalymultiplet relations that express the defect a - and c -anomalies in terms of the defect (mixed) ’tHooft anomalies for this U (1) R symmetry. Once the U (1) R symmetry is identified using thedefect a -maximization principle which we prove, this enables a non-perturbative pathway tothe conformal anomalies of strongly coupled defects. We illustrate our methods by discussinga number of examples including boundaries in five dimensions and codimension-two defectsin six dimensions. We also comment on chiral algebra sectors of defect operator algebrasand potential conformal collider bounds on defect anomalies. a r X i v : . [ h e p - t h ] J a n ontents a -Theorem 6 a -Maximization 15 a -maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 a -anomalies in SCFTs 24 N = 1 SCFTs . . . . . . . . . . . . . . . . . . . . . . . . 244.1.1 Boundary ’t Hooft anomalies from bulk fermions . . . . . . . . . . . . 254.1.2 Boundary conditions for free hypermultiplets . . . . . . . . . . . . . . 274.1.3 Boundaries of E n SCFTs . . . . . . . . . . . . . . . . . . . . . . . . . 314.1.4 Boundary SQCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Codimension-two defects in 6d SCFTs . . . . . . . . . . . . . . . . . . . . . 354.2.1 Codimension-two defects in free theories . . . . . . . . . . . . . . . . 364.2.2 Punctures in interacting SCFTs . . . . . . . . . . . . . . . . . . . . . 37 N = 1 gauge theories 44 B.1 5d Spinor conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44B.2 Hypermultiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.3 Vector multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Introduction and Summary
The theory space of quantum field theories (QFT) is immensely rich and diverse. It en-compasses a wide range of quantum dynamics, including renormalization group (RG) flows,phase transitions and critical phenomena, that take place in a large and elaborate zoo ofquantum systems. An enduring challenge is to identify and understand non-perturbativestructures in the geometry of the theory space.A particularly well-posed and important problem is the monotonicity of RG flows. This isintuitively clear since the RG procedure involves coarse-graining over short-distance physicsand therefore leads to a reduction in the effective degrees of freedom. In spacetime dimension d = 2, this intuitive picture was rigorously justified by the celebrated c -theorem [1], wherea height function, known as the c -function, was constructed over the theory space, whichdecreases monotonically under RG flows. A particular feature of the 2 d c -function is that itcoincides with the conformal anomalies at the critical fixed points of the RG flows, describedby conformal field theories (CFT). Such conformal anomalies are expected to “count” thedegrees of freedom in the CFTs, thus it is natural to study conformal anomalies in higherdimensions in an effort to extend the 2 d c -theorem.The conformal anomalies of CFTs are present for even d and important physical observ-ables that govern the CFT dynamics (e.g. through correlation functions of the stress-tensor).They are also known as the trace anomalies, since the symmetric, conserved, and tracelessstress-tensor T µν in a CFT T can develop an anomalous nonzero trace when the theory isplaced on a spacetime manifold with nontrivial metric ( M , g ) [2], A W T ≡ (cid:104) T µµ (cid:105) = 1(4 π ) d (cid:32) − ( − d aE d + (cid:88) i c i W i (cid:33) , (1.1)that solves the Wess-Zumino consistency conditions [3, 4]. Equivalently, the conformalanomalies contribute to the anomalous variation of the path integral, under the Weyl trans-formation g µν → e σ ( x ) g µν , δ σ log Z [ g ] = i (cid:90) M d d x (cid:112) | g | σ A W T . (1.2)Here E d is the Euler density in d dimensions, which is related to the Euler characteristic by χ ( M ) = π ) d/ (cid:82) M E d where χ ( S d ) = 2 for an even dimensional sphere, and W i are Weylinvariants of the Riemann curvature (that are not total derivatives) of scaling dimension We only include the contributions that cannot be removed by adjusting local counter-terms. . The c -anomaly in d = 2 is proportional to the a -coefficient in (1.1) and (1.2). It isthus natural to expect the a -anomaly to play the role of the 2d c -anomaly for general evenspacetime dimensions and the monotonicity of RG flows to be governed by an a -theorem [5].Decades after the work of [1], the a -theorem was finally proven in d = 4 [6–8]. Akey insight of [6, 7] is a conformal version of the conventional spurion analysis for globalsymmetries. Upon introducing a background dilaton field suitably coupled to the theory ofinterest, one can restore conformal symmetry along the RG flow. Anomaly matching thensuggests that the difference between the UV and IR conformal anomalies must be reproducedby the Weyl transformation of the spurious dilaton. The d = 4 a -theorem then follows fromunitarity constraints on the dilaton effective action. Subsequent efforts to generalize theargument of [6, 7] to higher dimensions were made in d = 6 [9–12]. Here we focus on adifferent extension, namely the a -theorem for conformal defects of dimension p = 4 in d > b -theorem for surface defects (i.e. p = 2)in [13] (see also [14]).Defects are an integral part of modern understanding of QFT. Familiar examples includeboundary conditions and quantum impurities (e.g. Wilson line of a probe particle in agauge theory). More generally defects define extended operators over submanifolds of thespacetime, which enrich the algebra of local operators. On the one hand, they participateactively in the bulk field theory dynamics, providing elegant formulations of (generalized)symmetries and behaving as non-local order parameters for bulk phase transitions [15]. Onthe other hand, they give rise to new phase transitions and critical phenomena localized onthe p -dimensional defect worldvolume. The incorporation of defects is clearly essential for aproper understanding of the theory space of QFTs.Similar to how CFTs correspond to bulk universality classes in the theory space, confor-mal defects describe universality classes of defect RG flows. The coupled bulk-defect criticalphase is also commonly referred to as the defect CFT (DCFT) (see [16] for a recent review).A p -dimensional conformal defect D in a d -dimensional CFT shares many kinematic featuresof a standalone p -dimensional CFT, for the obvious reason that they are both invariant un-der the SO ( p,
2) conformal symmetry. However a crucial difference of a defect D from astandalone CFT in p -dimensions is the generic absence a locally conserved p -dimensionalstress-tensor. Instead, the defect symmetry is inherited from the ambient CFT. From nowon we focus on the case p = 4, namely the conformal defect D corresponds to a boundary(interface) in a 5 d CFT or a higher codimension defect in d > There are no such Weyl invariants for d = 2, one for d = 4 and three for d = 6. A W T in the absence of defects (1.1), anomalous trace contributions A W D localized on the defectworldvolume Σ ⊂ M , (cid:104) T µµ ( x ) (cid:105) D = A W T + δ (Σ) A W D , A W D = 1(4 π ) ( − aE + cW ) + I ext ( C ( d ) , K ) . (1.3)As before, the same anomalies are captured by the anomalous variation of the DCFT parti-tion after coupling to background geometry, δ σ log Z D [ g, X ] = i (cid:90) M d d x (cid:112) | g | σ A W T + i (cid:90) Σ d z (cid:112) | h | σ A W D , (1.4)under a Weyl transformation of the ambient metric g µν ( x ) → e σ ( x ) g µν ( x ). Here z a arecoordinates on the submanifold Σ ⊂ M which is specified by the embedding functions X µ ( z a ). The induced metric on Σ is h ab = ∂ a X µ ∂ b X ν g µν which transforms as h ab ( z ) → e σ ( X ( z )) h ab ( z ) under the Weyl transformation.A few remarks are in order. In (1.3) and (1.4), E and W are the intrinsic Euler densityand quadratic Weyl invariant of Σ which also appear in the trace anomaly of a standalone4d CFT, and we refer to them as the intrinsic conformal anomalies of the defect D . Theyare given explicitly by the following combinations of Riemann curvatures on Σ, E = R abcd R abcd − R ab R ab + R = 14 (cid:15) abcd (cid:15) efgh R ef ab R ghcd ,W = R abcd R abcd − R ab R ab + 13 R = C abcd C abcd . (1.5)We will continue to use a and c to denote the corresponding conformal anomaly coeffi-cients of the DCFT. Note that these anomalies are present and universal regardless of bulkspacetime dimension (e.g. d can be odd). The full structure of conformal anomalies forthe DCFT is however much richer then that in a standalone 4d CFT, with extra contribu-tions corresponding to extrinsic conformal anomalies of the defect D , given by the last term I ext ( C ( d ) , K ) in (1.3). It contains additional diffeomorphism invariants that depend on theembedding Σ ⊂ M subject to the Wess-Zumino consistency condition [3,4], and in particularincludes Weyl invariants constructed from the (traceless) extrinsic curvature K µab and bulk We assume that the normal bundle of Σ is topologically trivial, which is obviously the case for theconformal defect in flat space. See [20] for general discussions of geometric invariants for submanifolds and their relations throughGauss-Codazzi type identities. C ( d ) µνρσ pulled back to Σ. The independent extrinsic conformal anomalies fora p = 4-dimensional defect have not been completely classified (see [21] for a partial list inthe d = 5 case ) but their explicit forms will not be important for this work. A main question of interest here is whether the defect a -anomaly in (1.3) decreases mono-tonically under defect RG flows. In Section 2, we find an affirmative answer by extendingthe work of [6,7] and invoking Lorentzian unitarity constraints on the defect dilaton effectiveaction, thus establishing the defect a -theorem. We also describe explicitly the defect dilatoneffective action for the simple RG flow between conformal Neumann and Dirichlet boundaryconditions in the d = 5 free scalar theory.If we know the defect a -anomaly of a UV DCFT or an IR DCFT at either ends of a defectRG flow, the defect a -theorem produces non-perturbative constraints on the RG trajectory.However such conformal anomalies are notoriously difficult to access, already in standaloneCFTs. For p = 4 DCFTs, our knowledge is even more limited: the defect a -anomalies wereonly known for the free scalar theory [25–27] and for the free fermion in d = 5 [25], and sofar no results for the defect c -anomalies have been obtained. We will address these issues inthis work. For d = 4 CFTs with N = 1 supersymmetry (SUSY), an elegant non-perturbativemethod was developed to solve for the a - and c -anomalies, known as a -maximization [28].The N = 1 SUSY relates in a simple way the conformal a - and c -anomalies to the ’t Hooftanomalies involving the U (1) R symmetry of the CFT [29]. The ’t Hooft anomalies are mucheasier to compute thanks to their robustness under deformations (which can break conformalbut preserve Poincar´e and U (1) R symmetries and possibly other flavor symmetries). Theonly subtlety is to identify the U (1) R symmetry, which can mix with other U (1) flavorsymmetries. This is accomplished by the a -maximization principle [28, 30, 31]. For the p = 4 conformal defects preserving the minimal N = 1 SUSY, in Section 3we show that the same relations between conformal anomalies and ’t Hooft anomalies in4d SCFTs [29] continue to hold for the DCFTs. The U (1) R superconformal R-symmetryin the DCFT descends from the (larger) R-symmetry and transverse rotation symmetry ofthe ambient SCFT. The relevant ’t Hooft anomalies amount to a modification of the Ward We thank Sergey Solodukhin for correspondence on this point. Related works on conformal invariants of p = 4 submanifolds include the Willmore energy in [22, 23]which generalizes the extrinsic Graham-Witten conformal anomaly for surface defects in [17]. See also [24]for a partial classification of submanifold conformal invariants for general p and d in [24]. We thank Yoshiki Sato for correspondence on this point. In [32], another way to identify the superconformal U (1) R symmetry was introduced, known as τ RR -minimization, by minimizing the R-current two-point function. The τ RR -minimization and a -maximizationare equivalent at the fixed point, but in practice the latter is more powerful since the ’t Hooft anomalies arewell-defined away from the CFT. d -dimensional U (1) R current J on the defect worldvolume Σ, (cid:104)∇ µ J µ (cid:105) D ⊃ δ (Σ)2(2 π ) (cid:63) Σ (cid:18) − k RRR F ∧ F + 124 k R tr R ∧ R (cid:19) , (1.6)where F = dA is the background U (1) R field strength and R is the Riemann curvature two-form. The anomaly coefficients above are simply related to the defect conformal anomaliesby a = 9 k RRR − k R , c = 9 k RRR − k R , (1.7)as in the standalone SCFT [29]. Furthermore in the presence of 4d conserved currents onthe defect, we prove a defect version of the a -maximization principle [28] (see also [32]) thatidentifies the superconformal U (1) R symmetry as a linear combination that may involvethese 4d currents. As a by-product, this defect a -maximization principle also leads to analternative non-perturbative proof of the defect a -theorem for certain supersymmetric RGflows, extending the results of [30, 33].In Section 4, to illustrate our methods, we apply the defect a -maximization principle toselected examples of 4d N = 1 DCFTs in 5d and 6d SCFTs, and determine their defectconformal anomalies. For free theories, our results are consistent with the previous answerson defect a -anomalies [25–27] obtained from the heat kernel method [34] and produce newconstraints on their defect c -anomalies. For interacting theories in 5d, we discuss boundaryconditions of 5d SCFTs, including a boundary version of the familiar 4d N = 1 SQCD andits boundary conformal anomalies. In 6d, we give a reinterpretation of the known resultsof the “conformal anomalies of punctures” in (generalized) class S constructions [35–37] asthe conformal anomalies of codimension-two defects as defined in (1.3). We end with a briefdiscussion of future directions in Section 5. a -Theorem In this section, we will prove the following defect a -theorem that constrains RG flows betweenconformal defects of dimension p = 4. Theorem 1 (Defect a -Theorem) For a unitary defect RG flow between two unitary con-formal defects D UV and D IR of dimension p = 4 , the corresponding defect a -anomalies satisfy a ( D UV ) > a ( D IR ) . (2.1)6n analogy to the different versions of a -theorems for 4d QFTs, the above would be the weak version of the defect a -theorem. A stronger version of the a -theorem requires an a -function defined on the entire theory space which has the desired monotonicity properties andcoincides with the conformal a -anomalies at the fixed points. The strongest a -theorem furtherdemands the RG flows to be gradient flows with respect to the a -function. Although thestronger (and strongest) a -theorem has not been completely proven, there is ample evidencefor its validity (see e.g. [33, 38–41]). Similarly one can formulate a stronger version of thedefect a -theorem below (analogously for the strongest version), but a proof is beyond thescope of this work. Closely related is the question whether scale invariance implies conformalinvariance for p = 4 defects. Substantial progress has been made in proving their equivalencein unitary interacting 4d QFTs [42–49], which should also have natural extensions to defects. Conjecture 1
Given a unitary CFT T of dimension d > , there exists a height function ( a -function) a ( λ i ) on the space of p = 4 unitary Poincar´e invariant defects in T , parametrizedby couplings { λ i } , such that under a defect RG flow parametrized by scale µ , µ ddµ a ( λ i ) = β j ( λ i ) ∂∂λ j a ( λ i ) ≥ . (2.2) Moreover the inequality is saturated at the fixed points λ i = ˆ λ i with β j (ˆ λ i ) = 0 , where thevalue of a coincides with the defect conformal a -anomaly of the fixed point DCFT a (ˆ λ i ) = a ( D ) . (2.3)We emphasize that the a -function here is subject to the local condition, i.e. the µ dependenceof a ( λ i ) comes entirely from the running couplings λ i . Here we prove the defect a -theorem (Theorem 1) by extending the method of [6, 7] (seealso [42]). We will see that apart from a few novelties due to the difference between a defectand a standalone CFT (e.g. the extra extrinsic anomalies in (1.3)), the arguments in [6,7,42]essentially carry through (which we explain to make it self-contained), and the theorem isestablished by a defect version of the optical theorem.We start by considering a defect RG flow from a UV conformal defect D UV to an IRconformal defect D IR . The defect conformal symmetry is explicitly broken in the defect field7heory (DFT) at an immediate scale but we can restore it by coupling to a non-dynamicaldilaton field τ ( z ) on the defect worldvolume Σ which transforms as τ → τ + σ under a localWeyl rescaling of the ambient metric g µν → e σ g µν . Denoting the UV DCFT action abstractly by S D UV , the DFT is generally described by arelevant deformation on the defect worldvolume Σ, S DFT = S D UV + (cid:90) Σ d z (cid:112) | h | (cid:88) O UV ∈D UV λ O UV O UV ( z ) . (2.4)After coupling to background dilaton, we have S DFT [ τ ] = S D UV + (cid:90) Σ d z (cid:112) | h | (cid:88) O UV ∈D UV Ω − ∆ O UV λ O UV O UV ( z ) , (2.5)with Ω ≡ e − τ , (2.6)and the Weyl invariance becomes manifest. To linearized order, the coupling takes the form S DFT [ τ ] ≡ S DFT + (cid:90) Σ d z (cid:112) | h | T ( z ) τ ( z ) + O ( τ ) , (2.7)where T ( z ) is an operator coming from the trace of the bulk stress-tensor T µµ ( x ) = δ (Σ) T ( z ),which is nontrivial away from the defect fixed points. Having reinstated the Weyl symmetry with a compensator field τ , the anomalous Weylvariation of the partition function Z DFT [ τ ] must be constant along the RG flow as a con-sequence of the Wess-Zumino consistency condition and thus completely determined by theconformal anomalies of the UV DCFT D UV . Consequently, near the IR end of the defectRG flow, the same Weyl variation must be reproduced by the effective action, which takesthe following form S eff = S D IR + S dilaton [ τ ] + (cid:90) Σ d z (cid:112) | h | (cid:88) O IR ∈D IR Ω − ∆ O IR λ O IR O IR ( z ) . (2.8)Here S D IR describes abstractly the IR DCFT and S dilaton [ τ ] is the dilaton effective actioncoming from integrating out massive modes along the flow. The coupling between D IR and As usual, one can think of conformal symmetry as the subgroup of Diff × Weyl that leaves the flat spacemetric invariant. Near the UV fixed point, T ( z ) is dominated by the relevant (or marginally relevant) operator O UV withthe largest scaling dimension ∆ O UV ≤ τ is contained in the last term above, which is controlled by irrelevant defectoperators O IR ( z ) in the IR DCFT.Comparing UV and IR asymptotics of the RG flow, the anomaly matching conditionboils down to δ σ log Z D UV = δ σ log Z D IR + iδ σ S dilaton [ τ ] , (2.9)which requires δ σ S dilaton [ τ ] = (cid:90) Σ d x (cid:112) | h | σ ∆ A W , (2.10)with ∆ A W ≡ A W D UV − A W D IR = 1(4 π ) ( − ∆ aE + ∆ cW ) + ∆ I ext ( C ( d ) , K ) , (2.11)from (1.3), placing strong constraints on the dilaton effective action.The solutions for S dilaton [ τ ] can be found similar to the analysis in [6]. In general it takesthe form S dilaton [ τ ] = S WZ [ τ ] + S inv [ τ ] . (2.12)The first term on the RHS is a cohomologically nontrivial solution to the Wess-Zuminoconsistency condition and commonly referred to as the Wess-Zumino (WZ) term, which canbe obtained from integrating the anomaly [3] S WZ = − (cid:90) dt (cid:90) d z (cid:112) | h | e − tτ ( z ) τ ∆ A W ( e − tτ ( z ) h ) . (2.13)We will be interested in the dilaton effective action for a flat defect on flat space, in whichcase we set g µν = η µν and split the spacetime coordinates into tangential and orthogonalcoordinates to the defect as x µ = ( z a , y i ).For the intrinsic defect conformal anomalies in (2.11), the integral was performed in [50]which gives S intWZ [ τ ] = (cid:90) Σ d z (cid:112) | h | (cid:18) − ∆ a (4 π ) (cid:18) τ E + 4 (cid:18) R ab − h ab R (cid:19) ∂ a τ ∂ b τ − ∂τ ) (cid:3) τ + 2( ∂τ ) (cid:19) + ∆ c (4 π ) τ W (cid:19) , (2.14)and reduces to the following simple form in flat space S intWZ [ τ ] flat = ∆ a π (cid:90) Σ d z (cid:0) ∂τ ) (cid:3) τ − ( ∂τ ) (cid:1) . (2.15)9he extrinsic anomalies on the other hand do not contribute to S WZ on flat space. Under aWeyl transformation, the bulk Weyl curvature and the extrinsic curvature transform as C ( d ) µνρσ → C ( d ) µνρσ e − σ , K µab → K µab + h ab N µν ∂ ν σ . (2.16)Here N ν µ is the projector to the normal directions of Σ defined by g µν = N µν + P µν , P µν ≡ ∂ a X µ ∂ b X ν h ab , (2.17)and P µν is the projector to the tangential directions of Σ. In the flat space with σ = tτ ( z )in (2.16), we see clearly ∆ I ext ( C ( d ) , K ) does not contribute to a WZ term.The dilaton effective action may also contain terms that are Weyl invariant, correspondingto homogeneous solutions of (2.10). They are captured by the second term S inv [ τ ] in (2.12).These terms can be constructed from the Weyl invariant worldvolume metricˆ h ab ≡ e − τ h ab , (2.18)using the corresponding curvature invariants as in [6, 7]. Such invariants again separateinto intrinsic and extrinsic types. For similar reasons as explained above for the WZ term,the extrinsic Weyl invariants do not play a role for the flat defect in flat space. With thisunderstanding, we focus on the intrinsic invariants given by S inv [ τ ] = (cid:90) d z (cid:113) | ˆ h | (cid:16) β + β ˆ R + β ˆ R + β ˆ E + β ˆ C abcd ˆ C abcd + O ( ∂ ) (cid:17) . (2.20)In the flat space limit, it gives the following interactions for τ up to the fourth derivativeorder, S inv [ τ ] flat = (cid:90) d z (cid:16) β Ω + 6 β ( ∂ Ω) + 36 β (cid:0) (cid:3) τ − ( ∂τ ) (cid:1) + O ( ∂ ) (cid:17) . (2.21)Putting everything together, from (2.15) and (2.21), we have the full defect dilaton effectiveaction on flat space up to fourth derivative order which takes the identical form as in [6, 7].After a redefinition [7], Ψ ≡ − Ω = 1 − e − τ , (2.22) Weyl invariant couplings between relevant operators in the IR DCFT and the dilaton of the form (cid:90) d z (cid:113) | ˆ h | Ω − ∆ O IR O IR , (cid:90) d z (cid:113) | ˆ h | Ω − ∆ O IR ˆ R O IR , (2.19)with ∆ IR < IR <
10e have S dilaton [ τ ] flat = (cid:90) d z (cid:18) β (1 − Ψ) + 6 β ( ∂ Ψ) + 36 β ( (cid:3) Ψ) (1 − Ψ) − ∆ a π (cid:18) ∂ Ψ) (cid:3) Ψ(1 − Ψ) + ( ∂ Ψ) (1 − Ψ) (cid:19) + O ( ∂ ) (cid:19) . (2.23)To prove the defect a -theorem, we would like explore unitarity constraints on the dilatoninteraction proportional to ∆ a . This can be achieved by studying the four-point amplitude A ( s, t, u ) of the probe dilaton Ψ( p )Ψ( p ) → Ψ( − p )Ψ( − p ) with external defect momenta p i and s = − ( p + p ) , t = − ( p + p ) , u = − ( p + p ) are the usual Mandelstam variables.To isolate the contribution from ∆ a , we work with the special kinematics such that thebackground dilaton is “on-shell” (cid:3) Ψ = 0 . (2.24)In this case, the dilaton interactions in (2.23) simplify drastically. In particular the dilatonamplitude A ( s, t ) at the fourth derivative order is completely determined by the interaction S dilaton [ τ ] ⊃ − ∆ a π (cid:90) d z ( ∂ Ψ) ⇒ A ( s, t ) ⊃ ∆ a π ( s + t + u ) . (2.25)Furthermore we focus on the forward limit p + p = p + p = 0, in which case the amplitudeis a function of s only and has the following small s expansion, A ( s ) = const + ∆ a π s + O ( s ∆ O IR − ) . (2.26)The corrections coming from the coupling between the IR defect D IR and defect dilaton τ though the irrelevant operator O IR in (2.8) are suppressed by s ∆ O IR − .We expect the amplitude A ( s ) to satisfy the Mandelstam analyticity of four-dimensionalamplitudes, namely it is analytic on the upper half plane (from causality) and has branchcuts along the real axis due to the massless states in the IR DCFT that are being exchanged.The positivity of ∆ a comes from a contour argument and standard dispersion relations foramplitudes.We start by considering the integral over a contour Γ in the upper-half complex s -planeas in Fig 1, 0 = − πi (cid:73) Γ ds ∂ s A ( s ) s = I Γ + I Γ + I Γ (2.27)which vanishes by analyticity. The contour Γ consists of three parts Γ , Γ , Γ and the integral11 e( s )Im( s ) ×− (cid:15) (cid:15) R − R Γ Γ Γ Γ Γ = Γ ∪ Γ ∪ Γ Figure 1: The integration contour Γ = Γ ∪ Γ ∪ Γ for the dispersion argument. Here thelimits (cid:15) → R → ∞ are implicit.splits accordingly as above. The part over the small semi-circle Γ gives, I Γ = − ∆ a . (2.28)The integral over Γ above the branch cuts yields, I Γ = 2 π (cid:90) ∞ (cid:15) ds ∂ s Im A ( s ) s , (2.29)where we have used crossing symmetry A ( s ) = A ( − s ) since u = − s in the forward limit,and the reality property of S-matrix A ( s (cid:63) ) = ( A ( s )) (cid:63) to combine the integrals over the twosegments of Γ . Finally the integral over the large semi-circle Γ at infinity vanishes I Γ = 0 , (2.30)since the large s behavior of A ( s ) is dominated by the least relevant UV perturbation in(2.5) which gives s ∆ O UV − for ∆ O UV < Therefore we arrive at the formula∆ a = 2 π (cid:90) ∞ (cid:15) ds ∂ s Im A ( s ) s = 4 π (cid:90) ∞ (cid:15) ds Im A ( s ) s , (2.31) A more careful argument shows that the same is true for marginally relevant deformations [42]. A ( s ) (and thus its imaginary part) in the IR (Im A ( s ) ∼ s > )and UV (Im A ( s ) ∼ s < ) asymptotics. These properties also ensure that the last integral in(2.31) is manifestly convergent.Applying the optical theorem to the forward scattering Ψ( p )Ψ( p ) → Ψ( p )Ψ( p ) on thedefect, ∆ a = 4 π (cid:90) ∞ (cid:15) ds sσ ( s ) s , (2.32)where σ ( s ) is the total cross-section for the scattering Ψ( p )Ψ( p ) → D IR which involveDCFT states both on the defect and in the bulk. Unitarity of the defect field theory requiresthe above integrand to be positive. Since the integral is convergent, we arrive at the desiredinequality ∆ a > , (2.33)that establishes Theorem 1. Here we discuss an explicit defect RG flow, between the Neumann and Dirichlet boundaryconditions of the d = 5 free scalar theory. In particular we will derive the nontrivial four-derivative interaction for the defect dilaton (2.25) which captures the change ∆ a in the defectconformal a -anomalies.The free scalar action on half space R , reads S = 12 (cid:90) y ≥ d x∂ µ Φ ∂ µ Φ , (2.34)with a boundary Σ at y = 0. We start with the Neumann boundary condition ∂ y Φ | Σ = 0 , (2.35)which admits a relevant deformation (boundary mass term) of the form S Σ = − (cid:90) d z m Φ . (2.36)By varying the full action S + S Σ with respect to Φ, we see the boundary condition gets See [51, 52] for recent discussions of the optical theorem for boundary field theories (and more generalnonlocal field theories). ∂ y Φ − m Φ | Σ = 0 . (2.37)which flows to the Dirichlet boundary condition Φ | Σ = 0 as m → ∞ .As explained in the last section, we introduce the defect dilaton τ to restore the defectconformal symmetry, S tot [ τ ] = 12 (cid:90) y ≥ d x∂ µ Φ ∂ µ Φ − (cid:90) Σ d z m Φ + 12 (cid:90) Σ d z m ΨΦ , (2.38)with Ψ defined as in (2.22). By integrating out Φ, we obtain the effective action for Ψ.Treating the last term in (2.38) as a perturbation, the propagator for Φ subjected to theboundary condition (2.37) can be found straightforwardly. We define the propagator after aFourier transformation in the defect coordinates z a asˆ G ( p, − p, y, y (cid:48) ) ≡ (cid:104) Φ( p, y )Φ( − p, y (cid:48) ) (cid:105) , (2.39)then the boundary condition (2.37) requires( ∂ y − m ) ˆ G ( p, − p, y, y (cid:48) ) (cid:12)(cid:12)(cid:12) y → + = 0 . (2.40)We will work in the Euclidean signature and Wick rotate back to Minkowski signature later.The solution to the equation of motion and consistent with the boundary condition isˆ G ( p, − p, y, y (cid:48) ) = e − p | y − y (cid:48) | p + | p | − m | p | + m e − p ( y + y (cid:48) ) | p | . (2.41)When the Φ’s are restricted to boundary (which is what we need for computing the defectdilaton effective action), the propagator is simplyˆ G ( p, − p, ,
0) = 1 | p | + m . (2.42)To extract the four-point interaction of Ψ in the IR effective action, we need to evaluatethe following boundary one-loop Feynman diagrams in the large m limit, F ( p , p , p , p ) ≡ (cid:90) (cid:89) i =1 (cid:0) d (cid:126)z i e i(cid:126)p i · (cid:126)z i (cid:1) (cid:104) Φ ( z , ( z , ( z , ( z , (cid:105) c = δ ( p + p + p + p )( I + I + I ) , (2.43)14ith I ≡ (cid:90) d k (2 π ) | k | + m )( | k + p | + m )( | k + p + p | + m )( | k − p | + m ) , (2.44)and its cyclic permutations I and I . In Appendix A, we perform the integral explicitlyand obtain F ( p , p , p , p ) | p i =0 = δ ( p + p + p + p ) (cid:18) π s + t + u m + O (cid:18) s m , t m (cid:19)(cid:19) , (2.45)which corresponds to a dilaton interface of the form S dilaton ⊃ − π ) (cid:90) d z ( ∂ Ψ) . (2.46)Compared to (2.23), we find that for the boundary RG flow in the free scalar theory∆ a = 1723040 . (2.47)The boundary a -anomalies for the free scalar were also computed from heat kernel methods[25, 53] (see also [34] for a review on these methods) a Dir = − , a Neu = 1746080 , (2.48)in agreement with what we found above. In Section 4.1.2, we will give a simple rederivationof the above. a -Maximization Conformal anomalies are important observables in CFTs but are generally hard to accessin interacting theories of dimension d >
2. This is partly because unlike in two dimensionswhere the conformal anomaly is simply determined by the two-point function of stress-tensor T µν , the conformal anomalies in higher dimensions d ≥ T µν . Alternatively one can in principledetermine the conformal anomaly of a CFT by computing its partition functions on curvedbackgrounds. However this is not feasible in practice for a general CFT. For conformal The defect conformal a -anomaly defined here is related to that in [25] by a there = − a here . d = 2 , , p = 4-dimensional defects, providing a non-perturbativetool to access the defect conformal a - and c -anomalies for defects preserving the minimalsupersymmetry.These are defects invariant under an N = 1 superconformal subalgebra of the full super-conformal symmetry of the bulk SCFT, su (2 , | ⊃ so (4 , × u (1) R , (3.1)which contains four Poincar´e supercharges and four conformal supercharges. The bosonicsubalgebra generates the conformal subgroup longitudinal to the defect and the U (1) R sym-metry that generally comes from a combination of the R-symmetry in the ambient SCFTand the transverse rotation symmetry (in d = 6 only). In this section, we will derive thefollowing universal relations between the defect conformal anomalies a and c , and the defect’t Hooft anomalies k RRR and k R that involve the defect U (1) R symmetry (see (3.5)). Theorem 2
The conformal a - and c -anomalies of a four-dimensional N = 1 superconformaldefect is completely determined by the coefficients k RRR and k R for the U (1) R and mixed U (1) R -gravity ’t Hooft anomalies, a = 9 k RRR − k R , c = 9 k RRR − k R . (3.2)These relations are identical to those satisfied by standalone 4d N = 1 SCFTs [29], eventhough the nature and origin of the anomalies are very different. Furthermore, we willprove a defect version of the a -maximization principle [28] that identifies the superconformal The detailed subalgebra embeddings can be found in [61]. (1) R symmetry through a simple algebraic procedure. Together, they provide a powerfulnon-perturbative tool to extract the defect conformal anomalies in strongly coupled systems. We start by describing the defect ’t Hooft anomalies associated to the defect U (1) R symmetry.In contrast with the case of a local 4d CFT, the Noether current J µ for the defect U (1) R symmetry is a d -dimensional conserved current, which satisfies the operator equation ∂ µ J µ ( x ) = 0 , (3.3)in flat space, everywhere including at the defect along Σ, but away from other operatorinsertions. This ensures that the U (1) R charge, defined by an integral of the current fluxthrough a codimension-one submanifold S , Q ≡ (cid:90) S (cid:63)J , (3.4)remains topological when S intersects with the defect Σ (so that the charge Q acts on thedefect modes).In the presence of other operator insertions, the current conservation law can be modifiedby contact-terms representing a (mixed) anomaly for the U (1) R symmetry. While this mod-ification can occur both as contact terms in the bulk (for even d ) and on the defect volume,here we will focus on the defect contributions. Upon coupling the bulk-defect system to back-ground metric g µν , U (1) R gauge field A µ and gauge fields B ia for additional abelian globalsymmetries U (1) i generated by locally conserved currents J ia on the defect, the anomalousWard identity for the U (1) R current can take the following general form (cid:104)∇ µ J µ (cid:105) D ⊃ δ (Σ)2(2 π ) (cid:63) Σ (cid:18) − k RRR F ∧ F + 124 k R tr R ∧ R − k Rij F i ∧ F j − k RRi F ∧ F i (cid:19) , (3.5)with F ≡ dA and F i ≡ dB i . The sums over the repeated i, j indices are implicit. Equiv-alently, the partition function of the defect field theory develops the following anomalous The abelian gauge fields A and B i are anti-Hermitian in this paper, which differs by a factor of i fromthose in [28]. A → A + dλ , δ λ log Z D = 12(2 π ) (cid:90) Σ d z λ (cid:18) − k RRR F ∧ F + 124 k R tr R ∧ R − k Rij F i ∧ F j − k RRi F ∧ F i (cid:19) , (3.6)which is captured by the Stora-Zumino anomaly descent procedure [62–64] from the followingdegree six anomaly polynomial I = − k R c ( F ) p ( T ) + k RRR c ( F ) + k Rij c ( F ) c ( F i ) c ( F j ) + k RRi c ( F ) c ( F i ) − k i c ( F i ) p ( T ) + k ijk c ( F i ) c ( F j ) c ( F k ) , (3.7)where we have also included general anomaly terms involving the flavor symmetry. Here thecharacteristic classes are defined as usual, c ( F ) = i π F, p ( T ) = − π ) tr( R ∧ R ) . (3.8)The same anomaly polynomial (3.7) implies that the p -dimensional defect flavor symmetrycurrents J ia receives an anomalous divergence (cid:104)∇ a J ai ( z ) (cid:105) D ⊃ π ) (cid:63) Σ (cid:18) − k Rij F ∧ F j − k RRi F ∧ F + 124 k i tr R ∧ R + 13 k ijk F j ∧ F k (cid:19) , (3.9)and the partition function varies accordingly under a gauge transformation B i → A i + dω i on Σ δ ω i log Z D = 12(2 π ) (cid:90) Σ d z ω i (cid:18) − k Rij F ∧ F j − k RRi F ∧ F + 124 k i tr R ∧ R + 13 k ijk F j ∧ F k (cid:19) . (3.10) Supersymmetry generally leads to constraints on the anomalies admissible in a given super-symmetric (defect) field theory. In particular, for a superconformal defect D in an ambientSCFT, since the stress-tensor T µν and the preserved R -symmetry current J µ are relatedby acting with the preserved supercharges Q , one naturally expects relations between thetrace anomaly T µµ and the ’t Hooft anomaly for the R -symmetry current, which define su-persymmetric anomaly multiplets . In the following we will establish the relations (3.2) for p = 4-dimensional N = 1 superconformal defects.18e will look for the supersymmetric completion of the anomalous Ward identity (3.5)and (cid:104) T µµ (cid:105) D ⊃ π ) ( − aE + cW ) , (3.11)or equivalently that of the anomalous variation (3.6) and δ σ log Z D ⊃ i (4 π ) (cid:90) d z (cid:112) | h | σ ( − aE + cW ) . (3.12)We focus on the terms involving background metric and U (1) R gauge field. The SUSYcompletion of (3.6) and (3.12) can be obtained by coupling to off-shell N = 1 supergravity onthe defect worldvolume Σ and imposing the Wess-Zumino consistency conditions involvingthe supersymmetry, U (1) R and Weyl transformation generators (and their commutators)acting on log Z D . Because all these variations δ ( · ) log Z D are local expressions on Σ, thisproblem is identical to the one solved in [65, 66] for standalone 4d theories, and gives riseto the super-Weyl anomaly for the DCFT. Here we follow the conventions of [67] which wasalso used [50] except that our abelian gauge fields are anti-Hermitian. The solution takesa simple form in the N = 1 superspace with chiral and anti-chiral Grassmann coordinates θ α , ¯ θ ˙ β . The metric and R-symmetry gauge field are bosonic components of the supergravitysuperfield H a ( z, θ, ¯ θ ), and the imaginary R-symmetry gauge parameter λ combines with the real Weyl transformation parameter σ into a chiral superfield δ Ω( z, θ ) satisfying δ Ω | θ =0 = σ + 23 λ . (3.13)The most general super-Weyl anomaly solving the Wess-Zumino consistency conditions isgiven by a chiral superspace integral together with its anti-chiral conjugate, δ Ω log Z D [ H a ] = 12(4 π ) (cid:90) Σ d zd θ E δ Ω A SW D + ( c.c ) , (3.14)where E is the chiral superspace measure satisfying E | θ =0 = (cid:112) | h | . The chiral anomalydensity A SW D is built from curvature superfields W αβγ , G a , R obtained from covariant super-derivatives D α , ¯ D ˙ α acting on H a , which contain the Weyl curvature, Ricci curvature andRicci scalar respectively. Moreover G a | θ =¯ θ =0 = iA a is identified with the U (1) R gauge The part of the anomaly that depends on the background flavor symmetry gauge fields has a separatesupersymmetric completion. The conventions of [67] differ from those of [68] used in [29] which employs a different set of torsionconstraints. A SW D = κ W αβγ W αβγ + κ Ξ (3.15)is a combination of the super-Weyl density W αβγ W αβγ and the super-Euler density Ξ, whichtakes the following form in the old minimal supergravity Ξ ≡ W αβγ W αβγ −
14 ( ¯ D − R )( G a G a + 2 R ¯ R ) . (3.16)To compare with the bosonic variations (3.6) and (3.12), we need the F-term components ofthese composite chiral superfields (see [67] and [50]) W αβγ W αβγ (cid:12)(cid:12) θ → W − F ab F ab + 2 i (cid:63) (tr R ∧ R − F ∧ F ) , Ξ | θ → E + 2 i (cid:63) (tr R ∧ R − F ∧ F ) , (3.19)where we have dropped terms involving other supergravity fields on the RHS.Thus we have from (3.15) and (3.12) c = − κ , a = κ , (3.20)and from (3.15) and (3.12) k RRR = 169 (3 κ + 5 κ ) , k R = − κ + κ ) . (3.21)Together we arrive at (3.2) as desired.Next let us consider the SUSY completion of the ’t Hooft anomaly (3.10) involving the 4dflavor symmetry current J ia . The gauge transformation parameter ω i is promoted to a chiral Unlike the super-Weyl density W αβγ W αβγ , the form of the super-Euler density Ξ in terms of the super-fields depends nontrivially on the chosen supergravity formulation (which is correlated with a choice of thesupercurrent multiplet in the 4d field theory [69, 70]). See [71, 72] for realizations of Ξ in new minimal andnon-minimal N = 1 supergravities. For example the first equality in (3.19) comes from D δ W αβγ = D ( α W βγδ ) + 34 (cid:15) δ ( α D λ W βγ ) λ = D ( α W βγδ ) + 34 (cid:15) δ ( α D λ W βγ ) λ , (3.17)and the following relations (only keeping terms dependent on metric and U (1) R gauge field) D ( α W βγδ ) | θ =0 = ( σ ab ) αβ ( σ cd ) γδ C abcd + . . . , D λ W αβλ | θ =0 = iD ( α ˙ α G β ) ˙ α | θ =0 = 43 ( σ ab ) αβ F ab + . . . , (3.18)from solving the torsion constraints and Bianchi identities (see Section 5.5.3 and 5.8.3 in [67]). δ Λ i with ω i = i Im δ Λ i | θ =0 . Focusing on the anomalous variations that dependonly on δ Λ i and supergravity superfields, we have the SUSY completion δ Λ i log Z D = κ π ) (cid:90) Σ d zd θ E δ Λ i W αβγ W αβγ + ( c.c ) . (3.22)Using (3.19) and comparing with (3.10), we arrive at the following relation between themixed U (1) i - U (1) R and U (1) i -gravity anomalies,9 k RRi = k i = 24 κ . (3.23) a -maximization Given a p = 4-dimensional N = 1 superconformal defect D , the defect U (1) R symmetry isgenerally generated by a d -dimensional conserved current of the following form, J tµ ≡ ˆ J µ + t i δ (Σ) δ aµ J ia . (3.24)Here ˆ J µ is a bulk current satisfying (3.3) and (almost) determined by the embedding of thedefect superconformal symmetry su (2 , |
1) in the bulk superconformal algebra. In particular,it is normalized such that the supercharges Q preserved by the defect D carries charges ± The ambiguities come from locally conserved currents J ia on the defect worldvolume Σ withmixing coefficients t i , whose symmetry charges commute with Q .Following [28], we define the trial conformal anomalies, in terms of the (mixed) ’t Hooftanomalies involving the U (1) R t symmetry generated by J tµ , a ( t ) ≡ k R t R t R t − k R t , c ( t ) ≡ k R t R t R t − k R t , (3.25)which coincide with (3.2) for the genuine superconformal U (1) R symmetry at t i = t (cid:63)i . Belowwe derive the defect version of the a -maximization principle that determines t (cid:63)i . Theorem 3 (Defect a -Maximization) The superconformal defect U (1) R symmetry J µ = ˆ J µ + t (cid:63)i δ (Σ) δ aµ J ia (3.26) Note that an anomaly of the form (cid:82) d zd θ E δ Λ i Ξ + ( c.c ) is forbidden by the Wess-Zumino consistencycondition [ δ σ , δ ω i ] log Z D = 0 since we require the Weyl anomaly to be invariant under U (1) i gauge transfor-mations. The sign (and normalization) of ˆ J µ is fixed by requiring its charge defined by ˆ R ≡ (cid:82) S d − (cid:63) ˆ J to appear in s determined by a local maximum t i = t (cid:63)i of the trial defect a -anomaly a ( t ) , ∂ i a ( t ) | t i = t (cid:63)i = 0 , ∂ i ∂ j a ( t ) | t i = t (cid:63)i < . (3.27) Moreover the defect conformal anomalies are given by a = a ( t (cid:63) ) , c = c ( t (cid:63) ) . (3.28)The first condition in (3.27) simply follows from the anomaly multiplet relations for ’t Hooftanomalies involving the U (1) i flavor symmetry (3.23). To derive the second condition in(3.27), we note that ∂ i ∂ j a ( t ) | t i = t (cid:63)i = 2716 k Rij . (3.29)In the following we will show that k Rij is negative definite as a consequence of unitarity andsupersymmetry on the defect D .To explore possible constraints on the ’t Hooft anomaly coefficient k Rij , we turn on flavorsymmetry background gauge fields B ia and look for the SUSY completion of the correspondinganomaly terms in (3.6) δ λ log Z D ⊃ − k Rij π ) (cid:90) Σ d z λF i ∧ F j . (3.30)Promoting B ia to a background vector superfield on Σ with field strength chiral superfield W iα , we have δ Ω log Z D ⊃ κ ij (4 π ) (cid:90) Σ d zd θ E δ Ω W iα W αj + ( c.c )= 1(4 π ) (cid:90) Σ d z (cid:112) | h | σ (cid:0) Re κ ij F iab F jab − Im κ ij (cid:15) abcd F iab F jcd (cid:1) + λ (cid:0) Im κ ij F iab F jab + Re κ ij (cid:15) abcd F iab F jcd (cid:1) + . . . . (3.31)Comparing with (3.30), we find Im κ ij = 0 , Re κ ij = − k Rij , (3.33) the anti-commutator of the supercharge Q and its Hermitian conjugate in radial quantization as {Q , Q † } ∝ ∆ − ˆ R + . . . where ∆ is the usual dilatation operator. Here we have assumed the absence of an exotic parity-even anomaly of the type ∇ µ J µ ∼ δ (Σ) C ij F iab F jab , (3.32) (cid:104) T µµ (cid:105) D ⊃ − δ (Σ) k Rij F iab F jab . (3.34)Now recall the two-point functions of the conserved currents J ia are fixed by conformalsymmetry to take the form (cid:104) J ia ( z ) J jb ( z ) (cid:105) D = τ ij ( ∂ δ ab − ∂ a ∂ b ) 1 | z | , (3.35)with positive definite coefficient τ ij from unitarity. The RHS suffers from a short distancesingularity which can be regularized using [74] R (cid:18) z (cid:19) = − (cid:3) log( µ z ) z , (3.36)and the conformal anomaly arises from the dependence of the regularization scheme on thescale parameter µ as in µ ∂∂µ R (cid:18) z (cid:19) = 2 π δ ( z ) . (3.37)Therefore we have µ ∂∂µ (cid:104) J ia ( z ) J jb ( z ) (cid:105) D = 2 π τ ij ( ∂ δ ab − ∂ a ∂ b ) δ ( z − z ) . (3.38)On the other hand, µ ∂∂µ (cid:104) J ia ( z ) J jb ( z ) (cid:105) D = (cid:104) (cid:90) M d x T µµ ( x ) J ia ( z ) J jb ( z ) (cid:105) D = − k Rij ( ∂ δ ab − ∂ a ∂ b ) δ ( z − z ) , (3.39)where the last equality follows from (3.34). Thus we conclude that k Rij = − π τ ij , (3.40)which is negative definite as desired.Before ending this section, let us make a few comments on the a -maximization procedureand its relation to the defect a -theorem. The a -maximization holds with respect to all U (1)flavor symmetry currents on the defect worldvolume Σ. In practice, we often do not directlydeal with the strongly coupled fixed point. Instead we infer the set of the U (1) symmetries for the DCFT. This would be a defect analog of the impossible anomaly discussed in [73]. a -function before maximizing. It can happen that there areaccidental symmetries that are missed in this way, which may lead to nonsensical answers forthe U (1) R symmetry and conformal anomalies (e.g. a naive violation of the unitarity boundfor certain defect operators). In such cases, we have to adjust the ansatz for the candidate U (1) R symmetry by including the accidental symmetries (e.g. from operators that hit theunitarity bound) and redo the a -maximization (see [30] for relevant discussions in 4d SCFTs).As explained in [28], the a -maximization principle almost implies the a -theorem for super-symmetric RG flows triggered by (marginally) relevant perturbations, since the maximizationprocedure is performed over a larger space of U (1) symmetries in the UV than in the IR.This was later made more precise in [30, 33] by constructing explicitly an a -function alongthe supersymmetric RG flow with the desired properties as in the strongest version of the a -theorem. A direct generalization of their construction leads to the defect a -theorem forsupersymmetric RG flows from (marginally) relevant defect perturbations. Once again, caseswith accidental symmetries must be treated with care [33]. a -anomalies in SCFTs N = 1 superconformal defects of dimension p = 4 exist in 5d and 6d SCFTs. TheseSCFTs are generally strongly coupled and do not have conventional perturbative Lagrangiandescriptions, which makes it especially challenging to study the defects thereof. In thefollowing, we will apply our non-perturbative methods developed in the last section to anumber of examples and determine their defect conformal anomalies exactly. N = 1 SCFTs
In 5d N = 1 SCFTs, the p = 4-dimensional superconformal defects appear either as half-BPS boundaries or interfaces. The defect U (1) R symmetry is identified with the Cartangenerator of the bulk SU (2) R symmetry, R = R d , (4.1)up to mixing with flavor symmetry currents on the defect. A half-BPS interface between two5d SCFTs T and T is related by the folding trick to a half-BPS boundary for the doubled See [61, 75] for a general classification of unitary superconformal defects based on the preserved (and T ⊗ ¯ T (where the second factor undergoes an orientation flip). For this reason, wewill focus on superconformal boundaries in 5d SCFTs. To determine the boundary conformal anomalies using our method requires the knowledgeof the boundary ’t Hooft anomalies involving the superconformal U (1) R symmetry. If therelevant DCFT admits a U (1) R preserving deformation to a free theory, one can hope todetermine the ’t Hooft anomalies from those of the free fields. In d = 5, such boundaryanomalies can come from bulk Dirac fermions (and complex two-forms). Let us consider a 5d Dirac fermion Ψ
Dirac on half space R , with a timelike boundary Σat y = 0, and suppose it has charge q under a U (1) global symmetry Ψ Dirac → e iqθ Ψ Dirac andalso transform in an irreducible representation ρ for an nonabelian global symmetry G . Thestandard U (1) × G -preserving boundary conditions for Ψ Dirac on Σ at y = 0 correspond to P ± Ψ Dirac | Σ = 0 , (4.2)where P ± ≡ (1 ± Γ y ) is a (anti)chiral projector on the boundary. The boundary (mixed)’t Hooft anomalies involving the U (1) symmetry are as summarized in Table 1.Fields Anomaly I d ψ d ρ q c ( F ) − d ρ qp ( T ) c ( F ) − T ρ qc ( F ) c ( F G ) + a ρ c ( F G )¯ ψ − d ρ q c ( F ) + d ρ qp ( T ) c ( F ) + T ρ qc ( F ) c ( F G ) − a ρ c ( F G )5 d P + Ψ Dirac | Σ = 0 − d ρ q c ( F ) + d ρ qp ( T ) c ( F ) + T ρ qc ( F ) c ( F G ) − a ρ c ( F G ) P − Ψ Dirac | Σ = 0 d ρ q c ( F ) − d ρ qp ( T ) c ( F ) − T ρ qc ( F ) c ( F G ) + a ρ c ( F G )Table 1: The ’t Hooft anomalies contributed by a 4d Weyl fermion ψ and its conjugate ¯ ψ ,and a 5d Dirac fermion Ψ Dirac with different boundary conditions. Both ψ and Ψ Dirac carrycharge q under the U (1) symmetry and transform in an irreducible representation ρ for thenonabelian G symmetry. The dimension, Dynkin index and cubic Casimir eigenvalue for ρ are denoted by d ρ , T ρ and a ρ respectively. The background gauge connections for U (1) × G are A and A G respectively. broken) symmetries. See earlier works [76, 77] for discussions of anomaly inflow to the boundary from bulk massless fermions,in the context of the E end-of-the-world brane in 11d supergravity. Here Γ y = i Γ Γ Γ Γ coincides with the standard 4d chirality matrix (see Appendix B for the spinorconventions). U (1) × G preserving mass deformation in thebulk S Dirac = (cid:90) R , d x ¯Ψ Dirac (Γ µ D µ − m )Ψ Dirac + 12 (cid:90) Σ ¯Ψ Dirac Γ y Ψ Dirac , (4.3)where ¯Ψ Dirac ≡ Ψ † Dirac i Γ as usual and the boundary term is necessary for the reality of theaction. Integrating out the massive Dirac fermion in the bulk generates a 5d Chern-Simonsterm for the background U (1) × G gauge fields A and A G , and Riemann curvature 2-form R in the effective action [79, 80], − sign( m )2 (cid:90) R , (cid:18) d ρ q π A ∧ F ∧ F + d ρ q π A ∧ tr( R ∧ R ) − T ρ q π A ∧ Tr ( F G ∧ F G ) + a ρ ( A G ) (cid:19) , (4.4)where the last term is the usual non-abelian Chern-Simons 5-form defined by d CS ( A G ) = − πc ( F G ) and the Chern-Simons level depends on the sign of m . In the above, d ρ isdimension of the representation ρ , T ρ is the Dynkin index and a ρ denotes the cubic Casimireigenvalue. For G = SU ( N ) and ρ = (cid:3) (the fundamental representation), d ρ = N , T ρ = and a ρ = 1.The Chern-Simons action (4.4) clearly contributes to the boundary ’t Hooft anomaliesthrough the inflow [83], but we also need to remember there maybe residual massless bound-ary modes from the massive 5d fermion. Indeed, the equation of motion(Γ y ∂ y + m + Γ a ∂ a )Ψ Dirac = 0 , (4.5)implies that a normalizable boundary massless mode Ψ Dirac ( y ) ∼ e −| m | y is possible ifΓ y Ψ Dirac | Σ = sign( m )Ψ Dirac | Σ , (4.6)whose contribution to the boundary ’t Hooft anomaly comes from the inflow ofsign( m ) (cid:90) R , (cid:18) d ρ q π A ∧ F ∧ F + d ρ q π A ∧ tr( R ∧ R ) − T ρ q π A ∧ Tr ( F G ∧ F G ) + a ρ ( A G ) (cid:19) , (4.7)as sign( m ) coincides with the chirality of the boundary massless fermion according to (4.6). See [78] for similar discussions of a 3d Dirac fermion on half space R . One may be cautious about the unquantized Chern-Simons level here as it would not be gauge invariantunder large gauge transformations. Here we emphasize that the relevant physical information is just containedin the infinitesimal gauge variation of (4.4) which is well defined, and can be verified, for example, by a directcalculation of the divergence of the 5d U (1) current using Feynman diagrams for the fermions satisfying theboundary conditions (4.2). Towards the end of this section, we will also provide other arguments that lead
26t is now straightforward to verify the entries in Table 1 based on the above. For example,with the boundary condition P + Ψ Dirac | Σ = 0, the full boundary anomaly is just given by(4.4) with the overall coefficient − since for m > m < P − Ψ Dirac | Σ = 0 and that of a 4d boundary chiral fermion ψ as follows. Let us putthe 5d Dirac fermion on a slab Σ × [0 , L ] with identical boundary conditions P − Ψ Dirac = 0at the two ends y = 0 and y = L . For small L and in the low energy limit, this is the sameas a chiral fermion on Σ. Thus the anomaly for a single boundary is half of that of a chiralfermion. Alternatively, starting with a boundary satisfying P + Ψ Dirac | Σ = 0, we can couplethe bulk fermion to a boundary chiral fermion ψ by (cid:82) Σ d z ¯Ψ Dirac P + ψ + ( c.c ). Integratingout ψ , this flips the boundary condition of the bulk Dirac fermion to P − Ψ Dirac | Σ = 0. Sincethe boundary conditions P ± Ψ Dirac | Σ = 0 have opposite anomalies by parity, we again reachthe same conclusion. We start by considering the simplest 5d SCFT defined by a free hypermultiplet which con-sists of four real scalars Φ iA and a symplectic-Majorana fermion Ψ A . The theory has an SU (2) R × SU (2) F symmetry and i = 1 , A = 1 , R , and consider superconformal boundary conditions onΣ at y = 0.With regard to the boundary N = 1 superconformal symmetry, the hypermultiplet splitsinto two chiral multiplets ( X, ψ X ) and ( Y, ψ Y ) on Σ, X = q , ψ X = P + Ψ , Y = q , ψ Y = P + Ψ . (4.9)The boundary superconformal U (1) R symmetry is identified the Cartan of SU (2) R symmetry to the same conclusions for these boundary anomalies. Nonetheless it is certainly desirable to understandthese anomalies for general bulk-defect coupled systems, from the modern perspective (see e.g. [81,82]) usinginvertible field theories in one higher dimension, with suitable generalizations. The symplectic-Majorana condition is (Ψ Aα ) ∗ = C αβ (cid:15) AB Ψ Bβ (4.8)where α, β = 1 , , , C αβ is the anti-symmetric 5d charge conjugation matrix.Consequently it captures the same independent degrees of freedom as a Dirac fermion Ψ Dirac ≡ Ψ .
27s in (4.1), under which the complex scalars (
X, Y ) carry charge +1 but the fermions ( ψ X , ψ Y )are uncharged.The simplest supersymmetric boundary conditions come from putting together the Neu-mann and Dirichlet boundary conditions for the scalars, and the standard U (1) R symmetricboundary conditions for the fermions (4.2) [84], B X [Φ] : Y | Σ = ψ Y | Σ = ∂ y ¯ X | Σ = 0 , B Y [Φ] : X | Σ = ψ X | Σ = ∂ y ¯ Y | Σ = 0 , (4.10)which amounts to setting one of the two chiral multiplets to zero identically. The correspond-ing boundary conformal anomalies are determined by the ’t Hooft anomalies as in (3.2). Thelatter can receive inflow contributions from charged fermions in the bulk as explained in thelast section. Here the boundary ’t Hooft anomalies vanish since the fermions are unchargedunder U (1) R . Consequently, the defect conformal anomalies all vanish, as in Table 2.In fact, the boundary conditions B X,Y [Φ] are special points on a CP conformal man-ifold of superconformal boundaries for a free 5d hypermultiplet, with the same vanishingdefect conformal anomalies. This boundary conformal manifold comes about from the bulk SU (2) f flavor symmetry acting on the boundary conditions (4.10) which preserves a U (1) f subgroup. Close to the B X [Φ] point, the SU (2) f rotation induces a marginal perturbation δ S B X [Φ] = ζ (cid:90) Σ d zd θX + ( c.c ) . (4.11)More general superconformal boundaries for the 5d hypermultiplet can be obtained by cou-pling B X [Φ] to a 4d N = 1 SCFT T d on Σ through a superpotential, B X [Φ] ⊕ T d with (cid:90) Σ d zd θX O d + ( c.c ) → B gen [Φ] , (4.12)where O d is a scalar chiral primary operator in T d of U (1) R charge R ( O d ) ≤ O d ) = R ( O d ) ≤ .If R ( O d ) = 1, the coupling in (4.12) is exactly marginal and the total boundary Note that in terms of the Dirac fermion Ψ we have ψ Y = ( P − Ψ ) ∗ . See Appendix B for further details. In general, flavor symmetry currents broken by a conformal defect give rise to exactly marginal couplingsfor the defect [75, 84]. This is because scalar chiral primary operators in 4d N = 1 SCFTs are absolutely protected if R < T d , a ( B gen [Φ]) = a ( T d ) , c ( B gen [Φ]) = c ( T d ) . (4.13)We emphasize that these are generally strongly coupled boundary conditions for the freehypermultiplet. In particular this includes the example of an E -invariant boundary condi-tion for 28 free hypermultiplets, obtained by an exactly marginal coupling to the 4d N = 1 SU (2) SQCD with N f = 4 on the boundary [84].If instead R ( O d ) <
1, the coupling between the boundary condition B X [Φ] and the SCFT T d is relevant and should flow to the new superconformal boundary B gen [Φ]. In the simplestscenario, T d is a free chiral multiplet φ of R-charge and we can take O d = φ . Then thesuperpotential deformation (cid:82) Σ d zd θφX simply imposes Dirichlet boundary condition on X while lifting the Dirichlet boundary condition on Y (by Y = φ ) and thus we have a boundaryRG flow [84] B X [Φ] ⊕ φ with (cid:90) Σ d zd θXφ + ( c.c ) → B Y [Φ] , (4.14)which is trivially consistent with the boundary a -theorem. We leave the investigation ofmore general boundary conditions for the free hypermultiplet that arise this way to futurework. From the above discussion, it should be clear that the zoo of interacting superconformalboundary conditions for the free hypermultiplets is rather rich, and it would be interestingto classify them from the bootstrap approach along the lines of [86].Finally let us comment on our results in relation to the free field defect conformal anoma-lies obtained in [25] from heat kernel computations [34]. The results for boundary a -anomaliesare tabulated in Table 2 and they are consistent with our findings. The fact that the Dirich-let and Neumann boundary conditions for a scalar Φ contribute opposite defect conformalanomalies is easy to understand. We take two scalars Φ and Φ on R , satisfying confor-mal Dirichlet and Neumann boundary conditions respectively, and then turn on an exactlymarginal coupling given by (cid:82) Σ d z Φ ∂ y Φ . After unfolding, this coupling identifies the twoscalars Φ and Φ living on R , and R , − respectively at Σ, and the original boundary corre-sponds to a transparent interface. Since the defect a -anomaly does not depend on marginalcouplings on the defect due to the Wess-Zumino consistency conditions [3, 4], the originalboundary must have vanishing total a -anomaly. Together with the difference a Neu − a Dir Note that for the relevant coupling in (4.12) to preserve a manifest U (1) R symmetry, the SCFT T d should have a U (1) flavor symmetry under which the operator O d is charged. a Neu and a Dir in [25].Following a similar argument for two fermions Ψ and Ψ with boundary conditions P + Ψ | Σ = P − Ψ | Σ = 0 and boundary marginal coupling (cid:82) Σ d z ¯Ψ Γ y P + Ψ , we conclude thatthe total a -anomaly again vanishes. Here the chiral and anti-chiral boundary conditionsare further related by a parity-reversal along Σ which does not affect the boundary a - or c -anomalies which are parity-even. Therefore the individual boundary a -anomalies for Ψ and Ψ must vanish, again consistent with the explicit computations in [25].The situation is less clear for the boundary conformal c -anomalies. The free field bound-ary c -anomalies have not been computed to the author’s knowledge. Nonetheless the fermioncases are restricted by the parity symmetry as in Table 2, and the vanishing c -anomaly forthe supersymmetric boundary B X [Φ] requires c Dir + c Neu + 2 c Ψ = 0 . (4.15)The precise values of the individual c -anomalies above should be accessible from the bulkstress-tensor two-point function in the presence of the boundary (see related discussions inlower dimensions in [87]). Fields a c d real scalar Weyl fermion photon chiral
148 124 vector
316 18 d ∂ y Φ | Σ = 0 c Neu Φ | Σ = 0 − c Dir P + Ψ | Σ = 0 0 c Ψ P − Ψ | Σ = 0 0 c Ψ B X,Y [Φ] 0 0Table 2: The conformal anomalies of 4d free fields and N = 1 supermultiplets, and theboundary conformal anomalies of 5d free fields and their supersymmetric completions.30 .1.3 Boundaries of E n SCFTs
Let us now discuss superconformal boundaries of interacting 5d N = 1 SCFTs. A par-ticularly well-studied set of examples known as the E n SCFTs for 0 ≤ n ≤ n ≥
1, the E n SCFT, upon a supersymmetricmass deformation, is described by an N = 1 SU (2) super-Yang-Mills theory coupled to n − E n SCFTs are expected to be their UV completions. The man-ifest global symmetry in the IR gauge theory is U (1) I × SO (2 n −
2) where the first factorcomes from the topological instanton current and the second factor is due to the funda-mental matter. This is enhanced to E n in the SCFT [88–90] (see also [92–97] for furtherevidences). The 5d N = 1 IR gauge theories have standard half-BPS boundary conditions preservingthe 4d N = 1 supersymmetry (see Appendix B for details). The hypermultiplet splits intotwo N = 1 chiral multiplets on Σ, and setting either to zero leads to the boundary conditions B X [Φ] and B Y [Φ] defined in the last section. Similarly the 5d vector multiplet V whichcontains a real scalar σ , a gauge field A µ and a gaugino λ iα , decomposes into one 4d N = 1vector multiplet v and one chiral multiplet φ of zero U (1) R charge. The supersymmetricNeumann and Dirichlet boundary conditions correspond to setting either v or φ to zero B N [ V ] : φ | Σ = 0 → F ya | Σ = P + λ | Σ = σ + iA y | Σ = 0 , B D [ V ] : v | Σ = 0 → A a | Σ = P − λ | Σ = D y σ | Σ = 0 . (4.16)Here we study the maximally symmetric boundary conditions for the IR gauge theories ofthe E n SCFTs coming from assigning B D [ V ] or B N [ V ] to the 5d SU (2) vector multiplet,and assigning B X [Φ] to all n − We expect them to bedescribed by certain strongly coupled superconformal boundary conditions for the SCFT,which we will define as B N [ E n ] and B D [ E n ] respectively. In the IR gauge theory, theseboundary conditions preserve the U (1) I × U ( n − ⊂ U (1) I × SO (2 n −
2) subgroup of thebulk symmetry. It would be interesting to understand the symmetry enhancement in the Here E n for n = 1 , , . . . , SU (2) , SU (2) × U (1) , SU (3) × SU (2) , SU (5) , SO (10) global symme-tries respectively. More general boundary conditions and interfaces for 5d N = 1 gauge theories were considered in [98],including a duality interface that maps one boundary to another while preserving the boundary ’t Hooftanomalies. At the fixed point, such a duality interface should correspond to a superconformal interface inthe 5d SCFT with vanishing a and c anomalies. This is strongly supported by a nontrivial superconformal index on S × HS which counts boundarylocal operators in protected representations of the boundary superconformal symmetry su (2 , |
1) [98]. B N [ E n ] and B D [ E n ] are matched by those of thegaugino λ in the IR gauge theory. Since λ has U (1) R charge − SU (2) gauge group, from Table 1, we find I ( B N [ V ]) = 14 c ( F ) − p ( T ) c ( F ) + c ( F ) c ( F G ) , I ( B D [ V ]) = − c ( F ) + 116 p ( T ) c ( F ) − c ( F ) c ( F G ) . (4.17)Note the mixed anomaly between U (1) R and the G = SU (2) gauge symmetry.For the Dirichlet boundary condition B D [ E n ], the bulk gauge symmetry SU (2) becomesan emergent global symmetry on the boundary, but it cannot mix with the U (1) R due to itsnonabelian nature. Consequently the ’t Hooft anomalies for the superconformal U (1) R canbe read off from (4.17), k RRR = k R = − , (4.18)and the boundary conformal anomalies follow from (3.2), a ( B D [ E n ]) = − , c ( B D [ E n ]) = − . (4.19)In the case of the Neumann boundary condition B N [ E n ], since the SU (2) gauge fieldsare dynamical on the boundary Σ, a mixed U (1) R - SU (2) anomaly would break the U (1) R symmetry explicitly. We can remedy this by introducing local degrees of freedom on theboundary. For example, we can couple the bare Neumann boundary condition B N [ E n ] to2 N f
4d chiral multiplets Q I that transform as doublets (with indices I = 1 ,
2) under the SU (2) gauge group, and denote the modified boundary condition as B N f N [ E n ]. These chiralmultiplets provide an additional U (2 N f ) flavor symmetry from locally conserved currentson Σ and the U (1) factor can mix with the U (1) R symmetry of the boundary. The su-perconformal U (1) R symmetry of B N f N [ E n ] is the unique combination that is free from a U (1) R - SU (2) anomaly (see Table 1), which requires assigning the following R-charge to thechiral multiplets, R ( Q I ) = 1 − N f . (4.20) The number of boundary fundamental chiral multiplets is chosen to be even to avoid the global Wittenanomaly [99]. Relatedly the SU (2) gauge theory in 5d has a discrete theta angle θ = 0 , π ( SU (2)) = Z . Here this theta angle is trivial θ = 0 for the E n theories. If θ = 1, there is a nontrivial inflow of theWitten anomaly to the boundary which must be cancelled for a Neumann type boundary condition (e.g. byintroducing one more fundamental chiral multiplet on the boundary). U (1) R ’t Hooft anomalies are k RRR = 32 − N f , k R = 32 − , (4.21)and the defect conformal anomalies are a ( B N f N [ E n ]) = 1532 − f , c ( B N f N [ E n ]) = 12 − N f . (4.22)Note that in the above we have assumed the absence of accidental U (1) symmetries that canalso mix with the U (1) R symmetry. One way to detect such phenomena is to check whetherunitarity bounds are obeyed by operators with the putative R-symmetry (4.20) [100, 101].Here the meson operator M = (cid:15) IJ Q I Q J is a gauge invariant scalar chiral primary operatoron the boundary whose conformal dimension is fixed by its R-charge,∆( M ) = 32 R ( M ) = 3 R ( Q ) = 3 − N f . (4.23)This is consistent with the 4d unitarity boundary ∆ ≥ N f ≥
2, which is a necessarycondition for our results (4.22) to be physically meaningful.Instead of adding fundamental 4d chiral multiplets on the boundary, one can also intro-duce other matter (or more generally a 4d N = 1 SCFT with U (1) global symmetries) tocancel the mixed U (1) R - SU (2) anomaly. In the presence of multiple U (1) symmetries freefrom this mixed anomaly, the boundary a -maximization procedure will be needed to pickout the superconformal U (1) R symmetry (see the next section for a simple example). Weleave this exercise to the interested readers. The N = 1 supersymmetric QCD (SQCD) is described by an N = 1 SU ( N ) super-Yang-Mills theory coupled to N f pairs of chiral multiplets ( Q I , ˜ Q I ) with I = 1 , , . . . , N transform-ing in the fundamental and anti-fundamental representations of SU ( N ). When the numberof flavors lie in the conformal window N ≤ N f ≤ N , the SQCD is expected to flow to a4d N = 1 SCFT [100, 101]. Here we shall describe a boundary analog of the SQCD theo-ries where the dynamical gauge field propagates in a 5d bulk, which provides candidates ofsuperconformal boundary conditions for the bulk SCFT in the UV.The relevant bulk theory is described by a 5d N = 1 SU ( N ) κ gauge theory with N ≥ κ . For 0 ≤ κ ≤ N , the UV completion is expected to be a 5d SCFT33 N,κ with U (1) I global symmetry and the IR gauge theory arises from a symmetric massdeformation coupled to the U (1) I current multiplet [91]. The U (1) I symmetry is realizedby the instanton current in the IR. Let us consider a supersymmetric boundary condition for T N,κ by assigning Neumann boundary condition B N [ V ] (see (4.16)) to the 5d SU ( N ) vectormultiplet (which emerge in the IR gauge theory description). The 5d gaugino contributesthe following boundary anomalies (from Table 1) I ( B N [ V ]) = N − c ( F ) − N − p ( T ) c ( F ) + N c ( F ) c ( F SU ( N ) ) + κ c ( F SU ( N ) ) . (4.24)Since the gauge fields are dynamical on the boundary Σ, we need to add additional 4d matterto cancel the SU ( N ) gauge anomalies as well as the mixed U (1) R - SU ( N ) anomalies. Oneway to achieve this is to couple B N [ V ] to N f chiral multiplets Q I and N f + κ chiral multi-plets ˜ Q I transforming in the fundamental and anti-fundamental representations of SU ( N )respectively. Note that a novelty compared to the 4d SQCD is unequal number of “quarks”and “anti-quarks” here, where the offset is due to the anomaly inflow from the 5d SU ( N ) κ Chern-Simons coupling. We refer to this boundary field theory as boundary SQCD and as-sume that it descends from a superconformal boundary condition B N f N [ T N,k ] for the 5d SCFT T N,k upon the supersymmetric U (1) I mass deformation. Below we will study the boundaryconformal anomalies for B N f N [ T N,k ].The boundary matter has U ( N f ) × U ( N f + κ ) global symmetry. In particular, it contains U (1) A axial and U (1) B baryon symmetries, familiar in the study of 4d SQCDs [100,101]. Wewill denote their generators by R A and R B respectively. The chiral multiplets have charges R A ( Q ) = R A ( ˜ Q ) = 1 , R B ( Q ) = − R B ( ˜ Q ) = 1 . (4.25)The U (1) R symmetry relevant for the superconformal boundary B N f N [ T N,k ], is generally acombination with parameters t A and t B , R t = R d + t A R A + t B R B (4.26)where R d is the R-symmetry inherited from the 5d bulk under which Q and ˜ Q are uncharged.In order for the corresponding R-current to be conserved in the presence of dynamical For special Chern-Simons level κ = N , the SCFT T N,κ develops an enhanced SU (2) flavor symmetryfrom instanton operators charged under U (1) I [95, 102]. U ( N ) gauge fields, we demand a vanishing mixed U (1) R - SU ( N ) anomaly, N N f − t A + t B ) + N f + κ − − t A + t B ) = 0 . (4.27)The ’t Hooft anomalies for the candidate U (1) R -symmetry follow from (4.24) and the bound-ary matter content, k R t R t R t = N −
12 + N f ( − t A + t B ) + ( N f + κ )( − − t A + t B ) ,k R t = N −
12 + N f ( − t A + t B ) + ( N f + κ )( − − t A + t B ) = N − N − . (4.28)Carrying out the boundary a -maximization subject to the constraint (4.27), we find that thetrial anomaly a ( t ) is maximized at t B = 0 , t A = 1 − Nκ + 2 N f , (4.29)and the boundary conformal anomalies are a ( B N f N [ T N,k ]) = 3( N + N − − N κ + 2 N f ) ,c ( B N f N [ T N,k ]) = 2 N + 5 N − − N κ + 2 N f ) . (4.30)Once again, unitarity bound on the boundary meson operator M = Q I ˜ Q I requires∆( M ) = 3 − Nκ + 2 N f ≥ , (4.31)thus we should choose N f such that 2 N f ≥ N − κ . (4.32)It would be interesting to understand the fate of the boundary SQCD beyond this range.We leave this to future investigation. Let us now discuss p = 4-dimensional superconformal defects in 6d SCFTs. In 6d N = (1 , N = 1 superconfor-35al symmetry. For 6d N = (2 ,
0) SCFTs, both half-BPS and quarter-BPS codimension-twodefects are present, preserving 4d N = 2 and N = 1 superconformal symmetries respec-tively. They play important roles in the class S construction of N = 2 SCFTs in fourdimensions [35, 37] as well as the N = 1 generalizations [103–105].Up to mixing with U (1) symmetries localized on the defect volume Σ, the U (1) R symme-try of the codimension-two defect is identified with the following combination of symmetrygenerators in the 6d N = (1 ,
0) superconformal algebra osp (6 ∗ | R = 23 (2 R d − M ⊥ ) , (4.33)where R d is the Cartan element of the 6d SU (2) R symmetry normalized to have integereigenvalues, and M ⊥ is the rotation generator in the transverse plane with eigenvalues ± when acting on spacetime spinors. In the free 6d SCFT described by a free N = (1 ,
0) hypermultiplet Φ with holomorphicscalars (
X, Y ) of scaling dimension ∆ = 2, a half-BPS superconformal codimension-twodefect can be defined by a scale invariant singularity of the form X ( x a , w ) ∼ α X w , Y ( x a , w ) ∼ α Y w , (4.34)where w is the complex coordinate for the transverse directions to the defect. The singularityis clearly invariant under the U (1) R symmetry (4.33). Similar defects can be defined in thefree N = (2 ,
0) SCFT using the hypermultiplet within the N = (2 ,
0) tensor multiplet.We note that the singularity (4.34) implies the existence of a dimension zero operator onthe defect worldvolume Σ that carries nontrivial spin under the transverse rotation. This issomewhat unconventional and indicates that the naive cluster decomposition fails on Σ [107].More generally, codimension-two defects in free theories can be classified by studyingboundary conditions for the conformally coupled free fields on
AdS × S with metric ds = R du + dz a u + R dθ , (4.35)which is related to flat space by a Weyl transformation. For free scalar fields this analysis This is an obvious generalization of the construction for 3d N = 4 hypermultiplet in [106] to higherdimensions. a -anomalies for the Dirichlet and Neumann bound-ary conditions were computed using the heat kernel method. Nontrivial superconformalcodimension-two defects in the free 6d SCFTs correspond to supersymmetric completions ofthese boundary conditions on AdS × S . In these cases, the conformal anomalies follow fromthe ’t Hooft anomalies as in (3.2), which can be determined by inflow from the Kaluza-Kleintower of fermions and two-forms (from the 6d tensor multiplet) upon reduction on S . Thissetup can also be extended to interacting 6d SCFTs (see for example [108]). We leave thestudy of such supersymmetric boundary conditions on AdS × S to future work. More interesting defects arise in interacting 6d SCFTs. Despite the lack of perturbativeLagrangians for such theories, the existence of various defects can be inferred by numer-ous constructions in string/M/F-theory, and by compactifying the 6d theory on compactmanifolds and reducing to lower dimensional theories where a Lagrangian can become avail-able. The most well-studied examples are half-BPS codimension-two defects in the 6d (2 , g [35, 37, 109]. The defects are characterized byhomomorphisms ϕ : su (2) → g , and so we will refer to them as D ϕ [ g ]. When the 6d SCFT iscompactified on a Riemann surface C with suitable twisting to preserve an su (2 , |
2) subal-gebra (which contains (3.1)). These codimension-two defects can be added without furtherbreaking the symmetry. They introduce punctures on the Riemann surface C and contributeintimately to various aspects of the resulting 4d theory. In particular, the codimension-twodefects are crucial to determining the ’t Hooft and conformal anomalies of the 4d SCFTs(see [109] for an extensive review). However a proper characterization of the conformalanomalies for defects was missing in these works, and the relations between the defect con-formal and ’t Hooft anomalies (3.2) were assumed. Furthermore the defect ’t Hooft anomalieswere mostly inferred from consistency checks within the class S construction, and a directderivation for the defect ’t Hooft anomalies was not available until recently [110–113].From the discussions in the previous sections, we now understand precisely what suchdefect anomalies mean in terms of the DCFT data (e.g. in (1.3)). They are physicallydifferent from the anomalies of standalone CFTs. For example the classes of anomalies aremuch richer and conventional unitarity constraints on the anomalies no longer hold ( a and The codimension-two defect also hosts nontrivial extrinsic conformal anomalies in addition to the a - and c -anomalies that depend on the extrinsic curvature. The conventional class S setup involves a direct productgeometry M = M × C for the 6d theory and consequently such extrinsic anomalies do not contribute.They will be important if we were to generalize the class S setup by including a nontrivial warp factor. can be negative in unitary DCFTs). Yet the defect anomalies still share many featuresthat we are familiar with in the case of standalone CFTs, such as a monotonicity a -theoremwhich we have proved in Section 2. Furthermore we have also established firmly the anomalymultiplet relation (3.2) and the a -maximization principle (see Theorem 3) for these selecteddefect anomalies with superconformal symmetry.In the following we will simply collect the recent results for defect ’t Hooft anomaliesfrom [113], and restate the results, which follow from (3.2), as the defect conformal anomaliesdefined in (1.3).We will focus on the case g = A N − for which the work of [113] applies. Here ρ isequivalent to a partition [ n i ] of N with N = n + · · · + n k and n i ≥ n i +1 >
0. The defect D [ n i ] [ A N − ] can be engineered by a single M5 brane intersecting N parallel M5 branes ina particular coincident limit. Alternatively, the same defect is described by N M5 branesprobing a Taub-NUT space TN k with k -centers that collide in a singular limit [35, 37, 114].Upon compactifying the 6d (2 ,
0) SCFT on T which gives rise to the 4d N = 4 super-Yang-Mills theory, this defect becomes a Gukov-Witten surface operator which has explicitLagrangian descriptions [115].The authors of [113] determined the defect ’t Hooft anomalies of D [ n i ] [ A N − ] from inflowin M-theory using the second description of the defect above. The results were given in adifferent parametrization of the anomaly polynomial I with k RRR = 227 ( n v − n h ) + 89 n v , k R = 23 ( n v − n h ) , (4.36)and for the defect D [ n i ] [ A N − ], ( n v − n h )([ n i ]) = 12 (cid:32) N − n (cid:88) i =1 s i (cid:33) ,n v ([ n i ]) = 16 N ( N + 1)(4 N − − n (cid:88) i =1 N − (cid:32) i (cid:88) j =1 s i (cid:33) . (4.37)Here [ s i ] with 1 ≤ i ≤ n is the dual (transpose) partition of [ n i ]. The defect a - and c -anomalies follow from (3.2), and coincide with their expected contributions to the 4d N = 2SCFT in the class S construction [109]. This comes from taking the difference between the “inflow” contribution and the “non-puncture” con-tribution from equations (6.2) in [113] and simplifying as explained therein. Note that with the enhanced N = 2 superconformal symmetry on the defect, a -maximization is trivial. D ϕ [ g ] given in [109], leading to the following expressions for the defectconformal anomalies, a ( D ϕ [ g ]) =2 ρ g · ρ g − ρ g · h + 548 dim g + 148 (rank g − dim g ) ,c ( D ϕ [ g ]) =2 ρ g · ρ g − ρ g · h + 112 dim g + 124 (rank g − dim g ) . (4.38)Here ρ g is the Weyl vector for g , h = ϕ ( σ ), and g is decomposed with respect to theeigenvalues of [ h, · ] as g = (cid:77) j ∈ Z g j . (4.39)To prove the formulas (4.38) for codimension-two defects in general (2 ,
0) SCFTs requires aderivation of the corresponding defect ’t Hooft anomalies, by extending the work of [113] tocases with an M-theory orientifold (for g = D n ), and by studying inflow in IIB string theorywith ADE singularities [116].Before ending this section, we note that beyond the family of the D ϕ [ g ] defects whichdefine regular (tame) punctures in the class S setup, the 6d (2 ,
0) SCFTs admit a much largerzoo of superconformal codimension-two defects that give rise to irregular (wild) punctureswhere the superconformal symmetry is emergent in the IR [37,117–122], as well as the twisteddefects (punctures) which are attached to codimension-one topological defects generatingthe outer-automorphism symmetry of certain (2 ,
0) theories [109, 123–129]. More recently,codimension-two defects in 6d N = (1 ,
0) SCFTs including the E-string theory have alsobeen analyzed [130–135]. The results about their contributions to the conformal anomaliesof the 4d SCFT in a generalized class S setup should again be interpreted as defect conformalanomalies in the sense explained here.In complementary to the rich landscape of examples we have for codimension-two defectsin 6d SCFTs, it would be interesting to understand and identify universal bounds on theirphysical data, much like what we have done in the case of standalone 4d SCFTs, using theconformal bootstrap approach (see [136] for a review). For example, one may wonder if thereis notion of minimal defect that minimizes certain ’t Hooft or conformal anomalies in a givenbulk SCFT. Since such defects can be used to engineer 4d SCFTs upon compactification,this information will also be relevant for the search of minimal 4d SCFTs that have beenexplored in [137–141]. 39
Discussions
In this paper, we have analyzed the anomalies of conformal defects (or DCFTs) of dimen-sion p = 4 in d -dimensional CFTs. We proved a defect analog of the 4d a -theorem whichstates that the defect conformal a -anomaly must decrease along unitary defect RG flowsconnecting UV and IR DCFTs. For conformal defects that preserve the minimal amount ofsupersymmetry, we established the anomaly multiplet relations between defect conformal a -and c -anomalies, and the ’t Hooft anomalies involving the superconformal U (1) R symmetry.The general ’t Hooft anomalies are determined by inflow from the bulk CFT, and the U (1) R symmetry is identified by the defect a -maximization principle which we have also derived.Together they provide a non-perturbative pathway to the conformal anomalies of stronglycoupled defects. To illustrate our methods, we examined a number of examples of defectsin 5d and 6d SCFTs. Here we conclude by discussing a few future directions beyond thosementioned in the main text. Defect correlation functions and defect chiral algebras
Conformal symmetry places stringent constraints on the correlation functions of local oper-ators. In conventional CFTs in dimension d ≥
4, the two- and three-point functions of thestress-tensor T ab is completely fixed by conformal symmetry and Ward identities, up to threeconstants [74], (cid:104) T ab ( z ) T cd (0) (cid:105) = c (cid:104)(cid:104) T ab ( z ) T cd (0) (cid:105)(cid:105) , (cid:104) T ab ( z ) T cd ( z ) T ef (0) (cid:105) = c (cid:104)(cid:104) T ab ( z ) T cd ( z ) T ef (0) (cid:105)(cid:105) c + a (cid:104)(cid:104) T ab ( z ) T cd ( z ) T ef (0) (cid:105)(cid:105) a + b (cid:104)(cid:104) T ab ( z ) T cd ( z ) T ef (0) (cid:105)(cid:105) b . (5.1)Here (cid:104)(cid:104)·(cid:105)(cid:105) denotes theory independent conformal structures. For d = 4, the coefficients a and c are nothing but the conformal anomalies defined in (1.1) for a 4d CFT.In the presence of a p -dimensional conformal defect D , the correlators of the bulk stress-tensor T µν are constrained by the SO ( p,
2) conformal symmetry and the d -dimensional Wardidentities. Following the logic of [74], we expect for p = 4, the defect c -anomaly to bedetermined by the defect two-point function (cid:104) T µν ( x ) T ρσ ( x ) (cid:105) D , and the defect a -anomaly bythe three-point function (cid:104) T µν ( x ) T ρσ ( x ) T λζ ( x ) (cid:105) D . However because of the extra transversedirections, there are now additional conformally invariant tensor structures and furthermoretheir coefficients are general functions of invariant cross-ratios. The structure of the defecttwo-point function (cid:104) T µν ( x ) T ρσ ( x ) (cid:105) D has been worked in [142–145]. If D is a conformal40oundary (i.e. d = p + 1), this is determined by a single function f ( ξ ) of the invariantcross-ratio ξ , ξ ≡ ( x − x ) y y . (5.2)For p = 4, the conformal c -anomaly should be determined by (a limit of) f ( ξ ) but theexplicit relation is still to be derived. For defects of higher codimensions, there is one moreindependent cross-ratio ξ (cid:48) ≡ y · y | y || y | , (5.3)and the number of independent tensor structures is two for d = p + 2 and seven for d − p > (cid:104) T µν ( x ) T ρσ ( x ) T λζ ( x ) (cid:105) D is much more complicated,with six cross-ratios in general and many tensor structures [147].Supersymmetry are known to produce new (differential) constraints on these tensor struc-tures. It would be interesting to explore the structure of stress-tensor multiplet correlationfunctions for superconformal defects. Furthermore, when sufficient supersymmetry is pre-served, it is possible to define a simpler but nontrivial subsector of the full operator algebrain the DCFT that is closed under OPE. In particular, in the case of a half-BPS codimension-two superconformal defect in the 6d (2 ,
0) SCFT, the chiral algebra defined in [148] has anatural extension to defect operators with respect to the su (2 , |
2) defect superconformalsymmetry. The resulting chiral algebras will be of a different nature. For example the ab-sence of a local stress-tensor multiplet on the defect implies that the corresponding chiralalgebra no longer contains a Virasoro subalgebra. Furthermore the bulk operators in the 6d(2 ,
0) SCFT also contain a chiral algebra subsector defined with respect to a different D (2 , osp (8 ∗ |
4) [149]. The interplay between theseprotected subsectors of bulk and defect operators will provide a wealth of information in the N = 2 supersymmetric DCFTs. We hope to report on this in the future. Bounds on defect conformal anomalies
As we have emphasized in the main text, despite sharing many features of the conformalanomalies of standalone CFTs, the defect conformal anomalies are physically distinct. Inparticular, in a unitary DCFT, both the a - and c -anomalies can be negative, and no lowerbounds have been identified. In light of these observations, perhaps we should look forbounds on the ratio ac of the defect conformal anomalies.For conventional CFTs, such bounds arise naturally in studying positivity constraints of One such relation was proposed in [87] but a counter-example appeared in [146]. ≥ ac ≥ . (5.4)If the CFT is superconformal, a stronger bound is achieved, depending on the amount ofsupersymmetry preserved, N = 1 : 32 ≥ ac ≥ , N = 2 : 54 ≥ ac ≥ . (5.5)In all cases, the upper and lower bounds are saturated by free vector and scalar theoriesrespectively, with appropriate supersymmetric completions. It would be very interesting toexplore a generalization of the collider bounds in [150] to cases with conformal boundaries ormore general conformal defects, by studying positivity constraints on energy correlators inthe presence of defect excitations. We emphasize that the simplest superconformal bound-ary condition for a 5d hypermultiplet has a vanishing defect c -anomaly (see Section 4.1.2).Therefore in order for such bounds to exist in d = 5, extra restrictions on the defect (bound-ary) need to be imposed. We have not observed similar issues in d = 6. Acknowledgements
The author thanks Nathan Agmon for collaborations on related topics. The author is alsograteful to Zohar Komargodski, Ken Intriligator and Yuji Tachikawa for reading and com-menting on a draft of the manuscript. The work of YW is supported in part by the Centerfor Mathematical Sciences and Applications and the Center for the Fundamental Laws ofNature at Harvard University.
A Boundary four-point amplitude in free scalar theory
Here we study the large m expansion of the one-loop Feynman diagram that computes thefour-point function of Φ ( z,
0) on the Neumann boundary of a free scalar field in d = 5, I = (cid:90) d k (2 π ) | k | + m )( | k + p | + m )( | k + p + p | + m )( | k − p | + m ) . (A.1)42e start by introducing Schwinger parameters for the propagators, I = (cid:90) d k (2 π ) (cid:90) (cid:89) i =1 ds i e − m (cid:80) i s i e − ( s | k | + s | k + p | + s | k + p + p | + s | k − p | ) . (A.2)Next we use the Laplace transform (cid:90) ∞ dt √ π se − s t e − tk t / = e − s | k | , (A.3)and obtain I = (cid:90) d k (2 π ) (cid:90) (cid:89) i =1 ds i e − m (cid:80) i s i (cid:90) (cid:89) i =1 dt i √ π e − ( t k + t | k + p | + t | k + p + p | + t | k − p | ) 4 (cid:89) i =1 s i e − s i ti t / i . (A.4)Performing the k integral, this gives I = 12 π (cid:90) (cid:89) i =1 ds i e − m (cid:80) i s i (cid:90) (cid:89) i =1 dt i (cid:89) i =1 s i e − s i ti t / i π ( (cid:80) i t i ) e t t s + t t t (cid:80) i ti , (A.5)after Wick rotating to Minkowski signature and imposing the “on-shell” condition p i = 0.Let us now expand I in the large m limit. We are particularly interested in thefour-derivative term which takes the following form as is clear from the symmetry of (A.5), I ⊃ α ( s + t ) + α stm . (A.6)Expanding the last exponential factor in (A.5), we obtain α = 12(4 π ) (cid:90) (cid:89) i =1 ds i e − (cid:80) i s i (cid:90) (cid:89) i =1 dt i (cid:89) i =1 s i e − s i ti t / i ( t t ) ( (cid:80) i t i ) . (A.7)Performing a change of variables s i → t i s i , this becomes α = 12(4 π ) (cid:90) ∞ (cid:89) i =1 ds i s i (cid:90) ∞ (cid:89) i =1 dt i (cid:89) i =1 e − ti ( s i +4 s i ) t i ( t t ) ( (cid:80) i t i ) . (A.8)The t i integral can be simplified using the following integration identity, from in [152]43we have corrected a typo there), (cid:90) ∞ n (cid:89) i =1 dt i e − (cid:80) i q i t i (cid:81) i t p i − i ( (cid:80) i t i ) r = (cid:81) i Γ( p i )Γ( r ) (cid:90) ∞ dx x r − (cid:81) i ( x + q i ) p i , (A.9)with q i , p i , r > (cid:80) i p i > r .Applying (A.9) to (A.8), we find α = 12(4 π ) (cid:90) ∞ (cid:89) i =1 ds i s i (cid:90) ∞ dx Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) Γ(4) x (cid:81) i =1 ( s i + s i + x ) (cid:81) i =3 ( s i + s i + x ) . (A.10)This last integral can be evaluated in Mathematica by first integrating s i and then x , giving α = 61645120 π . (A.11)A similar computation also determines α , α = 1107520 π . (A.12)Combining with the contributions from the other two one-loop diagrams I and I , wefind that the full four-point amplitude at the fourth derivative order is given by I + I + I ⊃ π s + t + u m . (A.13) B Supersymmetric boundaries for 5d N = 1 gauge the-ories B.1 5d Spinor conventions
The 5d Gamma matrices Γ µ satisfy the Clifford algebra(Γ µ ) αγ (Γ ν ) γβ + (Γ ν ) αγ (Γ µ ) γβ = 2 η µν δ αβ , (B.1)where α, β = 1 , , , tµ and conjugate Γ ∗ µ obey the same algebra and are related to Γ µ byΓ tµ = Cγ µ C − , − Γ ∗ µ = B Γ µ B − . (B.2)44ere C and B are charge conjugation matrices related by C = B Γ and satisfy C αβ = − C βα , ( C − ) αβ = − ( C ∗ ) αβ ≡ C αβ . (B.3)The symplectic-Majorana (SM) condition on a 5d spinor reads(Ψ αA ) ∗ = (cid:15) AB C αβ Ψ βB , (B.4)with the convention (cid:15) = (cid:15) = 1. B.2 Hypermultiplet
A 5d N = 1 hypermultiplet consists of four real scalars Φ iA , a SM fermion Ψ A and four auxil-iary real scalars F iA . Here i = 1 , A = 1 , SU (2) R and SU (2) F doublet indicesrespectively. These indices are lowered and raised by the invariant tensors (cid:15) ij , (cid:15) ij , (cid:15) AB , (cid:15) AB satisfying (cid:15) ij (cid:15) jk = δ ik and (cid:15) AB (cid:15) BC = δ AC . The fields are subject to the reality conditions(Φ iA ) ∗ = (cid:15) AB (cid:15) IJ Φ jB , (Ψ Aα ) ∗ = (cid:15) AB C αβ Ψ Bβ , ( F iA ) ∗ = (cid:15) AB (cid:15) IJ F jB . (B.5)The on-shell supersymmetry transformations are δ ξ Φ iA = − iξ i Ψ A , δ ξ Ψ A = Γ µ ξ i ∂ µ Φ iA , (B.6)where ξ iα is a SM spinor corresponding to the eight supercharges.We identity the 4d N = 1 superalgebra by the following projection ξ = P + ξ , (B.7)where P ± ≡ (1 ± Γ y ). Then it’s clear from (B.6) that the hypermultiplet splits into twochiral multiplets closed under δ ξ separately, X = (Φ , P + Ψ , ∂ y Φ ) , Y = (Φ , P + Ψ , ∂ y Φ ) . (B.8)Note that ∂ y Φ and ∂ y Φ coincide with the on-shell auxiliary field F and F respectively.This comes from the effective 4d superpotential (cid:82) dy (cid:82) d θX∂ y Y from the 5d Lagrangian ofa hypermultiplet on R , [84].The consistent boundary conditions preserving boundary N = 1 supersymmetry amounts45o setting a linear combination of X and Y to zero identically. They define the B X [Φ] and B Y [Φ] boundary conditions (and their rotated versions) in Section 4.1.2. B.3 Vector multiplet
A 5d N = 1 vector multiplet V contains a real scalar σ , a SM fermion λ iα , a gauge field A µ and three auxiliary scalars D ij with i, j indices symmetrized and satisfying( D ij ) ∗ = D ij = (cid:15) ik (cid:15) jl D kl . (B.9)The supercharges act on the vector multiplet fields as δ ξ A µ = iξ i γ µ λ i ,δ ξ σ = iξ i λ i ,δ ξ λ i = −
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