SSurface Defect, Anomalies and b -Extremization Yifan Wang , Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
Quantum field theories (QFT) in the presence of defects exhibit new types of anomalieswhich play an important role in constraining the defect dynamics and defect renormaliza-tion group (RG) flows. Here we study surface defects and their anomalies in conformal fieldtheories (CFT) of general spacetime dimensions. When the defect is conformal, it is charac-terized by a conformal b -anomaly analogous to the c -anomaly of 2d CFTs. The b -theoremstates that b must monotonically decrease under defect RG flows and was proven by cou-pling to a spurious defect dilaton. We revisit the proof by deriving explicitly the dilatoneffective action for defect RG flow in the free scalar theory. For conformal surface defectspreserving N = (0 ,
2) supersymmetry, we prove a universal relation between the b -anomalyand the ’t Hooft anomaly for the U (1) r symmetry. We also establish the b -extremizationprinciple that identifies the superconformal U (1) r symmetry from N = (0 ,
2) preserving RGflows. Together they provide a powerful tool to extract the b -anomaly of strongly coupledsurface defects. To illustrate our method, we determine the b -anomalies for a number ofsurface defects in 3d, 4d and 6d SCFTs. We also comment on manifestations of these defectconformal and ’t Hooft anomalies in defect correlation functions. a r X i v : . [ h e p - t h ] D ec ontents b -theorem and dilaton effective action 53 Defect anomalies and supersymmetry 8 b -anomaly from ’t Hooft anomalies and SUSY . . . . . . . . . . . 103.3 Gravitational anomalies and Chern-Simons counterterms . . . . . . . . . . . 113.4 Defect RG flow and b -extremization . . . . . . . . . . . . . . . . . . . . . . . 12 N = (0 , surface defects and anomalies 16 N = 2 Ising SCFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 N = 2 SQED and mirror symmetry . . . . . . . . . . . . . . . . . . . 204.2 Surface defects in 4d SCFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Surface defects in 6d SCFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Introduction and setup
A fundamental problem in the study of quantum field theories (QFT) is to uncover non-perturbative constraints on renormalization group (RG) flows. A powerful tool to tacklethis problem comes from anomalies for the relevant symmetries. In particular, the ’t Hooftanomaly for a symmetry G must be constant along G -symmetric RG flows, leading to non-trivial constraints on the low energy dynamics, known as the anomaly matching condition [1].The ’t Hooft anomalies can often be extracted without knowing the details of the QFT dy-namics, thanks to the robustness under small deformations of the theory, providing a quickand valuable diagnostic on candidate RG flows and infra-red (IR) phase diagrams from agiven ultra-violet (UV) description.Much subtler are the anomalies for accidental symmetries that only emerge at ends ofthe RG flow, the epitome of which is the conformal symmetry at RG fixed points and theassociated conformal anomalies in even spacetime dimensions. The hallmark of conformalsymmetry is the existence of a locally conserved, symmetric and traceless stress tensor T µν ,which gives rise to a conformal field theory (CFT). Importantly the traceless condition canbe violated by contact terms, or equivalently by certain curvature terms when the theory isplaced on curved manifolds, that solve the associated Wess-Zumino consistency conditions[2], (cid:104) T µµ (cid:105) = − ( − d/ aE d + (cid:88) i c i W i , (1.1)where E d is the Euler class in d dimensions normalized such that (cid:82) M E d = χ ( M ) with χ ( S d ) = 2 for an even dimensional sphere, and W i are Weyl invariants of the curvature.Equivalently, the trace anomaly contributes to the anomalous variation of the path integralunder the Weyl transformation g → ge σ where g is the metric on the spacetime manifold M , δ σ log Z [ g ] = (cid:90) M d d x √ g σ (cid:32) − ( − d/ aE d + (cid:88) i c i W i (cid:33) . (1.2)The coefficients a and c i are the conformal anomaly coefficients. Unlike the usual ’t Hooftanomalies, the conformal anomalies are only defined at the end points of RG flows anddo not match since conformal symmetry is broken along the flow. Instead they give riseto powerful inequality constraints on RG flows. Indeed it has long been expected that acombination of the conformal anomalies measures degrees of freedom in the CFT and thusshould decrease under RG flow which is intuitively an irreversible coarse-graining procedurethat produces IR dynamics from UV descriptions. For d = 2, the answer is affirmative as2hown in [3] where the only conformal anomaly coefficient is a = c d . In fact there exists alocally defined c -function on the space of 2d QFTs which coincides the conformal anomaliesat the fixed points and monotonically decreases along all RG flows, proving the so-called c -theorem which establishes the irreversibility of RG flows for d = 2 [3]. Shortly after itwas conjectured in [4] that the a -anomaly in (1.1) is the appropriate generalization of c d to even d ≥ a -theorem was postulated. It was only until recentlythe a -theorem was proven in d = 4 [5–7], and some progress has been made in extendingthe proof of [5, 6] to d = 6 with success in special cases [8–10]. Notably, by introducing adilaton to compensate for the broken conformal symmetry along the RG flow, the authorsof [5, 6] recasted the problem into one of the ’t Hooft anomaly matching type, and deducedthe a -theorem based on unitarity constraints on the dilaton effective action.The studies of QFTs are greatly enriched by the incorporation of defects, which are ubiq-uitous in nature and can come from boundaries, interfaces or higher-codimension impuritiesin quantum systems, and are just starting to be systematically explored. Local deformationson the defect worldvolume give rise to defect RG flows that produce an array of new defectcritical phenomena. A natural task is to identify the new defect anomalies that constrainthese defect RG flows and the defect phase diagram. The usual ’t Hooft anomalies havestraightforward generalizations for defect symmetries. The trace of the stress tensor can alsoreceive new anomaly contributions localized on the defect worldvolume at the critical point.Here we focus on the case of two dimensional defect, which can be a boundary or interfacein d = 3 or more generally a surface defect in higher dimensions. In this case, the most generaldefect ’t Hooft anomalies for continuous symmetries take the following form. Let us denotethe surface defect by D and its worldvolume submanifold by Σ. As we vary the full partitionfunction Z D [ A, e ] in the presence of a background gauge connection A for the defect globalsymmetry G as well as the vielbein and spin connection ( e, ω ) by gauge transformations δ λ A = dλ and δ θ ω = dθ + [ ω, θ ], we have δ θ,λ log Z D [ A, e ] ⊃ i (cid:90) Σ ( κ tr( λF ) + κ θR + κ tr( λ ) R + κ θ tr( F )) , (1.3)where F and R are the corresponding curvature two-forms. In the above, we have onlyincluded the anomalous variations intrinsic to the defect, which does not include the bulkanomalies that may be present without the defect. The defect ’t Hooft anomalies are capturedby the coefficients κ i (the last two in (1.3) correspond to mixed anomalies which are possibleif G contains abelian factors). Like their bulk counterparts, these defect ’t Hooft anomaliescan often be obtained by deforming the coupled system to a weakly interacting description3hanks to the topological nature of these anomalies.At the critical point of the bulk-defect coupled system, we have a conformal surface defectin the ambient CFT, and the full theory is sometimes referred to as defect CFT (DCFT). Aflat defect D preserves the so (2 ,
2) conformal subalgebra and is thus expected to share manyfeatures of 2d CFTs. However a crucial difference between the conformal defect and a localCFT is the generic absence of a locally conserved stress tensor on the defect. More generally,symmetries on D can, but do not necessarily, lead to locally conserved currents. This is ofcourse consistent due to the mild non-locality on the defect worldvolume. Thus apart fromthe familiar anomalies for 2d CFTs, we have the following anomalous contributions to theWard identities of bulk conserved currents. For an abelian global symmetry current J fµ inthe bulk, such defect anomalies take the form (cid:104)∇ µ J µf (cid:105) D ⊃ π δ (Σ) (cid:63) Σ ( ik ff F f + (cid:88) I (cid:54) = r ik fI F I + k fg R ) , (1.4)where F r and F I are curvatures for background abelian gauge fields. The coefficient k ff characterizes the pure anomaly associated to the symmetry of J rµ , while k fI and k fg aremixed anomalies with other abelian flavor and Lorentz symmetries. As for the stress tensor T µν , there is potentially a defect gravitational anomaly (cid:104)∇ µ T µa (cid:105) D ⊃ k g π δ (Σ) (cid:15) ab ∇ b R Σ , (1.5)and a defect conformal anomaly whose general form is given by [11–13] (cid:104) T µµ (cid:105) D ⊃ π δ (Σ) (cid:18) bR Σ + d ( K iab K abi − K i K i ) − d W abcd h ac h bd (cid:19) . (1.6)Here we have split the spacetime coordinates into directions tangential and normal to Σ as x µ = ( x a , y i ). We denote the induced metric on Σ by h ab , the Ricci curvature scalar on Σby R Σ , the extrinsic curvature by K iab , and the pull back of the bulk Weyl curvature to Σby W abcd . Correspondingly the Weyl variation of the path integral δ σ log Z D [ g ], compared to(1.2), receives defect localized contributions A Weyl D ≡ π (cid:90) Σ d x √ hσ (cid:18) bR Σ + d ( K iab K abi − K i K i ) − d W abcd h ac h bd (cid:19) . (1.7)In this paper, we will be mostly interested in the conformal (gravitational) anomalies b Here and below we only include anomalous contributions that purely depend on the metric, as the mixed k g . To make connection to known results about 2d CFTs, we define c L ≡ b − k g , c R ≡ b + k g , (1.8)which coincide with the chiral central charges of a local 2d CFT in the degenerate case wherethe bulk CFT is trivial and the stress tensor is only supported on the defect. Under defectRG flows, the gravitational anomaly k g stays constant. On the other hand the b -anomalyplays the role of the c -function for surface defects and the b -theorem states that b decreasesmonotonically under defect RG flows. The b -theorem is proven in [14] by employing a defectversion of the dilaton effective action method of [5, 6]. In Section 2, we revisit the defect b -theorem and present an explicit derivation of the dilaton effective action for the bound-ary RG flow of a free scalar field in d = 3, which also provides a check on the boundary b -anomalies obtained from heat kernel methods [15]. However a generic surface defect doesnot have Lagrangian descriptions and it has remained challenging to extract the b -anomaliesfor interacting defects, which we will overcome in this work. As we explain in Section 3, theconformal anomalies of a large class of interacting surface defects can be derived in termsof their ’t Hooft anomalies which are much easier to compute. This is made possible bythe presence of N = (0 ,
2) supersymmetry on the defect worldvolume Σ. The method relieson an defect version of the c -extremization principle [16, 17] which we prove here. We em-phasize that our results are completely non-perturbative and do not rely on any Lagrangiandescriptions. In Section 4, we apply our methods to selected examples of superconformalsurface defects in various spacetime dimensions. It is straightforward to reproduce recentresults of b anomalies based on large N holography [18–21] and supersymmetric localizationwhen the defect has enhanced supersymmetries [22]. We also describe examples for whichthe previous methods do not apply, such as the boundary b -anomalies of 3d N = 2 Ising andSQED SCFTs. We end with a short summary and discuss future directions in Section 5. b -theorem and dilaton effective action We start by reviewing the proof of the defect b -theorem [14] using the spurious dilaton [5, 6].Let us consider a defect RG flow from a UV conformal defect D . We formally restore thedefect conformal symmetry by introducing a non-dynamical dilaton field τ localized on thedefect worldvolume Σ, which shifts τ → τ + σ under local Weyl rescaling g µν → e σ g µν .Consequently the conformal (Weyl) anomaly remains constant for the full system along the anomalies are already captured by (1.4).
5G flow. In the IR, the effective action takes the following form, S eff = S D IR + S τ + . . . (2.1)where S D IR abstractly describes the IR DCFT, S τ is the dilaton effective action, and wehave omitted coupling between the IR DCFT and τ which is suppressed in the derivativeexpansion (higher than second order) [14]. Anomaly matching then demands δ σ log Z D = δ σ log Z D IR − δ σ S τ , (2.2)which fixes the form of S τ up to terms that are diffeomorphism and Weyl invariant [6, 14].In particular for a flat defect in flat space, we simply have S τ = ∆ b π (cid:90) Σ d x ∂ a τ ∂ a τ + (cid:90) Σ d x Λ e − τ . (2.3)where ∆ b ≡ b UV − b IR . On the other hand, since by construction τ couples to the defectlocal operator T in T µµ = δ (Σ) T for a flat defect, we have [6, 14]∆ b = 3 π (cid:90) Σ d x | x | (cid:104)T ( x ) T (0) (cid:105) (2.4)and unitarity (reflection positivity) requires ∆ b > b -anomalies from heat kernel methods at the defectfixed points.We start with the Euclidean free scalar action S = 12 (cid:90) y ≥ d x∂ µ Φ ∂ µ Φ , (2.5)on R with Neumann boundary condition ∂ Φ | Σ = 0 . (2.6)6e then turn on a relevant perturbation in the form of a boundary mass term∆ S = − (cid:90) d x m Φ . (2.7)This deforms the boundary condition to the mixed type ∂ Φ − m Φ | Σ = 0 , (2.8)which interpolates between the Neumann boundary condition at m = 0 and Dirichlet bound-ary condition Φ | Σ = 0 as m → ∞ .Now we introduce the defect dilaton τ to restore the defect conformal symmetry, S tot = 12 (cid:90) y ≥ d x∂ µ Φ ∂ µ Φ − (cid:90) Σ d x me τ Φ (2.9)and the goal is to derive the effective coupling for τ from integrating out Φ. Since theaction is quadratic, this can be easily accomplished by Wick contractions once we have thetwo-point function (propagator), G ( (cid:126)x, (cid:126)x (cid:48) , y, y (cid:48) ) ≡ (cid:104) Φ( (cid:126)x, y )Φ( (cid:126)x (cid:48) , y (cid:48) ) (cid:105) , (2.10)with the general boundary condition ∂ y Φ( (cid:126)x, y ) − m Φ( (cid:126)x, y ) | y → + = 0 . (2.11)The solution is given by a m -dependent linear combination of the free scalar propagator (inthe absence of boundary) between x and x (cid:48) , and that between its mirror image ¯ x ≡ ( x, − y )and x (cid:48) . Here we find it convenient to perform a Fourier transform in the (cid:126)x coordinates, andthe desired two-point function is the following linear combinationˆ 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 | (2.12)up to a prefactor that impose momentum conservation in the boundary directions. It satisfies( ∂ y − m ) ˆ G ( p, − p, y, y (cid:48) ) (cid:12)(cid:12)(cid:12) y → + = 0 (2.13)as desired. The effective action for τ can be determined by computing the boundary corre-7ators of Φ . For this purpose, we simply needˆ G ( p, − p, ,
0) = 1 | p | + m , (2.14)and the one-loop Feynman diagram in a large m expansion (cid:90) d xe i(cid:126)p · (cid:126)x (cid:104) Φ ( (cid:126)x, (0 , (cid:105) c = 2 (cid:90) d k (2 π ) | k | + m )( | k + p | + m ) = α + α p m + O (cid:18) p m (cid:19) . (2.15)In particular, the two-derivative term in the dilaton effective action is − α (cid:90) d x ∂ a τ ∂ a τ . (2.16)Explicit computation (see Appendix A) gives α = − π . (2.17)Compared to (2.3), we conclude for the boundary RG flow of a free 3d scalar field∆ b = 18 (2.18)This is indeed consistent with the fixed point values of the b -anomalies computed from heatkernel methods [14] b Dir = − , b Neu = 116 . (2.19)In comparison, a real 2d scalar on Σ contributes b = 1. Let us now consider unitary superconformal surface defects preserving N = (0 ,
2) supersym-metry. According to the classification of [23], such defects can exist in 3d
N ≥
2, 4d
N ≥
N ≥ ,
0) SCFTs. Familiar examples include boundaries and interfaces in 3d, as wellas surface operators in 4d and 6d SCFTs, sometimes preserving an enhanced superconformalsymmetry (see Section 4 for detailed examples). In all cases, the defect preserves a U (1) r symmetry of the bulk SCFT, which is identified with the R -symmetry of the N = (0 , sl (2 , R ) ⊕ osp (2 | , R ) . (3.1)8n this section, we will derive the following universal relation between the defect b -anomalyand the ’t Hooft anomaly of U (1) r given by k , b + k g k , (3.2)or equivalently c L = 3 k − k g , c R = 3 k . (3.3)Note that for standalone 2d N = (0 ,
2) SCFTs, these relations are automatically satisfied.In those case the anomaly coefficients appear in the OPE of 2d local stress tensor and R -symmetry currents, and (3.3) is a simple consequence of the SUSY Ward identities. Here wewill show that the same relation persists for defect anomaly coefficients, even when such 2dlocal conservation laws are no longer present. We start by reviewing the ’t Hooft anomalies of a general surface defect D (not necessarilyconformal). For the anomalies associated to diffeomorphism and abelian symmetries whichcouple to background gauge fields A I , the corresponding anomaly polynomial is given by, I = − k g p ( T ) + 12 k IJ c ( F I ) c ( F J ) = k g π tr R ∧ R − k IJ π F I ∧ F J (3.4)which determines the anomalous variation of log Z D by the usual descent procedure. Equiv-alently, the defect anomalies are produced by the inflow from the CS action S CS = i π (cid:90) M (cid:0) πk g CS g − k IJ A I dF J (cid:1) (3.5)to its boundary Σ = ∂ M . Here CS g is the gravitational Chern-Simons term satisfying d CS g = 1192 π tr R ∧ R . (3.6)Note that CS g is defined up to a shift by an exact 3-form, which corresponds to the Bardeen-Zumino counter-term on Σ which shifts between anomalies for diffeomorphism and Lorentzsymmetries [24].To ease subsequent comparisons to known results in the literature, our normalization issuch that for a complex right-moving Weyl fermion χ on Σ that carry charge q under the9 (1) symmetry, I ( χ ) = 1192 π tr R ∧ R − q π F ∧ F . (3.7) b -anomaly from ’t Hooft anomalies and SUSY For the superconformal defect D , it can happen that the bulk stress tensor T µν is related tosome symmetry current J µ by supercharges Q preserved by the defect (i.e. [ Q , D ] = 0). Insuch a scenario, the trace anomaly T µµ and ’t Hooft anomaly ∂ µ J µ are related by SUSY Wardidentities, as is well-known for bulk anomalies in even dimensions. For N = (0 ,
2) supercon-formal surface defects, the relevant symmetry current J µ is the one that generates the defectsuperconformal symmetry U (1) r . It is in general a linear combination that involves a bulk R -symmetry current J Rµ and possibly the current x [ i T j ] µ that generates rotation transverseto the defect (see [23] for explicit expressions). The current J µ is preserved by the defect (cid:104) ∂ µ J µ (cid:105) D = 0 , (3.8)in the absence of background couplings. Nevertheless, as reviewed previously, the U (1) r symmetry can have a ’t Hooft anomaly localized on the defect, (cid:104) ∂ µ J µ (cid:105) D = k π iF δ (Σ) . (3.9)When coupled to background gauge field A on the entire spacetime manifold M , it con-tributes to an anomalous variation under a gauge transformation λ on the defect worldvolumeΣ, δ λ log Z D = k π i (cid:90) Σ λF , (3.10)which is local on Σ. For the superconformal defect, the anomalous variations with respectto other symmetry transformations are governed by the SUSY completion.We first focus on the case without gravitational anomalies, namely k g = 0 for the defect D . The N = (0 ,
2) completion of (3.10) is fixed by SUSY and diffeomorphism invariance[25], δ Ω log Z D = i k π (cid:18)(cid:90) d xdθ + δ Ω R − − c.c. (cid:19) . (3.11) Strictly speaking we also assume that the right-moving U (1) r symmetry can be extended to a non-anomalous non-holomorphic U (1) A symmetry which is gauged by the background N = (0 ,
2) supergravity[25]. However this can be easily achieved by tensoring the defect D with an auxiliary N = (2 ,
2) anomaly-freeSCFT (e.g. the supersymmetric T sigma model) of c L = c R = 3 and k L = k R = 1. Denoting the generatorof the left-moving R-symmetry in this free N = (2 ,
2) SCFT by r L and the right-moving R-symmetry of the θ + is the Grassmann coordinate in the (0 ,
2) superspace, R − is the curvature superfield R − = − i θ + ( R Σ √ g + 2 F ) + fermions (3.12)and δ Ω a chiral superfield whose bottom component σ + iλ packages together the Weyl and U (1) r transformation parameters. Comparing (3.11) to (1.7), we conclude b = c L = c R = 3 k . (3.13) In the previous section, we focused on surface defect D with vanishing gravitational anomaly k g = 0. Here we will lift this restriction and arrive at the general result (3.3).For defects with k g ∈ Z , we can cancel the gravitational anomaly by introducing decou-pled free fields that respect the N = (0 ,
2) supersymmetry. The argument from the previoussection then proceeds for the total system without gravitational anomaly and the result(3.3) follows after subtracting off the free field contributions. More generally one can cancelthe gravitational anomaly by a supersymmetric version of the anomaly inflow mechanism.The CS term (3.5) has a supersymmetric generalization [26–30] that preserves 3d N = 2superconformal symmetry on a closed manifold, S sCS = 2 iκ (cid:90) M (cid:18) CS g − π B ∧ dB + fermions (cid:19) (3.14)where B is the 3d U (1) R gauge field. If M has boundary ∂ M = Σ, with suitable boundaryterms, we expect to preserve the half-BPS N = (0 ,
2) supersymmetry and B restricted toΣ is the identified with the U (1) r background gauge field A .Taking κ = − k g , the inflow from (3.14) shifts both the gravitational and the U (1) r full system by r , we identify the combination √ k + 1 r L − r as the generator for the non-anomalous U (1) A symmetry that is gauged in the N = (0 ,
2) supergravity. The rest of the argument is unaffected. Gauge fields are anti-Hermitian in this paper and compared to [25] F there = iF here . Note that gauge fields in the paper are anti-Hermitian. This leads to a relative minus sign in (3.14)compared to the expression in [29, 30]. Although we will not derive these boundary terms here, let us make a few comments. The Chern-Simonsterm (3.14) being superconformal naturally plays a role in N = 2 supergravity on AdS . In particular theyaccount for the gravitational anomalies of boundary N = (0 ,
2) 2d SCFTs. Bosonic parts of the boundaryterms were proposed [31, 32] but the full supersymmetric completion is unknown to the author’s knowledge.It would be desirable to derive the full boundary term from N = 2 supergravity, perhaps by extending thework of [33] for N = 1 supergravity. theshifts are k → k − k g , c R → c R − k g , c L → c L + k g . (3.15)Since the total system with the auxiliary M and Chern-Simons term (3.14) is free of grav-itational anomaly and respects N = (0 ,
2) supersymmetry, we conclude c R − k g (cid:18) k − k g (cid:19) (3.16)from (3.13) in the previous section, and the result (3.3) follows. b -extremization In the previous sections, we have shown that at the conformal fixed point of the defect fieldtheory, the conformal anomalies of an N = (0 ,
2) surface defect are completely determined byits U (1) r and gravitational anomalies as in (3.3). Since ’t Hooft anomalies are preserved alongsymmetric RG flows, it is natural to ask whether such relations can be used to determinethe b -anomaly of the IR conformal defect from an effective description of the defect atsome immediate scale along the flow. As we will show, this is accomplished by the b -extremization principle, which is a simple extension of the c -extremization principle thatapplies to standalone 2d N = (0 ,
2) theories.Let us consider a defect RG flow that preserves N = (0 ,
2) Poincar´e supersymmetry,a U (1) ˆ r R-symmetry under which the supercharges have charges ˆ r = ± q I that commute with the supercharges. In the absence ofaccidental symmetries along the defect RG flow, the IR superconformal U (1) r symmetry isgenerally a linear combination r = ˆ r + (cid:88) I t I(cid:63) q I . (3.17)We define the trial b -anomaly as a quadratic polynomial in the mixing parameters t I , b trial ( t I ) = 3( k ˆ r ˆ r + 2 (cid:88) I t I k ˆ rI + (cid:88) I,J t I t J k IJ ) − k g , (3.18)whose coefficients k ˆ r ˆ r , k ˆ rI , k IJ characterize the ’t Hooft anomalies among the U (1) ˆ r and A quick way to see this is to perform a constant Weyl transformation and (3.14) is clearly invariant.Under a general Weyl rescaling, the gravitational Chern-Simons action (cid:82) M CS g changes by a boundary termon Σ which corresponds to mixed Lorentz-Weyl anomalies (or mixed diffeomorphism-Weyl anomalies) [34]. q I . Its value at t I = t I(cid:63) yields the actual b -anomaly according to (3.3), b = b trial ( t I(cid:63) ) . (3.19)The b -extremization principle states that ∂b trial ( t I ) ∂t I (cid:12)(cid:12)(cid:12)(cid:12) t I = t I(cid:63) = 0 (3.20)which determines the value of t I(cid:63) uniquely and thus the IR b -anomaly follows.The proof of (3.20) is similar to that in the original work on c -extremization [16, 17] witha new ingredient that the current J µ for U (1) r R-symmetry is not locally conserved on thedefect worldvolume Σ. Let’s suppose we are at the IR N = (0 ,
2) superconformal fixed pointof a defect RG flow. The U (1) r current is generally a linear combination J µ ( x ) = J B µ ( x ) + δ ¯ zµ J D ¯ z ( (cid:126)x ) δ d − ( (cid:126)y ) (3.21)of some bulk current J B µ satisfying ∂ µ J B µ ( x ) = 0 , (3.22)and defect right-moving current J D ¯ z satisfying ∂ z J D ¯ z ( (cid:126)x ) = 0 . (3.23)When no such defect conserved currents exist on D , the U (1) r current J µ is irreducible ,namely its N = (0 ,
2) descendants generate an irreducible defect stress-tensor superconformalmultiplet. An obvious example of reducible J µ is when the DCFT contains a decoupled N = (0 ,
2) SCFT with its own stress-tensor multiplet whose primary is a locally conserved2d R-current.The condition (3.20) is equivalent to the vanishing of the mixed U (1) r anomaly k rI = 0for the superconformal defect. Since (cid:104) ∂ µ J µ ( (cid:126)x, (cid:126)y ) j Ia ( (cid:126)x (cid:48) ) (cid:105) = k rI π (cid:15) ab ∂ b δ ( (cid:126)x − (cid:126)x (cid:48) ) δ d − ( (cid:126)y ) , (3.24) In practice, we will work with the scheme such that k IJ is symmetric. Here we remind the readers that we split the bulk spacetime coordinates as x µ = ( x a , y i ) = ( (cid:126)x, (cid:126)y ) intodirections tangential and transverse to the defect worldvolume Σ. For surface defects, we will use x a and( z, ¯ z ) interchangeably. For related works on surface defect stress-tensor multiplets (not necessarily conformal), see for example[35, 36].
13t suffices to show that (cid:104) J µ ( (cid:126)x, (cid:126)y ) j Ia ( (cid:126)x (cid:48) ) (cid:105) = 0 , (3.25)for any defect conserved current j Ia ( (cid:126)x ).We start with the case where J µ is irreducible. The two-point functions of bulk anddefect primary vectors are highly constrained by the residual conformal symmetry [37, 38].For vector primary operators, the relevant two-point function is fixed up to two constants, (cid:104) J µ ( (cid:126)x, y ) j Ia ( (cid:126)x (cid:48) ) (cid:105) = c even I µa + c odd I µb (cid:15) ab s | (cid:126)y | d − , (3.26)where s µ ≡ ( (cid:126)x − (cid:126)x (cid:48) , (cid:126)y ) and I µa ≡ δ µa − s µ s a | s | (3.27)is the defect version of the familiar symmetric tensor coming from the Jacobian for inversiontransformation. Imposing the current conservation conditions, we conclude c even = c odd = 0 . (3.28)Now if J µ is reducible, we can write J µ ( x ) = J B µ ( x ) + δ ¯ zµ J D ¯ z ( (cid:126)x ) δ d − ( (cid:126)y ) (3.29)where J B µ and J D ¯ z are separately conserved, and J B µ is irreducible. By the previous arguments,it then suffices to show that (cid:104) J D ¯ z ( (cid:126)x ) j Ia ( (cid:126)x (cid:48) ) (cid:105) = 0 , (3.30)which follows from the explanations in [16,17] which we repeat below (in a somewhat differentway) for completeness.From N = (0 ,
2) superconformal symmetry, the defect U (1) r current J D ¯ z (a reduciblecomponent of the full U (1) r current) resides in a supercurrent multiplet [39], J D = J D ¯ z − iθ + G D ¯ z + − i ¯ θ + ¯ G D ¯ z + − θ + ¯ θ + T D ¯ z ¯ z (3.31)where G D ¯ z + , ¯ G D ¯ z + are right-moving defect supercurrents and T D ¯ z ¯ z is the defect stress-tensor.Note that all of these operators are conserved locally on the defect. If j Ia is left-moving,(3.30) is obvious by conformal symmetry. For right-moving currents j Ia , we denote them as We include a parity-odd structure which was not considered in [37, 38]. I ¯ z and consider the three-point function (cid:104) j I ¯ z (¯ z ) G D ¯ z + (¯ z ) ¯ G D ¯ z + (0) (cid:105) . (3.32)The above three-point function vanishes from inspecting the j I ¯ z ( z ) G D ¯ z + ( z ) OPE since byassumption the N = (0 ,
2) supercharges and thus the supercurrent G D ¯ z + are uncharged under j I ¯ z . On the other hand, from the z J D ¯ z (0) term in the G D ¯ z + (¯ z ) ¯ G D ¯ z + (0) OPE we conclude (cid:104) J D ¯ z j I ¯ z (cid:105) = 0 as desired. This completes the proof of the b -extremization principle.Before we end this section, let us comment on a caveat in the arguments above thatwas also present for c -extremization in [16, 17]. Here we have assumed that the currents j Ia are conformal primaries and whose right-moving and left-moving components are separatelyconserved, which follow from unitarity and the existence of a normalizable conformally in-variant vacuum for the DCFT. In particular this excludes the possibility of a non-compactcomplex scalar φ (which completes to a N = (0 ,
2) chiral multiplet) on the defect. Inthat case, the right-moving non-primary current ∂ ¯ z φ can mix with the U (1) r current in thepresence of a nontrivial background charge on the surface defect (i.e. due to the coupling q (cid:82) Σ d x √ hR Σ Re φ and its SUSY completion). The b -extremization principle continues tohold in this case with the understanding that the resulting b -anomaly is really the effectiveanomaly defined as b eff = b − h min , (3.33)where ¯ h min is the minimal ¯ L eigenvalue among the defect local operators, closely related tothe effective conformal charge c eff defined in [40]. In particular the proper generalization ofZamolodchikov’s c -theorem [3] to non-compact CFTs uses c eff [41–43]. Similarly we expect b eff to be the monotonic quantity under defect RG flows when the defect field theory isnon-compact.A further related subtlety when there is no normalizable conformally invariant vacuum inthe DCFT is the appearance of non-holomorphic conserved currents whose left and right mov-ing components are not separately conserved. This can also be illustrated in the context ofa free non-compact N = (0 ,
2) chiral multiplet [17]. As explained there, the c -extremizationprinciple (3.20) does not hold when j Ia is non-holomorphic and irreducible, and the sameproblem arises for the b -extremization of defect conformal anomaly. Nevertheless in practiceone can try to isolate such non-holomorphic currents and extremize b trial among the rest ofthe currents to determine the conformal b -anomaly.15 Examples of N = (0 , surface defects and anomalies There has been promising recent progress in understanding the conformal anomalies of N = (0 ,
2) superconformal surface defects, namely b and d , in (1.6). From Wess-Zuminoconsistency conditions, it is obvious that the b -anomaly cannot depend on marginal cou-plings on the surface defect. Furthermore it was shown in [44] that with N = (0 , b cannot depend on bulk marginal couplings either.For N = (0 ,
2) surface defects in 4d SCFTs, it was proven in [45] using SUSY Wardidentities that d = d . Given the relation between d and the displacement operator two-point function c D , and between d and the stress tensor one-point function h [46, 47], itwas explained in [45] how to determine the d , anomalies from knowledge of the chiralalgebra underlying the 4d N ≥ b -anomaly is however more elusive,and expected to enter in the two-point function of bulk stress tensor in the presence of thesurface defect. Alternatively, it can be accessed from the S partition function of the SCFTdecorated with the surface defect on S by inspecting the logarithmic dependence of the freeenergy on the sphere radius. When a localization formula [50, 51] for such a setup exists (i.e.for N = (2 ,
2) surface defects in 4d N = 2 SCFTs with gauge theory descriptions [52–55]),this was implemented in [22] to identify the b -anomaly. We will see how to recover theseresults easily using (3.3) which does not rely on the localization formulae.Conformal anomalies of N = (4 ,
4) superconformal surface defects in 6d (2 ,
0) SCFTshave been studied in [12, 18–22, 56–63]. In particular, the relation d = d was shownto persist for the half-BPS surface defects in 6d (2 ,
0) SCFTs [63], and their values aredetermined by the defect superconformal index [64] as explained in [22]. However the b -anomalies have only be obtained in the free theory [61] and in the large N limit fromholography [18–21]. Here we will give exact answers for b for general half-BPS surfacedefects in 6d (2 ,
0) SCFTs of arbitrary ADE types and the generalization to N = (0 , ,
0) SCFTs is straightforward.Finally little is known about the conformal anomalies of boundaries in 3d CFTs apartfrom some attempts from holography [21], thus we will be most pedagogical with this casein the following, to illustrate our method by determining the b -anomalies for several typesof simple N = (0 ,
2) superconformal boundaries. We leave the more sophisticated examplesthat involve non-abelian Chern-Simons-matter bulk SCFTs to future investigation. This is a common feature of the a -type conformal anomalies, whose Weyl variations are total derivatives. .1 Boundaries in 3d SCFTs In d = 3, the relevant N = (0 ,
2) superconformal defect is either a boundary or an interfaceof some 3d
N ≥ T and T is related by thefolding trick to the boundary in the tensor product theory T × ¯ T (the second factor involvesan orientation-reversal), we will focus on boundary defects here without loss of generality.For a 3d N = 2 bulk SCFT, the U (1) r symmetry of the half-BPS DCFT is identified withthe U (1) R symmetry of the bulk SCFT. The identification for N > U (1) R of an N = 2 subalgebra) and the detailed mapping can be found in [23].Examples of N = (0 ,
2) boundaries and interfaces can be found in [65–73]. We will followthe conventions of [73] here.
We start by considering the boundary conditions of the free 3d N = 2 SCFT made out ofa single chiral multiplet Φ which consists of a complex scalar φ and a Dirac fermion ψ ± whose U (1) r charges are r = and r = − respectively. The theory lives on the half space R , defined by y ≥ R , at y = 0.To study N = (0 ,
2) preserving boundary conditions, it is convenient to decompose thebulk degrees of freedom into representations of the N = (0 ,
2) subalgebra. Here the 3d chiralmultiplet Φ decomposes into a N = (0 ,
2) chiral multipletΦ = φ + θ + ψ + − iθ + ¯ θ + ∂ ¯ z φ , (4.1)and a Fermi multiplet Ψ = ¯ ψ − + θ + f − iθ + ¯ θ + ∂ ¯ z ψ − , (4.2)where f is an auxiliary field that satisfies f = ∂ y ¯ φ on-shell [73]. Here ± are indices for rightand left moving spinors as before. Both sub-multiplets have U (1) r charge r = .The basic supersymmetric boundary conditions of Φ d involve setting either Φ or Ψ tozero on Σ, corresponding to supersymmetric Dirichlet and Neumann boundary conditions[65, 70, 73]: B D [Φ d ] : Φ | Σ = 0 → φ | Σ = ψ + | Σ = 0 , (4.3)and B N [Φ d ] : Ψ | Σ = 0 → ∂ y φ | Σ = ψ − | Σ = 0 . (4.4)The 3d fermions contribute nontrivial boundary ’t Hooft anomalies and can be worked out17y considering mass deformations [73]. The results are summarized in Table 1.Fields Anomaly I d χ − − q c ( F ) + p ( T ) χ + 12 q c ( F ) − p ( T )3 d ψ + | Σ = 0 − q c ( F ) + p ( T ) ψ − | Σ = 0 q c ( F ) − p ( T ) k π AdA (cid:12)(cid:12) y ≥ − k c ( F ) Table 1: The ’t Hooft anomalies for 2d Weyl fermions χ ± , 3d Dirac fermions ψ ± with differentboundary conditions, and a classical 3d CS action on R , . Here the fermions carry charge q under the vector U (1) symmetry for which A is the background gauge connection.For either boundary conditions, since there are no extra currents on the boundary, the b -anomalies (or equivalently c L , c R ) are easily determined by (3.3) from the U (1) r and grav-itational anomalies, B D [Φ d ] : k = − , k g = − , B N [Φ d ] : k = 18 , k g = 12 . (4.5)The results are tabulated in Table 2. In particular we find perfect agreement with knownresults about b -anomalies of free scalar and fermion on R , [14, 74]. We emphasize thatunlike the conformal anomalies of standalone CFTs, the defect conformal anomalies do notneed to be positive. The N = (0 ,
2) Dirichlet and Neumann boundary conditions are related by supersymmet-ric defect RG flows [73]. This is achieved by coupling the boundary conditions with an extrafree 2d chiral multiplet C or Fermi multiplet Γ, and turning on superpotential deformations(known as flip from [67]) as follows, B D [Φ d ] ⊕ C with (cid:90) Σ d xdθ + C Ψ −→ B N [Φ d ] B N [Φ d ] ⊕ Γ with (cid:90) Σ d xdθ + ΦΓ −→ B D [Φ d ] . (4.6)From Table 2, we see clearly the defect b -theorem is obeyed for these simple RG flows. It would be interesting to see if there is a universal lower bound on the b -anomaly of conformal boundary(surface) defects. See some relevant discussions in [20]. c L c R d χ − χ + ϕ d ψ − | Σ = 0 −
14 14 ψ + | Σ = 0 − ∂ y φ | Σ = 0
18 18 φ | Σ = 0 − − B N [Φ d ] −
18 38 B D [Φ d ] − Table 2: The top entries give the conformal anomalies for 2d Weyl fermions χ ± , complexscalar ϕ and N = (0 ,
2) Fermi and chiral multiplets. The bottom entries give the boundaryconformal anomalies of 3d Dirac fermion ψ , complex scalar φ with basic boundary conditionsand their N = (0 ,
2) supersymmetric completions.Given two 3d chiral multiplets Φ d and Φ (cid:48) d with Dirichlet and Neumann boundary con-ditions respectively, there is a superpotential deformation that couples the two multipletstogether at the boundary B D [Φ d ] ⊕ B N [Φ (cid:48) d ] with (cid:90) Σ d xdθ + ΨΦ (cid:48) (4.7)which is exactly marginal, and amounts to an N = (0 ,
2) preserving rotation of the originalboundary conditions for (Φ d , Φ (cid:48) d ). Via the unfolding trick, we have Φ d and Φ (cid:48) d on R , and R , − respectively joined at Σ. Then the superpotential in (4.7) implements the identificationbetween (Φ , Ψ) and (Φ (cid:48) , Ψ (cid:48) ) along the interface at Σ, so that the total system is simply Φ d on R , with a transparent interface. This explains why B D [Φ d ] and B D [Φ d ] have opposite’t Hooft and defect conformal anomalies as in Table 1 and 2. N = 2 Ising SCFT
Let us now consider the 3d N = 2 Ising SCFT which is defined by a single 3d chiral multipletΦ d with bulk superpotential W = Φ d . Here the U (1) r charge of Φ d is r = .The simplest superconformal boundary condition is the Dirichlet boundary B D [Φ d ]. To see this, one uses the following boundary variations of the chiral multiplet action on the two half U (1) r and gravitational ’t Hooft anomalies, contributed bythe Dirac fermion ψ ± in the chiral multiplet, k = − , k g = − . (4.9)Thus from (3.3) we conclude c L = 13 , c R = − . (4.10)There are more interesting boundary conditions coming from coupling the Ising SCFT withDirichlet boundary condition to nontrivial N = (0 ,
2) SCFTs on the boundary but we willleave that to future work. N = 2 SQED and mirror symmetry
When the 3d SCFT has gauge theory descriptions, there is a plethora of interesting super-symmetric boundary conditions that are expected to flow to superconformal boundaries inthe IR [73]. We will study them more systematically in a future publication and focus on thesimplest example here, namely the N = 2 U (1) SQED with one chiral multiplet of charge1. Since this theory is mirror dual of a free chiral multiplet, we will also be able to makeconnections to boundary conditions of the free SCFT.The 3d N = 2 vector multiplet V d has components ( A µ , λ ± , σ, D ) where D is an auxiliaryfield. Here we consider the Dirichlet boundary condition [73] B D [ V d ] : A a | Σ = λ − | Σ = D | Σ = 0 , (4.11)together with B D [Φ d ] for the charged chiral multiplet. We will denote the full boundarycondition as B D [SQED].The 3d theory has a U (1) ˆ R symmetry under which the fermions in the vector and chiralmultiplets have charges ˆ R [ λ ± ] = 1 , ˆ R [ ψ ± ] = − . (4.12)In addition, there is a topological global symmetry U (1) T which maps to the flavor symmetry spaces [73] δS [ B D [Φ d ]] = (cid:90) Σ d xdθ + δ ΨΦ + c.c., δS [ B N [Φ (cid:48) d ]] = (cid:90) Σ d xdθ + Ψ (cid:48) δ Φ (cid:48) + c.c. . (4.8) R = ˆ R + 12 T , (4.13)such that the BPS monopole has R = saturating the 3d unitarity bound.A novelty of the Dirichlet boundary condition for gauge field B D [ V d ] is the presence ofadditional 2d global symmetry U (1) G that comes from the gauge symmetry in the bulk.Consequently, the U (1) R symmetry can mix with U (1) G in the presence of the Dirichletboundary, and the superconformal U (1) r symmetry generator will be a linear combination ofthe generators R and G . Below we will see how b -extremization fixes this linear combinationand thus the conformal defect anomalies in this case.We start by recalling the boundary ’t Hooft anomalies for U (1) R and U (1) G for the SQEDgiven in [73] I = I UV4 + 14 c ( F ˆ R ) −
14 ( c ( F G ) − c ( F ˆ R )) − k g p ( T ) . (4.14)where I UV4 = −
14 ( c ( F G ) − c ( F ˆ R ) − c ( F T ) c ( F G ) + 14 c ( F ˆ R ) −
14 ( c ( F T ) − c ( F ˆ R ) (4.15)comes from UV Chern-Simons couplings required for the duality between SQED and freechiral multiplet in the absence of a boundary. This includes a U (1) Chern-Simons cou-pling for the gauge field to ensure gauge invariance. The second term of (4.14) comes fromboundary anomalies of the gaugino λ ± , and the third term is due to the fermions ψ ± in thechiral multiplet.The last term in (4.14) encodes the boundary gravitational anomaly which can be ob-tained by giving large negative mass m < λ ± and ψ ± . Integrating out the fermions,we end up with a U (1) scalar QED which is free of gravitational anomalies. But we alsoneed to remember that the boundary condition λ − | Σ = ψ + | Σ = 0 supports an edge mode of ψ − for m <
0, which contributes to the boundary gravitational anomaly k g = − . (4.16)Equivalently we can consider large positive mass m > Here we have included a shift (the last term in (4.15)) compared to the expression in [73], to match with U (1) Maxwell-Chern-Simons theory coupled to scalars, which contributes k g = − U (1) is equivalent to the gravitational Chern-Simons action e − i (cid:82) CS g [75]. Now for m > λ + , thus we recover thesame total boundary gravitational anomaly as above.To summarize, the full boundary ’t Hooft anomalies for B D [SQED] are given by I = − c ( F G ) − c ( F G )( c ( F T ) − c ( F ˆ R ) −
14 ( c ( F T ) − c ( F ˆ R )) + 148 p ( T ) . (4.17)Let us now write down the trial b -anomaly (3.18) using these ’t Hooft anomalies in (4.17),with the candidate U (1) r symmetry generated by r = R + tG = ˆ R + T + tG with parameter t , b trial ( t ) = 3 k rr + 14 = 3( − t + t −
18 ) + 14 . (4.18)Extremizing with respect to t , we find that the superconformal U (1) r symmetry for theDirichlet boundary condition of SQED is r = ˆ R + 12 T + 12 G , (4.19)and the defect conformal anomalies are given by B D [SQED] : c L = 78 , c R = 38 . (4.20)In [73] it was proposed that under mirror symmetry, B D [SQED] is dual to B N [Φ d ] for thefree 3d chiral multiplet with an extra free Fermi multiplet on the boundary. Recall the defectconformal anomalies of a free 3d chiral multiplet, B N [Φ d ] : c L = − , c R = 38 . (4.21)We see the difference is precisely saturated by that of a 2d Fermi multiplet (see Table 2). There is a rich zoo of surface defects in 4d SCFTs preserving N = (0 ,
2) or a further enhancedsuperconformal symmetry, such as the half-BPS surface defects in N = 1 SCFTs [35, 76, 77]and those in N = 2 , our regularization scheme for the free chiral multiplet in Section 4.1.1. N = 4 super-Yang-Mills (SYM) [78], which is defined by a codimension-twosingularity in the SYM fields. The resulting defect enjoys small N = (4 ,
4) superconformalsymmetry psu (1 , | × psu (1 , |
2) in the IR which contains (3.1) as an N = (0 ,
2) subalgebra,thus we can determine the defect conformal anomalies from the ’t Hooft anomalies followingour general arguments in the previous section. For this purpose, it is convenient to use an al-ternative UV description of the same surface defect, as a 2d-4d system, that involves couplingthe 4d gauge theory with gauge group G on M to an auxiliary 2d field theory on Σ [78]. Typically the coupling is through gauging a G flavor symmetry on the defect. For SU ( N )SYM, the GW surface defects are labelled by a partition N = (cid:80) ni =1 k i . The correspondingauxiliary 2d theory is described by a 2d N = (4 ,
4) linear quiver gauge theory, with gaugegroup (except for the bold node which is a flavor symmetry) U ( p ) × . . . U ( p n − ) × SU ( p n ) and bifundamental hypermultiplets between each pair of consecutive nodes [84]. Here therank of the gauge nodes are p j = j (cid:88) i =1 k i , p n = N . (4.22)The ’t Hooft anomalies of the surface defect follow immediately from the field content ofthe auxiliary 2d theory. In particular there is no gravitational anomaly in this N = (4 , U (1) r is simply related to the (right-moving) superconformal SU (2) R anomaly by k = 2 k R . (4.23)Recall the 2d gauge theory has SU (2) L × SU (2) R × SU (2) I R-symmetry in the UV andthe superconformal R-symmetry on the Higgs branch is identified with SU (2) L × SU (2) R [85]. The chiral fermions in the theory, λ ± from the vector multiplet and ψ ± from thehypermultiplet, transform under SU (2) L × SU (2) R × SU (2) I × SU (2) F as( λ + , λ − ) : (2 , , , + ⊕ (1 , , , − , ( ψ + , ψ − ) : (1 , , , + ⊕ (2 , , , − , (4.24)where we have introduced SU (2) F to keep track of the global symmetry of a free hypermul-tiplet. Thus the right-moving SU (2) R superconformal R-symmetry receives anomaly from More precisely, the N = (4 ,
4) superconformal defect is described by the Higgs branch of the auxiliary2d theory [84]. See [85] for discussions on related subtleties in N = (4 ,
4) RG flows. For general gauge group G , the GW surface defects are labelled by the Levi subgroups L of G . Here for G = SU ( N ), we have L = S [ U ( k ) × · · · × U ( k n )] corresponding to the partition N = (cid:80) i k i . − and ψ + in the quiver gauge theory, k R = n − (cid:88) i =1 ( p i p i +1 − p i ) = 12 (cid:32) N − n (cid:88) i =1 k i (cid:33) . (4.25)Consequently from (4.23) we have determined the conformal anomalies of a general Gukov-Witten surface defect in SU ( N ) SYM b = c L = c R = 6 k R = 3 (cid:32) N − n (cid:88) i =1 k i (cid:33) . (4.26)GW surface defects in SYM with general gauge group G are labelled by Levi subgroups L ⊂ G . We will denote them by D L [ G ]. The auxiliary 2d theory can be described by a N = (4 ,
4) non-linear sigma model with hyperK¨ahler target space T ∗ ( G/ L ) [78], thus b = c L = c R = 6 k R = 3 (dim( G ) − dim( L )) . (4.27)We observe that for these surface defects, the defect conformal anomalies are nothing butthe usual conformal anomalies of the auxiliary 2d theory (in the IR conformal limit). Thisis not a coincidence. In the description of surface defects by 2d-4d systems, the 2d theoryis coupled to the 4d SCFT by bulk gauge fields, thus they decouple in the weak couplinglimit g YM →
0. Since b -anomalies of surface defects with N = (0 ,
2) supersymmetry do notdepend on bulk marginal couplings [44], they must coincide with the conformal anomalies ofthe 2d theory viewed as a standalone SCFT (in the IR). This gives a quick way to determinethe defect conformal anomalies for a large class of surface defects in 4d N = 2 conformalgauge theories, and easily reproduces the localization results found in [22]. Despite the non-Lagrangian nature of the 6d SCFTs, their string/M/F-theory constructionssuggest that they host interesting surface defects. For example, in M-theory, such surfaceoperators arise from two-dimensional M2-M5 intersections [86], possibly in the presence oftransverse singularities [87] and/or Horava-Witten walls [88]. They define half-BPS N =(0 ,
4) or N = (4 ,
4) surface defects in 6d N = (1 ,
0) and N = (2 ,
0) SCFTs respectively. Note that this is a different superconformal algebra compared to the small N = (4 ,
4) algebra preservedby a half-BPS surface defect in the 4d N = 4 SYM. This difference is crucial in determining the defectconformal anomalies from its ’t Hooft anomalies. sl (2 , R ) ⊕ osp (4 ∗ |
2) for the N = (0 ,
4) defect and osp (4 ∗ | ⊕ osp (4 ∗ |
2) for the N = (4 ,
4) case. In either case, the N = (0 ,
2) superconformalalgebra (3.1) is a subalgebra and thus our results from the previous sections apply. We willneed the following relation between the U (1) r generator of the N = (0 ,
2) subalgebra andR-symmetry generators in osp (4 ∗ | r = 2( R − I ) . (4.28)Here R and I are the Cartan generators of the su (2) R × su (2) I R-symmetry of the right-moving osp (4 ∗ | Then if we know the ’t Hooftanomalies for the surface defect, we can use (3.3) to determine the conformal anomalies. Notethat because of the enhanced R-symmetry (and supersymmetry), we don’t expect mixingwith global symmetries.For illustration, let us work out the anomalies for a class of N = (4 ,
4) surface defectsin the 6d (2 ,
0) theory labelled by an ADE Lie algebra g . The surface defect is in generalcharacterized by a weight vector λ ∈ Λ w ( g ), and we will refer to it by D λ [ g ]. Upon compact-ification on an S longitudinal to the defect, the bulk SCFT is described by 5d N = 2 SYMwith gauge algebra g and the surface defect D λ [ g ] corresponds to a half-BPS Wilson loop inthe representation with highest weight λ [60, 64].The ’t Hooft anomalies of the surface defect D λ [ g ] can be deduced by moving onto thetensor branch of the 6d SCFT since the R-symmetries and Lorentz symmetry are preserved.On a generic point of the tensor branch in a general 6d (1 ,
0) SCFT, the effective actiontakes the following schematic form S TB = 2 π (cid:90) η ij (cid:18) dB i ∧ (cid:63)dB j + B i ∧ I j (cid:19) + . . . (4.29)where B i denotes the self-dual 2-form field for each tensor multiplet, η ij is a symmetric,positive-definite and integral charge matrix, and I i ∧ B j where I i is a 4-form made of charac-teristic classes in background gauge fields and geometry is the Green-Schwarz term [90] thatplays an important role in the matching of 6d ’t Hooft anomalies on the tensor branch [91,92].On the tensor branch, the N = (0 ,
4) surface defect is expected to be described by a BPSself-dual string of charge Q i under the 2-form fields B i . From anomaly inflow [98, 99], the The factors of 2 in (4.28) are important. See Appendix B.3 of [89] for explicit expressions. See for example [93–97] for works on the self-dual strings in various 6d (1 ,
0) theories. I = 12 η ij Q i Q j ( c ( F L ) − c ( F R )) + η ij Q i I j . (4.30)Now let us come back to the particular case of D λ [ g ] defects in 6d (2 ,
0) SCFTs. Toapply (4.30), we note that here η ij is the Cartan matrix of g , the string charge satisfies λ = (cid:80) i Q i α i where α i are the simple roots of g , and the Green-Schwarz term takes a simpleform with [91, 99] I i = ρ i ( c ( F I ) − c ( F F )) (4.31)where ρ = (cid:80) i ρ i α i is the Weyl vector of g . Therefore we can write I [ D λ [ g ]] = 12 ( λ, λ )( c ( F L ) − c ( F R )) + ( λ, ρ )( c ( F I ) − c ( F F )) . (4.32)Note the absence of gravitational anomalies k g = 0 in this case. The N = (4 ,
4) surface defect(and the corresponding string on the tensor branch) has SO (4) L × SO (4) R R-symmetry, and SO (4) L = SU (2) L × SU (2) F wheres SO (4) R = SU (2) R × SU (2) I [99]. From (4.28) and(4.32), we can read off the ’t Hooft anomaly for the U (1) r symmetry of the N = (0 , k = ( λ, λ ) + 8( λ, ρ ) . (4.33)Consequently we obtain the conformal anomalies for the surface defect D λ [ g ] from (3.3), b = c L = c R = 3( λ, λ ) + 24( λ, ρ ) . (4.34)This agrees with the results of [21] for g = su ( N ) obtained from holographic entanglemententropy in the presence of the defect. In this paper, we have studied the defect analog of the 2d conformal anomalies, namelythe b -anomaly, for unitary conformal surface defects in CFTs. We revisited the defect b -theorem and provided an explicit example of the dilaton effective action for defect RG flowsin the free scalar theory. Such defect dilaton effective action played an important role inthe proof of [14]. We also investigated ’t Hooft anomalies that arise in the presence of asurface defect. For defects with N = (0 ,
2) superconformal symmetry, we derived a universal26elation between the b -anomaly and the ’t Hooft anomaly of U (1) r symmetry. Since thelatter is much more robust against deformations, this provides a shortcut to determinethe b -anomalies of strongly coupled conformal defects using weak coupling results that aretypically available after deformations. A potential subtlety arises when trying to identify thesuperconformal defect U (1) r symmetry away from the conformal fixed point. This is settledby the b -extremization principle that governs defect RG flows with N = (0 ,
2) supersymmetrywhich we have proved in this work. The b -extremization picks out the superconformal U (1) r symmetry among the symmetries preserved by an RG flow, whose ’t Hooft anomaly thendetermines the conformal b -anomaly, in analogue to the well-known c -extremization principlefor standalone 2d N = (0 ,
2) theories [16, 17]. To illustrate our method, we then set off todetermine the b -anomalies for a number of surface defects in 3d, 4d and 6d SCFTs. We nowdiscuss some future directions below. Holographic dual of b -extremization and b -anomalies The c -extremization principle of 2d N = (0 ,
2) SCFTs [16, 17] has an elegant geometricdual in the context of AdS/CFT [100–102], in terms of extremizing certain functionals ofoff-shell geometries in string/M/F-theory (in close analogy to the holographic dual of a -maximization [103] developed in [104, 105]). It would be interesting to develop an extensionthat applies for surface and general defect SCFTs. In particular for defects that correspondto branes wrapping submanifolds S of the internal manifold M int in the holographic dual,the candidate functional will involve the embedding of S ⊂ M int .For product geometries, namely when M int is trivially fibered over the AdS base, theextremization problem appears to be trivial. For example, for 4d N = 1 SCFTs dual totype IIB string theory on AdS × SE with a Sasaki-Einstein (SE) internal manifold, aclass of half-BPS N = (0 ,
2) surface defects correspond to D3 branes wrapping
AdS × S ,where the S ⊂ SE is required to be a closed orbit of the Reeb vector field ξ to preservesupersymmetry [81]. With a single probe D3-brane, one naturally expects the conformal b -anomaly of the dual surface defect to be proportional to the size (cid:96) of S . The Reeb vector ξ realizes SE as a principal U (1) bundle over a K¨ahler-Einstein base manifold Z , and theorbits are simply the S fibres labelled by points on Z . Since the Reeb fibres are geodesicsand have the same length (cid:96) [105, 106], the D3 brane wrapping these fibres should give rise toa family of surface defects with identical b -anomalies, potentially related by marginal defect Here for simplicity we have assumed that SE is a regular Sasaki-Einstein manifold but the statementshere hold with small modifications for quasi-regular
Sasaki-Einstein manifolds. We refer the readers to[105, 106] for background material on these concepts. It would be interesting to investigate such an N = (0 ,
2) defect conformalmanifold.
Defect anomalies from defect correlation functions
Both ’t Hooft anomalies and conformal anomalies are physical observables of a given CFT.Although they are often defined as contact term modifications of symmetry Ward identities,the anomalies contribute unambiguously to the correlation functions of local (and sometimesextended) operators at separate points. For anomalies in the absence of defects, the con-nection between their contact-term and separated-point-correlation manifestations is wellestablished, while for defect anomalies, this is yet to be fully developed.For surface defects, it is straightforward to see that the two-point function of bulk con-served current J µ for a U (1) J symmetry has a unique parity-odd structure that is possiblein d = 3 with the defect along y = 0, (cid:104) J µ ( x ) J ν ( x (cid:48) ) (cid:105) odd D = f ( v ) s i(cid:15) µρσ X ρ I σν , (5.1)where s ≡ x − x (cid:48) and v ≡ ( x − x (cid:48) ) ( x − x (cid:48) ) + 4 yy (cid:48) (5.2)is the invariant cross ratio under the residual conformal symmetry. The details on the tensorstructure can be found in Appendix B. Since the defect OPE limit corresponds to v →
1, wenaturally expect f (1) to encode the boundary U (1) J anomaly k JJ . A simple computationof the two-point function for a 3d Dirac fermion and comparison to Table 1, leads to thefollowing conjecture f (1) = 12 π k JJ . (5.3)A similar exercise can be done for the parity-odd contribution to the stress-tensor two-pointfunction which should relate to the boundary gravitational anomaly. These parity-odd defecttwo-point functions have also been studied in special kinematic regime in the momentumspace [107], where connections to bulk Chern-Simons contact terms [29, 30] and defect ’tHooft anomalies were made. It would be desirable to compare with the position spaceapproach here.As for the surface defect b -anomaly, it was conjectured to be determined by the parity-even part of the stress-tensor two-point function (cid:104) T µν ( x ) T αβ ( x (cid:48) ) (cid:105) D in the defect OPE limit Here we work with the Euclidean CFT obtained from Wick rotation, and thus the extra i factor. → N results for theDirichlet boundary of the 3d O ( N ) model as explained in the recent work [109]. Defects of dimensions p (cid:54) = 2Finally it will be interesting to investigate anomalies and constraints on RG flows for de-fects of other longitudinal dimensions, namely p (cid:54) = 2. In particular, the a -theorem of [5, 6]has an immediate generalization to p = 4 defect RG flows which will appear in [110]. The a -maximization principle of [103] also has a natural extension to the superconformal defectsimilar to what we studied here [110]. The story in odd defect dimensions is qualitativelydifferent due to the absence of such a -type conformal anomalies. Instead it has been con-jectured that the defect free energy F D plays the role of the monotonic function under RGflows, which have passed a number of tests (see [111] for a recent summary). For half-BPSsuperconformal boundaries (interfaces) in 4d N = 2 SCFTs, there is also a proposal of aboundary version [112] of the F -maximization principle [113] for 3d N = 2 SCFTs. Theproofs for these conjectures remain largely open in d > Acknowledgements
The author thanks Nathan Agmon for collaboration on related topics. The author alsothanks Zohar Komargodski for interesting comments on the draft. The work of YW issupported in part by the Center for Mathematical Sciences and Applications and the Centerfor the Fundamental Laws of Nature at Harvard University.
A Boundary Feynman diagram
We would like to evaluate the integral I ( p, m ) ≡ (cid:90) d k (2 π ) | k | + m )( | k + p | + m ) . (A.1)We proceed by introducing the Schwinger parameters s , s and rewrite the integal as I ( p, m ) = (cid:90) d k (2 π ) (cid:90) ∞ ds ds e − ( s | k | + s | k + p | ) − m ( s + s ) . (A.2)29ext using the Laplace transform, e − s | k | = (cid:90) ∞ dt √ π e − tk se − s t t / , (A.3)we have I ( p, m ) = 14 π (cid:90) d k (2 π ) (cid:90) ∞ ds ds s s e − m ( s + s ) (cid:90) ∞ dt dt ( t t ) e − t k − t | k + p | e − s t − s t . (A.4)Performing the k integral and rescaling the variables s i → s i /m, t i → t i /m , we have I ( p, m ) = 1(4 π ) (cid:90) ∞ ds ds s s e − ( s + s ) (cid:90) ∞ dt dt ( t t ) e − s t − s t t + t e − t t t t p m . (A.5)Now for the p m term has coefficient after a change of variables t i → /t i , α = − π ) (cid:90) ∞ ds ds s s e − ( s + s ) (cid:90) ∞ dt dt e − s t − s t ( t t ) ( t + t ) . (A.6)We further make the change of variables t = t + t and u = t t + t , α = − π ) (cid:90) ∞ ds ds e − ( s + s ) s s (cid:90) ∞ dt (cid:90) due − t ( s u + s (1 − u )) (cid:112) u (1 − u )= − π (cid:90) ∞ ds ds e − ( s + s ) s s ( s + s ) . (A.7)where we first do the t integral followed by the u integral. The leftover integral can be doneeasily again by a change of variables similar to what we have used for t i , yielding α = − π (cid:90) ∞ ds (cid:90) du se − s u (1 − u ) = − π . (A.8) B Parity-violating boundary two-point function
The two-point function of bulk conserved currents in a general d -dimensional Euclidean CFTwith a conformal boundary condition D has a unique parity-even structure [114, 115] (cid:104) J µ ( x ) J ν ( x (cid:48) ) (cid:105) even D = 1 s d − (cid:18) π ( v ) I µν ( s ) − d − v∂ v π ( v ) ˆ I µν ( s ) (cid:19) , (B.1)30here s ≡ x − x (cid:48) and v ≡ ( x − x (cid:48) ) ( x − x (cid:48) ) + 4 yy (cid:48) (B.2)or equivalently ξ ≡ ( x − x (cid:48) ) yy (cid:48) with v = ξξ + 1 (B.3)define the invariant cross-ratio under the residual O ( d,
1) conformal symmetry. The tensorstructure in (B.1) involves the familiar Jacobian factor of inversion I µν ( x ) = δ µν − x µ x ν x , (B.4)and its modification due to the boundary defect,ˆ I µν ( x ) ≡ δ µν − X µ X (cid:48) n , X µ ≡ v (cid:18) ys s µ − δ µ (cid:19) , X (cid:48) µ ≡ v (cid:18) − y (cid:48) s s µ − δ µ (cid:19) , (B.5)which transform nicely under bilocal O ( d,
1) actions on x and x (cid:48) [114]. Finally π ( v ) is ageneral function of the cross-ratio v subject to regularity constraints in the bulk OPE limit v → v → d = 3, another tensor structure becomes possible that uses the (cid:15) µνρ tensor (cid:104) J µ ( x ) J ν ( x (cid:48) ) (cid:105) odd D = f ( v ) s i(cid:15) µρσ X ρ I σν ( s ) , (B.6)which has the right O ( d,
1) transformation properties and satisfies current conservation. Notethat another similar tensor structure built out of I µν , X µ , X (cid:48) ν is not independent (cid:15) µρσ X ρ I σν ( s ) = (cid:15) ν ρσ X (cid:48) ρ I σµ . (B.7)As discussed in the main text, we expect the function f ( v ) in the defect OPE limit v → U (1) J symmetry, f (1) = αk JJ (B.8)for some theory independent constant α .To fix α , let us consider the free Dirac fermion ψ on R , with conformal boundaryconditions ψ + | Σ = 0 or ψ − | Σ = 0 which we refer to as D + and D − respectively. Here the31uclidean gamma matrices are chosen to be γ = ( σ , σ , σ ) (B.9)and ± labels the eigenvalues of γ which coincides with the chirality on the 2d boundary.The free fermion theory has a U (1) J global symmetry generated by J µ = ¯ ψγ µ ψ , (B.10)under which ψ have charge +1. The U (1) J symmetry is clearly preserved by the D ± boundaryconditions.According to general discussions in [73] (see Table 1), the D ± boundary conditions con-tribute a boundary ’t Hooft anomaly which is detectable in the presence of nontrivial U (1) J background gauge field, k JJ [ D ± ] = ∓ . (B.11)Now we compute (cid:104) J µ ( x ) J ν ( x (cid:48) ) (cid:105) D ± explicitly using the free fermion propagator [116] (cid:104) ψ ( x ) ¯ ψ ( x (cid:48) ) (cid:105) D ± = i π (cid:18) γ · ( x − x (cid:48) ) | x − x (cid:48) | ± γ γ · (¯ x − x (cid:48) ) | ¯ x − x (cid:48) | (cid:19) (B.12)where ¯ x ≡ ( x, − y ) is the reflection of x across the boundary at y = 0.Performing the Wick contraction, we obtain after some algebra (cid:104) J µ ( x ) J ν ( x (cid:48) ) (cid:105) D ± = − π ) s ( − v ) I µν + 4 v X µ X (cid:48) ν ± iv (cid:15) µαβ X α I βν )= 18 π s ((1 − v ) I µν + 2 v ˆ I µν ∓ iv (cid:15) µαβ X α I βν ) . (B.13)Compared to (B.1) and (B.6), we find for the boundary conditions D ± , π ( v ) = 18 π (1 − v ) , f ± ( v ) = ∓ π v . (B.14)Using (B.11), we thus conclude in (B.8), α = 12 π . (B.15) Note π ( v ) computed for free Dirac fermion in [115] contains a a typo. For Dirac fermion in d -dimensions, π ( v ) ∝ − v d − up to a constant. eferences [1] G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking , NATO Sci. Ser. B (1980) 135.[2] J. Wess and B. Zumino, Consequences of anomalous Ward identities , Phys. Lett. B (1971) 95.[3] A. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2DField Theory , JETP Lett. (1986) 730.[4] J. L. Cardy, Is There a c Theorem in Four-Dimensions? , Phys. Lett. B (1988)749.[5] Z. Komargodski and A. Schwimmer,
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