JHEP06(2019)123 Springer June 6, 2019 June 15, 2019 June 25, 2019 : , : : c Received Accepted Published = 2 theories that are obtained by and Emily Nardoni N b Published for SISSA by https://doi.org/10.1007/JHEP06(2019)123 [email protected] , Ruben Minasian SCFTs. In our approach, the contributions to the ’t a S [email protected] , . = 2 class 3 N 1904.07250 Federico Bonetti, The Authors. a c Anomalies in Field and String Theories, M-Theory, Supersymmetric Gauge

We derive the anomaly polynomials of 4d , [email protected] -flux in the 11d geometry. This computation provides a top-down derivation of these 4 [email protected] 3400 North Charles Street, Baltimore,Institut MD de 21218, Physique U.S.A. Th´eorique,Universit´eParis Saclay, CNRS, CEA, F-91191, Gif-sur-Yvette, France Mani L. Bhaumik InstituteUniversity for of Theoretical Physics, California, Department Los of Angeles,E-mail: Physics CA and 90095, Astronomy, U.S.A. Department of Physics and Astronomy, Johns Hopkins University, b c a Open Access Article funded by SCOAP Keywords: Theory ArXiv ePrint: nomials for the 4d Hooft anomalies due toG boundary conditions at thecontributions punctures that utilizes are the determined geometric entirelyprevious definition by field-theoretic of the arguments. field theories, complementing the Abstract: wrapping M5-branes on a Riemann surfaceinflow with in arbitrary regular the punctures, corresponding using anomaly M-theory setup. Our results match the known anomaly poly- Ibrahima Bah, Anomaly inflow for M5-branes onsurfaces punctured Riemann JHEP06(2019)123 3 5 34 32 12 I 35 12 37 I fibers 26 14 11 4 20 S 10 term 40 44 23 24 3 8 ) 32 4 23 E 9 21 – 1 – 29 SCFTs 40 12 20 S I for a puncture term 4 8 term 37 E X ) 3 30 10 ) ∧ 4 34 g,n 4 6 3 E 33 = 2 class E (Σ inf N 6 I 12 I away from punctures 8 4 E 2 A.5 Descent formalismA.6 and integration along Computation of A.1 Conventions for characteristicA.2 classes Global angularA.3 forms Bott-Cattaneo formula A.4 Derivation of 6.2 Anomalies of6.3 the Relating inflow6.4 data to Young Matching diagram CFT data and inflow results 5.1 Computation of5.2 the ( Computation of the 6.1 Summary of inflow anomaly polynomial 3.3 The bulk contribution to anomaly inflow 4.1 Warm-up:4.2 reformulation of a non-puncture Local geometry and form 2.2 Four-dimensional anomalies2.3 from integrals of Inclusion of punctures 3.1 Normal bundle3.2 to the M5-brane The stack form 2.1 Anomaly inflow and the class D Review of Gaiotto-Maldacena solutions E Proof of matching with CFT anomalies B Evaluation of the integralC for the Free ( tensor anomaly polynomial A Global angular forms, Bott-Cattaneo formula, and 7 Conclusion and discussion 5 Puncture contributions to anomaly inflow 6 Comparison with CFT expectations 4 Introduction of punctures 3 M5-brane setup Contents 1 Introduction 2 Outline of computation JHEP06(2019)123 ] ]. 9 N 16 = 2 ]; in (1.1) – N 14 13 , 12 ], building 5 , ], building on 4 2 depends on the , 1 In an interacting S can preserve 2 used in the compact- S g,n was first computed in [ . S = 2 theories were identified ) α N P ( were constructed in [ -dimensional QFT can be organized CFT 6 d S I ) depends on the choice of boundary α n =1 P X α ]. ( ]. ], 11 ) + , CFT 6 18 ]. , I 17 – 2 – g,n 10 7 21 , , = 2 theories of class – , which is the worldvolume theory on a stack of (Σ 5 1 5 , , N − 4 4 19 , N CFT 6 9 -punctured Riemann surface Σ = 2 theories were first constructed in [ I A n , = N = 1 theories of class g , and contains information about the ’t Hooft anomalies of α N CFT 6 P The can be decomposed as a sum of a “universal” or “bulk” term, I ) depends on the surface only through its Euler characteristic, 1 , and is insensitive to the choice of boundary conditions at the program, in which 4d QFTs are obtained by dimensional reduction n g,n CFT S 6 − I = 1 theories in [ (Σ 1) N CFT 6 − I ]. Strong evidence for the existence of these SCFTs is the construction of g 8 gravity duals. The holographic duals of the – 2( 6 N − ]. A large class of 3 = 4 is obtained by compactifying on a torus with no twist. + 2)-form known as the anomaly polynomial, which is a polynomial in the curva- ) = N d = 1 supersymmetry. ], and for the The individual puncture contribution The 6-form ’t Hooft anomaly polynomial for a 4d theory of class ’t Hooft anomalies provide crucial insight into the properties of QFTs, and are es- g,n 4d Throughout, we refer to anomalies in background (rather than dynamical) gauge or gravity fields as ’t 9 N 1 2 (Σ conditions at the puncture the flavor symmetry associatedand to anomaly matching it. arguments [ These contributions can be obtained byHooft S-duality anomalies. χ punctures. This contribution forusing S-duality, the and can be computed6d by theory integrating over the the 8-form Riemann anomaly polynomial surface of [ the and of individual terms for each puncture [ The bulk term The geometric naturegeometrically of engineered anomalies constructions. makes them especially amenableparent to 6d computation theory, on in theification, genus- and on theanomaly boundary polynomial conditions for the 6d theory at the punctures. The total SCFT, anomalies are related toa central free charges theory, by they thethe directly superconformal degrees specify of algebra freedom the [ in matterin a content. a QFT. The ( Thus, anomalies they oftures a provide of background a gauge measure and of gravitational fields associated to global symmetries [ on work in [ their large- in [ pecially useful observables in the study of strongly coupled theories. of a 6d SCFT,case generically of with the 6d a (2,0) partialM5-branes. theory topological of Depending type twist. on the Inor choice this of work twist, we thework focus theories in on of [ the class Geometric engineering has become afield standard tool theories, for constructing especially andstrongly exploring in quantum coupled their QFTs strong inof coupling four M5-branes regimes. dimensions on is aframework of realized A Riemann the class in large surface M-theory with class by defects. of wrapping generically a These stack constructions fit in the larger 1 Introduction JHEP06(2019)123 ], 3 = 2 22 -flux (2.1) along 4 N G 12 . In sec- I ) and the 0 . The 11d 6 g, g,n ]. The quan- W is a Riemann (Σ 28 – g,n , we briefly review CFT 23 6 I 2.1 to compute the puncture we provide an overview of fibration over Σ 2 5 4 S , decomposes as are “filled” with new geometries 6 , 6 6 W M ) below. Upon integrating NW 2.8 , we consider the case without punctures, ⊕ 6 ), ( 2.2 ]. The present work is a follow up to [ , which is an we compare the total inflow result with the 9 6 2.7 TW 6 – 3 – M 12 is external 4d spacetime and Σ = , restricted to I 4 6 from their geometric construction via M5-branes. 11 small disks from the Riemann surface, together with W W | 1 M n − 11 N is related via standard descent relations to the classical , up to the decoupling of center-of-mass modes. A 1 TM 12 − I , where N -flux. The latter encode all data about the punctures. A g,n 4 is devoted to the discussion of the local geometry and Σ G punctures. In section 4 , by excising ), for any regular puncture. Our analysis is inspired and motivated × 6 n coincident M5-branes with a smooth 6d worldvolume α 4 M P ( N W on a suitable 6d space . The class with = 12 12 CFT g theories of type 6 6 I I I S W 0) theory of type , surrounding the M5-brane stack, one recovers the 8-form anomaly polynomial of , we outline a two-step procedure for introducing punctures. Firstly, one constructs theories with regular punctures. In this section we provide a summary and overview fibers on top of them. Secondly, the “holes” in 4 4 S In this work we study 4d theories obtained by considering an M5-brane stack with Anomaly cancellation for M5-branes in M-theory was analyzed in [ The outline of the rest of the paper is as follows. In section The main goal of this paper is a first-principles derivation of the anomalies of the S 2.3 S = 2 class 2.1 Anomaly inflowConsider and a the stack class of tangent bundle of the ambient space by integrating tion a modified version of the supported by non-trivial the 6d (2 worldvolume surface of genus and argue that the 6-form anomaly polynomial of the resulting 4d theory can be computed cancelled by a classical inflowthis from mechanism the and 11d argue ambient that space. itacteristic In can class section be neatly summarized byanomalous introducing variation a of 12-form the char- the 11d action, see ( Our goal is an anomaly-inflowclass derivation of the ’tof Hooft the anomaly strategy polynomial used of in 4d the main computations intum this anomaly generated paper. by the chiral degrees of freedom localized on the M5-brane stack is known CFT anomaly polynomial. Infuture the directions. conclusion we Some summarize technical our aspectstogether findings of with and our useful discuss derivation background are relegated material. to the appendices, 2 Outline of computation the main strategy used inwe the describe computation in of greater the detailanomaly inflow the inflow. anomaly M5-brane polynomial. Section setup, In andconfiguration section we near discuss a the puncture. bulkcontribution These contribution to data to anomaly are inflow. used in In section section N Using anomaly inflow inpuncture M-theory, term we determine bothby the the bulk term holographic dualswhere of the results these of theories the [ computation and main features of the derivation were presented. JHEP06(2019)123 . . 5 r B R y (2.3) (2.4) (2.5) (2.6) (2.2) A , and y 6 and the AB A δ transverse NW y 5 = R 2 r for the explicit fibration over the ). This is achieved 4 fibers of A.2 S , 2.2 5 4 , R . E 5 6 dy W ∧ 4 inside → dy ) should only be considered as a of the M5-brane stack. Next, we 10 N . ∧ , which is an  2.2  M 4 3 S . The classical variation of the action . of the M5-brane stack, the 11d space- = 3 dy → ) must be improved in two respects in ,  4 C r vol 4 ∧ E B 2.2 ∧ N. 2 at surrounding the origin of the ∧ is no longer invariant under diffeomorphisms of radius dy = ], ( 4 10 denoting the radial coordinate  df  4 S ∧ r is constructed with the coordinates M ,S B 26 – 4 – E B 1 , = ) π N df 4 \ 4  is isomorphic to a small tubular neighborhood of 4 S dy 25 E 11 π B , and approaches 0 monotonically as we increase Z ) 6  = 2 2 y \ dG M ( 4 ), with 5 = r 11 ( NW . We refer the reader to appendix on r dG 6 f M ( M = 0 is modified to ∂ π N δ S 4 1 at NW ≡ − = 2 dG 10 on 4 ] 5, are local Cartesian coordinates in the M dG AB [ are the tangent bundle and normal bundle to the M5-brane stack, with a multiple of the global angular form, ,..., 6 4 is the volume form on the ) is thus replaced by , S . 4 is equal to 6 4 S . From this point of view, the M5-brane stack sits at the origin of the = 1 2.2 , which encode the five directions transverse to the stack. The normal bundle f NW E vol W 11 6 , A N 6 M acquires a non-trivial boundary , NW A 11 TW y M After excising a small tubular neighborhood The second step is to gauge the SO(5) action of the normal bundle. This requires that In the first step, we regularize the delta-function singularity in ( The M5-brane stack acts as a singular magnetic source for the M-theory 4-form flux ) is the standard 5d delta function. The relation ( inside y . The Bianchi identity 6 ( 4 5 time worldvolume The M-theory effective action and gauge transformations of the M-theory 3-form The closed and SO(5) invariantSO(5) 4-form connection Θ expression of Let us stress that, in our notation, we absorb the factor The 4-form vol directions, normalized to integrate to 1. we replace by excising a smallintroduce tubular a neighborhood radial bump function The function The relation ( where δ schematic expression. As explainedorder in to [ implement anomaly inflow. the branes. G respectively. The normal bundle W fibers of admits an SO(5) structure group.on the It brane; induces this an is SO(5) identified action with onto the the R-symmetry degrees of of the freedom quantum field theory living on where JHEP06(2019)123 0 4 ]. g, S 26 (2.7) (2.8) (2.9) , is the (2.11) (2.12) (2.10) 25 decoup 8 I . ) denote its Pon- A.5 11 ). 2.9 TM , together with one free ( 1 i − p , N . i 0) theory, and A ) , 8 12 . 11 I , X stems from the fact that, upon , and 12 ∧ 11 I 12 is external 4d spacetime, and Σ TM 4 ], = 0 I ( M = 4 2 , E is implicit in ( . p 26 W , − (0) 11 4 (1) 10 12 10 4 I is related via descent to a formal 12-form I originates from the topological couplings I 25 decoup 8 − M 4 E I 10 2 S 12 (1) 10 ) ∧ Z M I + I 4 11 Z 0) theory of type , where = , d – 5 – E , 0 = g, CFT 8 ∧ TM (0) 11 I ( Σ 4 M I π 1 for a review of the derivation, based on [ inflow 8 δ E 2 cancels against the quantum anomalies of the chiral + p × δS I h 1 6 4 = A.4 1 − 0) tensor multiplet. without punctures. In such a setup, the structure group W 192 , inflow (1) 8 10 inflow 8 = g I I I = = d 6 12 8 I transverse to the M5-brane stack, we obtain the inflow anomaly W X 4 S ) makes use implicitly of the fact that descent and integration over 2.11 is the tangent bundle to 11d spacetime is the 8-form characteristic class is the anomaly polynomial of the interacting (2 is a 10-form proportional to the gauge parameters. By virtue of the Wess- 8 11 X (1) 10 CFT 8 I TM I ), 0) tensor multiplet, related to the center of mass of the M5-brane stack. We may , The anomaly polynomial The relevance of the 12-form characteristic class under such a transformation takes the form 2.9 M 2.2 Four-dimensional anomalies fromThe integrals discussion of of theM5-brane previous worldvolume subsection is isis readily a specialized Riemann to surface the of case genus in which the 6d (2 then write where anomaly polynomial of a free (2 Notice that ( commute. We offer an argument for the previous statementdegrees in of appendix freedom oninteracting degrees the of M5-brane freedom stack. of the In 6d (2 the IR, the latter are organized into the integrating it along the polynomial of the 6d theory living on the stack [ where tryagin classes. Let us stress that a pullback to We refer the readerIn ( to appendix characteristic class, We are adopting a standardpower descent of notation, the with variation the parameter.in superscript the The M-theory (0), class effective (1) action, indicating and the is given by where Zumino consistency conditions, the quantity S JHEP06(2019)123 4 4 ). to be S S α 2.6 . . We =0 (2.18) (2.16) (2.17) (2.13) (2.14) (2.15) P . The n SO(2), 6 0 around as is an g, is an 3.1 M α × D =0 , . . . , n =0 n 6 n 6 bulk 6 M = 1 M M . The connection α 0 g, Σ punctures. Let × 4 n SO(3) or SO(2) W denote a small disk on the × α with , D ) g . 4 for the 4d field theory is given by S ) to the case of a compactification below. . . and a 6d space, denoted . = 0 × , 4 0 4 ). Let by excising the small disk α 3.1 2.15 12 g, g,n inflow 6 W , W I D fibration over I ) generically fails in the case of emergent Σ Σ 4 of genus ( =0 balances against the contributions of an 4 puncture is located, for 2.13 n = 1 supersymmetry, respectively. A more → 6 decoup =0 6 W n . We can present the space =1 S → n → I 6 [ g,n th α α M 2.16 10 N and an internal part with legs on Σ M α P + ∪ =0 Z M inflow 4 6 n bulk 6 6 as in ( – 6 – I is a background gauge field for the continuous might not capture all the anomalies of the CFT. = W → M M , CFT 6 bulk 6 6 =0 I fiber on top of it. It follows that = 2 or n 6 M → → =0 , , inflow + 6 ) is now an 4 n inflow 6 NW 6 4 M I 4 N = I S S S M 2.6 fibration over =0 inflow 6 n 6 I =0 n 6 . We stress that ( M is reduced from SO(5) to SO(2) 0 M 6 g, NW ). We should bear in mind that, in analogy with the uncompacti- as an decomposes as the product of 2.9 introduced in ( , 10 , 0 10 10 M g,n g, M M , together with the denotes the space obtained from α P given in ( bulk 6 12 M I Our starting point is the space We argue that the inflow anomaly polynomial The space 0) tensor multiplet on Σ , where each point fibration over Σ the point on the Riemann surface where the Riemann surface, centered around the point symmetries in the IR, in which case 2.3 Inclusion ofLet punctures us now outline aof general an strategy for M5-brane extending stack ( on a Riemann surface Σ interacting CFT as well as of decoupling modes, The decoupling modes are(2 precisely those arising from the compactification of a free 6d with fied case, the inflow anomaly polynomial It is fixed bycan the now supersymmetry regard conditions of M-theory, as discussed in section The topology of the above fibration encodes the information originally contained in ( global symmetries of theoff, 4d field the theory. space Whenemphasize these that background we gauge are fieldsfibration considering are over a Σ turned setup with no punctures. The space for compactifications preserving 4d detailed explanation of this point is found in section splits into an externalexternal part part with of legs on the connection on of the normal bundle JHEP06(2019)123 ). an and 2.18 ), the (2.23) (2.19) (2.21) (2.22) (2.20) α 6 not X is 2.14 theories. fibers, and α 6 4 S X S , 6 . In this work we near the puncture, 10 . M 4 , M 12 E ) = 2 class ). First of all, we stress ) with a new geometry I α α 6 P N ), the space -flux configuration, which , ( X 4 2.21 2.17 Z G , 2.13 4 , and the individual contribu- . In analogy with ( in ( inflow inflow 6 8 ) = ) and captures the anomalies of W I I 4 α bulk bulk S 6 6 . P n =1 g,n ( 2.11 X α × M M Σ α . 6 , together with Z α 4 X α 6 ) + D inflow = 6 W X n =1 onto [ g,n α 12 → α I 6 ∪ (Σ . Moreover, the gluing must preserve all the 10 X ,I bulk 6 – 7 – M is given by ( bulk 6 bulk 12 . In contrast with ( M 6 inflow 6 I α Z I M → -flux configuration are constrained by several con- , , associated to M P 4 6 = = inflow 8 G bulk , 6 ) = I 6 M α 6 12 M is performed by first integrating along the , both in the bulk of the Riemann surface and near each I g,n Z M . The geometry X ) implies that the total inflow anomaly polynomial can be 4 6 ) is supported by a non-trivial as a space with boundaries. Accordingly, one has to assign 10 E (Σ M bulk ) = Z 6 M and its 2.19 2.19 bulk 6 M g,n = α 6 inflow on 6 in ( I M fibration over external spacetime X in ( 4 (Σ . The class is understood as a globally-defined object on 6 6 E α 6 bulk contribution g,n inflow 12 6 M M X I I ] constrains the gluing of inflow 6 I 6 in 17 4 M is an configurations that describe regular punctures for E 10 4 E M below is devoted to describing all the relevant features of the geometries The local geometry Several comments are in order regarding the decomposition ( The structure of The class To introduce punctures, we replace each term smoothly onto the bulk geometry that encodes the puncture data. We denote the resulting space as 4 fibration over a 2d base space. α α 6 6 4 sistency conditions. AsX mentioned earlier, werelevant symmetries must of the be problem able (includingtion the to correct amount glue of the supersymmetry).associated local Sec- geometry where the integration over then integrating on Σ the 6d (2,0) SCFT that lives on a stack of flat M5-branes. that one should think of suitable boundary conditions atNotice the also punctures that for [ the connection in the fibration ( where one has turn out to be enough to determine the inflowwritten anomaly as polynomial. a sumtions of of a punctures, associated to encodes the details of theS puncture at construct local expressions for puncture, which are then constrained by regularity and flux quantization. These conditions 10d space Each local geometry is encoded in the class X Smoothness of JHEP06(2019)123 . 6 is fiber (3.2) (3.3) (3.4) (3.5) (3.1) NW SO(2) 4 for the SO(3) N S 1 . φ fibration N S , which is depending = 2 super- M5-branes 2 iφ e S n g,n 2 N N × µ 1 − ). We proceed to S wraps a Riemann 1 6 p 2.13 W r as an decomposes according to = 6 4 5 SO(3). Accordingly, 0) theory on Σ S W ) and ( , i y × n . are identified with the + 2.4 6 − 4 denote the Chern root of , . n 1) punctures. In particular, we recall NW − g,n n SO(3) , g Σ , y N 4d T 2( and structure group SO(2), and ⊕ n − ⊕ with 2 = 1 + ˆ 4 R 2 g ˆ t ) , and is necessary to preserve 4d ) = 1]. This is readily achieved by the following ). 3 – 8 – SO(2) , introduced in ( − , y TW g,n g,n N [0 =0 = = 2 supersymmetry in four dimensions, in which = n 2.21 6 + (ˆ and satisfies = (Σ ˆ n 6 ∈ 2 ˆ t χ 6 M N ) of genus 2 µ with fiber 5: = y TW ˆ t 6 NW g,n depending on external spacetime. The part of ˆ + (ˆ W reduces from SO(5) to SO(2) g,n n punctures. The tangent space to R-symmetry of the 4d theory. We refer the symbol ,..., 2 Σ ) suggests a presentation of the ) 6 n Z 1 fibration R y is denoted = 1 3.3 4 (ˆ . This identification amounts to a topological twist of the parent ˆ t S NW for the 2-sphere defined by the second relation. The isometries of A ) according to ( and structure group SO(3). Let ˆ with g,n − , 2 Ω 3 g Σ g,n A S R , y T 3 (Σ , 2 , theory compactified on Σ 1 ) in the absence of punctures and in the presence of punctures, respectively. ˆ . This enables us to compute the bulk contribution to the inflow anomaly y is a bundle over 1 inflow 6 and of the ]. The angular directions in the fibers of − of genus I 2 4 r µ bulk N 2.18 6 denotes the part of ˆ E g,n A = SO(2) M is fixed to be 4d ), ( 3 ). We are interested in the case in which the worldvolume N n , 0) 2 , , g,n The decomposition ( We consider setups preserving 1 2.1 2.13 are related to the SU(2) y 2 Ω parametrization of We use the symbol S 6d (2 symmetry [ in ( over an interval with coordinate a bundle with fiber We can write where ˆ on Σ the structure group of decomposes in a direct sum, where The Chern root of The 11d tangentto space ( restricted tosurface the Σ M5-brane worldvolume decomposes according the class analyze polynomial 3.1 Normal bundle to the M5-brane stack This section is devoted towrapping the description a of Riemann the surface M-theorythe setup Σ properties of a of stackimplementing of the a normal partial bundle topologicalessential to twist to the preserve of supersymmetry the M5-branes in parent in four 6d dimensions. this (2 We scenario, then discuss and the its properties of role in 3 M5-brane setup JHEP06(2019)123 . , is π A 4 d and E (3.9) (3.6) (3.7) (3.8) (3.12) (3.10) (3.11) = ), then F SO(2) 2.5 is N = 0. ,( is closed and 4 µ A . It is natural E 2 . Regularity of is a closed form E 4 can be split into S 4 bulk 6 has periodicity 2 E 3 F M φ reduces to a multiple Ω 2 e shrinks for and 2 Ω A S . , fibration . The 2-form ) φ of the form 4 . F , the 4-form 4 g,n S .  A.2 SO(3) symmetry, one verifies that + E (Σ π 2.3 . Both Σ × F . The field strength of 2 χ n F reads , γ . − . = φ Ω 2 − = SO(2) (1) = 1 e F − A N Σ π π ∧ = 1 2 2 F Dφ (0) = 0. The normalization of according to R-symmetry in four dimensions. As apparent 2 connection, the form Ω 2 dφ γ E r e is the global, SO(3) invariant angular form for ∧ Ω 2 – 9 – g,n , e , γ = 2 Ω = Σ Ω 2 φ S dγ e Z 4 in the bulk of the Riemann surface, i.e. away from Z SO(3) A  = 1, while the 2-sphere E Dφ 4 = N + N µ E π Σ . Throughout this work, the angle F (0) = 0 2 = ) and use a factorized A φ γ 2 E g,n can be found in appendix = 3.3 Σ Ω 2 Z A e shrinks demands . We normalize shrinks for 2 2 Ω Ω becomes proportional to the volume form on an S S 1 φ 4 away from punctures S ) is constrained by regularity conditions. If we turn off all E µ corresponds to the U(1) 4 ( 1 γ φ E S = up to the exterior derivative of a globally-defined 3-form. γ Ω 2 e bundle. If we turn off the ) is identified with the Chern character ˆ ∧ π ), the circle is the total connection for the bundle 2 E (2 connections, ), we have A / (1) = 1. To summarize, 3.5 SO(3) γ F The gauge-invariant differential for the angle N 3.4 By writing down all possible terms compatible with SO(2) in the region where 3 SO(3) 4 E fixes given by The function N The explicit expression for gauge-invariant. We can write Let us explain ourthe notation. The form of the volume form on 3.2 The form In this section wepunctures. discuss the As form per theinvariant general under discussion the of action subsection ofto exploit the the structure decomposition group ( of the where the first termto is ( the internal piece, and the second is the external piece. Thanks where and an internal part, withspacetime. legs We on use the the notation Riemann surface, and a part with legs along external The isometry of from ( circle parametrized by the angle JHEP06(2019)123 1 φ α 6 R S ) is X ) to π (2 (3.14) (3.15) (3.16) (3.13) g,n / normal 4 denotes . ), which (Σ G i for more 2 Ω 4 2 S C 2.5 ) R r 1 R-symmetry SO(3) ) are the first × c inflow . A.6 6 = 1, where r N I i 4 ( g,n µ 2 ) SO(3) − ) 4d N 4 ( n 1 (ˆ p TW − ( ), in constraining the local ) 1 over any 4-cycle must be 4 , at fixed p 4 in the bulk of the Riemann 2 Ω , 4 h S 4 R 2 r 1 E TW SO(3) c ( E 1 ). 4 N , p N c ( are straightforward in the bulk of − ) 1 ) 2.4 4 p then takes the form E g,n ], which are not relevant in our setup. The g,n ) = 29 ) + (Σ (Σ 4 ) χ to have mass dimension 3. The coupling of an inflow 6 ). In the integral above, Σ SO(3) 3 I N χ 1 TW 24 C N ( SO(3) ( = 1 − 1 3.12 N p 4 ( h R 2 – 10 – 1 E c ]. We refer the reader to appendix in the path integral measure. The quantity p 4d 2 r 1 Ω ). The quantities 3 c ˆ 30 S n 4d C ) are given in terms of the 4d Chern classes as ) × , p ˆ 3.4 n R ), and ( N r 1 i N 3 e g,n ) c Σ N − SO(3) Z g,n ). The derivation follows standard techniques, and makes 3.11 ) 3 N = 2 ( N (Σ g,n 1 ), ( χ 4d p 2.22 ˆ (Σ n ) (4 3.9 1 χ 48 . This analysis can be carried out separately for each puncture. , up to the effects discussed in [ g,n and 4 A trivial example of a flux quantization condition is ( 1 ), ( − 12 C (Σ counts the total number of M5-branes in the stack. A more interesting 2.3 4d 4 is integrally quantized then follows from ( 3.8 χ 4 n 4 ) = is a shorthand notation for the second Chern class of the 4d SU(2) 1 6 E was introduced in ( E − g,n R 2 4d c is realized with a factor n (Σ 3 ) = C is a shorthand notation for the first Chern class of the 4d U(1) g,n inflow 6 r 1 I (Σ c The quantities ˆ Let us stress that, in our conventions, the integral of We take the components of the 3-form potential 4 inflow 6 I In this section we discussintroduced punctures in and section analyze the properties of the local geometries M2-brane to an integer for anyfact 4-cycle that the flux of 4 Introduction of punctures where bundle, while R-symmetry bundle. The bulk contribution to The notation ˆ Pontryagin classes of thebundle, tangent respectively. bundle to external spacetime, and the use of a resultdetails. of The Bott result and reads Cattaneo [ 3.3 The bulkIn contribution the to previous anomaly section inflow wesurface. have fixed We a are local thereforeanomaly expression in inflow, for defined a in position ( to compute the bulk contribution which follows from ( the 4-cycle obtained by combiningshrinks. the Even Riemann though flux surface quantization and the conditions Riemann for surface, theypuncture will geometries play an and essential flux role configurations. in section integrally quantized. simply states that example of flux quantization is the relation JHEP06(2019)123 , 2 Ω is α 4 on S X D Σ (4.5) (4.2) (4.3) (4.4) (4.1) × . For A 1 φ . Σ for S A Σ 4 A X with a new , − 4 α 6 S that will prove dφ . The disk X plot (a). 4 4 × , which is in turn = S 1 E 4 for D × ) plane, described by X 6 X for punctures, we first D , µ , is realized as an Σ 6 = 2 r is defined at the center of 4 corresponds to a region of X 6 , Dφ S and seek a reformulation of Σ dχ ) X D ) A 4 2 Ω 2 S S µ ( 2 × 2 β , , in the vicinity of a puncture. − , and write ) R 1 ds α 4 D is simply 2 (1 Ω 2 , E ≤ 6 S µ 2 Σ 2 = ( r X ). As usual, µ 2 6 + dβ , , defined by , a non-trivial puncture can be described ) Dβ 2 + β . ds ≤ , β X χ Σ 2 β 2 Σ 2 0 r + r 2.3 µ ( 2 µ R Dφ is a half strip in the ( φ 0 to ensure that fibration over a 4d space ) U − 2 dµ µ – 11 – 2 Ω = ) + µ ) + 1 → = 4 , 3 S Our starting point is χ − 0 Σ Σ X B + and r ( ( A 2 2 2 Σ ) in the form ≥ Σ , i.e. we set r + (1 Σ ds ds dr 4.1 r 2 2 µ ) = ) = ) = 6 4 3 − from the Riemann surface and replacing dµ B 1 X X ( ( ( is fibered over the Riemann surface with a connection constant, which is the shaded region in figure D 2 2 2 + . We also provide a reformulation of the class 0 1 φ 3 2 r ds ds ds S R goes to zero as non-puncture dβ interval. The line element on can be recast as an U 2 Σ r µ 4 , with S + 0 2 Σ × . In order to gain insight into the properties of < r dr 6 can be taken to be of the form plot (a). More precisely, the interior of the disk D Σ X r 1 D = ) = 6 6 X X ( Let us introduce a new angular coordinate 2 ds We can rewrite the line element ( see figure the form the disk where the function the disk. The 2d space spanned by We have recalled that simplicity, we have temporarily turned off all external connections. The connection a circle fibration over beneficial in the discussion of genuine punctures. Geometry for the non-puncture. parametrized by standard polar coordinatesfibration ( over the by removing a smallgeometry disk analyze the case ofthis a trivial geometry that isthat best suited for generalizations to non-trivial spaces. We show and so on. We demonstratesuitable that circle the fibration, puncture and data we are analyze encoded the in form4.1 monopole of sources for a Warm-up: reformulationAccording of to a the non-puncture strategy outlined in section Therefore in what follows, we omit the puncture label JHEP06(2019)123 , Σ 1), r , 1) is (4.7) (4.8) (4.6) plot ) = 0 circle , 1 , µ ) = 1 on = 0 for Σ plot (a). plot (a), 0), which Dβ 0. Second r ) = (0 , µ ( 1 . U 1 ) = (0 Σ ≥ ≥ L ) plot (a). , µ r parametrized 2 ( Σ ρ , µ 1 Σ µ 3 r L Σ . Figure = 0, as opposed r 0, B , r are related to − 2 is related to the / η ≥ max (1 , η Dβ , 1), with = 1 , η 2 2 , ρ ) ) ) ≥ µ . The latter is in turn 2 1) where the by U 4 , , µ η 4 max ) X plot (b). First of all, the η 2 X 1 1 + ) = (0 − 2 dχ / − η 2 + (1 + ) = (0 , µ 1 1) and ( ρ Σ circle shrinks is ( 2 Σ 2 Σ . The factor 1 ) strip. Moreover, , µ r r r ) plane with + ( 3 + ≤ ( Σ 2 , µ r 2 R 1 2 = ρ Dβ 1) is a specific example of a general µ Σ ρ, η , 2 β r dη + p ≤ . The coordinates semi-axis. The corner ( over the 3d base space + π = 2 0 0), the thick black line in figure η max , η dρ ) = (0 ≥ ( ,R L dχ = (TN) near ( Σ = 0. Finally, the union of the dot-dashed blue = 0 2 2 , µ V – 12 – µ Σ , r 2 axis. The dot-dashed blue line is mapped to the β η ds Σ r (which is specified in + r R − η 2 1 = 0 fibration over a 4d space dV , V , η µ 3 2 Dβ Ω 2 = R 1 S ) and µ 4 − 1), i.e. on the dot-dashed blue line. Similarly, ) near ( has periodicity 2 on the 0), i.e. on the dashed red line. − X V − ∗ ( ≤ 1 β fibration has a monopole source. We write the Taub-NUT 4.8 ≥ 2 = near this point can be modeled by a single-center Taub-NUT µ p max as an Σ ds η 4 Σ ≤ 6 Dβ , r r X = 0 X dDβ , (TN) = = L dχ , L η = 1 , while the dashed red line is mapped to 2 is smooth in the interior of the ( ρ has a discontinuity at the point ( − µ ds = 0 in the 3d base space L ) strip, the only point where the max fibration with connection L Σ η dβ , µ r 1 β Σ S = ≤ are standard cylindrical coordinates on shrinks on the locus ( r circle = 0) shrinks on the loci ( η 2 χ Ω ). We can make the following observations: χ , S Dβ η ≤ dβ , ρ , µ, χ the locus ( on the locus ( to correspond to the dot-dashedrespectively. blue line and the dashed red line in figure where the dot-dashed blue line and the dashed red line meet in figure ) half strip is mapped to the quadrant in the ( Σ The The function In the ( The The coordinate change ( We see that r , µ (i) by Σ (ii) (iv) r (iii) ( of all, the thickline black and line the is dashed mappedmapped red to to line the is point mappedregion to 0 the as verified bysimplicity. comparing class of maps with the qualitative features depicted in figure where fact that, in our conventions, µ space, showing that the metric as shrinks. The metric on by ( We have reinterpreted written as an where we have introduced JHEP06(2019)123 on on η µ (4.9) away (4.10) (4.11) (4.12) (4.13) 4 E ) we have introduced in the new . If we turn φ fibration over D 3.11 g Dβ 2 Ω . ⇢ AAAB4nicbZDLSsNAFIZP6q3WW9Wlm2ARXEhJWkGXBTcuK9gLtKFMJqfN0MlkmJkUSugLuBDEhS58IR/BtzGp2bT1X32c/1z4jy8508ZxfqzS1vbO7l55v3JweHR8Uj096+o4URQ7NOax6vtEI2cCO4YZjn2pkEQ+x54/fcj93gyVZrF4NnOJXkQmgo0ZJSYvDVUYj6o1p+4sZW+CW0ANCrVH1e9hENMkQmEoJ1oPXEcaLyXKMMpxURkmGiWhUzLBlERazyN/YV9FxIR63cuL/3mDxIzvvZQJmRgUNGvJvHHCbRPbeQw7YAqp4fMMCFUsu2zTkChCTRZ2dZNGQSLUN3YwY1Iv2UuXn1tUsuzuetJN6DbqbrPuPt3WWo3iC2W4gEu4BhfuoAWP0IYOUAjhFT7g0wqsF+vNev9rLVnFzDmsyPr6BUkjixc= for S ) φ and U , the form A with ( (dashed, grey) are also 2 Σ Dχ 3.2 E r (1 + ) with the desired properties, . AAAB4XicbZC7TgJBFIbP4A3xhlrabCQmmBiyC4WWJDaWGOWSwIbMDgeYMDs7mZklIRsewMLEWGjhE/kIvo0LbgP4V1/Ofy75T6AEN9Z1f0hua3tndy+/Xzg4PDo+KZ6etUwUa4ZNFolIdwJqUHCJTcutwI7SSMNAYDuY3C/89hS14ZF8tjOFfkhHkg85ozYtPZWD636x5FbcpZxN8DIoQaZGv/jdG0QsDlFaJqgxXc9V1k+otpwJnBd6sUFF2YSOMKGhMbMwmDtXIbVjs+4tiv953dgO7/yESxVblCxtSb1hLBwbOYsUzoBrZFbMUqBM8/Syw8ZUU2bTrKubDEoaorlxBlOuzJL9ZPm4eSHN7q0n3YRWteLVKt5jtVSvZl/IwwVcQhk8uIU6PEADmsBgBK/wAZ+EkRfyRt7/WnMkmzmHFZGvX7uSihk= (b) . The shaded regions in both ρ, η 3 Σ N γ , parametrized by cylindrical on the horizontal axis, and r 3 R on the horizontal axis, and ). In light of the results of the ρ R Σ ) to ( ) = → r , and 4 , µ  3.11 µ Σ ρ, η X r , thus obtaining the quantity π ( AAAB4nicbZDLSsNAFIZPvNZ6q7p0EyyCCylJFXRZcOOygr1AG8rJ5LQdOpmEmUmhhL6AC0Fc6MIX8hF8G5OaTVv/1cf5z4X/+LHg2jjOj7WxubW9s1vaK+8fHB4dV05O2zpKFKMWi0Skuj5qElxSy3AjqBsrwtAX1PEnD7nfmZLSPJLPZhaTF+JI8iFnaPJSnwwOKlWn5ixkr4NbQBUKNQeV734QsSQkaZhArXuuExsvRWU4EzQv9xNNMbIJjijFUOtZ6M/tyxDNWK96efE/r5eY4b2XchknhiTLWjJvmAjbRHYeww64ImbELANkimeXbTZGhcxkYZc3aZIYkr62gymP9YK9dPG5eTnL7q4mXYd2vebe1Nyn22qjXnyhBOdwAVfgwh004BGa0AIGY3iFD/i0AuvFerPe/1o3rGLmDJZkff0CMuCLCA== ⌘ . 2 g Dβ max → , φ Dβ AAAB7HicbZDLSsNAFIYnXmu9VV26GSyCCylJFXRZcOOygr1AE8vJ9LQdOjMJM5NiCXkLF4K40IXv4iP4Nqa1m7b+q4/znwv/CWPBjXXdH2dtfWNza7uwU9zd2z84LB0dN02UaIYNFolIt0MwKLjChuVWYDvWCDIU2ApHd1O/NUZteKQe7STGQMJA8T5nYPPSk48WuqmvJZXwnHVLZbfizkRXwZtDmcxV75a+/V7EEonKMgHGdDw3tkEK2nImMCv6icEY2AgGmII0ZiLDjJ5LsEOz7E2L/3mdxPZvg5SrOLGoWN6Se/1EUBvRaSDa4xqZFZMcgGmeX6ZsCBqYzWMvbjKoQKK5pL0xj82Mg3T2w6yYZ/eWk65Cs1rxrirew3W5Vp1/oUBOyRm5IB65ITVyT+qkQRjR5JV8kE9HOS/Om/P+17rmzGdOyIKcr1+l8o9k ⌘ 1 β W A L Dχ . can be reformulated as an constant. − − 4 ) strip, with ,W 0 − given by ( fibration over S r π i inside ) quadrant, with dχ 2 , µ in terms of the 1-forms As explained in section 1 β 2 × L Dχ E dβ Σ – 13 – ,S 2 S ) r = 4 ρ, η E D Y = Dχ U  X , with = d 0 Dχ 6 g Dβ → ) with = ⌃ (solid, grey) and lines of constant . In order to match the above (1 + X with ). < r 6 r AAAB5nicbZDLSsNAFIZP6q3WW9Wlm2ARXEhJqqDLghuXFe0F2lBOpqft0JkkzEwKJfQVXAjiQhe+jo/g25jGbNr6rz7Ofy78x48E18ZxfqzCxubW9k5xt7S3f3B4VD4+aekwVoyaLBSh6vioSfCAmoYbQZ1IEUpfUNuf3C/89pSU5mHwbGYReRJHAR9yhiYtdVS/98RHEvvlilN1Mtnr4OZQgVyNfvm7NwhZLCkwTKDWXdeJjJegMpwJmpd6saYI2QRHlKDUeib9uX0h0Yz1qrco/ud1YzO88xIeRLGhgKUtqTeMhW1CexHFHnBFzIhZCsgUTy/bbIwKmUkDL2/SFKAkfWUPpjzSGXtJ9r15Kc3uriZdh1at6l5X3cebSr2Wf6EIZ3AOl+DCLdThARrQBAYCXuEDPq2x9WK9We9/rQUrnzmFJVlfvxOJjLg= 2 η µ Σ 3.9 E X − , r dχ should be replaced everywhere by 1 ρ D.18 h → , dχ 2 Ω S N γ AAAB4XicbZDLTsJAFIZP8YZ4Q126aSQmmBjSwkKXJG5cYpRLAg05HQ4wYTptZqYkpOEBXJgYF7rwiXwE38aC3QD+qy/nP5f8x48E18Zxfqzc1vbO7l5+v3BweHR8Ujw9a+kwVoyaLBSh6vioSXBJTcONoE6kCANfUNuf3C/89pSU5qF8NrOIvABHkg85Q5OWnsp43S+WnIqzlL0JbgYlyNToF797g5DFAUnDBGrddZ3IeAkqw5mgeaEXa4qQTXBECQZazwJ/bl8FaMZ63VsU//O6sRneeQmXUWxIsrQl9YaxsE1oL1LYA66IGTFLAZni6WWbjVEhM2nW1U2aJAakb+zBlEd6yV6yfNy8kGZ315NuQqtacWsV97FaqlezL+ThAi6hDC7cQh0eoAFNYDCCV/iAT4tZL9ab9f7XmrOymXNYkfX1C7oWihg= (a) ) = ), 5 ρ, η ( ). for the non-puncture. ), is provided in ( Y ), respectively. We are thus led to consider the ansatz 4 are functions of ρ, η ρ, η, χ 4.8 AAAB33icbZC7SgNBFIbPxluMt6ilzWAQLCTsRkHLgI1lAuYCyRLOTk6SIbMXZmYDYUltIYiFFj6Sj+DbuFm3SeJffZz/XPiPF0mhjW3/WIWt7Z3dveJ+6eDw6PikfHrW1mGsOLV4KEPV9VCTFAG1jDCSupEi9D1JHW/6uPQ7M1JahMGzmUfk+jgOxEhwNGmp6QzKFbtqZ2Kb4ORQgVyNQfm7Pwx57FNguESte44dGTdBZQSXtCj1Y00R8imOKUFf67nvLdiVj2ai171l8T+vF5vRg5uIIIoNBTxtSb1RLJkJ2TIDGwpF3Mh5CsiVSC8zPkGF3KRJVzdpCtAnfcOGMxHpjN0ke9uilGZ31pNuQrtWdW6rTvOuUq/lXyjCBVzCNThwD3V4gga0gAPBK3zAp4XWi/Vmvf+1Fqx85hxWZH39ArFDiYU= 1 µ AAAB4XicbZDLSsNAFIbP1Futt6pLN8EiuJCSVEGXBTcuK9oLtKFMpqft0JkkzKVQQh/AhSAudOET+Qi+jWnMpq3/6uP858J/glhwbVz3hxQ2Nre2d4q7pb39g8Oj8vFJS0dWMWyySESqE1CNgofYNNwI7MQKqQwEtoPJ/cJvT1FpHoXPZhajL+ko5EPOqElLTz1p++WKW3UzOevg5VCBXI1++bs3iJiVGBomqNZdz42Nn1BlOBM4L/WsxpiyCR1hQqXWMxnMnQtJzViveovif17XmuGdn/AwtgZDlrak3tAKx0TOIoUz4AqZEbMUKFM8veywMVWUmTTr8iaNIZWor5zBlMc6Yz/JHjcvpdm91aTr0KpVveuq93hTqdfyLxThDM7hEjy4hTo8QAOawGAEr/ABn4SRF/JG3v9aCySfOYUlka9fin2Kpg== E ) indicates that 4.11 , which is in turn a non-trivial W 4 . The plot on the left depicts the ( , 4.4 X ), ( Y We have shown that the space on, ( 4.10 An example of a class of coordinate transformations from ( 5 φ where to set different from ( previous section, we seek ain re-writing ( of In particular, we must replace The form from punctures takes the form ( coordinates ( In the above discussion, weA have not included the external connection (b) also shows thecoordinates shaded ( region corresponding to the interior ofa the space disk Figure 1 the vertical axis. Linesincluded. of The constant plot onthe the vertical right depicts axis. the We ( plots include correspond the to image the of subregion lines of constant JHEP06(2019)123 is are 2 E (4.17) (4.19) (4.20) (4.18) (4.14) (4.15) (4.16) W ). Notice along the shrinks at and 4 2 Ω E 4.15 S Y , the form . As we can see, . We utilize the . dη , π max max bulk ) 6 2 dβ η ), and η M is regular everywhere, = max η dW to , η for non-trivial punctures. contains the volume form ρ ρ, η, χ − W − 6 . 4 across , Ω 2 η 3 φ X E π ( e ), by virtue of ( L 2 F R δ , . ) max . Y axis at 1) → 4.17 axis across the monopole location. − ) strip, or equivalently the entire η 4 LW max η X , µ = ( + 0) = 0 = 0 for any Σ < η < η r → , ρ, η N. Y ) and that function singularities, =0 ( ( η > η 1 β ρ L dχ , . In contrast, δ

− 3.9 ) = − for max π , and follows from the fact that η 2 Dχ max 4 dβ – 14 – ,S in the form ) ,W E , η = 4 vanish at = 2 for a puncture . E (0 X 0N for 0 6 η 4 W dη , dL X Dβ W ( E L dW → ) 0) = 0 6 + and The reformulation of the non-puncture geometry of sec- ρ, X max ) = ( is continuous along the η Y , η Y → , − ), repeated here for the reader’s convenience: (0 function is simply the jump of are discontinuous along the W dL 2 η Ω = 0. ( δ LW Y 4.9 S + δ Y η are regular in the entire ( + ) are a potential source of Y N U dY is piecewise constant, and dL = ( L Y is discontinuous at . Recall the factorized form ( and = (+ 2 γ E fibration is of the form and Y γ is again parametrized by cylindrical coordinates ( =0 1 β ρ axis, 3

S dY R , and determines the correct gluing prescription of η 6 dY X provides a natural starting point for the construction of a genuine puncture ge- , which shrinks at function singularities cancel against each other in ( 2 Ω ) quadrant. It is worth noting that Even though δ 4.1 S = 0. The ρ, η The space η Geometry for ation puncture. ometry same fibration structure ( 4.2 Local geometryWe and are now form in a positionIn to discuss this the section geometry and wenon-trivial the show form 4-cycles that of all the puncture geometry data are encoded in the fluxes of where the prefactor of the the also that the function smooth there. To check this, we write The terms This is necessary toproportional ensure to regularity of on ( Finally, we observe that both In particular, because both Along the JHEP06(2019)123 . L 1 in a max , the η η 3 (4.23) (4.24) (4.21) (4.22) = R = 1 copies has self- η p η 1 − in the base ) we imme- , a CP k max 4.22 η circle 6 orbifold singularity. χ , each = 1. ) is piecewise constant a surrounding a 1 k ) and ( , k 1. In analogy with the  k a 1 monopoles, located at Z ρ, η k / , the g TN ( . > 4 . 4.23 ≥ − L a . We introduce the notation a R = 1, p a max k is modeled by a single-center η =: p max 4 η a ` X = − η > η < η < η p 1 +1 − a Combining ( . a ` η b η > η 7 k = is measured by the discontinuity of the a   the space axis for p = + − a a b X η a a := 0. The function η η η η circle. This allows us to glue the local puncture – 15 – 0 for = = = η η η = = φ . The latter has an a i a ` fibration has only one unit-charge monopole source η ’s form a linear chain with intersection number +1 L k η 1 h 1 β a is a decreasing sequence of positive integers. As a final ` S = ’s. CP ), we have 1 =1 denotes a small 2-sphere of radius ) = 0 for p a ) = π ) than in the non-puncture case. In the base space } CP 2 2 a , η a , η dDβ S ` ρ, η, χ ρ, η (0 2 . The last monopole location is identified with a A crucial aspect of the generalization from the non-puncture to { ) quadrant is a region analogous to the shaded region in figure (0 ( S L L L Z with charge . If ρ, η a is a non-negative integer. a 2, and the − η k a , . . . , p − . We now consider several monopoles and allow for charges greater denote the resolved Taub-NUT space. In = k η a = 1 max k = 0) coincides with the η a g TN ), = a Dβ axis, with jumps across each monopole location 1, the sequence η , η η ≥ a . Let k 1 ) = (0 The charge of the monopole at In the non-puncture case, the It is also interesting to exploreThe more sign general is possibilities, inferred in from which the the non-puncture case. gluing involves Supersymmetry a demands non-trivial that all monopole charges CP 6 7 ρ, η intersection number between distinct, neighboring identification of circles. We briefly comment on thiscarry point the in same the sign. conclusion. a genuine puncture isnon-puncture the case, possibility in of theTaub-NUT a vicinity space monopole of TN charge This singularity admitsof a minimal resolution in terms of a collection of Since remark, the non-puncture geometry is recovered byOrbifold setting singularities. where the quantity diately derive the important relation connection across the base space spanned by ( We also demand This condition guarantees that,(i.e. setting along the geometry to the bulk of the Riemann surface in a straightforward way. ( For uniformity of notation, we also define along the relevant portion of the ( plot (b). The unshaded region outside is identifiedlocated with the at bulk of thethan Riemann one. surface. More precisely, we consider a configuration with but with a more general JHEP06(2019)123 : . ) π =1 =1 +1 p a p a a } } circle along a (4.28) (4.25) (4.26) (4.27) a near a , η )) and = 0 at p a χ . {C satisfy 4 {C η A ) E a 4.16 Dβ a,I . Crucially, A b p ω ( , the C a . At the point Y , repeated here 1 to denote the B . It follows that 2 is shrinking. We 2 a = − E A a 2 Ω a a has periodicity 2 max B A S , i χ ,  ) for a ). W B π is constructed as follows. ) , . . . , k η > η 1, we enforce 2 g Dβ a 4.12 2.5 +1 C − a for = 1 0, see figure ` W 1 I φ , − − , S ) , see ( a L ρ > k . π a,I , . . . , p ( N p 2 on the r.h.s. denotes the entries of b ω Dχ , we must again demand that both ) was derived under the assumption su + ( IJ 4 ) = C a Y = 1 = 0, Y E = + k  a in the interior of the interval ( ( h ) = 0 (which follow from ( 4.28 − a η d 4 by combining the arc a 1, the 4-cycle su . IJ a − E 4 B ). = − = A C 4 ( at the monopole locations implies that flux p S = . ) and the ansatz ( 2 C S 4 Y R with E a,J N π 4.16 , we first have to study the non-trivial 4-cycles E ∼ = , with b ω 2 3.9 dχ a 4 a – 16 – p B , . . . , p E ∧ C ∧ C )). While ( i = 0, is shrinking. At point a,I = 1 ) = 0, b W ω = 1, and we have recalled that a ) a ..., 4.21 Let us now discuss the structure of the form a , we use the symbol Dβ B cycles resolving the singularity. The forms k Ω 2 a . We denote the corresponding 4-cycle as ( +1 + . We then obtain e k 1 = 0, as in ( a through g TN ` Ω 2 2 Ω W and the symbol η dχ e Z 4 max S g CP TN − R E . Below we construct two families of 4-cycles, denoted a ∧ +1 6 L ). For each 2 , i.e. at a a circle in the base is nothing but ` X E k η > η + ( ( . χ = = su Y 4 =1 h dβ p a E d . The same construction can be repeated by selecting a point ) vanish at } 2 Ω ), the ∧ a . In order to ensure regularity of S in the base Ω 2 ρ, η for a puncture. ( {C e 1, it is verified that it also holds for (which follows from ( 4.22 4 4.2 a Y . As we shall see, regularity of axis, and an arbitrary point C − E ) quadrant, pick an arbitrary point Z +1 =1 η circle a by setting p a ` = } χ ρ, η a a 4 ) and axis in the region A , . . . , p E ) = {B To compute the flux of In order to analyze the properties of In the resolved geometry η a a ρ, η C , the ( A = 1 ( Z a In the second step,In we the used finalL step, we utilized a This observation allows us towe fix must a choose uniform the orientation convention convention that for ensures all 4-cycles point the by virtue of ( In the ( along the A thus obtain a 4-cyclein with the the base, topology and of an in the puncture geometry and configurations are labelled by a partition of The 4-cycles where the dots represent terms associatedin to the subsection flavor symmetry of theW puncture, discussed the Cartan matrix of The form puncture. We assume the factorizedfor form convenience, ( Poincar´edual 2-forms to the where there is no sum over JHEP06(2019)123 , , 1 2 Ω B S and along p × (4.29) (4.30) (4.31) , . . . , p a must be N S carry the 4 , depicted as follows. = 2 a = 2 = E . The bubble C a S a a a a , Y . B C C B 1 R N − = + 4 ]. As a result, we can is shrinking there, and a E , . . . , p 1 2 Ω , η C S 1 shrinks at the location of R ) shows that = 1 through all the − . ) is arbitrary, we learn that 1 β a 4 η S 1 +1 ⇢ AAAB4nicbZDLSsNAFIbP1Futt6pLN8EidCElsYIuC25cVrAXaEOZTE6boZNMmJkUSugLuBDEhS58IR/BtzGp2bT1X32c/1z4jxcLro1t/5DS1vbO7l55v3JweHR8Uj0962qZKIYdJoVUfY9qFDzCjuFGYD9WSENPYM+bPuR+b4ZKcxk9m3mMbkgnER9zRk1eGqpAjqo1u2EvZW2CU0ANCrVH1e+hL1kSYmSYoFoPHDs2bkqV4UzgojJMNMaUTekEUxpqPQ+9hXUVUhPodS8v/ucNEjO+d1MexYnBiGUtmTdOhGWklcewfK6QGTHPgDLFs8sWC6iizGRhVzdpjGiI+tryZzzWS3bT5ecWlSy7s550E7o3DafZcJ5ua6168YUyXMAl1MGBO2jBI7ShAwwCeIUP+CQ+eSFv5P2vtUSKmXNYEfn6BUYhiw0= 4.14 E ) holds for any choice of a − , η . a , a p 4.28 η form part of the 4-cycle AAAB6nicbZDLSsNAFIZP6q3WW9Wlm2ARupCSqKDLohuXFewFmlAm05N26CQZZiaFEvISLgRxoQtfxkfwbUxiNm39Vx/nPxf+4wnOlLasH6Oysbm1vVPdre3tHxwe1Y9PeiqKJcUujXgkBx5RyFmIXc00x4GQSAKPY9+bPeR+f45SsSh81guBbkAmIfMZJTorOU5A9FT5yX06EqN6w2pZhcx1sEtoQKnOqP7tjCMaBxhqyolSQ9sS2k2I1IxyTGtOrFAQOiMTTEig1CLwUvOiuLjq5cX/vGGs/Ts3YaGINYY0a8k8P+amjsw8jjlmEqnmiwwIlSy7bNIpkYTqLPTyJoUhCVBdmuM5E6pgNyk+mNay7PZq0nXoXbXs65b9dNNoN8svVOEMzqEJNtxCGx6hA12gIOAVPuDT4MaL8Wa8/7VWjHLmFJZkfP0CWl6OoA== B +1 a with the topology of an a max ]. We can also construct a 4-cycle a , . . . , p B η a a a AAAB6nicbZDLSsNAFIZP6q3WW9Wlm2ARupCSqKDLohuXFewFmlAm05N26CQZZiaFEvISLgRxoQtfxkfwbUxiNm39Vx/nPxf+4wnOlLasH6Oysbm1vVPdre3tHxwe1Y9PeiqKJcUujXgkBx5RyFmIXc00x4GQSAKPY9+bPeR+f45SsSh81guBbkAmIfMZJTorOU5A9FT5yX06IqN6w2pZhcx1sEtoQKnOqP7tjCMaBxhqyolSQ9sS2k2I1IxyTGtOrFAQOiMTTEig1CLwUvOiuLjq5cX/vGGs/Ts3YaGINYY0a8k8P+amjsw8jjlmEqnmiwwIlSy7bNIpkYTqLPTyJoUhCVBdmuM5E6pgNyk+mNay7PZq0nXoXbXs65b9dNNoN8svVOEMzqEJNtxCGx6hA12gIOAVPuDT4MaL8Wa8/7VWjHLmFJZkfP0CRCmOkQ== B . . In contrast with the case S = 0. Moreover, we can check that the A a = 0. However, is then simply obtained as 2 , η = Ω a = 1 B 1 , we can construct a 4-cycle S η η p a S AAAB6nicbZDLSsNAFIZP6q3WW9Wlm2ARupCSqKDLghuXFe0FmlBOpqft0EkyzEwKpfQlXAjiQhe+jI/g25jUbNr6rz7Ofy78J5CCa+M4P1ZhY3Nre6e4W9rbPzg8Kh+ftHScKEZNFotYdQLUJHhETcONoI5UhGEgqB2M7zO/PSGleRw9m6kkP8RhxAecoUlLnheiGTEU9lMPe+WKU3MWstfBzaECuRq98rfXj1kSUmSYQK27riONP0NlOBM0L3mJJolsjEOaYaj1NAzm9kV2Ua96WfE/r5uYwZ0/45FMDEUsbUm9QSJsE9tZHLvPFTEjpikgUzy9bLMRKmQmDb28SVOEIelLuz/hUi/Yny0+OC+l2d3VpOvQuqq51zX38aZSr+ZfKMIZnEMVXLiFOjxAA5rAQMIrfMCnJawX6816/2stWPnMKSzJ+voFnNuOIQ== N. a − B a < η < η AAAB5HicbZDLSsNQEIYn9VbrLerSTbAIXUhJVNBlwY3LCvYCbSgnJ5P22JML50wKJfQNXAjiQhe+j4/g25jWbNr6rz7mnwv/eIkUmmz7xyhtbG5t75R3K3v7B4dH5vFJW8ep4tjisYxV12MapYiwRYIkdhOFLPQkdrzx/dzvTFBpEUdPNE3QDdkwEoHgjPJSuz/xY9IDs2rX7YWsdXAKqEKh5sD87vsxT0OMiEumdc+xE3IzpkhwibNKP9WYMD5mQ8xYqPU09GbWRchopFe9efE/r5dScOdmIkpSwojnLbkXpNKi2JoHsXyhkJOc5sC4Evlli4+YYpzyuMubNEYsRH1p+ROR6AW72eJ3s0qe3VlNug7tq7pzXXceb6qNWvGFMpzBOdTAgVtowAM0oQUcnuEVPuDTCIwX4814/2stGcXMKSzJ+PoF6bOMCA== η = . . . AAAB5HicbZDLSsNQEIYn9VbrLerSTbAIXUhJVNBlwY3LCvYCbSgnJ5P22JML50wKJfQNXAjiQhe+j4/g25jWbNr6rz7mnwv/eIkUmmz7xyhtbG5t75R3K3v7B4dH5vFJW8ep4tjisYxV12MapYiwRYIkdhOFLPQkdrzx/dzvTFBpEUdPNE3QDdkwEoHgjPJSuz/xY9IDs2rX7YWsdXAKqEKh5sD87vsxT0OMiEumdc+xE3IzpkhwibNKP9WYMD5mQ8xYqPU09GbWRchopFe9efE/r5dScOdmIkpSwojnLbkXpNKi2JoHsXyhkJOc5sC4Evlli4+YYpzyuMubNEYsRH1p+ROR6AW72eJ3s0qe3VlNug7tq7pzXXceb6qNWvGFMpzBOdTAgVtowAM0oQUcnuEVPuDTCIwX4814/2stGcXMKSzJ+PoF6bOMCA== . . . a p 1 a p axis. The circle AAAB4nicbZDLSsNAFIZPvNZ6q7p0EyxCF1ISFXRZcOOygr1AG8pkctoOnSTDzEmhhL6AC0Fc6MIX8hF8G5OaTVv/1cf5z4X/+EoKQ47zY21sbm3v7Jb2yvsHh0fHlZPTtokTzbHFYxnrrs8MShFhiwRJ7CqNLPQldvzJQ+53pqiNiKNnmin0QjaKxFBwRnmpj8QGlapTdxay18EtoAqFmoPKdz+IeRJiRFwyY3quo8hLmSbBJc7L/cSgYnzCRpiy0JhZ6M/ty5DR2Kx6efE/r5fQ8N5LRaQSwohnLZk3TKRNsZ3HsAOhkZOcZcC4Ftllm4+ZZpyysMubDEYsRHNlB1OhzIK9dPG5eTnL7q4mXYf2dd29qbtPt9VGrfhCCc7hAmrgwh004BGa0AIOY3iFD/i0AuvFerPe/1o3rGLmDJZkff0CL96K/g== ⌘ a and – 17 – p AAAB6nicbZDLSsNAFIZP6q3WW9Wlm2ARupCSqKDLihuXFewFmlAm05N26CQZZiaFEvISLgRxoQtfxkfwbUxiNm39Vx/nPxf+4wnOlLasH6Oysbm1vVPdre3tHxwe1Y9PeiqKJcUujXgkBx5RyFmIXc00x4GQSAKPY9+bPeR+f45SsSh81guBbkAmIfMZJTorOU5A9FT5yX06IqN6w2pZhcx1sEtoQKnOqP7tjCMaBxhqyolSQ9sS2k2I1IxyTGtOrFAQOiMTTEig1CLwUvOiuLjq5cX/vGGs/Ts3YaGINYY0a8k8P+amjsw8jjlmEqnmiwwIlSy7bNIpkYTqLPTyJoUhCVBdmuM5E6pgNyk+mNay7PZq0nXoXbXs65b9dNNoN8svVOEMzqEJNtxCGx6hA12gIOAVPuDT4MaL8Wa8/7VWjHLmFJZkfP0CQquOkA== A AAAB5nicbZC7SgNREIZnvcZ4i1raLAYhhYRdFbQM2FhGMBdIljB7MkkOOXvhnNlAWPIKFoJYaOHr+Ai+jZu4TRL/6mP+ufCPHytp2HF+rI3Nre2d3cJecf/g8Oi4dHLaNFGiBTVEpCLd9tGQkiE1WLKidqwJA19Ryx8/zP3WhLSRUfjM05i8AIehHEiBnJXaXWLspTjrlcpO1VnIXgc3hzLkqvdK391+JJKAQhYKjem4TsxeipqlUDQrdhNDMYoxDinFwJhp4M/sywB5ZFa9efE/r5Pw4N5LZRgnTKHIWjJvkCibI3sexe5LTYLVNAMUWmaXbTFCjYKzwMubDIUYkLmy+xMZmwV76eJ7s2KW3V1Nug7N66p7U3Wfbsu1Sv6FApzDBVTAhTuowSPUoQECFLzCB3xaI+vFerPe/1o3rHzmDJZkff0CV8+M3g== AAAB6nicbZDLSsNAFIZP6q3WW9Wlm2ARupCSqKDLihuXFewFmlAm05N26CQZZiaFEvISLgRxoQtfxkfwbUxiNm39Vx/nPxf+4wnOlLasH6Oysbm1vVPdre3tHxwe1Y9PeiqKJcUujXgkBx5RyFmIXc00x4GQSAKPY9+bPeR+f45SsSh81guBbkAmIfMZJTorOU5A9FT5yX06EqN6w2pZhcx1sEtoQKnOqP7tjCMaBxhqyolSQ9sS2k2I1IxyTGtOrFAQOiMTTEig1CLwUvOiuLjq5cX/vGGs/Ts3YaGINYY0a8k8P+amjsw8jjlmEqnmiwwIlSy7bNIpkYTqLPTyJoUhCVBdmuM5E6pgNyk+mNay7PZq0nXoXbXs65b9dNNoN8svVOEMzqEJNtxCGx6hA12gIOAVPuDT4MaL8Wa8/7VWjHLmFJZkfP0CWOCOnw== AAAB5nicbZDLSsNAFIZP6q3WW9Wlm2ARupCSWEGXBTcuK9gLtKFMpqft0EkyzJwUSsgruBDEhS58HR/BtzGt2bT1X32c/1z4j6+kMOQ4P1Zha3tnd6+4Xzo4PDo+KZ+etU0Ua44tHslId31mUIoQWyRIYldpZIEvseNPHxZ+Z4baiCh8prlCL2DjUIwEZ5SVun0kNkhUOihXnJqzlL0Jbg4VyNUclL/7w4jHAYbEJTOm5zqKvIRpElxiWurHBhXjUzbGhAXGzAM/ta8CRhOz7i2K/3m9mEb3XiJCFROGPGvJvFEsbYrsRRR7KDRykvMMGNciu2zzCdOMUxZ4dZPBkAVoru3hTCizZC9Zfi8tZdnd9aSb0L6pufWa+3RbaVTzLxThAi6hCi7cQQMeoQkt4CDhFT7g05pYL9ab9f7XWrDymXNYkfX1C24TjO0= A ⌘ ⌘ +1 axis. We introduce the notation η η > η y a a , . . . , p η 1 β AAAB6HicbZDLTsJAFIZP8YZ4Q126aSQmLJS0aqJLEjcuMZFLhIacDgcYmU6bmSkJafoOLkyMC134Nj6Cb2NBNoD/6sv5zyX/8SPBtXGcHyu3tr6xuZXfLuzs7u0fFA+PGjqMFaM6C0WoWj5qElxS3XAjqBUpwsAX1PRHd1O/OSaleSgfzSQiL8CB5H3O0GSlpw4Z7CZ44abdYsmpODPZq+DOoQRz1brF704vZHFA0jCBWrddJzJegspwJigtdGJNEbIRDijBQOtJ4Kf2WYBmqJe9afE/rx2b/q2XcBnFhiTLWjKvHwvbhPY0jN3jipgRkwyQKZ5dttkQFTKTRV7cpEliQPrc7o15pGfsJbP/pYUsu7ucdBUalxX3quI+XJeq5fkX8nACp1AGF26gCvdQgzowkPAKH/BpPVsv1pv1/teas+Yzx7Ag6+sXMwmNUA== AAAB6HicbZDLSsNAFIZP6q3WW9Wlm2ARCkpJVNBlwY3LCvaCbSgn09N27GQSZiaFEvIOLgRxoQvfxkfwbUxrN239Vx/nPxf+40eCa+M4P1ZubX1jcyu/XdjZ3ds/KB4eNXQYK0Z1FopQtXzUJLikuuFGUCtShIEvqOmP7qZ+c0xK81A+mklEXoADyfucoclKTx0y2E3w3E27xZJTcWayV8GdQwnmqnWL351eyOKApGECtW67TmS8BJXhTFBa6MSaImQjHFCCgdaTwE/tswDNUC970+J/Xjs2/Vsv4TKKDUmWtWRePxa2Ce1pGLvHFTEjJhkgUzy7bLMhKmQmi7y4SZPEgPSF3RvzSM/YS2b/SwtZdnc56So0LivuVcV9uC5Vy/Mv5OEETqEMLtxAFe6hBnVgIOEVPuDTerZerDfr/a81Z81njmFB1tcvMA+NTg== ⌘ ⌘ η 3 S , = 2 inside the interval ( vanishes at for for 0 for a a ] with Y ] on the A 1 > a ) is consistent. Indeed, ( , η a For ) provides the boundary condition , η Z Z y ) deserves further comments. First of all, since 1 ] and obtain a 2-cycle ∈ − ∈ 4.28 . a a 4.27 a p 4.28 η . We thus recover the expected relation . The desired 4-cycle y y , η =1 1 p a 2 , which is part of the 4-cycle shrinking at the endpoint − } max a a ) = ) = a S η , η , η = 0, because not {B (0 (0 η > η 0 is still a closed 4-cycle. y and [ Y Y 1 1 β B has finite size in the entirety of [ S . A generic profile of monopoles. The arcs circle is 2 Ω ) is piecewise constant along the axis for S 1 β The identification ( The computation ( S η ρ, η , and in particular for the non-puncture. In that case, ( ( a as a bubble insince figure by combining the interval [0 the therefore The 4-cycles Consider the interval [ the monopole sources, butcombine has finite size in the interior of [ For any puncture, supersymmetry requiressame that sign. the flux It of follows that Notice that orientation we chose ink ( the quantized and the locationY of Figure 2 denotes the 2-cycle JHEP06(2019)123 p ), = 4.16 a (4.37) (4.34) (4.35) (4.36) (4.32) (4.33) across , since ) term 4 a L ), ( η S , , however, . We have ) to = 0. As a 4 − dη . Y π E ) 0 4.15 η a y ( 4.36 η δ ) are a potential . − ) by selecting the ) η ) to the case of the . Consistency with 1 , , ( 4.17 , − a ) that δ a =1 a 4.26 k are piecewise constant ) k p a 4.32 a in ( , a } , η ` = 0 a Y a 4.16 is an increasing sequence w (0 0 w − dY {B − =1 has periodicity 2 W and p a 1 . +1 } a between consecutive monopole β − − ). Specializing ( must be equivalent to L ` and a N a ) ( , y 1 w a , w a W { p B =1 k dL − 4.30 + , η a X , a Z a (0 y in terms of = . . w ∈ is inferred from the jump of a b a W 1 y k =0 k , see ( a , and that is computed from ( ) = ρ − b = a η 2

a Ω a N ` w w S w =1 ). We know from ( π flux quantization. Moreover, supersymmetry p a − = 2 a The quantities p Dβ =1 =1 − } b 4 X ) evaluates to + a p X a – 18 – a +1 ∧ y ). Notice that 4.23 E . a = = {B ` axis. The terms ( a 4.32 N , since η y N 4.16 2 dW a dη , dL E w ) ∧ , w a Ω 2 + a η . e a 1 w . In that case, the cycle a − − B must have the same sign for all B a η function at a given Z y ( 4 max ) = δ η δ a E − − integrates to 1 over ) 1 a , η ) that the jumps in the values of = Ω 2 = ) to express the values of y − 1 e a (0 4 η y E 4.32 through the cycles ) respectively. We can achieve a cancellation of each factor, W and partition of 4.35 0 = a − 4 B 4 a = 1, E Z y 4.21 Dβ E ( p p =1 ), ( a X axis, with jumps at the location of the monopoles. The total form function singularities in η δ = 4.29 ) by demanding =0 ρ

To argue in favor of our orientation convention, we specialize ( It follows from ( The flux of 4.17 , see ( dY a Moreover, we have alsowe established thus that obtain a crucial sum rule for the flux data where in the lastresult, step we can we use made ( use of ( The normalization of each η in ( must be regular everywhere along the source of where the last relation followsof from positive ( integers. Regularity of along the locations must be integers,demands by that virtue the of fluxthe of non-puncture case requires that this sign must be positive. In conclusion, we can write also chosen an orientation for non-puncture, the latter is the onlywe non-trivial immediately 4-cycle in see the that non-puncture the geometry. r.h.s. From ( of ( terms with one We have recalled that JHEP06(2019)123 23) , 6) 2) 2) , , , . , 17 3 3 4 5 , , , , N not E axis does (4.38) (4.39) η 4 X )=(6 )=(2 )=(1 ) = (12 3 3 3 3 ’s at each ` 1 ] along the , ,k a ,w 2 ,N 2 ` 2 2 ). Moreover, . ), which have , CP , η does not yield ,k 1 circle does 1 b ,w ,N ` 1 ` AAACAnicbVDLSsNAFJ3UV62vqEs3Q4vQQihJ62sjFNy4rGAf0IYwmd62QycPZiaFErpz4be4EMSFLvwBP8G/MY3ZtPVs7uGe++AcN+RMKtP80XIbm1vbO/ndwt7+weGRfnzSlkEkKLRowAPRdYkEznxoKaY4dEMBxHM5dNzJ3ULvTEFIFviPahaC7ZGRz4aMEpW0HL1Y7gPnjmXgtNayWq/gW1y+Mi6NWsXRS2bVTIHXiZWREsrQdPTv/iCgkQe+opxI2bPMUNkxEYpRDvNCP5IQEjohI4iJJ+XMc+f43CNqLFe1RfM/rRep4Y0dMz+MFPg0GUm0YcSxCvDCJh4wAVTxWUIIFSz5jOmYCEJVEsbyJQk+8UAaeDBloUy5HafJzguJd2vV6Tpp16pWvWo9XJQaZpZCHp2hIiojC12jBrpHTdRCFD2jV/SBPrUn7UV7097/RnNatnOKlqB9/QLf25P3 1 − ( β 1 1 1 k 4.33 a ) AAAB+XicbZDLSgMxGIX/qbdab6OuxE2wCC0MZdIKuhEKblxWsBdohyGTpm1o5kKSKZSh+CwuBHGhC5/CR/BtnKmzsfUswsd//iSc40WCK23b30ZhY3Nre6e4W9rbPzg8Mo9POiqMJWVtGopQ9jyimOABa2uuBetFkhHfE6zrTe8yvztjUvEweNTziDk+GQd8xCnR6cg1zypTF1to6tazo1FFt6iCrYZVr7pm2a7ZS6F1wDmUIVfLNb8Gw5DGPgs0FUSpPrYj7SREak4FW5QGsWIRoVMyZgnxlZr73gJd+kRP1KqXDf/z+rEe3TgJD6JYs4CmK6k3igXSIcrSoSGXjGoxT4FQydOfEZ0QSahOO/j7kmIB8Zmy0HDGI7VkJ1kWuiil2fFq0nXo1Gu4UcMPV+WmlbdQhHO4gApguIYm3EML2kDhCV7gHT6MxHg2Xo2339WCkd85hT8yPn8A6ZiQCA== ( 4.32 k S w N 1 η AAAB9XicbZDLSsNAFIZP6q3WW1RcuQkWoYVQklbUjVBw47KCrYU2hMn0tB06uZCZtJTQR3EhiAtd+Bw+gm/jtGbT1h8OfJz/nBn+40WcCWlZP1puY3Nreye/W9jbPzg80o9PWiJMYopNGvIwbntEIGcBNiWTHNtRjMT3OD57o/u5/zzGWLAweJLTCB2fDALWZ5RI1XL1s9LEtc2JW1VVK9+VqmbNvC67etGqWAsZ62BnUIRMDVf/7vZCmvgYSMqJEB3biqSTklgyynFW6CYCI0JHZIAp8YWY+t7MuPSJHIpVb978z+sksn/rpCyIEokBVSPK6yfckKExT2b0WIxU8qkCQmOmfjbokMSESpV/+SWBAfFRmEZvzCKxYCddHHNWUNnt1aTr0KpW7FrFfrwq1s3sCnk4hwsogQ03UIcHaEATKKTwCh/wqU20F+1Ne/8bzWnZziksSfv6BcTwj4k= AAAB/HicbZBNSwJBHMZn7c3sbatjHYYkUBDZWQO7BEKXTmKQJuiyzI5/dXD2hZ1ZQRY79Fk6BNGhDn2HPkLfpl3bi9pzGH78n//M8DxOILhUhvGj5TY2t7Z38ruFvf2DwyP9+KQj/Shk0Ga+8MOuQyUI7kFbcSWgG4RAXUfAozO5Tf3HKYSS+96DmgVguXTk8SFnVCUjWz8vNW1SwU3bTI9aGd/gEjErpF4xa2VbLxpVYyG8DiSDIsrUsvXv/sBnkQueYoJK2SNGoKyYhoozAfNCP5IQUDahI4ipK+XMdeb40qVqLFe9dPif14vU8NqKuRdECjyWrCTeMBJY+TgNiAc8BKbELAHKQp78jNmYhpSppIbllyR41AVZwYMpD+SCrXjR6byQZCerSdehY1ZJrUrur4oNI2shj87QBSohguqoge5QC7URQ8/oFX2gT+1Je9HetPe/1ZyW3TlFS9K+fgHIBZBt ( ( Dβ , while the 2-forms is continuous along contains non-trivial − ⇥ is zero on the a , b 4 1 S w = 1, the space L AAAB7HicbZDLSsNAFIZPvNZ6q7p0M1gEF6UkKuiy4MZlBXuBNobJ9LQdOpmEmUmlhL6FC0Fc6MJ38RF8G6c1m7b+q4/znwv/CRPBtXHdH2dtfWNza7uwU9zd2z84LB0dN3WcKoYNFotYtUOqUXCJDcONwHaikEahwFY4upv5rTEqzWP5aCYJ+hEdSN7njBpbenoOPNI1PEJNRoEXlMpu1Z2LrIKXQxly1YPSd7cXszRCaZigWnc8NzF+RpXhTOC02E01JpSN6AAzGmk9icIpOY+oGeplb1b8z+ukpn/rZ1wmqUHJbIv1+qkgJiazQKTHFTIjJhYoU9xeJmxIFWXGxl7cpFFSG7VCemOe6Dn72fyH06LN7i0nXYXmZdW7qnoP1+VaJf9CAU7hDC7AgxuowT3UoQEMFLzCB3w60nlx3pz3v9Y1J585gQU5X79u6Y6Q w X a,I above. 1 − b ω WL k b ) defines a partition of exemplifies the translation ∧ + w 4.2 3 ( . Y π 2 a,I 4.37 = 1, a a =1 2 k b k F b X (because p . We highlighted the decomposition 1 a ⇥ × p = − w 2 =1 k a ] combined with a . , I a X 1 k AAAB7HicbZC7TgJBFIbP4g3xhlraTCQmFoTsoomWJDaWmMglgZXMDgeYMDu7mZnFkA1vYWFiLLTwXXwE38YBtwH8qy/nP5f8J4gF18Z1f5zcxubW9k5+t7C3f3B4VDw+aeooUQwbLBKRagdUo+ASG4Ybge1YIQ0Dga1gfDf3WxNUmkfy0Uxj9EM6lHzAGTW29PTcq5Ku4SFqMu5Ve8WSW3EXIuvgZVCCTPVe8bvbj1gSojRMUK07nhsbP6XKcCZwVugmGmPKxnSIKQ21nobBjFyE1Iz0qjcv/ud1EjO49VMu48SgZLbFeoNEEBOReSDS5wqZEVMLlCluLxM2oooyY2Mvb9IoqY1aJv0Jj/WC/XTxw1nBZvdWk65Ds1rxrirew3WpVs6+kIczOIdL8OAGanAPdWgAAwWv8AGfjnRenDfn/a8152Qzp7Ak5+sXceqOkg== of the 2-cycle w w introduced above ( ` a = N k 3 4 a , η p are field strengths of 4d external con- =1 p k = 23 and associated Young diagram. The w a X =2 ` = X p a ⇥ p } + + N a,I 3 a allows us to include additional terms in b b N F a is the collection of resolution it attains the value AAAB7HicbZC7TgJBFIbP4g3xhlraTCQmFoTsiomWJDaWmMglgXUzOxxgwuzsZmYWQwhvYWFiLLTwXXwE38YBtwH8qy/nP5f8J0wE18Z1f5zcxubW9k5+t7C3f3B4VDw+aeo4VQwbLBaxaodUo+ASG4Ybge1EIY1Cga1wdDf3W2NUmsfy0UwS9CM6kLzPGTW29PQcVEnX8Ag1GQXVoFhyK+5CZB28DEqQqR4Uv7u9mKURSsME1brjuYnxp1QZzgTOCt1UY0LZiA5wSiOtJ1E4IxcRNUO96s2L/3md1PRv/SmXSWpQMttivX4qiInJPBDpcYXMiIkFyhS3lwkbUkWZsbGXN2mU1EYtk96YJ3rB/nTxw1nBZvdWk65D86riVSvew3WpVs6+kIczOIdL8OAGanAPdWgAAwWv8AGfjnRenDfn/a8152Qzp7Ak5+sXdOuOlA== w k 4 ω , ), the quantity {S 4 a b a X η ∧ w X F – 19 – 1 a = π 4.35 − =1 F 2 b a X η , we can use 23) ) to infer p =2 = , p a X a 2) 6) 2) a , , , 17 k = + , N 3 3 5 4.37 a , , , a Ω 2 w e ) = ∧ 3 =1 1, introduced at the end of section a X a 2 , this quantity is the line charge density in the Gaiotto-Maldacena puncture )=(1 )=(2 )=(6 ) = (12 is the Poincar´edual in , η E > 3 3 3 3 D = ` a a , ). The quantities ,k = ,w )(0 ,N k 2 ω 2 In the case of the non-puncture, i.e. AAAB+3icbZDNSsNAFIUn/tb6F3XZzWARXJSSWEE3hYIbV1LB/kATw8102g6dScLMpFJCFj6LC0Fc6MKH8BF8G9OaTVvP6uOeO3M5x484U9qyfoy19Y3Nre3CTnF3b//g0Dw6bqswloS2SMhD2fVBUc4C2tJMc9qNJAXhc9rxxzczvzOhUrEweNDTiLoChgEbMAI6G3lm6Q7XsaNi4SVQt9PHGn7yADsVPPbAM8tW1ZoLr4KdQxnlanrmt9MPSSxooAkHpXq2FWk3AakZ4TQtOrGiEZAxDGkCQqmp8FN8JkCP1LI3G/7n9WI9uHYTFkSxpgHJVjJvEHOsQzzLh/tMUqL5NAMgkmWXMRmBBKKzFhZ/UjQAQVUF9ycsUnN2k3mlaTHLbi8nXYX2RdWuVe37y3KjkrdQQCV0is6Rja5QA92iJmohgp7RK/pAn0ZqvBhvxvvf6pqRvzlBCzK+fgGgA5MA 2 N 2 ` 4 is an increasing sequence of positive integers, see ( , fiber direction. Let us stress once more that the ,k 1 E ,w 1 4.26 ,N ` 1 WL k 1 ( =1 = 3 monopole sources with prescribed ( p a w N Dβ ( + p } 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 ( a Y thus reads w = 0, and therefore the first interval [0 ( At the monopole location ) and the sum rule ( { 4 0 8 E η is as in ( ]. . An example of a flux configuration for max 9 2 ) into a Young diagram, in the conventions used throughout this work. E are integer and positive. It follows that the relation ( axis. a The existence of non-trivial 2-cycles in It is worth noting that, thanks to ( η 4.37 η > η As explained in appendix k 8 axis with the where nections. The 2-form solutions [ monopole source with The total 2-cycles. First of all,the there topology of are a the 2-sphereη 2-cycles and are obtained byshrink combining at the interval [ a 2-cycle. The second class of 2-cycles in If we choose the lastfor monopole Flavor symmetry. not admit any non-trivial 2-cycles.and/or As monopole soon charges as greater we than consider one, more however, than the one geometry monopole source the Recall that all which can be equivalently encoded inof a Young ( diagram. Figure Figure 3 configuration has of the Young diagram in rectangular blocks of dimensions JHEP06(2019)123 , ). 6 ’s. 1 X (5.4) (5.1) (5.2) (5.3) ) are 4.26 (4.40) CP . 2. (The 2 contains 4.39 # ) plane is 4 ≥ , . E as in ( a,I 4 in ( a b ρ, η ω i 2 k X ) E ∧ 1 . a,I − b F φ π 2 a π a,I 2 , 2 b F w F a 1 ∧ F − ), with on the ( − =1 2 a a I X π k , w 2 g Dβ 4.39 p dW =1 )( are not visible in the super- ∧ a X a ). In this section we compute = 1 ∧ ` F + ] π a Ω 2 a 2 2 2.9 G e Dχ w ) ω 2 Ω ∧ ∧ − S . a Z a π in turn. For notational convenience, puncture to the total inflow anomaly WL # F 2 N ) dW 12 th ( a + I p =2 a α k ’s at each monopole with ∧ a X ` ( given by ( 1 , Y i ) " U 2 [( 12 ) d CP ∧ I p 2 =1 ). The Bott-Cattaneo formula, reviewed in ) + 3 Y a E 1 SO(3) that saturate the integral over the 6d space = 1.) The 4d connections 4 " – 20 – WL that saturate the integration over 4.9 − N 3 X a 3 S a ( 3 + ) Z k ) 1 w is readily performed. (Recall that they both have 4 ), with 2 = p Y E E − term near a puncture is given in ( 1 4 ( β F +3 h 3 a 4 correspond to the remaining U(1) factors. , see ( 3 , 3 2.22 G d throughout the rest of this section. 4 = ) ) w . Combining all elements, we arrive at E χ a 4 ( 4 2 3 X α F ) E X 2 a B E ( ` Z Ω 2 4 e 2 3 X ( h 2 − Z Ω , the contribution of the S p ) =1 integrate to zero. It follows that Z = a X 3 2.3 Ω 2 are in one-to-one correspondence with the Cartan generators of the ) φ π e ) is given by ( 2 ) captures the Cartan subalgebra of the full flavor symmetry group 2 , while the SO(3) F α ), considering the two terms in E F a,I ( N P − 4 b ( ( F fibration over G 4.39 1 X = 2.22 p Z 2 Ω .) The integral of the 2-form 3 1 4 , implies ) S π inflow 6 2 is understood to be zero if I = E ( I 3 4 A.3 ) 4 X Z E ) factor in ( a are the Poincar´eduals of the resolution The states associated to the non-Cartan generators of 6 k X Z a,I b The integration over theperiodicity angles 2 discussed in detail in appendix To proceed, we isolate the terms in ( while even powers of The total expression for the form Our task is to identifywhich the is terms in an ( appendix polynomial the integral in ( we suppress the puncture label 5.1 Computation of the ( For the purpose of computingall ’t Hooft necessary anomaly information. coefficients, however, the form 5 Puncture contributions toAs anomaly explained in inflow section The connections SU( gravity approximation, since they originate from M2-branes wrapping the resolution sum over interpreted as background gauge fieldsMore for precisely, the ( flavor symmetry associated to the puncture. ω JHEP06(2019)123 ) 4 is e X 4 (5.7) (5.8) (5.5) (5.6) 3.15 X reduces b π 4 F 2 i ..., ). We thus X ) 1 ∧ + − are associated a 2 π a is evaluated at # 4.25 F 2 w π a,I b,J 2 − a,b b b ω orbifold singularity F WL 2 a K a ∧ k w + ’s, as well as over the =2 1 Z a p π )( / Y X F 2 a a,b , 4 ` , ) CP . We use the notation monopole. It follows that . r 1 R a a a,I c k b,J b,J w , b ( ω th 6 b b b,J ω , a, a . ) requires full control over the π − b,J b a ω ω g TN − . ) 2 ∧ K b a F ω a 1 ∧ ∧ 5.6 k is the one with legs along external has an ). The integration over N π ( − π a,J a,I a a ∧ =1 ( b,J b 2 2 4 b 2 su X J IJ b b ω ω ω ). In this case, integration over F k a F E in ( X ` ) ) ) π C a,I 4.38 ∧ ) p 2 ∧ b =1 5.2 a and is performed using ( F b X a ) N 2. The singularity is modeled by a single- π a its orbifold singularities. b,J π a,I ( WL WL WL k F 2 p 2 =2 ) + 3 b F a, b,J ≥ ) a X a,b 1 ( + + + all ) δ K − a , and the resolution g TN a ) 3 b,J a k − k Y Y Y ( ( a ( ( ( – 21 – w + 2 a, a,I 4 4 4 S su = ( IJ ) in terms of 4d Chern classes, b and term K X X X − π ) C K integral in ( F 1 2 Z Z Z 3 a by virtue of ( 8 1 1 . As a result, the quantity − 4 b,J =1 a,b w =1 b b − ( ∧ a = = = − =1 X , X J ( SO(3) b k a X ) K X J ) ) X k a N k 2 a π I,J N = ∧ 1 ` a,b p F 2 ( a,I =1 a b,J b,J by resolving ( . We refrain from a discussion of these coefficients. − b 1 are computed as follows. The 2-forms X 2 4 a ( ( =1 K ). , a a p , which can be resolved to h N I 4 X ) K a, ) k a,b π E a ω p =2 k X K K p p a X a,I =1 =1 (2 b,J =1 ( ( a p a X X / , r 1 ) X K =2 φ c a,b p r 1 R 2 ) and c F X c " a,I a,b π ) that the puncture geometry ( 12 r 1 ’s of the orbifold singularity at the ∧ c 1 (2 + K − + 6 / φ 4.2 π φ WL CP 2 F = 2 F + 3 3 = ) − Y 4 ), and gives a factor ( a 4d E = − ( n is only non-zero for , η 3 6 ) ) X 4 Z E b,J ( ( , 6 ) ) = (0 Let us summarize the final result of the computation of this subsection, using ( The coefficients Let us now turn to the second X a,I Z ( ρ, η 5.2 Computation ofRecall the from section at the location of eachcenter monopole Taub-NUT of space charge TN for the space obtained from to express ˆ have A computation of theintersection coefficients pairing among thenormalization 2-cycles of the 2-forms to the resolution K ( to an integration over the resolved orbifold We have used the factspacetime, that the only relevant part of where the coefficients are saturated by considering terms quadratic in the 2-forms JHEP06(2019)123 . , 2 φ E 4d A n (5.9) ). We (5.11) (5.15) (5.12) (5.13) (5.14) (5.10) . This is . 3.4 ) 4d 4 in the term n e X Ω 2 T e ( is a local model 1 4 localizes onto the p ), we compute e ) , . X 4 .... 2 e ) X ) a , k ) + 4d A.10 WL 2 introduced in ( 4 ) , , n 4 e (ˆ + X g TN 4d e 4d X 1 2 ). We can now compute the n T T Y . ˆ n ) = 0, we can write ( ( ( SO(2) T 4 1 1 1 2 4 4 ) (ˆ 4.4 N p e e 4 p . The integral in the directions e is saturated by ) + X X X i 6 e 4 + a Z X 2 T that is shifted with +ˆ k T 2 e Ω ) X ( X λ T i 2 S 1 ( 2 ⊕ 4d T g TN c − λ 1 ) before the 4d gauging is turned on. ( n Z p 1 (ˆ — see ( 4d 4 + over p a := 1 4 e n λ . 1 φ X − 8 (ˆ N SO(3) λ = − . The integral over , ) T tot 2 , and we have selected the only piece of X N =: − 2 β p 2 circle is gauged with the 4d connection 4d Ω =1 . Since = ) ), with the dots representing the remaining ) ∧ 2 a X 4 ˆ n S 4 ⊕ 2 χ λ 4 e tot 2 π ) e SO(3) X 4 X λ E − – 22 – T N ) = T tot 2 SO(3) , λ ( ( 4 = 2 λ = 1 TW 1 + ( e ( N p 4d X p 1 ( φ 2 2 ˆ 1 n = λ T ) ) F p ( 1 2 6 in terms of 1 ) + , including the contribution from the gauging with ˆ tot 1 tot 1 4 p W 4 χ | λ λ + ) ) + e X 4 11 λ ), while the integral on TW T = ( = ( 4 ( := 1 WL e TW X TM p tot tot ( h ) ) + T 1 tot 1 4 4 ( p 1 λ e e 1 Y h X X 96 p ( r 1 4 T T c − e ( ( X 1 2 Z = p p 1 48 ), and standard formulae for Pontryagin classes ( 8 of the monopoles, denote the Chern roots of = X a ) is the first Pontryagin class of ) that it is the sum of Chern roots η 8 2.10 2 4 . It follows that 4 λ e X X = , E is the spacetime component of the Chern root of ), ( 5.11 T 1 η ∧ ( λ 4 in 1 4d 5.9 p is saturated by n E Ω 2 6 e 4 We are now in a position to integrate Let With this notation, the relevant decomposition of the 11d tangent bundle, restricted X e X ∧ Z 2 We have already performed thewhich integral is over relevant, i.e.positions the part with E We have selected theterms, terms which with will one not be important for the followingof discussion. where Using ( due to the definition ofshifted the Pontryagin angle classes for In order to account for this fact, we shift the Chern roots of where ˆ see from ( In our geometry, the U(1) associated to the to the brane worldvolume, is The above expression is motivatedof by the the cotangent fact bundle that to the the resolved Riemann space surface in the vicinity of the puncture. JHEP06(2019)123 ), , a  (6.1) (6.2) , η a (5.17) (5.16) k R 2 a c b N π r 1 F c 2 ) = (0 Γ. Exploiting 1 6 / ) 4 ∧ α − R ρ, η P a ) π ( 1 , F 2 on the l.h.s. ’s of − i at ( 2 a 2 α a,b ) w a inflow v r 1 K c ). A minor comment n N − =2 2 4 ( a − p 5.6 ). X w  a,b − ). ) ], and comparing with the value . a )( 4 R 2 k + 3.15 31 a  c ) are evaluated making use of ` a 4 a g TN π k TW π b,J puncture to the inflow anomaly a,J ( 2 . T b w 2 − b F 1 ( F a ) 1 p th g TN − p 4 ∧ k using ( ∧ r 1 α T a c a ( π R 2 takes the value 1 N π F 2 c a,I TW 1 ( p 12 2 ( b ) , F 9 a ) = 2 1 r 1 , ` ) a ) are defined in ( p c for a generic ALF space based on a − b,J a k k h ( α WL k 3 χ ( r 1 a, Z ) ) of the P ) + c ( r 1 + / α su g TN IJ 1 K c 4 a P ( 1 – 23 – − C T k Y ( R 3 a ( − =1 1 3 1 a , using equation (23) in [ 1 b flavor w X J =1 s , k  − p N a Z ) X a − k inflow I,J 6 p k α =1 p , 3 =1 a I b X a , P a X inflow 6 a ( w ) g TN N I k ( p α =2 Z 1 24 ) and the findings of the previous sections. The result reads a a X 2 a P p =1 ` in ( N a X inflow = 2 3 ) ) ], one reads out the integral of 2.22 − + 2  8 p h =1 b,J 31 a X n flavor  ( X , p r 1 =1 a, 1 2 a X c − ∧ K v 4 , inflow n 6 ] gives the Euler characteristic E ) = ) = ) = I 6 α α α a,b 31 X + P P P ] for ALF resolutions of K Z ( ( ( ) = ( α 31 P ( flavor inflow inflow v , ) n h ). The integrals of the individual classes n inflow 6 inflow 6 I − I 4.38 v given in equation (33) in [ Equation (12) in [ n 9 χ ( about our notation isthe in order. above equations. We have reinstated Strictly the speaking, puncture each label punctureself-duality comes of with curvature, specializing its toof local Γ = data, and on The coefficients where the anomaly coefficients are given in terms of the quantized flux data as 6.1 Summary ofWe inflow can anomaly assemble polynomial thepolynomial, contribution making use of ( 6 Comparison with CFTIn expectations this section we firstwe summarize then the prove that total it result matches for with the the inflow CFT anomaly expectation. polynomial, and In conclusion, we obtain where we have expressed the result in terms of see ( the results of [ We exploited the fact that the quantity JHEP06(2019)123 as (6.9) (6.7) (6.8) (6.3) (6.4) (6.5) (6.6) a )) (see ) is the T m . from the F ) G α α ( . . 2 P (SU( ) ( 2 can be written )) F c a G F − k ( CFT h generators G 2 n F ch n =1 (SU( )) = r 1 G X α 2 c ]. Here, ch . c m ) G r 1 12 c k ) + N a − (SU( g,n − N 2 R 2 3 2 c (Σ . r 1 N b p =1 c T a X N, v CFT h a ) )(4 2 n . n T ) factor in the flavor symmetry group, g,n g,n a a − = =2 k ). N  (Σ (Σ h → − N ) 2 χ χ n 4 R CFT h − 1 2 1 6 π + a,J 2 b = TW v F SCFTs 2 tr ; for instance, ch ( ) n ) = ) = – 24 – 1 F a ∧ − S p k (2 ( G g,n g,n , n r 1 = 1 π ). = 2 SCFT with flavor symmetry , and so on. We prefer to omit the label c 12 a,I ) 2 inflow SU b α a (Σ (Σ α F ab ` k 1 = N δ 12 P ) = 2 superconformal algebra [ , ( 6.50 a F c α a k − ( G k has two contributions to their anomalies, which we denote N inflow inflow v )–( k 3 ) su IJ , n CFT v ) h S = 2 class α r 1 C n n p c correspond to the number of vector multiplets and hypermul- 1 ( 6.46 ), and =1 n =1 parameters as − N − h h 1 3 X α a v X n h n k  I,J n n ) a ( + h ) + v N n and n g,n and p − =1 v (5 a X v v n (Σ 1 n 24 r 1 n c ). The flavor central charge is defined in terms of the CFT v = − = ( n a = 2 theory of class A.1 = N CFT 6 I CFT v An For the sake of completeness, we also restate the bulk contribution of the Riemann In the piece related to flavor symmetry, we expect an enhancement of the first term to n in terms of the The parameters tiplets respectively when thenumber theory of is vector free, multiplets andcharges and otherwise as hypermultiplets. can be These regarded are as related an to effective the SCFT central This structure follows from2-form the part of theappendix Chern character for The anomaly polynomial of anyin 4d the form We would now like toresults compare are these summarized expressions in with the ( anomalies of the 4d6.2 SCFT. Our Anomalies of the surface to the anomalies: The corresponding flavor central charge is r.h.s. ’s of the above relations in order tothe avoid second cluttering Chern the class expressions. of the full non-Abelian SU( the r.h.s. ’s we should write JHEP06(2019)123 ρ , 1 − (6.18) (6.16) (6.17) (6.10) (6.11) (6.14) (6.15) (6.12) (6.13) N A to the 4d . = F j g k e G i SCFTs can be e p boxes. Consider = condition for the j X S c . For . i N g , N +  2 1 → ) depend on the local 1 j . = 2 k e + j − j 6.9 f k e 2 (2) 1 i N N of the Riemann surface, =1 N − su e p = 1 3 i j X j 2 1 : X χ ]. . The partition is given as , − ρ = 3 e p − = 3 1) j , i e `  e ` N 2 1 2 − i . , i 4 3 e =1 ρ N ,..., # j X + N )  i i − k e )( ) . , = k e = 1 ( i 2 i i 0 i e e ` g,n g,n U N e N N ≡  e e p p e (Σ (Σ p =1 =1 =1 . i i Y X i X χ χ e p +1 i =1 , which encodes the " e i p X 1 2 i 1 2 1 2 – 25 – e = , with e ` N S i k e − − e − − ` N N, = = = 2 1 F = ) ) = ) = ) = ) = , i − α α , encoded in a Young diagram with i G k e ( e +1 P P g,n g,n +1 N N e p ( ( i ) it is also useful to introduce the notation e ` CFT SU e α ] by integrating the 6d (2,0) anomaly polynomial over the − (Σ (Σ N k P − CFT v CFT ( = +1 i 18 ) i n , e ` CFT v CFT e p h e ) 7 N n e n rows of length h CFT v,h N = n − n i e p − i k e v − e N n v , ( function in the dual quiver description [ n are defined as 1 ( = 2 puncture is labeled by an embedding is the commutant of the embedding β − i i k e e F N N G − i e N = i The puncture contribution to the ’t Hooft anomalies of the class A regular e ` vanishing of the stated in terms of this data as follows: Notice the relation 2 In order to write down CFT, where The quantities A puncture corresponding to this partition contributes a flavor symmetry Riemann surface without punctures.puncture data, The which we remaining will terms now in review. ( is one-to-one with a partitiona of Young diagram with These were computed in [ The bulk terms are proportional to the Euler characteristic JHEP06(2019)123 ], ) of ) in i 19 as (6.19) , k e is the (6.24) (6.25) (6.26) (6.20) (6.21) (6.22) (6.23) i 9 e N a 4.37 ` . case in [ . p ) n . 0 A ] g, 1 associated to a free − (Σ p h , . . . , w χ w . n 2 ) axis has charge 1 2 , − , defined by sequentially num- 2 i p . v e i η − N w p n ) = 1 a − p = ` 2 ( ]. These are related to the N . 1 ( on the p . e p integers =1 i , . . . , w , i w X a ,..., a ) η 1 ` N − = − free tensor ) , i ,
+ 1 a ) 1 e 1 p . Let us recast the sum rule ( h w ’th box) [ 1 N − a i n +1 − a − ` − − a a a w i ` 2 − w ( w 10 ) , a set of , v N − − a ]. ) ∈ 3 ` n = determined by the Young diagram a e p a ( ( e ` η i ` 20 − w – 26 – in decreasing order. The exponent of ( N 3 (height of = for more details. a − N p =1 i ` a 4 ,..., a X is specified by giving the lengths of its rows. More , k ,..., C 1 in terms of = ) 1 6 for all w 2 i = 0 Y e ` a p − axis for pole structure g, , = k 2 N η 1 i w e ` p ) (Σ ( lengths 2 . a χ 1) ` a ` ( 1 2 of the Young diagram that describes the CFT is ` − , i = i 1 − e ` i w (2 e ` ) = 1 N =1 ) as a partition of ` along the i X distinct ) to express L = [( 6.22 boxes in the Young diagram, starting with 1 in the upper left corner and increasing 4.23 Y monopoles. The monopole located at N free tensor v p n = value of i e ` The map to the rows It will also be useful to note the following expressions for Another common notation uses the 10 bering each of the from left to right across a row such that Then, the sequence Equivalently, we can write We are using aprecisely, notation we in list which the number of rows with length where we used ( the form We can interpret ( The map between the data ofa the profile Young diagram with and the inflow data is as follows. Consider These expressions can be found byof dimensional a reduction of single the M5-brane 8-form anomaly — polynomial see appendix 6.3 Relating inflow data to Young diagram data the flavor group. Thesewith contributions the were general computed ADE explicitly formula for derived the in [ tensor multiplet reduced on a Riemann surface without punctures: The last equation is the mixed flavor-R-symmetry contribution due to a factor SU( JHEP06(2019)123 23) , 6) 2) 2) , , , 17 . 3 3 5 , , , , 5 (6.31) (6.32) (6.28) (6.29) (6.30) (6.27) )=(6 )=(2 )=(1 ) = (12 3 3 3 3 ` i . , ,k ,w 2 ,N 2 = ` 2 2 . , i a ,k 1 ` e ,w ,N ` 1 N 1 2 = 4 in figure AAACAnicbVDLSsNAFJ3UV62vqEs3Q4vQQihJ62sjFNy4rGAf0IYwmd62QycPZiaFErpz4be4EMSFLvwBP8G/MY3ZtPVs7uGe++AcN+RMKtP80XIbm1vbO/ndwt7+weGRfnzSlkEkKLRowAPRdYkEznxoKaY4dEMBxHM5dNzJ3ULvTEFIFviPahaC7ZGRz4aMEpW0HL1Y7gPnjmXgtNayWq/gW1y+Mi6NWsXRS2bVTIHXiZWREsrQdPTv/iCgkQe+opxI2bPMUNkxEYpRDvNCP5IQEjohI4iJJ+XMc+f43CNqLFe1RfM/rRep4Y0dMz+MFPg0GUm0YcSxCvDCJh4wAVTxWUIIFSz5jOmYCEJVEsbyJQk+8UAaeDBloUy5HafJzguJd2vV6Tpp16pWvWo9XJQaZpZCHp2hIiojC12jBrpHTdRCFD2jV/SBPrUn7UV7097/RnNatnOKlqB9/QLf25P3 ( 1 1 ` ` k × AAAB+XicbZDLSgMxGIX/qbdab6OuxE2wCC0MZdIKuhEKblxWsBdohyGTpm1o5kKSKZSh+CwuBHGhC5/CR/BtnKmzsfUswsd//iSc40WCK23b30ZhY3Nre6e4W9rbPzg8Mo9POiqMJWVtGopQ9jyimOABa2uuBetFkhHfE6zrTe8yvztjUvEweNTziDk+GQd8xCnR6cg1zypTF1to6tazo1FFt6iCrYZVr7pm2a7ZS6F1wDmUIVfLNb8Gw5DGPgs0FUSpPrYj7SREak4FW5QGsWIRoVMyZgnxlZr73gJd+kRP1KqXDf/z+rEe3TgJD6JYs4CmK6k3igXSIcrSoSGXjGoxT4FQydOfEZ0QSahOO/j7kmIB8Zmy0HDGI7VkJ1kWuiil2fFq0nXo1Gu4UcMPV+WmlbdQhHO4gApguIYm3EML2kDhCV7gHT6MxHg2Xo2339WCkd85hT8yPn8A6ZiQCA== ( w N N. ) ⇥ AAAB9XicbZDLSsNAFIZP6q3WW1RcuQkWoYVQklbUjVBw47KCrYU2hMn0tB06uZCZtJTQR3EhiAtd+Bw+gm/jtGbT1h8OfJz/nBn+40WcCWlZP1puY3Nreye/W9jbPzg80o9PWiJMYopNGvIwbntEIGcBNiWTHNtRjMT3OD57o/u5/zzGWLAweJLTCB2fDALWZ5RI1XL1s9LEtc2JW1VVK9+VqmbNvC67etGqWAsZ62BnUIRMDVf/7vZCmvgYSMqJEB3biqSTklgyynFW6CYCI0JHZIAp8YWY+t7MuPSJHIpVb978z+sksn/rpCyIEokBVSPK6yfckKExT2b0WIxU8qkCQmOmfjbokMSESpV/+SWBAfFRmEZvzCKxYCddHHNWUNnt1aTr0KpW7FrFfrwq1s3sCnk4hwsogQ03UIcHaEATKKTwCh/wqU20F+1Ne/8bzWnZziksSfv6BcTwj4k= AAAB/HicbZBNSwJBHMZn7c3sbatjHYYkUBDZWQO7BEKXTmKQJuiyzI5/dXD2hZ1ZQRY79Fk6BNGhDn2HPkLfpl3bi9pzGH78n//M8DxOILhUhvGj5TY2t7Z38ruFvf2DwyP9+KQj/Shk0Ga+8MOuQyUI7kFbcSWgG4RAXUfAozO5Tf3HKYSS+96DmgVguXTk8SFnVCUjWz8vNW1SwU3bTI9aGd/gEjErpF4xa2VbLxpVYyG8DiSDIsrUsvXv/sBnkQueYoJK2SNGoKyYhoozAfNCP5IQUDahI4ipK+XMdeb40qVqLFe9dPif14vU8NqKuRdECjyWrCTeMBJY+TgNiAc8BKbELAHKQp78jNmYhpSppIbllyR41AVZwYMpD+SCrXjR6byQZCerSdehY1ZJrUrur4oNI2shj87QBSohguqoge5QC7URQ8/oFX2gT+1Je9HetPe/1ZyW3TlFS9K+fgHIBZBt N ⇥ ( ( , 1 ) 1 = − 1 a AAAB8XicbZC7SgNBFIbPxluMt42WNoNBsJCwGwUtAzaWEcwFsssyOzlJhsxemJlNCCEPYiGIhRY+iY/g2ziJ2yTxrz7Ofy78J0wFV9pxfqzC1vbO7l5xv3RweHR8YpdPWyrJJMMmS0QiOyFVKHiMTc21wE4qkUahwHY4elj47TFKxZP4WU9T9CM6iHmfM6pNKbDLk8AlnuYRKuKhEIEb2BWn6ixFNsHNoQK5GoH97fUSlkUYayaoUl3XSbU/o1JzJnBe8jKFKWUjOsAZjZSaRuGcXEZUD9W6tyj+53Uz3b/3ZzxOM40xMy3G62eC6IQsUpEel8i0mBqgTHJzmbAhlZRpk311k8KYmrTXpDfmqVqyP1s+cl4y2d31pJvQqlXdm2rt6bZSd/IvFOEcLuAKXLiDOjxCA5rAYAKv8AGflrJerDfr/a+1YOUzZ7Ai6+sXJWyQEg== w ) as 1 w w = 1) . . − i N 2 6.13 a e ` 1 2 . We highlighted the decom- k e w i w AAAB93icbZDLSsNAFIYn9VbrLerChZvBIlTQkkRBlwU3LivYCzQhTKan7eDkwsykpYQ8iwtBXOjCx/ARfBunNZu2/quP85+Zw/8HCWdSWdaPUVpb39jcKm9Xdnb39g/Mw6O2jFNBoUVjHotuQCRwFkFLMcWhmwggYcChEzzfz/zOGIRkcfSkpgl4IRlGbMAoUXrkmye1ie9cTXz7AruKhSCxC5z7jm9Wrbo1F14Fu4AqKtT0zW+3H9M0hEhRTqTs2VaivIwIxSiHvOKmEhJCn8kQMhJKOQ2DHJ+HRI3ksjcb/uf1UjW48zIWJamCiOoV7Q1SjlWMZ9lwnwmgik81ECqYvozpiAhClW5g8ScJEdFpL3F/zBI5Zy+b15lXdHZ7OekqtJ26fV13Hm+qDatooYxO0RmqIRvdogZ6QE3UQhTl6BV9oE9jarwYb8b732rJKN4cowUZX79l9JHV ( e ` , N, w , ) + a and shows the example considered 3 , = ` ` = 0 , N a 1 4 as ` 1 =1 ⇥ i 1) p a − . − ) k e } . 2 3 # − a w ) a ,N N w a ,...,N w N , this time listed with repetitions. The k ( (4 ( 3 . Figure Y =1 = 1 2 1 6 1 ∈ { = i p w U 1 AAAB93icbZDLSsNAFIYn9VbrLerChZvBIlTQkrSCLgtuXFawF2hCmExP26GTCzOTllDyLC4EcaELH8NH8G2c1mza+q8+zn9mDv/vx5xJZVk/RmFjc2t7p7hb2ts/ODwyj0/aMkoEhRaNeCS6PpHAWQgtxRSHbiyABD6Hjj9+mPudCQjJovBZpTG4ARmGbMAoUXrkmWeVqVe/mXq1K+woFoDEDnDu1T2zbFWthfA62DmUUa6mZ347/YgmAYSKciJlz7Zi5c6IUIxyyEpOIiEmdEyGMCOBlGngZ/gyIGokV7358D+vl6jBvTtjYZwoCKle0d4g4VhFeJ4N95kAqniqgVDB9GVMR0QQqnQDyz9JCIlOe437ExbLBbuzRZ1ZSWe3V5OuQ7tWtevV2tNtuWHlLRTRObpAFWSjO9RAj6iJWoiiDL2iD/RppMaL8Wa8/60WjPzNKVqS8fULaoeR2A== k e ( i / i w − − ( with the p =1 in terms of if Y ) simplify to a a , " , k ) = ) = k N – 27 – i 2) S e ` 6.18 , non non = 1 2 = 1 = a e p , P P =1 1 i )–( 2 X 0k if ( ( F , 6) 2) 5 = ( G ,...,N , , , a CFT CFT v 3 5 6 6.16 , ` The Young diagram data that labels a non-puncture (no is reformulated in terms of =6 ` = ) , , , =1 n ) i h e i p 3 1 e ` k e n = 1 a , p − w )=(2 )=(6 )=(6 e v p p p N, e ` ` n =3 ( a = p AAAB/3icbZDNSsNAFIUn9a/Wv6hLFw4WwUUpSSvqplBw47KC/QETymRy2w6dJOPMpFJCFy58FheCuNCFj+Aj+DamNZu2ntXHPXfmco4nOFPasn6M3Mrq2vpGfrOwtb2zu2fuH7RUFEsKTRrxSHY8ooCzEJqaaQ4dIYEEHoe2N7ye+u0RSMWi8E6PBbgB6YesxyjR6ahrHotaFTu4hJ2HmPgYO4/MB824D1jgGr7omkWrbM2El8HOoIgyNbrmt+NHNA4g1JQTpe5tS2g3IVIzymFScGIFgtAh6UNCAqXGgTfBpwHRA7XoTYf/efex7l25CQtFrCGk6Urq9WKOdYSnGbHPJFDNxykQKll6GdMBkYTqtIn5nxSEJABVwv6ICTVjN5nVOimk2e3FpMvQqpTtarlye16sW1kLeXSETtAZstElqqMb1EBNRNEzekUf6NN4Ml6MN+P9bzVnZG8O0ZyMr1+xDpQZ w ( e p ,..., ,..., ,...,w 1 1 p =1 1 e ` ` X a ( ( w AAACknicbVHbattAEF0pvSTqJW6ah0JflpoWGYSRnDYthZSUPDQPfUihvoBlxGo1jpespEU7ijFCP5W/yCfkb7qW3Uvszr6cnTPnDDMTKyk0+v6dZe88ePjo8e6e8+Tps+f7rRcHA52XBYc+z2VejGKmQYoM+ihQwkgVwNJYwjC+Olvyw2sotMizn7hQMEnZZSamgjM0qah1486jwKNhkqP26DxSHfqOnlC35x15xx0ahk6oZizDPK3CuUgAhUyAhiBlFNTUXYE/+ub72+LY++D1Ggt3S0r/Sjaof9uoemXlGK/Gbfk6Uavtd/0m6DYI1qBN1nERtW5NL16mkCGXTOtx4CucVKxAwSXUTlhqUIxfsUuoWKr1Io1r+jZlONOb3DL5P25c4vTTpBKZKhEybkoMNy0lxZwu104TUQBHuTCA8UKYzpTPWME4muPcd9KQsRTMapJroXSDJ1Vz6doxswebk26DQa8bHHV7P963T/31FnbJa/KGuCQgH8kpOScXpE+49cr6Yn2zzu1D+7P91T5bldrWWvOS3Av7+y91g71w ( = 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 N , reformulating the partition of . The example of figure non-puncture : 3 Before going on, we pause to go through several examples of puncture profiles, mapping We can identify the monopole charge . Therefore we can equivalently rewrite the flavor symmetry ( non-puncture : a For this case, the CFT answers ( The corresponding inflow data is the Young diagram data to thepunctures. inflow data We and draw computing the the anomaly corresponding contributions Young of diagrams the Example for 1: the case non-puncture. of flavor symmetry) is: In this way,over number-of-rows the are variables relatedsame length. by that taking run into over account the number-of-monopoles multiplicity and of rows those of that the run When this is nonzero, it correspondsk to a location of a monopole, and equals a corresponding is exactly the sequencetotal of number lengths of of rows all isin rows equal figure of to the quantity Figure 4 position of the Young diagram in rectangular blocks of dimensions ( JHEP06(2019)123 n = 4 N (6.39) (6.37) (6.38) (6.34) (6.35) (6.36) (6.33) . . , . + 1 1) i = 1 = 1) − . = 1 2 i N 1) . , e N , k 2 = − ), we obtain 2 . 2 N , N, w = 1) = 1 1 2 6.2 N
1 1 ( , w = k N − 2 1 ( = 1) 1 ) = − − = 1 1 , SU(4), U(1), SU(2). ), we observe agreement up to 3 − ∅ , ,...,N 1 max N w N ) = 6.6 P =2 ( k e i 4 ( e ` N,N N, = 1) , max , 1 2 1 6 ), ( , ), which for the maximal puncture sim- 2 = ) P ( inflow v = 0 1 6.5 , ` N = 2 n 2 ) = ) = 6.18 1 − = CFT v e ` = 2 ( non non 2 n )–( ) = – 28 – 1 P P N, k + 1, or in other words, the number of punctures ` ,...,N ( ( The puncture that preserves the maximal flavor ( 2 , ,N max χ ) = = 6.16 1 k e P The puncture profile that preserves the minimal flavor 1 ( , , − → inflow v inflow = 2 max ) n χ 1 P h N = 1 ( = 2 , ` n N inflow 1 = p ( ) k e − h e p ( = 1 CFT n v ) ) is known as a maximal puncture. In this case the tilde’d p h n − ( n N v − n ( v ): n ( = SU( N = U(1) : = U(1) : F F 1. This is exactly the behavior of a non-puncture, whose only contribution SU( G F G − G n . The Young diagrams corresponding to the four discussed examples, drawn with → n The CFT answers are given by ( We can compare with the inflow answer. Plugging in to ( (1) terms. Equivalently, in terms of the inflow data: Example 3: minimal puncture. symmetry of U(1) corresponds to a Young diagram with In comparison, the inflow result is variables from the geometry since there is both one monopole and one row, and are given by: plify to Comparing with theO bulk inflow answersExample ( 2:symmetry maximal of puncture. variables that denote the Young diagram data are exactly equivalent to the un-tilde’d This has the net effectfrom of shifting is “filling” a hole on the Riemann surface. Figure 5 boxes. From left to right, the preserved flavor symmetry is: JHEP06(2019)123 (6.45) (6.46) (6.47) (6.51) (6.52) (6.42) (6.43) (6.44) (6.48) (6.49) (6.50) (6.40) (6.41) ) gives . 6.9
. , . 2) via: ) . + 3 1 2 = 2 N N/ N 1 + . = − 2 ) = 0 2 2 2 n e N N 5 4 N min , 4 5 , 6 − ) P N, w 2 ( 0 − , , g, = ) ) = 3 N/ ) = 1 2) CFT h (Σ N g,n = . n rect χ 4 rect N/ 1 1 2 P , . (Σ P , ( e ( ,N 1 6 N . 1 2 χ ) + = ( 2 1 2 − 2 ) = 0 ) = = 0 e ` min 0 0 CFT v v inflow N/ , ) ) = ) = 0 , we can preserve SU( , g, g, P a α α ( ) = ) = 0 ) = 2 punctures on the surface `ala ( k 2) = N P P (Σ (Σ ( ( 1 n g,n g,n min N/ CFT SU( N/ k P inflow h , n , n (Σ (Σ ( = = CFT v CFT h 2 1 2 , k + 1 n n 2 2 e N ` ) + k e – 29 – CFT v CFT h ( a CFT v 4 1 ), we see that our results for the bulk anomalies free tensor v free tensor h k 2 For even n n , using the mapping discussed in the previous sub- N/ ) + ) + n n , n N α α , inflow SU( E = 1 2 ) for the minimal puncture are 1 4 ) = = = P P 6.11 = 0 , n ) + ) + k ( ( 1 1 − = 2 g,n g,n ), we once again find rect k e )–( e ) = 6.18 p ( CFT v CFT h P + 1) ( (Σ (Σ , ` inflow v inflow h ) = n n 6.2 )–( min n n 6.10 N ( + + P rect = 1 ( 1 2 inflow v inflow h inflow P 6.16 p ) ( − n n 2) : h CFT v inflow v inflow h n n n n N/ CFT ) with ( ) = − ) h v 6.6 ) + n min n 2) : ( P − ( )–( = SU( min N/ v P F ( n 6.5 ( = 2 monopoles, each with monopole charge 1. CFT G ) p h = SU( n v inflow n F − v G The CFT anomalies ( n ( We see thatcontribution the of inflow a computation singlethe free exactly punctures. tensor cancels multiplet the over the CFT Riemann computation, surface up that does to not the see We prove these relations insection. appendix Then, adding up the contribution of all Our results for anomalies due to a single puncture on the surface can be summarized as Comparing ( can be summarized as and the inflow puncture anomalies are 6.4 Matching CFT and inflow results or equivalently, in terms of the inflow data: For this case, the CFT puncture anomalies are Example 4: rectangular diagram. Plugging the inflow data into ( There are JHEP06(2019)123 , . - 4 a 12 — hol 4 hol I 6 G and con- G -flux N 4 6 M S G M -flux con- 4 , the holo- , and 6 G N M with an arbitrary g,n for more details. In other D ]. These solutions are warped with the connections of exter- near the puncture. 9 with regular punctures is given 4 6 1 E M − fibration over the Riemann surface, , supported by a non-trivial N theories obtained from compactifica- 4 A hol 6 S S M symmetry results from a non-trivial mixing is an r coincides with the topology of ]. The interplay between M5-brane geometric that labels a regular puncture is derived from on a Riemann surface Σ 6 5 – 30 – 1 , N M hol 6 4 − = 2 class in the region of N M N 4 A theory of type supergravity obtained by reducing M-theory on G , which is the class 4 5 ) relevant to the inflow procedure — which gives results S 4 E E , AdS 6 M = 2 irregular punctures. In that case, we expect a more subtle = 2 class N . This observation is particularly interesting in light of the fact that, N with an internal 6d manifold . The topology of 0) theory of type . The latter is a smooth geometry supported by non-trivial N , 6 along non-trivial 4-cycles encode all the discrete data of the class 5 hol 4 4 M G G = 1 constructions such as [ AdS . ]. 0 2 N circle with a global U(1) isometry on the Riemann surface (which is necessarily g, 1 φ S We believe that the methods of this paper can be generalized to treat a larger class Our inflow analysis has interesting connections to holography. At large The inflow anomaly polynomial is obtained by integrating the characteristic class central charges of the CFT. Anomaly inflow thus provides a route to the exact central interplay between bulk andsetups with puncture. irregular punctures, Thisof the intuition the 4d U(1) is motivateda by sphere) [ the fact that, in supergravity solutions that describeincluding the near-horizon geometryengineering, of anomaly a inflow, and stack holography of warrants M5-branes, further investigation. of punctures. For instance,flux it configuration would be for interesting to identify the local geometry and thanks to superconformal symmetry, thec ’t Hooft anomaly coefficients arecharges, related which to in the turn containto non-trivial the information effective about action higher-derivativeThis of corrections circle the of ideas admits natural generalizations to other holographic setups based on 11d nal spacetime turned off.words, the We classical refer solution the toprovides two-derivative reader a supergravity — to local which appendix expression iscal for valid properties at the of large metric the andthat pair flux are ( that exact is in representative of the topologi- graphic dual of an by the Gaiotto-Maldacena solutionsproducts of of 11d supergravity [ configuration is equivalent in coholomogy to but in the presencethe of fluxes punctures of it acquiresstruction. a In richer particular, structure. theregularity partition and The of flux topology quantization of of to the decoupling modessurface from Σ a free (2,0) tensor multiplet compactifiedover on the the space Riemann figuration. In the absence of punctures In this work wetion have of considered the 4d 6dnumber (2 of regular punctures.Hooft anomalies from We the havethat corresponding provided anomaly M5-brane a inflow setup. from first-principles the More derivation M-theory precisely, bulk of we cancels their have exactly shown against ’t the CFT anomaly, up 7 Conclusion and discussion JHEP06(2019)123 ]. ). = 1 17 , . We 1 11 N − [ N ]. In each 0) SCFTs ρ, η, χ, β S A , ) puncture, 33 p, q 0) theory of type , ], or (1 we constructed for 6 32 = 1 class X , however, is different. N spanned by ( 4 bulk 6 X in the bulk, respectively. M governs the anomalies of 4d 0) theory of type 2 Ω , S 12 I onto in the bulk. For a ( , whose anomalies were derived via 6 N 2 X Ω ) punctures in S D p, q U(1) R-symmetry, which is twisted with onto the space × are rotated in a non-trivial way before being 2 punct – 31 – S 2 punct S 0) theory of type , and the azimuthal angle of also governs the anomalies of many other lower-dimensional β 12 + I is trivially identified with φ U(1) R-symmetry in the bulk of the Riemann surface. We expect × 2 punct type, and 2d SCFTs from M5-branes wrapped on four-manifolds. It is S ) puncture is described by the same local geometry 0) E-string theories, whose anomalies are studied in [ S , p, q is a fibration of a 2-sphere 6 ], (1 and the azimuthal angle of X = 2 punctures. The gluing prescription of 28 χ N ) puncture preserves locally an SU(2) with a different 6d SCFT that can be engineered in M-theory using M5-branes. For 1 Finally, we emphasize that our description of punctures is different from and comple- We also envision generalizations of our approach to a broader class of M-theory/string Our strategy can also be applied to regular ( p, q − = 2 theories obtained from compactification of the 6d (2 N Alessandro Tomasiello, and Yifan WangThe for work interesting conversations of and IB correspondence. ported and in FB part is byAspen supported ERC Center in Grant for 787320 part Physics,part - by supported of QBH NSF by this NSF Structure. grant work. grant PHY-1820784. We PHY-1607611, gratefully for RM acknowledge hospitality is the during sup- Acknowledgments We would like toIntriligator, David thank Kaplan, Jacques Jared Distler, Kaplan,tone, Zohar Greg Thomas Moore, Komargodski, Dumitrescu, Raffaele Craig Savelli, Simone Lawrie, Sakura Mario Giacomelli, Sch¨afer-Nameki,Jaewon Song, Mar- Yuji Ken Tachikawa, us to address afield wider theories. class of questions involving anomalies in geometrically engineered One can also considerType generalizations IIB/F-theory and of (massive) this Type framework IIA. to other branementary constructions to in previous methodsdeveloped here that is use more more readily field-theoretic generalizable tools. in M-theory and Indeed, string the theory, approach thus allowing example, one may consider theinflow (2 in [ describing M5-branes probing an ALEcase, singularity, with a anomalies single analyzed characteristic in class [ and would of govern the many anomalies lower-dimensional theories of obtained both via the parent dimensional 6d reduction theory, of the former. N expect that the same class theories obtained fromtheories the of same class parent theorynatural to in conjecture that six this framework dimensions,A still holds including if we 4d replace the 6d (2 In the usual case, the angle identified with the angle theory constructions. Our findings reveal that the class A( respect to the SU(2) that a regular ( regular The space JHEP06(2019)123 ]. 30 (A.7) (A.3) (A.4) (A.5) (A.6) (A.1) (A.2) . This can . u π F F 2 = . 1  c 4 so 12 F I ... . ) bundle can be written 2 tr  2 ) + M ) 2 − u c (2 2 ... 2 F ) . , such that . so 2 + so − A j (tr 2 F         2 1 λ c i . c = . − . ( λ . (tr u + 2 1 2 u  1    2 F iA c 4 0 i η LW ⇢ AAAB4nicbZDLSsNAFIbP1Futt6pLN8EiuCglsYIuC25cVrAXaEOZTE6boZNMmJkUSugLuBDEhS58IR/BtzGp2bT1X32c/1z4jxcLro1t/5DS1vbO7l55v3JweHR8Uj0962qZKIYdJoVUfY9qFDzCjuFGYD9WSENPYM+bPuR+b4ZKcxk9m3mMbkgnER9zRk1eGqpAjqo1u2EvZW2CU0ANCrVH1e+hL1kSYmSYoFoPHDs2bkqV4UzgojJMNMaUTekEUxpqPQ+9hXUVUhPodS8v/ucNEjO+d1MexYnBiGUtmTdOhGWklcewfK6QGTHPgDLFs8sWC6iizGRhVzdpjGiIum75Mx7rJbvp8nOLSpbdWU+6Cd2bhtNsOE+3tVa9+EIZLuASrsGBO2jBI7ShAwwCeIUP+CQ+eSFv5P2vtUSKmXNYEfn6BUdVixE= ( ) plane depicted in figure h (2 LW 2 ρ, η = 0. Moreover, + W d η Y dW . LW = 0 for 2 h 3 ( ρ := 3 N − = 3 1 in the ( = R AAAB6HicbZDLTsJAFIZP8YZ4q7p0M5GYuCCkVRNdkrhxiUYuERoyHQ4wMp02M1MS0vAOLkyMC134Nj6Cb2OL3QD+qy/nP5f8x48E18ZxfqzC2vrG5lZxu7Szu7d/YB8eNXUYK4YNFopQtX2qUXCJDcONwHakkAa+wJY/vs381gSV5qF8NNMIvYAOJR9wRk1aeuoG1IwYFeShZ5edqjMXWQU3hzLkqvfs724/ZHGA0jBBte64TmS8hCrDmcBZqRtrjCgb0yEmNNB6Gvgzcpbd08teVvzP68RmcOMlXEaxQcnSltQbxIKYkGRhSJ8rZEZMU6BM8fQyYSOqKDNp5MVNGiUNUFdIf8IjPWcvmf9vVkqzu8tJV6F5UXUvq+79VblWyb9QhBM4hXNw4RpqcAd1aAADCa/wAZ/Ws/VivVnvf60FK585hgVZX78uuI1Q i axis. The limiting value of – 38 – 2 1 = 0 for localized at the puncture. R ) spoils the separation between bulk and puncture contribu- e ω ≈ − η 3 1 ) R W e ω 4 ∂ 1 AAAB5HicbZDLSsNQEIYn9VbrLerSTbAILkpJVNBlwY3LCvYCbSgnJ5P22JML50wKJfQNXAjiQhe+j4/g25jWbNr6rz7mnwv/eIkUmmz7xyhtbG5t75R3K3v7B4dH5vFJW8ep4tjisYxV12MapYiwRYIkdhOFLPQkdrzx/dzvTFBpEUdPNE3QDdkwEoHgjPJSuz/xY9IDs2rX7YWsdXAKqEKh5sD87vsxT0OMiEumdc+xE3IzpkhwibNKP9WYMD5mQ8xYqPU09GbWRchopFe9efE/r5dScOdmIkpSwojnLbkXpNKi2JoHsXyhkJOc5sC4Evlli4+YYpzyuMubNEYsRF2z/IlI9ILdbPG7WSXP7qwmXYf2Vd25rjuPN9VGrfhCGc7gHC7BgVtowAM0oQUcnuEVPuDTCIwX4814/2stGcXMKSzJ+PoF6ueMDA== , ω . . . ) into the sub-intervals ( AAAB4nicbZDLSsNAFIZPvNZ6q7p0EyyCi1ISFXRZcOOygr1AG8pkctoOnSTDzEmhhL6AC0Fc6MIX8hF8G5OaTVv/1cf5z4X/+EoKQ47zY21sbm3v7Jb2yvsHh0fHlZPTtokTzbHFYxnrrs8MShFhiwRJ7CqNLPQldvzJQ+53pqiNiKNnmin0QjaKxFBwRnmpj8QGlapTdxay18EtoAqFmoPKdz+IeRJiRFwyY3quo8hLmSbBJc7L/cSgYnzCRpiy0JhZ6M/ty5DR2Kx6efE/r5fQ8N5LRaQSwohnLZk3TKRNsZ3HsAOhkZOcZcC4Ftllm4+ZZpyysMubDEYsRFOzg6lQZsFeuvjcvJxld1eTrkP7uu7e1N2n22qjVnyhBOdwAVfgwh004BGa0AIOY3iFD/i0AuvFerPe/1o3rGLmDJZkff0CMRKLAg== ⌘ WL AAAB5HicbZDLSsNAFIZP6q3WW9Wlm2ARXJSSqKDLghuXFewF2lAm05N27CQZZk4KJfQNXAjiQhe+j4/g25jWbNr6rz7Ofy78x1dSGHKcH6uwsbm1vVPcLe3tHxwelY9PWiZONMcmj2WsOz4zKEWETRIksaM0stCX2PbH93O/PUFtRBw90VShF7JhJALBGWWlVg+J9d1+ueLUnIXsdXBzqECuRr/83RvEPAkxIi6ZMV3XUeSlTJPgEmelXmJQMT5mQ0xZaMw09Gf2RchoZFa9efE/r5tQcOelIlIJYcSzlswLEmlTbM+D2AOhkZOcZsC4Ftllm4+YZpyyuMubDEYsRFO1BxOhzIK9dPG7WSnL7q4mXYfWVc29rrmPN5V6Nf9CEc7gHC7BhVuowwM0oAkcnuEVPuDTCqwX6816/2stWPnMKSzJ+voFU6CLpg== ⌘ E , ) plane. We also depict the boundary 1 max 1 + ω AAAB7HicbZDLSsNAFIYnXmu9VV26GSyCi1ISFXRZcOOygr1AE8vJ9LQdOjMJM5NiCXkLF4K40IXv4iP4Nqa1m7b+q4/znwv/CWPBjXXdH2dtfWNza7uwU9zd2z84LB0dN02UaIYNFolIt0MwKLjChuVWYDvWCDIU2ApHd1O/NUZteKQe7STGQMJA8T5nYPPSk48WuqmvJZXwnHVLZbfqzkRXwZtDmcxV75a+/V7EEonKMgHGdDw3tkEK2nImMCv6icEY2AgGmII0ZiLDjJ5LsEOz7E2L/3mdxPZvg5SrOLGoWN6Se/1EUBvRaSDa4xqZFZMcgGmeX6ZsCBqYzWMvbjKoQKKp0N6Yx2bGQTr7YVbMs3vLSVeheVn1rqrew3W5Vpl/oUBOyRm5IB65ITVyT+qkQRjR5JV8kE9HOS/Om/P+17rmzGdOyIKcr1+kJI9e ⌘ e ω max along the small arc tends to a finite value as the arc gets ρ, η d ) interval. As a result, R Y , η a 1 ∂ ( is not a viable choice, and we must use = h Z ω , η 2 1 ) along the 1 e ω ω = − W d 2 a max in the ( η ω 3 , η ) R , one finds R a Z ,η in the shaded region 1 consists of two arcs and two segments. The form max − 2 a η ω R ( ∂ Z η > η , one may consider 1 ω . The region , which is already accounted for separately in our discussion. The small arc is 3 ) 4 Crucially, the integral of Let us split the interval (0 0 with fixed E Instead of are constant in each ( 11 → In this case,ρ however, we get a non-zero contribution from theA vertical contribution from segment, the since, verticaltions taking segment to of the the integral. limit Therefore, closer to the interval (0 is extracted as follows. Y constant. It follows that the integralthat receives the contributions from contribution the from two the arcsarc. only. large arc The does interpretationto not is go ( the to following. zeroidentified as The with we large the increase contribution the arc to size represents ( of the the bulk contribution The boundary horizontal segment gives zero, because segment. This can be seen noticing that, at Figure 6 (counterclockwise) orientation. Let us integrate JHEP06(2019)123 . . a a ` N a from (B.9) (B.6) (B.7) (B.8) w (B.10) (B.11) (B.12) (B.13) → 1 axis. − ω . η a ) R has a finite 1 N LW − 2 . 2 a ) = +  are continuous w ) . Y 1 Y − a . LW − 2 a 2 a 1 + LW w w is positively oriented Y ≤ , and + W dτ )( − R a a a a . 2 a τ ∂ ` Y τ w ∂ w a a 2 ≤ w ) → )( dτ , 1 a and 2 ) − ` 2 a − a W . a a , a LW R , W ) w 1 ( N a R a ( ` + − via = 1. The first term gives clearly O 2 ) interval must be − a a R =1 = , a a ` ( a Y a a a = N τ 2 τ τ τ a , η O i N ) + 3 τ , L 1 ( 2 a τ − ) + − a 1], but it is suppressed by the explicit − 2 + 3 ( R a ) + 3 ` , 2 1 to a a a 1 1 η = a 1 − − N a dτ + LW − 3 p a dτ i i 1 a = w = ) + 3 + 2 2 a – 39 – 1 ) ) R a 2 ( − 0, because both τ ) a − Y W dτ in the ( 3 ( a 3 a = reads a R is constant, τ w → w 1 LW LW ∂ Y ( W + LW ω a Y 2 − 2 a 3 + + ) a ` R 3 + a w − Y Y w , ρ ( ( Y ( h = h ( LW = 2 a 2 a a τ ` W 1 τ + a − W 2 3 a a R  η η Y − W ∂ = = + p η η =1 h 3 ( we thus have from below, 3 a X i a a τ ) a a η − ∂ − η R Y = = = = , we notice that = const, the form → η 1 1 1 + 3 a ω η 0, ω ω R (even though their derivatives have a discontinuity). In order to analyze the . In summary, we do not expect any localized contributions to Z → a a LW η are constant depending on monopole data. The quantity ( a R 2 (2 R = , 2 1 η a One might wonder if the integral on the small arc can pick up contributions localized LW h This quantity hasfactor a of non-zero integralmonopole on sources. [ where value as At leading order in and this quantity goesacross to zero as second term in This quantity has to be integrated from Restricted on if considered counterclockwise, which induces the negative orientation alongat the the monopoles. Let us introduce coordinates We conclude that Notice that an additional minus sign originates from the fact that It follows that the constantAs value a of result, Recall that, as JHEP06(2019)123 . , is 4d  2 n  2 S ) (C.5) (C.1) (C.2) (C.3) (C.4) (D.2) (D.1) 2 2 + ˆ ds ˆ t , , dx 5  − ], and we 2 + 9 2 1 . = . dx AdS  n dx . ∧ ) ( . i 4 1  2 is ˆ ) D 2 ) 2 e dx .  = 2 superconformal 4d TW ) ( D + 6 n 1 e 2 ). Making use of the SO(2) N . p y D dx 0 ∂ 1 r 1 SO(3) N TW dy g, x c ( ) + (ˆ 1 2  N ∂ 1 4 = 0 (Σ 1 p 12 D − ⊕ − χ y y − 1 TW are determined in terms of the ∂ − dv ( ) 3 1 6 v ) , we arrive at ) − p ˆ t e λ SO(2) r 1 6 2 D dx c N ) − ( 2 − free tensor h D NW v x e ) y ( 1 3 = ∂ 2 1 ( , we have equivalently  + y 6 p y ∂ ) 1 2  0 v,h SO(3) − dφ 1 4 − g, , n n = is the corresponding volume form, the angle N 1 ) (1 ( 4 ( is the metric on the unit-radius 0 2 (Σ 1 y ) + S is replaced with g, p χ 5 + 6 – 40 –  1 2 and the 1-form ˆ t 2 (Σ ,NW ) + 2 S , and the Chern root of 2 AdS e λ 4d χ − , vol ˆ t e λ 0 , v TW 2 6 ˆ n ( ds g, 1 2 ds ) R 2 2 − S is e c λ Σ e p D − − 6 r 1 0 y 3 T ) − c − g, y = e ) ( ⊕ ) D y ∂ 0 Σ 2 d 6 y 4 y g, T ) ∂ − SO(3) v , the factor + (Σ ) given in terms of N 0 (1 NW TW 5 ( + χ g, ( decompose as 1 y ) via 2 1 2 6.7 p = free tensor v 2 6 p 2 AdS dφ − n 6  ( with no punctures. The starting point is the 6d anomaly polynomial = 4d , x  ds ), and collecting all terms linear in = 1 n , and the function 0 1 ), we get the final result, 48 NW 4 π g, ∧ 2 ˆ e λ TW  , 6 2 = 6 y, x − e − λ S ( A.10 3.15 2 e 8 h e I D vol ˆ TW t 3 free tensor 6 48 / I = κ 2 κ ⊃ D is a normalization constant, 8 = = 2 I κ 2 11 The most general solution to 11d supergravity preserving 4d GM 4 ds has periodicity 2 G the metric on the roundφ unit-radius function where clarify their connection with themain inflow text. setup in the presence of punctures discussed insymmetry the takes the form D Review of Gaiotto-MaldacenaIn solutions this appendix we briefly review the Gaiotto-Maldacena (GM) solutions [ identifications ( In the parametrization ( Making use of ( Upon integration over Σ The bundles As usual, the Chern root of In this appendix we dimensionally reduceRiemann the surface anomaly polynomial Σ of a single M5-brane on a C Free tensor anomaly polynomial JHEP06(2019)123 . 2 2 Ω Σ S S A , -flux y/N 1 ), we × φ (D.6) (D.7) (D.3) (D.4) (D.5) 4 1, the S 1 φ G part , S D.4 is a sym- y g > β in ( D directions. The πκ N . ], with the round internal circle in the GM 2 ) with all external x = 8 φ ,N puncture, it is useful , 3.9 1 ). By a similar token, ) encodes the detailed x th π is identified with GM 4 α 2 2 α 3.7 in ( µ G β . , x . It follows that 4 4 puncture is described by a α 1 S ), ( E sin x N Z th , ]). Σ 3.6 = ) in the α 9 r ), and . α 2 y = , , x . D.1 α 1 α 2  = 0 2 x x , cfr. ( ) ) . The D dv 2 2 Σ in ( 2 − e x y 2 A D 2 y 2 2 is identified with the ∂ − − S N 3 − 1 y φ 2 1 parametrize a hyperbolic space of constant 2 + + = in cohomology x S 2 2 ) for N shrinking at 2 ) is v D x parametrizes the interval [0 − y 1 ) is interpreted as the near-horizon geometry of – 41 –  φ 4 ( , , in accordance with our conventions for 2 y S D.3 1 D.3 1 (1 − 2 x β , x 4 x − ∂ D.1 2 ) E = via π + cos N 3 with the integer counting the number of M5-branes in = bundle, D y 1 β circle Σ e 2 + x , 2 r N ∂ 2 1]. Clearly, φ Σ surrounding the M5-brane stack is written as an π = (8 in the GM solution is identified with the GM 4  , y r = 2 4 direction on top of the point ( d κ SO(2) G v α 1 S ) y N x v is identified with the angular form − + 1 is a reminder that all 4d connections are switched off. , the directions x GM 4 dφ 4 ( interval [0 on the G D surrounding the M5-brane stack. From the function  E is identified with the round µ 4 = 0, and the 2 A S 2 Ω S is required to satisfy the Toda equation y S π context, the metric in ( vol . In the case of a Riemann surface with no punctures and genus D ), 2 κ S x , = D.1 1 x π GM 4 2 In order to describe a Riemann surface with punctures, one has to allow for suitable In the inflow setup, the G shrinking at 2 to introduce polar coordinates In a sufficiently smallmetry. neighborhood Thus, of in the the puncture, study a rotation of of local the puncture angle geometries one assumes an additional U(1) singular sources in thesource Toda that equation is ( aprofile delta-function of localized the at source astructure in of point the the ( puncture. In studying the local geometry near the the GM 4-form flux 4d connections turned off. More precisely, where the bar over fibration over the solution ( Furthermore, the connection of the connection In order to identify thethe quantity stack, we needquantization to (which choose are different from the conventions of [ negative curvature. The Riemann surfaceof is this realized hyperbolic as space. usual TheS by coordinate taking a discrete quotient parametrize the compute relevant solution to the Toda equation ( With this choice of In the class a stack of M5-branesdinates wrapping a compact Riemann surface, parametrized by local coor- The function JHEP06(2019)123 ) η ,  ρ, η 2 ( (D.8) (D.9) a (D.14) (D.10) (D.11) (D.12) (D.13) ` V dχ a ), delta- η 2 ) and the = ρ + , y D.9 V ¨ b Σ V ) along the r k ˙ η − V b ( as η 2 ˙ is identified with λ V 1 , V ρ 2 − =1 1 b , a X η + ≥ = , 2 ) is then the continuous a ρ . a η V. dη ( 2 η ) = 0) = 0. , λ ∂ 0 + ˙  V 2 ,N 2 = , k b ρ, η  Σ dρ ( k p + ( , r ) and a new function a  V . 00 dχ  p = determined by 00 ) with additional U(1) symmetry is b < η X V . ¨ 0 log ρ, η ˙ V V V dβ ) ˙ via N ) V = = 0 2 D.3 ¨ ) corresponding to a regular puncture. − ··· V  ˙ a η 0 V V , and so on, and we introduced ˙ V ` ( + ρ, η ˙ V < − 2 η V 2 ( λ 2 e V = 0. After the transformation, B¨acklund ∆ 2 2 ∂ ˙ ˙ η ˙ 2 S V isometry of the full solution. V V V, y ∂ , ρ 0 − ds a < η − ) + → = ˙ ρ ∂ ρ 1 V 0 η dβ – 42 – V ) are written in terms of ρ 00  V  = e ∆ e ∆ = (2 , V ) < η ) = lim + , ρ ∂ D.1 ) is mapped to the source-free, axially symmetric η 2 ¨ ( V V 0 bona fide p ( < η < η ρ ρ ) is related to . Crucially, for a generic punctured Riemann surface λ dχ ∂ shrinks at + 1 e − η ∆ with the angle around the axis. D.3 β ( − 5 00 ρ ∂ 1 ρ 2 ˙ ˙ , y a λ V V β β , V S , η 2 = ρ 2 2 AdS e a + ∆ ˙ ˙ 2(2 k V ) V φ ds = a a 2 4 + η η D =  ), the extend to a e − 3 = 0 and with non-trivial charge density profile − p =1 χ , / 2 Σ a  X 1 η ρ r V ) are traded for new coordinates ( D.1 d (  not = a 00 ∧ , y e ` ∆ parametrizes the axis of cylindrical symmetry, while V 2 ] identifies the correct form of Σ N ˙ are integers. The corresponding charge profile r S 9 + V 2 η ( a a  k D vol 3 N N η > η / κ 2 = ( κ and D = 2 = a ) = η η 2 11 ( A puncture is described by a suitable source for the Laplace equation ( The 11d metric and 4-form flux ( The analysis of solutions to the Toda equation ( GM 4 λ ds G where piecewise linear function satisfying The analysis of [ Suppose the puncture is labelled by the partition of function localized at axis. The charge density profile where we used the notation In the presentation ( this condition is translated into the boundary condition The coordinate the distance from the axis, and The source-free TodaLaplace equation equation for ( this symmetry does best performed by means offunction the transformation. B¨acklund The coordinates ( determined implicitly by the relations rotation symmetry associated to JHEP06(2019)123 as , axis  β 2 ) η + (D.20) (D.18) (D.19) (D.15) (D.16) (D.17) 1 η φ . In this . 1  − k = ) quadrant 2 1 η ) χ η a ρ, η η + ( = ) are integrally − 2 π ρ ) can be regarded N η (2 , ), is readily read off ), with / p i . Second of all, in the ) , but one verifies that + ( D.8 a 3 − circle in the base space 1 , 2 GM 4 k 4.20 2 R ρ χ 2 G ) − ) is regular, up to orbifold ρ, η, χ y 1 ), located along the , p ) ) takes the form η a 1 ) strip to the ( − η − /k + . 2 D.10 , µ 2 Σ − ρ, η  η ( ) r Σ 2 N 2 ) r y − a M p + ( 2 ρ, η, χ, β η ) 1 ) to ( 2 − (1 1 ρ − /k 2 1 ) + , y 1 /k Σ . a k η 2 Σ r p Σ N r ( r  2 , k ¨ 0 . Indeed, one verifies that the coordinate V 1 1 + ( a ˙ 2 − V k 1 η = /k 2 − according to ( ( ). ˙ ρ (1 V ˙ = with this source and satisfying the boundary 0. It follows that the V V 2 1 2 2+2 M p 2 k h D.5 − Σ – 43 – V → fibration, introduced in ( r + = p 1 ρ =1 4 β , y a , ρ ensures that all fluxes of . The 3d base space is axially symmetric. Because a X η S 2 L 1 / y = k fibration over the 3d space ( + 1 V − ) k 4.1 1 1 1 ρ β η  ,y S 2 2 in the four directions ( /k /k  Σ ) ) as below ( r 2 2 Σ Σ ( log a 1 1 r r for the 1 k η η D − e Z − N L , corresponding to the non-puncture, we recover the expected ) ) log − + / 1 + 1 a = 1, corresponding to a partition of the form π 4 N η η η ) tends to 1 as p R ) = = = ¨ − and visualized in figure V + ( + ( η = (8 1 ). η 2 2 − ) has the qualitative features depicted in figure ρ, η η ( axis in a smooth way. This was the crucial point in the discussion of κ ( reads ρ ρ ˙ 4.1  V V η D.4 a p p D (2 = 0) = 0 reads k D.18 = 1, + + ) we recognize an 1 2 ˙ 1 V/ 1 1 k ρ, η η η ( as in ( ), . The connection D.10 − + ) := . Moreover, the form of V := 0. The explicit solution for a a D η η η 0  , k 4.1 η D.10 a = = η Let us now relate the puncture GM solutions to our inflow setup. First of all, as The simplest case is ( η Σ r M the quantity 2 shrinks along the section from ( discussed in section transformation ( metric in ( in the general discussionof of backreaction section effects, its metric deviates from the flat metric on If we choose function already anticipated by ouras notation, a specific the transformation realization B¨acklund of ( the coordinate change from the ( and the function with inverse at quantized, if we set situation the coordinate transformation relating ( The 11d metric determined by thissingularities choice of of the form condition where where JHEP06(2019)123 . ) η is a a η 4 , η . At 1 (E.4) (E.1) (E.2) (E.3) a = E satisfy (D.21) (D.22) − w η a η ) derived = ) W 1 i . − 2 a 4.24 and 1 2 w . In conclusion, Y . + − in the vicinity of i 2 a 1 2 4 k e i w 4.2 E e is piecewise constant + N ), and infer )( 2 . a L e p ` =1 . is continuous along the N i 1 2 X a 4.12 0 1 2 1 2 w ˙ along the interval ( V e − ∆ ), ( ) = − − ˙ L LW we also see a direct match of ). First, let us evaluate V α  a a 2 3.9 i + ) in section a P k − N ( e η N a ( Y 6.50 η , a . N = 4.38 − ` a CFT ) with the class = a ) to ( )–( 2 ) k a ), one can verify that p e η =1 ∆ goes to zero at the positions h a a X w N / ) + n 0 N 1  6.48 1 2 ˙ D.10 D.10 V ) = − − D.20 3 a a p without reference to the IR geometry. In e =1 p axis, and ˙ =1 v V . The value of a X i X w , η a n is consistent with the fact that, in the GM η ,W ) = η = α . Then, we can replace − − 4.2 00 (0 a a – 44 – i P 3  η a = V ( k e ) it is easy to verify that a N W ) + ( i w 2 e η k ∆ ( α e ˙ N = = a V 2 a P CFT and i ` ( 2 a ) N D.15 e p =1 e h 2 3 i N a w X 1 6 n =  w inflow − − ) p =1 ) with the expression ( v h a X , and LW n n a are all integer. Using k + − D.14 ) = a ), which matches exactly with the general relation ( v η α = ) + ( Y n P i α ), established earlier in the absence of punctures, is also valid for ( in ( ( k e P D.14 ( , a a D.6 N CFT v w n is only nonzero at the location of a monopole, which occurs at inflow = i ) as in ( k e i is piecewise constant along the h without reference to the fully backreacted picture. ) + a n α axis, with jumps located at ` Y P η − 4.2 ( v n ( inflow v We can also match the GM 4-form flux in ( n and the sum simplifies to Next, we wish to evaluate The quantity that location E Proof of matchingIn with this CFT appendix we anomalies explicitly prove the results ( solutions, the locations the expression of the identification ( puncture geometries. Crucially, evennon-trivial, if and encodes all the 4d data connections that are label turned the off, puncture. the class axis. Moreover, one verifies thatThis the means quantity that, in the GM solutions, Of course, the identification of Using these explicit expressions, togetherthe with ( general properties discussedparticular, in section along the is given by in section the puncture. It is straightforward to compare ( Using this explicit expression and ( JHEP06(2019)123 (E.7) (E.9) (E.5) (E.6) (E.8) (E.10) (E.11) (E.12) a N ]. a . k . This gives: i 1 6  where possible in 2 − SPIRE )] a ) a IN ` 1 . . ) w − ][ 1 ] a a a − 1 2 − w a i , w w ( + 1 a ` − −  , . . . , w a a a 1 1 2 (1 + 2 + w . 1 N − [ ) claimed in the main text. The w a a a − 1 2 2 ∈ k a N w is zero. ) + [ 1 6 1 N 6.49 i = − − k e 1 2 ) = a − ) follows from the aforementioned fact i 2 arXiv:0904.2715 α ) N − [ 1 P N a , i ) ( as − ` 1  6.18 a . Thus we have shown a ) and ( . − a w i − N +1 ), and perform the sum over +1 1 2 e CFT v 1 a N a a n − − 6.48 ` N + a w − E.6 X a – 45 – w ) and ( 2 − i (2012) 034 +1 (1 + 2 N = ) for all ) + e a a 2 i ( a N N a ` α ` 6.4 a 1 2 ( 08 N w ` P p = =1 ., and elsewhere ( = 1 2 1 6 a X a − p ), pull out a factor of ( =1 − 1 1 a k a X  i − − k −  ( a a as a sum over 6 1 , ) ) + ), which permits any use, distribution and reproduction in E.4 = a inflow v p =1 1 = 1 JHEP ` i N w ) that 1 2 i a − X n − , − k e  a a + − − 2 i = = w a a a N e 4.38 ) = N ) give the results ( w N N − and − ) into ( − a a a E.4 = = 2 dualities ( w i E.7 N N E.12 ( CC-BY 4.0 a ( N e 2 ` N ) and ( a  = ` = 2 This article is distributed under the terms of the Creative Commons N i 1 6 e p E.5 e =1 N N − i X  ) and ( , a − p =1 a X w E.3 = ) = i E.4 ( D. Gaiotto, [1] Attribution License ( any medium, provided the original author(s) and source are credited. References Together, ( matching of the flavor central chargesthat ( at Open Access. where in the second line we used These sums simplify to Next, we substitute ( order to make use of the first equality in ( We now re-write the sum over It follows from ( To do this, first note the useful relation JHEP06(2019)123 , 07 , , , JHEP ]. , JHEP JHEP JHEP , ] , , -branes B 500 = 2 = 2 5 SPIRE (1984) 269 N IN -branes N M 5 ][ dualities M -branes on Riemann B 234 (2010) 099 5 (1998) 543 ]. ]. = 1 -branes and anomaly M Nucl. Phys. 11 5 -branes sources , N 6 M D arXiv:1203.2930 ]. SPIRE B 526 SPIRE [ Nonperturbative formulas for IN ]. IN JHEP , ][ ]. bundle over a Riemann surface Nucl. Phys. hep-th/9907107 ]. SPIRE , superconformal field theories 2 [ IN T SPIRE ][ = 2 IN SPIRE SCFTs from SPIRE [ IN Nucl. Phys. N New Seiberg dualities from d IN ]. ]. , ][ 4 (1986) 233] [ ][ (2013) 1340006 (2000) 665 Nilpotent orbits and codimension-two defects of 171 ]. SPIRE SPIRE 17 ]. ]. ]. 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