The singularity structure of scale-invariant rank-2 Coulomb branches

Journal of High Energy Physics, May 2018

Abstract We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 \( \mathcal{N}=2 \) superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kähler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the U(1) R symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.

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The singularity structure of scale-invariant rank-2 Coulomb branches

Accepted: May The singularity structure of scale-invariant rank-2 Philip C. Argyres 0 1 3 4 6 Cody Long 0 1 3 5 Mario Martone 0 1 2 3 0 persymmetry , Supersymmetric Gauge Theory 1 Boston , MA 02115 , U.S.A 2 Physics Department, University of Texas , Austin , USA 3 PO Box 210011, Cincinnati OH 45221 , U.S.A 4 Walter Burke Institute for Theoretical Physics, California Institute of Technology , USA 5 Department of Physics, Northeastern University 6 Physics Department, University of Cincinnati We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 N = 2 superconformal eld theories. Only a dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kahler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the U(1)R symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classi cation of Coulomb branch geometries at all ranks. Conformal Field Theory; Di erential and Algebraic Geometry; Extended Su- - Austin TX 78712, U.S.A. Pasadena CA 91125, U.S.A. 1 Introduction and summary of results 2 Topology of Coulomb branch singularities for rank-2 SCFTs 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 4.3 3 A few concrete examples Basic ingredients of SK geometry Some regularity assumptions Complex scaling action and orbits in rank-2 4.2 SK geometry near V CB scaling dimensions when V is unknotted 4.4 Lagrangian eigenspaces of U(1)R monodromies 5 CB operator dimensions from U(1)R monodromies 6 Summary and further directions A Review of rank-1 scale-invariant SK geometries B Analytic form of the SK section near V n f0g C Sp(4, R) conjugacy classes D Sp (4, Z) characteristic polynomials Introduction and summary of results A striking prediction from the study of the geometry of Coulomb branches (CBs) of 4d N = 2 superconformal eld theories (SCFTs) [1{5] is that the spectrum of scaling dimensions of the CB operators of rank-1 theories can take only one of eight rational values. This fact can be understood in terms of simple considerations involving the topology of the locus of metric singularities on the CB, positivity of the special Kahler metric on the CB, and the electric-magnetic (EM) duality monodromies around the singularities. In the rank-1 case, { 1 { where the CB is 1 complex dimensional, the argument is particularly simple, because the metric singularity is a single point and all other points on the CB are related by the action of the spontaneously broken dilatation and U(1)R symmetries. The answer turns out to be closely related to Kodaira's classi cation of degenerations of elliptic bers of elliptic surfaces [6, 7]. In this paper we will generalize this argument to the rank-2 case, where the CB is 2 complex dimensional. The metric singularities now become a collection of complex curves, which are particular orbits of the combined holomorphic action of the dilatation and U(1)R symmetries of the microscopic SCFT. The EM duality monodromies around these singularities form a representation of the fundamental group of the non-singular part of the CB satis es an integrability condition which was trivially satis ed in the rank-1 case, and so the topological, algebraic, and geometric ingredients in the rank-2 case are considerably more intricate than in the rank-1 case. It may be worth noting that the analog for the rank-2 case of Kodaira's classi cation of singular elliptic bers is the quite complicated classi cation [8] of singular genus-2 hyperelliptic bers; however, this classi cation is only over a 1-dimensional base and does not incorporate any of the SK constraints, and is therefore insu cient for our purposes. We will show that, at least to compute the spectrum of CB operator dimensions, one can bypass most of the intricacies of topology and details of Sp (4; Z) conjugacy classes. The key is to recognize that EM duality monodromies around cycles which are U(1)R orbits have special properties. In particular, the SK section, i.e. the set of special coordinates and dual special coordinates, lies in an eigenspace of these monodromies, which includes a lagrangian subspace of the space of electric and magnetic charges, and the associated eigenvalues have unit norm. This, together with a determination of the nite list of possible characteristic polynomials of the relevant EM duality monodromies, restricts the set of allowed CB dimensions to rational numbers satisfying some simple equations. Furthermore, this set is nite if it is assumed that all CB dimensions are greater than or equal to 1. This latter assumption follows from unitarity if the CB coordinates are vevs of CB chiral operators in the SCFT, a su cient condition for which is that the CB chiral ring is freely generated [9]. rank-1 scaling dimensions. The resulting list of 24 allowed rank-2 CB scaling dimensions is given in table 1. The dimensions greater than one range from 12=11 to 12, and, of course, the list includes the 8 In addition to this concrete result on the spectrum of CB scaling dimensions, we develop a set of tools which will be useful for constructing all possible scale invariant rank-2 CB geometries. Our key results are: the algebraic description of the possible varieties, V, of CB singularities in (2.11); the computation of the possible topologies of V given in (2.18); the factorized description of the local EM duality monodromy linking components of V in terms of Sp(2; Z) = SL(2; Z) matrices given in (4.9); the fact that the SK section is an eigenvector of U(1)R monodromies with unit-norm eigenvalue (4.17); the lagrangian eigenspace property (4.30) and fact that all eigenvalues have unit norm (4.31) of the generic { 2 { (knotted) U(1)R monodromy; and the interrelations of the three topologically distinct U(1)R monodromies recorded in tables 2{4. It may be helpful to put what we do here in the broader context of the program of systematically classifying CB geometries initiated in [10{14] for the rank-1 case. At its core, this program relies on a two step process, each one in principle generalizable to rank-r theories: (i) Classify the complex spaces that can be interpreted as CBs of SCFTs. These are metrically singular spaces which are SK at their regular points ,and which have a well-de ned action of the microscopic N = 2 superconformal symmetry algebra. HJEP05(218)6 (ii) Further classify the possible mass or other relevant deformations of the set of geometries obtained in step (i). These are complex deformations preserving an SK structure and satisfying various other physical consistency requirements, described in [10]. This paper presents rst results in the rank-2 case towards realizing step (i). We emphasize that nding the spectrum of rank-2 CB dimensions is not by itself a classi cation of scaleinvariant rank-2 CB geometries. For instance, despite the niteness of the list of allowed scaling dimensions, it is not obvious that the set of distinct scale-invariant geometries is nite. We do not attempt to address step (ii), the analysis of deformations, which seems considerably more challenging than step (i). Looking beyond rank-2, we note that it is possible to generalize many of the arguments in this paper to arbitrary rank N = 2 SCFTs [15]. In particular these generalizations can be used to show that all the CB operators of N = 2 SCFTs have rational scaling dimensions and, for a given rank, only a nite and computable set of possibilities is allowed. The outline of the rest of the paper is as follows: section 2 analyzes the topology of the set of singularities in the CB. We denote the CB by C, and its subset of metrically singular points by V . The set of metrically regular points, M = C n V, is a 2-dimensional SK manifold. After a brief review of the essential elements of SK geometry, we motivate some regularity assumptions which amount to assuming that C ' C2 as a complex space, and that V does not have accumulation points in transverse directions. We then introduce the holomorphic Cf action on C induced by dilatations and U(1)R transformations of the underlying SCFT. We conclude section 2 with the analysis of the topology of V, which can be the nite union of arbitrarily many Cf orbits, by computing the fundamental group of M. Section 3 illustrates the arguments of section 2 by analyzing examples of the simple case of rank-2 lagrangian SCFTs. In particular, we show how to work out the topological structure of V in these cases from familiar physical considerations. Section 4 is concerned with the connection between the topology of the singularity locus V and the EM duality monodromies around various cycles linking V . This connection is forged by the SK geometry of M. The central role is played by , the SK section, which is the 4-component vector of special coordinates and dual special coordinates varying { 3 { holomorphically on M,1 and su ering EM duality monodromies around V showing that regularity of the SK metric on M and the SK integrability condition imply that derivatives of span a lagrangian subspace of the charge space. We then argue that has a well-de ned nite limit almost everywhere on V, and that locally only two of its components can vanish identically along V . This implies that the EM duality monodromy around a small circle linking a component of V must have a simple factorized form in terms of Sp(2; Z) monodromies, and allows us to complete an argument, started in section 2, showing that the scaling dimensions of the two CB coordinates are commensurate. With commensurate coordinates, the orbits of the U(1)R action on the CB are closed, and is an eigenvector with a unit-norm eigenvalue of the EM duality monodromy around such orbits. Furthermore, for a generic such orbit, the eigenspace in which takes values is shown to contain a lagrangian subspace of the charge space. These somewhat technicalsounding results provide a tight set of relations between the topology of V, its associated monodromies, and the scaling action on the CB. Section 5 applies the results of section 4 to derive the main result of the paper: the full list of possible scaling dimensions of Coulomb branch operators of scale invariant rank2 theories, collected in table 1, and a set of correlations among the conjugacy classes of the three di erent types of U(1)R monodromies, recorded in tables 2{4. To derive the latter results some detailed information about the conjugacy classes of Sp (4; Z) is used. We conclude in section 6 with a summary of the likely next steps required in pursuit of constructing all scale-invariant rank-2 CB geometries. The paper is completed by four appendices collecting both some known and some original technical results. Appendix A reviews the construction of rank-1 scale invariant geometries, which we aim to generalize. Though we do not strictly need it for any of the arguments of this paper, in appendix B we derive the analytic form of the SK section in the vicinity of a point of V n f0g in terms of the Jordan block decomposition of the monodromy matrix around V . Its explicitness may be helpful for making the reader's understanding more concrete. Appendix C collects some useful results about conjugacy classes of Sp(4; R), reviewing generalized eigenspaces and some symplectic linear algebra along the way. Finally, appendix D describes the EM duality group, Sp (4; Z), and derives the possible characteristic polynomials of their elements with only unit-norm eigenvalues. Some elementary properties of cyclotomic polynomials are reviewed there as well. 2 Topology of Coulomb branch singularities for rank-2 SCFTs U(1)2 gauge charge sectors. In this section we will describe the topology of the set of metric singularities V in a rank-2 CB C . The metrically-regular points of the CB, M := C n V, form a special Kahler (SK) manifold, which we assume to be 2-complex-dimensional. In section 2.1 we review the essential elements of SK geometry. In general how, or even whether, the complex structure of M extends to C is not clear from physical rst principles. In this paper we will therefore make the simplifying 1Integer linear combinations of its components give the N = 2 central charges in various low energy { 4 { assumption that the complex (but not metric) structure of M extends smoothly through C (this assumption has physical implications, which are discussed below). Together with the assumption that the microscopic eld theory is a SCFT, this will show that as a complex manifold, C = C2. Also, as we explain in section 2.2, we do not know how to rule out, from rst principles, sets of metric singularities V which are dense in C, and so we also assume that V has no such accumulation points. In section 2.3 we describe the holomorphic Cf action of the combined (spontaneously broken) dilatation and U(1)R symmetries on the CB of a SCFT. We then classify the Cf orbits of points in C, which in our rank-2 case coincide with possible irreducible components of V. In the case that a certain class of \knotted" orbits occur as components of V, we deduce that the scaling dimensions of the CB operators must be commensurate. In section 2.4 we describe the topology of V in more detail. Speci cally, we compute 1(M) explicitly in terms of a simple set of generators and relations, using the results of a recent knot group computation [16]. To see the connection to knot groups (which are the fundamental groups of the complements of knots in S3), note that by dilatation invariance it is enough to consider X := V \ S3, where S3 is a three sphere of radius centered at the origin of C = C2. Then X is a deformation retract of V, which is a 1-real-dimensional manifold embedded in the 3-sphere | i.e., a knot | and 1(M) is the knot group of this knot. We show that X is a torus link | a real curve which wraps a torus, T 2, p times around one cycle and q times around the other, with ` parallel and disconnected components. Unknotted circles, wrapping `0 times around the inside and ` times the outside of the torus, are allowed as well. Examples of such X 's are shown in 1 gures 1, 2 and 3. The importance of 1(M) is that the main arithmetic constraint on the SK geometry of C arises from the fact that the EM duality monodromies of C form a representation of 1(M). The other main constraint is a geometric one, arising from the existence of an SK metric on C, and will be discussed in section 4. These are the ingredients needed for constructing all rank-2 SCFT CB geometries via analytic continuation, generalizing the rank-1 classi cation. 2.1 Basic ingredients of SK geometry M On the CB C of vacua of a rank-r 4d N = 2 SUSY QFT, the manifold of generic points C is described by a free N = 2 U(1)r gauge theory in the IR. In particular, in this continuous set of vacua all elds charged with respect to the r massless vector multiplets are massive. Combinations of the vevs of the complex scalars of the U(1) vector multiplets are good complex coordinates on M, and the kinetic terms of the scalars de ne a Kahler metric on M. Low energy N = 2 supersymmetry implies the existence of an SK structure on M, which relates adjoint-valued (i.e., neutral) scalars to the U(1) vector elds. The main ingredients are the charge lattice and its Dirac pairing, and the N = 2 central charges, in terms of which the SK geometry of M is completely determined. There are various other formulations of SK geometry; a paper that describes the main formulations, and is explicit about the equivalence of the various formulations, is [ 17 ]. { 5 { HJEP05(218)6 where d is the exterior derivative on M.3 Some consequences of this condition will be explored in section 4 below. The BPS mass of a dyon with charge vector p is jZpj, where is the central charge. Here pT denotes the dual pairing V V ! C. The SK section also determines the Kahler geometry of M. For instance, the Kahler h d ^; d i = 0 ; Zp := p T ; K = ih ; i ; The charge lattice is a rank 2r lattice, Z2r, of the electric and magnetic U(1)r charges of the states in the theory, along with the Dirac pairing hp; qi 2 Z for charge vectors p; q 2 Z2r. The Dirac pairing is non-degenerate, integral, and skew bilinear. The electricmagnetic (EM) duality group is the subgroup of the group of charge lattice basis changes, Sp (2r; Z) GL(2r; Z) which preserves the Dirac pairing.2 It is convenient to introduce a complex \charge space" V := C Z2r ' C2r, and to extend h ; i to V by linearity. The central charge is encoded as a holomorphic section of a rank 2r complex vector bundle : E ! M with bers V (the linear dual of the charge space) and structure group Sp (2r; Z). We will call the \SK section"; its 2r complex components can be thought of as the r special coordinates and r dual special coordinates on M. V inherits a Dirac pairing and Sp (2r; Z) action from that on V . The SK section is not unique: two and 0 related by 0 = M T for M 2 Sp (2r; Z) de ne the same special Kahler The SK section satis es a further condition, which we will call the SK integrability sections geometry M. condition: potential on M is given by (2.1) (2.2) (2.3) from which the metric can be readily obtained. The consequences of demanding regularity of the Kahler metric on M will be discussed in section 4. Finally, the condition that be a holomorphic section of E , and that E has structure group Sp (2r; Z), simply means that is a holomorphic vector eld locally on M, and that the analytic continuation of along any closed path in M will give a monodromy, ! M T , with M is trivial in 1(M), then M 2 Sp (2r; Z). By continuity, and since Sp (2r; Z) is discrete, if = I. Thus the monodromies M = M[ ] only depend on the homotopy class [ ] of , and M[ ] give a representation of 1(M) in Sp (2r; Z). 2The reason for the Sp (2r; Z) notation is that we are allowing more general Dirac pairings than the canonical \principally polarized" one. This generality is important, for instance, if one wants to describe \relative" eld theories which appear naturally in rst principles [13, 18] and class-S [ 19 ] constructions of N = 2 eld theories. Sp (2r; Z) is discussed in appendix D, but since the facts that Sp (2r; Z) Sp(2r; R) and that Sp (2r; Z) GL(2r; Z) are the only facts we will use about Sp (2r; Z) in this paper, the distinction between Sp (2r; Z) and the more familiar Sp(2r; Z) EM duality groups will not play any role. in which the Dirac pairing is given by the canonical syrmmplaetcrtiiccesfoArmij :J==@( 01i=0@1u)j and Ir, T = where = BA 1 for A, B the r 3In a basis of V then (2.1) is equivalent to couplings and theta angles. r matrix of U(1)r gauge { 6 { The CB C is the metric completion of the SK manifold M whose points correspond to vacua with r massless vector multiplets and a mass gap for all other elds charged under the low energy U(1)r gauge group. We will call the points of C n M | which, by de nition, are at a nite distance in the metric on M | the singularities of the CB, and denote the set of all singular points by V C . These are the vacua where some states charged under the U(1)r gauge group become massless. Note that C need not be singular as a complex space at V; however, it will have metric singularities (non-smooth or divergent metric invariants) at all points of V, re ecting the breakdown of the description of the low energy e ective action in terms of free vector multiplets. In fact, in general it is not obvious that C need even inherit a complex structure at all. Even if C is assumed to be a complex analytic space, such spaces can be quite complicated. We propose to bypass possible \strange" behaviors by assuming: The complex structure of M extends through V to give a complex manifold C. (2.4) This is certainly a stronger assumption than is needed to perform the following analysis; a discussion of weaker assumption will appear elsewhere [ 20 ]. In the case of a SCFT, this assumption has a clear physical interpretation: it implies that the (reduced) CB chiral ring of the SCFT is freely generated (see [9] for a discussion of the low energy consistency of this assumption). In [9] it was also shown that CBs of SCFTs with non-freely generated chiral rings can have intricate complex singularities which can be separating and non-equidimensional | thus making C not even topologically a manifold | but are not disallowed by any physical requirements. Thus while it is conjectured that all N = 2 SCFT CB chiral rings are freely generated, we do not know of a physical reason for this to be true. Even with the assumption that C is a complex manifold, there are only a limited number of general things that can be physically inferred about the topology and analytic geometry of the set of metric singularities V C on the CB. If a state in the theory with charge q 6= 0 becomes massless at a point where Zq = 0, then there will be charged massless states in the spectrum of the e ective theory everywhere on the locus Vq := fu 2 C j Zq(u) = 0g. This follows since if there were a wall of marginal stability transverse to the Zq = 0 locus for the BPS state with charge q to decay, say, to states with charges p and m, then charge conservation and marginal stability imply Zq = Zp + Zm and arg(Zq) = arg(Zp) = arg(Zm). Therefore Zq = 0 implies Zp = Zm = 0. The set of all metric singularities V will be the union4 of the Vq subsets, V = S q2 Vq, for q running over some subset, , of charges in the EM charge lattice . Since the equation de ning Vq is linear in q, all q 2 can be taken to be primitive vectors in . However need not be a sublattice of , since if there are BPS states with charges p and q in the spectrum, there need not be a BPS state with charge p + q in the spectrum, as the states with charges p and q in the spectrum need not be mutually BPS. 4If Vq itself has disconnected components, then it may be possible that only some of these components are in V, since then walls of marginal stability may prevent BPS states with charge q from being in the spectrum of the e ective theory at other components. { 7 { Since the section, , is a locally holomorphic function on M, so is Zq = qT , and therefore Vq is a complex codimension one locus in C. However, because Zq is not analytic on C (it has branch points along Vq, re ecting its multivaluedness associated with its having non-trivial EM duality monodromy around Vq), Vq is not obviously an analytic subspace of C. In particular, a given Vq might have accumulation points where it becomes dense in C, and, if the cardinality of is in nite, then the union of the Vq could conceivably also accumulate densely in C. For example, if u is one of the r complex coordinates on C, one could imagine a central charge which behaves like Zq = psin( =u). This has zeros (and branch points) at the hyperplanes u = 1=n for n integer, and is dense around the u = 0 hyperplane. If a state of charge q were in the spectrum, then V would include all these hyperplanes. Furthermore, by including the u = 0 hyperplane in V (for instance if there were another state of charge p in the spectrum with central charge, say, Zp = u1=3), then every point in M = C n V has an open neighborhood with jZqj > 0 and jZpj > 0, and so has a consistent low energy interpretation as a theory of free massless vector multiplets. Of course the above toy example is not a full- edged SK geometry at its regular points. In particular, we suspect that there is no set of EM duality monodromies and compatible SK metric on M consistent with Zq having an essential singularity at u = 0. Since we do not know how to prove it, we will assume that such behavior does not occur. In particular, we will assume that Any complex curve in C transverse to V intersects V in a set of points with no accumulation point. (2.5) complex co-dimension 1 in C. topology and geometry of V. If V were an analytic subset of C, this would essentially be the de nition of it being of We now add superconformal invariance to the mix, thereby greatly constraining the 2.3 Complex scaling action and orbits in rank-2 For the remainder of the paper we focus on CBs of N = 2 SCFTs. In particular, we will therefore only need to characterize those V which are invariant (as a set, not pointwise) under superconformal transformations. Conformal invariance, together with N = 2 supersymmetry, implies that there is a Cf action on the CB which arises as follows: scale invariance implies a smooth R+ action on C, arising from the action under dilation D, with an isolated xed point at the unique superconformal vacuum, O 2 C. N = 2 superconformal invariance implies that, in addition, there exists a U(1)R global symmetry. On the Coulomb branch the vevs of chiral scalars spontaneously break both D and U(1)R, and their respective D and U(1)R charges are proportional. This means that the R+ D-action and the R U(1)R action5 on C combine to give a holomorphic Cf action on C, which we denote by P 7! P for 2 Cf and P 2 C. 5Note that we do not require that the U(1)R action is a circle action, but only an R = Sf1 action. This is equivalent to not requiring that the scaling dimensions of the coordinates on C be rational. In the end, however, we will only nd solutions in which the dimensions are all rational. { 8 { Here Cf denotes the universal cover of C , e.g., the Riemann surface of y = ln . We will call this Cf action on C the complex scaling action on the CB. We normalize the Cf action so that quantities with mass dimension 1 scale homogeneously with weight one in . Let us specialize now to the case of a 2 complex dimensional CB. Take u := (u; v) to be a vector of complex coordinates on an open set around O. Without loss of generality we will take O = (0; 0). In a neighborhood of O there exists a continuous complex scaling action on C which exponentiated to get an action of the form xes O. The scaling action can then be linearized around O, and then 1 , then (2.6) corresponds physically to a scaling action on the two complex scalar operators around O on the CB which is not reducible. Such non-reducible representations of the conformal algebra were shown in [21] to not occur in unitary CFTs. Therefore M in (2.6) is diagonalizable, M = 0 u 0v , giving the Cf action This corresponds physically to the existence, in the spectrum of the SCFT/IRFT theory at the vacuum O, of a basis of CB scalar operators for which the scaling action reduces to that of two primary elds with de nite scaling dimensions equal to u and v. Conformal invariance demands that these scaling dimensions be real and positive, and, since we have assumed via (2.4) that the CB chiral ring is freely generated, unitarity implies that they are also both greater than or equal to 1 (see [9] for a discussion): u 1 and v v implies that any neighborhood of O can be analytically 2 using the exponentiated action (2.7). Thus, as a complex space, C = C2, and (u; v) 2 C2 are complex coordinates vanishing at the superconformal vacuum and diagonalizing the scaling action. Complex scaling orbits and singularities. Since dilatations and U(1)R transformations are symmetries of the SCFT, the complex scaling action (2.7) on the CB must Thus V will be unions of orbits Vi of this Cf action, and we write There are three qualitatively di erent 1-dimensional orbits of this complex scaling action: (a) the orbit through the point (u; v) = (1; 0), (b) the orbit through the point (u; v) = (0; 1), and (c) the orbit through a point (u; v) = (!; 1) for ! 6= 0. Type (a) is the submanifold V1 := fv = 0 & u 6= 0 g ' C consisting of the v = 0 plane minus the origin. plane minus the origin. Type (b) is the submanifold V0 := fu = 0 & v 6= 0 g ' C consisting of the u = 0 { 9 { Type (c) orbits are the non-zero solutions to the equation V! := fu = ! v u= v g for a given ! 2 C . Thus we can denote all the possible complex scaling orbits by V! by allowing ! 2 P f0g [ C [ f1g. We will call orbits of types (a) or (b) \unknotted" orbits, and orbits of type (c) \knotted" orbits, for reasons which will become clear in section 2.4. 1 ' Now assume that a knotted orbit V!, ! 2 C , is a component of the set of singularities u and v are not commensurate, then V! does not satisfy our second regularity condition (2.5). For instance, the intersection of V! with the curve u = ! has an accumulation point unless u= v 2 Q, i.e., unless u and v are commensurate. Furthermore, when u and v are commensurate then the general variety of singularities is V = f0g [i2I V!i for some index set I. A necessary condition for the !i not to have an accumulation point in P1 is that I must be a nite set; that is jIj < 1. Actually, it is interesting to note that while the regularity assumption (2.5) is needed to deduce that the number of components V!i is nite, it is not needed to deduce that u and v are commensurate, so long as there is a knotted component (i.e., one with ! 2 C ). The argument is as follows: if u= v 2= Q, then V! with ! 2 C is dense in a 3-real-dimensional submanifold of C. This is easy to see, for instance, by foliating C by 3-spheres related by dilatations. The intersection of the 3-sphere with V! xes juj and jvj, and imposes the linear constraint = ( u= v) on the phases ei and ei of u and v, respectively. Thus V! \ S3 is this line wrapping the \square" torus, T 2 = f( ; ) j + 2 and + 2 g. If the slope u= v of this line is irrational, then the line does not close, and is dense everywhere in T 2. V! is thus dense in the 3-manifold, T3, which is the orbit of this T 2 under dilatations (this bit of analytic geometry will also be used in section 2.4, where it is explained in more detail.) Now pick any point P 2 T3 which is not on V!. Then, because V! is dense in T3, every open neighborhood of P intersects V!. Thus there is no open neighborhood of P with central charges bounded away from zero, and so P cannot be consistently interpreted as a regular point on the CB | i.e., as having a low energy description as a theory of free and massless vector multiplets. Thus V! cannot be a component of V for incommensurate v.6 This should be contrasted with the example given in the paragraph above (2.5). u We have therefore learned that if u and v are commensurate, then the singularity set can be any union of the point at the origin with a nite number of distinct Cf orbits V! (knotted or not), while if u and v are incommensurate, the singularity set can only be a union of the origin with either or both unknotted orbits (V0 and V1). We will see eventually, in section 4.3, that in the case where only unknotted orbits are present in V, the CB geometry factorizes into that of two decoupled rank-1 SCFTs. Since the scaling dimensions of the CB parameters of rank-1 SCFTs are already known to be rational, we will thereby learn that in all cases u and v are commensurate. So from 6There is a way to avoid this conclusion: all points of T3 could be in V. This can happen if the uncountably in nite number of orbits V! consisting of all ! with xed norm j!j are part of V. This would violate the regularity assumption (2.5). HJEP05(218)6 now on we will write p q := v u V! [ f0g is thus the algebraic variety described by the equation and, algebraically, V is described by the curve in C = C2: up = !vq ; V = <u`0 ` Y(up j=1 !j vq) v`1 = 0 ; 9 = ; for 8 : p; q 2 Z + with gcd(p; q) = 1 : (2.9) (2.10) (2.11) where the !j 2 C are all distinct. Here `0 and ` 1 are either 0 or 1, depending on which unknotted orbits are present, and ` is the number of knotted orbits in V . In particular, V n f0g is a smoothly embedded 1-dimensional complex submanifold of C with `0 + ` + ` 1 disconnected components. 2.4 We now describe the (point set) topology of how V is embedded in C. A given knotted component, V!j with !j 2 C , is homeomorphic to the curve X(p; q) := fu simply by continuously mapping !j to 1 in C . Likewise, the set of ` such distinct compop = vqg C 2 nents is homeomorphic to the curve X(p; q)` := fup` = vq`g mapping each !j to e2 ij=` along paths which do not intersect in C . C2 simply by continuously To see the topology of X(p; q), intersect it with S3 := fjujp + jvjq = 2e g for 2 R, which are a family of topological 3-spheres foliating C2 n f0g. Note that di erent 's are related by dilations (i.e., 2 C \ R+). We then see that X(p; q) \ S3=0 is a \deformation retract" of X(p; q) n f0g in C2. Therefore 1(C2 n X(p; q)) ' 1(S03 n (X(p; q) \ S03)). denote X(p; q) \ S03 := K(p; q). K(p; q) is a one real-dimensional curve given by Therefore it is enough to analyze the topology of X(p; q) \ S03 in S3. Henceforth we will 0 K(p; q) = n(u; v) 2 C2 j u = ei ; v = ei with p = q mod 2 o : e i for ; 2 Rg, embedded in S03, and winds p times around one cycle (the Thus K(p; q) is a knot in S03 which lies on the 2-torus T 2 := f(u; v) 2 C2 j u = ei ; v = direction) and q times around the other cycle (the or u direction). A similar construction shows that 1(C2 n X(p; q)`) ' 1(S03 n K(p; q)`), where K(p; q)` is the link with ` components, each of which is homeomorphic to the K(p; q) torus knot, but the jth component is translated along the direction by 2 j=(p`). Thus K(p; q)` = n(u; v) 2 C2 j u = ei ; v = ei with p = q mod 2 =`o : (2.13) Finally, the intersections K0 := V0 \ S03 and K1 := V1 \ S03 are the circles (or \unknots") K0 = n(u; v) 2 C2 j u = 0; K1 = n(u; v) 2 C2 j u = 21=pei ; v = 21=qei v = 0 with with (K1) circles. The solid gray torus is there for visualization purposes. We denote the total link consisting of a torus link together with unknots by L(p;q)(`0; `; `1) := (K0)`0 [ K(p; q)` [ (K1)`1 : (2.15) Here we are using a notation where (K0)`0 := K0 if `0 = 1, and := ? if `0 = 0, and similarly for (K1 )`1 . Similarly, ` = 0 means that there is no torus link component. Thus, for example, L(p;q)(0; `; 0) = K(p; q)`, and L(p;q)(0; 0; 1) = K1. These links are relatively easy to visualize. For example, gure 1 depicts an L(1;6)(1; 1; 1) link with the K(1; 6) knot in red on the surface of a solid gray torus (the torus is present \z-axis" in green. The three dimensions are the stereographic projection of S03 to R purely for visualization), the K0 threading the interior of the torus in blue, and K1 as the 3 with the point at in nity being (u; v) = ( 21=p; 0) and origin being (u; v) = (+21=p; 0). Thus the green line goes through the point at in nity, so is topologically a circle. The fundamental group of C n V. The fundamental group of the metrically smooth part of the CB M, with V given in (2.11) is 1(M) = expression is known as the knot group of the link (2.15). 1(S03 n L(p;q)(`0; `; `1)). The last One can compute the knot group using the groupoid Seifert-van Kampen theorem [16]. For clarity, we rst describe the result in the case with a single torus knot and no unknots. It is ; 1 j 0p = 1 i q : Here the fundamental group has been given as a set of generators, 0 and a single relation, 0p = 1 q . This is the classic result for a torus knot found from a simple application of the Seifert-van Kampen theorem [ 22 ]. The 0 and in the example of a K(1; 6) knot in gure 2. The relation, 0 = 1 cycles are shown 16, is obvious in this simple case. (2.16) 1, subject to γ0 γ ∞ interior of the donut, while 1 threads the hole of the donut. f2 f1 f1 cycle links the rst strand in the direction of the 0 cycle, while f2 links the rst two strands. The 0 and 1 cycles, as in gure 2, are not shown. The generalization to the case of a torus link, K(p; q)`, is quite non-trivial, but thanks to the analysis in [16] we have the following result: 1(M) = h 0; f1; f2; : : : ; f`; 1 j 0pfj = fj 1q ; f` = 1 i : (2.17) There are ` 1 additional generators, fj for j = 1; : : : ; ` 1, and ` relations. It is convenient to add an `th additional generator, f`, simply to make the set of relations look more uniform, but then we must impose f` = 1. The fj generators correspond to cycles which loop individual strands of the link, as shown in gure 3 for the case of a K(1; 2)3 link. In [16] the general result with unknots was found to be: 1(M) = h 0; 0; f1; : : : ; f`; 1 ; 1 j 0 0 = 0 0 ; 1 1 = (K1) circles. The 0 cycle links only the K0 unknot, while 1 links only the K1 unknot. The 0 and 1 cycles, as in gure 2, are not shown. The two generating cycles associated with the unknots are depicted in gure 4. Note that if `0 or ` 1 (or both) are zero, indicating the absence of one or both of the unknot singularities, then the general result (2.18) holds but with additional relations setting 0 or 1 (or both) equal to the identity. A set of consistent EM duality monodromies around the components of V must form a representation of 1(M) in Sp (4; Z) (the EM duality group). The EM duality monodromy around a given component of V largely determines the analytic form of the section of special coordinates on the CB near V ; we will explain this in section 4 below. A representation of 1(M) in Sp (4; Z) is then arithmetic \data" constraining the possible global form of the CB geometry: it provides the boundary conditions that an analytic continuation of from the vicinity of one component of V to that of another must satisfy. The rest of this paper is aimed at sorting out the ingredients necessary for performing this analytic continuation. 3 A few concrete examples Since the discussion in the previous section might appear quite abstract, we will now illustrate the singularity structure of a few CBs with some familiar (i.e., lagrangian) rank2 SCFTs. This will provide a direct physical interpretation of the topology of V C . In particular, we will analyze the singularity structure of two well-known rank-2 theories: SU(3) gauge theory with a single massless hypermultiplet in the adjoint representation, and SU(3) gauge theory with six massless hypermultiplets in the fundamental representation. These examples are particularly illuminating, given that the singularity structure of these two theories realize all the possible distinct topologies discussed above, namely unknots, single (p; q) knots, and a (p; q) link. The moduli space of a lagrangian theory can be explicitly constructed from its eld content. N = 2 gauge theories are described in terms of N = 1 super elds by a chiral eld strength multiplet W = W aT a, and a chiral multiplet = aT a, both transforming The ak's are not gauge invariant, and the residual gauge action on (3.1) corresponds to the Weyl group of the gauge Lie algebra, which is just the group of permutations of the ak. The gauge-invariant coordinates on C are the algebraically independent Weyl invariant combinations of the ak's, u := 6 1 X ak ; 2 k v := 1 2 a1a2a3 ; where the overall normalization of u and v is arbitrary, and has been chosen to simplify the expressions below. in the adjoint representation of the gauge group, which form an N = 2 vector multiplet, and chiral multiplets QiI and QeiI in representations of the gauge group RQ and RQ, which form a hypermultiplet. The index a runs over a gauge Lie algebra basis, I = 1; : : : ; dim RQ is the hypermultiplet gauge representation index, and i is a avor index; i distinguishes di erent hypermultiplets in the same representation RQ. We begin by describing some generalities about SU(3) CBs. The CB is parametrized by the vacuum expectation values of A, the complex scalar in . To simplify notation we use the symbol A in place of hAi where it will not be confusing. Upon eliminating the auxiliary elds, the N = 2 lagrangian contains a scalar potential V Tr [A; Ay] 2, which implies that the Coulomb vacua are parametrized by A taking value in the complexi ed Cartan subalgebra, and so can all be simultaneously diagonalized by a gauge rotation. In particular, for SU(3) we can write: 0 a1 A 3 k=1 X ak = 0 : (3.1) (3.2) (3.3) (3.4) 1 := 1 1 0 ; 2 := 0 1 1 ; and the dot product is the matrix trace. Then the Weyl chamber conditions correspond to setting a1 a2 a3. The CB vev (3.1) generically breaks the gauge group to U(1)2 unless two of the ak's coincide, in which case one of the two U(1)'s is enhanced to an SU(2). This happens precisely at the boundary of the Weyl chamber which is given by those values of A for which A 1;2 = 0. The theory also contains N = 1 superpotential terms W Qei Qi, where the T a's act in the appropriate representation on (Qei; Qi). When A acquires a vev, the superpotential generates masses for the hypermultiplets; in particular, for the fermionic components (which to make notation easier we will also indicate with Qei and Qi) the mass term is of the form We can x the Weyl group redundancy in the description (3.1) by restricting the ak's to a single Weyl chamber by setting A 1;2 0, where 1;2 are the SU(3) simple roots. In the matrix notation of (3.1), the simple roots can be represented by mI eiI A Q I QiI : The I run over the weight vectors of the representation RQ. Thus on an interior point of the Weyl chamber, unless A I = 0 for some I, all hypermultiplets are massive, and the e ective theory on the CB is a free U(1)2 theory. As stated previously, the singular locus V 2 C is parametrized by those (u; v) for which extra massless states charged under the U(1)'s appear in the theory. From the discussion above we see this happens for those values of A such that (a) A (b) A I = 0: some component of the hypermultiplets become massless, or 1;2 = 0: W bosons associated with the extra unbroken SU(2) become massless, restoring an SU(2) gauge symmetry. SU(3) with 1 adjoint hypermultiplet In this example the theory contains only one hypermultiplet, transforming in the adjoint representation of SU(3). In fact, this theory has an enhanced N = 4 supersymmetry. The weight vectors of the representation of the hypermultiplet obviously coincide with the roots of the Lie algebra, and therefore along the (singular) subvariety where one of the two U(1)'s is enhanced to a non-abelian SU(2), some components of the hypermultiplet also become massless. Before analyzing the e ective IR theory along this subvariety, we write it explicitly in terms of the coordinates (u; v) on C: A1SU(2) A2SU(2) 1 = 0 or 2 = 0 or A1SU(2) = A2SU(2) = a a 2a 2a a a 9 = ; =) u3 = v2 : (3.5) In the notation introduced in section 2.3, the hypersurface u3 = v2 (minus the origin) is a knotted Cf orbit of type (c), and it is topologically equivalent to K(2; 3). The components of the hypermultiplets which are massless along (3.5) transform in the adjoint representation of the unbroken SU(2), and are uncharged under the other U(1) factor. It follows that the e ective theory along (3.5) is an N = 4 SU(2) gauge theory with a decoupled free U(1) factor. The existence of a SCFT all along (3.5) implies the presence of metric singularities all along the hypersurface. It follows that in this case V is topologically equivalent to L(2;3)(0; 1; 0). 3.2 SU(3) with 6 fundamental hypermultiplets This case is slightly more subtle. The hypermultiplets transform in the fundamental representation of SU(3) whose weights are 1 = ; ; I = aI , I = 1; 2; 3, and therefore components of the hypermultiplets become massless if any of the ak's vanish. Note that since we are working in a speci c Weyl chamber, the only possibility for an ak to vanish away from the SCFT vacuum at the origin is: A0 2 = 0 or A0 = B 0 =) v = 0 : (3.6) (3.7) 0 a a 1 C A The hyperplane above is again one of the Cf orbits previously analyzed, speci cally a type (a) unknotted orbit. It is straightforward to analyze the e ective IR description of the theory along (3.7). The gauge group is fully broken to U(1)2, which can be chosen in such a way that the extra massless components are 6 massless hypermultiplets with charge 1. In this case this is an IR free theory with massless matter all along (3.7), and we thus expect metric singularities along this sublocus. Thus this provides a component of the singular locus, V0, which is topologically equivalent to the link L(2;3)(0; 0; 1). Note that this topological description misses the algebraic multiplicity of the singularity which can instead be inferred from the SW curve of the theory [23{25], where it is found to be of multiplicity 6. This extra piece of information re ects the fact that 6 charge-1 hypermultiplets are becoming massless there, so the coe cient of the beta function of the U(1) gauge factor they are charged under is 6. e SU(3) theory. Now let us focus on those regions with an enhanced SU(2) symmetry and the corresponding e ective theory. It can be explicitly seen from (3.5) that away from the origin, none of the a's vanish along this subvariety, thus below the energy scale a all the hypermultiplets are massive. The IR theory is a product of a pure SU(2) gauge theory with a decoupled free U(1). Because the pure SU(2) is an asymptotically free theory, determining the location of the singular subvariety is trickier. It is in fact well-known that the SU(2) con nes at some scale SU(2), and no massless W-bosons arise in the IR. However, this theory still has a non trivial singularity structure; by appropriately tuning the CB vev of the pure SU(2) gauge theory, either a dyon or a monopole can become massless. This is the celebrated result [1] that the pure SU(2) theory has singularities at a2 = 2 SU(2), where a are the vevs of the vector multiplet complex scalar in the SU(2) Cartan subalgebra. e Let us now turn to the implications of this observation for the singularity structure of the Notice that for A We rst need to relate ea, parametrizing the IR SU(2) CB vev, with the ak's in (3.1). 1 = 0 (A 2 = 0) the IR SU(2) is embedded in the top left (bottom right) 2 (a2 2 corner of the SU(3) matrices. Thus by inspection ea = (a1 a2)=2 (ea = a3)=2). Next, observe that SU(2), the con ning scale for the pure SU(2) gauge factor, is proportional to the value of a in (3.1). This can be seen as follows. The strong coupling scale for an asymptotically free theory is de ned as / expf2 i SU(2)( )g, where is an arbitrary scale at which the running gauge coupling of the SU(2) e ective gauge factor has value SU(2)( ). In the UV, the SU(3) theory is a SCFT, and so its gauge coupling, , is an exactly marginal coupling which therefore does not run with scaling. Therefore at the scale a where the SU(3) is Higgsed to SU(2) U(1), the SU(2) e ective coupling is : SU(2)(a) = . Therefore SU(2) / a e2 i . Now let us go back to the study of the singular variety of the N = 2 SU(3) SCFT. Con nement of the SU(2) implies that the region in (3.5) is no longer singular as there are no extra massless BPS states there. Instead we expect a massless dyon and a massless monopole to enter the theory at ea 2 = 2 SU(2) which translates to the loci of adjoint 0a(1 + ) a(1 ) 0a(1 + i ) a(1 i ) a(1 2a 1 C A ) 1 C A and and Ai1 = B 02a Ai2 = B a(1 + i ) scalar vevs: A1 = B or A2 = B 02a 2a 1 C A 1 C A a(1 i ) (3.8) (3.9) where = e2 i . The singular subvarieties above can be parametrized in terms of (u; v) We call the union of these two components of the singular region VSU(2), and it is topologically equivalent to 2 parallel K(2; 3) knots or an L(2;3)(0; 2; 0) link. Thus the singular CB locus of the SU(3) with six fundamentals SCFT is the union of the Cf orbits described above: V = V0 [ VSU(2). It is topologically equivalent to an L(2;3)(0; 2; 1) link. This result agrees with the more straightforward analysis of [23{25] in which the SW curve for this theory is constructed and the discriminant locus computed explicitly. 3.3 Other rank-2 lagrangian SCFTs A similar analysis can be performed for the other lagrangian rank-2 SCFTs. There are quite a few possibilities. In fact for each one of the semisimple rank-2 gauge algebras | SU(3), SO(5) = Sp(4), SU(2) SU(2), and G2 | there are many allowed choices for hypermultiplet representations giving vanishing beta function for the gauge coupling. The analysis of the singular geometries for all of these theories contains ingredients similar to the discussion just outlined above, and thus we will not present it in detail. Still it is worth pointing out a few distinct features which we learn from the study of the CB geometries of lagrangian SCFTs: The singular locus VgN =4 of the CB geometries for theories with enhanced N = 4 supersymmetry and gauge Lie algebra g are topologically L(2;n)(0; 1; 0) links, where n is the highest dimension of the Casimir of Weyl(g). Furthermore the CB in this case is an orbifold Cg the x points of the supersymmetry. N =4 = C2= , where = Weyl(g), and VgN =4 corresponds to action. This is not the case for theories with only N = 2 ones with only N beginning [1, 2]. In rank-2, as implied by the previous observation, the scale invariant limit of the CB geometry is sensitive to supersymmetry enhancement. The singularity structure of theories with the same gauge group but enhanced N = 4 are distinct from the = 2. In rank-1 this was known not to be the case since the But, as in rank-1, many distinct rank-2 lagrangian SCFTs share the same scale invariant CB geometry. For a given gauge group, there are multiple choices of hypermultiplet representation which give N = 2 SCFTs. In particular for SU(3), in addition to the two cases presented above, the theory with one hypermultiplet in the fundamental and one in a two-index symmetric representation is also a SCFT. This theory has the same CB geometry as does the theory with six fundamentals.7 This is also the case for SO(5) = Sp(4) gauge algebras where there are a few di erent representation assignments giving rise to N = 2 SCFTs, all of which have singular loci topologically equivalent to L(2;4)(0; 2; 1), as is readily obtained from their SW curves [26, 27]. The last point suggests that to fully distinguish the di erent SCFTs purely from the analysis of their CB geometries we need to study the allowed mass deformations of the scale invariant geometries. This turned out to be a very fruitful e ort in rank-1 [10{14], but many of the techniques that worked there do not seem to generalize to rank-2. We will not make any attempt to study mass deformations here but hope to study this problem in the future. 4 SK geometry of the Coulomb branch in rank-2 In this section we will discuss constraints on the CB geometry that arise from demanding a regular special Kahler metric at all points of M. In particular, after a brief review of the SK metric and integrability condition in section 4.1, we will see in section 4.2 how the physical condition that the CB metric be regular in directions parallel to the singularity V gives strong constraints on the possible EM duality monodromy around a path linking V. In section 4.3 we will use the results of section 4.2 to nd the spectrum of possible dimensions f u; vg of CB coordinates in the case where V has no knotted components. In particular, we show that the problem essentially factorizes into a product of rank-1 geometries, and so the allowed values of u;v are just those of the rank-1 CBs, recorded in table 5 of appendix A. These eight possible values are rational, and so u= v are also rational. This then completes the argument started in section 2.3 that the CB scaling dimensions are commensurate. An important ingredient in the argument of section 4.3 is the use of monodromy around cycles which are orbits of the U(1)R symmetry action on the CB. Such monodromies necessarily have an eigenvalue with unit norm. We call these \U(1)R monodromies" and explore them further in section 4.4. We will indicate U(1)R monodromies with a fancy M . Since we have determined that the CB scaling dimensions are commensurate, there will be closed U(1)R orbits through every point of the CB. This, together with the SK integrability condition and regularity of the CB metric, implies that the eigenspace of the unit-norm eigenvalue of a U(1)R monodromy M must contain a lagrangian subspace of the charge space, V ' C4. This puts a strong constraint on the conjugacy class of M . In 7We thank Y. Lu for pointing this out to us. with respect to the symplectic decomposition V = S? v Sv. However, because it is a U(1)R monodromy, we learn in addition from (4.17) and (4.13) that M0 has an eigenvector with eigenvalue Clearly the eigenvalue of the I block in (4.18) is 1. The possible eigenvalues of unit norm of the Mu block are expf2 in=kg for k 2 f1; 2; 3; 4; 6g and any integer n. This is a simple property of SL(2; Z) matrices, derived in (A.3) in appendix A. Since unitarity and the assumption (2.4) imply possible values of u are u 1 | see the discussion above (2.8) | we learn that the Note that this is precisely the set of allowed CB dimensions for rank-1 theories, recorded in table 5 of appendix A. The argument of the last paragraph applies equally well to the V1 singularity and the 1 monodromy just by everywhere interchanging the roles of u and v, giving the symplectic decomposition V = Su Su? in which M1 = I Mv for some Mv 2 Sp(2; Z). But it is not clear yet how the Su subspace de ned at V1 is related to the Sv subspace de ned at V0, and so the result is that the possible values of v also lie in the same set appearing in (4.20). An immediate consequence of this is that u and v are commensurate (since they are, in fact, rational separately). Recall that we showed in section 2.3 that u and v were commensurate if there were any knotted components of V. We have now shown that they are also commensurate when there are no knotted components. Thus in all cases the CB dimensions are commensurate. As we will discuss in the next subsection, this implies that the U(1)R orbits through any point in the CB is closed, and gives a powerful constraint on the possible structure of U(1)R monodromies. Before we explain that, we outline an argument showing that, in fact, the CB geometry with only unknotted singularities necessarily factorizes, and so describes the CB of two decoupled rank-1 SCFTs or IRFTs. We do not give the full details of the argument, since it is technical in the IRFT case; we do, however, provide the basic analytic ingredients for making the argument in appendix B. Factorization of the CB geometry for unknotted singularities. If W0 V is the subspace spanned by the electric and magnetic charges of states becoming massless at V0 as in (4.7), then Sv W0ann by (4.8). In the case that W0 is 2-dimensional, then, in fact, Sv = W0ann, as remarked below equation (4.8). But that means, by (4.7), that the fv? components of in the eigenbasis decomposition (4.12) vanish on V0: fv?(0; v) = 0 : (4.21) Now 1(M = C n V) is very simple in this unknotted setting: it is generated by loops, 0 and 1, linking V0 and V1, respectively, which commute: 0 1 = 1 0. This can be visualized as in gure 4 without the red knot.16 The EM duality monodromies M0 and M1 around 0 and 1, respectively, therefore commute [M0; M1] = 0 : But since M0 and M1 commute, they have common eigenspaces17 and since their symplectic structures also have to match, we must have either (i): Su = Sv ; or (ii): Su = Sv? : The other four possibilities, i.e., that Su is the span of one si 2 Sv and one sj? 2 Sv?, cannot be realized because those spans are lagrangian, not symplectic, subspaces of V . In case (i) we have, by the same reasoning that led to (4.21), that fv?(u; 0) = 0 as well. In this case the only non-vanishing components of at V0 and V1 are fv 2 Su = Sv. But these are the eigenspaces of the I factor of both the M0 and M1 monodromies. Therefore M0 and M1 must both have eigenvalue = +1. This implies by (4.19) and its analog for v that u = v = 1. But this is a free eld theory describing two massless vector multiplets, and so, in fact has no singularities at all. In other words, in this case the potentially non-trivial SL(2; Z) parts of the M0;1 monodromies are trivial: Mu = Mv = I. Case (ii) is less trivial. Now the same reasoning implies that in addition to (4.21), we must have fv(u; 0) = 0 : In this case the non-vanishing components of at V0 is fv 2 Sv and at V1 is fv? 2 Su. These are now the eigenspaces of the Mu and Mv SL(2; Z) factors of the M0 and M1 monodromies, respectively. Therefore, acting on these eigenspaces, the U(1)R monodromies are M0 = Mu and M1 = Mv, inside the V1 and V0 singularities, respectively. Furthermore, this restricted SL(2; Z) monodromy problem in the two singularity components is equivalent to the rank-1 monodromy problem for analyzed in appendix A. Thus, we nd that fv(0; v) = rank-1 (v) for elliptic SL(2; Z) monodromy Mv, fv?(u; 0) = rank-1 (u) for elliptic SL(2; Z) monodromy Mu. (4.25) Given the boundary conditions (4.21), (4.24), and (4.25), it is trivial to perform the analytic continuation to nd that fv(u; v) = fv(v) and fv?(u; v) = fv?(u) for all (u; v) 2 C2. Together with (4.12) and the fact that fv and fv? are valued in symplectic complements, the Kahler potential (2.3) for this geometry is K = ihf v(u); fv(u)i + ihf v?(v); fv?(v)i, and so the geometry factorizes into a direct product of rank-1 SCFT CB geometries. This argument made the assumption that the subspaces W0;1 spanned by the charges of states becoming massless at V0;1, respectively, were both 2-dimensional. This is equivalent to assuming that there are simultaneously electrically and magnetically charged states 16Indeed, since this a (very) degenerate case of the general torus link, its knot group is given by (2.18) with the identi cations 0 = 1 , 1 = 0, and the fj = 1. 17In the case where they have generalized eigenspaces, coming from non-trivial Jordan blocks, the subspace corresponding to a sum of blocks of a given eigenvalue of one matrix will split into a sum of Jordan block subspaces of the commuting matrix, even though their generalized eigenvector bases may not coincide. (4.22) (4.23) (4.24) becoming massless at each singularity and so that each is described by a rank-1 interacting SCFT, as found above. If, instead, one or both of W0;1 were 1-dimensional, the argument given above for the fv and fv? boundary conditions (4.21) and (4.24) breaks down. Physically, only electrically charged states become massless at one or both of the singularities, describing rank-1 IRfree theories (IRFTs) instead of SCFTs. In this case the Mu;v SL(2; Z) monodromies are of parabolic type, meaning they have non-trivial Jordan blocks, and the behavior of near the singularity is more complicated, as outlined at the end of appendix A. This case can be systematically analyzed by solving directly for the analytic structure of in the vicinity of a component of V in terms of the generalized eigenvector (Jordan block) decomposition of its monodromy. We record this analytic form for in appendix B. Though we will make no further use of this analytic form in this paper, it will presumably be useful for future e orts to construct all scale-invariant CB geometries by analytic continuation from their boundary values at the locus V of metric singularities. Lagrangian eigenspaces of U(1)R monodromies Consider a point, P , on the CB which is not on either of the unknotted Cf orbits. This through this point is the set fu = ei' is a point with coordinates u = (u ; v ) 2 C 2 with u 6= 0 and v 6= 0. The U(1)R orbit u ; ' 2 Rg, where the Cf action, \ ", is given by (2.7). As long as u and v are commensurate, this orbit forms a closed path. To see this, de ne the positive coprime integers p and q by q=p = u= v as we did before in (2.9), and de ne the real number s := q u = v p : Then the smallest positive value of ' such that ei' u = u is easily checked to be ' = 2 =s. Thus p;q := fu = u eiqs'; v = v eips' ; ' 2 [0; 2 =s)g ; describes a simple closed path in the CB. Note that this path is homotopic to the U(1)R orbits through other points in a small enough neighborhood of u . By our argument on U(1)R monodromies in the last subsection, (4.17) holds: (u ) is an eigenvector of the U(1)R monodromy Mp;q 2 Sp (4; Z) around p;q with an eigenvalue of unit norm: Mp;q (u ) = (u ) with = expf2 i=sg : Since p;q is homotopic to nearby U(1)R orbits, it follows that (4.28) holds not just at u but in a whole open neighborhood of u . Then taking the u-derivatives of (4.28) gives Mp;q d d with = expf2 i=sg ; in this neighborhood. Writing d eigenspace of Mp;q. Recall from the discussion in section 4.1 that (4.26) (4.27) (4.28) (4.29) regularity of the Kahler metric and the SK integrability condition imply that @u and @v span a lagrangian subspace of V . Thus we learn: The eigenspace of Mp;q contains a lagrangian subspace. (4.30) This constraint greatly restricts the allowed conjugacy class of the Mp;q 2 Sp (4; Z) monodromy. Appendix C lists the Sp(4; R) conjugacy classes. Using this list it is a simple matter to nd the ones with a unit norm eigenvalue whose eigenspace contains a lagrangian subspace; these are listed in (C.8). It turns out that these are matrices all of whose eigenvalues have unit norm. Since the Sp (4; Z) conjugacy classes are subsets of Sp(4; R) conjugacy classes, this is also true of all Sp (4; Z) elements that satisfy (4.30). So even though only a single unit-norm eigenvalue of Mp;q is required by virtue of its being a U(1)R monodromy, nevertheless: All of the eigenvalues of Mp;q have unit norm. (4.31) 5 CB operator dimensions from U(1)R monodromies We now combine the constraints on U(1)R monodromies derived in the previous sections with some simple topology of the U(1)R orbits to derive a nite set of possible scaling dimensions, f u; vg, for the CB operators. First, note that there are three distinct classes of U(1)R orbits in C2 n f0g. We have met them all in the last section, but we reproduce them here: 0 := 1 := p;q := u = 0; v = v eips'; u = u eiqs'; v = 0; u = u eiqs'; v = v eips'; ' 2 0; ' 2 0; ' 2 0; 2 ps 2 2 qs s ; ; ; where u and v are non-zero complex numbers. Here we are parameterizing, as before, the commensurate CB dimensions by u := qs; v := ps; p; q 2 N; gcd(p; q) = 1; s 2 R+: C2 n f0g. De ne 0 , 1, and p;q are homotopic to, respectively, the K0, K1 unknots, and the K(p; q) torus knot introduced in section 2.4. They depend on a choice of base point P = (u ; v ) 2 It is easy to see that p;q is in the knotted orbit V!, while 0 lies inside the unknotted complex scaling orbit V0, and 1 inside V1. Consider a general rank-2 SCFT CB, C = C2. As explained in 2, the subvariety, V, of metric singularities of C is a nite union of distinct V! complex scaling orbits: V = [j V!j . ! = up=vq 2 P1 : (5.1) (5.2) (5.3) All p;q with ! 2= f!j g [ f0; 1g are homotopic in C n V. This is easy to see since ! takes values in P1, so we can continuously deform a p;q with one value of ! to another by following a path in P1 that avoids the nite number of !j points as well as the ! = 0 and ! = 1 points. Note, however, that deforming ! continuously to 0 or to 1 is not a homotopy since the unknotted 0 and 1 orbits have a di erent topology than the p;q knots. This is re ected in the way the periodicity of the ' coordinate in (5.1) jumps discontinuously at ! = 0 and ! = 1. In fact, from these periodicities it is easy to see that as ! ! 0 or 1, p;q is homotopic to a path that traverses 0 or 1 an integer number of times: HJEP05(218)6 p;q ( 0) p ( 1 )q : Thus 0 , 1, and p;q represent three distinct homotopy equivalence classes of U(1)R orbits in M = C n V, the manifold of metrically regular points of the CB. Denote the U(1)R monodromies su ered by upon continuation around 0 , 1, and p;q by M0, M1 , and Mp;q, respectively. Then the unit-norm eigenvalue property of U(1)R monodromies (4.17) implies that M0 (0; v) = exp(2 i=ps) (0; v) ; M1 (u; 0) = exp(2 i=qs) (u; 0) ; Mp;q (u; v) = exp(2 i=s) (u; v) ; M0q = M1 p = Mp;q : (5.4) (5.5) (5.6) (5.7) (5.8) for all (u; v) 2 C n V and with u 6= 0 and v 6= 0. Also, the homotopy relations (5.4) imply As discussed at length in the previous section, the SK section, , has a nite, nonzero, and continuous limit as it approaches any point of V n f0g, the locus of metric singularities away from the origin (it is not analytic there | it has branch points | but its limit is still well-de ned). Thus, in particular, the above statements (5.5){(5.6) about the M0 and M1 monodromies hold even if the u = 0 or v = 0 planes are in the singular locus. Because the Mp;q monodromy applies to U(1)R orbits in all of the regular points of the CB minus the u = 0 and v = 0 planes, it satis es the conditions (4.30) and (4.31) derived in the last section, which stated that its exp(2 i=s) eigenspace must be at least two-dimensional and contain a lagrangian subspace. In appendix D we derive the list of possible eigenvalues that Sp (4; Z) matrices satisfying these conditions can have. In fact, in that appendix we determine the characteristic polynomials of these matrices. The characteristic polynomials are invariants of the conjugacy classes of Sp (4; Z), but typically to each characteristic polynomial there can exist many conjugacy classes. A list of all Sp(4; Z) conjugacy classes with only unit-norm eigenvalues (what we called \elliptic-elliptic type" in appendix C) can be extracted from [8, 29]; the subset of such conjugacy classes with no non-trivial Jordan blocks is nite. In the notation for the characteristic polynomials introduced in appendix D, there are only ve which can correspond to matrices with a lagrangian eigenspace: [14], [24], [32], Possible CB scaling dimensions of rank-2 SCFTs fractional integer with the assumption that the CB chiral ring is freely generated. [42] and [62]. A characteristic polynomial of the form [N #] has eigenvalues expf 2 i=N g. Comparing this to (5.7) it follows that + C for N 2 f1; 2; 3; 4; 6g and C 2 Z: (5.9) This implies s is rational and therefore the CB dimensions u and v are rational. However this does not constrain them to lie in a nite set since there is an in nite set of allowed values for s, due to the freedom in choosing C 2 Z in (5.9). Because the M0 and M1 monodromies only apply to orbits in the u = 0 and v = 0 planes, respectively, and not to an open set in the CB, the conditions (4.30) and (4.31), which were so restrictive for the Mp;q monodromy, do not apply. But because of the homotopy relations (5.8) and because all the eigenvalues of Mp;q have unit norm, it follows that all the eigenvalues of M0 and M1 , not just the one associated with the eigenspace in which lies, have unit norm. This allows the classi cation of their possible characteristic polynomials as products of cyclotomic polynomials. Using this, in appendix D we show that the characteristic polynomials of M0;1 can be one of nineteen possibilities, listed in (D.2). This determines the set of possible eigenvalues that these monodromies can have. Writing these eigenvalues in the form exp(2 iB=A) where A > B, A; B 2 N, and gcd(A; B) = 1 gives a nite list of possible (A; B) pairs (there are 24 possible pairs). Calling (A0; B0) and (A1; B1 ) the pairs corresponding to the eigenvalues of the M0 and M1 monodromies, respectively, we read o from (5.5) and (5.6) that 1 ps and 1 qs with C1; C0 2 N: (5.10) The unitarity bounds together with (5.2) imply the left sides of these equations are less than or equal to one, which in turn implies that C0 = C left with a nite set of 24 allowed scaling dimensions for 1 = 0 in (5.10). We are therefore u;v. The list of allowed values for u and v, separated into fractional and integers values, is reported in table 1, while in tables 2, 3, and 4 we collect the details of the monodromy assignments for the di erent values of u;v. It is important to stress that we have not imposed all the constraints implied by our topological arguments. For instance, we have only listed here the possible set of values either u or v can take. A simultaneous assignment of u and v from this list determines s which then also has to satisfy (5.9). Not all pairs do satisfy this condition: of the 300 possible distinct assignments of table 1, only 244 satisfy this constraint. u and v from the list of 24 possible values in v or Mp;q 1 1; 12 3; 32 1; 12 4; 43 2; 23 1; 13 6, 65 2; 25 1; 15 23 , 130 3, 35 4 2 1 2 2 1 3 1 2 4 1 3 6 u;v, and Mp;q U(1)R monodromies that are compatible with a given M0;1 monodromy. The last two columns give the values of the s, p, q parameters which can be realized by simultaneous solutions for both u and v. How to use this information to deduce the allowed pairs of ( u; v) values is explained in the text. We record in tables 2{4 the detailed monodromy data which characterizes each allowed pair ( u; v) of CB operator dimensions. By scanning the tables one determines the possible eigenvalue classes of the various U(1)R monodromies compatible with a given pair As an illustration of how to use the tables, suppose a CB geometry has Mp;q monodromy in eigenvalue class [14]. Now take a speci c instance of the v unknot monodromy, say M0 = [224] appearing in the fth row of table 3, which has this value of Mp;q. Then the possible values of v are 4, 2, or 4=3, with respective s values 1, 1=2, or 1=3, and p = 4. In the case where, say, v = 4=3, thus s = 1=3 and p = 4. Then the possible values of u have to have the same values of s, Mp;q, and a coprime q. These can be determined by scanning the tables. For instance, u = 4=3 with M1 = [42] appearing the bottom line of the fourth row of table 2 is not allowed because, though it has s = 1=3, it has q = 4 which u = 5=3 with M1 = [5] appearing in the is not coprime to p = 4. On the other hand, rst row of table 4 is allowed since q = 5. and leave this analysis for the future. Here we will not make any attempt to study the implications of these extra constraints, HJEP05(218)6 Mp;q [123] [124] [126] [223] [224] [226] v or 1; 13 , 12 1; 13 , 14 1; 15 , 16 12 , 13 , 14 1; 12 , 13 2; 23 , 25 1; 12 , 15 32 , 43 , 34 , 130 3, 32 , 34 , 35 13 , 14 , 18 , 19 1; 12 , 14 , 15 12 , 13 , 19 , 110 Finally, all known examples of rank-2 SCFTs in [30{37] have CB dimensions which are in the list derived here, though there are entries in our list which do not appear (yet) in any known example. An earlier attempt at a classi cation of rank-2 SCFT CBs by one of the authors and collaborators [25, 27] reports some examples with dimensions not appearing in table 1; however it turns out these con icting examples are not consistent CB geometries (the geometries in [25, 27] which are incorrect are those with fractional powers of the CB vevs appearing their SW curves; as a result their EM duality monodromies are not in Sp (4; Z)). 6 Summary and further directions In this paper we took a rst step towards generalizing the successful story of the classi cation of N = 2 SCFTs rank-1 theories [10{14] to arbitrary ranks. We illuminated how the special Kahler structure, and in particular the Sp (4; Z) monodromy action, is intricately tied with the globally de ned complex scaling action on the CB. This strongly constrains the scaling dimensions u and v of the CB operators. We obtained the striking result that only a nite list of rational scaling dimensions is allowed for u and v. The allowed values are listed in table 1. In particular the maximum allowed mass dimension of rank-2 CB parameters is = 12. Rank-2 U(1)R monodromy classes and scaling dimensions (III) v or 5; 52 , 53 , 54 8; 83 , 85 , 87 1; 12 , 13 , 14 2; 23 , 25 , 27 1; 13 , 15 , 17 2; 23 , 27 , 29 1; 13 , 17 , 19 2; 25 , 27 , 121 1, 15 , 17 , 111 56 , 265 , 365 , 565 34 , 145 , 241 , 343 23 , 130 , 134 , 232 6, 65 , 67 , 161 10 2 4, 45 , 47 , 141 3, 35 , 37 , 131 Using an extension of these arguments, a similar result can be obtained for arbitrary ranks, and will be reported on elsewhere [15]. Aside from this concrete result on the spectrum of CB scaling dimensions, we have developed a set of tools which we believe will be key to constructing all possible scale invariant rank-2 CB geometries. Our key results are: the algebraic description of the possible varieties, V, of CB singularities in (2.11); the computation of the possible topologies of the V C given in (2.18); the factorized description of the local EM duality monodromy M V linking components of V in terms of Sp(2; Z) matrices given in (4.9); the fact that the SK section is an eigenvector of U(1)R monodromies with unit-norm eigenvalue (4.17); the lagrangian eigenspace property (4.30) and fact that all eigenvalues have unit norm (4.31) of the generic (knotted) U(1)R monodromy; and the interrelations of the three di erent U(1)R monodromies recorded in tables 2{4. The next steps towards the goal of constructing all scale-invariant rank-2 CB geometries are likely: 1. Analyze the implications of the U(1)R monodromy conditions found in this paper. Here we only analyzed the compatibility of the eigenvalues for these matrices, but these conditions imply also that that the associated eigenspaces need to coincide. Presumably this is a non-trivial constraint which imposes further restriction on the allowed pairs of scaling dimensions ( u; v). 2. Investigate the constraints coming from the relationship between the factorized form (4.9) of monodromies linking single components of V and the U(1)R monodromies (which, in some sense, link all the components at once). These monodromies are related by the knot group (2.18) which re ects the presence of unknots and/or multiple component of the torus links. In the analysis of this paper, the allowed CB operator dimensions we found only depended on the integers (p; q) characterizing the U(1)R orbit but did not depend on the number or type of components in V . But the expression for the knot group re ects the existence of all the components of V and should be re ected in further constraints on the allowed monodromies and thus on the allowed scaling dimensions. Two longer-term generalizations of the current project are to extend our considerations to higher rank CBs, and to characterize the mass (or other relevant) deformations of the scale-invariant geometries considered here. For the higher-rank generalization, one potential technical hurdle is that, to the best of our knowledge, the full classi cation of non-hyperbolic conjugacy classes of Sp(2r; Z) for r 3 is not known. It is also currently unclear to us whether the full list of these conjugacy classes is actually needed | the partial results of this paper only required coarser and more easily obtained information about the EM duality group. While the approach to non-scale-invariant geometries by deformation of scale-invariant ones was fruitful in the rank-1 case [10{14], it is already apparent from the structures found in this paper that most tools that worked in rank-1 are not generalizable in a straightforward way to higher ranks. On the other hand, we are also not aware of any insurmountable obstacle for the implementation of such a program in rank-2. Acknowledgments It is a pleasure to thank Y. Lu for collaborating in the early stages of the project and for sharing with us useful insights. Furthermore we would like to thank J. Distler, B. Ergun, I. Garc a-Etxebarria, J. Halverson, B. Heidenreich, D. Kulkarni, M. Lotito, D. Regalado and F. Yan for helpful comments and discussions. PA is supported in part by DOE grant DE-SC0011784 and by Simons Foundation Fellowship 506770. CL is supported by NSF grant PHY-1620526. MM is supported by NSF grant PHY-1151392. A Review of rank-1 scale-invariant SK geometries Topology. By the assumption that the CB chiral ring is freely generated, in the rank-1 case it has a single generator, and therefore C ' C as a complex space. Choose a complex coordinate u on C such that a singularity is located at u = 0. The complex scale symmetry gives a holomorphic Cf action on C with u = 0 as a xed point. It is a conformal isometry of the metric on M coming from the combination of the actions of the U(1)R and dilatation generators on the CB. Thus this action is simply 2 Cf; (A.1) the special coordinates emanating from the origin, so that action of M across the cut. ( 1= u u) = (u). Thus where u is the mass scaling dimension of u. Unitarity bounds for 4d CFTs plus the assumption that the CB chiral ring is freely generated imply Since M ' Cnf0g, its fundamental group is generated by a path that circles once around u = 0 counterclockwise, and there is only a single non-trivial monodromy, M 2 Sp (2; Z), corresponding to analytic continuation along this path. We can thus describe (u) 2 C 2 by a holomorphic eld on the u-plane minus a cut is continuous on M except for a \jump" by the linear Geometry. Since the central charge has mass dimension 1, it transforms under (A.1) as (u) = u1= u ! ; for some 2 C : (A.2) gives ds2 0 < Im < 1. Here we have chosen an overall complex constant factor to set the normalization of the bottom component to 1. The SK integrability condition is trivially satis ed. The metric With (A.2) this ( Im )ju(1= u) 1j2dudu. Positivity and well-de nedness of the metric imply Duality. For rank-1, any Dirac pairing can be written up to a GL(2; Z) change of basis, as hp; qi = pi(J )ij qj with J M T J M = J becomes simply ad bc = 1. Thus Sp(2; Z) = SL(2; Z). symplectic form J = ( 01 10 ). The EM duality group Sp (2; Z) are those M 2 GL(2; Z) such that M T J M = J . Therefore Sp (2; Z) is actually independent of the choice of : for all , Sp (2; Z) = Sp(2; Z) = fM 2 GL(2; Z)jM T J M = J g. Write M = ac db , then As we follow a path u0 ! e2 iu0, ! M , and therefore e2 i= u ( 1 ) = M ( 1 ) for some M 2 SL(2; Z). Therefore, M must have an eigenvalue, = exp(2 i= u), with j j = 1. J for some positive integer , where J is the usual The characteristic equation of M is 2 have unit norm if j Tr M j conjugacy classes of SL(2; Z) satisfying this trace condition: ( Tr M ) + 1 = 0, since det M = 1, so both roots 2; otherwise neither does. It is easy algebra to list all the Tr M = 2 Tr M = 1 Tr M = 0 Tr M = Tr M = 1 2 M M M M M T n ST or (ST ) 1 S or S 1 ST or ( ST ) 1 T n = +1 = e i =3 = e i =2 = e 2i =3 = 1 (A.3) Here T := ( 10 11 ) and S := 01 01 . In the rst and last line n is an integer. When n 6= 0 these are called parabolic conjugacy classes. All the other cases are called elliptic conjugacy classes. Since = e2 i= u and since u 1 we immediately read o the list of allowed values of u, M , and , shown in table 5. The value of is determined by solving for the eigenvector 3 1 2 S I ST S 1 I T n T n ( ST ) 1 e2i =3 3=2 4=3 6=5 (ST ) 1 ei =3 i any e2i =3 i ei =3 any i1 i1 of M normalized as in (A.2). The rst seven entries are scale-invariant singular geometries ( at cones, in this case), and correspond to elliptic conjugacy classes. The eighth entry corresponds to an identity monodromy matrix, and therefore to no singularity (a regular point). The two entries below the dotted line correspond to parabolic conjugacy classes of SL(2; Z). For the parabolic classes there is no scale-invariant solution for since = i1. So we should look for solutions by including the leading corrections to scaling. So, e.g., expand (u) = u+ 0u (u= ) 0 + 1u ln 1 (u= ), where the j are 2-component vectors of exponents correlated with the entries of the j 2 C2, and parabolic case we look for a solution to (e2 iu) = T n (u). We nd 0 = (0 0), 1 = (1 0) is an arbitrary mass scale. In the u = 1 and 1 = ( 2ni 0). Thus for the T n monodromies we nd = u 1 + 2ni ln u ! 1 : (A.4) For this solution the metric is ds2 = n 4 ln uu2 + 2 dudu. Note that as juj ! 0, 1, so the metric is positive-de nite in the vicinity of u = 0 only for n > 0. This metric has a mild non-analyticity at u = 0 with 2 opening angle there and positive curvature away from u = 0. Thus the T n monodromies for n 2 Z+ give sensible geometries. They correspond physically to IR-free N = 2 QED, for example with n charge-1 massless hypermultiplets. is the Landau pole. A similar story goes for the T n monodromies. They give positive de nite metrics for n 2 Z+, corresponding to IRFTs such as SU(2) with n + 4 massless fundamental hypermultiplets. B Analytic form of the SK section near V n f0g Here we record the analytic form of the SK section, , in the vicinity of any regular point P of V, the variety of metric singularities in a rank-r CB. Since P is a regular point of V, we can pick local complex coordinates (u?; uk) on the CB vanishing at P such that u? = 0 describes V locally and uk are r 1 coordinates such that @uk are tangent to V at P . Then depends analytically on uk, at least in a neighborhood of P . For instance, in the rank-2 SCFT case, we can describe u? and uk explicitly and also nd the explicit analytic dependence of on uk. This is because in this case the ukdependence is determined by the complex scale symmetry; it will not be true for ranks greater than 2. The SK section transforms as in (4.14) under the complex scaling action. Linearizing around = 1, (4.14) becomes a di erential equation with general solution with b complex analytic in y except at the CB singularities, since complex analytic on the CB minus its singularities. Consider the vicinity of a regular point of V, that is of a point P 2 V n f0g. Say P is a point on a V! component of V with ! 2 P1. Let u be the coordinates of P , so y = u v u= v = !1=p (for some choice of the pth root). Expanding around P we have from (B.1) (with a slight abuse of notation) = (v + uk)1= v b(u?) ; uk := v v ; u? := y y : (B.2) Returning now to the general-rank case, we will suppress the uninteresting analytic dependence of on the uk coordinates, and focus on the interesting non-analyticities in its dependence on u?. For ease of typing, we will from now on write u for u?. By assumption, there is a CB singularity at u = 0 around which su ers an EM duality monodromy, M 2 Sp (2r; Z). Thus upon continuing for 0 2 , encircling u = 0, along a closed path u( ) = u0ei = v1= v (y) ; b y := uv u= v ; (B.1) (e2 iu) = M (u) : By writing = Pj fj (u)vj where vj is a (generalized) eigenbasis of the monodromy matrix M , it is simple to determine from (B.3) the analytic behavior of fj (u) around u = 0. Explicitly, a complex change of basis brings M to Jordan normal form, (B.3) (B.5) (B.6) M M j Mj C4 = M Cnj where j Mj = BBB CC 2 GL(nj ; C) ; (B.4) where the index j labels the di erent Jordan blocks each with eigenvalue j . This basis fv1(j); : : : ; vn(jj)g, unique up to an overall normalization, of each Cnj subspace thus satis es M v(j) = j vk(j) + v(j) k k 1 v0(j) := 0 : Writing (B.3) in this basis then determines the analytic form of (u) to be 0 B B B B j 1 C j 1 C C C C A (u) = X u j gj (u) X vk(j)( j ) k 1 Pnj k j ln u 2 i nj k=1 where gj(u) 2 C is analytic in u in a neighborhood of u = 0, j is de ned in terms of j by expf2 i jg = j with 0 Re j < 1 ; P`(x + 1) = P`(x) + P` 1(x) : The gj are taken to be analytic around u = 0, and in particular to not have any poles, because (u) cannot diverge as u ! 0 (this was argued in section 4.2). If the jth Jordan block has both Re j = 0 and non-constant polynomial dependence on ln u, then niteness of (u) as u ! 0 implies the stronger condition that gj(u) must vanish as u ! 0. The recursion relation (B.8) does not determine the constant term of each polynomial, and these constants can be chosen independently for each Jordan block. If we de ne c` := `! P`(0), then the rst six polynomials are (B.7) (B.8) 4! P4 = c0x4 + (4c1 5! P5 = c0x5 + (5c1 c0)x + c2 3c0)x2 + (3c2 6c0)x3 + (6c2 3c1 + 2c0)x + c3: 12c1 + 11c0)x2 + (4c3 6c2 + 8c1 6c0)x + c4 10c0)x4 + (10c2 30c1 + 35c0)x3 + (10c3 30c2 + 55c1 50c0)x2 + (5c4 10c3 + 20c2 30c1 + 24c0)x + c5 : The main properties to take away from (B.6) are that: the eigenvalue j of each Jordan block determines the leading (fractional) powers, u j , appearing in ; a Jordan block of size nj will contribute logarithms in u up to order lnnj 1; unless a Jordan block has eigenvalue j = 1, its contribution to will vanish at u = 0; and, if j = 1 for a Jordan block with nj > 1, then its contribution to (0) is non-zero only if only its 1-eigenvector contributes (i.e., all the cj coe cients in the logarithmic polynomials vanish except for cnj 1). There are further interesting constraints on (B.6) that come from incorporating the simple factorized form of the Sp (2r; Z) linking monodromy found in (4.9), with the properties of SL(2; Z) conjugacy classes described in appendix A, the conditions (4.3) for the positivity of the Kahler metric near V, and the conditions (4.6) for the nondegeneracy of the metric components parallel to V at P . C Sp(4, R) conjugacy classes Here we summarize following [38] the conjugacy classes of Sp(4; R) and some of their properties. We then use this knowledge to deduce in which of those conjugacy classes an Mp;q EM duality monodromy associated to an U(1)R orbit (as described in section 4.4) can appear. Even though Sp(4; R) is not the EM duality group, Sp (4; Z), we discuss it here because the description of its conjugacy classes is substantially easier than that of Sp (4; Z). Since Sp (4; Z) is a subgroup of Sp(4; R) (as explained in appendix D), the conjugacy classes of Sp (4; Z) are subsets of the conjugacy classes of Sp(4; R), and that turns out to provide enough information for our purposes. Generalized eigenvectors and Jordan blocks. First, recall the de nition of an `generalized eigenvector, or `-eigenvector for short. An `-eigenvector, v`, with eigenvalue of a square matrix M is a non-zero vector for which (M I)`v` = 0 for some positive integer `, but not for ` 1. If ` = 1, then it is a regular eigenvector. If ` > 1, then v` m := (M I)mv` for m < ` is an (` m)-generalized eigenvector. Thus if there is an `-generalized eigenvector, the associated eigenvalue must have multiplicity at least equal to `. A series of such `-generalized eigenvectors with eigenvalue and 1 n correspond to an n n Jordan block when M is put in Jordan normal form. The matrix for an n n Jordan block is shown in (B.4). spaces of all Jordan blocks with eigenvalue . We de ne the generalized eigenspace with eigenvalue to be the direct sum of the Properties of Sp(2r, R). The following properties are true for Sp(2r; R) for all r, and so for the case of interest here, r = 2. They are explained in standard texts on symplectic degenerate skew-symmetric matrix. We can choose a basis of R2r so that geometry. Sp(2r; R) is the group of M 2 GL(2r; R) such that M T J M = J for J a non(C.1) (C.2) J = where I is the r A; B; C; D r r real matrices, then r identity matrix. It follows that if M = ( CA DB ) 2 Sp(2r; R) with AT C = CT A ; BT D = DT B ; AT D CT B = I : Some basic but non-trivial properties of M 2 Sp(2r; R) are that M is similar to M 1 , det(M ) = 1, the eigenvalues of M occur in reciprocal pairs, and complex eigenvalues f 1; 2 ; 2 1; 1 1g,with each i either complex of norm one or non-zero real. have unit norm. Thus the set of eigenvalues of any M 2 Sp(4; R) are always of the form Another basic property involves the symplectic-orthogonality of generalized eigenvectors. De ne the symplectic pairing of two vectors in C2r by hu; vi = uT J v. If u and v are 1-eigenvectors of M 2 Sp(2r; R) with eigenvalues hu; vi = hM u; M vi = hu; vi. It follows that hu; vi = 0 unless and , respectively, then = 1, i.e., eigenvectors of non-reciprocal eigenvalues are symplectic-orthogonal. This property generalizes to the statement that whole generalized eigenspaces of non-reciprocal eigenvalues are symplecticorthogonal. Recall some de nitions from symplectic geometry. If W is a linear subspace of C2r, the symplectic complement of W is the subspace W ? := fv 2 C2rjhv; wi = 0 for all w 2 W g. It satis es (W ?)? = W and dim W + dim W ? = 2r. Then W is symplectic if W ? \ W = f0g. This is true if and only if h ; i restricts to a nondegenerate form on W . Thus a symplectic subspace is a symplectic vector space in its own right. W is isotropic if W W ?. This is true if and only if h ; i restricts to 0 on W , i.e., if and only if all vectors in W are symplectic orthogonal. Finally, W is lagrangian if W = W ?. A lagrangian subspace is an isotropic one whose dimension is r. Every isotropic subspace can be extended to a lagrangian one. Then the symplectic orthogonality of generalized eigenspaces implies that if W the generalized eigenspace associated with eigenvalue then W is isotropic if W W1= is symplectic, and W 1 are each symplectic. = is 1, Sp(2, R) conjugacy classes. Next, recall the structure of Sp(2; R) ' SL(2; R) conjugacy classes. An M 2 Sp(2; R) has eigenvalues f ; 1 g with 2 C and either j j = 1 or 2 R. The matrices can be divided into three sets: \hyperbolic" if j j 6= 1 in which case it is similar to a diagonal matrix, \parabolic" if 1 and M is similar to a 2 2 Jordan block form (i.e., has a 1-eigenvector and a 2-eigenvector), and \elliptic" if j j = 1 and is similar over the complex numbers to a diagonal matrix (i.e., has two 1-eigenvectors). Note that M = I are special cases of the elliptic class. The eigenvalues together with whether in the case of = 1 there is a generalized eigenvector or not gives a classi cation of all the Sp(2; R) conjugacy classes. The Sp(2; Z) ' SL(2; Z) Sp(2; R) conjugacy classes are a re nement of these conjugacy classes. In particular, only eigenvalues of the form ei k=3 or ei k=2 are allowed in the elliptic cases and each value (together with its inverse) corresponds to at most two separate conjugacy classes; the two parabolic classes corresponding to 1 in the real case each split into an in nite series of conjugacy classes in the integer case; and only rational eigenvalues are allowed in the hyperbolic cases, and there is a complicated pattern of how many conjugacy classes correspond to a given eigenvalue. Sp(4, R) conjugacy classes. The Sp(4; R) conjugacy classes have a similar, though inevitably more complicated, description in terms of their eigenvalues and whether or not there are `-generalized eigenvectors than in the Sp(2; R) case. We will adapt the hyperbolic/elliptic nomenclature of the Sp(2; R) case to this case by classifying M as: \hyperbolic-hyperbolic" (HH) if j ij 6= 1 for both i = 1 and 2; \hyperbolic-elliptic" (HE) if j 1j 6= 1 and j 2j = 1; and \elliptic-elliptic" (EE) if j ij = 1 for both i = 1 and 2. We further subdivide these classes by the size of their Jordan blocks when put in Jordan normal form by a complex change of basis. We will list these sizes as subscripts when they are larger than one: these are the analogs of the parabolic-type elements of Sp(2; R). One then easily nds that only the following cases are allowed in Sp(4; R): (HH) (HE) (EE) (HH)2;2 (HE)2 (EE)2 (EE)2;2 (EE)4 : (C.3) It takes considerably more work [38] to describe the di erent Sp(4; R) conjugacy classes realizing these cases. Given the 2 2 block structure imposed by the choice (C.1) of J , there are three useful ways of combining 2 2 matrices into 4 4 matrices: the usual direct sum, , what we HJEP05(218)6 will call the upper direct sums, with 1, and the interlaced direct sum, . If A = a b and B = ( pr qs ), then these sums are de ned by c d A 0a b B := BBBc d 1 C C ; p qC A r s A 0a b B := BBBc d 1 C C ; p qC A r s A 0a B := BBBc C ; C A s where 1 and the empty entries are all 0. The interlaced sum, , is the one that respects the symplectic structure: A B is in Sp(4; R) if and only if A and B are in Sp(2; R). This is because J = J2 J2 where J2 is the 2 2 symplectic structure. It follows easily from (C.2) that A B is in Sp(4; R) if and only if B = A T , and A B is in Sp(4; R) if and only if B = A T and the upper left entry of A vanishes. Next, de ne the following six types of 2 2 matrices: Ha = Hea = P P E = Ee = a 0 ! 0 cos sin 0 sin cos 1 a 2 R and a 6= 0; 1; a 2 R and a 6= 0; 1; 2 f 1g; 0 2 f 1g; 2 f 1g; 0 < : Note that Ha, P , and E are representatives of the hyperbolic, parabolic, and elliptic Sp(2; R) conjugacy classes, and that Pe in Sp(2; R). Note also that E0 = E = I. and Ee are conjugate to P and E , respectively, Then representatives of all the Sp(4; R) conjugacy classes are [38] M 2 (HH)2;2 M 2 (EE)2;2 M 2 (EE)4 Note that A B B A, and similarly for M M M M M M M M Ha Hea Ha Ha E E P P Hb; Hea T ; E ; P ; E P ; P P T or E e E T ; or P . e P T or E e e E T ; (C.5) (C.6) Sp(4,R) conjugacy classes with lagrangian 1-eigenspaces. In sections 4.3 and 4.4 we showed that the EM duality \knot" monodromies, Mp;q, must be an element of Sp(4; R) with an eigenvalue of unit norm. From our discussion above, this means the monodromy cannot be of any of the (HH)-types shown in the rst two lines of (C.6). In addition, we showed that the unit-norm eigenvalue must have a 1-eigenspace of dimension 2 or greater. It is not hard to read o from (C.6) that the only possible conjugacy classes with this property are M M M M Ha E0 E0 P E0 E P P or or or or P Ha E E E ; E P ; P T : Furthermore, we also showed in section 4.4 that if the 1-eigenspace is 2-dimensional it must be lagrangian, i.e. for any basis f 1 ; 2g we require h 1; 2i = 0. We check this for the above list. First, for the conjugacy classes with representatives given by interlaced sums, , if both eigenvectors are from the same summand, then they will have h 1; 2i 6= 0. So only interlaced sums for which eigenvectors for the same eigenvalue come from both sides of the interlaced sum can give lagrangian eigenspaces. For the remaining direct sum cases, it is easy to check that for M = Ee E T , 1T = (0 0 i 1) and 2T = (e i 1 0 0) are an eigenbasis of the ei eigenspace satisfying h 1; 2i = 0; and for M = Pe Pe T , a basis of the eigenspace is 1T = (0 0 1) and 2T = ( 1 0 0) which also satis es h 1; 2i = 0. Therefore the list of Sp(4; R) conjugacy classes of \knot" monodromies that can appear in scale-invariant singularities are: M 2 (EE)2 M 2 (EE)2;2 M M M E E0 P E P1 P or or or E E P e E T P 1; P T : or E E or E E T HJEP05(218)6 (C.7) (C.8) Note that all these conjugacy classes are of \elliptic-elliptic" type, so, in particular, only have eigenvalues on the unit circle in the complex plane. D Sp (4, Z) characteristic polynomials matrix Properties of Sp (4, Z). The EM duality group, Sp (2r; Z), is the subgroup of GL(2r; Z) preserving the Dirac pairing on the charge lattice. If we write hp; qi := pi(J )ij qj for p; q 2 Z2r, then Sp (2r; Z) is de ned to be the set of M 2 GL(2r; Z) such that = J . By a GL(2r; Z) change of basis any non-degenerate, integral, skewsymmetric quadratic form can be put in the form J where is a diagonal r r = diagf 1; 2; : : : ; rg with the i positive integers such that ij i+1. The set f ig is a GL(2r; Z)-invariant characterization of the form. Sometimes J is called a polarization. If = Ir then J is a principal polarization, and Sp (2r; Z) = Sp(2r; Z), the \usual" EM duality group. If any of the ratios i+1= i are not perfect squares, then Sp (2r; Z) is not 0 0 isomorphic to Sp(2r; Z) as a group. However, all pairings can be brought to principal form within GL(2r; R), so all Sp (2r; Z) are isomorphic to a subgroup of Sp(2r; R). Since the latter fact together with the fact that Sp (2r; Z) GL(2r; Z) are the only facts we will use about Sp (2r; Z) in this paper, the distinction between Sp (2r; Z) and the more familiar Sp(2r; Z) EM duality group will not play any role. Possible eigenvalues of elliptic-elliptic elements of Sp (4, Z). Following an argument from [29] we can easily determine the possible eigenvalues of type (EE)n elements M 2 Sp (4; Z). These have eigenvalues of unit norm. Their characteristic polynomials, PM (x), have integer coe cients since M 2 GL(4; Z) is a matrix with integer entries. Polynomials irreducible over the integers whose roots have unit norm and have integer coe cients are the cyclotomic polynomials n(x) = Y gcd(m;n)=1 x e2 im=n ; and satisfy degree( n) = '(n) = n Qprimes pjn(1 and counts the number of primitive nth roots of unity. Since the degree of PM (x) is 4, it can can only be a product of n of degrees less than 4. If n has a prime divisor greater than 5 or if it has more than two distinct prime divisors then '(n) > 4. So the only possible n are in the list n 2 f1; 2; 3; 4; 5; 6; 8; 10; 12g, which have degrees dn 2 f1; 1; 2; 2; 4; 2; 4; 4; 4g, respectively. From this and the fact that Sp(2r; R) eigenvalues always appear in reciprocal pairs, we can read o the 19 possible characteristic polynomials of type (EE)n elements: p 1) which is Euler's totient function, (D.1) (D.2) (D.3) where we have introduced the notation [224]; [8]; [226]; [10]; [12]; Qiniri := Q ( ni )ri : i We will also use the symbol [X] to denote the set of elements M characteristic polynomials are [X]. Note that if M 2 [X], then any M 0 conjugate to M is also in [X], so [X] is a union of conjugacy classes. The eigenvalues of any M 2 [X] are simply read o as the primitive ni-th roots of unity each with multiplicity ri. Note that dimensions of the generalized eigenspaces are f4g for the entries in the rst line of (D.2), f2; 2g for those in the second line, f2; 1; 1g for those in the third, and 2 Sp (4; Z) whose f1; 1; 1; 1g for the last line. Possible conjugacy classes of Sp (4, Z) with lagrangian 1-eigenspaces. By comparing with (C.8) we determine which Sp(4; R) conjugacy classes with a lagrangian 1eigenspace can occur in Sp (4; Z) and what are their characteristic polynomials. M 2 (EE) : M 2 (EE)2 : M 2 (EE)2;2 : E2 =3 Ee2 =3 E0 E0 P1 E0 2 [14] E2 =3 2 [32] E T e2 =3 2 [32] P1 2 [14] P1 2 [14] P 2 [14] Pe1 E E =2 Ee =2 E P 1 E 2 [24] E =2 2 [42] Ee =T2 2 [42] P 1 2 [24] P 1 2 [24] T Pe 1 2 [24]: E =3 Ee =3 E =3 2 [62] Ee =T3 2 [62]: So only elements in [14], [24], [32], [42], and [62] have lagrangian 1-eigenspaces. Possible orders of elliptic-elliptic elements of Sp (4, Z). We can easily determine the characteristic polynomials of arbitrary powers of any M 2 [X ]: [1222] [3 4]12 [123] [4 6]6 [1222] [4 6]12 (D.4) [14] [14] [12]6 [24] [12]12 [14] : Here we are using a notation where [X ]a [Y ] means that M a [Y ] does not imply that if Mi 2 [X ] then Qia=1 Mi 2 [Y ]: this is only necessarily 2 [Y ] if M 2 [X ]. Note that true if all Mi are equal. 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Philip C. Argyres, Cody Long, Mario Martone. The singularity structure of scale-invariant rank-2 Coulomb branches, Journal of High Energy Physics, 2018, 86, DOI: 10.1007/JHEP05(2018)086