Singularity Structure of N=2 Supersymmetric Yang-Mills Theories: A Review
OOctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4
International Journal of Modern Physics Ac (cid:13)
World Scientific Publishing Company
SINGULARITY STRUCTURE OF N = 2 SUPERSYMMETRIC YANG-MILLS THEORIES: A REVIEW ∗ JIHYE SOFIA SEO
Ernest Rutherford Physics Department, McGill University,3600 rue University, Montreal, QC H3A 2T8, CanadaandCentre de recherches math´ematiques, Universit´e de Montr´ealC.P. 6128, succ. centre-ville, Montr´eal, Qu´ebec, H3C 3J7, [email protected]
Received Day July 2013Revised Day Month 2013In this review, we consider the case where electrons, magnetic monopoles, and dyons be-come massless. Here we consider the N = 2 supersymmetric Yang-Mills (SYM) theorieswith classical gauge groups with a rank r , SU ( r + 1), SO (2 r ), Sp (2 r ), and SO (2 r + 1).which are studied by Riemann surfaces called Seiberg-Witten curves. We discuss physicalsingularity associated with massless particles, which can be studied by geometric singu-larity of vanishing 1-cycles in Riemann surfaces in hyperelliptic form. We pay particularattention to the cases where mutually non-local states become massless (Argyres-Douglastheories), which corresponds to Riemann surfaces degenerating into cusps. We discussnon-trivial topology on the moduli space of the theory, which is reflected as monodromyassociated to vanishing 1-cycles. We observe how dyon charges get changed as we movearound and through singularity in moduli space. Keywords : Seiberg-Witten curve; pure supersymmetric Yang-Mills theories; (maximal)Argyres-Douglas singularity; dyon charges of vanishing 1-cycles.PACS numbers:11.15.Tk, 11.30.Pb ∗ Partially based on a seminar given at Imperial College in January 2012 and a colloquium atGwangju Institute of Science and Technology in April 2013.1 a r X i v : . [ h e p - t h ] J u l ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Contents
1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. N = 4 SYM: gluon scattering amplitudes at hadron colliders . . . . 31.2. N = 2 SYM: massless magnetic monopole . . . . . . . . . . . . . . . 51.3. Plan of the review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72. Essentials of N = 2 SYM and Seiberg-Witten Geometry . . . . . . . . . 72.1. Review of N = 2 Seiberg-Witten theory and geometry . . . . . . . 82.2. Review of hyperelliptic curves y = f ( x ) . . . . . . . . . . . . . . . . 123. First Look at Seiberg-Witten Curves for SU ( r + 1) and Sp (2 r ) . . . . . 153.1. Seiberg-Witten geometry for pure N = 2 SU ( r + 1) theories . . . . 153.2. Seiberg-Witten curve for pure Sp (2 r ) theories and root structure . . 164. Electric and Magnetic Charges of Massless Particles . . . . . . . . . . . . 174.1. Rank 1 examples: how to read off monodromies of the Seiberg-Wittencurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.1. Review of SU (2) monodromy . . . . . . . . . . . . . . . . . . 184.1.2. Sp (2) monodromy . . . . . . . . . . . . . . . . . . . . . . . . 204.1.3. Moving on to the higher rank case . . . . . . . . . . . . . . . 224.2. N = 2 SU ( r + 1) theory . . . . . . . . . . . . . . . . . . . . . . . . 224.3. Monodromies of pure N = 2 Sp (2 r ) theories . . . . . . . . . . . . . 265. Argyres-Douglas Loci: Massless Electron & Massless Magnetic Monopole 295.1. Singularity structure of Sp (4) = C : detailed look on BPS spectra . 295.2. Maximal Argyres-Douglas theories and dual Coxeter number . . . . 376. D´ej`a Vu: Singularity Tools: Exterior Derivative & Double Discriminant . 396.1. Exterior derivative detects coexistence of multiple massless BPSdyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2. Factorization of double discriminant, and order of vanishing . . . . 407. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1. Motivation
Understanding of physics often advances through consideration of extreme and sin-gular situations. We address lots of questions in extreme limits, with very small orvery large values of density, temperature, velocity, mass etc. These are not only fortheoretical curiosity; instead often it turns out to be a golden mine for new dis-covery and applications. Neutron stars and big bang are at extremely high density.Superconductivity and superfluidity is studied at extremely low temperature.Some extremes may simplify the situation enough to provide ideal setting tofocus on the essence of the system. For example, ideal gas law assumes no interactionamong particles. Many freshmen-level classical mechanics problems assume thatfriction vanishes and spring is massless.Some extremes pose us such a big challenge that it takes a paradigm shift toovercome the huddle. For example, consider the thought experiment in classicalmechanics of the escape velocity of the satellite. If the gravity is so strong thenctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4
Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories the escape velocity approaches the speed of light. This was the first encounter(in thought) with black holes. Very careful consideration and study of the highestpossible velocity, the speed of light, gave birth to special relativity, and we learnedto think of space and time together as one combined object, spacetime.Even in more contemporary setting, singularities in physics deserves seriousattention. It may serve as a warning signal: for example, it may occur when wehave integrated out massive fields which are in fact massless. UV divergences infield theory urge us to look for a better theory at higher energy. Understandingsingularity is a cornerstone to solving field theory problem, just as imagining anextreme situation gives us an often correct intuition for classical mechanics problem.We often started to think about collision between particles, where their masses areequal or very different.Mathematics, especially geometry has been a faithful and fruitful language indescribing physical system. Gravity - Einstein’s General Relativity - is best de-scribed in the language of differential geometry. What about other forces in nature?Electromagnetic, weak, and strong forces are formulated in terms of gauge theorieswith gauge groups U (1), SU (2) and SU (3) respectively. As reflected in Ref. 1, thesegauge theories are well-described in another field of mathematics, so-called fiberbundle theory.In studying the singularity of physical system, geometry is particularly useful.Physical singularity is reflected in geometry as mathematical singularity. There ex-ists a famous dictionary between geometry and physics for gravitational singularity.A black hole in physics will appear as a geometric singularity, that is a puncturein a spacetime fabric. Later we will discuss physical singularity associated withboth electrons and magnetic monopoles having zero mass. So far there is no knownLagrangian for this system. Thanks to the close relationship with geometry, how-ever, this can be studied in terms of geometry. Recently Ref. 3 has revealed strangebehavior of moduli space near the singularity by careful observation of geometricsingularity of Seiberg-Witten curve associated to the physical system.In this article, we will review physical and geometric singularity of Yang-Millstheories, which have close relationship with electromagnetic and nuclear forces, butwith multiple supersymmetries. The amount of supersymmetry is denoted by N ,the number of supercharges. Roughly it means that 2 N particles form a set (asupermultiplet, as we will discuss in more detail later.) in which all the physicalqualities are identical to one another except for the spin. We will motivate thesupersymmetric Yang-Mills theories by looking at N = 4 and N = 2 theories in thisintroductory section, and the rest of the article will focus on N = 2 supersymmetricYang-Mills theories. N = 4 SYM: gluon scattering amplitudes at hadron colliders
One may ask “Fine. I buy that gauge theories are important because they describenuclear and electroweak forces. But why should anyone care about supersymmet- ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO ric gauge theories, when LHC did not observe any superpartners yet?”. First werecall that supersymmetry provides an attractive and graceful exit out of manyserious paradoxical situations. It plays a crucial role in resolving the issues of hi-erarchy problem, unification of coupling constants of gauge theories, mismatch ofcosmological constant, etc. In other words, supersymmetry has been a best friendto theorists, who would like to make theoretical sense and feel aesthetic harmonyout of observed experimental facts.However, supersymmetry also makes contributions for experiments. One of themost important inputs of supersymmetry recently is computation of scattering am-plitudes of gluons. Computation is much easier when theory has supersymmetry.At the tree level, gluon scattering amplitudes agree between supersymmetric andnon-supersymmetric theories. Therefore, easier computation in supersymmetric the-ories can provide useful results for non-supersymmetric and more realistic theories,at least to the leading order. Results on scattering amplitudes in maximally su-persymmetric gauge theories ( N = 4), obtained by many string theorists, are im-plemented into the tools such as BlackHat, used by experimentalists at hadroncolliders. Though massless, gluons are responsible for carrying lots of energy awayfrom the collision process, and it is a big plus to understand their scattering ampli-tudes. Supersymmetric Yang-Mills theories are even more relevant in this LHC era,with or without supersymmetry detection.In past several years there has been a dramatic progress (almost at an exponen-tial rate) in computation of gluon scattering amplitudes in N = 4 (maximal) super-symmetric gauge theories. N = 4 supersymmetric Yang-Mills theories are specialin that the 3-point function of gluons can be written down purely out of symme-try argument. Having so much supersymmetry, the theory enjoys superconformalsymmetry and its dual superconformal symmetry. Conformal symmetry means onecan forget about lengths. One does not even need to know the Lagrangian. Onecan write down S-matrix purely from the symmetry and consistency consideration,with no need for Feynman diagrams. As nicely reviewed in a recent paper Ref. 5,to build n -point function the 3-point function are put together like lego blocks byamalgamation and projection operators. While postponing manifestation of uni-tarity and locality, scattering amplitudes manifest dual conformal supersymmetryand Yangian symmetry, which would remain opaque in evaluation of each Feynmandiagram. Geometry is a bias-free place to look for symmetries in physics.Gluons being massless, their 4-dimensional null (light-like) momenta enjoy am-phibian lifestyle: both Lorentzian and twistor spaces provide a natural habitat todescribe kinematics. In lieu of Feynman diagrams, scattering amplitude can be or-ganized by much simpler Hodges diagrams in twistor space, as nicely reviewedin Ref. 10. Using a higher dimensional version of Cauchy’s theorem, this can bewritten as partial sum of residues at isolated singularities. Scattering amplitudesin maximally supersymmetric gauge theories are given as a contour integral over actober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories Grassmannian a . Supersymmetric Yang-Mills theories thrive in a close relationshipwith geometry.Another motivation to consider N = 4 SYM is an inseparable bond betweensupersymmetric gauge theory (SYM) and supergravity (SUGRA). Amplitudes forSYM and SUGRA have tight kinship: supergravity amplitudes can be written interms of SYM amplitudes, roughly speaking. One of the hot issues is the questionof UV finiteness of supergravity: is supergravity a valid theory by itself, or do wemust recruit string theory (or other candidates of quantum gravity) to make thesupergravity consistent at arbitrarily high energy? The answer has been elusive, butsupersymmetric gauge theory might be able to help.More excitement in N = 4 supersymmetric Yang-Mills theories can be found inRef. 5 and its referecences. Now we will switch to less supersymmetric ones, N = 2supersymmetric Yang-Mills theories for the rest of the review. Motivation for N = 2 SYM: massless magnetic monopole
Fruitful symbiosis between physics and geometry, which we observed for generalrelativity and N = 4 SYM, holds true for N = 2 SYM as well. The guest of honorfor N = 2 supersymmetric Yang-Mills theory is a Riemann surface. Its most famouspersona is as a Seiberg-Witten curve:
11, 12 in a teamwork with Seiberg-Witten one-form, it encodes lots of information about the physical theories, as we will delvedeeper in the rest of the review. This Riemann surface also serves as a spectralcurve of integrable system. Seen from the 4-dimensional field theory perspective, on which this review willmainly focus, this curve does not live inside the spacetime. One may regard itas an auxiliary object or a bookkeeping device, which happens to encode lots ofuseful information. This review will focus more on what to learn out of a given SWgeometry, rather than how to obtain such geometry to begin with. So far the bestway to understand the origin of SW geometry seems to be string theory.In string theory settings (although we won’t discuss them in depth here), theSeiberg-Witten geometry is closely related to extended objects in string theoriesand M-theory. Ref. 14 interprets that the 4-dimensional field theory comes fromwrapping M5-brane on the Seiberg-Witten curve. In M-theory, M5 brane is a soli-tonic object spanning 5 spatial and 1 temporal directions, carrying a conservedcharge. Just as we consider world-line of a point particle traveling in time, we canconsider 6-dimensional theory on the world-volume spanned by time evolution ofM5-brane. However if we let M5-brane to wrap a 2-dimensional Riemann surfaceand further assume that the Riemann surface is small compared to other directionsin the spacetime, then we will have only 4 remaining directions effectively. This typeof 4-dimensional theories are discussed in Refs. 14, 15. There are also interpreta-tions of Seiberg-Witten curve in terms of non-critical and anti-self dual strings in a Grassmannian is a manifold which is a generalization of a projective space. A simple example ofprojective spaces is a sphere. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO type IIA, IIB, and heterotic string theories, again appearing as wrapping extendedsolitonic objects on appropriate cycles, as reviewed in Ref. 16.Thanks to Seiberg-Witten theory,
11, 12 many N = 2 supersymmetric Yang-Millstheories can be equivalently written as a Riemann surface (written down as a for-mat of hyperelliptic curve) called Seiberg-Witten curve with a one-form. Studyingsingularity on geometry-side provides us a powerful microscope probing singularityof physical theory. Degeneration and monodromy of hyperelliptic curves translatesinto massless fields and their dyon charges in Seiberg-Witten theories b . A highersingularity coming from collision of milder singularity gives us an exotic theory withmassless electron and massless monopole, so-called Argyres-Douglas theory. It de-fies Lagrangian description: when Lagrangian mechanics turns its back on us, wehave all the more reason to seek the friendship with geometry.Let us pause for a moment and remind ourselves why we need to study mag-netic monopoles, especially why the light ones. Despite many experimental claimsand findings, magnetic monopole (massive or massless) is something we have notobserved in a concrete manner yet. In the set of Maxwell’s equations, magneticmonopoles naturally arise if one tries to manifest the hypothetical electro-magneticsymmetry, and introduces magnetic sources just like electric sources. There are twomain reasons why the magnetic monopole must exist beyond a theorist’s fancifulimagination, as pedagogically reviewed in Ref. 17 by D. Milstead and E. J. Wein-berg. Grand Unified Theory (GUT) (of electromagnetism and nuclear forces) pre-dicts existence of magnetic monopole as shown in Refs. 18, 19. The mere existenceof magnetic monopole explains and necessitates quantization of electric charge. For the lack of experimental evidence of monopole, we tend to blame its highmass. In the present Universe, we expect the magnetic monopole to be very heavy- its mass energy is near that of a bacterium or near kinetic energy of a runninghippo, which is a lot larger compared to other elementary particles. Big bang alsoprovides an excuse for its absence in experimental data, arguing that the finitenumber density of magnetic monopole diluted out as the universe evolves. Howeverif we trace back the history of universe, then spontaneously broken symmetries(such as GUT and supersymmetry) are restored and magnetic monopole mighthave been not so heavy. Particles which are partners under supersymmetry andelectromagnetic duality can be thought of babies who were born as identical twins,but as time goes on, who grow into adults with different physical qualities.Supersymmetric Yang-Mills theory is a promising place to learn about quantumfield theories. First, supersymmetry allows computation. Second, some propertieswe find in supersymmetric theories often still hold even in non-supersymmetricquantum field theories. In some sense supersymmetric field theories provide fruitfuland fertile toy models to learn about realistic quantum field theories. b A dyon refers to a particle which potentially carries both electric and magnetic charges. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4
Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories Plan of the review
In section 2, we revisit essential elements of N = 2 supersymmetric Yang-Millstheories and Seiberg-Witten geometry, and prepare ourselves with necessary toolsfor studying singularity. In section 3, we study Seiberg-Witten curves for SU ( r + 1)and Sp (2 r ) SYM and discuss the root structure of those families of hyperellipticcurve, in preparation for section 4 where we compute dyon charges of masslessstates. Section 5 deals with higher singularity with massless electron and monopole(Argyres-Douglas theories) among N = 2 supersymmetric Yang-Mills theories withclassical gauge groups SU ( r + 1), SO (2 r ), Sp (2 r ), and SO (2 r + 1). In section 6, werevisit the tools to capture singularity to learn more about the singularity structure.We conclude with open questions in section 7.
2. Essentials of N = 2 SYM and Seiberg-Witten Geometry Many wonderful reviews
16, 21–23 exist on N = 2 SYM, while this review is morefocussed on the Seiberg-Witten geometry and singularity of those theories. Here wewill only pinpoint the aspects that are necessary for understanding the main ideaof the review. The Lagrangian of N = 2 supersymmetric Yang-Mills theory can bewritten elegantly in N = 1 or N = 2 superspace language, where supersymmetry ismore manifest. Here we will write it down in a spelled-out fashion in 4-dimensionalspacetime language as below: L = − g (cid:90) d x Tr (cid:20) F µν F µν (cid:21) + θ π (cid:90) d x Tr i F µν ˜ F µν − (cid:90) d x
12 Tr (cid:2) φ + , φ (cid:3) + (fermions) . (1)Here A µ and φ transform as a vector and a scalar (respectively) of Lorentz group SO (3 ,
1) of spacetime. Both are adjoint representation of the gauge group G . Thechoice of gauge group also determines structure constants f abc and generators T a , A µ = A aµ T a , [ T a , T b ] = f abc T c ,F µν = ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ] , ˜ F µν = 12 (cid:15) µνρσ F ρσ (2)and g, θ are real-valued coupling constants. If we did not have supersymmetry, thenonly the first term of Eq. (1) would appear in the Lagrangian, as the Yang-Millsaction. For example, the action for weak and strong nuclear forces can be writtendown by choosing G = SU (2) , SU (3) and taking only the first term of Eq. (1).Having N = 2 extended supersymmetry dictates that scalar, spinors, and vectormust transform together forming a N = 2 supermultiplet c . Instead of keeping track c More specifically it is called N = 2 vector multiplet for it contains a vector. The space whosecoordinates are the scalar components of vector multiplets is called a Coulomb-branch of modulispace. The scalars of another N = 2 supersymmetry representation, hypermultiplet, form a Higgsmoduli space. A supermultiplet is a representation of a supersymmetry algebra. More details aregiven in excellent reviews, Refs. 22, 23. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO of all component fields in a given supermultiplet, we can save our effort and restrictour attention to the term one of them only. Here it is enough to consider the partwith the scalar φ only which is the third term of Eq. (1), focussing on vacuumstructure of a N = 2 supersymmetric Yang-Mills theory. In Eq. (1) the second termgives instanton number and the last term denote terms involving fermions.However we won’t make usage of this Lagrangian any more in this review, be-cause Seiberg and Witten proposed another ‘geometric’ way to study N = 2 SYMtheories,
11, 12 which we now turn to.
Review of N = 2 Seiberg-Witten theory and geometry
In late 1990s, Seiberg and Witten made a profound discovery on N = 2, d = 4supersymmetric Yang-Mills theory with gauge group G = SU (2),
11, 12 giving a hugeimpact both on physics and mathematics. After two dozen years, Gaiotto blew anew life into research on N = 2 superconformal theories recently, by discoveringa plethora of new theories with often surprising features, which can be all lego-edfrom simple building blocks .Simply speaking, Seiberg and Witten proposed a powerful dictionary betweenphysics and geometry for N = 2 theories. Seiberg-Witten geometry comes in pack-age with Seiberg-Witten (SW) curve and Seiberg-Witten (SW) differential 1-form.The SW curve is a complex curve, or a real 2 dimensional Riemann surface, whosegenus is equal to the rank r of the gauge group (such as SU ( r + 1), Sp (2 r ), SO (2 r ),and SO (2 r + 1)) for the supersymmetric Yang-Mills theories (that is, with no mat-ters added). It is also equal to the complex dimension of moduli space. Recall thatthe moduli are to be understood as parameters controlling the theory and the sub-sequent SW geometry. If the gauge group had rank 3, then the corresponding SWcurve may look like the Riemann surface in Fig. 1, with genus 3.Physics ↔ Geometrysupersymmetric Yang-Mills theory ↔ Riemann surfacerank of gauge group ↔ genus(BPS) particles ↔ (some) 1-cycles (3)In pure Seiberg-Witten theory the dimension of the moduli space (or the numberof moduli/parameters) is also equal to the genus e , which, in turn, is equal to therank of the gauge group. At a generic point in the moduli space, the SW curve issmooth and all the 1-cycles are non-vanishing as in Fig. 1. However, we could moveto a less generic location in the moduli space where we have vanishing 1-cycles asin Fig. 2 and Fig. 3. d All these methods study theories with low energy effective action. This review also deals withthose only. e One may introduce matters into the Seiberg-Witten theory: By pure
SW theory, they mean lackof matter, and it will be our focus on this review. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4
Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories
Note that on a SW curve, we can draw various 1-cycles as in Fig. 1. Here wehave chosen a particular set of symplectic basis 1-cycles, α i ’s and β i ’s. The only ruleto keep for the choice of symplectic basis cycles is that the intersection numbersmust satisfy: β i ◦ α j = δ ij . (4)The intersection number is an anti-symmetric ( α j ◦ β i = − δ ij ) and bilinear (lineardependence on both arguments) operation among 1-cycles f . Each 1-cycle has anorientation (as seen by the arrow in figures), and the intersection number comeswith a sign g .As anticipated from Eqn. (3), 1-cycles of the Riemann surface correspond tophysical particles h , and a choice of symplectic basis 1-cycles assigns electric andmagnetic charges to the particles. For given i , α i and β i denote electric and magneticcharge (respectively) for the i ’th U (1) inside the gauge group G .However, the choice is certainly not unique, and we could modify the choice by α (cid:48) i ≡ β i , β (cid:48) i ≡ − α i (5)for given i only, and this still preserves the symplectic property of Eqn. (4). Inphysics this corresponds to the electromagnetic duality on i ’th U (1) charge. Anotherimportant fact is that the intersection number (being a scalar) is invariant underelectromagnetic dualities of Eq. (5), and in general, under symplectic transformation(re-choice of symplectic basis 1-cycles).Some of 1-cycles correspond to physical states (stable BPS/supersymmetricdyon), with quantized electric and magnetic charges. As shown Fig. 1, any 1-cyclecan be written in terms of basis 1-cycles α i ’s and β i ’s with integer coefficients, withdyonic charges superposed. These integer coefficients exactly correspond to amount f For later convenience, the choice of overall sign for intersection number chosen to match that ofRef. 16 and is opposite of that of Ref. 24. g This is an algebraic intersection number, as opposed to a geometric intersection number. h Only some of 1-cycles, which pass the test of wall-crossing formulas, correspond to stable BPS(supersymmetric) particles. Our main focus here is for massless states. With an assumption thatthe massless states are stable, we can consider 1-1 mapping between 1-cycles and BPS particles,with restriction to the massless sector. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO of electric and magnetic charges of each U (1). Two 1-cycles ν and ν in Fig. 1 canbe written as follows: ν = β + β , ν = − α + β . (6)Physical interpretation of this would be that, if they corresponded to BPS dyons,then the first object ( ν ) behaves as a magnetic monopole for both the first and thesecond U (1)s, and the second object ( ν ) carries the same charge as a bound stateof a positron and magnetic monopole of the third U (1).Seiberg-Witten geometry contains lots of (if not all) information about the the-ory. The physical information is stored not only in SW curves, but also in the SW1-form. The SW curve and SW 1-form work together, and without each other theylose meaning, just like a needle and a thread. The SW curve provides 1-cycles overwhich to integrate the SW 1-form. Then we obtain complex number which is mean-ingful physically (central charge). By integrating Seiberg-Witten differential 1-form λ SW over 1-cycle ν , we obtain a complex number. For the purpose of this review,we are only interested in its magnitude, which is the mass of the particle M ν = (cid:12)(cid:12)(cid:12)(cid:12)(cid:73) ν λ SW (cid:12)(cid:12)(cid:12)(cid:12) . (7)Since we are focussing on physical singularity associated with massless particles,Eqn. (7) provides the most important piece of information for the purpose of thisreview, among what we learn from the Seiberg-Witten geometry. Assuming λ SW isfree of delta-function behavior, vanishing of 1-cycle ν signals existence of masslessBPS state (with dyonic charge given by ν ) since its mass given in Eqn. (7) vanishes i .Therefore, study of vanishing 1-cycles can teach us about massless BPS states inthe system. We therefore assume that:Singularity loci of SW curve ⊂ Singularity loci of SW theory.
Fig. 2. Vanishing 1-cycles of genus-3 Riemann surface. All these 3 cycles are mutually local, sinceintersection numbers all vanish.
Now let us imagine tuning various parameters (moduli) for the gauge theory ofFig. 1, to force some 1-cycles to vanish. In Fig. 2, we have three vanishing 1-cycles i If the integrand λ SW has a delta-function type singularity, integrating it over an infinitesimalinterval may give a finite value to Eqn. (7). ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories α , α , and β which do not intersect each other. They correspond to an electronwith respect to the first U (1), another electron with respect to the second U (1), anda magnetic monopole with respect to the third U (1). All three particles are massless.If we operate an electromagnetic duality on the third U (1), then the third particlewill be renamed into a massless electron with respect to the third U (1). All themassless particles are mutually local , in that they can be treated as pure electrons(carrying no magnetic charge) in some choice of symplectic basis 1-cycles (i.e. after acertain series of performing electromagnetic dualities). This is possible only because(if and only if, in fact) the corresponding 1-cycles have vanishing intersection numberwith one another. An equivalent mathematical statement is this: if all the 1-cyclesin a certain set have zero intersection number with one another, then they can bewritten in terms of linear combination of α i ’s with no need for β i terms. We willsoon explain why we call them local , (near Eqn. (8)) after explaining non-localitynow. Fig. 3. Mutually non-local vanishing cycles of genus-3 Riemann surface. Their intersection num-ber is non-zero.
Two 1-cycles α and β vanish in Fig. 3, and they correspond to an electronand a magnetic monopole, both charged with respect to the third U (1), and withzero mass. No matter how one may try to redefine electric and magnetic charges byelectromagnetic dualities and so on, it is never possible to make both of them intoelectric particles at the same time. That is because the two vanishing 1-cycles havenon-zero intersection number α ◦ β = − (cid:54) = 0, regardless of choice of symplecticbases. If a set of 1-cycles were able to be written as electric particles (in terms of α i ’s only), then they must have had zero intersection number with one another.We will pause briefly here to explain naming of locality versus non-localityfor vanishing 1-cycles (massless particles, equivalently). As nicely reviewed in Ref.16, 1-cycles transform under monodromy action, as one moves around on a non-contractible loop, surrounding a singularity, in moduli space (changing the modulivalues accordingly). If the singularity is where a 1-cycle ν vanishes, then the other1-cycle γ gets transformed according to this Picard-Lefshetz formula, as explainedin Ref. 16 M ν : γ → γ − ( γ ◦ ν ) ν. (8)ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
For a 1-cycle γ which does not intersect with ν i.e. γ ◦ ν = 0, no change will bemade on γ under monodromy. (However, any 1-cycles which intersect with ν willbe shifted as one goes around the singularity loci associated with vanishing of the1-cycle ν , as we will explain below.)This gives a motivation for the naming: if two cycles have zero intersectionnumber, when one vanishes, the other cycle does not get affected. Or in physicslanguage, two particles can be written as purely electric ones at the same time.When one becomes massless the other does not change its charges. In some sense,they do not need to care about each other, and they are mutually local . It is possibleto write down Lagrangian for those theories, by adding each local pieces.However, now assume that two cycles have non-zero intersection number. Whenone vanishes, the other cycle receives a monodromic shift. In physics language, thetwo particles are mutually non-local and they cannot be written as purely electricones at the same time. When one particle becomes massless, it is ambiguous howto assign charge to the other particle. The dyon charge of the second particle is nota single-valued function of moduli near the singularity locus where the first particlebecomes massless.In general, there is no known Lagrangian for these systems. However, in Ref.2, this exotic theory (so-called Argyres-Douglas theory) has been discovered andstudied, inside moduli space of SU ( r + 1) SW theories. Since there is no Lagrangiandescription yet (if not never), studies are conducted by careful analysis of scalingdimensions near the singularity loci of Seiberg-Witten geometry which can be writ-ten as a hyperelliptic curve equipped with 1-form. Recent key developments in thisdirection can be found in Refs. 3, 25 among others. Now we turn to review geometryof hyperelliptic curves. Review of hyperelliptic curves y = f ( x ) So far in this section, we discussed Riemann surface with 1-cycles which potentiallycould collapse. For the purpose of this review, the Riemann surfaces of our interestcan be written as an algebraic variety given by y = f ( x ), which include hyperellipticcurves.Since we need to deal with singularity as well, let us begin by recalling a few fun-damental facts about singularity of algebraic varieties. Let us consider an algebraicvariety given by F ( x, y, z, . . . ) = 0. This is an object embedded inside a bigger space, ambient space whose coordinates are x, y, z, . . . . It is singular if exterior derivative dF = 0 vanishes, or in other words if all the partial derivatives vanish, namely ∂F∂x = ∂F∂y = · · · = 0.The exterior derivative d is written in terms of the partial derivatives withrespect to all the coordinates of the ambient space. Since the Riemann surface isembedded in an ambient space whose coordinates are x and y , the exterior derivativectober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories is given as d = dx ∂∂x + dy ∂∂y . (9)Later, we will consider algebraic variety embedded inside the moduli space whosecoordinates are complex-valued moduli u i ’s, then the exterior derivative will ac-cordingly be d = (cid:80) i du i ∂∂u i .In simpler words, at each point on a given surface (algebraic variety) embeddedin a bigger space, an ambient space , we can consider tangent space. However, ifthe surface develops singularity, then the tangent space suddenly changes there( dF = 0). Singularity of an algebraic variety (super-elliptic curve) F ≡ y n − f ( x ) = 0 , n ≥ ∂F∂x = − ∂f∂x = 0 and ∂F∂y = ny n − = 0. Therefore the singularityis at where y = 0 = f ( x ) = ∂f∂x . In order for f ( x ) and ∂f∂x to have a common root,it is equivalent to demanding f ( x ) to have a degenerate root. We will now see thatit happens if and only if f ( x ) has vanishing discriminant ∆ x f = 0.Discriminant of a polynomial f n ( x ) = (cid:81) ni =1 ( x − e i ) is given in terms of its rootsas ∆ x ( f n ( x )) = (cid:89) i
3. First Look at Seiberg-Witten Curves for SU ( r + 1) and Sp (2 r ) Among hyperelliptic curves given in Eqn. (13), here we will consider a few familiesonly, which are Seiberg-Witten curves for SYM with SU ( r + 1) and Sp (2 r ) gaugegroups. In the parameter space of the hyperelliptic curves, these will form subspaceswith dimension almost halved l . Here we will focus on ‘root structure’, in otherwords, potential degeneracy of branch points. For both cases, we note that thecurve is factorized into two polynomials which never share roots. Branch points willbe divided into two mutually-exclusive sets where multiplicity may happen onlywithin each group. Each set of branch points will be assigned with a name and acolor, therefore enabling bi-coloring (green and purple) of coming figures in thisreview. Seiberg-Witten geometry for pure N = 2 SU ( r + 1) theories The SW curve and SW 1-form for pure SU ( r + 1) of Ref. 26 are rewritten as y = f SU ( r +1) = f + f − , λ S W = − dx log (cid:18) −
12 ( f + + f − ) − (cid:112) f + f − (cid:19) , (14) k By isolated points, we roughly mean that they form a discrete set and are separated. For example,the points do not congregate to form a line or a plane. l Recall from the definition of hyperelliptic curve in Eqn. (13) that we allow the coefficient of eachpower of x to vary, all independently from each other. For most generic hyperelliptic curve, theoverall power of f ( x ) equals to the number of parameters. By assigning the hyperelliptic curve arole of SW curve for certain gauge groups, we no longer have the full freedom of varying all ofthem. As we will see soon the number of independent parameters is r while f ( x ) has the power2 r + 1 , r + 2 etc. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO where f ± are given in terms of r gauge invariant complex-valued moduli u i ’s as: f ± ≡ x r +1 + r (cid:88) i =1 u i x r − i ± Λ r +1 . (15)Note that we are not allowing the full 2 r + 1 degrees of freedom of hyperellipticcurves of Eqn. (13). Instead we get to vary r (same as the rank of the gauge group)moduli u i ’s only while Λ is a fixed non-zero constant.In discussing singularity of hyperelliptic curve, we will consider collision amongthe branch points of f ( x ). Therefore, it is convenient to give names to the roots of f ± , as f + ≡ r (cid:89) i =0 ( x − P i ) , f − ≡ r (cid:89) i =0 ( x − N i ) . (16)Since Λ (cid:54) = 0, f ± can never vanish at the same time. Therefore f + and f − cannever share a root, and there is no vanishing 1-cycle mixing these two sets of roots.More explicitly, the discriminant of the SU ( r + 1) SW curve factorizes into ∆ x f SU ( r +1) = (2Λ r +1 ) r +2 ∆ x f + ∆ x f − . (17)In other words, in order for f SU ( r +1) ( x ) to have a degenerate root, f + or f − itselfshould have a degenerate root. This justifies binary color coding in figures for branchpoints and vanishing 1-cycles. On the x -plane, only P i ’s (or N i ’s) can collide amongthemselves.At discriminant loci ∆ x f SU ( r +1) = 0 and near the corresponding vanishing 1-cycle, the 1-form of (14) is regular λ S W = − dx log (cid:0) ± Λ r +1 (cid:1) , near f ± = ∆ x f ± = 0 , (18)confirming that the singularity of the SW curve is indeed the singularity of the SWtheory, as promised earlier above Fig. 2. SW curve for pure Sp (2 r ) theories and root structure Now consider a slightly different hyperelliptic curve, y = f Sp (2 r ) = f C f Q , λ = a dx √ x log (cid:18) xf C + f Q + 2 √ xyxf C + f Q − √ xy (cid:19) , (19)with f C and f Q defined as: f C ≡ x r + r (cid:88) i =1 u i x r − i , f Q ≡ xf C + 16Λ r +2 , (20)with r (again same as the rank of the gauge group) gauge invariant complex moduli u i ’s. This can be easily obtained by taking no-flavor limit of Ref. 27.Observe in (20) that f C = f Q = 0 is possible only if Λ = 0. In a quantum theorywe demand Λ (cid:54) = 0, so f C and f Q can never share a root. For any choices of moduli,ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories f C and f Q can never vanish at the same time. Just similarly to the SU ( r + 1) casein Eqn. (17), the discriminant of the Sp (2 r ) SW curve also factorizes as ∆ x f Sp (2 r ) = (16Λ r +2 ) r ∆ x f C ∆ x f Q . (21)Again, in order for f Sp (2 r ) ( x ) to have a degenerate root, f C or f Q itself should havea degenerate root.We can study multiplicity of zeroes for f C and f Q separately without worryingabout their roots getting mixed. Again, just as in the SU case, when we drawvanishing cycles and collision of branch points, we can use binary coloring. Thebranch points and vanishing cycles are all grouped into two mutually exclusivegroups (for C and Q respectively.).In order to give new names to two sets of branch points, let us introduce C i ’sand Q i ’s as given in f C = r (cid:89) i =1 ( x − C i ) , f Q = r (cid:89) i =0 ( x − Q i ) . (22)At discriminant loci ∆ x f Sp (2 r ) = 0, near the corresponding vanishing 1-cycle, the1-form of (19) becomes infinitesimally small, far from becoming a delta function.This confirms that a singularity of the SW curve is indeed a singularity of the SWtheory, again confirming the claim near Fig. 2.
4. Electric and Magnetic Charges of Massless Particles
So far, we discussed existence of massless particles in N = 2 theories. In this section,we will discuss electric and magnetic (dyonic) charges of these massless particles.Recall that the massless state was associated with a vanishing 1-cycle of Riemannsurface, from Eqn. (3) and Eqn. (7). Dyonic charges can be read off by decomposinga 1-cycle into symplectic basis 1-cycles, as discussed near Eqn. (6).In the moduli space, there will be complex codimension 1 loci with a vanishing1-cycle and it creates nontrivial topology on the moduli space with monodromydetermined by the dyonic charge of the vanishing 1-cycle, as in Eqn. (8).In this section, first we will look at dyon charges of the massless particles for thefamous and simpler rank 1 case, and then move to higher rank cases reviewing theresults obtained in Ref. 24, with focus on SU ( r + 1) and Sp (2 r ) case. Rank 1 examples: how to read off monodromies of theSeiberg-Witten curves
For pure SYM, the rank of gauge group r equals to the genus of the SW curveand the number of complex moduli u i ’s, as one might recall from SW geometry of SU ( r +1) and Sp (2 r ) gauge theory given in Eqns. (14) and (19). Here we will warm-up by considering their rank 1 cases, which has one complex modulus u for a genus-1curve. Therefore the moduli space is a complex plane (real 2-dimensional surface),ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO which we denote as u -plane. There exists non-trivial topology on the moduli space,created by existence of singular points on it.By singular points on moduli space of SW curve, we mean the values of moduliwhich make the SW curve singular (i.e. with vanishing 1-cycles). Recall that it isequivalent to having zero discriminant of hyperelliptic curve. Demanding this singlecomplex condition on the moduli space, we will have a complex codimension-1loci in the moduli space as a solution set. The modulus u being the only parametercontrolling the properties of a genus-one SW curve, vanishing discriminant conditionwill fix u to possible isolated (separated) values.First we will locate the singular points by discriminant condition, and thenconsider monodromy properties around each of them, by reading off dyon chargesof vanishing 1-cycle, in the spirit of Eqn. (8). Starting from a generic place in modulispace (a reference point u ∗ on the moduli surface), we make non-contractible loopsaround each singular point (where discriminant vanishes), and consider monodromyalong each path, associated with the singularity surrounded inside.Here we will discuss monodromy of rank r = 1 cases of SU ( r + 1) SW curvegiven in Eqn. (14) and Sp (2 r ) SW curve given in Eqn. (19), which we will call SU (2)and Sp (2) curves. Since SU ( r + 1) and Sp (2 r ) gauge groups are identical at rank1, these two distinct curves in fact describe the same physical theories and indeedtheir monodromy properties match up with each other. Historically both curveswere called SU (2) SW curves, but we will call one of them Sp (2) curve, because ithas nice generalization for Sp (2 r ) SW theories.After absorbing some powers of two’s into Λ for convenience, Sp (2) curve be-comes y = x (cid:18) x ( x − u ) + 14 Λ (cid:19) , (23)as first given in Ref. 12. On the other hand, SU (2) curve y = ( x − u ) − Λ = ( x − u + Λ )( x − u − Λ ) , (24)follows from Eqn. (14), as first proposed by Ref. 26. It is straightforward to checkthat discriminant vanishes at u = ± Λ for both SW curves for pure SU (2) = Sp (2)theory given by (23) and (24). Now we will turn to finding out which 1-cycle of SWcurve vanishes at u = ± Λ .4.1.1. Review of SU (2) monodromy As explained in Ref. 26, SU (2) curve has massless dyon and monopole at twodifferent locations in the moduli space m . The curve in Eqn. (24) has four branch m In fact the moduli space has an extra singular point at u = ∞ as reviewed in Ref. 16. However itis not associated with a particular vanishing 1-cycle, and its monodromy can be inferred from theknowledge of other singular points purely from consistency requirement. Therefore in this reviewwe won’t discuss the singular points at infinity in moduli space. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories points N = − N = (cid:112) u + Λ , P = − P = (cid:112) u − Λ . (25)At a generic value of modulus u , they are all distinct. As we vary u toward twospecial values u → ± Λ , different pairs of branch points will collide: N and N collide as u → − Λ and P and P collide as u → Λ . Fig. 4. Vanishing cycles for a pure SU (2) Seiberg-Witten theory with its SW curve y = ( x − u ) − Λ = ( x − u + Λ )( x − u − Λ ). The branch points are drawn on the x -plane for varyingvalues of the modulus u . From the left, u takes the u ∼ Λ , u ∼ u ∼ − Λ in the three figuresdrawn here. Fig. 4 denotes vanishing 1-cycles, associated to collision of those branch points.The branch points are drawn on the x -plane for varying values of the modulus u .From the left, u takes the u ∼ Λ , u ∼ u ∼ − Λ in the three figures drawn here.On the left of Fig. 4 ( u ∼ Λ ), two purple branch-points P and P come close toeach other. A 1-cycle drawn in purple denotes the corresponding vanishing 1-cycle,which goes through 2 branch cuts. Half of it is solid line, the other half is dashedline. Recalling that hyperelliptic curves are double-sheet fibration over an x -plane,we can take solid lines to be on the upper sheet ( y = (cid:112) f ( x )) and dashed lines tobe on the lower sheet ( y = − (cid:112) f ( x )). Each time a 1-cycle meets a branch-cut, ithas to switch from solid to dash and vice versa.Similarly, on the right of Fig. 4 ( u ∼ − Λ ), two green branch-points N and N come close to each other. A 1-cycle drawn in green denotes the correspondingvanishing 1-cycle, which goes through 2 branch cuts. In the center of Fig. 4 ( u ∼ u = ± Λ .By counting their intersection number, we can read off from the figure thatthese two 1-cycles have mutual intersection number 2. We are allowed to choose anappropriate symplectic basis, so that we can they can be written as β and β − α ( β ◦ ( β − α ) = − J. S. SEO
Fig. 5. Vanishing cycles for a pure SU (2) Seiberg-Witten theory with its SW curve y = ( x − u ) − Λ = ( x − u + Λ )( x − u − Λ ). As u varies on u -plane on green and purple paths given,the branch points will move along the path given in respective colors on x -plane. Note that theleft side of the figure denotes the moduli space and the right side denotes the SW curve embeddedin an ambient space whose coordinates are x, y . The topological information of Fig. 4 is summarized in Fig. 5. On the left,moduli space is given. Green and purple paths denote the choice of how to vary themodulus u . As u varies on the paths given, the branch points will move followingthe paths drawn on x -plane, as on the right of Fig. 5. For higher ranks of SU ( r + 1)SW theory in subsection 4.2, we will omit figures like Fig. 4 (which is a procedurehow one obtains the information about vanishing 1-cycles) and only display figuressimilar to Fig. 5, which contains full topological information of vanishing 1-cyclesand trajectories in the moduli space.Note that what trajectory each cycle takes does matter. It is important not onlywhich two branch points are connected, but also through what trajectory they areconnected. This should be clear from simple counting: with finite number of branchpoints ( N ), we can choose a finite number of pairs of branch points ( N ( N − ),however we have infinite (countable) number of distinct 1-cycles (as points on the( N − Sp (2) monodromy Starting from the Sp (2) curve of Eqn. (23), shift x by x → x + u to obtain y = ( x + u ) (cid:18) x ( x + u ) + 14 Λ (cid:19) , (26)which identifies with expression given in Ref. 12. Then perform x → /x, y → x − y transformation to obtain y = x (1 + 2 ux ) (cid:0) ux + Λ x (cid:1) , (27)ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories whose four branch points are O ∞ = 0 , C = − u , Q = − (cid:16) u + (cid:112) − Λ + u (cid:17) ,Q = − (cid:16) u − (cid:112) − Λ + u (cid:17) . (28)At generic value of u , all four branch points are separated, but as we vary u intoappropriate values, Q and Q collide with each other. Fig. 6. Vanishing cycles for pure Sp (2) Seiberg-Witten theory with a SW curve y = x (cid:0) x ( x − u ) + Λ (cid:1) . The branch points are drawn on the x -plane for varying values of the modulus u . Fig. 6 shows a related animation. In the center figure, all branch-points are sep-arated, and two purple 1-cycles (not vanishing here) are drawn for later conveniencewith two different thickness. On the left and right figures of Fig. 6, Q and Q col-lide with each other. These figures preserve topological information about branchcuts, branch points, and 1-cycles. On the left, a thick 1-cycle vanishes, which is thesame 1-cycle as in the center figure. On the right figure, a thin 1-cycle vanishes, andit is to be identified with the thin 1-cycle in the center figure.Topological information of Fig. 6 is summarized in Fig. 7 along with the choiceof trajectory on the moduli space. On u -plane, there are two singularities, anddepending on which singularity the path surrounds, Q and Q collide along differentpath. In Fig. 7, it is denoted with two different thickness of lines.Again from the trajectories of branch points on the right of Fig. 7, we observethat the two 1-cycles have mutual intersection number 2. After an appropriate choiceof symplectic basis, they can be identified as β and β − α , or a magnetic monopoleand dyon, matching the result for SU (2) case (the same theory physically) above.ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Fig. 7. Vanishing cycles for pure Sp (2) Seiberg-Witten theory with a SW curve y = x (cid:0) x ( x − u ) + Λ (cid:1) . discriminant vanishes at u = ± Λ . As u varies on u -plane on thick andthin paths given, the branch points will move along the path in respective thickness on x -plane,forming vanishing 1-cycles. Moving on to the higher rank case
Now we move on to cases with higher and arbitrary rank r of pure SYM with SU ( r +1) and Sp (2 r ) gauge groups. Now we have r complex moduli u , . . . , u r . Thereforethe moduli space is a real 2 r -dimensional space. Discriminant loci are various real(2 r − SU ( r + 1) and Sp (2 r ) SW theories confinedon a certain moduli plane, which is chosen for the ease of computation. N = 2 SU ( r + 1) theory Here we discuss monodromy of the SU ( r + 1) curve given in Eqn. (14). Again themoduli space has r complex-valued coordinates u i ’s. Instead of considering the fullreal 2 r -dimensional moduli space, we consider its real 2-dimensional slice, a u r − -plane, after fixing all other r − r − u r to a constant where its phase is chosen carefullyas below u = · · · = u r − = 0 , u r / Λ r +1 ∈ i R (29)On such a hyperplane of moduli space, the discriminant simplifies into∆ x f ± = ( − r ] r r ( u r − ) r +1 + ( − r +12 ]( r + 1) r +1 ( u r ± Λ r +1 ) r , (30)ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories at whose vanishing loci the SW curve degenerates.Note the symmetries on the moduli slice: there is Z r +1 symmetry from rotationof the phase on u r − . Also there is a Z symmetry for ∆ x f ± , associated with complexconjugation of all u i ’s and switching between ∆ x f + and ∆ x f − , due to fixed phaseof u r as decided in Eqn. (29). In Eqn. (30), note that u r is a constant and only u r − is a variable. Since u r − has power r + 1, there are r + 1 solutions to each of∆ x f + = 0 and ∆ x f − = 0. Those solutions have equal magnitude, and the phasesare equally distanced among one another.If we recall from Eqn. (17), that ∆ x f ∼ ∆ x f + ∆ x f − , it follows ∆ x f ∼ ∆ x f + ∆ x f − = 0 has 2 r + 2 solutions on u r − -plane. Therefore we can single out2 r + 2 points, arranged on a circle, on u r − -plane, where discriminant of the SWcurve vanishes. This is depicted for rank 9 case in Fig. 8 as eighteen marked pointsin purple and green. Each singular point on u r − -plane is responsible for vanishingof a 1-cycle and associated monodromy of Seiberg-Witten curve. Fig. 8. Singularity structure of SU (10) curve on a moduli slice of the u -plane given by u = u = · · · = u r − = 0 , u r = const . Here r = 9 case is drawn. As we go around each singular pointon the u -plane, we have a vanishing cycle on the x -plane as shown in Fig. 9. We pick a reference point at origin of u r − -plane, so that we manifest Z r +1 and Z symmetries on x -plane as well. When u r − = 0, the branch points are spreadctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO on a circle on x -plane: N i ’s (and P i ’s) are even distributed on a circle with Z r +1 symmetry among themselves, and there is Z symmetry between N i ’s and P i ’sbecause of u r Λ r +1 is a pure imaginary number (or zero). Rank 9 case is depicted inFig. 9: branch points on x -plane are drawn as green and purple marked points, ata reference point ( u r − = 0) on the moduli slice given by Eqn. (29). Fig. 9. Drawn here are vanishing cycles for SU (10) at a moduli slice, a u -plane given by u = u = · · · = u = 0 and u Λ ∈ i R ∪ { } . As we vary u r − , approaching each of 2 r + 2 singular points on u r − -plane, thena corresponding 1-cycle will vanish on x -plane, and we read off its dyon charge.On the u r − -plane, starting from the origin as the reference point, we make a non-contractible loop surrounding each singular point on u r − -plane (as shown in Fig. 8).ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories As we vary u r − , we observe the vanishing cycles on the x -plane, as in Fig. 9 for rank9 case: the branch cuts are drawn connecting the branch points of correspondingcolors.Also, Fig. 10 (we stretched out the x -plane, opening up the circle on the x -planeinto a line) is given for general ranks. Vanishing 1-cycles satisfy ν Pi ◦ ν Pi +1 = ν Ni ◦ ν Ni +1 = − , ν Pi ◦ ν Ni = 2 , ν Pi ◦ ν Ni +1 = − , (31)while all other intersection numbers vanish. Fig. 10. Vanishing 1-cycles for SU ( r + 1) curveFig. 11. A particular choice of symplectic basis cycles for SU ( r + 1) curve To read off the dyon charges of massless states, we choose a set of symplecticbases. All choice is equivalent to up to electromagnetic duality. With a symplecticbasis given in Fig. 11, we are choosing α i cycles to go around each branch cutconnecting P i and N i branch points. We are choosing β i cycles to connect between P and P r branch points. In the Fig. 11, to make it convenient to generalize toarbitrary ranks, we rearranged the branch cuts on the x -plane into a line. Withctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO such a choice of symplectic bases, we can write down the vanishing 1-cycles as: ν P = β , ν N = β + r (cid:88) i =1 α i + α , ν Pr = − β r + r − (cid:88) i =1 α i , ν Nr = − β r − α r ,ν Pi = β i +1 − β i − α i , ν Ni = β i +1 − β i + α i +1 − α i , i = 1 , · · · , r − . (32)We will end this subsection by how this analysis (mostly based on Ref. 24) fits inthe existing literature. Recall that Seiberg-Witten curves and massless dyon chargeswere much-studied for low rank cases. Original Seiberg-Witten paper
11, 12 studiedthe curves, massless dyons (monodromies), and some aspects of singularity aspectsfor SU (2) theory with and without matter. Refs. 26, 29 have discussed six vanishingcycles of the SW curve for pure SU (3) on a slightly different moduli slice from theone chosen here. Vanishing 1-cycles of SU ( n ) SW curve were also studied in Refs.30, 31. There are lots of recent developments in obtaining BPS spectra (includingmassive ones) as discussed in Refs. 32, 33, 34, 35 for example. Monodromies of pure N = 2 Sp (2 r ) theories In the last subsection we discussed pure SU ( r + 1) SW theories and computed theirmonodromies. Here we will continue the similar analysis for Sp (2 r ) SW theories.Seiberg-Witten curves for pure Sp (2 r ) theory is given by y = x (cid:32) r (cid:88) i =1 u i x i (cid:33) (cid:32) r (cid:88) i =1 u i x i + x r +1 (cid:33) (33)after appropriate coordinate transformations of no-flavor limit of Ref. 27 and settingΛ (cid:54) = 0 to satisfy 16Λ r +2 = 1 without loss of generality.Vanishing cycles are computed in Ref. 24 in a certain moduli slice for r ≤ u -plane given by fixing u = · · · = u r − = 0 and setting u r to be a fixed small number. Choose the origin u = 0 asa reference point. Up to rank 6, if we choose u r to be small enough then branchpoints on the x -plane are arranged such that all the Q i ’s are surrounding origin O ∞ (a branch point at x = 0 in Eqn. 33), and all the C i ’s are surrounding all the Q points. Vanishing 1-cycles have the following non-zero intersection numbers: ν Qi ◦ ν Qi +1 = ν Ci ◦ ν Ci +1 = 1 , ν Qi ◦ ν Ci = ν Ci ◦ ν Qi − = 2 , i = 1 , . . . , r,ν Qr ◦ ν Q = 3 , ν Q ◦ ν Cr = ν C ◦ ν Qr = − . (34)Rank 6 case is depicted in Fig. 12. Unlike the SU ( r + 1) case, it is difficult to findan exact method to obtain the vanishing cycles for arbitrary high ranks of Sp (2 r ).Instead, we compute the vanishing cycles in some patches of the moduli space forlow ranks, and read off a pattern to conjecture for general ranks. Ref. 24 conjecturesthat it is always possible to choose u r to be small enough such that all the Q i pointsare inside the ν C cycles, such that (34) holds, for any rank r .ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories Sp (12) curve at a moduli slice, which is given by a u -plane of u = · · · = u r − = 0 , u r = 1 / We write down these 2 r + 1 vanishing cycles ν Ci = − β i + β i +1 + α i +1 , ν Qi = ν Ci − α i + α i +1 , i = 1 , · · · , r − ,ν Cr = β − β r − r (cid:88) i =2 α i , ν Q = β + 2 α , ν Qr = ν Cr + β + α − α r . (35)in terms of symplectic basis given in Fig. 14. We are choosing α i cycles to go aroundeach branch cut connecting Q i and C i branch points. We are choosing β i cycles toconnect between Q and Q r branch points. Fig. 13. Vanishing 1-cycles for Sp (2 r ) curve ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Fig. 14. A particular choice of symplectic basis cycles for the Sp (2 r ) curve ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories
5. Argyres-Douglas Loci: Massless Electron & Massless MagneticMonopole
In previous section we studied vanishing 1-cycles at a complex codimension 1 lociof the moduli space. Demanding one 1-cycle to vanish takes away 1 complex degreeof freedom, and therefore such loci have one dimension less than the full modulispace. In this section, we consider singularity loci of moduli space with codimension2 (or more), where we demand 2 (or more) 1-cycles to vanish. This section willdiscuss degeneration of Seiberg-Witten curves so that mutually intersecting 1-cyclesvanish at the same time. Such geometry (reviewed in subsection 2.1) corresponds toArgyres-Douglas theories, containing massless dyons which are mutually non-local(they cannot turn into pure electric particles by any electromagnetic dualities) asstudied in Ref. 2.In this review, our first encounter with Argyres-Douglas theory will be in thecontext of Sp (4) SW theory. Now we take a closer look at various singularity lociwith complex-codimensions 1 and 2 inside the moduli space. Singularity structure of Sp (4) = C : detailed look on BPSspectra Similarly to the previous section, we compute the dyon charges of vanishing 1-cyclesfor a SW curve, for Sp (4) gauge group. Instead of confining ourselves on a modulislice, we will consider a set of moduli slices. We will observe how vanishing cycleschange as we change the choice of hypersurface.Since the gauge group has rank 2, Sp (4) SW theory has a 2 complex (4 real)dimensional moduli space whose coordinates are two complex moduli u ≡ u , v ≡ u . Instead of full 4 real dimensional moduli space, we will take a 3 real dimensionalsubspace by fixing the phase of v such that v is real, as in the left side of Fig. 15.This choice was made so that the moduli subspace contains all interesting singularpoints inside, where multiple 1-cycles vanish at the same time n .As we vary magnitude of v , we consider singularity structure on each u -plane:we locate where the curve degenerates, and we compute the corresponding dyoncharge for vanishing 1-cycle.On the right of Fig. 15, each of the five u -planes, marked with (a) to (e), is a sliceof the moduli space at different magnitude of v . The purple and green curves runningvertically are denoted by Σ Ci ’s and Σ Qi ’s. These are singularity loci in the modulispace with complex codimension 1. This is where the corresponding 1-cycle vanishes(i.e. a dyon becomes massless), and it is captured by the vanishing discriminant ofthe curve (one complex condition).Interesting things happen when discriminant loci Σ’s intersect inside modulispace, at complex codimension 2 loci, forming a worse singularity. There, two 1- n As we will discuss later, all solutions to the vanishing double discriminant given in Eqn. (36)satisfy that v is real. This is a necessary, but not sufficient, condition for two 1-cycles to vanish. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Fig. 15. A subspace inside the moduli space of pure Sp (4) Seiberg-Witten theory. Vanishingdiscriminant loci Σ Q and Σ Q (cid:48) are drawn with two different types (big and small) of dashed lines.They get interchanged at a cusp point (red). cycles vanish at the same time. In other words, massless dyons coexist. For example,at a blue point labelled ‘node’ in the right of Fig. 15, Σ Q and Σ C intersect, andthat is where 1-cycles denoted by ν Q and ν C degenerate at the same time.When two 1-cycles vanish at the same time, the SW curve (embedded in theambient space whose coordinates are x, y ) degenerates into either cusp or node form.The shape of intersection loci of Σ’s also takes cusp or node form respectively insidethe moduli space. Each leads to different kind of singularity, mutually non-local andlocal massless dyons. This is not tied to Sp (4) gauge group, and similar phenomenaoccur for pure SU (3) theory, too. When Σ’s (discriminant loci) intersect each other, it is seen as colliding of sin-gular points on the corresponding u -plane, the moduli slice. For example, on thethree u -planes marked by (a), (c), and (e), there are five singular points where Σ’sctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories pierce through. On the two u -planes marked by (b) and (d), two of the singularpoints are on top of each other, so we see only 4 separate points on u -plane.Recall from subsection 2.2 that collision of branch points on the x -plane wascaptured by vanishing of discriminant operator with respect to x , ∆ x acting on f . Similarly, when the singular points collide on the u -plane, it is captured byanother discriminant operator with respect to u , ∆ u acting on ∆ x f . In other wordsdemanding vanishing discriminant ∆ x f = 0 and double discriminant ∆ u ∆ x f = 0bring out all the candidates for having two vanishing 1-cycles. However among this,only those which satisfy d ∆ x f = 0 truly qualifies as a singularity loci of discriminantloci of hyperelliptic curve. The points where ∆ u ∆ x f = ∆ x f = 0 and d ∆ x f (cid:54) = 0 iswhere the discriminant loci is smooth but it appears singular on a particular sliceof moduli space. We will elaborate more near Fig. 19 later.In the case of Sp (4) SW theory, vanishing double discriminant gives one con-straint as below: ∆ u ∆ x f Sp (4) = 2 v ( v − ) (2 v − ) , (36)whose roots v = (cid:40) , α i , √ α j (cid:41) (37)are marked by seven dots in the left of the Fig. 15. One of the solutions of Eqn.(36) is v = 0, but it does not translate into having two massless dyons. The othersix points correspond to having two massless BPS dyons. In subsection 6.1 we willdiscuss how d ∆ x f = ∆ x f = 0 is equivalent to having two massless BPS dyons. v = 0 does not satisfy that relation, but the rest 6 does (with proper choice of u value).Note the degeneracy of roots to the double discriminant of Eqn. (36). The roots3 α i and √ α j have degeneracy 3 and 2 respectively. In the next section we willreview that this is a universal criterion for having mutually non-local and localmassless dyons. Actually vanishing of double discriminant is a necessary conditionfor having multiple massless dyons, but not a sufficient condition as we will see insubsection 6.1.For each of u -planes marked by (a) to (e) of Fig. 15, we have drawn a corre-sponding x -plane in Fig. 16 displaying the vanishing cycles on the x -plane, for eachmoduli slice. Let us have a closer look staring from the top slice marked as (a). (a) choose a u -plane where | v | > √ , then vanishing discriminant loci of the SWcurve (marked as Σ Ci ’s and Σ Qi ’s in Fig. 15) intersect the u -plane at fiveseparate points u Ci ’s and u Qi ’s. Each of the 5 singularity points on the u -plane ( u Ci ’s and u Qi ’s) is responsible for one vanishing 1-cycle ( ν Ci ’s and ν Qi ’s with corresponding choice of sub-/super- scripts). By observing therelative trajectory of branch points on the x -plane (with help of plotting inctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Fig. 16. Singularity structures at slices of the moduli space for pure Sp (4) Seiberg-Witten theory. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories Mathematica), we can read off the 5 vanishing cycles as below: ν Q = β , ν Q = − β + β + α + 4 α , ν Q = − β − β − α − α ,ν C = − β + β + 2 α + 3 α , ν C = − β − β − α . (38) (b) As we change the moduli | v | so that | v | = √ , singularity points on the u -plane, u Q and u C , now collide. Corresponding 1-cycles ν Q and ν C vanishsimultaneously at the point given by u = u Q = u C and | v | = √ . Howeverthese two cycles are mutually local as seen from Eqn. (38) or Fig. 16:responsible four branch points on the x -plane collide pairwise ( C ↔ C , Q ↔ Q ). The SW curve degenerates into a node form, y ∼ ( x − C ) ( x − Q ) × · · · . Singularity loci (of vanishing ∆ x f ) Σ Q and Σ C intersect at | v | = √ , with node-like crossing. Note that the node-like singularity appearsboth in the ambient space and the moduli space of SW curve. (c) We can change the moduli | v | further to enter the range 3 < | v | < √ . Unlike(b), u Q and u C are separated again restoring back to (a). Dyon charges ofthe vanishing cycles for (a), (b), and (c) stay the same as given by Eqn.(38). (d) As we change the magnitude of the moduli v , to satisfy | v | = 3, discriminantloci Σ Q and Σ Q intersect on the u -plane, forming a cusp-like singularityinside the moduli space. Two singular points u Q and u Q collide on the u -plane, and the vanishing cycles ν Q and ν Q merge into something which isno longer a 1-cycle. Now it is not possible to tell apart vanishing 1-cycles o .Now three branch points Q , Q , and Q collide at the same time on the x -plane. Two cycles ν Q and ν Q become massless at the same time, but theyare mutually non-local as clear from Eqn. (38). The SW curve degeneratesinto a cusp form y ∼ ( x − a ) ×· · · , giving Argyres-Douglas theory, with twomutually non-local massless BPS dyons. Note that the cusp-like singularityappears both in the ambient space and the moduli space of SW curve. (e) After passing Argyres-Douglas loci of (d), we again have 5 separated singularpoints on the u -plane. Especially u Q and u C are separated. However notethat the BPS dyon charges of vanishing cycles changed from the cases of(a), (b), and (c) given in Eqn. (38) as we go through cusp-like (or Argyres-Douglas) singularity of (d). Instead of ν Q , we have a new vanishing cycle ν Q (cid:48) = − β − α − α = ν Q + ν Q . (39) o Before we identified a vanishing 1-cycle by collision of 2 branch points. Their trajectory formed a1-cycle, which goes through 2 branch cuts. Half of it was solid line, the other half was dashed line,in figures here. Recall that each time a 1-cycle meets a branch-cut, it has to switch from solid todash and vice versa. Now we have 3 branch point colliding together. Now we the trajectory has 3pieces involving 3 branch cuts, and there is no consistent way of assigning dash and solid for theodd number of pieces. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
In observing the right side of Fig. 16, we realize that the set of vanishing 1-cycleshave changed in going from (c) to (e), through (d). In order to check whether this“jump” is something intrinsic - geometrically and physically - to this system, wecan conduct a few tests. One quick test we perform here is to choose alternativepaths on moduli slice and check how it may affect vanishing 1-cycles. Instead of thepaths on the left of (e) in Fig. 16, we can change a new set as given in Fig. 17.
Fig. 17. Alternative choice for trajectory in moduli slice
The new path still surrounds the same singularity, however its relative locationwith respect to other singularities changed. Two trajectories in the moduli space γ Q (cid:48) in Fig. 16 and γ Q (cid:48) e in Fig. 17 both surround only u Q (cid:48) , a singular point, on the modulislice. In other words, they form non-contractible loops surrounding a singular locusΣ Q (cid:48) in Fig. 15 and nothing else. However the key difference is their relative locationwith respect to two other singularities Σ Q and Σ C in Fig. 15 (or γ Q and γ C in Fig.16 and Fig. 17.In order to answer how this change may affect the assignment of dyon charges ofvanishing 1-cycles, let us consider following cartoon in Fig. 18. On the same mod-uli slice, we have a few different trajectories drawn, surrounding singular points ofvanishing discriminant of the curve. We have drawn two other bunches of singu-lar points in the top and the bottom, to denote other singularities whose spatialrelationship with all the trajectories depicted remain unchanged. Both γ b and γ c surround a singular point u b only. We can associate a monodromy matrix for eachclosed loop in the moduli space, to denote how the 1-cycles get transformed amongthemselves, in the spirit of Picard-Lefshetz formula given in Eqn. (8). Trajectories γ b and γ a can be combined, in that order, to give a trajectory γ d , and similarlytrajectories γ a and γ c can be combined to give a trajectory γ d . Therefore we havethe following relations among the monodromy matrices (The matrix on the rightctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories will be operated first.) M a M b = M c M a = M d . (40)For a SW curve of genus 1, the monodromy matrix is given in Ref. 16 as M ( g,q ) = (cid:18) qg q − g − gq (cid:19) (41)for a closed loop in moduli space, which surrounds a singularity associated withvanishing 1-cycle gβ + qα (or a massless dyon of charge ( g, q )). If dyon has vanishingcharge or if dyon is absent, then the monodromy matrix reduces to the identity.For higher genus case which we are considering here, we can arrange the dyoncharges to contain only α , β with vanishing contribution from other α i ’s and β i ’s,by symplectic transformation (or electromagnetic dualities). In other words, thedyon charge will look like( (cid:126)g, (cid:126)q ) = ( g , g , . . . , g r ; q , q , . . . , q r )= ( g , , . . . , q , , . . . ,
0) = ( g, , . . . , q, , . . . ,
0) = gβ + qα (42)This is equivalent to saying that we can arrange the monodromy matrices M a and M b of Fig. 18 to take the following form M ( g, ,..., q, ,..., = qg q − g − gq
00 0 0 , (43)which contains Eqn. (41) as a left-top 2 × × J. S. SEO
Again by symplectic transformation with no loss of generality, we can assign thedyon charges of γ a and γ b to be (0 , a ) and ( b, , a ) ◦ ( b,
0) = aα ◦ bβ = − ab. (44)Please note we assign the dyon charges to the trajectories γ ’s but not to the singularpoints u ’s themselves.From Eqn. (40) and Eqn. (41), we have monodromy matrices as given below: M a = M (0 ,a ) = (cid:18) a (cid:19) = (cid:18) − a (cid:19) − ,M b = M ( b, = (cid:18) − b (cid:19) = (cid:18) b (cid:19) − ,M c = M a M b M − a = (cid:18) − a b a b − b a b (cid:19) = M ± ( b, − a b ) . (45)Note that there is an ambiguity for the overall sign of dyon charge associated to M c , because Eqn. (41) is invariant under ( g, q ) → ( − g, − q ).For fun, we can also examine M d and attempt (and fail) to interpret it in termsof dyon charges. From Eqn. (40), monodromy matrix for γ d is given as M d = M a M b = (cid:18) − a b a − b (cid:19) (46)whose trace matches that of Eqn. (41) only for ab = 0. Therefore M d cannot beinterpreted as a singularity of a single vanishing 1-cycle for ab (cid:54) = 0.We can interpret the result of Eqn. (45) and Fig. 18 as following:(1) The dyon charge of a singular point u b depends on choice of trajectory in themoduli slice.(2) If the trajectory passes through another singular point u a (but not surroundingit by a closed loop), then dyon charge of u b gets shifted by multiple of dyoncharge of u a .(3) If we chose the first sign in Eqn. (45), then this relation becomes( b, → ( b, − [( b, ◦ (0 , a )] (0 , a ) = ( b, − a b ) , (47)with a close agreement with Eqn. (8).We can now write down general formula for how dyon charge gets changed. In Fig.18, for each trajectories γ a , γ b , γ c , let us say vanishing 1-cycles are ν a , ν b , ν c . Thentheir relationship is given as similar to Eqn. (47) as ν c = ν b − ( ν b ◦ ν a ) ν a . (48)Going back to the question of spectra jump for Sp (4) related to Fig. 17, nowwe can use the techniques we learned above, especially that of Eqn. (48). First, byctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories inspecting Fig. 17 and Fig. 18, we can plug in ν a = ν Q , ν b = ν Q (cid:48) , ν c = ν Q (cid:48) e intoEqn. (48) to obtain ν Q (cid:48) e = ν Q (cid:48) − ( ν Q (cid:48) ◦ ν Q ) ν Q = ν Q (cid:48) − ν Q = ν Q , (49)where the last equality was given from Eqn. (39). If somehow we can argue that γ Q (cid:48) e of Fig. 17 is more natural choice of trajectory in moduli space than γ Q (cid:48) of Fig. 16,then we can undo the spectra jump as we move from (c) to (e) in Fig. 15. It maysuggest that we can avoid the spectra jump (in dyon charges of massless states) ifwe choose a different path on moduli slice. However for the moment we cannot sayconclusively what will be the most natural and consistent choice of trajectory paths.It will be interesting to check this by focussing on massless sector of wall-crossingformulas of BPS spectra given in Refs. 36, 37, 32. Maximal Argyres-Douglas theories and dual Coxeter number
So far we considered Argyres-Douglas theories with two massless dyons. Here, wewill consider maximal
Argyres-Douglas theories, where maximal number of dyonsbecome massless and are mutually non-local. This is achieved by bringing the maxi-mal number of branch points on x -plane, so that f ( x ) will have a root with maximaldegeneracy. Let us recall the SW curve for SU ( r + 1) gauge group given Eqn. (14)and Eqn. (15). y = f SU ( r +1) = f + f − , f ± ≡ x r +1 + r (cid:88) i =1 u i x r − i ± Λ r +1 . (50)If we have u r = ∓ Λ r +1 while all other u i ’s vanish, then f ± = x r +1 holds and f ( x ) has a root x = 0 with maximal degeneracy.
2, 16
It is straightforward to find2 maximal Argyres-Douglas points in moduli space of pure SU ( r + 1) and SO (2 r )SW theories.Taking no-flavor limit of Ref. 27, we have SW curve for pure SO (2 r ) as y = C SO (2 r ) − Λ r − x = C SO (2 r ) , + C SO (2 r ) , − (51)with C SO (2 r ) , ± = C SO (2 r ) ± Λ (2 r − x ,C SO (2 r ) ( x ) ≡ x r + s x r − + · · · + s r − x + ˜ s r , (52)in agreement with Ref. 26. Some of the monodromy properties for pure SO (2 r ) werestudied in Ref. 38 with emphasis on the SO (8) example.Maximal Argyres-Douglas points for the SO (2 r ) will be two moduli points givenby s r − = Λ ± (2 r − , s i = 0 , i (cid:54) = r − C SO (2 r ) , ∓ = x r to have a root with maximal order of vanishing. Justas in the SU ( r + 1) case, this computation is straightforward, partially thanks to Z ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO symmetric structure between C SO (2 r ) , ± and between f ± in the SW curve, which islacking in the Sp (2 r ) and SO (2 r + 1) cases. Scaling behavior at maximal Argyres-Douglas points for SU ( r + 1) and SO (2 r ) SW theory was studied in Ref. 3 recently,with focus on the shape of moduli space in the neighborhood of those theories. Morespecifically, the quantum Higgs branch appears p .Locating maximal Argyres-Douglas (AD) points for Sp (2 r ) and SO (2 r + 1)SYM involves more algebra, and the number of maximal Argyres-Douglas pointsequals to the dual Coxeter number of the gauge group. For pure Sp (2 r ) theory, Ref.28 proposes r + 1 candidates for maximal Argyres-Douglas theories, while Ref. 24proposes 2 r − SO (2 r + 1)theory.Recall, from Eqn. (19) and Eqn. (20), the SW curve for Sp (2 r ) SYM is y = f Sp (2 r ) = f C f Q , f C = x r + r (cid:88) i =1 u i x r − i , f Q ≡ xf C + 16Λ r +2 . (54)Maximal AD points occur when we bring all roots of f Q = 0 together. This happensat r + 1 points in moduli space, where the curve develops A r singularity. Thisoccurs when the moduli take the following values: u i = (cid:18) r + 1 i (cid:19) ( − Q ) i , (55)with Q given by Q = − exp (cid:18) πir + 1 k (cid:19) (16) r +1 Λ , k ∈ Z . (56)This forces the f Q to have a root with maximal degeneracy as f Q = ( x − Q ) r +1 . (57)It also follows that C i ’s, the roots of f C are given as below: { C i } = (cid:26) Q (cid:18) − exp (cid:18) πir + 1 k (cid:19)(cid:19)(cid:27) , k ∈ Z , k / ∈ ( r + 1) Z . (58)Given that Eqn. (56) allowed for r + 1 different choices of phase for Q , we have Z r +1 symmetric r + 1 points on the moduli space, where maximal Argyres-Douglassingularity occurs.Similarly, 2 r − SO (2 r + 1) SYM werelocated in Ref. 24. The curves for SO (2 r + 1) and SO (2 r ) SYM are almost similar,but SO (2 r +1) case is much harder to solve for the maximal Argyres-Douglas points.We again take no-flavor limit of the curve of Ref. 27 to obtain the curve for pureSW theory,
26, 40 y = f SO (2 r +1) ( x ) = C SO (2 r +1) − Λ r − x = C SO (2 r +1) , + C SO (2 r +1) , − (59) p Some of the best place to learn about Higgs branch are Refs. 22, 23. ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4
Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories with C SO (2 r +1) , ± = C SO (2 r +1) ± Λ (2 r − x,C SO (2 r +1) ( x ) ≡ x r + s x r − + · · · + s r − x + s r . (60)Two polynomials C SO (2 r +1) , ± will share a root if x = 0. This, however, doesnot give much mileage for bringing maximal number of branch points together for f SO (2 r +1) ( x ). This root does not have high enough order of vanishing. Instead wewill work on having C SO (2 r +1) , + (or equivalently C SO (2 r +1) , − ) to have a root withmaximal order of vanishing. Since we have only r degrees of freedom, we can onlybring together r + 1 branch points, and the maximal order of vanishing is r + 1.The best we can do is to bring C SO (2 r +1) , + into the following form: C SO (2 r +1) , + = ( x + b ) r +1 ( x r − + u x r − + u x r − + · · · + u r − x + u r − ) (61)when the moduli s i ’s satisfy s k = ( − b ) k (2 r − r − k − r ! k !( r − k )! , (62)with b given by b = ω k r − (cid:20) ( − r +1 (2 r − r )!! (cid:21) / (2 r − Λ , k ∈ Z . (63)Here ω m is m ’th root of unity. Eqn. (63) allows 2 r − b . Inturn there are 2 r − r − Z r − symmetric among themselves, in the moduli space of pure SO (2 r + 1) SYM where maximal Argyres-Douglas theory occurs.
6. D´ej`a Vu: Singularity Tools: Exterior Derivative & DoubleDiscriminant
So far we encountered a few tools to detect singularity of the SW curves. In subsec-tion 2.2, we discussed various tools for singularity search, namely exterior derivative d in ambient space, and discriminant. In Eqn. (36), we saw that considering doublediscriminant ∆ u ∆ x f ( x ) gives candidates for interesting singularities with 2 masslessdyons. Now let us write down general rules in a systematic way.We detected singularity of a Riemann surface embedded in the ambient spacewhose coordinates are x, y by taking an exterior derivative, or by demanding all thepartial derivatives with respect to x and y to vanish. Similarly, when an algebraicvariety in moduli space forms a singularity, it is captured by demanding the exteriorderivative to vanish inside the moduli space. Recall that the vanishing discriminantloci of the SW curve form a complex codimension 1 algebraic variety inside themoduli space, which is given by ∆ x f = 0. Therefore it follows that d ∆ x f = 0(where d is taken inside the moduli space) will pinpoint us to where ∆ x f = 0 formsa singularity, namely where discriminant loci intersect themselves. That is wherewe have multiple vanishing 1-cycles (massless BPS dyons).ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
As explained in subsection 2.2, the exterior derivative d can be written interms of the partial derivatives with respect to all the coordinates of the ambi-ent space. Therefore the exterior derivative inside the moduli space is given as d = (cid:80) ri =1 du i ∂∂u i . One might think that demanding the d = 0 actually reduces r degrees of freedom, since we demand all the r partial derivatives to vanish. However,∆ x f = d ∆ x f = 0 indeed contains codimension 2 solutions (instead of codimension r + 1).When the ∆ x f = 0 loci become singular (where ∆ x f = d ∆ x f = 0 holds), doublediscriminant also vanishes (∆ u ∆ x f ( x ) = 0). In other words, singularity is seen fromthe viewpoint of moduli slices (parallel u -planes) as well. However the converse doesnot hold. Something might appear singular on certain moduli slices but it can besmooth in the full ambient space (moduli space). In other words, ∆ x f = ∆ u ∆ x f = 0is a necessary but not sufficient condition for having ∆ x f = d ∆ x f = 0. Exterior derivative detects coexistence of multiple masslessBPS dyons
The vanishing discriminant condition of the SW curve ∆ x f = 0 defines an algebraicvariety Σ. Since ∆ x f is written only in terms of moduli u i ’s (without x and y ), Σis an algebraic variety embedded inside the moduli space, denoting moduli loci ofmassless BPS states. When this algebraic variety Σ self-intersects, two or more BPSstates become massless, which occurs when we demand ∆ x f = d ∆ x f = 0.An heuristic example is depicted in Fig. 19. Inside a moduli subspace, we drawa figure-eight-like object, which is analogous to ∆ x f = 0 loci as in Fig. 15. We markvarious u -planes with (a) to (e). The number of singular points changed on each u -planes. When the singular points collide on the u -plane we have ∆ u ∆ x f = 0, forexample on slices (b) and (d). Slice (b) is true singularity, while (d) is not. Thuswe see that ∆ x f = ∆ u ∆ x f = 0 is a necessary but not sufficient condition to have∆ x f = d ∆ x f = 0.Fig. 15 and Fig. 19 look very similar to each other in that it displays the singu-larity structure inside moduli space. Both shows the collision of discriminant lociand formation of higher singularity - (b) and (d) of Fig. 15 and (b) of Fig. 19. Themain difference is this: Fig. 19 contains an example where ∆ x f = ∆ u ∆ x f = 0 holdsbut d ∆ x f (cid:54) = 0 on its part (d). Factorization of double discriminant, and order of vanishing
Massless dyons coexist at complex-codimension-2 loci where both discriminant andits exterior derivative vanish. There the curve looks like either of following two: Cusp
The curve y = ( x − a ) ×· · · has a cusp-like singularity. Vanishing discriminant∆ x f = 0 locus also intersects with cusp-like singularity in moduli space. There∆ u ∆ x f = 0 also holds, with order of vanishing 3. Two massless dyons aremutually non-local.ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories
Node
The curve y = ( x − a ) ( x − b ) × · · · has a node-like singularity. Vanishingdiscriminant ∆ x f = 0 locus also intersects with node-like singularity. There∆ u ∆ x f = 0 also holds, with order of vanishing 2. Two massless dyons aremutually local.Order of vanishing of each root of ∆ u ∆ x f tells us the type of singularities. Inorder to justify that, we discuss roots of ∆ u ∆ x f = 0. When ∆ x f = 0 and d ∆ x f = 0hold, each root of vanishing double discriminant ∆ u ∆ x f = 0 corresponds to twomassless dyons with appropriate combinatoric meaning.ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Double discriminants of SU ( r + 1) and Sp (2 r ) factorize as: ∆ u ∆ x f SU ( r +1) = ∆ u ∆ x ( f + f − ) = (cid:16) v (2 r +2) + · · · (cid:17) = (cid:16) v ( r +1) + · · · (cid:17) (cid:0) v r +1 + · · · (cid:1) (cid:0) v r +1 + · · · (cid:1) × (cid:16) v ( r +1)( r − / + · · · (cid:17) (cid:16) v ( r +1)( r − / + · · · (cid:17) ≡ P N II ) ( N III ) ( P III ) ( N II ) ( P II ) , (64)∆ u ∆ x f Sp (2 r ) = ∆ u ∆ x ( f Q f C ) = (cid:16) v (2 r +1) + · · · (cid:17) = (cid:16) v r ( r +1) + · · · (cid:17) ( v r + · · · ) (cid:0) v r +1 + · · · (cid:1) × (cid:16) v r ( r − / + · · · (cid:17) (cid:16) v ( r +1)( r − / + · · · (cid:17) ≡ QC II ) ( C III ) ( Q III ) ( C II ) ( Q II ) , (65)where u ≡ u , v ≡ u denote the two moduli among r complex moduli. The sub-scripts II and III denote the order of vanishing - each corresponding to node andcusp like singularity.
Case I:
As an example, the first factor in second line of Eqn. (64) is(
P N II ) ≡ (cid:16) v ( r +1) + · · · (cid:17) . (66)This corresponds to having two pairs of branch points on the x -plane collide eachother pairwise, where each pair is P i type and N i type respectively. The curvedegenerates into a node-like singularity y = ( x − P i ) ( x − N j ) × · · · . (67)Number of choices for choosing one pair of P i ’s and N i ’s is given as: (cid:18) r + 11 (cid:19) = ( r + 1) , (68)which is exactly the power of v in (66). Case II:
The first factor in the third line of Eqn. (64) is( N II ) ≡ (cid:16) v ( r +1)( r − / + · · · (cid:17) , (69)and this corresponds to the scenario where two pairs of N -type branch points onthe x -plane collide each other pairwise. The curve degenerates into a node-likesingularity y = ( x − N i ) ( x − N j ) × · · · . (70) Case III:
The second factor in the second line of Eqn. (64)( N III ) ≡ (cid:0) v r +1 + · · · (cid:1) (71)ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 Singularity Structure of N = 2 Supersymmetric Yang-Mills Theories is related to having three N -type branch points on the x -plane collide all together.The curve degenerates into a cusp-like singularity y = ( x − C i ) × · · · . (72)Similarly, we can understand other factors of Eqn. (64) and Eqn. (65).
7. Conclusion
In this review, we observed the relevance and importance of supersymmetric Yang-Mills theories. We discussed N = 2 SYM with classical gauge groups, with particularattention to their singularity structure associated with massless states. We trans-lated the physics questions into the language of Seiberg-Witten geometry, so that ahyperelliptic curve equipped with a 1-form encodes physical information of SYM.Physics ↔ Geometrysupersymmetric Yang-Mills theory ↔ Hyperelliptic curverank of gauge group ↔ genusparticle ↔ ↔ (cid:12)(cid:12)(cid:72) λ (cid:12)(cid:12) massless states ↔ singularity(vanishing 1-cycle at ∆ x f = 0)massless ↔ worse singularity e − & magnetic monopole such as cusp (73)With this dictionary in mind, we examined singularity loci of families of hy-perelliptic curves which are associated with pure SW theories. At discriminant loci∆ x f = 0 of the SW curve, we have vanishing 1-cycles. We identified BPS dyoncharges of all the 2 r + 1 and 2( r + 1) vanishing cycles respectively for pure Sp (2 r )and SU ( r + 1) SW curves.When discriminant loci form a singularity inside the moduli space ( d ∆ x f =0), multiple massless dyons coexist. Here the ‘double discriminant’ also vanishes(∆ u ∆ x f = 0). Note however that the converse does not hold. If order of vanishingof roots to the double discriminant is high (equal to 3), then vanishing 1-cyclescoexist and intersect: we are at the Argyres-Douglas loci.On top of many open questions proposed in Ref. 24, it will be interesting tostudy the behavior of the SW curve near the Argyres-Douglas loci, extending theworks of Ref. 3, which discusses appearance of quantum Higgs branch at maximalArgyres-Douglas points of SW theories. It will be also interesting to understandthe results reviewed here in the context of wall-crossing and quiver mutation inRefs. 36, 37, 32. Finally, the analysis of monodromy may benefit from making moreconnection to the braiding procedure in knot theory.ctober 5, 2018 16:55 WSPC/INSTRUCTION FILE jihyeSEOjuly4 J. S. SEO
Acknowledgments
It is a great pleasure to thank Murad Alim, Philip Argyres, Heng-Yu Chen, KeshavDasgupta, Hoyun Jung, Dong Uk Lee, Andy Neitzke, Jihun Park, Alfred Shapere,Yuji Tachikawa, Donggeon Yhee, and Philsang Yoo for helpful discussions. LongChen, Pedro Liendo, Chan-Youn Park, and especially Philsang Yoo gave invalu-able feedback on the manuscript. The author benefited from encouragement fromHoward, Mini, Sun-young Park, and Daniel Tsai. This review is dedicated to thememory of Arthur G.The work is supported in part by NSERC grants. All the figures are created bythe author using the program Inkscape with TeXtext.
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