[PDF] Singularity structure and massless dyons of pure N=2, d=4



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3- LE MONOPOLE DISCRIMINANT

Le monopole discriminant parfaitement produit donc la même quantité que des entreprises en concurrence parfaite ayant le mêmem coût marginal agrégé La différence fondamentale est qu’en concurrence parfaite, il n’y a qu’un seul prix d’équilibre : toutes les « unités » produites sont vendues au même prix Le monopole discriminant



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+ Un monopole discriminant fera les m^emes choix, avec des prix p h = v h et p l = v l 27/59 Sans discrimination, le monopole fera payer soit v l soit v h Il fera payer v h si v h c 1 >2v l c 2 ()c 2 c 1 >2v l v h: En particulier si v est \tr es grand", mais v l >c c 1, il y aura production d’une seule unit e alors qu’il aurait et e e



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Singularity structure and massless dyons of pure N=2, d=4

Discriminant x: discriminant with respect to x f n(x) = Xn i=1 a ix i = a n Yn i=1 (x e i) x (f n(x)) = a2n 2 n Y i

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IntroductionA

r,Crcurves uxfConclusion

Singularity structure and massless dyons of pure

N= 2;d= 4theories withArandCr

Jihye Seo

Disclaimer: all gures are mouse-drawn, not to scale.

Based on works w/ Keshav Dasgupta

Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)1 / 23

IntroductionA

r,Crcurves uxfConclusion

Introduction

Seiberg-Witten and Argyres-Douglas theories

I Seiberg-Witten curves, massless dyons (low ranks), and some aspects of singularity in moduli space were studied for

ISU(2) w/o and w/ matter[Seiberg Witten 94]

ISU(n)[Klemm Lerche Yankielowicz Theisen 94, Argyres Faraggi 94],

SU(n) w/o and w/ matter[Hanany Oz 95]

ISO(2r)[Brandhuber-Landsteiner 95]SO(2r+1)[Danielsson Sundborg

95], SO with matter[Hanany 95]

I The hyper-elliptic curvey2=f(x)degenerates into a cusp form y

2= (xa)m

whenm3branch points collide onx-plane. For SU(n) SW curves, this give exotic theories, discovered and studied[Argyres Douglas 95], where we havemutually non-localmassless dyons. (Under symplectic transformation, you cannot bring everything purely

electronic.)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)2 / 23

IntroductionA

r,Crcurves uxfConclusion

Introduction

Main idea

Want to build Gaiotto'sN= 2theory (see Yuji's talk) in F theory I

N= 2Sp(2r)SW theories viarD3-branes

I compute massless dyon charges for pureSU(r+ 1);Sp(2r) I study wall crossing I

provide plethora of Argyres-Douglas (AD) theoriesphysics$geometrymassless d.o.f.$singularityxf= 0exotic theories (i. e. AD)$uxf= 0Double discriminant captures higher singularity of hyper-elliptic

curves.Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)3 / 23

IntroductionA

r,Crcurves uxfConclusion Math review: discriminantsx;uand singularity of curves

Discriminant

x: discriminantwith respect tox f n(x) =nX i=1a ixi=ann Y i=1(xei) x(fn(x)) =a2n2nY iOrderofvanishing (multiplicity of roots)2$x= 0Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)4 / 23

IntroductionA

r,Crcurves uxfConclusion Math review: discriminantsx;uand singularity of curves

Recipe for locating Argyres-Douglas loci

I

Start with hyper-elliptic Seiberg-Witten curve

y

2=f(x;u;v;)

I

Demandingxf= 0anduxf= 0gives two massless

BPS dyons.[Argyres Plesser Seiberg Witten '95]

I Checkorderofvanishing (o.o.v.) of each solution to uxf= 0 I If o.o.v.3, Argyres-Douglas loci: The hyperelliptic curve degenerates into a cusp-like singularityy2= (xa)3 and two mutually non-local dyons become massless.(checked up

to rank 5)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)5 / 23

IntroductionA

r,Crcurves uxfConclusion A

1=C1: original SW curve and massless monopole/dyon, F theory picture of D3/O7

Review of monodromy ofSU(2) =Sp(2)SW curve

How did they get massless monopole and dyon?

y

2= (x2u)24Start at a generic point in moduli space, then branch points on

x-plane are separated. As you move around a singular locus in u-plane, a pair of branch points approach & change with each

other. Their trajectory gives vanishing cycle.Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)6 / 23

IntroductionA

r,Crcurves uxfConclusion A

1=C1: original SW curve and massless monopole/dyon, F theory picture of D3/O7

F-theoretic (quantum) 3/7 brane picture ofSp(2)

[Sen '96] [Banks-Douglas-Seiberg '96]&[Vafa 96]X6789 X 0123
X 45

D3-braneHiggs branch

Coulomb branch

(0, 1) seven-brane (1, -1) seven-braneTwo (p;q) 7-branes are monopole and dyon in the original SW theory.I

PutrD3-branes as probes to getSp(2r)gauge theory.

[Douglas Lowe Schwarz 96]Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)7 / 23

IntroductionA

r,Crcurves uxfConclusion

Seiberg-Witten curves for pureArandCr

SW curve for pureSU(r+ 1) =Ar

From[Klemm Lerche Yankielowicz Theisen 94]

y

2=fSU(r+1)xr+1+u1xr1+u2xr2++ur

22r+2
=f+f f xr+1+u1xr1+u2xr2++urr+1 f +rY i=0(xPi)frY i=0(xNi)

Note thatfdo not share roots (f+f= 2r+16= 0).

On the x-plane, onlyPi's (orNi's) can collide among themselves.

!Discriminant factorizesxfSU(r+1)= #(xf+)(xf)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)8 / 23

IntroductionA

r,Crcurves uxfConclusion

Seiberg-Witten curves for pureArandCr

SW curve for pureSp(2r) =Cr

By taking no-

avor limit of[Argyres Shapere 95], obtain y

2=fSp(2r)

rY a=1 x2a! xrY a=1 x2a+ 2r+2! =fCfQ f CrY a=1 x2a=rY i=1(xCi) =xr+u1xr1+u2xr2++ur f

QfCx+ 2r+2=rY

i=0(xQi)

Similarly as in SU(r+1) curve,xfSp(2r)= #(xfC)(xfQ)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)9 / 23

IntroductionA

r,Crcurves uxfConclusion

Identify massless dyons ofArandCrcurves atxf= 0

Vanishing cycles of SW curve for pure SU(r+1)

SU(3) [Klemm Lerche Y ankielowiczTheisen 94]

Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)10 / 23

IntroductionA

r,Crcurves uxfConclusion Identify massless dyons ofArandCrcurves atxf= 0ForSU(r+ 1), in a region of moduli space given by u

1==ur2= 0

u r=r+12I; monodromy satises

Pi\Pi+1=Ni\Ni+1= 1

Pi\Ni=2

Pi\Ni+1= 2

all other intersection numbers

vanish.Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)11 / 23

IntroductionA

r,Crcurves uxfConclusion

Identify massless dyons ofArandCrcurves atxf= 0

Pi=i+1ii

Ni=i+1i+i+12i

i= 1;;r1 P0=1

Pr=r+r1X

i=1 i

N0=1+rX

i=1 i+1

Nr=r2rJihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)12 / 23

IntroductionA

r,Crcurves uxfConclusion

Identify massless dyons ofArandCrcurves atxf= 0

Sp(2r)monodromyIn a moduli region byu2==ur1= 0andur= const: choose

small enoughurto keepQ\C= 0(checked up to rank 5).Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)13 / 23

IntroductionA

r,Crcurves uxfConclusion

Identify massless dyons ofArandCrcurves atxf= 0

With the same choice of symplectic basis asSU(r+ 1)before, the vanishing cycles are written as Q 0=1

Qr=rrX

i=1 ir

Cr=1rrX

i=2 i Q i=i+1ii

Ci=i+1i+i+1

i= 1;;r1whose non-vanishing intersection numbers come from only Q i\Q i+1= 1; Qr\Q

0=Ci\Ci+1=1

The spectra change as we move in moduli space. (C2example)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)14 / 23

IntroductionA

r,Crcurves uxfConclusion Example: Singularity structure of Sp(4)=C2Example:

Sp(4) =C2

xf= 0at which intersect at

uxf= 0Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)15 / 23

IntroductionA

r,Crcurves uxfConclusion Example: Singularity structure of Sp(4)=C2Two cyclesQ 0andQ

2vanish at two dierent moduli lociuQ

0and u Q

2respectively.Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)16 / 23

IntroductionA

r,Crcurves uxfConclusion Example: Singularity structure of Sp(4)=C2As we vary in a moduli space (whereuQ

0anduQ

2intersect

tangentially), we hit a locus where they vanish simultaneously. The curve degenerates intoy2(xa)3 . (Argyres-Douglas)

The red curve on the right does not give a well dened cycle.Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)17 / 23

IntroductionA

r,Crcurves uxfConclusion

Example: Singularity structure of Sp(4)=C2NowuQ

0anduQ

2

0are separated, butuQ

0runs toward another

singularity locusuC2. (Q 2 0=Q 0+Q

2; Not related by symplectic transformation)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)18 / 23

IntroductionA

r,Crcurves uxfConclusion Example: Singularity structure of Sp(4)=C2This time, singularity lociuQ

0anduC2intersect (node-like crossing).

Vanishing cyclesQ

0andC2seem mutually non-local, but the

curve looks likey2= (xa)2(xb)2 and it is not

Argyres-Douglas form. ()generalized AD)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)19 / 23

IntroductionA

r,Crcurves uxfConclusion

Example: Singularity structure of Sp(4)=C2NowuQ

0anduC2are separated, but the vanishing cycles did not

change, unlike (b!c)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)20 / 23

IntroductionA

r,Crcurves uxfConclusion uxfcaptures higher singularity Massless dyons coexist at a codim-2Clocifxf= uxf= 0g [Argyres Plesser Seiberg Witten '95], where the curve looks like either of following two: I y2= (xa)3 I curve has cusp -like singularity (Argyres-Douglas)

Ixf= 0locus also intersects atuxf= 0(o.o.v3) with

cusp -like singularity (tangential intersection)

Itwo massless dyons are mutually non-local.

I y2= (xa)2(xb)2 I curve has no de -like singularity

Ixf= 0locus also intersects atuxf= 0(o.o.v2) with

node -like singularity Itwo massless dyons are mutually local, with some exception (generalized AD: non-Argyres-Douglas & non-local loci)

occurring atSU(5");Sp(4")Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)21 / 23

IntroductionA

r,Crcurves uxfConclusion uxfcaptures higher singularity

Generalized Argyres-Douglas locicurve degen.ny

2= (xa)3 y

2= (xa)2(xb)2o.o.v ofux32

shape of curvecuspnode shape ofx= 0cuspnode intersectionmutually non locallocalnon-local nameArgyres-DouglasMLgen AD I generalized Argyres-Douglas: non-Argyres-Douglas &

non-local loci (9conformal limit?)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)22 / 23

IntroductionA

r,Crcurves uxfConclusion

Conclusion and Future Directions

Conclusion

I Discriminantxf= 0of the SW curvey2=f: identied all

2r+ 1and2(r+ 1)massless dyons forSp(2r)andSU(r+ 1)

I At double discriminantuxf= 0: massless dyons coexist. If order of vanishing3, then Argyres-Douglas. (checked up to rank 5)

Questions still remain...

I

Behaviour in the Argyres-Douglas neighborhood?

I Rankrcurve classication as rank 2 of[Argyres Crescimanno

Shapere Wittig '05]

[Argyres Wittig '05] I Global behaviour in moduli space: wall crossing[ShapereVafa99,

GaiottoMooreNeitzke09]Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)23 / 23

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