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
Monopole
+ 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
Leçon 5 : MARCHES DE MONOPOLE
III - Monopole discriminant à plusieurs marchés 1 - Mécanismes de fonctionnement Dans l’objectif d’exploiter pleinement les opportunités du marché, le monopole peut appliquer une politique de discrimination, qui consiste à vendre le même bien à des prix différents, selon les différentes clientèles
1 MICROECONOMIE
Dossier révision oral économie 3/4 Microéconomie – le marché en monopole 2 L'équilibre d'un marché en monopole discriminant a Exercice « modèle » Sur son territoire, une entreprise détient le monopole de la production d’un bien Q pour lequel la demande est de la forme : Q = -P/5 + 24 1
Chapitre I Le monopole
A Recette totale du monopole Lorsque l’élasticité-prix de la demande est supérieure à1en valeur absolue, une hausse de la quantitévendue se traduit par une baisse du prix sur le marché mais cette diminution est plus que compensée, en termes relatifs, par l’accroissement relatif des ventes La recette totale du monopole s’accroît
Partie 2 : l’intervention de l’Etat
Monopole le monopole est la situation où il existe un seul offreur sur le marché face à plusieurs demandeurs Ex : le marché de l’eau potable, de phosphate et le raffinage du pétrole Le monopole simple : le monopoleur fixe un prix unique ONEP Le monopole discriminant : le monopoleur offre le même service à des prix différents
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
[PDF] exercice corrigé sur l'equilibre de monopole
[PDF] rapport de stage fle master 1
[PDF] monopole discriminant et pouvoir de marché
[PDF] maximisation du profit définition
[PDF] les temps du récit tronc commun
[PDF] rapport de stage enseignement français algerie
[PDF] les temps du récit et leurs valeurs
[PDF] rapport de stage fle m1
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[PDF] fiche théorème de pythagore
[PDF] rapport de stage enseignement primaire algerie
[PDF] rapport de stage enseignement du français
[PDF] rapport de stage tisec pdf
[PDF] versification fiche de revision
IntroductionA
r,Crcurves uxfConclusionSingularity 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 uxfConclusionIntroduction
Seiberg-Witten and Argyres-Douglas theories
I Seiberg-Witten curves, massless dyons (low ranks), and some aspects of singularity in moduli space were studied forISU(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 Sundborg95], SO with matter[Hanany 95]
I The hyper-elliptic curvey2=f(x)degenerates into a cusp form y2= (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 purelyelectronic.)Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)2 / 23
IntroductionA
r,Crcurves uxfConclusionIntroduction
Main idea
Want to build Gaiotto'sN= 2theory (see Yuji's talk) in F theory IN= 2Sp(2r)SW theories viarD3-branes
I compute massless dyon charges for pureSU(r+ 1);Sp(2r) I study wall crossing Iprovide 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 curvesDiscriminant
x: discriminantwith respect tox f n(x) =nX i=1a ixi=ann Y i=1(xei) x(fn(x)) =a2n2nY iIntroductionA
r,Crcurves uxfConclusion Math review: discriminantsx;uand singularity of curvesRecipe for locating Argyres-Douglas loci
IStart with hyper-elliptic Seiberg-Witten curve
y2=f(x;u;v;)
IDemandingxf= 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 upto 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 A1=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?
y2= (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 eachother. 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 A1=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 0123X 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.IPutrD3-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 uxfConclusionSeiberg-Witten curves for pureArandCr
SW curve for pureSU(r+ 1) =Ar
From[Klemm Lerche Yankielowicz Theisen 94]
y2=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 uxfConclusionSeiberg-Witten curves for pureArandCr
SW curve for pureSp(2r) =Cr
By taking no-
avor limit of[Argyres Shapere 95], obtain y2=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 fQfCx+ 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 uxfConclusionIdentify 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 u1==ur2= 0
u r=r+12I; monodromy satisesPi\Pi+1=Ni\Ni+1= 1
Pi\Ni=2
Pi\Ni+1= 2
all other intersection numbersvanish.Jihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)11 / 23
IntroductionA
r,Crcurves uxfConclusionIdentify massless dyons ofArandCrcurves atxf= 0
Pi=i+1ii
Ni=i+1i+i+12i
i= 1;;r1 P0=1Pr=r+r1X
i=1 iN0=1+rX
i=1 i+1Nr=r2rJihye SeoMcGill UniversitySingularity structure and massless dyons of pureN= 2; d= 4theories withSU(r+ 1)andSp(2r)12 / 23
IntroductionA
r,Crcurves uxfConclusionIdentify massless dyons ofArandCrcurves atxf= 0
Sp(2r)monodromyIn a moduli region byu2==ur1= 0andur= const: choosesmall 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 uxfConclusionIdentify massless dyons ofArandCrcurves atxf= 0
With the same choice of symplectic basis asSU(r+ 1)before, the vanishing cycles are written as Q 0=1Qr=rrX
i=1 irCr=1rrX
i=2 i Q i=i+1iiCi=i+1i+i+1
i= 1;;r1whose non-vanishing intersection numbers come from only Q i\Q i+1= 1; Qr\Q0=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 atuxf= 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 0andQ2vanish at two dierent moduli lociuQ
0and u Q2respectively.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 (whereuQ0anduQ
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 uxfConclusionExample: Singularity structure of Sp(4)=C2NowuQ
0anduQ
20are separated, butuQ
0runs toward another
singularity locusuC2. (Q 2 0=Q 0+Q2; 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 lociuQ0anduC2intersect (node-like crossing).
Vanishing cyclesQ
0andC2seem mutually non-local, but the
curve looks likey2= (xa)2(xb)2 and it is notArgyres-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 uxfConclusionExample: 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 singularityIxf= 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 singularityGeneralized 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 uxfConclusionConclusion and Future Directions
Conclusion
I Discriminantxf= 0of the SW curvey2=f: identied all2r+ 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...
IBehaviour in the Argyres-Douglas neighborhood?
I Rankrcurve classication as rank 2 of[Argyres CrescimannoShapere 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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