Confinement and Duality in Supersymmetric Gauge Theories

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Scuola Normale Superiore

Classe di Scienze

Corso di Perfezionamento in Fisica

Confinement and duality in

supersymmetric gauge

theories

Tesi di Perfezionamento

Simone

GIACOMELLI

Relat or e:

_{Prof. Kenichi KONISHI}

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Acknowledgements

I would like t o t hank first of all my advisor K enichi Konishi for all his support during t hese years, st art ing from my undergraduat e course, and for having int roduced me t o t his beaut iful field of research. He const ant ly encouraged me and act ively cont ribut ed in t he development of t his project . He will always be of great inspirat ion for me and I really owe him a lot .

T his t hesis includes most of my work as a graduat e st udent at Scuola Nor-male. It is a pleasure for me t o acknowledge t his beaut iful inst it ut ion t hat gave me so much during t he past eight years, providing a st imulat ing envi-ronment in which my passion for science has grown const ant ly. I would also like t o t hank professor August o Sagnot t i for many precious comment s and suggest ions.

I am grat eful t o all my collaborat ors and t o various colleagues, in part icular Alessandro Tanzini, Giulio Bonelli, Michele Del Zot t o and Sergio Cecot t i. I really benefit ed from many illuminat ing discussions wit h t hem. I would also like t o t hank Yuji Tachikawa for many helpful comment s on my work. I acknowledge t he Cent er for t he Fundament al Laws of Nat ure of Har-vard University, T he Scuola Int ernazionale Superiore di St udi Avanzat i and IPMU, where part of t his work was done, for hospit ality.

I would also like t o t hank my family and my friends for all t heir support and for encouraging me. A special t hank goes t o my girlfriend Anna.

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Introduction

Gauge symmet ry is ubiquit ous in physics and is one of t he key ingredient s underlying t he dynamics of element ary part icles. T he St andard Model, t he t heory which unifies t he present -day part icle physics is described by a la-grangian invariant under SU(3)_{×SU(2)×U(1) local transformations, where} SU(2)_{× U(1) is the gauge group associated to electroweak forces and SU(3)} is t he gauge group associat ed t o st rong int eract ions (t he color group)

From a t heoret ical point of view, t he model for st rong int eract ions (QCD) remains very challenging, despit e t he fact t hat it has been formulat ed al-most fourty years ago. One of t he al-most remarkable propert ies of t he model is asympt ot ic freedom: t he t heory is weakly coupled at very high energy, making it t ract able wit h st andard pert urbat ive t echniques in t hat regime, whereas it becomes st rongly int eract ing at low energies. Since t here is no general t echnique t hat allows t o approach a st rongly coupled field t heory, it has so far been impossible t o follow analyt ically t he flow from high t o low energies (RG flow) and underst and in a precise way t he propert ies of t he t heory in t he st rongly coupled regime.

Bot h from an experiment al and t heoret ical (and numerical) point of view, t here are st rong indicat ions t hat QCD in t he infrared limit is charact erized by confinement and chiral symmet ry breaking and t hat t hese two phenomena are deeply relat ed, being originat ed by t he same mechanism.

Confinement means t hat t he element ary fields in QCD charged under t he color group (t he quarks) t end t o form bound st at es at low energies since t heir mut ual int eract ion becomes st ronger and st ronger and t hese bound st at es (hadrons) become t he effect ive dynamical variables in t he infrared. Chiral invariance refers inst ead t o t he symmet ry t hat QCD acquires in t he limit of massless quarks. T his is a very reasonable approximat ion as long as just t he two light est quarks are considered. Furt hermore, observat ions suggest t hat t his approximat e symmet ry of nat ure is spont aneously broken by a non vanishing condensat eh ¯ i 6= 0. Neither of these properties can be explained in t he framework of st andard pert urbat ion t heory and t hus must be relat ed t o some st ill unknown nonpert urbat ive effect . Underst anding precisely t he infrared dynamics of QCD is t hus one of t he most challenging open problems in field t heory.

Due t o t he complexity of t he t heory of st rong int eract ions it is import ant 1

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t o approach t he problem st art ing from simpler models t hat can help under-st anding t he main feat ures and t he mechanism underlying confinement and symmet ry breaking. In t his respect supersymmet ric non-abelian gauge t he-ories play a leading role and have been for decades a source of ideas for t he explorat ion of t he st rongly-coupled gauge dynamics.

T here is a dist inguished subclass of supersymmet ric gauge t heories, namely t hose wit h ext ended supersymmet ry (_{N = 2 theories) in which it is almost} always possible t o find a dual weakly coupled descript ion (elect ric-magnet ic duality), allowing t hus t o det ermine t he st ruct ure of t he low energy effect ive act ion and consequent ly t he infrared dynamics of t he t heory (Seiberg-Wit t en solut ion) [1, 2]. One of t he most remarkable byproduct s of t his const ruct ion is t he presence of massless monopoles and dyons which are best described as solit ons in t he original descript ion of t he t heory (t he descript ion used t o analyze t he t heory in t he UV). T he t heory in t he infrared can be essent ially underst ood in t erms of t hese dist inguished part icles.

T he above ment ioned models are just dist ant relat ives of QCD and many key propert ies are different ; for inst ance_{N = 2 theories are not confining, as} opposed t o QCD. Nevert heless, just breaking soft ly ext ended supersymmet ry t o_{N = 1, one can still use many of the properties derived for the undeformed} t heory but at t he same t ime flows t o a model in which confinement is ex-pect ed. T he out come is t hat t he massless part icles of t heN = 2 theory condense, leading t o confinement via t he ’t Hooft -Mandelst am mechanism [3]-[6]: as t he condensat ion of elect rically charged part icles in a superconduc-t or confines magnesuperconduc-t ically charged objecsuperconduc-t s [7], superconduc-t he condensasuperconduc-t ion of magnesuperconduc-t ic monopoles in t hese t heories form a sort of dual superconduct or in which elect rically charged part icles are confined. Furt hermore, when quark fields are int roduced in t he t heory t hese massless part icles acquire flavor charges via t he Jackiw-Rebbi effect [8]-[11], t hus explaining what is t he relat ion (at least for t his class of t heories) between confinement and chiral symmet ry breaking.

We can see from t he above discussion t hat t he st udy of supersymmet ric models is ext remely helpful in addressing problems essent ially relat ed t o t he det ails of t he dynamics since we have no t ools t o address t hem direct ly in QCD, where all t he const raint s coming from supersymmet ry are not avail-able. T he idea is t hus t o learn as much as possible about st rong dynamics in t he supersymmet ric case and t hen t o figure out how analogous mechanisms can be at work in t he QCD case [12, 13] (see also [14]-[19]).

Let us discuss now t he pict ure t hat has emerged so far from t he st udy of supersymmet ric t heories (we will discuss mainly t he SU(N ) case) wit h soft ly broken N = 2 supersymmetry. In the pure gauge case (in the absence of quarks) it has been shown t hat t he t heory dynamically abelianizes: despit e t he lagrangian is invariant under a non-abelian group, t he effect ive t heory in t he infrared is invariant just under it s maximal abelian subgroup due t o quant um correct ions [20] (see also [21]). T he st udy of t hese vacua is by now

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Contents st andard since a weakly coupled descript ion can always be found making use of elect ric-magnet ic duality, which is well underst ood in t he abelian case.

More int erest ing is t he sit uat ion in t he t heory wit h flavors, where vacua which do not abelianize in t he infrared indeed exist [22, 23]. Also in t his case confinement is realized via t he ’t Hooft -Mandelst am mechanism, but t his t ime it is less clear how t he elect ric-magnet ic duality works: in t he abelian case it just amount s t o a change of variables in t he pat h int egral whereas in t he non-abelian case t his procedure cannot be carried out .

T he most est ablished examples of such a duality are Seiberg duality in N = 1 SQCD [24] and its analogue for the models we have been discussing so far, called K ut asov duality [25]. In bot h cases t he duality requires t he int roduct ion of a new set of dynamical variables, which include magnet ic variables of a non-abelian kind, and involves a change in t he gauge group. T he validity of t hese dualit ies is st ill a conject ure but t hey have anyway remarkably passed numerous checks and led t o many significant result s (see e.g. [26]-[32] and [33] for a review).

Many aspect s of t hese t heories are st ill unclear and deserve furt her in-vest igat ions, especially due t o t he fact t hat t he observed degeneracies in t he hadron spect rum suggest t hat t he right answer for QCD is not abelianiza-t ion (see e.g. [34]-[36] and references abelianiza-t herein). Moabelianiza-t ivaabelianiza-t ed by abelianiza-t hese consid-erat ions, in t his t hesis we t ry t o bet t er underst and t he propert ies of soft ly broken N = 2 SQCD and t he possible mechanisms leading t o confinement and chiral symmet ry breaking.

T his t hesis is based on [37]-[41] and is organized as follows: t he first two chapt ers cont ain some review mat erial which is used in t he main body of t he t hesis (part of chapt er 1 appears also in [39]). Chapt er 1 cont ains some basic mat erial, including a brief review of supersymmet ry represent at ions and supersymmet ric field t heories in four dimensions, which is t he case of int erest for us. We t hen analyze in some det ail t he propert ies of _{N = 2} t heories and describe t he original argument given by Seiberg and Wit t en in [1] for t he det erminat ion of t he infrared effect ive act ion for SU(2) SYM. Chapt er 2 collect s more recent and advanced result s about _{N = 2 theories} which are relevant for t he present work. We discuss in part icular t he brane realizat ion of t hese models in type I IA/ M-t heory and t he more recent six-dimensional const ruct ion. Chapt ers 3-5 are based on my original work and summarize my result s. Every chapt er cont ains an int roduct ory sect ion in which I explain in det ail t he set up.

Chapt er 3 is based on [37, 39] and is devot ed t o t he st udy of N = 2 SQCD, soft ly broken by a mass t erm for t he chiral field in t he adjoint represent at ion of t he gauge group. T he basic t echniques t hat have been used in earlier works on t he subject are essent ially t he classical analysis using t he equat ions of mot ion and t he nonpert urbat ive one using Seiberg-Wit t en t heory. Looking at t he det ails of t he t heory, it t urns out t hat t he first approach is reliable just for very large masses of t he flavors whereas t he second, alt ough in principle

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always applicable, can be used just in t he small mass limit , due t o t echnical complicat ions t hat make t he met hod almost int ract able in t he general case. We approach t his problem using t he generalized Konishi anomaly and t he Dijkgraaf-Vafa superpot ent ial. T his met hod has been applied t o similar models and some of our result s are very similar. Anyway, what has not been emphasized in previous works is t hat t his t echnique allows t o ext ract more informat ion t han ot her met hods about t he generic mass case, making it possible t o follow t he vacua from weak t o st rong coupling. As a Byproduct , we are able t o shed new light on t he relat ion between semiclassical and quant um vacua, which involves an infrared duality very close t o Seiberg duality inN = 1 SQCD [42].

Anot her int erest ing result is t hat for a part icular value of t he mass some of t he vacua merge in a superconformal point , signalling t he t ransit ion be-tween Higgs and confining phase. Underst anding t he low energy dynamics at t his point is a nont rivial t ask, since t he t heory is st rongly int eract ing and t he Seiberg-Wit t en curve becomes singular, making it difficult t o ext ract any precise informat ion. A very similar class of singular point s has been consid-ered from a different perspect ive in [43]. T he cont ent of t his chapt er can be seen as a preliminary work t hat allows t o underst and t he following st eps of my analysis. From t his work one can ext ract a number of nont rivial result s such as t he pat t ern of dynamical flavor symmet ry breaking.

Chapt er 4 is based on [38] which is devot ed t o t he analysis of singular point s (where t he Seiberg-Wit t en curve degenerat es) in t he moduli space of N = 2 SQCD, with particular attention to the points which are not lifted by t he_{N = 1 perturbation and are thus relevant for the study of confinement} in t hese models. As I ment ioned before, it is difficult t o underst and t he infrared dynamics in t hese cases, since t he SW curve does not lead direct ly t o a weakly coupled act ion and t he t heory is oft en int rinsically int eract ing. In order t o shed new light on t his problem it is necessary t o make use of t he most recent result s onN = 2 superconformal theories [44, 45].

For our purposes it is part icularly import ant t he analysis performed in [46], in which t he aut hors argued t hat t he low-energy physics at t he above ment ioned singular point s in SU(N ) SQCD involves two (possibly free) su-perconformal sect ors. In [38] I essent ially generalized t o SO and USp SQCD t his analysis, recovering an analogous st ruct ure. As we will see, it t urns out t hat t his observat ion is fundament al in order t o get new insight about t he mechanism underlying confinement in SO and USp t heories, in which typically t he relevant point s in t he moduli space are singular.

T he second part of t he chapt er cont ains a revised version of my cont ri-but ion t o [41] and is devot ed t o explain how it is possible t o make use of t he result s present ed in [47], in which t he aut hors define a broad class of N = 2 SCFTs exploiting their type IIB superstring realization, to derive informat ion about t he BPS spect rum of IR fixed point s in N = 2 SQCD. T his is done by carefully mat ching t he SW curves associat ed t o t he t

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heo-Contents ries const ruct ed in t he above ment ioned paper. In t he last sect ion I discuss t he propert ies of some of t he infrared fixed point s in quiver gauge t heories, always exploit ing t he const ruct ion present ed in [41].

In chapt er 5 we come t o our final payoff: t he result s of chapt er 4 allow us t o explicit ly st udy t he mechanism underlying confinement in USp and SO t heories (in some cases), where t he st andard ’t Hooft -Mandelst am mecha-nism does not seem t o work. Indeed we will see t hat , alt hough confinement is realized as expect ed by t he condensat ion of magnet ically charged object s, t he det ails of t he confining mechanism are rat her unusual, making it possible t o reproduce all t he semiclassical expect at ions which might seem quit e hard t o combine at first looking [34].

T his can be also import ant for t he st udy of confinement in t he cont ext of QCD, where t he assumpt ion t hat t he underlying mechanism is given by t he ’t Hooft -Mandelst am scenario is known t o lead t o various difficult ies (doubling of t he meson spect rum, excess in t he number of Goldst one bosons...) [36]. T his chapt er is based on [39, 40] and is devot ed t o t he st udy of t he low-energy dynamics at singular point s in USp(2N ) and SU(N ) SQCD wit h four flavors, SO(2N ) SQCD wit h two flavors and SO(2N + 1) SQCD wit h one flavor.

We are st ill not able t o approach t he general case and t he rest rict ion on t he number of flavors comes from t he fact t hat in t hese special cases we are able, exploit ing t he analysis performed in chapt er 4, eit her t o iden-t ify a weakly coupled dual descripiden-t ion or iden-t o find a descripiden-t ion in iden-t erms of a st rongly int eract ing t heory t hat we know well enough t o ext ract t he infor-mat ion we are int erest ed in (such as chiral condensat es). In all t he cases our result s are perfect ly consist ent wit h t he semiclassical result s obt ained using t he t echniques present ed in chapt er 3 (pat t ern of dynamical flavor symmet ry breaking, number of vacua...).

At t he init ial st age of my PhD I also t ried t o apply t he ideas emerging from supersymmet ric gauge t heories t o nonsupersymmet ric Yang-Mills t he-ory; in part icular I focused on t he Faddeev-Niemi decomposit ion [48]. In t his paper t he aut hors claim t hat SU(2) Yang-Mills t heory is on-shell equiv-alent t o an abelian t heory wit h a unit t hree vect or and a complex scalar as mat t er fields. T his reformulat ion aims at rewrit ing t he t heory in t erms of a set of variables suit ed t o invest igat e t he scenario of dynamical abelianiza-t ion in Yang-Mills abelianiza-t heory, as is abelianiza-t he case in supersymmeabelianiza-t ric abelianiza-t heories, and has received much at t ent ion in t he past years especially for lat t ice simulat ions.

However, t his reformulat ion does not allow t o recover t he non abelian Gauss’ const raint s of t he original t heory. At a closer inspect ion I found out in collaborat ion wit h Jarah Evslin t hat t heir claim is act ually wrong and some solut ions of t he Faddeev-Niemi equat ions do not solve t he above men-t ioned consmen-t rainmen-t s and men-t hus are nomen-t solumen-t ions of Yang-Mills equamen-t ions [49]. I t hen formulat ed in collaborat ion wit h Jarah Evslin, K enichi Konishi and Albert o Michelini a similar decomposit ion for SU(3), suit ed for invest igat ion

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of t he non-abelian scenario (we rewrit e t he t heory in t erms of SU(2)_{× U(1)} variables) [50].

In order t o avoid t he problems we found in t he original proposal by Faddeev and Niemi, we const ruct ed a decomposit ion such t hat only a par-t ial gauge fixing is implied, wipar-t houpar-t affecpar-t ing par-t he Gauss’ conspar-t rainpar-t s. Our paramet rizat ion includes all t he solut ions of Yang-Mills equat ions of physical int erest such as monopoles, Wit t en’s generalized inst ant ons [51] and merons. Furt hermore, since only a part ial gauge fixing is implied, our formula can be used t o perform pat h int egral comput at ions. Finally, we reproduce t he no-go t heorem on non-Abelian monopoles (t he so-called t opological obst ruct ion) in t he pure Yang-Mills t heory [52]-[55]. Also, we show t hat t he knot -solit ons discussed by Faddeev and Niemi in [56, 57] do not exist if t he syst em does not dynamically Abelianize. T hese t opics have not been included in t he present t hesis.

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Chapter 1

Supersymmetric field theories

and the Seiberg-Witten

solution

In t his chapt er we briefly review t he basic concept s of supersymmet ric field t heories and t he Seiberg-Wit t en solut ion of _{N = 2 gauge theories. This} is not a comprehensive review of supersymmet ry and for furt her det ails t he reader is referred t o t he many good reviews available in t he lit erat ure. T hose which are closer t o my select ion of t opics are [58, 59]. In t he first two sect ions we int roduce t he supersymmet ry algebra and describe how t o build supersymmet ric lagrangians. Most of t he mat erial is t aken from [60, 61]. T his is t he st art ing point for t he analysis performed by Seiberg and Wit t en in [1] which is t he cent ral t opic of sect ion t hree. In sect ion four we review how t he Seiberg-Wit t en solut ion allows t o shed light on confinement and summarize t he available result s in t he lit erat ure on t his t opic. Since t his is t he cent ral t heme of t he present work, t he mat erial in t his sect ion represent s t he st art ing point for our analysis.

1.1 Supersymmetric theories in four dimensions

1.1.1 Notation and conventions

We will make use of t he not at ion adopt ed in [61]. Our convent ion for t he Minkowsky met ric is µν = diag(1;−1; −1; −1) and we will indicate left and

right spinors wit h dot t ed and undot t ed indices respect ively. In order t o raise and lower indices we will use t he ant isymmet ric t ensor " ,

"αβ = "α ˙˙β = 0 1 −1 0 = { 2:

Let us now define

( µ)α ˙α≡ (1; ~);

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(¯µ)αα˙ =−( µ)α ˙α= "α ˙˙β"αβ( µ)_{β ˙}_β = (1;−~): Wit h t hese convent ions, Lorent z t ransformat ions are generat ed by

( µν)β_α = 1 4 h _µ α ˙β¯ ν ˙ββ_{− ( ↔ )}i_; (¯µν)α_β˙_˙ = 1 4 h ¯µ ˙αβ ν_{β ˙}_β − ( ↔ )i:

For t he product of spinors we will use t he following convent ions: = α α =− α α = α α= ;

¯_{¯ = ¯}_α_˙_¯α˙ _{= ¯ ¯}_;

( )†= ¯α˙¯α˙ = ¯ ¯= ¯¯:

In t his not at ion mat rices, Dirac and Majorana spinors can be writ t en as follows: µ₌ 0 µ ¯µ ; D = α ¯α˙ ; M = α ¯α˙ : T he following ident it ies hold:

µ_¯ν ₌ ν_¯µ _; ₍ µ_¯ν ₎†_{= ¯}_¯ν µ_¯; µ_¯₌_{− ¯}_¯µ _; ₍ µ_¯₎†₌ µ_¯;

1.1.2 Supersymmetry algebra

T he SUSY algebra can be writ t en as follows [62]: {QAα; ¯Q_βB˙ } = 2 _{α ˙}µ_βPµ AB; {QAα; QBβ} = 2 √ 2"αβZAB; { ¯QαA˙ ; ¯Q_βB˙ } = 2 √ 2"_{α ˙}_˙_βZ_AB∗ : (1.1)

Q and ¯Q are t he generat ors of supersymmet ry t ransformat ions (supercharges) and t ransform as operat ors of spin1=2 under t he Lorent z group. Indices A and B run from 1 t o N, where N is t he number of supersymmet ries. T he supersymmet ry charges commut e wit h P2, so all t he st at es in a given repre-sent at ion of t he SUSY algebra have t he same mass. T he operat ors Z e Z∗, ant isymmet ric in t he A; B indices, are called cent ral charges. Clearly t hey can be nonzero only if we have at least two supersymmet ries. T heir presence will be relevant below, when we will discuss t heories wit h eight supercharges. A field t heory will be supersymmet ric if t he set of it s st at es fall in represen-t arepresen-t ions of represen-t he algebra (1.1). We will now briefly review represen-t he properrepresen-t ies of represen-t he SUSY algebra represent at ions, specializing t o t he four-dimensional case.

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1.1 Supersymmetric theories in four dimensions M assive ir r educible r epr esent at ions

Let us discuss first t he case wit hout cent ral charges. In t he massive case we can go t o t he rest frame, in which Pµ= (M ; 0; 0; 0) and define

aA_α = QA_α=√2M ; (aA_α)†= ¯QαA˙ =

√ 2M : T he SUSY algebra t hen becomes

{aA1;(aB1)†} = AB; {aA2;(aB2)†} = AB;

T he vacuum_{|Ωi is defined by the relation a}A

α|Ωi = 0 and the representation

can be const ruct ed applying t he operat ors(aA_α)† t o t he vacuum. If |Ωi has spin zero, it is easy t o see t hat we can const ruct 2N_mdist inct st at es applying m raising operat ors. T he represent at ion t hus has dimension

2N X m=0 2N m = 22N:

and t he maximum spin in t he mult iplet is N =2. For example, an _{N = 1} mult iplet describes22 _{= 4 st at es,}

|Ωi; a†_α|Ωi; √1 2"

αβ_a† αa†β|Ωi:

T he spin cont ent is(0)_{⊕(1=2)⊕(0). If the vacuum |Ω}si has instead spin s the

represent at ion will include(2s+1)22N st at es. For inst ance, in t heN = 1 case t he SUSY mult iplet includes part icles wit h spin(s)_{⊕(s+1=2)⊕(s−1=2)⊕(s).} In any case t he number of bosonic and fermionic degrees of freedom is t he same.

M assless ir r educibl e r epr esent at ions

In t he massless case we can go t o a reference frame in which Pµ= E (1; 0; 0; 1)

and reduce t he SUSY algebra t o: {QAα; ¯Q_αB˙ } = 0 0 0 4E A B:

Since t he ant icommut at or of QA_α and ¯QαA˙ is a posit ive operat or, from t he

relat ion_{QA

1; ¯Q˙1B} = 0 we can conclude that both QA1 and ¯Q˙1Aannihilat e all

physical st at es. T he algebra t hus cont ains only N nontrivial supercharges. We can now define as before t he raising and lowering operat ors

aA= 1 2√M Q A 2; (aA)†= 1 2√M ¯ QA_˙2: 9

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T he SUSY algebra can t hus be rewrit t en in t he form:

{aA;(aB)†} = AB; {aA; aB} = 0; {(aA)†;(aB)†} = 0: T his is a Clifford algebra wit h 2_{N generators and the representation will} have dimension2N_{. T he operat ors a}A_{increase t he helicity of massless st at es}

by 1=2 whereas t he operat ors (aA)† decrease it by t he same quant ity. We define as before t he vacuum as t he st at e_{|Ωi annihilated by all the operators} aA_{. We are free t o assume t hat it is an eigenst at e of J}

z. T he ot her st at es

in t he mult iplet are generat ed applying t he operat ors(aA)† t o |Ωi. In this way we can build N_mst at es applying m lowering operat ors, all wit h helicity Jz|Ωi − m=2. Notice that the representations obtained in this way are not

necessarily CPT invariant so, if we want t o build a physical t heory we must add t he CPT conjugat e st at es.

Indicat ing wit h t he helicity of t he vacuum, let us not ice t hat forN = 1 a massless represent at ion cont ains a Majorana spinor and a complex scalar if = 1=2; if = 1 we get a Majorana spinor and a massless vect or. For _{N = 2} and = 1=2 t he mult iplet includes two Majorana spinors and two complex scalars, i.e. two copies of t he corresponding_{N = 1 multiplet. If} = 1 we get a massless vect or, two majorana spinors and a complex scalar, which is t he cont ent of t heN = 1 multiplets with = 1 and = 1=2. For N = 4 and = 1 t he represent at ion is CPT selfconjugat e and cont ains a massless vect or, four spinors and t hree complex scalars. T his is t he field cont ent of t heN = 2 multiplets with = 1 and = 1=2 t oget her. We cannot have more t han 16 supercharges wit hout int roducing gravity.

C ent r al char ges and B P S st at es

Let us focus on t he N even case: modulo a unit ary t ransformat ion we can assume t hat t he cent ral charge mat rices are of t he form Z = " ⊗ D with D a diagonal mat rix. We can t hus focus on t he_{N = 2 case and defining now}

aα = 1 2 Q1_α+ "αβ(Q2β)† ; bα = 1 2 Q1_α− "αβ(Q2β)† ; t he algebra (1.1) reduces t o {aα; a†_β} = αβ(M + √ 2Z ); _{bα; b†_β} = αβ(M − √ 2Z ): (1.2) We can t hus immediat ely deduce t he relat ion M ≥√2_{|Z | (in particular all} massless st at es have Z = 0). When M =√2|Z |, either {a; a†_{} or {b; b}†_{} in}

(1.2) are zero and t he dimension of t he represent at ion decreases, since t he st at es in t he mult iplet are annihilat ed by a subset of t he supercharges. In t he case N = 2, a reduced (or BPS) massive mult iplet has t he same dimension as a massless one.

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1.1 Supersymmetric theories in four dimensions

1.1.3 Superspace and superfields

In order t o writ e down t he lagrangian for supersymmet ric t heories it is conve-nient t o adopt t he superspace and superfield formalism. We will now briefly review t hese t opics.

Sup er space

T he superspace is defined adding ant icommut ing Grassmann variables α_{; ¯} ˙ α

t o t he spacet ime coordinat es xµ and can t hus be ident ified wit h t he set of t riples(x; ; ¯) (we will from now on focus on t heN = 1 superspace in four dimensions). We indicat e wit h e ¯¯ t he expressions α

α =−2 1 2 and

¯_α_˙¯α˙ _{= 2¯}

˙1¯˙2. Int roducing auxiliary ant icommut ing paramet ers e ¯ we

can rewrit e (1.1) in t erms of Q e ¯ ¯Q, obt aining an algebra which involves commut at ion relat ions only. We can now writ e a SUSY t ransformat ion wit h paramet ers and ¯ just exponent iat ing:

G(x; ; ¯) = eı(−xµPµ+θQ+¯θ ¯Q)_:

We t hus obt ain

G(0; ; ¯)G(x; ; ¯) = G(xµ+ { µ¯ − { µ¯; + ; ¯ + ¯);

so under a SUSY t ransformat ion ; ¯ t he superspace coordinat es t ransform as:

xµ→ xµ+ { µ¯ − { µ¯; → + ;

¯ → ¯+ ¯:

(1.3) We can now easily writ e t he supercharges as different ial operat ors on t he superspace: Qα= @ @α − { µ α ˙α¯α˙@µ; Q¯α˙ =− @ @¯α˙ + { α µ α ˙α@µ: (1.4)

Let us finally int roduce t he super derivat ives Dα= @ @α + { µ α ˙α¯α˙@µ; D¯α˙ =− @ @¯α˙ − { µ α ˙α α@µ: (1.5)

which sat isfy t he relat ion {Dα; ¯Dα˙} = −2{ α ˙µα@µ. We int roduce t hem

be-cause t hey ant icommut e wit h Q and ¯Q and t his will make it easier t o writ e down supersymmet ric lagrangians.

Sup er fi elds

A superfield is a funct ion defined on t he superspace F(x; ; ¯) (defined in t erms of it s expansion in powers of t he ant icommut ing variables)

F(x; ; ¯) = f (x) + ’ (x) + ¯ (x) + m(x) + ¯¯n(x) + µ¯vµ(x)

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+ ¯¯(x) + ¯¯ (x) + ¯¯d(x):

A supersymmet ry t ransformat ion act s on super fields as F = ( Q + ¯ ¯Q)F . From t his we can read t he t ransformat ion propert ies of t he various compo-nent s. Not ice t hat t he variat ion of t he highest compocompo-nent of a superfield is always a t ot al derivat ive. T he spacet ime int egral of t his component will t hen be invariant under supersymmet ry t ransformat ions. T his is basically how supersymmet ric lagrangians are const ruct ed, as we will now see.

Clearly every funct ion of superfields is a superfield and since t he algebra of superfields is closed under SUSY t ransformat ions t hey give a represent a-t ion of a-t he SUSY algebra. However, a-t his will be reducible and in order a-t o obt ain irreducible represent at ions we have t o impose some const raint s. T he most common (and t he only ones we will use) are:

• ¯Dα˙Φ = 0 which defines C hir al Sup er fi elds:

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1.1 Supersymmetric theories in four dimensions Not ice t hat V3 _{= 0 in t his gauge. T he field st rengt h is t hen defined by}

Wα=−1 4D¯ 2_D αV; W¯α˙ =−1 4D 2_D_¯ ˙ αV; (1.7)

which is a gauge invariant chiral superfield.

T he generalizat ion t o t he non-abelian case is st raight forward: we will have V = VaTa (Ta’s are t he group generat ors), t he gauge t

ransfor-mat ions are given by

e−2V → e−ıΛ†e−2VeıΛ;

whereΛ = ΛaTa and t he field st rengt h can be writ t en as

Wα=

1 8D¯

2_e2V_D

αe−2V: (1.8)

It t ransforms in t he expect ed way: Wα → e−ıΛWαeıΛ

1.1.4 Lagrangian _{N = 1 field theories}

I t his sect ion we will explain how t o build lagrangians for _{N = 1 theories} describing chiral and vect or mult iplet s. T hese const it ut e t he building blocks for gauge t heories lagrangians, in part icular t hose wit hN = 2 supersymme-t ry, which represensupersymme-t supersymme-t he ssupersymme-t arsupersymme-t ing poinsupersymme-t of supersymme-t he Seiberg-Wisupersymme-t supersymme-t en analysis. T he basic idea is t o consider t he highest component of a suit able superfield. C hir al mult iplet s

T he kinet ic t erms for t he fields in t he SUSY mult iplet can be int roduced considering t he highest component ofΦ†_iΦj. Neglect ing t ot al derivat ives and

summing over i = j , we get t he lagrangian

L = Φ†iΦi|θ2_θ¯2 = @_µA†_i@µA_i+ F_i†F_i− { ¯_i¯µ@_µ _i:

T his lagrangian describes as expect ed a scalar and a spinor massless fields. It also describes an auxiliary field F , which can be eliminat ed by means of t he equat ions of mot ion.

Mass and int eract ion t erms can be added int roducing a superpot ent ial. T he most general superpot ent ial compat ible wit h renormalizability is

L _{= Φ}† iΦi|θθ ¯θ ¯θ+ 1 2mijΦiΦj+ 1 3gijkΦiΦjΦk+ iΦi _θθ+ h:c: : (1.9) We can rewrit e everyt hing in t erms of an int egral over superspace

L = Z d4 Φ†_iΦi+ Z d2 _{W(Φ) +} Z d2¯ ¯_W(Φ†): 13

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T his is not t he most general lagrangian compat ible wit h supersymmet ry. We can also consider t he highest component of K(Φ; Φ†_{) where K is called}

K ähler p ot ent ial and should sat isfy t he const raint ¯K(zi;z¯j) = K (¯zi; zj).

At t he classical level any supersymmet ric t heory has a U(1) simmet ry called R-simmet ry, which act s on chiral superfields as

RΦ(x; ) = e2ınαΦ(x; e−ıα ); RΦ†(x; ¯) = e−2ınαΦ†(x; eıα¯):

n is called R-charge. Since d2 _{→ e}−2ıαd2 , t he R-charge of t he superpot en-t ial _{W should be one.}

Sup er sy m m et r ic gauge t heor ies

T he st andard kinet ic t erm for t he abelian mult iplet is L = 1 4 Z d2 WαWα+ Z d2¯ ¯Wα˙W¯α˙ : (1.10)

Similarly, t he SUSY analog of Yang-Mills (SYM) t heory is described by t he lagrangian L = 1 4g2 Z d2 WαWα+ Z d2¯ ¯Wα˙W¯α˙ ; (1.11)

where g is t he coupling const ant . If we want t o include t he t erm we can simply consider L = 1 8 I m T r Z d2 WαWα ; where = =2 + 4 {=g2.

If we want t o include mat t er fields t he minimal coupling can be imple-ment ed int roducing t he t erm

Z

d4 Φ†e−2VΦ

for any mat t er field. Int eract ion t erms can be simply included using t he superpot ent ial as before.

1.1.5 _{N = 2 gauge theories}

SY M t heor y

As we have explained above, t he field cont ent of a N = 2 vect ormult iplet (Aµ; ; ; ) is equivalent t o t hat of two N = 1 mult iplet s, one vect or

mul-t iplemul-t (Aµ; ) e one scalar mult iplet ( ; ). T heN = 2 pure gauge theory is

t hus equivalent t o aN = 1 gauge theory with a chiral multiplet in the ad-joint represent at ion. T he relat ive normalizat ion is fixed by t he requirement

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1.1 Supersymmetric theories in four dimensions t hat t he two fermions should ent er symmet rically in t he lagrangian. Using t he receipt given previously, we can immediat ely writ e down t he lagrangian for N = 2 SYM: L = ₈1 ImTr Z d2 WαWα + 2 Z d4 Φ†e−2VΦ : (1.12) If we rewrit e it in t erms of t he component fields and eliminat e t he two auxiliary fields using t heir equat ions of mot ion, we get

L = 1 g2Tr −1 4FµνF µν_{+ g}2 32 2FµνFe µν_{+ (D} µ )†(Dµ )− 1 2[ †_; _]2 −{ µDµ¯ − { ¯ ¯µDµ − { √ 2[ ; ] †_{− {}√2[¯; ¯] : (1.13) T he scalar pot ent ial of t he t heory is t hus

V = −_2g1₂Tr[ †; ]2: (1.14) At t he classical level t he vacuum configurat ions can be found minimizing t he pot ent ial (1.14). We can immediat ely see from t he above formula one of t he charact erizing propert ies of field t heories wit h ext ended supersymmet ry: t he presence of flat direct ions. T he pot ent ial at t ains it s minimum when t he field commut es wit h it s hermit ian conjugat e and can t hus be diagonalized. T he set of gauge inequivalent vacua can be paramet rized by t he eigenvalues of (or more precisely t heir gauge invariant combinat ions Tr k_{) and is called}

moduli space.

I ncluding m at t er fi elds

Mat t er mult iplet s coupled t o t he_{N = 2 gauge multiplet can also be included.} T hey are described by t he so called hypermult iplet s, which can be con-st ruct ed using twoN = 1 chiral multiplets Q and eQ†in t he same represent a-t ion of a-t he gauge group. We can for insa-t ance describe_{N = 2 SQCD adding} Nf hypermult iplet s in t he fundament al represent at ion. T he lagrangian can

be writ t en down simply adding t o (1.12) t he t erms (for SU(N ) gauge t heo-ries) L = Z d4 Q†_ie−2VQi+ eQie2VQe†i + Z d2 √2 eQiΦQi+ miQeiQi + h.c.

T he t erm √2 eQiΦQi is linked t hrough N = 2 SUSY to the coupling of the

hypermult iplet wit h t he vect or superfield V . 15

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C ent r al char ges in _{N = 2 gauge t heor ies}

We have seen t hat gauge t heories wit h ext ended supersymmet ry always in-clude a scalar field in t he adjoint represent at ion. It s vacuum expect at ion value generically breaks t he gauge group down t o t he maximally abelian subgroup. All t he t heories wit h t his property cont ain in t heir spect rum magnet ic monopoles and dyons [63]-[65]. As shown by Olive and Wit t en in [66], t he cent ral charge of SUSY represent at ions in t hese t heories is propor-t ional propor-t o propor-t he elecpropor-t ric and magnepropor-t ic charges. T heir compupropor-t apropor-t ion leads propor-t o propor-t he following result for SU(2) SYM t heory

Z = a(ne+ nm); =

2 + 4 {

g2 ;

where a is t he vev of . If we include mat t er fields in various represent at ions t his formula should be modified adding t he flavor charges Si:

Z = a(ne+ nm) + 1 √ 2 X miSi;

where mi are t he masses of t he mat t er fields. T his result holds for t heories

wit h gauge group SU(2), which has rank one. In t he general case we will have one elect ric and one magnet ic charges for each Cart an generat or.

1.2 _{N = 2 SYM and low energy effective action}

As we have seen t he classical moduli space (Coulomb branch) of SYM t heory can be paramet rized by t he vev of t he scalar field in t he vect or mult iplet , which can be supposed t o be diagonal, more precisely by t he gauge invariant combinat ions of it s eigenvalues Tr kwit h k = 1; : : : ; rank G. If we int roduce mat t er fields in t he t heory t he Coulomb branch is just a submanifold of t he whole moduli space, which also includes t he so-called Higgs branch. It is paramet rized by t he vev of t he scalar fields in t he mat t er hypermult iplet s, whereash i = 0. On the Higgs branch the gauge group is generically com-plet ely broken. T he Higgs and Coulomb branches can also int ersect along t he so called mixed branches, on which t he vev of is different from zero. A key property of t he Higgs branch (dict at ed by ext ended supersymmet ry) is t hat it is a hyperkähler manifold and is not modified by quant um correct ions [22]. T his is not t rue for t he Coulomb branch. T he purpose of t his sect ion is t o underst and how t he Coulomb branch is modified by quant um correct ions. For definit eness we will rest rict our discussion t o SU(N ) gauge t heories.

1.2.1 Breaking of the R-symmetry

We will now discuss how quant um correct ions break t he classical R-symmet ry of t he t heory. T his will play an import ant role in t he ot her chapt ers.

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1.2_{N = 2 SYM and low energy effective action} We can rewrit e t he lagrangian for _{N = 2 SYM theory with gauge group} SU(Nc) in t erms of t he Dirac spinor D =

¯

const ruct ed using t he spinors in t he vect or mult iplet

L =_g1₂Tr −1₄FµνFµν+ g2 32 2FµνFe µν_{+ (D} µ )†(Dµ )− 1 2[ †_; _]2 +{ ¯D µDµ D+ { √ 2[ ¯D; 1 + 5 2 D] †_{− {}√_{2[ ¯} D; 1₋ 5 2 D] : One can t hen easily check t hat t he following t ransformat ion

U(1)_R: _→ ′ = e2ıα _; D → D′ = eıαγ5;

¯_D _{→ ¯}′

D = ¯Deıαγ5;

is a symmet ry of t he classical t heory but is broken at t he quant um level by t he chiral anomaly: @µJ5µ=−8πNc2FµνFeµν. From t he pat h int egral represent at ion

we find infact Z [d ′][d ¯′_D][d _D′ ] exp_{{S[ ′; ¯_D′ ; ′_D; Aµ]} = Z [d ][d ¯D][d D] exp{{S[ ; ¯D; D; Aµ]} exp{−{ 4Nc }; (1.15) where = _32π12 R d4_xF

µνFeµν is t he inst ant on number. T he t ransformat ions

U(1)_R which leave t he pat h int egral invariant sat isfy t he relat ion = 2πn_4N

c

where n = 1; : : : ; 4Nc. T he residual symmet ry group is t hen Z2Nc, which

is furt her broken in each vacuum by t he vev of . If we include mat t er hypermult iplet s t he corresponding spinors cont ribut e t o t he anomaly. For SQCD wit h Nc colors and Nf flavors, which is t he case of int erest for us, t he

unbroken subgroup is Z2Nc−Nf.

1.2.2 Low energy effective action

We have seen t hat semiclassically t he t heory abelianizes on t he Coulomb branch due t o t he vev_{h i = diag(a}1; : : : ; aN), wit hPiai= 0. If ai−aj ≫ Λ

for all i and j, t his breaking occurs at very high energy where t he t heory is weakly coupled. Below t hat scale t he t heory becomes abelian and t he coupling const ant decreases at lower energies. T he t heory is t hus weakly coupled at all scales and t he semiclassical pict ure is reliable. T he gauge mult iplet s associat ed t o broken gauge generat ors acquire mass of orderh i by t he Higgs mechanism and at low energy t he dynamics will be encoded in an abelian effect ive act ion, writ t en in t erms of t he vect or mult iplet s associat ed wit h t he Cart an generat ors. We expect t his pict ure t o remain valid at t he quant um level however, a pert urbat ive comput at ion will not be reliable in

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t he inner region of t he moduli space, where quant um correct ions become import ant . T he quest ion is t hen how one can det ermine t he effect ive act ion encoding t he infrared dynamics.

As a preliminary st ep, one can not ice t hat ext ended supersymmet ry im-poses st rong const raint s on t he st ruct ure of t his effect ive act ion, which must have t he following form, as found in [67] using t heN = 2 superspace for-malism: L = 1 8 Im Z d2 Fab(Φ)WaαWαb+ 2 Z d4 (Φ†e−2V)aFa(Φ) : (1.16) In t he previous formula_Fa(Φ) = @F=@Φa; Fab(Φ) = @2F=@Φa@Φb andF

is a holomorphic funct ion called pr ep ot ent ial. T he Seiberg-Wit t en solu-t ions allows solu-t o desolu-t ermine isolu-t exacsolu-t ly.

Let us now focus on t he case Nc = 2, t he generalizat ion being st raight

-forward. We will t hen haveh i = a 3. At t he classical level t he prepot ent ial

assumes t he formF = 1

2 a2 and it receives pert urbat ive correct ions only at

t he one loop level. T his can be det ermined using R-symmet ry [68]: as we have seen t he group U(1)_R is broken by t he chiral anomaly and under it s act ion t he lagrangian changes as

L = −

4 2FµνFe µν_:

We can rewrit e t his result in t erms of t he prepot ent ial applying t he R-symmet ry t ransformat ion t o t he effect ive act ion (1.16), obt aining

L = − 8 Im h aF′′′(a)( eFµνFµν+ {FµνFµν) i :

Comparing t he above equat ions we find_F′′′(a) = _aπ2ı and int egrat ing we get t he one loop correct ion t o t he prepot ent ial

Fone−loop(a) =

{ 2 a

2_ln a2

Λ2; (1.17)

WhereΛ is t he dynamical scale of t he t heory.

T his descript ion of t he effect ive act ion in t erms of t he superfieldsΦ and Wα is appropriat e in t he semiclassical region (for u = hTr 2_{i ≫ Λ}2_{) but}

cannot be valid globally on t he moduli space. In order t o see t his one can not ice t hat unit arity requires t he kinet ic t erm t o be posit ive definit e, which in t urn implies t hat

Im ≡ Im@

2_F

@a@¯a = 2 + { 4 g2

should be posit ive. On t he ot her hand, sinceF is holomorphic, its imaginary part is harmonic and consequent ly cannot be posit ive everywhere on t he

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1.2_{N = 2 SYM and low energy effective action} complex plane. T he solut ion t o t his problem is t hat our choice of coordinat es is valid only on a region of t he moduli space and when we approach a point where Im (a) = 0 we should adopt a different set of coordinat es ˜a such t hat Im˜(˜a) > 0 in a neighbourhood of t hat point . Let us now st udy how t o implement t his change of coordinat es.

1.2.3 Duality transformations

Let us defineΦD t o be t he field dual t oΦ and t he funct ionFD(ΦD) t o be

t he dual of t he prepot ent ial F(Φ) through the equations

ΦD =F′(Φ); FD′ (ΦD) =−Φ: (1.18)

Using t hese relat ions in (1.16) we can rewrit e t he second t erm as follows Im Z d4xd2 d2¯Φ†F′(Φ) = Im Z d4xd2 d2¯(−F_D′ (ΦD))†ΦD = Im Z d4xd2 d2¯Φ†_D_F_D′ (ΦD):

We t hus see t hat t his t erm is invariant under t he duality t ransformat ion (1.18). Let us now analyze t he t erm _F′′(Φ)Wα_W

α. Remember t hat Wα

cont ains t he abelian field-st rengt h Fµν = @µAν − @νAµ for some Aµ. T his

const raint can be implement ed imposing t he Bianchi ident ity @νFeµν = 0;

t he corresponding const raint in t he superspace formalism is ImDαWα = 0.

Insert ing t his in t he pat h int egral we can now int egrat e eit her wit h respect t o V or Wα, imposing t he condit ion ImDαWα= 0 by means of an auxiliary

vect or superfield VD which plays t he role of a Lagrange mult iplier:

Z [dV ] exp { 8 Im Z d4xd2 F′′(Φ)WαWα ≃ Z [dW ][dVD] exp { 8 Im Z d4x Z d2 F′′(Φ)WαWα+1 2 Z d2 d2¯VDDαWα : T he second t erm can now be rewrit t en as

Z d2 d2¯VDDαWα=− Z d2 d2¯DαVDWα= Z d2 D¯2(DαVDWα) = Z d2 ( ¯D2DαVD)Wα=− 1 4 Z d2 (WD)αWα;

where we have defined t he dual of W by means of t he relat ion (WD)α =

−14D¯2DαVDand we have exploit ed t he chirality of t he field st rengt h ¯Dβ˙Wα=

0. Int egrat ing explicit ly wit h respect t o W we finally get Z [dVD] exp { 8 Im Z d4xd2 − 1 F′′_(Φ)W α DWDα : 19

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We have t hus rewrit t en t he lagrangian (1.16) in t erms of dual variables. T he generalized coupling (a) is replaced in t hese variables by ₋_{τ (a)}1 . T his t ransformat ion is t he SUSY count erpart of elect ric-magnet ic duality: t he t ransformat ion W _{→ W}D implies Fµν → eFµν. Not ice t hat

FD′′(ΦD) =−

dΦ dΦD

=− 1 F′′_(Φ);

and t his implies

D(aD) =−

1 (a);

where we have defined aD =hΦDi. Substituting in the lagrangian we finally

get 1 8 Im Z d4x Z d2 _F_D′′W_DαWDα+ Z d2 d2¯Φ†_D_F_D′ (ΦD) : (1.19) Not ice t hat t he met ric induced on t he moduli space

ds2= Im dad¯a= Im(daDd¯a) = {

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1.2_{N = 2 SYM and low energy effective action} Since t he funct ional int egral is invariant if t he act ion changes by an int eger mult iple of 2 , we can conclude t hat t he t ransformat ions (1.23) are in t he duality group if a∈ Z. Their effect is to shift the angle by 2 a. Equations (1.18) and (1.23) t oget her generat e t he duality group SL(2; Z). It is easy t o verify t hat t he act ion of t his group on t he generalized coupling const ant is

→ a_c _{+ d}+ b;

where ad_{− bc = 1 and a; b; c; d ∈ Z. For N}c > 2 t he duality group which

leaves t he met ric invariant is Sp(2Nc− 2; R).

1.2.4 Masses of monopoles and dyons

We have seen t hat in t his class of models monopoles and dyons arise in a very nat ural way and t heir mass is bounded below by M _≥ √2_{|Z |, where} Z is t he cent ral charge. St at es sat urat ing t his bound are called BPS (see [69, 70]) and are organized in short mult iplet s of t heN = 2 SUSY algebra. Semiclassically Z = a(ne+ nm) and t he purpose of t his sect ion is t o achieve

an exact formula, writ t en in t erms of t he low energy effect ive quant it ies. Let us suppose t hat t he effect ive t heory cont ains a mat t er hypermult iplet described by t he chiral superfields M ; fM . When t he vev a of t he field is different from zero t he mult iplet becomes massive . If it s elect ric charge is ne (and it s magnet ic charge zero), t he int eract ion t erm assumes necessarily

t he form _√

2neMΦ fM :

by supersymmet ry. We can t hus conclude t hat it s cent ral charge is Z = nea.

If we have inst ead a magnet ic monopole wit h charge nm, we can reduce

ourselves t o st udy a syst em equivalent t o t he previous one by means of a duality t ransformat ion and conclude t hat t he cent ral charge in t his case is Z = aDnm. T his argument implies t hat for a generic dyon wit h charges

(ne; nm) we will have t he formula

Z = ane+ aDnm: (1.24)

Comparing t he two equat ions we find t hat semiclassically aD = a. T he

generalizat ion of t his formula t o a rank r t heory is simply given by

Z = aine,i+ aiDnm,i; i = 1; : : : ; r (1.25)

where ai are local coordinat es on t he moduli space and ne,i; nm,i are t he

charges wit h respect t o t he various U(1) subgroups. We conclude remarking t hat , since t he cent ral charge det ermines t he mass of t he various part icles in t he spect rum, t he duality t ransformat ions discussed previously should not modify t he value of t he cent ral charge. If we organize t he paramet ers aD

and a in a vect or v, under t he act ion of a duality t ransformat ion v → M v. In order t o leave Z invariant we should t hen impose t he t ransformat ion rule w_{→ wM}−1 where w= (nm; ne).

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1.2.5 β function for the effective U(1) theory

In t he next sect ion we will make use of t he one-loop expressions for a and aD

t o comput e t he monodromy mat rices. We will now see how we can det ermine t hem st art ing from t he bet a funct ion of t his t heory. If we have Weyl fermions wit h charges Qf and complex scalars wit h charges Qs (wit h respect t o t he

U(1) gauge group), t heir cont ribut ion t o t he bet a funct ion of t he t heory is:

(g)≡ dg d = g3 16 2  X f 2 3Q 2 f+ X s 1 3Q 2 s   :

Since t he bet a funct ion is posit ive t he t heory is infrared free and at very low energies we expect t he one-loop approximat ion t o be reliable. If we indicat e wit h b t he coefficient of t he t erm g3 and set = g2=4 , we can writ e

d d

1

=−8 b:

Since in a hypermult iplet we have two Weyl fermions and two complex scalars wit h t he same charge Q, it s cont ribut ion will be

b= 1 16 2Q 2₂ ·2₃ + 2_·1 3 = 1 8 2Q 2_:

Set t ing now = {= we obt ain d d =−

1 Q2:

If we now ident ify t he energy scale wit h a and set Q= 1 we are led t o t he equat ion

≈ −{ lna Λ:

If t he hypermult iplet we are considering becomes massless at let ’s say u0,

we will havelim_u→u0a(u) = 0. We are now free t o choose t he paramet

riza-t ion in such a way riza-t hariza-t a_{≈ c(u − u}0) wit h c∈ C. Since = da_daD we obt ain

int egrat ing

a(u)_{≈ c(u − u}0); aD(u)≈ aD(u0)− { c(u_{− u}0) ln u_{− u}0 Λ : (1.26) If inst ead t he hypermult iplet describes a monopole becoming massless at u0,

we havelimu→u0aD(u) = 0. Performing t he above comput at ion in t erms of

magnet ic variables we find

D ≈ −

{ lnaD

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1.3 The Seiberg-Witten solution Recalling now t hat D =−_dada_D we find

aD(u)≈ c(u − u0);

a(u)_{≈ a(u}0) + {c(u− u0) lnu− u0

Λ :

(1.27)

Not ice t hat t he dual coupling const ant t ends t o zero when t he monopole becomes massless, i.e. for u_{→ u}0. T he t heory can t hus be analyzed using

st andard t echniques once we adopt t he magnet ic descript ion.

1.3 The Seiberg-Witten solution

1.3.1 Monodromies and singularities of the moduli space

In t he previous sect ion we have seen t hat we can paramet rize t he moduli space wit h t he coordinat e u = _htr 2_{i and for large u the theory can be}

described using t he paramet er a defined by h i = 1

2a 3. However, t his

descript ion is not globally valid and at st rong coupling we should perform a duality t ransformat ion. It is t hus convenient t o int roduce t he vect or v= (aD(u); a(u))T. One of t he key ingredient s of t he Seiberg-Wit t en solut ion is

t he presence of singular point s: t he funct ions aD(u) and a(u) are not

single-valued and if we loop around a singular point t hey undergo a nont rivial t ransformat ion which can be convenient ly described in t erms of a mat rix (monodromy mat rix) act ing on t he v vect or. We will now explore t his point . M onodr omy at infi nit y

Let us st art from t he semiclassical region, where t he t heory can be analyzed by means of a st andard one-loop comput at ion. Using t he formula for t he prepot ent ial (1.17), and recalling t hat aD = @F=@a we get

aD(u) = {a 1 + ln a 2 Λ2 ; u→ ∞: (1.28)

Looping now around t he point at infinity in t he moduli space (u → e2πı_u)

we find a→ −a; aD → { (_−a) 1 + lne 2πı_a2 Λ2 =_−aD+ 2a: (1.29)

T he monodromy mat rix at infinity is t hen aD a → M∞ aD a ; M_∞= −1 2 0 ₋₁ : (1.30) 23

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St r uct ur e of t he m oduli space

Since t here is a singularity at infinity we necessarily have at least anot her one in t he u plane (ot herwise every loop would be cont ract ible and we would not have any monodromy) and t he associat ed monodromy should be in SL(2; Z) as we have seen before. We cannot act ually have only two singular point s in t he moduli space, as shown in [1]. If t his were t he case t he monodromy associat ed t o t he second singular point would be equal t o M_∞. On t he ot her hand t his t ransformat ion act s t rivially on t he met ric (1.20), which would t hen be globally expressible as a harmonic funct ion Im (a) on t he moduli space. As we have seen t his is in cont radict ion wit h t he requirement of posit ivity. We can t hus conclude t hat we need at least t hree singular point s. In [1] Seiberg and Wit t en assume t hat t his “minimal” choice is indeed correct . We will now see how t his assumpt ion leads t o a self-consist ent pict ure and allows t o det ermine explicit ly t he prepot ent ial.

Let us suppose t hat we have exact ly two singular point s (plus t he singu-lar point at infinity st udied before). First of all, due t o t he Z=2Z symmet ry of t he t heory which act s on t he moduli space sending u in−u, we can imme-diat ely say t hat t hese two singularit ies should be locat ed at opposit e point s in t he moduli space, which we can assume t o be u = _{±1. What is then} t heir physical int erpret at ion? T he key point is t hat at t he singularit ies t he effect ive descript ion in t erms of t he lagrangian (1.16) is not adequat e. Since we are dealing wit h a Wilsonian effect ive act ion, in which all massive fields (whose mass depends on u) are int egrat ed out (see [71, 72]), t he most nat ural explanat ion of t he singularit ies is t hat some of t he massive fields act ually be-come massless at t hese point s due t o st rong quant um correct ions. As argued in [1], t hese massless mult iplet s cannot be ident ified wit h gauge vect ormult i-plet s; t hey must t hen be BPS hypermult ii-plet s and t he only such mult ii-plet s in t he spect rum of our t heory are t he solit onic monopoles and dyons. Fol-lowing t his int erpret at ion we can now safely assume (wit h a suit able choice of convent ions) t hat one of t he singularit ies is due t o a massless monopole.

We are essent ially proposing t hat t he infrared dynamics at t he singularity is described by a SQED, in which t he massless elect ron is act ually a magnet ic monopole in t erms of t he original UV variables of t he t heory. Let us now see how we can det ermine t he monodromy mat rix at t he singularity.

M onodr om ies in t he st r ongly coupled r egion

Let us analyze t he monodromies at finit e u. Assuming t hat t he singularity at u= 1 is due t o a massless monopole, we can conclude from (1.24) t hat aD

must vanish t here and, t urning t o t he dual descript ion as explained in t he previous sect ion, we find equat ion (1.27). Looping around t he singularity count erclockwise in t he moduli space u_{− 1 → e}2πı(u_{− 1) we then find the}

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1.3 The Seiberg-Witten solution t ransformat ion law:

aD a → M1 aD a = aD a− 2aD ; M1 = 1 0 −2 1 ; (1.31) where M1 is our monodromy mat rix.

In order t o det ermine t he monodromy mat rix at t he second singularity u =−1, it is enough to observe that a loop around the point at infinity is equivalent t o t he composit ion of a loop around t he point u=_{−1 and another} one around u= 1. T his t ells us t hat M_∞ is equivalent t o t he composit ion of M1 and M−1. We can t hus conclude t hat t he following relat ion holds:

M_∞= M1M−1: (1.32)

We t hen immediat ely get

M₋₁ = −1 2 −2 3 : (1.33)

In order t o det ermine t he charges of t he part icle responsible for t his singu-larity we can proceed as follows [58]: T he monodromy due t o a dyon wit h charges(nm; ne) can be comput ed det ermining first of all t he duality t

rans-format ion which t urns it int o an elect ron(0; 1). Under an arbit rary SL(2; Z) t ransformat ion

α β γ δ

we have t he fol-lowing relat ions

aD a → aD+ a aD+ a ; nm ne → nm− ne − nm+ ne : Imposing t hat t he result ing dyon has charges(0; 1), we can det ermine t he duality t ransformat ion. In t hese variables t he monodromy associat ed t o t he dyon is equal t o t hat of an elect ron, which we can easily det ermine from

(1.26):

1 2 0 1

:

Going back t o t he original variables we find t he monodromy mat rix 1 + 2nmne 2n2e −2n2 m 1− 2nmne : (1.34)

Comparing now wit h (1.33) we conclude t hat t he dyon becoming massless at u=_{−1 has charges (1; −1).}

1.3.2 Solution of the model

We will now see how t he st ruct ure of t he quant um moduli space described above allows us t o det ermine explicit ly a(u) and aD(u) int erpret ing t hem as

periods of a suit able ellipt ic curve. 25

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M oduli space and ellipt ic cur ves

We have seen t hat t he moduli spaceM of SU(2) SYM is the complex plane wit h singularit ies at 1,_{−1 and ∞. We can parametrize it using the coordinate} u equal t o (semiclassically)_htrΦ2_{i and it is characterized by a Z}

2 symmet ry

t hat act s as u → −u. The quantities of interest for us a and aD can be

expressed as (many-valued) funct ions of u.

T he first key observat ion is t hat t he duality we have st udied before im-plies t hat we can const ruct a flat SL(2; Z) bundle V over _{M and the pair} (aD(u); a(u)) can be int erpret ed as a holomorphic sect ion of V . We have t he

following monodromies around 1,_{−1 and ∞:} M_∞= −1 2 0 ₋₁ ; M1= 1 0 −2 1 ; M₋₁= −1 2 −2 3 : We can not ice t hat t he monodromy mat rices generat e t he group Γ(2) of mat rices in SL(2; Z) congruent t o t he ident ity modulo 2, and t hat C_{\ −1; 1} coincides wit h t he quot ient of t he upper half plane H by Γ(2), where t he group act ion is defined by

→ a + b c + d; ∈ H ; a b c d ∈ Γ(2): (1.35) We can now est ablish a link wit h t he t heory of algebraic curves not icing t hat t he space H =Γ(2) also paramet rizes t he family of ellipt ic curves [73]

y2 = (x_{− 1)(x + 1)(x − u):} (1.36) T he idea is t hen t o associat e t o every point u in t he moduli space_{M a} genus one Riemann surface Eu det ermined by t he above equat ion. T he curve

becomes singular whenever two of t he branch point s in t he x plane coincide and t his precisely happens for u= 1;_{−1; ∞. Let us notice that (1.36) has a} Z₄ _{symmet ry which act s as u}_{→ −u; x → −x; y → ±{y. However, only a Z}₂ subgroup act s nont rivially on t he u plane. T hese are precisely t he propert ies charact erizing our t heory t hat we discussed before.

T he first de Rham cohomology group Vu = H1(Eu; C) of any t orus Eu

has dimension 2 and can be t hought t o as t he space of meromorphic (1,0)-forms wit h vanishing residues on Eu. We can t hen const ruct a vect or bundle

having as base space C_{\ {−1; 1} and as fibers V}u; it can be locally t rivialized

choosing two cont inuously varying cycles 1; 2 on Eu, in such a way t hat

t heir int ersect ion number is one, and int egrat ing over t hem a represent at ive of t he equivalence classes in H1(Eu; C). T he crucial point is t hat t his

bun-dle can be ident ified wit h V . Our sect ion (aD; a) can t hen be writ t en as

! = a1(u)!1+ a2(u)!2, where ! 1 and ! 2 are two independent element s in

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1.3 The Seiberg-Witten solution on Eu 1; 2 as before and set

aD = I γ1 ! ; a= I γ2 ! : (1.37)

Furt hermore, if we ident ify t he periods p1; p2 of t he t orus Eu, defined as

pi =

I

γi

dx

y i = 1; 2 wit h daD=du and da=du respect ively, we find

(u) = daD da = daD=du da=du = p1 p2 : (1.38)

T he posit ivity condit ion Im (u) > 0 follows now from t he second Riemann relat ion. T his ident ificat ion allows us t o find a solut ion which sat isfies all t he physical const raint s described above. Conversely, assuming t o have found a solut ion (u), we can det ermine for every value of u t he associat ed ellipt ic curve Eτ, and consequent ly it s periods. Since(aD; a) and (daD=du; da=du)

t ransform in t he same way under t he act ion of t he group SL(2; Z), t he family of ellipt ic curves det ermined by (u) has t he same monodromies as (1.36). We can t hus conclude t hat t hey coincide and t hat t he given (u) funct ion coincides wit h t he one provided by t he Seiberg-Wit t en solut ion.

From equat ion (1.37) and from t he definit ion of periods we can see t hat (1.38) is aut omat ically sat isfied if we impose t he relat ion

d!

du = f (u) dx

y :

All we need t o do now is t o det ermine f(u), mat ching t he asympt ot ic expan-sion of our solut ion wit h t he behaviour of aD and a in a neighborhood of t he

point s 1,−1 e ∞. Expanding at first order as in [1] we find that f = −√2=4 does t he job. Int egrat ing in u we can det ermine ! and t hus t he fundament al relat ions (wit h a suit able choice of periods 1; 2)

a= √ 2Z 1 −1 √ x_{− u} √ x2_{− 1}dx; (1.39) aD = √ 2Z u 1 √ x− u √ x2_{− 1}dx: (1.40)

1.3.3 Solution for SQCD with classical gauge groups

T he SW cur ves for N = 2 SQCD

T he idea of encoding t he infrared effect ive act ion in an auxiliary family of algebraic curves can be applied t o a wide class of _{N = 2 models, including}

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SYM t heory wit h any gauge group and any mat t er cont ent . We will now list in Table 1.1 t he curves for SQCD wit h classical gauge groups (t his is t he class of models we will be concerned about in t his t hesis) which were found in [74]-[80] using argument s similar t o t he one we have just reviewed. We refer t o t hese papers for t he det ailed derivat ion.

Gauge group SW curve SU (N ) y2 = P2 N(x)− 4 2N −NfQNf i=1(x + mi) U Sp(2N ) xy2 = [xPN(x) + 2 2N −Nf+2Q Nf i=1mi]2− 4 4N −2Nf+4Q Nf i=1(x− m 2 i) SO(2N ) y2 = xP2 N(x)− 4 4N −2Nf−4x 3QNf i=1(x− m 2 i) SO(2N + 1) y2 = xP2 N(x)− 4 4N −2Nf−4x 2QNf i=1(x− m 2 i)

Table 1.1: The SW curves for N = 2 SQCD with Nf flavors. PN(x) is a

monic poynomial of degree N . In the SU(N ) case the coefficient of the term xN−1 is set to zero. Turning it on gives the curve for U(N ) SQCD. There is no such constraint in the other cases.

Singular p oint s in t he m oduli space

We have seen t hat t here are (complex) codimension one singular submani-folds of t he Coulob branch where some BPS st at es become massless. T he Coulomb branch has dimension equal t o t he rank of t he gauge group so, when t he group is different from SU(2), it is possible t hat two (or more) such singular submanifolds int ersect , leading t o singular point s in which two (or more) different st at es become massless at t he same t ime. If t he Dirac product between t hese st at es is zero (i.e. t hey are relat ively local), t he low energy dynamics can be described in t erms of a local lagrangian and it is possible t o find a duality frame in which all t he st at es have zero magnet ic charges. We t hus have in t he infrared an abelian t heory wit h a bunch of elect rons.

If t he st at es are relat ively nonlocal such a duality frame does not exist and t he low energy t heory cannot be described by a local lagrangian. Such singularit ies usually signal t he presence of an int eract ing IR fixed point and are ubiquit ous inN = 2 theories. They are usually referred to as Argyres-Douglas (AD) point s, since Argyres and Argyres-Douglas described for t he first t ime such singularit ies in [81]. T heir analysis focuses on SU(3) SYM t heory, whose SW curve is

y2 = (x3_{− ux − v)}2_{− 4Λ}6:

T hey observed t hat set t ing u = 0 and v =_±2Λ3_{, t he curve degenerat es as}

y2 = x3(x3± 2Λ3_{), and in a neighbourhood of x = 0 can be approximat ed}

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1.4 Confinement in softly broken_{N = 2 gauge theories} simult aneously become massless at t hese point s, signalling t he presence of an infrared fixed point . T his analysis has been soon ext ended t o SU(2) SQCD [82] and t hen t o more general models [83, 84].

An import ant point I would like t o st ress is t hat it is not always enough t o check t hat in a neighbourhood of t he singular point t here are relat ively nonlocal st at es becoming light er and light er t o infer t hat t he low energy t heory at t he singular point is nonlocal: typically, when we include flavors, t here are point s in t he Coulomb branch where t he non-abelian gauge sym-met ry is part ly rest ored. Formally t he corresponding monopoles and dyons, which indeed are included in t he spect rum in a neighbourhood of t he singu-lar point , become massless. T his has been checked explicit ly for t he singusingu-lar point s in SU(3) and USp(4) SQCD wit h 4 flavors [85, 86]. T his however does not imply t hat t he t heory becomes nonlocal. We will infact propose a local lagrangian descript ion for t he singular point of USp(2N ) SQCD wit h four flavors in chapt er 5. T hese singular point s will play an import ant role lat er.

1.4 Confinement in softly broken

N = 2 gauge

the-ories

One of t he most import ant out comes of t he Seiberg-Wit t en solut ion is t hat it allows t o underst and t he phenomenon of confinement in a subclass of four dimensional gauge t heories, as explained by t he aut hors in [1]. Since t his will be a cent ral t opic in t his t hesis, let us review t he argument .

1.4.1 Confinement in SU(2) SYM theory

As we have remarked in t he int roduct ion,N = 2 gauge theories are just dis-t andis-t reladis-t ives of QCD and many key properdis-t ies indeed differ. For insdis-t ance, t hese models do not exhibit confinement . However, we can achieve a confin-ing t heory just makconfin-ing t he adjoint chiral mult iplet massive, t hus breakconfin-ing ext ended supersymmet ry. As long as t he mass is small we can underst and t hese models as pert urbat ions of t heN = 2 theory, whose behaviour in the IR is explicit ly under cont rol t hanks t o t he SW solut ion.

Focusing on t he by now familiar SU(2) SYM case, if we t urn on t he superpot ent ial t erm TrΦ2 t he degeneracy of vacua charact erizing t he_{N = 2} t heory disappears and we are left wit h two vacua, as can be seen e.g. using Wit t en’s index [87]. We can t ake int o account t he effect of t his pert urbat ion in t he IR adding t o t he effect ive lagrangian t he superpot ent ial U (it can be act ually shown t hat t his superpot ent ial is exact also for large [1]).

At a generic point in t he moduli space, where t he only mult iplet appear-ing in t he effect ive act ion is t he abelian vect ormult iplet , t his superpot ent ial has no minimum and t he corresponding vacuum is lift ed by t he_{N = 1}