dynamic in the University

SUSY  N=1  ADE  Dynamics     DK,  J  Lin     arXiv:  1401.4168,  1402.5411     See  also  J  Lin’s  talk     IntroducJon   In  the  last  twenty  years  there  has  been  important   progress  in  supersymmetric  field  theory  At  the  same   Jme,  many  qualitaJve  and  quanJtaJve  phenomena   remain  mysterious  Today,  I’d  like  to  discuss  an   example  of  this,  which  involves  a  class  of  theories  that   naturally  generalizes  SQCD  and  follows  an  ADE   classificaJon     Outline   •   A      1        (N  Seiberg,  1994)   •   A        k      (DK,  A  Schwimmer,  N  Seiberg,  1995)   •   D        k        (J  Brodie,  1996)   •   ADE    (K  Intriligator,  B  Wecht,  2003)     •   E        7          (DK,  J  Lin,  2014)   A1 N=1  SQCD  is  a  gauge  theory  with  gauge  group    SU          (N          c  )    ,   and      N      f      flavors  of  chiral  superfields  that  transform  in   the  fundamental  representaJon  of  the  gauge   group,      Q,          Q  ˜        The  low  energy  dynamics  of  this  theory   varies  with      N      f    ,    N      c      as  follows:         free electric conformal 1/3 free magnetic 2/3 runaway Nc / Nf Although  the  gauge  coupling  runs  with  the  scale,  one   can  think  of  the  discrete  parameter     x = Nc /Nf   as  a  `t  Hoob  coupling  that  measures  the  strength  of                     gauge  interacJons    in  the  infrared  (compare  to  the  `t   Hoob  coupling              of  N=4  SYM,  and  to  the  discrete   coupling      N        c    /k            of  CS  theory)       •  For  x1/2;  thus   adding  W  to  the  Lagrangian  leads  to  a  non-­‐trivial   fixed  point     For  k>2,  the  superpotenJal  is  superficially   irrelevant,  however  it  turns  out  that  for  sufficiently   large  x,  gauge  interacJons  reduce  its  dimension   enough  that  it  become  relevant  in  the  IR  for  all  k       A  stable  supersymmetric  vacuum  only  exists  in  the   range     x  k   erm constraints of the superpotential (1.5) are •  Quantum  constraints  on  chiral  operators:  the  F-­‐term   termconstraints   constraints ooff  tthe superpotential (1.5) are khe  D s{X, uperpotenJal   are     X = Y-­‐series   ; Y } = Xk = Y ; {X, Y } = tors are constructed from dressed quarks, ⇥Q, where ⇥ = ⇥(X, Y   e adjoint fields, which satisfies the constraints (1.9) Superficially ators are constructed from dressed quarks, ⇥Q, where ⇥ = ⇥(X, Naively,   one  can  use   these   to  construct   chiral   operators   toof   the infinite set t he   f orm      Q⇥  e    which      lj    Q          w ith       the constraints (1.9) Superficial he adjoint fields, satisfies d to the infinite set ⇥lj = X l Y j ; l = 1, · · · , k ; j = 1, 2, · · · ⇥lj = X l Y j ; l = 1, · · · , k ; j = 1, 2, · · · e set (1.10) isincompaJble   actually further truncated to d auality,   finite one, since Y = This   looks   with   Brodie’s   according   3j=1,  2,  3  s2hould  survive   k k to   w hich   o nly   o perators   w ith   term constraints (1.9) one has Y = Y · Y = Y · X = X he set (1.10) is actually further truncated to a finite one, since Y dex j in (1.10) runs only over the 3values j = 1, 2, 3, ink agreeme F-term constraints (1.9) one has Y = Y · Y = Y · X = X k odie’s onlyruns requires mesons these ndex jduality in (1.10) only over the with values j =quantum 1, 2, 3, innumber agreem erodie’s of theduality corresponding truncated set of dressed only requires mesons with thesequarks quantum numbe For  odd  k  it’s  actually  OK,  since  one  can  use  the  F-­‐term   equaJons  to  conclude  that       k k Y = Y · Y = Y · X = X · Y = Y =0     For  even  k,  the  situaJon  is  more  puzzling  On  the  one   hand,  at  least  classically  the  constraint      Y      3      =        0          is  not   valid,  but  on  the  other  it  is  required  by  the  duality       Brodie    proposed  that  in  that  case,  the  constraint   appears  quantum  mechanically,  although  its  origin  is   not  well  understood       ADE   The  understanding  of  the  above  theories  improved   significantly  aber  the  advent  of  a-­‐maximizaJon  (by  K   Intriligator  and  B  Wecht)  in  2003  These  authors   classified  all  possible  fixed  points  that  can  be  obtained   in  N=1  supersymmetric  gauge  theory  with  SU(N)  gauge   group  and  majer  in  the  fundamental  and  adjoint   representaJons     rared behavior for non-zero Nf one must take Na = (or smaller) ors of [8] considered models with two adjoint chiral superfields X and They  showed  that  such  fixed  points  have  an  ADE   W = W (X, Y ), and a tunable number of fundamentals Nf Inter classificaJon:   at non-trivial fixed points correspond to superpotentials with an AD   b O b A b D b E Ak Dk+2 E6 E7 E8 WO b=0 WAb = TrY 2 WD = TrXY b WEb = TrY WAk = Tr(X k+1 + Y ) WDk+2 = Tr(X k+1 + XY ) WE6 = Tr(Y + X ) WE7 = Tr(Y + Y X ) WE8 = Tr(Y + X ) b A, b D, b E) b are fixe naturally split into two classes The first four (O, •  The  ADE  classificaJon  is  due  to  gauge  dynamics   b •  The      O,        A,  b        D,  b        E  b          theories  are  interesJng,  but  we  will   not  discuss  them  further  today     •  The    A      k    ,    D        k      theories  are  those  reviewed  above     •   Our  goal  in  the  rest  of  this  talk  will  be  to  try    to   understand  the  excepJonal  theories       The E7 theory is again N = SQCD with two adjoint chiral superfields X, Y , b E W = TrY + TrY X h the superpotential 3 (4 is determines the R-charges of the fields to be those listed in Table The correspondi The  transformaJon  properJes  of  the  various  gauge   ble theory  fields  under  the  symmetries  are:   gle-particle index (1.5) is given by eq (3.2) but with rX , rY , rQ taking the values fro Field SU (Nc ) SU (Nf ) SU (Nf ) U (1)B Q f f 1 e Q f f V adj 1 0 X adj 1 Y adj 1 U (1)R 1 Nc Nf 1 Nc Nf Table 5: The field content of the E7 electric theory x e Q (1, Nf , 1, ) ory is again N = SQCD with two adjoint chiral superfields X •  The  superpotenJal  for  the  4 adjoints  is   X (1, 1, 0, ) otential   W = TrY + TrY X3 Y (1, 1, 0, ) the of the fields to bethat   those listedfin Table •  R-charges  The  F-­‐term   constraints   follow   rom   this  5 The corres ntial (2.1) leads to a truncation of the chiral ring The equations are  (3.2) but with rX , rY , rQ taking the val ndex superpotenJal   (1.5) is given by  eq et Y = X3 X Y + XY X + Y X = (Nc ) and SUchose (Nf ) a convenient SU (Nf ) relative U (1)Bnormalization U (1)R of ectedSU D-terms •  Classical  chiral  meson  operators  take  the   an appropriate normalization of the Kahler potential) As 1inNcthe f f 1 e form      Q⇥Q                    ,  with     Nf ect an important role to be played by the dressed quarks ⇥(X, Nc Y f 1n f 1 Nf ⇥ = X , Y X n , XY X n , Y XY X n adj 0 •  One  can  show  that  at  large  coupling  the  UV   variables  in  terms  of  which  the  theory  is   defined  must  break  down,  like  in  the  other   examples     •  We  assume  that  the  strong  coupling  region  is   governed  by  a  dual  descripJon  similar  to  the   other  cases   The  quantum  numbers  of  the  dual  fields  are   taken   to  bassume e:   the existence of a magnetic dual with gauge group SU (Ne ) = Thus, we again SU (↵N   N ) for an unknown integer ↵, and the fields   e c f c Field SU (Nc ) SU (Nf ) SU (Nf ) U (1)B q f f qe f f fc Nc /N adj 1 0 adj 1 adj 1 f f 2rQ + rj Ve e X Ye Mj , j = 1, ↵ fc N c /N U (1)R 1 ec 1N Nf ec 1N Nf Table 6: The “conjectured” field content of the E7 magnetic theory Here rj are the U (1)R charges of ⇥j and, as before, we not place any constraints on •  The  rank  of  the  dual  gauge  group  must  take  the   general  form       ec = ↵Nf N Nc   where    ↵          is  the    number  of  gauge  singlet  mesons  in   the  magneJc  theory  This  follows  from    SU            (N        f      )  3      `t   Hoob  anomaly  matching    This  number,  as  well  as   the  R-­‐charges  of  these  mesons,    r    j    ,  are  kept  free         •  To  determine  them,  we  demand  that  the   superconformal  indices  of  the  electric  and   magneJc  theories  coincide     •  In  general,  these  indices  are  very  complicated   funcJons  of  the  chemical  potenJals,  but  Dolan   and  Osborn  observed  that  they  simplify   significantly  in  the  large  N  Veneziano  limit                                                                                                (See  J  Lin’s  talk)       ctions (1.9) of the electric and magnetic single-letter indic This  gives  a  constraint  of  the  form   ↵   ↵ X + t9 + t9 + + t rj t = 1   + t9 t3 t9 t9 + t9 + t9 j=1 ith (2.12) gives n = 15 and α = 30 Thus, duality relat   N ) which   to the magnetic one SU (30N − N ) nconly be satisfied with finite ↵ if every root of the denomi f c ↵ = 30 determines      ↵,          r    j        One  finds                                  ,   if a aof duality ofset   theof   sort incthe and D hthat a root numerator, are ↵thbAe   roots of series unity and    cthe ertain      r    j  which  found ,  which   an   thought   of   isH ust impose on fthe (2.7) aonstraint   quantum of as  isarising   rom   applying   the       constraint this in fact thespectrum case when ↵ c= 30.4 The r.h.s is the orm   trj , with theaYrjXcoinciding with the meson spectrum + bXY X =   her support for the picture proposed in [7] nstants thatfull   are not determined by the above consideratio to   t he   l ist   o f   o perators     ng the index to the level of the constraint, one encounte the  infinite set of mesons to a finite set, which is (uniqu n the D series, as discussed in Appendix B y Thus,  we  conclude  that  the  dual  of  a    SU              (N          c    )    theory     has  gauge  group    SU            (30N                    f                N      c    )      This  proposal   saJsfies  a  number  of  detailed  consistency    condiJons:     •  There  are  precisely  30  mesons,  and  the  list  of    r      j      is   such  that    one  can  write  a  magneJc  superpotenJal   for  the  magneJc  meson  fields           •  `t  Hoob  anomaly  matching  is  non-­‐trivially  saJsfied     •  PotenJal  unitarity  violaJons  are  resolved   Open  problems   •  E        6    ,    E        8      :      we  saw  that    the    E        7      theory  has  a  very   similar  structure  to  the  A  and  D  series  ones  Using   the  superconformal  index  one  can  show  that    this   cannot  be  the  case  for  the  remaining  excepJonal   theories  Thus,  in  these  cases  there  must  be   qualitaJve  new  elements  What  are  they?   •  In  some  of  the  theories  we  found  that  there  must  be   quantum  constraints  on  the  chiral  ring  Can  one   derive  them?   •  The  D  and  E  series  seem  to  involve  some  type  of   matrix  singularity  theory,  which  is  important  for   studying  deformaJons  of  the  adjoint  superpotenJal   How  does  it  work?   •  Can  one  relate  the  dynamical  ADE  structure  that   arises  in  these  theories  to  a  geometric  or  algebraic   ADE  structure,  e.g  by  embedding  these  theories  in   string  theory?   ... (2.7) aonstraint   quantum of as  isarising   rom   applying   the       constraint this in fact thespectrum case when ↵ c= 30.4 The r.h.s is the orm   trj , with theaYrjXcoinciding with the meson...  unresolved   complicaJons in the  analysis  of the  vacuum   structure  of the  theory in the  presence  of  general   deformaJons  of the  superpotenJal   erm constraints of the superpotential... understand the  excepJonal  theories       The E7 theory is again N = SQCD with two adjoint chiral superfields X, Y , b E W = TrY + TrY X h the superpotential 3 (4 is determines the R-charges of the