Critical Solitons in Gauge Theories ⇔ Srings /D branes/Dualities M. Shifman Theoretical Physics Institute, University of Minnesota GGI-2006 “New Directions Beyond the Standard Model in Field & String Theory” ★ A. Yung, ... ★ The first GGI Workshop M. Shifman
Can there be ANY symmetry 1970’ s: YES! between bosons and fermions? Golfand & Likhtman, 71 Wess & Zumino, 73 x � 2 = 0 y “fermion” direction In 1+3 dimensions of the superspace → { t , x , y , z ; � i { t , x , y , z } − � } � − → M. Shifman
E. Witten: Supersymmetry, if it holds in nature, is part of the quantum structure of space and time. In everyday life, we measure space and time by numbers, “It is three o’clock, the elevation is ten meters,” and so on. Numbers are classical concepts, known to humans since long before quantum mechanics. The discovery of quantum mechanics changed our understanding of almost everything in physics, but our basic way of thinking about space and time has not yet been affected. Showing that nature is supersymmetric would change that, by revealing a quantum dimension of space and time, not measurable by ordinary numbers. This quantum dimension would be manifested in the existence of new elementary particles, which would be produced in accelerators and whose behavior would be governed by supersymmetric laws. M. Shifman
E = mc 2 Cultural icon of the 20th century � , Q � } = 2 � µ { ¯ Of the 21st ? Q ˙ �� P µ ˙ M. Shifman
? L = − 1 µ � G µ � a + i ¯ 4 g 2 G a � / D � b 2 f gluon gluino ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ supersymmetric SUSY Yang-Mills gluodynamics M. Shifman
brane quark T~ 1/gs string In some 20 years we were very successful in producing a raw first draft of the world from string theory. It turned out to be no- toriously difficult to pass to the second draft. This has not yet been done. -- E.Witten Duality between ST & YM ↓↓↓ YM must support domain walls of D-brane type & non-Abelian strings ending on the walls and trapping flux sources !!!! M. Shifman
✷✷✷ Why do we need SUSY & “stringy” ideas? ✷✷✷ A tool for solving otherwise unsolvable problems of strong coupling dynamics Costs nothing; Enormous progress since mid-1990’ s! ▲▲▲▲▲▲▲▲▲ Comes at a price: not quite QCD, but close ... M. Shifman
Critical = BPS saturated (Bogomol’nyi, Prasad, Sommerfeld) ✫ BEFORE SUSY Topological charges = central charges (Witten, Olive, 1976) ✵ {Q, Q} = P + C If C=0, all Q’ s are broken. ✵ If C ≠ 0, some Q’ s may survive! 1/2 BPS, 1/4 BPS, .... M (or T) ≡ C ✵ In many instances C’ s are exactly calculable ✵
✷✷✷ Non-Abelian Strings ✷✷✷ Abrikosov-Nielsen-Olesen string: Abelian ✵ Gauge group = U(1); Electric charge condenses; Magnetic flux is trapped in a tube and quantized; No internal degrees of freedom besides position of the tube center (e.g. Seiberg-Witten solution ⇒ ANO string) ✵ Non-Abelian strings: Assume the BULK theory has a global symmetry G unbroken in the vacuum. Assume G → H on the string; Coset G / H of orientational moduli. (Hanany-Tong, nongauge; Auzzi et al. B gauge set-up) antimonopole monopole
✷ Basic bulk theory: N =2 SQCD with U(N) gauge and Nf = N ✷ Example: U(2), two flavors; ✵ Parameters: m1 = m2, Fayet-Iliopoulos ξ � 1 µ ν + 1 Z g 2 | ∂ µ a | 2 + ¯ q A ∇ µ q A + ¯ d 4 x 4 g 2 F 2 q A q A ∇ µ ¯ ∇ µ ¯ ∇ µ ˜ S = ˜ + g 2 + g 2 2 � � 2 + 1 2 ( | q A | 2 + | ˜ � √ � | q A | 2 − | ˜ q A | 2 − ξ � ˜ q A | 2 ) � q A q A � � � � a + 2 m A � � , 8 2 � x 1 0 Large SU(2) → U( 1 ) qA circle k = √ξ /2 ! 0 1 String z axis x x 0 ei α y
U ( N ) gauge × SU ( N ) flavor → SU ( N ) global Color-flavor locked vacuum Weak coupling in the bulk ! � ≫ � 2 ) ✷ ( If Non-Abelian Strings: ✷ � 1 [ SU ( N ) × U ( 1 ) / Z N ] � = 0 1 0 ... 0 y Flux =1/N Abrikosov 0 1 ... 0 � � Tension =1/N Abrikosov string � string = � x ......... ... 0 0 ... e i �
SU(2)/U( 1 ) = CP( 1 )~O(3) sigma model classically gapless excitation g2 of the bulk theory is matched by g2 of the 2D sigma model, and so do Λ ’ s; 2D theory gets strongly coupled; mass gap generated; 2 vacua. Kink = trapped monopole M ➾ ➾ 1/2 magnetic flux 1/2 magnetic flux What does that mean in the dual (QCD) language?
Gluelump (non-SUSY version) : 2-string world sheet symmetric string antisymmetric string gluelump time gluelump “boundary”
✷✷✷ Branes/ Domain walls ✷✷✷ SUSY gluodynamics (1996, G. Dvali +MS) ✵ N vacua labeled by < λλ >=-6 N Λ 3 exp(2 π ik/N ) Im < > �� x {Q α Q β }= Σαβ N( Δ < λλ >)/8 π 2 elementary x x wall x x Re < > �� k-wall quantum anomaly x x x Twall= N Λ 3 N vacua for SU(N) ~ 1/gs D brane, Witten ‘97
Acharya & Vafa, from wrapped D brane + duality: ✵ World-volume theory = U(k) gauge theory (k=1 for elementary wall); ✵ Field content of N = 2; Level-N Chern-Simons term breaks N = 2 to N = 1; ✵ # of distinct k-walls = N!/k!(N-k)! Confirmed in field theory (Ritz, Vainshtein+MS) Stack of k noninteract. walls must support U(k) gauge fields!
Basic Elements of the Construction ( N =2 bulk): Elementary Domain wall ★ ( m1 ≠ m2 ) implementation of DS idea $%&'()*+,$%'& 0 1 0 " # Two edges (domains E ) of ! ! # " the width ~ √ 1/ ξ are separated by a broad middle band M of the width R ~ Δ m/(g2 ξ ). Text The tension T = Δ m ξ . . 0 Moduli: / z0 and σ ⇐ relative - ! !"# ! !"# ! ! phase between φ 1 and φ 2 σ dualizes 3D photon a la Polyakov
(boojums) Wall-string junctions ★★ ( SY,ST,ASY ) "Boojum" comes from M L.Carroll's children's book "Hunting of the Snark." 1/4 BPS Apparently, it is fun to hunt a snark, but if the snark turns out to be a boojum, you are in trouble! Condensed matter physicists adopted the name to describe solitonic objects of the wall-string junction type in helium-3. Also: The boojum tree (Mexico) is the strangest plant imaginable. For most of the year it is leafless and looks like a giant upturned turnip. G.Sykes, found it in 1922 and said, referring to Carrol ``It must be a boojum!" The Spanish common name for this tree is Cirio, referring to its candle-like appearance. Monopole=dual charge M. Shifman
Polyakov’s 3D confinement ★★★ ( ASY ) V = β 2 ξ ∆ W = β Q 1 ˜ m cos 2 σ + O ( β 3 ) Q 2 − ˜ Q 1 Q 2 � � √ breaks N = 2 to N = 1 2 vac II v a c I I string b string a v a c I vac I vac I String from the bulk vac II vac I
4D 3D !"## '%(&$) ++ ++ '%(&$) - - - - "$%& ! !"## 8 supercharges 4 supercharges walls & flux tubes SQED; CS M. Shifman
World-volume theory on the wall: � 2 � 2 T w 2 ( ∂ n z 0 ) 2 − 1 = 1 2 e 2 ( ∂ n a 2 + 1 ) 2 − 1 � � F ( 2 + 1 ) F ( 2 + 1 ) mn mn 4 e 2 4 e 2 ! $ # " In addition, the same logarithm ! ! ! ! " ! domain wall flux tube ! & % ! '(& "!!! # # = e 2 ! x i $ # ☺☺ 2 + 1 F 2 + 1 " !" !!! ! 0 i r 2 2 π Z r f Z r f ( F 0 i ) 2 2 π rdr = πξ r = πξ ∆ m ln r f 1 dr E G ( 2 + 1 ) = 2 e 2 ∆ m . r 0 r 0 r 0 2 + 1
T, ASY !"## "(%' ! $%&'() ☺ String: (ne , ns)=(+1 ,+1) ☺ Antistring: (ne, ns)=(-1,+1) $%&'() '()) !"#$%& n e = + 1 , incoming flux , n e = − 1 , outgoing flux n s = + 1 , string from the right , n s = − 1 , string from the left (%"$ ! !"#$%& M. Shifman
L >> l L = 2 πξ a − ≡ 1 � � a ( 2 ) 2 + 1 − a ( 1 ) !"## ℓ √ √ 2 + 1 2 2 '%(&$) n ≡ 1 � � A ( 1 ) n − A ( 2 ) A − S~ √ n 2 '%(&$) l Crucial tests: √ "$%& ! !"## S 2 � a − � = 2 πξ ℓ m s = ☺ √ m= 2 πξ L / s = 2 πξ ( L − ℓ ) If 2 then m ˜ ☺ “real mass” − 1 mn F − mn + 1 2 e 2 ( ∂ n a − ) 2 + | D n s | 2 + s | 2 � 2 ss − 2 ( m − a − ) 2 ¯ � 2 − 2 a 2 � ˜ s − e 2 � | s | 2 − | ˜ � � D n ˜ − ¯ s ˜ ˜ 4 e 2 F − s
Physics of the world-volume theory √ 1 3D ferm. e 2 ¯ ψ i � D ψ + ¯ ψψ +( m − a − ) ¯ ψ i � ˜ � � λ − i � ∂λ − + ¯ ˜ D ˜ ψ − a − ¯ ψ ˜ ˜ ψ 2 part 4 supercharges 1 � � Induced CS sign ( a )+ sign ( m − a ) ε nmk A − n ∂ m A − k 4 π SUSY D = D � � | m − a − | − | a − | 2 π ( m − 2 a − ) 2 π After k + e 2 2 e 2 ( ∂ n a − ) 2 − 1 1 mn ) 2 + 1 integrating out 8 π 2 ( 2 a − − m ) 2 4 e 2 ( F − 2 πε nmk A − n ∂ m A − S and S~ M. Shifman
Approximation not applic. if l is close Quantum W-antiW V(l) interaction to 0 or L Approximation applic. on plateau � a − � = m ℓ = L 2 , 2 L R l Stabilization! L/2 m a = e 2 Classical W-antiW π ≪ m s interaction Infinite rigidity of strings; induces CS M. Shifman
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