Kobayashi-Maskawa Institute for the Origin of Particles and the Universe M. Shifman W.I. Fine Theoretical Physics Institute, University of Minnesota Non-Abelian strings in supersymmetric Yang-Mills: 4D-2D correspondence A. Yung, A. Gorsky, ... IPMU, December 3, 2012 W . Vinci, M. Nitta M. Shifman 1 Wednesday, December 5, 12
★ Discovery of non-Abelian strings in supersymmetric Yang-Mills and applications ★ ★ Beyond supersymmetry M. Shifman 2/ IPMU Wednesday, December 5, 12
Superconductor of the 2 nd kind Cooper pair condensate Abrikosov (ANO) vortex (flux tube) → B →→ S S N → N → magnet magnetic flux magnet Abelian ☚ DUAL MEISSNER EFFECT (Nambu-’ t Hooft-Mandelstam, ∼ 1975) M. Shifman 3 Wednesday, December 5, 12
☞ The Meissner effect: 1930s, 1960s Superconductor of the 2 nd kind Cooper pair condensate Abrikosov (ANO) vortex (flux tube) → B →→ S S N → N → magnet magnetic flux magnet Abelian ☚ DUAL MEISSNER EFFECT (Nambu-’ t Hooft-Mandelstam, ∼ 1975) M. Shifman 3 Wednesday, December 5, 12
condensed QCD magnetic string ✺ ✺ monopoles ✺ ✺ ✺ Qualitative explanation of color ✺ ✺ ✺ confinement: Dual Meissner effect: ✺ ✺ ✺ ✺ ✺ ✺ QCD ✺ vacuum ✺ ✺ ✺ ✺ Hanany, Strassler, Zaffaroni ’97 SW=Abelian strings, “wrong” confinement... M. Shifman 4 Wednesday, December 5, 12
condensed QCD magnetic string ✺ ✺ monopoles ✺ ✺ ✺ Qualitative explanation of color ✺ ✺ ✺ confinement: Dual Meissner effect: ✺ ✺ ✺ • ‘t Hooft, 1976 ✺ ✺ ✺ QCD ✺ vacuum ✺ ✺ ✺ ✺ Hanany, Strassler, Zaffaroni ’97 SW=Abelian strings, “wrong” confinement... M. Shifman 4 Wednesday, December 5, 12
condensed QCD magnetic string ✺ ✺ monopoles ✺ ✺ ✺ Qualitative explanation of color ✺ ✺ ✺ confinement: Dual Meissner effect: ✺ ✺ ✺ • ‘t Hooft, 1976 ✺ ✺ • Mandelstam ✺ QCD ✺ vacuum ✺ ✺ ✺ ✺ Hanany, Strassler, Zaffaroni ’97 SW=Abelian strings, “wrong” confinement... M. Shifman 4 Wednesday, December 5, 12
condensed QCD magnetic string ✺ ✺ monopoles ✺ ✺ ✺ Qualitative explanation of color ✺ ✺ ✺ confinement: Dual Meissner effect: ✺ ✺ ✺ • ‘t Hooft, 1976 ✺ ✺ • Mandelstam ✺ QCD ✺ • Nambu vacuum ✺ ✺ ✺ ✺ Hanany, Strassler, Zaffaroni ’97 SW=Abelian strings, “wrong” confinement... M. Shifman 4 Wednesday, December 5, 12
condensed QCD magnetic string ✺ ✺ monopoles ✺ ✺ ✺ Qualitative explanation of color ✺ ✺ ✺ confinement: Dual Meissner effect: ✺ ✺ ✺ • ‘t Hooft, 1976 ✺ ✺ • Mandelstam ✺ QCD ✺ • Nambu vacuum ✺ ✺ ✺ ✺ ✭ Non-Abelian theory, but Abelian flux tube ☹ Hanany, Strassler, Zaffaroni ’97 SW=Abelian strings, “wrong” confinement... M. Shifman 4 Wednesday, December 5, 12
"...[monopoles] turn to develop a non-zero vacuum expectation value. Since they carry color-magnetic charges, the vacuum will behave like a superconductor for color-magnetic charges. What does that mean? Remember that in ordinary electric superconductors, magnetic charges are connected by magnetic vortex lines ... We now have the opposite: it is the color charges that are connected by color-electric flux tubes." G. 't Hooft (1976) M. Shifman 5 Wednesday, December 5, 12
N =2 ⇒ add the second gluino + add a scalar gluon φ a (a complex scalar field in the adjoint) V( φ a ) = | ε abc φ b φ c | 2 In the vacuum φ 3 ≠ 0 while φ 1 = φ 2 =0 ⇒ SU(2) gauge → U(1) ⇒ Georgi-Glashow model ⇒ ‘t Hooft-Polyakov monopoles If | φ 3 | ≫ Λ , then monopoles are very heavy! M. Shifman 6 Wednesday, December 5, 12
☺ First demonstration of the dual Meissner effect: Seiberg & Witten, 1994 ☺ • gluons+complex scalar superpartner • two gluinos • Georgi-Glashow model built in analytic continuation M. Shifman 7 Wednesday, December 5, 12
☺ First demonstration of the dual Meissner effect: Seiberg & Witten, 1994 ☺ • gluons+complex scalar superpartner • two gluinos • Georgi-Glashow model built in SU(2) → U(1), monopoles ➔ analytic continuation M. Shifman 7 Wednesday, December 5, 12
☺ First demonstration of the dual Meissner effect: Seiberg & Witten, 1994 ☺ • gluons+complex scalar superpartner • two gluinos • Georgi-Glashow model built in SU(2) → U(1), monopoles ➔ Monopoles become light if | φ 3 | ≾ Λ ➔ At two points, massless! analytic continuation M. Shifman 7 Wednesday, December 5, 12
☺ First demonstration of the dual Meissner effect: Seiberg & Witten, 1994 ☺ • gluons+complex scalar superpartner • two gluinos • Georgi-Glashow model built in SU(2) → U(1), monopoles ➔ Monopoles become light if | φ 3 | ≾ Λ ➔ At two points, massless! N=1 deform. forces M condensatition ➔ analytic continuation M. Shifman 7 Wednesday, December 5, 12
☺ First demonstration of the dual Meissner effect: Seiberg & Witten, 1994 ☺ • gluons+complex scalar superpartner • two gluinos • Georgi-Glashow model built in SU(2) → U(1), monopoles ➔ Monopoles become light if | φ 3 | ≾ Λ ➔ At two points, massless! N=1 deform. forces M condensatition ➔ U(1) broken, electric flux tube formed ➔ analytic continuation M. Shifman 7 Wednesday, December 5, 12
☺ First demonstration of the dual Meissner effect: Seiberg & Witten, 1994 ☺ • gluons+complex scalar superpartner • two gluinos • Georgi-Glashow model built in SU(2) → U(1), monopoles ➔ Monopoles become light if | φ 3 | ≾ Λ ➔ At two points, massless! N=1 deform. forces M condensatition ➔ U(1) broken, electric flux tube formed ➔ ☹ ☹ Dynamical Abelization ... dual Abrikosov string analytic continuation M. Shifman 7 Wednesday, December 5, 12
M. Shifman 8 Wednesday, December 5, 12
☞ Non-Abelian Strings, 2003 → Now M. Shifman 8 Wednesday, December 5, 12
“Non-Abelian” string is formed if all non- Abelian degrees of freedom participate in dynamics at the scale of string formation 2003: Hanany, Tong Auzzi et al. Yung + M.S. classically gapless excitation SU(2)/U(1) = CP(1) ∼ O(3) sigma model M. Shifman 9 Wednesday, December 5, 12
Prototype model � 1 � 2 + 1 ( F µ ν ) 2 + 1 � � d 4 x F a | D µ a a | 2 = S µ ν 4 g 2 4 g 2 g 2 2 1 2 Tr ( ∇ µ Φ ) † ( ∇ µ Φ ) + g 2 �� 2 + g 2 � 2 � � � � � 2 1 Φ † T a Φ Φ † Φ + Tr Tr − N ξ 2 8 √ � 2 + 1 ✓ ϕ 11 ϕ 12 i θ ◆ � a a T a Φ + Φ � � µ ν ˜ 32 π 2 F a F a µ ν + 2Tr 2 M , Φ = � � ϕ 21 ϕ 22 � ✓ m 0 ◆ M = 0 − m Basic idea: • Color-flavor locking in the bulk → Global symmetry G; Φ = √ ξ × I • G is broken down to H on the given string; • G/H coset; G/H sigma model on the world sheet. M. Shifman 10 Wednesday, December 5, 12
Prototype model � 1 � 2 + 1 ( F µ ν ) 2 + 1 � � d 4 x F a | D µ a a | 2 = S µ ν 4 g 2 4 g 2 g 2 2 1 2 Tr ( ∇ µ Φ ) † ( ∇ µ Φ ) + g 2 �� 2 + g 2 � 2 � � � � � 2 1 Φ † T a Φ Φ † Φ + Tr Tr − N ξ 2 8 √ � 2 + 1 ✓ ϕ 11 ϕ 12 i θ ◆ � a a T a Φ + Φ � � µ ν ˜ 32 π 2 F a F a µ ν + 2Tr 2 M , Φ = � � ϕ 21 ϕ 22 � U(2) gauge group, 2 flavors of (scalar) quarks ✓ m 0 ◆ M = SU(2) Gluons A a μ + U(1) photon + gluinos+ photino 0 − m Basic idea: • Color-flavor locking in the bulk → Global symmetry G; Φ = √ ξ × I • G is broken down to H on the given string; • G/H coset; G/H sigma model on the world sheet. M. Shifman 10 Wednesday, December 5, 12
✭ ANO strings are there because of U(1)! ✭ New strings: π 1 (U(1) × SU(2)) nontrivial due to Z 2 center of SU(2) z ✓ 1 0 ◆ ξ e i α ANO p 0 1 string x T=4 π ξ y α ✓ e i α 0 ◆ Non-Abelian p ξ 0 1 T U(1) ±T 3SU(2) x 0 ← string center in perp. plane T=2 π ξ SU(2)/U(1) ← orientational moduli; O(3) σ model M. Shifman 11 Wednesday, December 5, 12
✭ ANO strings are there because of U(1)! ✭ New strings: π 1 (SU(2) × U(1)) = Z 2 : rotate by π around 3-d axis in SU(2) → -1; another -1 rotate by π in U(1) π 1 (U(1) × SU(2)) nontrivial due to Z 2 center of SU(2) z ✓ 1 0 ◆ ξ e i α ANO p 0 1 string x T=4 π ξ y α ✓ e i α 0 ◆ Non-Abelian p ξ 0 1 T U(1) ±T 3SU(2) x 0 ← string center in perp. plane T=2 π ξ SU(2)/U(1) ← orientational moduli; O(3) σ model M. Shifman 11 Wednesday, December 5, 12
Worldsheet theory ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ S 2 Global SU(2) is gone! U(1) remains intact Two vacua= 2 degenerate strings ⇢ 2 CP(1) model with φ∂ µ φ − ( ∆ m ) 2 ¯ ∂ µ ¯ φφ � Z twisted mass S = d 2 x + fermions ( 1 + ¯ g 2 φφ ) 2 M. Shifman 12 Wednesday, December 5, 12
∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ Yung + M.S. = kink Z string junction 2 Hanany, Tong B B 3 3 z B B Evolution in dimensionless parameter m 2 / ξ M. Shifman 13 Wednesday, December 5, 12
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