Flux tubes, domain walls and orientifold planar equivalence - PowerPoint PPT Presentation

Flux tubes, domain walls and orientifold planar equivalence Agostino Patella CERN GGI, 5 May 2011 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 1 / 19

Introduction Orientifold planar equivalence Orientifold planar equivalence OrQCD SU ( N ) gauge theory ( λ = g 2 N fixed) with N f Dirac fermions in the antisymmetric representation is equivalent in the large- N limit and in a common sector to AdQCD SU ( N ) gauge theory ( λ = g 2 N fixed) with N f Majorana fermions in the adjoint representation if and only if C-symmetry is not spontaneously broken. Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 2 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Z  Z ff e − N 2 W ( J ) = D A D ψ D ¯ − S ( A , ψ, ¯ ψ ) + N 2 J ( x ) O ( x ) d 4 x ψ exp N →∞ W Or ( J ) = N →∞ W Ad ( J ) lim lim Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Z  Z ff e − N 2 W ( J ) = D A D ψ D ¯ − S ( A , ψ, ¯ ψ ) + N 2 J ( x ) O ( x ) d 4 x ψ exp � O ( x 1 ) · · · O ( x n ) � c , Or � O ( x 1 ) · · · O ( x n ) � c , Ad = lim lim N 2 − 2 n N 2 − 2 n N →∞ N →∞ Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Z  Z ff e − N 2 W ( J ) = D A D ψ D ¯ − S ( A , ψ, ¯ ψ ) + N 2 J ( x ) O ( x ) d 4 x ψ exp � O ( x 1 ) · · · O ( x n ) � c , Or � O ( x 1 ) · · · O ( x n ) � c , Ad = lim lim N 2 − 2 n N 2 − 2 n N →∞ N →∞ N →∞ � 0 | O | 0 � Or = N →∞ � 0 | O | 0 � Ad lim lim Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Z  Z ff e − N 2 W ( J ) = D A D ψ D ¯ − S ( A , ψ, ¯ ψ ) + N 2 J ( x ) O ( x ) d 4 x ψ exp � O ( x 1 ) · · · O ( x n ) � c , Or � O ( x 1 ) · · · O ( x n ) � c , Ad = lim lim N 2 − 2 n N 2 − 2 n N →∞ N →∞ N →∞ � a | O | b � Or = N →∞ � a | O | b � Ad lim lim Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Z  Z ff e − N 2 W ( J ) = D A D ψ D ¯ − S ( A , ψ, ¯ ψ ) + N 2 J ( x ) O ( x ) d 4 x ψ exp � O ( x 1 ) · · · O ( x n ) � c , Or � O ( x 1 ) · · · O ( x n ) � c , Ad = lim lim N 2 − 2 n N 2 − 2 n N →∞ N →∞ N →∞ � a | O | b � Or = N →∞ � a | O | b � Ad lim lim N →∞ � a | e − tH | b � Or = N →∞ � a | e − tH | b � Ad lim lim Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD B OSONIC C- EVEN F ERMIONIC C- EVEN B OSONIC C- ODD F ERMIONIC C- ODD ~ w ­ OrQCD B OSONIC C- EVEN B OSONIC C- ODD C OMMON S ECTOR Z  Z ff e − N 2 W ( J ) = D A D ψ D ¯ − S ( A , ψ, ¯ ψ ) + N 2 J ( x ) O ( x ) d 4 x ψ exp � O ( x 1 ) · · · O ( x n ) � c , Or � O ( x 1 ) · · · O ( x n ) � c , Ad = lim lim N 2 − 2 n N 2 − 2 n N →∞ N →∞ N →∞ � a | O | b � Or = N →∞ � a | O | b � Ad lim lim N →∞ � a | e − tH | b � Or = N →∞ � a | e − tH | b � Ad lim lim Symmetries inside the common sector are the same in AdjQCD and OrientiQCD. Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

Introduction Overview Overview Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 4 / 19

Center symmetry Overview Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 5 / 19

Center symmetry Symmetry (mis)matching Symmetry (mis)matching A center transformation around a compact dimension ˆ z is a local gauge transformation, that is periodic modulo an element of the center Z N . A µ ( x ) → Ω( x ) A µ ( x )Ω † ( x ) + i Ω( x ) ∂ µ Ω † ( x ) ψ ( x ) → R [Ω( x )] ψ ( x ) 2 π ik Ω( x + L ˆ z ) = Ω( x ) e N 2 π ik tr W → e tr W N Even- N OrQCD: Z 2 Odd- N OrQCD: – AdQCD: Z N Only Z 2 maps the common sector into itself OPE ⇒ OrQCD( N = ∞ ) has at least a Z 2 symmetry Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 6 / 19

Center symmetry Symmetry (mis)matching Symmetry (mis)matching Z N symmetry � tr W � = 0 � tr ( W 2 ) � = 0 � tr ( W 3 ) � = 0 . . . � tr ( W N ) � � = 0 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19

Center symmetry Symmetry (mis)matching Symmetry (mis)matching Z N symmetry Z 2 symmetry � tr W � = 0 � tr W � = 0 � tr ( W 2 ) � = 0 � tr ( W 2 ) � � = 0 � tr ( W 3 ) � = 0 � tr ( W 3 ) � = 0 . . . . . . � tr ( W N ) � � = 0 � tr ( W N ) � � = 0 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19

Center symmetry Symmetry (mis)matching Symmetry (mis)matching Z N symmetry Z 2 symmetry OrQCD( N = ∞ ) N � tr W � Or = 1 1 � tr W � = 0 � tr W � = 0 N � tr W � Adj = 0 � tr ( W 2 ) � = 0 � tr ( W 2 ) � � = 0 N � tr ( W 2 ) � Or = 1 1 N � tr ( W 2 ) � Adj = 0 � tr ( W 3 ) � = 0 � tr ( W 3 ) � = 0 N � tr ( W 3 ) � Or = 1 1 . . N � tr ( W 3 ) � Adj = 0 . . . . � tr ( W N ) � � = 0 � tr ( W N ) � � = 0 . . . Is Z N a symmetry for OrQCD( N = ∞ )? Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m T ℓ is an operator with defined angular momentum ℓ � 0 | T 0 | 0 � � = 0 � 0 | T 1 | 0 � = 0 � 0 | T 2 | 0 � � = 0 . . . Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 1 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 0 . 5 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 0 . 4 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 0 . 3 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 0 . 2 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 0 . 1 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

Center symmetry A quantum mechanical analogy A quantum mechanical analogy p 2 x + p 2 + 1 x x 2 + 1 y 2 m ω 2 2 m ω 2 y y 2 H = 2 m 5 4 3 2 1 1.5 1 0.5 0 -1.5 0 -1 -0.5 -0.5 0 0.5 -1 1 1.5-1.5 Vacuum probability distribution in coordinate-space for � = 0 . 05 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19