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 2 SU(N) gauge theory (λ = g N fixed) with Nf Dirac fermions in the antisymmetric representation is equivalent in the large-N limit and in a common sector to AdQCD 2 SU(N) gauge theory (λ = g N fixed) with Nf 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 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19 Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x lim WOr(J) = lim WAd(J) N→∞ N→∞ Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19 Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x hO(x ) ··· O(xn)i hO(x ) ··· O(xn)i lim 1 c,Or = lim 1 c,Ad N→∞ N2−2n N→∞ N2−2n Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19 Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x hO(x ) ··· O(xn)i hO(x ) ··· O(xn)i lim 1 c,Or = lim 1 c,Ad N→∞ N2−2n N→∞ N2−2n lim h0|O|0iOr = lim h0|O|0iAd N→∞ N→∞ Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19 −tH −tH lim ha|e |biOr = lim ha|e |biAd N→∞ N→∞ Symmetries inside the common sector are the same in AdjQCD and OrientiQCD. Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x hO(x ) ··· O(xn)i hO(x ) ··· O(xn)i lim 1 c,Or = lim 1 c,Ad N→∞ N2−2n N→∞ N2−2n lim ha|O|biOr = lim ha|O|biAd N→∞ N→∞ Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19 Symmetries inside the common sector are the same in AdjQCD and OrientiQCD. Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x hO(x ) ··· O(xn)i hO(x ) ··· O(xn)i lim 1 c,Or = lim 1 c,Ad N→∞ N2−2n N→∞ N2−2n lim ha|O|biOr = lim ha|O|biAd N→∞ N→∞ −tH −tH lim ha|e |biOr = lim ha|e |biAd N→∞ N→∞ Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19 Introduction Orientifold planar equivalence Gauge-invariant common sector AdQCD BOSONIC C-EVEN FERMIONIC C-EVEN BOSONIC C-ODD FERMIONIC C-ODD ~ w ­ OrQCD BOSONIC C-EVEN BOSONIC C-ODD COMMON SECTOR 2 Z Z ff e−N W(J) = DADψDψ¯ exp −S(A, ψ, ψ¯) + N2 J(x)O(x)d4x hO(x ) ··· O(xn)i hO(x ) ··· O(xn)i lim 1 c,Or = lim 1 c,Ad N→∞ N2−2n N→∞ N2−2n lim ha|O|biOr = lim ha|O|biAd N→∞ N→∞ −tH −tH lim ha|e |biOr = lim ha|e |biAd N→∞ N→∞ 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 ˆz is a local gauge transformation, that is periodic modulo an element of the center ZN . † † Aµ(x) → Ω(x)Aµ(x)Ω (x) + iΩ(x)∂µΩ (x) ψ(x) → R[Ω(x)]ψ(x) 2πik Ω(x + Lˆz) = Ω(x)e N 2πik tr W → e N tr W Even-N OrQCD: Z2 Odd-N OrQCD: – AdQCD: ZN Only Z2 maps the common sector into itself OPE ⇒ OrQCD(N = ∞) has at least a Z2 symmetry Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 6 / 19 Z2 symmetry OrQCD(N = ∞) htr Wi = 0 1 1 htr WiOr = htr WiAdj = 0 N N htr (W2)i 6= 0 1 2 1 2 3 htr (W )iOr = htr (W )iAdj = 0 htr (W )i = 0 N N . 1 3 1 3 . htr (W )iOr = htr (W )iAdj = 0 . N N htr (WN )i 6= 0 . Is ZN a symmetry for OrQCD(N = ∞)? Center symmetry Symmetry (mis)matching Symmetry (mis)matching ZN symmetry htr Wi = 0 htr (W2)i = 0 htr (W3)i = 0 . htr (WN )i 6= 0 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19 OrQCD(N = ∞) 1 1 htr WiOr = htr WiAdj = 0 N N 1 2 1 2 htr (W )iOr = htr (W )iAdj = 0 N N 1 3 1 3 htr (W )iOr = htr (W )iAdj = 0 N N . Is ZN a symmetry for OrQCD(N = ∞)? Center symmetry Symmetry (mis)matching Symmetry (mis)matching ZN symmetry Z2 symmetry htr Wi = 0 htr Wi = 0 htr (W2)i = 0 htr (W2)i 6= 0 htr (W3)i = 0 htr (W3)i = 0 . htr (WN )i 6= 0 htr (WN )i 6= 0 Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19 Center symmetry Symmetry (mis)matching Symmetry (mis)matching ZN symmetry Z2 symmetry OrQCD(N = ∞) htr Wi = 0 htr Wi = 0 1 1 htr WiOr = htr WiAdj = 0 N N htr (W2)i = 0 htr (W2)i 6= 0 1 2 1 2 3 3 htr (W )iOr = htr (W )iAdj = 0 htr (W )i = 0 htr (W )i = 0 N N . 1 3 1 3 . htr (W )iOr = htr (W )iAdj = 0 . N N htr (WN )i 6= 0 htr (WN )i 6= 0 . Is ZN 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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y T` is an operator with defined angular momentum ` h0|T0|0i 6= 0 h0|T1|0i = 0 h0|T2|0i 6= 0 ... Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19 Center symmetry A quantum mechanical analogy A quantum mechanical analogy p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 5 4 3 2 1 1.5 1 0.5 -1.5 0 -1 0 -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 Center symmetry A quantum mechanical analogy A quantum mechanical analogy p2 + p2 1 1 H = x y + mω2x2 + mω2y2 2m 2 x 2 y 2 2 lim |ψ0(x)| = δ (r) ~→0 lim h0|T`|0i = 0 for ` 6= 0 ~→0 The vacuum is invariant under rotations, but the Hamiltonian is not! Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19 Center symmetry Two-point functions of Polyakov loops Two-point functions of Polyakov loops † lim hP(x)P (y)ic 6= 0 N→∞ AdQCD hP(x)P(y)ic = 0 † lim hP(x)P (y)ic 6= 0 N→∞ OrQCD lim hP(x)P(y)ic = 0 N→∞ hReP(0)ReP(x)ic,Or = hReP(0)ReP(x)ic,Ad † † hP(0)P(x)ic,Or + hP(0)P(x) ic,Or = hP(0)P(x) ic,Ad Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 9 / 19 Open strings Overview Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 10 / 19 † hP(x)P (y)iYM ∝ partition function of YM coupled to a static quark in x and a static antiquark in y = X −βVn(|x−y|) = mne open-string states Open strings External charges External charges » – A 1 A † A A G (x)|open − stringi ≡ DiE (x) − ψ T ψ(x) |open − stringi = ρ (x)|open − stringi g2 i R ext Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 11 / 19 † hP(x)P (y)iYM ∝ partition function of YM coupled to a static quark in x and a static antiquark in y = X −βVn(|x−y|) = mne open-string states Open strings External charges External charges » – A 1 A † A A G (x)|open − stringi ≡ DiE (x) − ψ T ψ(x) |open − stringi = ρ (x)|open − stringi g2 i R ext Agostino Patella (CERN) Tubes and