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 walls GGI, 5/5/11 11 / 19 Open strings External charges External charges

† 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

» – 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 bosonic oriented = bosonic oriented σ˜b = σb fermionic unoriented = fermionic oriented σ˜f = σf

Open strings Bosonic and fermionic open strings Bosonic and fermionic open strings

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 12 / 19 = bosonic oriented = σb fermionic unoriented = fermionic oriented σ˜f = σf

Open strings Bosonic and fermionic open strings Bosonic and fermionic open strings

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 bosonic oriented σ˜b

Oriented bosonic open string in OrQCD

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19 = bosonic oriented = σb = fermionic oriented = σf

Open strings Bosonic and fermionic open strings Bosonic and fermionic open strings

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 bosonic oriented σ˜b fermionic unoriented σ˜f

Unoriented fermionic open string in OrQCD

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19 Open strings Bosonic and fermionic open strings Bosonic and fermionic open strings

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 bosonic oriented = bosonic oriented σ˜b = σb fermionic unoriented = fermionic oriented σ˜f = σf

Oriented open string in AdQCD

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19 Open strings Bosonic and fermionic open strings Bosonic and fermionic open strings

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 bosonic oriented = bosonic oriented σ˜b = σb fermionic unoriented = fermionic oriented σ˜f = σf

Supersymmetry (Nf = 1) SUSY implies degeneracy between bosons and fermions. The external charges explictly break SUSY. However SUSY breaking is a boundary effect.

σb = σf

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19 Open strings Equality of bosonic and fermionic string tensions in SYM Bosonic and fermionic string tensions in SYM

The Hamiltonian. Z » – 1 A A 1 A A i Adj 3 H = E E + B B + λγ¯ iD λ d x 2g2 i i 2g2 i i 2g2 i

The supercharges in the (on-shell) de Wit-Freedman formalism.

A A [S, Ai (x)] = γiλ (x)

A 1 A µ ν {Sα, λ (x)} = − F (x)[γ , γ ]αβ β 4 µν [S, GA(x)] = 0 0 {S, ¯S} = 2(γ H − γkΠk) X † α (S ) Sα = 4H α

The supercharges and the Hamiltonian do not commute in general.

Z [S, H] = γ0λAGA d3x

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 13 / 19 Open strings Equality of bosonic and fermionic string tensions in SYM Bosonic and fermionic string tensions in SYM

Z Z X † α 0 A A 3 0 A A 3 (S ) Sα = 4H [S, H] = γ λ G d x = γ λ ρext d x α

Consider the eigenstates in the string sector.

H |B, ni = (σbR + O(R0)) |B, nib H |F, ni = (σf R + O(R0)) |F, nif

Choose a bosonic state |B, ni.

b X † α X ˛˙ 0˛ ˛2 4σ R = hB, n| (S ) Sα |B, ni = ˛ F, n ˛ Sα |B, ni˛ α α,n0

At least one fermionic state |F, n0i and one index α exist with the following property: √ ˙ 0˛ “ ” F, n ˛ Sα |B.ni = O R

Computing the matrix element of [S, H]... 0 „ « Z b f hF, n | Sα |B, ni 1 1 ˙ 0˛ 3 A A (σ − σ ) + O = F, n ˛ d x (γ0λ )αρ |B, ni R1/2 R3/2 R3/2 ext

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 14 / 19 Domain walls Overview

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 15 / 19 Domain walls Deconfined phase Deconfined phase

iv Effective potential for the Polyakov loop W = e 1N

V (eiv) Ad = 0 N2 V (eiv) π2 1 Or = − (π2 − [2v mod 2π]2)2 N2 24β4 24π2β4

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 16 / 19 Domain walls Domain wall, ’t Hooft loop and Dirac string Domain wall, ’t Hooft loop and Dirac string

σ(β) ∝ k(N − k) N for k = : σ(β) ∝ N2 2

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19 Domain walls Domain wall, ’t Hooft loop and Dirac string Domain wall, ’t Hooft loop and Dirac string

2πik WBC: P(x1, x2, x3 + L) = e N P(x1, x2, x3) h i −βH −L1L2σ(β) Zwall = lim trL3,WBC e ∝ e L3→∞

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19 Domain walls Domain wall, ’t Hooft loop and Dirac string Domain wall, ’t Hooft loop and Dirac string

» 4πi R – −L H tr E4Yk dx1dx2 −L L σ(β) Zwall = lim trβ,TBC e 3 e g ∝ e 1 2 L3→∞ 2πiY 2πik e k = e N

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19 Domain walls Domain wall, ’t Hooft loop and Dirac string Domain wall, ’t Hooft loop and Dirac string

−L HDirac(L ,β) −L EDirac(L ,β) −L L σ(β) Zwall = h0|e 2 1 |0i ∝ e 1 2 = e 1 2

2 The operator HDirac(L1, β)/N is in the common sector and has a well defined large-N limit. σ (β) σ (β) lim Or = lim Ad N→∞ N2 N→∞ N2

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19 Domain walls Hamiltonian for the Dirac string Hamiltonian for the Dirac string

2 a N X 2 N X HDirac = tr E~(x) +

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 18 / 19 Conclusions Conclusions

In the large-N limit, OrQCD has a Z2 center symmetry. However its vacuum (in the confined phase) is symmetric under ZN transformations.

Orientifold planar equivalence holds in a larger sector than the particle sector.

In the open-string sector, orientifold planar equivalence holds nontrivially both for bosonic and fermionic strings.

Orientifold planar equivalence holds also for the domain wall interpolating between the two vacua P/N = ±1 (in the deconfined phase).

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