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

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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

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SLIDE 1

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

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SLIDE 2

Introduction Orientifold planar equivalence

Orientifold planar equivalence

OrQCD SU(N) gauge theory (λ = g2N fixed) with Nf 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 = g2N 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

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SLIDE 3

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

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SLIDE 4

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 e−N2W(J) = Z DADψD ¯ ψ exp  −S(A, ψ, ¯ ψ) + N2 Z J(x)O(x)d4x ff lim

N→∞ WOr(J) =

lim

N→∞ WAd(J) Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

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SLIDE 5

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 e−N2W(J) = Z DADψD ¯ ψ exp  −S(A, ψ, ¯ ψ) + N2 Z J(x)O(x)d4x ff lim

N→∞

O(x1) · · · O(xn)c,Or N2−2n = lim

N→∞

O(x1) · · · O(xn)c,Ad N2−2n

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

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SLIDE 6

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 e−N2W(J) = Z DADψD ¯ ψ exp  −S(A, ψ, ¯ ψ) + N2 Z J(x)O(x)d4x ff lim

N→∞

O(x1) · · · O(xn)c,Or N2−2n = lim

N→∞

O(x1) · · · O(xn)c,Ad N2−2n lim

N→∞0|O|0Or =

lim

N→∞0|O|0Ad Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

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SLIDE 7

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 e−N2W(J) = Z DADψD ¯ ψ exp  −S(A, ψ, ¯ ψ) + N2 Z J(x)O(x)d4x ff lim

N→∞

O(x1) · · · O(xn)c,Or N2−2n = lim

N→∞

O(x1) · · · O(xn)c,Ad N2−2n lim

N→∞a|O|bOr =

lim

N→∞a|O|bAd Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

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SLIDE 8

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 e−N2W(J) = Z DADψD ¯ ψ exp  −S(A, ψ, ¯ ψ) + N2 Z J(x)O(x)d4x ff lim

N→∞

O(x1) · · · O(xn)c,Or N2−2n = lim

N→∞

O(x1) · · · O(xn)c,Ad N2−2n lim

N→∞a|O|bOr =

lim

N→∞a|O|bAd

lim

N→∞a|e−tH|bOr =

lim

N→∞a|e−tH|bAd Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

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SLIDE 9

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 e−N2W(J) = Z DADψD ¯ ψ exp  −S(A, ψ, ¯ ψ) + N2 Z J(x)O(x)d4x ff lim

N→∞

O(x1) · · · O(xn)c,Or N2−2n = lim

N→∞

O(x1) · · · O(xn)c,Ad N2−2n lim

N→∞a|O|bOr =

lim

N→∞a|O|bAd

lim

N→∞a|e−tH|bOr =

lim

N→∞a|e−tH|bAd

Symmetries inside the common sector are the same in AdjQCD and OrientiQCD.

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 3 / 19

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SLIDE 10

Introduction Overview

Overview

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

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Center symmetry

Overview

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

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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 ZN. Aµ(x) → Ω(x)Aµ(x)Ω†(x) + iΩ(x)∂µΩ†(x) ψ(x) → R[Ω(x)]ψ(x) Ω(x + Lˆ z) = Ω(x)e

2πik N

tr W → e

2πik 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

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SLIDE 13

Center symmetry Symmetry (mis)matching

Symmetry (mis)matching

ZN symmetry tr W = 0 tr (W2) = 0 tr (W3) = 0 . . . tr (WN) = 0

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19

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SLIDE 14

Center symmetry Symmetry (mis)matching

Symmetry (mis)matching

ZN symmetry tr W = 0 tr (W2) = 0 tr (W3) = 0 . . . tr (WN) = 0 Z2 symmetry tr W = 0 tr (W2) = 0 tr (W3) = 0 . . . tr (WN) = 0

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19

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SLIDE 15

Center symmetry Symmetry (mis)matching

Symmetry (mis)matching

ZN symmetry tr W = 0 tr (W2) = 0 tr (W3) = 0 . . . tr (WN) = 0 Z2 symmetry tr W = 0 tr (W2) = 0 tr (W3) = 0 . . . tr (WN) = 0 OrQCD(N = ∞) 1 N tr WOr = 1 N tr WAdj = 0 1 N tr (W2)Or = 1 N tr (W2)Adj = 0 1 N tr (W3)Or = 1 N tr (W3)Adj = 0 . . . Is ZN a symmetry for OrQCD(N = ∞)?

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 7 / 19

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SLIDE 16

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

Tℓ is an operator with defined angular momentum ℓ 0|T0|0 = 0 0|T1|0 = 0 0|T2|0 = 0 . . .

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 17

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 1

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 18

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 0.5

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 19

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 0.4

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 20

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 0.3

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 21

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 0.2

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 22

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 0.1

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 23

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

  • 1.5
  • 1
  • 0.5

0.5 1 1.5-1.5

  • 1
  • 0.5

0.5 1 1.5 1 2 3 4 5

Vacuum probability distribution in coordinate-space for = 0.05

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 24

Center symmetry A quantum mechanical analogy

A quantum mechanical analogy

H = p2

x + p2 y

2m + 1 2mω2

xx2 + 1

2 mω2

yy2

lim

→0 |ψ0(x)|2 = δ2(r)

lim

→00|Tℓ|0 = 0

for ℓ = 0 The vacuum is invariant under rotations, but the Hamiltonian is not!

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 8 / 19

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SLIDE 25

Center symmetry Two-point functions of Polyakov loops

Two-point functions of Polyakov loops

AdQCD lim

N→∞P(x)P†(y)c = 0

P(x)P(y)c = 0 OrQCD lim

N→∞P(x)P†(y)c = 0

lim

N→∞P(x)P(y)c = 0

ReP(0)ReP(x)c,Or = ReP(0)ReP(x)c,Ad P(0)P(x)c,Or + P(0)P(x)†c,Or = P(0)P(x)†c,Ad

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 9 / 19

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SLIDE 26

Open strings

Overview

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 10 / 19

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SLIDE 27

Open strings External charges

External charges

GA(x)|open − string ≡ » 1 g2 DiEA

i (x) − ψ†TA Rψ(x)

– |open − string = ρA

ext(x)|open − string Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 11 / 19

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SLIDE 28

Open strings External charges

External charges

GA(x)|open − string ≡ » 1 g2 DiEA

i (x) − ψ†TA Rψ(x)

– |open − string = ρA

ext(x)|open − string Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 11 / 19

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SLIDE 29

Open strings External charges

External charges

P(x)P†(y)YM ∝ partition function of YM coupled to a static quark in x and a static antiquark in y = = X

  • pen-string

states

mne−βVn(|x−y|) GA(x)|open − string ≡ » 1 g2 DiEA

i (x) − ψ†TA Rψ(x)

– |open − string = ρA

ext(x)|open − string Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 11 / 19

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SLIDE 30

Open strings Bosonic and fermionic open strings

Bosonic and fermionic open strings

ReP(0)ReP(x)c,Or = ReP(0)ReP(x)c,Ad P(0)P(x)c,Or + P(0)P(x)†c,Or = P(0)P(x)†c,Ad

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19

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SLIDE 31

Open strings Bosonic and fermionic open strings

Bosonic and fermionic open strings

ReP(0)ReP(x)c,Or = ReP(0)ReP(x)c,Ad P(0)P(x)c,Or + P(0)P(x)†c,Or = P(0)P(x)†c,Ad bosonic oriented ˜ σb Oriented bosonic open string in OrQCD

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19

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SLIDE 32

Open strings Bosonic and fermionic open strings

Bosonic and fermionic open strings

ReP(0)ReP(x)c,Or = ReP(0)ReP(x)c,Ad P(0)P(x)c,Or + P(0)P(x)†c,Or = P(0)P(x)†c,Ad bosonic oriented ˜ σb fermionic unoriented ˜ σf Unoriented fermionic open string in OrQCD

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19

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SLIDE 33

Open strings Bosonic and fermionic open strings

Bosonic and fermionic open strings

ReP(0)ReP(x)c,Or = ReP(0)ReP(x)c,Ad P(0)P(x)c,Or + P(0)P(x)†c,Or = P(0)P(x)†c,Ad bosonic oriented = bosonic oriented ˜ σb = σb fermionic unoriented = fermionic oriented ˜ σf = σf Oriented open string in AdQCD

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 12 / 19

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SLIDE 34

Open strings Bosonic and fermionic open strings

Bosonic and fermionic open strings

ReP(0)ReP(x)c,Or = ReP(0)ReP(x)c,Ad P(0)P(x)c,Or + P(0)P(x)†c,Or = P(0)P(x)†c,Ad bosonic oriented = bosonic oriented ˜ σb = σb fermionic unoriented = fermionic oriented ˜ σf = 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

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SLIDE 35

Open strings Equality of bosonic and fermionic string tensions in SYM

Bosonic and fermionic string tensions in SYM

The Hamiltonian. H = Z » 1 2g2 EA

i EA i +

1 2g2 BA

i BA i +

i 2g2 ¯ λγiDAdj

i

λ – d3x The supercharges in the (on-shell) de Wit-Freedman formalism. [S, AA

i (x)] = γiλA(x)

{Sα, λA

β(x)} = − 1

4 FA

µν(x)[γµ, γν]αβ

[S, GA(x)] = 0 {S, ¯ S} = 2(γ0H − γkΠk) X

α

(S†)αSα = 4H The supercharges and the Hamiltonian do not commute in general. [S, H] = Z γ0λAGA d3x

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 13 / 19

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SLIDE 36

Open strings Equality of bosonic and fermionic string tensions in SYM

Bosonic and fermionic string tensions in SYM

X

α

(S†)αSα = 4H [S, H] = Z γ0λAGA d3x = Z γ0λAρA

ext d3x

Consider the eigenstates in the string sector. H |B, n = (σbR + O(R0)) |B, nb H |F, n = (σf R + O(R0)) |F, nf Choose a bosonic state |B, n. 4σbR = X

α

B, n| (S†)αSα |B, n = X

α,n′

˛ ˛˙ F, n′˛ ˛ Sα |B, n ˛ ˛2 At least one fermionic state |F, n′ and one index α exist with the following property: ˙ F, n′˛ ˛ Sα |B.n = O “√ R ” Computing the matrix element of [S, H]... (σb − σf ) F, n′| Sα |B, n R1/2 + O „ 1 R3/2 « = 1 R3/2 ˙ F, n′˛ ˛ Z d3x (γ0λA)αρA

ext |B, n Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 14 / 19

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SLIDE 37

Domain walls

Overview

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

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SLIDE 38

Domain walls Deconfined phase

Deconfined phase

Effective potential for the Polyakov loop W = eiv1N VAd(eiv) N2 = 0 VOr(eiv) N2 = π2 24β4 − 1 24π2β4 (π2 − [2v mod 2π]2)2

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 16 / 19

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SLIDE 39

Domain walls Domain wall, ’t Hooft loop and Dirac string

Domain wall, ’t Hooft loop and Dirac string

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

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19

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SLIDE 40

Domain walls Domain wall, ’t Hooft loop and Dirac string

Domain wall, ’t Hooft loop and Dirac string

WBC: P(x1, x2, x3 + L) = e

2πik N P(x1, x2, x3)

Zwall = lim

L3→∞ trL3,WBC

h e−βHi ∝ e−L1L2σ(β)

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19

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SLIDE 41

Domain walls Domain wall, ’t Hooft loop and Dirac string

Domain wall, ’t Hooft loop and Dirac string

Zwall = lim

L3→∞ trβ,TBC

» e−L3He

4πi g

R tr E4Yk dx1dx2

– ∝ e−L1L2σ(β) e2πiYk = e

2πik N

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19

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SLIDE 42

Domain walls Domain wall, ’t Hooft loop and Dirac string

Domain wall, ’t Hooft loop and Dirac string

Zwall = 0|e−L2HDirac(L1,β)|0 ∝ e−L1EDirac(L2,β) = e−L1L2σ(β) The operator HDirac(L1, β)/N2 is in the common sector and has a well defined large-N limit. lim

N→∞

σOr(β) N2 = lim

N→∞

σAd(β) N2

Agostino Patella (CERN) Tubes and walls GGI, 5/5/11 17 / 19

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SLIDE 43

Domain walls Hamiltonian for the Dirac string

Hamiltonian for the Dirac string

HDirac = a2N 2λ X

x

tr E(x)

2 +

N 2λa X

  • ℜetr (1 − zU) + HF

z = ( −1 if goes around the Dirac string 1 elsewhere

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

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SLIDE 44

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