Recent Results in Large-N Lattice Gauge Theories

Recent results in large-N lattice gauge theories

Marco Panero

Department of Physics and Helsinki Institute of Physics University of Helsinki

Lattice 2012, The XXX International Symposium on Lattice Field Theory Cairns, Australia, 26 June 2012 Introduction Recent results Concluding remarks

Outline

Introductory reviews on large-N QCD: • E. Br´ezinand S. R. Wadia, The Large N Expansion in Quantum Field Theory and Statistical Physics: From Spin Systems to 2-Dimensional Gravity, World Scientiﬁc, Singapore, 1993 • Y. Makeenko, hep-th/0001047 • M. Teper, 0912.3339 • B. Lucini and M. P., in preparation

At this conference, parallel talks relevant for this topic are presented by M. Garc´ıaPérez,A. González-Arroyo, M. Hanada, M. Honda, D. Kadoh, L. Keegan, M. Koreń,J.-W. Lee, R. Lohmayer, F. Negro, M. Okawa and P. Orland. Introduction Recent results Concluding remarks

Outline

1 Introduction

2 A selection of physical results

3 Concluding remarks Introduction Recent results Concluding remarks

QCD in the ’t Hooft limit

Consider a generalization of QCD with SU(N → ∞) gauge group • Take g → 0, with λ = g 2N fixed, to have a perturbatively smooth limit • Keep track of the number of independent color indices in Feynman diagrams through double-line notation for propagators • Dominance of planar diagrams without dynamical quark loops • Terms proportional to different powers of 1/N can be arranged in a topological series ∞ ∞ X 2−2h−b X n A = N c(h,b),nλ h,b=0 n=0 Analogous to a loop expansion in Riemann surfaces for string theory, upon replacing 1/N → gs • This also holds according to the conjectured holographic correspondence: In the large-N limit, loop effects on the string side become negligible (see also plenary talk by Hanada) Introduction Recent results Concluding remarks

QCD in the ’t Hooft limit

Consider a generalization of QCD with SU(N → ∞) gauge group • Take g → 0, with λ = g 2N fixed, to have a perturbatively smooth limit • Keep track of the number of independent color indices in Feynman diagrams through double-line notation for propagators • Quark: fundamental rep. ⇒ single line • Gluon: adjoint rep. ⇒ double line • Dominance of planar diagrams without dynamical quark loops • Terms proportional to different powers of 1/N can be arranged in a topological series ∞ ∞ X 2−2h−b X n A = N c(h,b),nλ h,b=0 n=0 Analogous to a loop expansion in Riemann surfaces for string theory, upon replacing 1/N → gs • This also holds according to the conjectured holographic correspondence: In the large-N limit, loop effects on the string side become negligible (see also plenary talk by Hanada) Introduction Recent results Concluding remarks

QCD in the ’t Hooft limit

g g g g g g N N N N N N N g g g g g g N N N N g N g g g 6 5 2 3 g g 6 3 3 g N = N λ g 6N 4= N λ3 g N = λ • Terms proportional to diﬀerent powers of 1/N can be arranged in a topological series ∞ ∞ X 2−2h−b X n A = N c(h,b),nλ h,b=0 n=0 Analogous to a loop expansion in Riemann surfaces for string theory, upon replacing 1/N → gs • This also holds according to the conjectured holographic correspondence: In the large-N limit, loop eﬀects on the string side become negligible (see also plenary talk by Hanada) Introduction Recent results Concluding remarks

QCD in the ’t Hooft limit

A wealth of phenomenological implications from large-N counting rules

Assuming that the large-N limit of QCD is confining: • The spectrum consists of infinitely many stable glueballs and√ mesons, with masses O(1) and interactions suppressed by powers of 1/ N: large-N QCD is a theory of weakly coupled hadrons • Exotica (e.g. tetraquarks, molecules, et c.) are absent • The OZI rule is exact • Loop effects in the chiral Lagrangian are suppressed by 1/N • The axial anomaly is suppressed by 1/N, and the square of the η0 mass is O(1/N) • Baryons can be interpreted as solitons of the theory, with masses O(N) • Quantitative predictions for baryon-meson couplings, baryon masses, magnetic moments, et c. from consistency conditions based on unitarity • Implications for the QCD phase diagram—quarkyonic matter (McLerran and Pisarski, 2007)? • Further implications for high-energy QCD (evolution equations, hadronic cross-sections, parton distributions and structure functions, large-N Standard Model, . . . ) Introduction Recent results Concluding remarks

A wealth of phenomenological implications from large-N counting rules

N N g g

N g g

4 2 2 N g N = λ • Loop eﬀects in the chiral Lagrangian are suppressed by 1/N • The axial anomaly is suppressed by 1/N, and the square of the η0 mass is O(1/N) • Baryons can be interpreted as solitons of the theory, with masses O(N) • Quantitative predictions for baryon-meson couplings, baryon masses, magnetic moments, et c. from consistency conditions based on unitarity • Implications for the QCD phase diagram—quarkyonic matter (McLerran and Pisarski, 2007)? • Further implications for high-energy QCD (evolution equations, hadronic cross-sections, parton distributions and structure functions, large-N Standard Model, . . . ) Introduction Recent results Concluding remarks

A wealth of phenomenological implications from large-N counting rules

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules imply that vev’s of products of gauge-invariant operators are dominated by disconnected contributions ⇒ Factorization of vev’s of physical operators hO1O2i = hO1ihO2i + O(1/N) The analogy with a classical limit can be made explicit by constructing appropriate coherent states (Yaffe, 1982) • Factorization leads to volume independence • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules imply that vev’s of products of gauge-invariant operators are dominated by disconnected contributions ⇒ Factorization of vev’s of physical operators • Factorization leads to Eguchi-Kawai volume independence: the lattice theory can be formulated in an arbitrary small volume provided center symmetry is unbroken • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules imply that vev’s of products of gauge-invariant operators are dominated by disconnected contributions ⇒ Factorization of vev’s of physical operators • Factorization leads to Eguchi-Kawai volume independence: the lattice theory can be formulated in an arbitrary small volume provided center symmetry is unbroken • But center symmetry does get broken in a small volume in the continuum limit • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules imply that vev’s of products of gauge-invariant operators are dominated by disconnected contributions ⇒ Factorization of vev’s of physical operators • Factorization leads to Eguchi-Kawai volume independence: the lattice theory can be formulated in an arbitrary small volume provided center symmetry is unbroken • But center symmetry does get broken in a small volume in the continuum limit; fixes: • Quenched EK (Bhanot, Heller and Neuberger)—but see (Bringoltz and Sharpe) • Twisted EK • Add dynamical adjoint fermions • Double-trace deformations • Partial reduction • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • But center symmetry does get broken in a small volume in the continuum limit; fixes: • Quenched EK • Twisted EK (González-Arroyo and Okawa; Teper and Vairinhos, Azeyanagi et al., Bietenholz et al., Garc´ıaPérezet al.) 0.56

0.55

0.54 σ √

/ 0.53 MS Λ 0.52 SU(N) 0.51 TEK N=841 0.515(3) + 0.34(1)/N 2 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 1/N 2 • Add dynamical adjoint fermions • Double-trace deformations • Partial reduction • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • But center symmetry does get broken in a small volume in the continuum limit; fixes: • Quenched EK • Twisted EK • Add dynamical adjoint fermions (Kovtun, Unsal¨ and Yaffe)—see also (Hollowood and Myers; Azeyanagi et al.; Catterall, Galvez and Unsal;¨ Bringoltz, Koreńand Sharpe; Cossu and D’Elia; Hietanen and Narayanan; Lee, Hanada and Yamada; Okawa et al.)

1.0 b κf 0.9

0.8 4 Z Z Z Z Z 4 ZN 5 4 3 2 1 ZN 0.7 Z1 Z2

0.6

0.5

0.4 κc 1st order 0.3 Hysteresis crossover or bulk region 0.2 transition

0.1 Z 4 N κ 0 0.04 0.080.12 0.16 0.20 0.24 • Double-trace deformations • Partial reduction • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • But center symmetry does get broken in a small volume in the continuum limit; ﬁxes: • Quenched EK • Twisted EK • Add dynamical adjoint fermions • Double-trace deformations (Unsal¨ and Yaﬀe; Ogilvie et al.; Vairinhos; Hanada et al.)

bN/2c 1 X X n 2 SYM −→ SYM + an|tr(L (~x))| N3 t ~x n=1 • Partial reduction • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • But center symmetry does get broken in a small volume in the continuum limit; ﬁxes: • Quenched EK • Twisted EK • Add dynamical adjoint fermions • Double-trace deformations • Partial reduction (Kiskis, Narayanan and Neuberger)

b=0.348, L=6, N=47

1.5 1 < k < 10 0.099(16)k+0.34(12)-0.55(19)/k

1 m(k)

0.5

0 2 4 6 8 k

• Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Factorization, volume reduction and large-N equivalences

“You can hide a lot in a large-N matrix” —Stephen Shenker • Large-N counting rules lead to factorization of vev’s • Factorization leads to EK volume independence if center symmetry is unbroken • Volume reduction can be interpreted as a large-N “orbifold” equivalence: Projection using a discrete subgroup of the global symmetries of the theory (Kovtun, Unsal¨ and Yaffe) • Orbifold equivalences at large N also relate theories with different field content—e.g., orientifold planar equivalence (Armoni, Shifman and Veneziano—see also numerical studies by Lucini et al.) • Finally, orbifold projections are also relevant for lattice supersymmetry (Catterall, Kaplan and Unsal;¨ see also Tsuchiya et al., Nishimura et al.) Introduction Recent results Concluding remarks

Outline

1 Introduction

2 A selection of physical results

3 Concluding remarks Introduction Recent results Concluding remarks

Results in 4D

• SU(N) is a confining theory in the large-N limit (see, e.g., Meyer and Teper hep-lat/0411039) • Confining flux tubes behave like Nambu-Goto strings • Glueball masses have a smooth dependence on N • Well-behaved scale-dependence of the coupling • The deconfinement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

• SU(N) is a confining theory in the large-N limit • Confining flux tubes behave like Nambu-Goto strings 10 9 8 E 7 σ √ 6 5 4 3 2 1 0 1.5 2 2.5 3 3.5 4 l√σ Torelon spectrum in SU(5) (Athenodorou et al., 1007.4720)

See also: Lucini and Teper, hep-lat/0107007; Lohmayer and Neuberger, 1206.4015; Mykk¨anen,in progress • Glueball masses have a smooth dependence on N • Well-behaved scale-dependence of the coupling • The deconﬁnement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

• SU(N) is a confining theory in the large-N limit • Confining flux tubes behave like Nambu-Goto strings • Glueball masses have a smooth dependence on N 3

2.5

1.5 am

0.5 Ground states Excited states [Lucini,Teper,Wenger 2004] 0 ++ -+ +- ++ +- -+ -- ++ +- -+ ++ +- -+ -- A1 A1 A2 E E E E T1 T1 T1 T2 T2 T2 T2 YM spectrum (Lucini, Rago and Rinaldi, 1007.3879) • Well-behaved scale-dependence of the coupling • The deconﬁnement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

5.5

λ 5 imp SU(2)

4.5 SU(3) SU(4) SU(6) 4 SU(8)

3.5 0 0.1 0.2 0.3 0.4 0.5

a√σ Bare coupling in pure YM (Allton et al., 0803.1092)

• The deconﬁnement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

Numerical results One-loop β -function Two-loop β -function 2.5

2 2 g

1.5

1 1 10 E [GeV] SU(4) YM, SF scheme (Lucini and Moraitis, 0805.2913) • The deconﬁnement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

N = 2 N = 3 0.8 N = 4 N = ∞

0.6

γm

0.4

0.2

0 0 10 20 30 40 2 g N

Mass anomalous dimension in QCDN with nf = 2 2S fermions (DeGrand, Shamir and Svetitsky, 1202.2675) • The deconﬁnement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

0.7

T c  0.6 √σ

0.5

2 fit: 0.5949(17) + 0.458(18)/N

0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 2 1/N (Lucini, Rago and Rinaldi, 1202.6684)

See also: Lucini, Teper and Wenger, hep-lat/0307017 and hep-lat/0502003; Piemonte et al., in progress • The equation of state appears to have only a trivial dependence on N • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

• SU(N) is a confining theory in the large-N limit • Confining flux tubes behave like Nambu-Goto strings • Glueball masses have a smooth dependence on N • Well-behaved scale-dependence of the coupling • The deconfinement temperature has a smooth dependence on N • The equation of state appears to have only a trivial dependence on N 1

ε ε 0.8 / SB

0.6 p/pSB

0.4

0.2 SU(3) SU(4) SU(6) 0 1 2 3 4 T/T c (Datta and Gupta, 1006.0938) See also: Bringoltz and Teper, hep-lat/0506034, M.P., 0907.3719; Mykk¨anenet al., 1202.2762 • Topological susceptibility and θ-dependence • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

0.03

χ / σ 2

0.02

0.01 simulation results large- N extrapolation

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 2 1 / N (Del Debbio et al., 2002) See also: Lucini and Teper, hep-lat/0110004, hep-lat/0401028, Cundy et al., hep-lat/0203030, Panagopoulos et al., 1109.6815 • Quenched mesonic spectrum • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

4 σ 

√ 3 / SU( ∞ ) m SU(2) SU(3) 2 SU(4) SU(5) SU(6) 1 SU(7) SU(17)

0   π f /√N ρ f /√N a a b π∗ ρ∗ * * * π ρ 0 1 1 a0 a1 b1 (Bali et al., in progress) See also: Del Debbio et al., 0712.3036, Bali and Bursa, 0806.2278, Hietanen et al., 0901.3752 • Quenched baryonic spectrum Introduction Recent results Concluding remarks

Results in 4D

SU(5) SU(7)

(DeGrand, 1205.0235) Introduction Recent results Concluding remarks

Results in 3D

Much like in 4D: • SU(N) is a confining theory in the large-N limit (Teper, hep-lat/9804008) • Confining flux tubes behave as Nambu-Goto strings (Athenodorou et al., 1103.5854; Caselle et al., 1102.0723; Mykkänen,in progress) • Glueball masses have a smooth dependence on N (Johnson and Teper, hep-ph/0012287; Meyer, hep-lat/0508002) • The equation of state depends only trivially on N (Caselle et al., 1105.0359 and 1111.0580) Introduction Recent results Concluding remarks

Results in 3D

Much like in 4D: • SU(N) is a confining theory in the large-N limit (Teper, hep-lat/9804008) • Confining flux tubes behave as Nambu-Goto strings (Athenodorou et al., 1103.5854; Caselle et al., 1102.0723; Mykkänen,in progress) 20 19 18 17 16 15 14 E 13 √σ 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 l√σ • Glueball masses have a smooth dependence on N (Johnson and Teper, hep-ph/0012287; Meyer, hep-lat/0508002) • The equation of state depends only trivially on N (Caselle et al., 1105.0359 and 1111.0580) Introduction Recent results Concluding remarks

Results in 3D

7 M ♦ ♦ √σ 6 ♦

5 ⋆ r r r 4

3 ♦ 0−− 2 r 0++ 1

0 0 0.05 0.1 0.15 0.2 0.25

1/N 2 • The equation of state depends only trivially on N (Caselle et al., 1105.0359 and 1111.0580) Introduction Recent results Concluding remarks

Results in 3D

0.16 SU(2) 0.14 SU(3) SU(4) 0.12 SU(5) SU(6) holographic model, for α = 3 / 2

- 1 ) ] 0.1 2 N ( 3

T 0.08 / [ ∆ 0.06

0.04

0.02

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 T T / c Introduction Recent results Concluding remarks

Results in 2D

Several exact results are known; in particular: • The continuum spectrum of large-N QCD in 2D was computed by ’t Hooft in 1974 • In 1979, Gross and Witten found a third-order transition in the lattice theory • The spectral density of Wilson loops was studied by Durhuus and Olesen in 1981

In general, a 2D world can be a useful laboratory for QCD toy models (see, e.g., works by Narayanan, Neuberger and Vicari; Orland et al., . . . )

Recently, the eigenvalue density of Wilson loops in 2D has been studied by Lohmayer, Neuberger and Wettig; similar studies have also been done in 4D (Lohmayer and Neuberger)

Various groups (e.g. Bringoltz; Galvez, Hietanen and Narayanan, et c.) have addressed the problem of 2D large-N theories at ﬁnite chemical potential Introduction Recent results Concluding remarks

Outline

1 Introduction

2 A selection of physical results

3 Concluding remarks Introduction Recent results Concluding remarks

Concluding remarks

Lattice studies of gauge theories in the large-N limit are theoretically very appealing, numerically tractable, and interesting for a very broad community. Introduction Recent results Concluding remarks

Concluding remarks

Lattice studies of gauge theories in the large-N limit are theoretically very appealing, numerically tractable, and interesting for a very broad community (as you can see). Introduction Recent results Concluding remarks

Concluding remarks

Lattice studies of gauge theories in the large-N limit are theoretically very appealing, numerically tractable, and interesting for a very broad community.

During the last ﬁfteen years, numerical simulations in this ﬁeld have given conclusive answers to various long-standing questions. However, many other issues are still open, and waiting for your involvement. Introduction Recent results Concluding remarks

Concluding remarks

Lattice studies of gauge theories in the large-N limit are theoretically very appealing, numerically tractable, and interesting for a very broad community.

From my personal point of view, particularly promising research directions for further numerical studies at large N include: • Simulations with dynamical fermions, in various representations • Finite temperature/ﬁnite density; comparisons with perturbative computations, with holography, or with eﬀective models • Topological properties (see, e.g., Lucini et al., hep-lat/0401028, hep-lat/0502003; Panagopoulos and Vicari, 0803.1593, 1109.6815; D’Elia and Negro, 1205.0238) • Large-N equivalences and volume reduction