The spectrum of 2+1 dimensional Yang-Mills theory on a twisted spatial torus

Abstract

We compute and analyse the low-lying spectrum of 2+1 dimensional SU(N ) Yang-Mills theory on a spatial torus of size l × l with twisted boundary conditions. This paper extends our previous work [1]. In that paper we studied the sector with non-vanishing electric flux and concluded that the energies only depend on the parameters through two combinations: x = λN l/(4π) (with λ the 't Hooft coupling) and the twist angleθ defined in terms of the magnetic flux piercing the two-dimensional box. Here we made a more complete study and we are able to condense our results, obtained by non-perturbative lattice methods, into a simple expression which has important implications for the absence of tachyonic instabilities, volume independence and non-commutative field theory. Then we extend our study to the sector of vanishing electric flux. We conclude that the onset of the would-be large-volume glueball states occurs at an approximately fixed value of x, much before the stringy torelon states have become very massive.

The two-dimensional twisted reduced principal chiral model revisited

“…[30]. As mentioned in the introduction, for the 2+1 dimensional case [35]- [36] we observed that this choice avoids the presence of symmetry-breaking phase transitions at N large. However, even if no transition is present it is still very important for practical purposes to reduce the size of the finite N corrections.…”

Prospects for large N gauge theories on the lattice

“…Finally, I will comment below upon two results that have come up from the combination of adiabatic continuity and large N volume independence: emergent fermionic symmetry [84], and singular large N limits in connection to non-commutative gauge theories [85][86][87].…”

“…We have explored in detail this issue for the case of a 2+1 dimensional Yang-Mills theory defined on a twisted T 2 × R [85,87]. The problem appears already at a perturbative level.…”

“…A different issue is whether one can smoothly take the singular large N limit described above, leading to any value of the non-commutativity parameter without encountering tachyonic instabilities along the way. This turns out to be possible for an uncountable set of irrational values ofθ with, however, zero measure [87]. In particular, one possible sequence corresponds to takinĝ N i = F i and k i = F i−2 , with F i the ith-number in the Fibonacci sequence, leading to a valueθ = (3 − √ 5)/2 [86].…”

“…The energy of states with electric flux (e, 0) for N = 17 and k = 3 compared with the prediction of eq. (4.3) with χ 0 = 0.6, √ σ /λ = 0.213 and φ 0 (z) = sin(πz)/π[87].The Golden Ratio and Non-Commutative gauge theories[García Pérez, Koren, González-Arroyo & Okawa] A different question -singular large N limitslim t e x i t s h a 1 _ b a s e 6 4 = " o 4 8 Y Y c 3 F 0 U / r C P S r 1 g 2 r x U q L I R c = " > A A A C O H i c bV B N i x N B E O 1 Z v 9 b 4 F f X o p T E I n s J M X N C L s O B F L 7 q C 2 V 1 I h 6 G m U 5 M U 6e k e u m u U M M z P 8 u L P 8 C Z e P C j i 1V 9 g J z s H 3 f V B F Y / 3 q u i u V 9 S G A q f p l 2 T v 0 u U r V 6 / t X x / c u H n r 9 p 3 h 3 X v H w T V e 4 1 Q 7 4 / x p A Q E N W Z w y s c H T 2 i N U h c G T Y v 1 i 6 5 + 8 R x / I 2 Xe 8 q X F e w d J S S R o 4 S v n w j T J U 5 S 0 p 4 + z S 0 3 L F 4 L 3 7 I B X Z k j e d K j 3 o V h X g 5 T q n r n 0 d m 3 w u e 5 n J L F A q X i F D 1 0 5 U T V 0 + H K X j d A d 5 k W Q 9 G Y k e R / n w s 1 o 4 3 V R o W R s I Y Z a l N c 9 b 8 E z a Y D d Q T c A a 9 B q W O I v U Q o V h 3 u 4 O 7 + S j q C x k 6 X w s y 3 K n / r 3 R Q h X C p i r i Z A W 8 C u e 9 r f g / b 9 Z w + W z e k q 0 b R q v P H i o b I 9 n J b Y p y Q R 4 1 m 0 0 k o D 3 F v 0 q 9 g h g K x 6 w H M Y T s / M k X y f F k n D 0 Z T 9 4 e j A 5 f 9 X H s i w f i o X g s M v F U H I q X 4 k h M h R Y f x V f x X f x I P i X f k p / J r 7 P R v a T f u S / + Q f L 7 D 9 2 c r t M = < / l a t e x i t >Can we reach any value of the NC parameter at large N?NO, only for an uncountable zero-measure set…”