MASS GAP AND FLOER HOMOLOGY IN LARGE-N YANG- MILLS Glueball and meson propagators of any spin in large-N QCD Nucl. Phys. B 875 (2013) 621[hep-th/1305.0273] and Yang-Mills mass gap, Floer homology (v1), glueball spectrum and conformal window in large-N QCD (v2) hep-th/1312.1350 (v2 to appear shortly)
M.B. SNS-Pisa and INFN-Roma1 CERN Theory Group - April 29 (2014)
Tuesday, April 29, 14 In this talk we show that the mass gap of large-N SU(N) Yang-Mills (or SU(N) QCD in the pure glue sector) can be controlled by a Topological Field Theory ( TFT ) underlying large-N YM
Tuesday, April 29, 14 By means of the TFT, we find an answer for the mass gap and the ASD glueball propagator
↵ i.e. the two-point correlator of OASD = Tr(F↵ F ) ↵ ↵ ↵ F = F ⇤F X ↵
1 ⇤F = ✏ F ↵ 2 ↵ in Euclidean or ultra-hyperbolic signature
i ⇤F = ✏ F ↵ 2 ↵ in Minkowski that is compatible with everything that we know presently about large-N YM, both in the infrared numerically by lattice gauge theory and experimentally by Particle Data Group, and more importantly in the ultraviolet by first principles as we will show momentarily
Tuesday, April 29, 14 The answer in Euclidean or ultra-hyperbolic signature in large-N YM is: ↵ OS = TrF↵ F X↵ ↵ OP = Tr(F ⇤F↵ ) X↵
ip x 4 O (x)O (0) e · d x h ASD ASD iconn Z 2 4 6 4 2 2 1 k g ⇤ 2p4 1 g ⇤ = k W = k W + infinite contact terms ⇡2 p2 + k⇤2 ⇡2 p2 + k⇤2 W W kX=1 kX=1 1 C(0) (p2)+0< O (0) > + infinite contact terms ⇠ ADS N ASD
p2 4 log log ⇤2 (0) 2 2p 1 1 MS 1 CASD(p )= 2 p2 1 2 p2 + O 2 p2 ⇡ 0 " 0 log ⇤2 0 log ⇤2 ! log ⇤2 # MS MS ✓ MS ◆
Tuesday, April 29, 14 The ASD correlator in the TFT needs an exact non- perturbative scheme for the large-N beta function, in such a way that the canonical coupling does not diverge at the infrared Landau pole of the Wilsonian or of the perturbative coupling M.B. JHEP 05(2009)116
g3 + 1 g3 @ log Z @g 0 (4⇡)2 @ log ⇤ 3 5 = = 0g 1g + @ log ⇤ 1 4 g2 ··· (4⇡)2 @gW 3 3 2 0 5 = 0gW @g 0g + 2 g @ log ⇤ = (4⇡) + 4 2 2 @ log ⇤ 1 (4⇡)2 g ··· @ log Z 2 0gW 2 = =2 0g + 2 4 @ log ⇤ 1+c g2 ··· = g3 + 0 g5 0 g5 + 0 W 0 (4⇡)2 (4⇡)2 ··· = g3 g5 + 1 5 0 1 ··· 0 = 1 11 (4⇡)2 3 0 = (4⇡)2 3 1 34 = 1 (4⇡)4 3
Tuesday, April 29, 14 Euler-MacLaurin formula, in order to extract the large- momentum asymptotics (Migdal, decades ago ...)
1 1 1 Bj j 1 Gk(p)= Gk(p)dk @k Gk(p) j! k=k1 k1 j=1 kX=k1 Z X h i
Tuesday, April 29, 14 The answer in Minkowski in large-N YM is:
m2 = k⇤2 ; k =1, 2 k QCD ···
Exact linearity, as opposed to asymptotic linearity, is as a strong statement as it sounds very unlikely even at large-N, but ...
Tuesday, April 29, 14 The prediction of the TFT agrees sharply with
SU(8) lattice YM computation by Meyer-Teper (2004) on the largest lattice (16^3 * 24), presently closest to continuum, i.e. with the smallest value of YM coupling (beta=2N/(g_YM)^2=45.5) ++ m0 ⇤ rs = m0++ rs = rps =1.42(11) + m0 rps = m0++ TFT:
r = r = p2=1.4142 s ps ···
Tuesday, April 29, 14 Spectrum of large N massless QCD
0 1 2 = = = 6 s s s 0 2 s= s= 5 4 s= 4 n or 3 k s=1 2
s= 3 1
s= 5 0 0 1 2 3 4 5 6 2 2 m L QCD Tuesday, April 29, 14
ê Plot from M.B. hep-th/1308.2925
The lattice data are taken from Meyer-Teper SU(8) hep-lat/0409183 for glueballs (red)
and from Lucini-Teper, Lucini-Teper-Wenger Lucini-Rago-Rinaldi hep-lat/1007.3879 for glueballs (green)
Bali-Bursa-Castagnini-Collins-Del Debbio-Lucini-Panero SU(17) hep-lat/1304.4437 for mesons (yellow)
Tuesday, April 29, 14 Based on the exact linearity of the glueball spectrum of the TFT, and on the previous plot, we have conjectured that there exists a Topological String Theory in large-N ‘t Hooft limit of massless QCD dual to the TFT, such that
the glueball and meson spectrum of the masses squared is exactly linear at the leading 1/N order
Tuesday, April 29, 14 The actual glueball (green) and meson leading Regge trajectories for any flavor (other colors) implied by Particle Data Group and BES collaboration versus the TFT theory
6 2 2 (black) ⇤W = (1505 MeV) m 5 f0(2100) =1.397(008) mf0(1500) 4
s 3
2
1 1 1 m2 m2 = ⇤2 (n + s ) s,n PGB 2 W 2
0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2 2 2 m !mPGB "W
Tuesday, April 29, 14
! "# But the most remarkable powerful check of the TFT is the asymptotic behavior for large- momentum of the ASD correlator because
it agrees with the following constraints that we know with certainty about YM mass gap at large-N by first principles
Tuesday, April 29, 14 Problem of the Yang-Mills Theory as formulated by A. Jaffe and E. Witten for the Clay Math. Inst. (2000)
Yang-Mills is massless in perturbation theory, but massless gluons are not observed in scattering asymptotic states
Thus pure YM must admit only massive states above the vacuum: the glueballs
Existence of the mass gap, i.e. exponential bound, in terms of the RG-invariant scale, for every gauge-invariant Euclidean correlation function and every compact gauge group
Tuesday, April 29, 14 Mass gap in SU(N) YM has IR and UV nature
1 1 2 2 2 0 mgap = const ⇤ exp( 2 )( 0gYM) (1 + ...) 2 0gYM Mass gap is zero to every order of perturbation theory Mass gap is not a strong coupling problem, since strong coupling does not allow us to remove the cutoff Mass gap is a weak coupling problem that needs an amazing asymptotic accuracy as the coupling vanishes when the cutoff diverges Mass gap is presently, in the full generality of Jaffe-Witten formulation, for any gauge invariant correlation function and any compact gauge group, almost hopeless in my opinion
Tuesday, April 29, 14 Easier problem: mass gap in the large-N limit of SU(N) YM Same features discussed above, but free theory at next to leading order 1/N
N 4 2 Z = A exp ( d xtrf (F↵ ) ) (G. ‘t Hooft 1974) 2g2 Z Z because of the following remarkable simplifications. In the pure glue sector:
2 n < (x ) (x ) (x ) > N O1 1 O2 2 ···On n conn⇠ thus at the leading 1/N order: 1 1 < trF 2 (x )... trF 2 (x ) >= N ↵ 1 N ↵ k X↵ X↵ 1 1 < trF 2 (x ) >...< trF 2 (x ) > N ↵ 1 N ↵ k X↵ X↵
Tuesday, April 29, 14 At next to leading 1/N order, because of the vanishing of the interaction associated to 3 and multi-point correlators,
two-point correlators are an infinite sum of free fields satisfying the the Kallen-Lehmann representation (A. Migdal, 1977):
(s) 2 (s) (s) ip x 4 1 (s) p↵ < 0 (0) p, n, s >0 (x) (0) conne · d x = P | |O | | hO O i (s) 2 (s)2 Z n=1 mn p + mn X (s) (s) p↵ (s) < 0 (0) p, n, s, j >= e ( ) < 0 (0) p, n, s >0 |O | j m |O | p p p e(s)( ↵ )e(s)( ↵ )=P (s) ↵ j m j m m j X
Tuesday, April 29, 14 Let we start with the following simple
but fundamental question What is the large momentum behavior of two-point correlators of any integer spin s in pure Yang-Mills, in QCD and in n=1 SUSY QCD with massless quarks, or in any confining asymptotically free gauge theory massless in perturbation theory ?
(s) (s) ip x 4 (x) (0) e · d x =? hO O iconn Z For example:
(s) = Tr(F 2 ), ¯ ↵ ,T , O ↵ ↵ ···
Tuesday, April 29, 14 The answer is simple but not completely trivial. We have found it by standard methods: Perturbation Theory + Asymptotic Freedom + Renormalization Group + Some non-trivial subtlety . . . (s) (s) ip x 4 (x) (0) e · d x hO O iconn Z 2 0 p 1 log log( 2 ) 0 (s) p↵ 2D 4 1 1 ⇤QCD 1 P p 2 1 2 + O( 2 ) ⇠ p log( p ) 2 log( p ) log( p ) " 0 ⇤2 ✓ 0 ⇤2 ⇤2 ◆# QCD QCD QCD up to a polynomial in momentum, i.e. a contact term, i.e. a distribution supported at x=0 in coordinate space (this is the first subtlety) that must be subtracted; p 2 2 P ( s ) ↵ is the projector obtained substituting m = p p p↵ P (s) in the massive projector of spin s m (this is the second subtlety) Tuesday, April 29, 14 Definitions:
(s) @ log Z 2 (s) (g)= = 0g + O log µ ···
@g (g)= = g3 g5 + @ log µ 0 1 ···
Tuesday, April 29, 14 Therefore, at the leading large-N order it must hold:
(s) 2 1 p < 0 (0) p, n, s >0 P (s) ↵ | |O | | (s) 2 (s)2 n=1 mn p + mn X 2 0 p 1 log log( 2 ) 0 (s) p↵ 2D 4 1 1 ⇤QCD 1 P p 2 1 2 + O( 2 ) ⇠ p log( p ) 2 log( p ) log( p ) " 0 ⇤2 ✓ 0 ⇤2 ⇤2 ◆# QCD QCD QCD up to contact terms Fundamental question: Which are the constraints on the residues and the poles that follow from this asymptotic equality ? Oddly, neither Migdal nor other people found any answer (for deep reasons in the case of Migdal, that I will discuss possibly at the end of the talk) We will answer this question today, after 36 years !
Tuesday, April 29, 14 The answer to the fundamental question is the following Asymptotic Theorem:
(s)2D 4 (s)2 1 (s)2 (s) (s) ip x 4 1 (s) p↵ mn Zn ⇢s (mn ) (x) (0) conn e · d x P hO O i ⇠ (s) 2 (s)2 Z n=1 mn p + mn X (s)2 1 (s)2 (s) p↵ 2D 4 1 Zn ⇢s (mn ) = P p + p 2 (s)2 ··· n=1 p + mn X (s)2 (s) p↵ 2D 4 1 Z (m) 2 P p 2 2 dm + ⇠ p m(s)2 p + m ··· Z 1 2 0 p 1 log log( 2 ) 0 (s) p↵ 2D 4 1 1 ⇤QCD 1 P p 2 1 2 + O( 2 ) ⇠ p log( p ) 2 log( p ) log( p ) " 0 ⇤2 ✓ 0 ⇤2 ⇤2 ◆# QCD QCD QCD
1 (s)2 1 (s)2 1 2 2 2 f(mn ) f(mn )dn = f(m )⇢s(m )dm ⇠ (s)2 n=1 1 m1 X Z Z (s) g(mn ) (s) (s) (s) (s) (g) Z Z (m )=exp O dg n ⌘ n (g) Zg(µ) m(s)2 0 n log log 2 0 (s)2 1 1 ⇤QCD 1 Zn (s)2 1 2 (s)2 + O( (s)2 ) ⇠ mn mn mn " 0 log 2 0 log 2 log 2 # ⇤QCD ✓ ⇤QCD ⇤QCD ◆
Tuesday, April 29, 14 Specialize the RG estimate to scalar and pseudoscalar glueball propagators
(g) 2 (g) 2 ip x 4 tr F (x) tr F (0) e · d x h gN ↵ ↵ gN ↵ ↵ iconn 2 log log p R P P ⇤2 = C p4 1 1 1 MS + O 1 S p2 2 p2 2 p2 0 log 0 log log " ⇤2 ⇤2 ! ⇤2 # MS MS ✓ MS ◆
g2 g2 ip x 4 tr F F˜ (x) tr F F˜ (0) e · d x h N ↵ ↵ ↵ N ↵ ↵ ↵ iconn 2 log log p R P P ⇤2 = C p4 1 1 1 MS + O 1 PS p2 2 p2 2 p2 0 log 0 log log " ⇤2 ⇤2 ! ⇤2 # MS MS ✓ MS ◆
g2 2 g2 2 ip x 4 tr F (x) tr F (0) e · d x h N ↵ ↵ N ↵ ↵ iconn 2 log log p R P P ⇤2 = C p4 1 1 1 MS + O 1 ASD p2 2 p2 2 p2 0 log 0 log log " ⇤2 ⇤2 ! ⇤2 # MS MS ✓ MS ◆
Tuesday, April 29, 14 Perturbative check: the 3-loop computation by Chetyrkin et al.
2 2 N 2 1 4 p2 2 p2
73 f0 = 3(4⇡)2 f f ⇡2 =(37631 242 ⇣(2) 110⇣(3)) 1 1 3 54 3 (4⇡)4 2 = 2 11 0 3(4⇡)2 2f = 313 f = 313 2 (4⇡)4 ) 2 2(4⇡)4 3f = 121 f = 121 f = 2 3 3(4⇡)4 ) 3 9(4⇡)4 ) 3 0
f =(37631 110⇣(3)) 1 ) 1 54 (4⇡)4
Tuesday, April 29, 14 Perturbative check: the 3-loop computation by Chetyrkin et al.
2 (N 1) 4 p2 2 p2
˜ 97 f0 = 3(4⇡)2 f˜ =(51959 110⇣(3)) 1 1 54 (4⇡)4 2 = 2 11 0 3(4⇡)2 2f˜ = 1135 f˜ = 1135 2 3(4⇡)4 ) 2 6(4⇡)4
Tuesday, April 29, 14 Check of the RG estimate (M.B. and S. Muscinelli, JHEP 08 (2013) 064 [hep-th/1304.6409])
g2 2 g2 2 ip x 4 tr F (x) tr F (0) e · d x h N ↵ ↵ N ↵ ↵ iconn 1 p4 2 2 2 2 R P=(1 2 ) 2 P2g (p ) 2g (µ ) N 2⇡ 0 + a +˜a 1 g4(p2) a +˜a 1 g4(µ2) + O(g6) 0 0
2 g2(p2)=g2(µ2) 1 g2(µ2) log p 0 µ2 2 2 g4(µ) log p + 2g4(µ2) log2 p + 1 µ2 0 µ2 ··· Tuesday, April 29, 14 An interesting aside: In the past years several proposals for the glueball propagators have been advanced based on AdS String/ Gauge Theory correspondence
UV test for glueball propagators: AdS String/ Gauge Theory correspondence versus the TFT
TFT (QCD), Polchinski-Strassler or Hard Wall (QCD), Soft Wall (QCD), Klebanov-Strassler background (n=1 cascading SUSY QCD)
Tuesday, April 29, 14 Polchinski-Strassler (Hard Wall)
p K1( ) p 2 2 ip x 4 4 µ 4 2 µ
Soft Wall
2 2 2 ip x 4 4 µ
Tuesday, April 29, 14 Klebanov-Strassler: n=1 cascading SUSY QCD
3 3 @g 2 g = (4⇡) @ log ⇤ 1 2 g2 (4⇡)2
2 2 2 ip x 4 4 3 p
Tuesday, April 29, 14 In this talk we show that the mass gap of large-N SU(N)Yang-Mills can be controlled by a TFT underlying YM According to Witten (talk at Simons Center 2012) every gauge theory with a mass gap should contain a possibly trivial TFT in the infrared
The canonical example (math) is twisted n=1 SUSY YM that leads to Donaldson invariants
The canonical example (physics) is the gluino condensate in n=1 SUSY YM In these examples the TFT is associated to cohomology in function space, that allows us to solve the TFT
Tuesday, April 29, 14 Cohomological localization in QFT
In Math In QFT Duistermaat-Heckman 1982 Witten 1992-1994 Atiyah-Bott 1982 Nekrasov 2003 Bismut 1985-1986 Pestun 2007 Localization on the fixed points QO =0 of a torus action: QSSUSY =0
1 n 2⇡ n 1 1 2 Q↵ =0 ! exp ( ✏H)= ( ) Pf (! @ H ) n! ✏ P P Z 2 Z XP Q =0 Cohomological interpretation: O exp( S )= O exp( S tQ↵) SUSY SUSY d! =0 Z Z d ! = (! + d↵) O exp( S tQ↵) dt SUSY Z 2 Z Z d =0 = OQ↵ exp( S tQ↵) SUSY exp( ! td↵)= exp( !) Z M M = Q(O↵ exp( S tQ↵)) = 0 Z Z SUSY Tuesday, April 29, 14 Z This cannot work in pure YM because of the lack of the would-be cohomology in function space, i.e. the lack of SUSY We have constructed a trivial TFT at N=infinity in pure large-N YM, that computes (trivial) homology classes of certain Lagrangian submanifolds immersed in space-time, defined by the v.e.v. of twistor Wilson loops in the adjoint representation that are trivial, i.e v.e.v.=1, at all scales in the large-N limit ˆ ˆ ˆ (B ; Lww)=P exp i (B )zdz +(B )z¯dz¯ 1 ZLww 1 lim < Tr (Bˆ ; Lww) >=1 N N !1 N
Tuesday, April 29, 14 Thus our TFT is completely trivial at N=infinity, not only in the IR according to Witten prediction, but also in the UV: a much more specific feature, that will turn out to be crucial
Why are we interested in such a trivial subsector (actually a subalgebra) of large-N YM ?
Because we will show that the trivial homology classes defined by twistor Wilson loops are localized at N=infinity on the critical points of a certain quantum effective action by a new field-theoretical version of Morse-Smale-Floer homology
Tuesday, April 29, 14 The critical points turn out to be singular instantons with a lattice of singularities supported on the aforementioned Lagrangian submanifolds of the TFT
(2) F (z)= µˆ (p) (z z ) ↵ ↵ p p X
Besides, we will show that in the continuum limit for the
lattice field µˆ ↵ (p) fluctuations are self-consistently suppressed at the leading 1/N order, and control the mass gap of large-N YM at the next-to-leading order
Tuesday, April 29, 14 More precisely, the critical points (for TWL in the fundamental representation) turn out to be a kind of surface operators supported on the Lagrangian submanifolds, with holonomy valued in Z(N), labelled by k=1,2...
Though the TFT is trivial at the critical points, the fluctuations around the critical points are non-trivial and control the joint spectrum of an infinite tower of scalar pseudoscalar glueballs. Thus Z(N) magnetic condensation realizes in the TFT a version of ‘t Hooft duality !
Tuesday, April 29, 14 ‘t Hooft e/m duality (1978-1981): if a YM theory with only fields in the adjoint representation has a mass gap, then either the Z(N) magnetic charge condense (confinement phase) or the Z(N) electric charge condense (Higgs phase)
Tuesday, April 29, 14 Roughly speaking, we replace in the TFT cohomological localization on field space with a new field theoretical localization that is a version of Morse-Smale homology on submanifolds of space-time
Given a non-degenerate Morse function f on a compact manifold M, Morse inequality holds:
N H (M) f i i X Smale gradient flow:
@xi @f(x) = gik @t @xk
Tuesday, April 29, 14 The arcs associated to the gradient flow between critical points whose Morse index differs by 1can be used to define a complex that is isomorphic to the usual singular homology
of the compact manifold M (Smale (1961-62)). In infinite dimension Floer homology replaces the Morse function with a classical field theory over M and defines, by gradient flow, new homology groups that are topological invariants of M Tuesday, April 29, 14 1
Our TFT associates critical points of a quantum effective action to the nodes of certain Lagrangian submanifolds, by means of a new version of Makeenko-Migdal loop equation for twistor Wilson loops that we call holomorphic loop
equation 1
Tuesday, April 29, 14 We recall that Makeenko-Migdal loop equation (1979) is a
Schwinger-Dyson equation for ordinary Wilson loops:
(x, y; A)=P exp i A↵dx↵ ZLxy
S Tr (e YM (A; L )) A =0 A (x) xx Z ↵
Tuesday, April 29, 14 N S dx < Tr( YM (x, x; A)) >= ↵ 2g2 A (x) ZLxx ↵ i dx dy (4)(x y)
The expectation value, <...>, can be combined with the matrix trace, Tr, to define a new normalized trace, TR. Thus the problem is to find a solution, A, uniformly for all loops, with values in a certain operator algebra with normalized trace, TR(1)=1. Such a solution is called the master field (Witten 1980)
Tuesday, April 29, 14 The holomorphic loop equation is Schwinger-Dyson equation for twistor Wilson loops in new variables
1 < Tr( (B ; L(1)) (B ; L(2))) > zz µ(z,z¯) zz N 1 dw 1 1 = < Tr (B ; L(1)) >< Tr (B ; L(2)) > ⇡ z w zw wz ZLzz N N
iF (B )=µ zz¯
1 1 1 µ = F + (F + iF ) (F + iF ) 01 2 02 03 2 02 03
1= (F (A) µ ) µ ↵ ↵ ↵ Z
Tuesday, April 29, 14 Recall that v.e.v. of twistor Wilson loops is trivial in the adjoint representation (at large-N it factorizes into the fundamental and conjugate representation)
1 lim < Tr (Bˆ ; Lww) > =1 N Locally, we Ndeform!1 Nthe M.-M. loop to a loop with zero cusp angle:
Tuesday, April 29, 14 1
Globally:
Tuesday, April 29, 14 2
Tuesday, April 29, 14 Regularize in gauge-invariant way the cusped holomorphic loop equation for real points (or in Minkowski space-time): z i(z + i✏) ! + 1 1 = P i⇡ (z w ) z w + i✏ z w + + + + + + M + 1 w˙ +(scusp) 1 w˙ +(scusp ) dw+(s) (z+(scusp) w+(s)) = + + 2 w˙ (scusp) 2 w˙ (scusp ) Z | + | | + | 1 1 = =0 2 2 Tuesday, April 29, 14 Since Gamma does not get quantum corrections at N=infinity because of the localization of the holomorphic loop equation around the cusps, it must already contain quantum corrections, and therefore it must imply the correct beta function of YM and the mass gap. We check all these properties by direct computation! Tuesday, April 29, 14 The TFT involves a change of variables: Non-SUSY Nicolai map in pure YM from 3=4-1 components of gauge connection (-1 is due to gauge fixing) to 3 components of the ASD part of curvature F = F F˜ ↵ ↵ ↵ 1 F˜ = ✏ F ↵ 2 ↵ ↵ 2 2 ˜ Tr(F↵ )=Tr(F↵ ) /2+Tr(F↵ F↵ ) 2 16⇡ NQ N 2 4 Z = exp tr (F ) d x A 2g2 4g2 f ↵ Z Z 1= (F µ ) µ ↵ ↵ ↵ Z 2 16⇡ NQ N 2 4 Z = exp tr (µ ) d x (F µ ) A µ g2 4g2 f ↵ ↵ ↵ ↵ Z Z Tuesday, April 29, 14 Since the change to the ASD variables is already a non- standard tool we proved with A. Pilloni JHEP 09 (2013) 039 hep-th/1304.4949 the identity of the one-loop1PI effective action in the original and in the ASD variables in any gauge of YM, QCD, n=1 SUSY YM and any theory that extends YM Tuesday, April 29, 14 In the TFT the ASD variables are restricted to a lattice of surface operators, that defines a hyper-Kalher reduction on a dense set in function space of NC YM (M.B. 2011) (2) 1 1= ( i[Dˆ , Dˆ ] µˆ (p) (z z ) ✓ 1)ˆ µˆ (p) ↵ ↵ p ↵ ↵ Z p p X (2) Y µˆ (z,z¯)= µˆ (p) (z z ) ↵ ↵ p p 1 X 2 N f(z , z¯ )ˆµ (p) f(z,z¯)ˆµ (z,z¯)d z D p p ↵ ! ↵ p Z Tuesday, April 29, 14 X Self-consistent check of the localization at the nodal points of the TFT, i.e. that Gamma does not get quantum corrections at N=infinity ! 1= (F (A) µ ) µ ↵ ↵ ↵ Z (2) µ (z)= µˆ (p) (z z ) ↵ ↵ p p the 3 Hermitean matricesX of residues of the ASD field at the point p must commute for the ASD equations (Hitchin equations) to admit a solution. Thus the ASD residues can be diagonalized by the same local unitary gauge transformation Math. proof C. Simpson (1990), Biquard-Bloac (2001), Phys. proof Gukov-Witten Mochizuki (2002) (2006)using Nahm equations Tuesday, April 29, 14 Why Gamma is exact at large-N ? There are only 3(N-1) fluctuating fields around the singular divisor of surface operators, as opposed to the 3(N^2- 1) smooth fluctuations in perturbation theory Thus quantum fluctuations are suppressed1/N around the trivial topological sector and the theory at the next to leading 1/N order is free, as it must be in the large-N limit of YM Tuesday, April 29, 14 Now we construct the TFT analytically Twistor Wilson loops are first defined in U(N) YM on non- commutative space-time, for large non-commutativity: Tr (Bˆ ; Lww)= N 1 Tr P exp i (Aˆz + Dˆ u)dz +(Aˆz¯ + Dˆ u¯)dz¯ N ZLww Dˆ u = @ˆu + iAˆu 1 [@ˆu, @ˆu¯]=✓ 1 z = x0 + ix1 ˆ uˆ =ˆx2 + ixˆ3 [ˆu, u¯]=✓1 z¯ = x0 ix1 u¯ˆ =ˆx ixˆ 2 3 NC YM in the limit of large non-commutativity is equivalent to SU(N)YM in large N limit on commutative space-time Tuesday, April 29, 14 V.e.v. of twistor Wilson loops is trivial 1 1 < Tr (Bˆ ; Lww) > = < Tr (Bˆ1; Lww) > N N N1 N lim < Tr (Bˆ ; Lww) > =1 ✓ N !1 N Hint: at lowest order in perturbation theory 1 1 Tuesday, April 29, 14 NC twistor loops are gauge equivalent for large theta to the following loops of the large-N (Morita equivalent) commutative gauge theory 1 1 Thus they are supported on Lagrangian submanifolds of twistor space of complexification of Euclidean space-time Triviality proof is based on vanishing of coefficients of propagators: 2 1 z˙z¯˙ + i z˙ z¯˙ =0 2 1 z˙z¯ + i z˙ z¯˙ =0 2 1 zz¯˙ + i z z¯˙ =0 Tuesday, April 29, 14 Wilson loop in NC YM theory, translations can be reabsorbed by gauge transformations = modern Eguchi- Kawai reduction 1 ˆ 1 ˆ ˆ trN TrNˆ (A; Lww)=trN TrNˆ P exp (@↵ + iA↵)dx↵ N N ZLww ˆ Uˆ(x)=ex↵@↵ ˆUˆ ˆ ˆ ˆ 1 ˆ ˆ 1 A↵ = U(x)A↵U(x) + i@↵U(x)U(x) ˆUˆ ˆ ˆ ˆ 1 @↵ = U(x)@↵U(x) (Aˆ; L )=P exp i ( i@ˆ + Aˆ )dx yz ↵ ↵ ↵ ZLyz 1 Uˆ(y) (Aˆ; Lyz)Uˆ(z) = P exp i ( i@ˆU + AˆU )dx ↵ ↵ ↵ ZLyz 1 1 1 = P exp i (Uˆ(x)Aˆ Uˆ(x) iU(x)@ˆ U(x) + i@ Uˆ(x)Uˆ(x) )dx ↵ ↵ ↵ ↵ ZLyz 1 = P exp i Uˆ(x)Aˆ↵Uˆ(x) dx↵ ZLyz = P? exp i A↵(x)dx↵ ZLyz Tuesday, April 29, 14 Reduction to a limit of finite rank surface operators (hyperfiniteness) by Morita duality (change of basis) for rational non-commutativity in area units of U(N) YM on NC torus with periodic b.c. to U(N N^)YM on commutative torus with twisted boundary conditions The twisted theory contains an untwisted sector: U(N) embedded diagonally as U(N)1^ in U(NN^) The untwisted sector plays a special role because may carry zero momentum on the torus and therefore may condense Tuesday, April 29, 14 NC Eguchi-Kawai reduction, and Morita duality from U(N) to U(NN^)/ Z(N N^) gauge theory with twisted boundary conditions on a commutative torus with side L/N^ and U(1) ‘t Hooft flux 2⇡il.x/Lˆ A(ˆx, y)= al(y)e l Z2 X2 . Mlˆ 1 l2 Mlˆ 1l2/2 2⇡il x/NLˆ A0(x, y)= al(y)V U ! e l Z2 X2 2⇡ UV = !VU Nˆ( )2 =2⇡✓ ˆ ! = e2⇡i/N ⇤ Mˆ 2⇡✓ = L2 ˆ ¯ Nˆ A0(xj + L/N)= jA0(xj) j ⇤ N =( )2L2 2 2⇡ r 1 =1N U ⇤ Mˆ ⇥ Nˆ =( )2L2 =1 V 2⇡ ˆ 2 N ⇥ N Tuesday, April 29, 14 3 ASD equations in resolution of identity can be written as an infinite dimensional Hitchin system in NC YM: 1 1 B⇢ = A + ⇢D + ⇢ D¯ =(Az + ⇢Du)dz +(Az¯ + ⇢ Du¯)dz¯ 1 0 1 iF ✓ 1=µ = µ + ⇢ n ⇢n¯ B⇢ ⇢ i@ D¯ = n A i@¯ D =¯n A 1 1= n n¯ µ ( iF µ ✓ 1) ( i@ D¯ n) ( i@¯ D n¯) ⇢ B⇢ ⇢ A A Z ZC⇢ The holomorphic gauge (this is a change Bz¯0 =0 of variables): i@z¯Bz0 = µ0 ⇢ =1 B1 = B 2 µ N8⇡ N4 0 2 4 Z = n n¯ µ0 exp( Q Tr (µ ) +4Tr (nn¯)d x) µ g2 g2 f f Z ZC1 0 Z 1 ( iF µ ✓ 1) ( i@ D¯ n) ( i@¯ D n¯) A A¯ D D¯ B A A Tuesday, April 29, 14 The new holomorphic loop equation 2 1 N8⇡ N4 2 4 <...>= Z n n¯ µ0... exp( Q Tr (µµ¯)+Tr (n +¯n) d x) g2 g2 f f Z ZC1 Z 1 µ ( iFB µ ✓ 1) ( i@AD¯ n) ( i@¯AD n¯) A A¯ D D¯ µ0 Tr (e (B0; L )) µ0 =0 µ (z,z¯) zz Z 0 1 dw Tuesday, April 29, 14 Effective action from the holomorphic loop equation Holomorphic/anti-holomorphic fusion (Cecotti-Vafa 1991) Classical YM action ASD variables 2 n n¯ µ0e | | ZC1 2 N8⇡ N 2 4 = n n¯ µ0 exp( Q Tr (µ )d x) | g2 4g2 f ↵ C1 W W ↵= Z X6 Z 1 ⇤ 1 µ 2 . . nb 2 2 Det ( A ↵ iadµ )Det( A)( ) Det ! ↵ 2⇡ µ0 | Jacobian of F.P. determinant Zero Nicolai map Feynman gauge modes Jacobian of holomorphic gauge Tuesday, April 29, 14 Effective action from the holomorphic loop equation Holomorphic/anti-holomorphic fusion Classical YM action ASD variables 2 n n¯ µ0e | | ZC1 2 N8⇡ N 2 4 = n n¯ µ0 exp( Q Tr (µ )d x) | g2 4g2 f ↵ C1 W W ↵= Z X6 Z 1 ⇤ 1 µ 2 . . nb 2 2 Det ( A ↵ iadµ )Det( A)( ) Det ! ↵ 2⇡ µ0 | anomalous dimension beta function glueball potential Tuesday, April 29, 14 Effective action restricted to a lattice of surface operators Z = A A¯ D D¯ ( iF µ (2)) ( i@ D¯ n (2)) ( i@¯ D n¯ (2)) | B p p A p p A p p p p p ZH X X X ˆ n 4NN 2 (np) (¯np) n b 2 exp ( tr Tr ((µ n +¯n ) +4n n¯ )) (⇤p✓) b !0 2 µ n n¯ g2 N Nˆ p p p p p (µ ) p ^ p ^ p| W p p p X Y Tuesday, April 29, 14 Large-N pure Yang-Mills canonical beta function (M.B. 2008) After rescaling of zero modes (same technique as in NSVZ) 2 2 2 2 8⇡ k(N k)g 8⇡ k(N k)g 2 2 1 2 = 2 +2k(N k)g log g + g k(N k) log Z 2g (⇤) 2gW (⇤) 2 But now an anomalous dimension occurs because of the non-trivial Jacobian of change to ASD variables (Nicolai map) ! 1 1 4 1 2 = 2 + 2 log g + 2 log Z 2gW (⇤) 2g (⇤) (4⇡) (4⇡) Tuesday, April 29, 14 3 g3 @ log Z @g 0g + 2 = (4⇡) @ log ⇤ @ log ⇤ 1 4 g2 (4⇡)2 Remark: Z factors are different in UV and IR due to different cutoff occurring because of gluing rules of local systems Global constraint: gluing rules that arise by local systems UV IR Tuesday, April 29, 14 Density of surface operators: ⇢ = (2)(z z ) p0 glueball potential Xp0 2 (2) N 0 = d z (z z ) 2 p0 Z Xp0 ⇤ N =( )2L2 = (2)(x x )2d4x 2 2⇡ p Z Assuming translational invariance: 8NNˆ = d2ud2z⇢2tr Tr (µµ¯)+ d2ud2z⇢2 log (µ) 2 g2 N Nˆ | | W Z Z 1/2 . . . 2 log Det ( iµ )Det( ) 1 A ↵ ↵ A µ= (µ iµ ) 2 01 03 2 2 2 2 2 2 d ud z⇢ n [µ0] log ⇤ d u⇢ log Pf (!0) b | | Z Z glueball kinetic term Tuesday, April 29, 14 k(N k)Nˆ 2(4⇡)2 10 1 ⇤ = (1 g2 log ) d2ud2z⇢2 k 2g2 W 3 (4⇡)2 M W Z ⇤ 2k(N k)Nˆ 2 d2ud2z⇢2 log + ... M Z ˆ 2 2 1 11 1 ⇤ 2 2 2 = k(N k)N (4⇡) ( 2 2 log ) d ud z⇢ + ... 2gW 3 (4⇡) M 1 Z 2 g2 ⇤e 0 W = k(N k)Nˆ 2(4⇡)2( log ) d2ud2z⇢2 + ... 0 M 1 Z 2 g2 ⇤e 0 W = Nˆ log d2ud2zM4 + ... 0 M Z ⇤ = Nˆ log W d2ud2zM4 + ... 0 M Z The subtraction point M is chosen at the scale of the action: Nˆ M 4 = k(N k)(4⇡)2⇢2 N as in Veneziano-Yankielowicz SUSY effective action At the critical points: ⇤ ⇤ k =4M 3 log W M 3 = M 3(4 log W 1) = 0 M M M 4 1 4 M = e ⇤W Tuesday, April 29, 14 and the density squared of surface operators scales as: 2 1 4 ⇢ (Nkˆ ) ⇤ k ⇠ W The (local part) of the effective action is negative at the critical points because of AF as for VY SUSY effective action Nˆ d2ud2zM4 0 4 Z Consequences of scaling of density: All critical points have the same renormalized action (this is due to our choice of subtraction point as in VY. Physical interpretation: the whole Z(N) condenses) Tuesday, April 29, 14 Jacobian to holomorphic gauge µp Hitchin locus !0 (µp) p 1 1 !0 ^ | = ^ ^ = Det(!0) 2 (µ ) ^ µ G (µ )2 p p p0 p p p p p p Y ^ ^ Y ^ Y Q Glueball potential d4x⇢2 log (µ) 2 = d4x⇢2 log (µ µ ) 2 | | | ↵ | Z Z ↵X> The glueball potential is the logarithm of the square of a holomorphic function. Thus second derivative vanishes but at the zeros of the holomorphic function Tuesday, April 29, 14 Mass gap arises despite vanishing of density at large N^, because the vanishing is compensated by large N^ degeneracy and singular nature of the glueball potential 2 2 2 @ 2 @ 2 Mij = log (µ) = log µ↵ µ @µi@µ¯j | | @µi@µ¯j | | ↵X> @ 1 1 = ↵i i)+c.c. @µ¯j µ↵ µ µ↵ µ ↵> X @ 1 1 = + c.c. @µ¯j µi µ µ↵ µi i X X @ 1 1 = + + c.c. @µ¯j µi µ µi µ i X X @ 1 = + c.c. @µ¯j µi µ =i X6 =2⇡ ( (2)(µ µ ) (2)(µ µ ) )+c.c. i ij i j =i X6 =2⇡ (2)(µ µ ) (2)(µ µ )(1 )+c.c. i ij i j ij ij =i 6 Tuesday, April 29, 14 X @ 1 (2) = ⇡ ij (zj) @z¯i zj 2 = diag(2⇡(k N)/N , 2⇡k/N) Critical points: p k N k M 2 =2⇡ (2)(0)((Nkˆ 1) (1 )) = 2⇡ (2)(0)(Nkˆ 1 ) Mass matrix: ij | {z ij } |ij{z }ij ij ij NNˆ (2)(0) = (2⇡)2 Mass Term: ˆ 2 2 2 ˆ 2 2 kNN 2 2 2 N20 ⇤W N N (2⇡) 2 ⇢k d zd utrkTrˆ ( µ µ¯)= d utrkTrˆ ( µ µ¯) 2⇡ N Nˆ(N k)2⇡ N Z Z Kinetic term: ˆ 5NN8 2 2 trN TrNˆ ( µp(u, u¯) µ¯p0 (v, v¯)) 2 2 d ud v 2 2 2 3(4⇡ ) ( zp(u, u¯) zp (v, v¯) + u v ) p=p 0 X6 0 Z | | | | 40NNˆ tr Tr ( µ(u, u¯) µ¯(v, v¯)) = d2zd2w ⇢2 d2ud2v N Nˆ 3(4⇡2)2 k ( u v 2 + u v 2)2 Z Z | | | | 2 4 2 40N 02(2⇡) N trN Trˆ ( µ(u+,u ) µ(v+,v )) N 2 2 du+du dv+dv 2 2 ! 3(4⇡ ) k(N k) 2(u+ v+ + i✏) (u v + i✏) Z 2 2 2 20N 02(2⇡) N = du+du trN Trˆ ( µ(u+,u ) @+@ µ(u+,u )) 3k(N k) N Z Tuesday, April 29, 14 Twistor loops are supported on Lagrangian submanifolds of twistor space that project to Lagrangian submanifolds of(complexified) space-time Tuesday, April 29, 14 We can choose the fiber of the twistor fibration so that the Lagrangian submanifold analytically continued to Minkowski is a plane diagonally embedded 1 Tr (Bˆ ; Lww) Tr P exp i (Aˆz+ + i Dˆ u)dz+ +(Aˆz + i Dˆ u¯)dz N ! N ZLww 1 1 (z,z,i¯ z,i z¯) (z+,z , z+, z ) ! ˆ ˆ ˆ ˆ 1 ˆ Tr (B ; Lww) Tr P exp i (Az+ + i Du+ )dz+ +(Az + i Du )dz N ! N ZLww iF = µ (2)(z z )+ µ (u, u¯) (2)(z z (u, u¯)) B p p p p p X X Correlators of surface operators supported on Lagrangian submanifold form a massive trajectory linear in k Tuesday, April 29, 14 Terms quadratic in density dominate the asymptotic loop expansion in power of density of effective action at large N N^ pure poles since non-local terms are suppressed Glueballs kinetic term can only arise by radiative corrections since classical YM action is ultralocal in ASD Field iF = µ (2)(z z )+ µ (u, u¯) (2)(z z (u, u¯)) B p p p p p X X Counterterm quadratic in ASD “field of residues” and in density of surface operators This counterterm is local on diagonal Lagrangian submanifold in Minkwoski Tuesday, April 29, 14 Effective action 2 2 N N20 ↵0 2 k( µ, µ)=2⇡ du+du trkTrˆ ( µ @+@ µ)+ du+du ⇤W trkTrˆ ( µ µ) Nˆ 2 (N k) k N N Z Z Tuesday, April 29, 14 ASD glueball propagator (ultra-hyperbolic signature) 1 2 4 8 2 4 i(p+x +p x+) k N k N 4gk⇤W (2⇡) d xe Tuesday, April 29, 14 ASD propagator (Minkowski) 1 2 4 4 2 4 i(p+x +p x+) k N k N 4gk⇢k(2⇡) d xe Tuesday, April 29, 14 To summarize: We can forget about loop equations. The TFT is a way to test long-standing ideas about mass gap and confinement in YM, by a change of variables in the YM functional integral and by restricting to a lattice of parabolic singularities Fluctuations of YM in these new variables and around the singularities become locally Abelian, and thus we can compute reliably 1/N fluctuations around the Z(N) critical points Tuesday, April 29, 14 Outlook Extension to QCD in the Veneziano limit: already done in v2 to appear, conformal window follows SUSY extension unlikely because there are no glueballs in ASD channel in Minkowski (M. Shifman 2011, using the Nicolai map !) because of cancellations due to SUSY A twistorial string of YM ? Likely, since twistor Wilson loops are supported on Lagrangian submanifolds of twistor space Twistorial A-model (Neitzke,Vafa 2004) Chern Simons +NC Eguchi-Kawai + stringy instantons = Z(2) surface operators (M.B. 2008) the string dual should provide Higher spin Regge trajectories String S-matrix ? Likely, old fashioned string program. No interesting twistor string Wilson loops because of triviality . . . Tuesday, April 29, 14 Can we solve large-N QCD in ‘t Hooft limit, perhaps only for the S-matrix ? We will see ... Tuesday, April 29, 14 NC EGUCHI-KAWAI REDUCTION Tuesday, April 29, 14 Eguchi-Kawai reduction (1982), Gonzalez Arroyo - Korthals Altes (1983), Minwalla-Ramsdonk-Seiberg (1999), Makeenko (2000), Szabo (2001), Douglas-Nekrasov (2001), Dhar- Kitazawa (2001), Alvarez-Gaume’-Barbon (2002)... NC YM is equivalent to a matrix model with rescaled action, because translations can be absorbed into gauge transformations Operator/function correspondence: a@ˆ a@ˆ [ˆx↵, xˆ ]=i✓↵ 1 e ˆ (x)e = ˆ (x + a) d ˆi j ij ˆ d k ikxˆ ikx @ (ˆx )= 1 (x)= d e e (2⇡) d d Z (2⇡) 2 Pf(✓)Trˆ fˆ = d xf(x) ˆ d ˆ f = d xf(x) (x) d Z 2 ˆ ˆ ˆ d Z (2⇡) Pf(✓)Tr( (x) (y)) = (x y) fˆgˆ = f ˆ? g d d i d x(f ? g)(x)=d xf(x)g(x) (f ? g)(x)=f(x)exp( @↵✓↵ @ )g(y) 2 x y |y=x Z Z i ↵ ↵ f (x ) ? ... ? f (x )=exp( @ i ✓ @ )f (x )...f (x ) 1 1 n n 2 x xk 1 1 n n i Tuesday, April 29, 14 a@ˆ a@ˆ e ˆ (x)e = ˆ (x + a) @ˆi(ˆxj)= ij1 d d (2⇡) 2 Pf(✓)Trˆ fˆ = d xf(x) Z d d (2⇡) 2 Pf(✓)Trˆ ( ˆ (x) ˆ (y)) = (x y) ddx(f ? g)(x)=ddxf(x)g(x) Z Z N ddxtr (F ? F )(x) 2g2 N ↵ ↵ Z N d 1 2 = (2⇡) 2 Pf(✓)tr Trˆ ( i[@ˆ + iAˆ , @ˆ + iAˆ ]+✓ 1) 2g2 N ↵ ↵ ↵ N 2⇡ d 1 2 = Nˆ( ) tr Tr ( i[@ˆ + iAˆ , @ˆ + iAˆ ]+✓ 1) 2g2 ⇤ N Nˆ ↵ ↵ ↵ 2⇡ d d Nˆ( ) =(2⇡) 2 Pf(✓) ⇤ Tuesday, April 29, 14< T r (B0; Lw+z+ ) > ⇡ L z+ w+ Z z+z+ The principal part does not contribute, since it is supported 1 < Tr (B0; Lz w ) >=0 on open loops, N + + ,the cusp does not, since backtracks. This is Nthe LOCALIZATION of the loop equation: N ZLww ZLww 2 =2 dz dz¯( +i ) N N ZLww ZLww =0 ¯ < T r (B0; L ) > µ (z,z¯) zz ⇡ z w zw wz 0 ZLzz Hint: the holomorphic loop equation resembles for the cognoscenti Dijkgraaf-Vafa holomorphic matrix model for the glueball superpotential in N=1 SUSY gauge theories (2002)