Effective string models for confining LGTs.

Transcription

1 Effective string models for confining LGTs. M. Caselle STRONGnet Workshop; October 2011

2 Plan of the Talk A brief history of the effective string picture of confinement String corrections in Wilson loop expectation values. String corrections in Polyakov loop correlators. Flux tube thickness. 1

3 A brief history of effective string picture of confinement Nielsen-Olesen, t Hooft, Wilson, Polyakov, Nambu... Conjecture: Two color sources in a confining gauge theory are bound together by a thin flux tube, which can fluctuate like a massless string. Seminal ideas: large N limit use of duality to understand confinement 3d gauge Ising model described by a fermionic string (Polyakov) use LGT to test the conjecture 2

4 1980 Lüscher, Symanzik, Weisz, Münster.. Universal signatures of the quantum fluctuations of the confining string: Universal quantum corrections ( Lüscher term ) to the potential. V (R) σr (d 2)π 24R Logarithmic increase of the flux tube thickness. These effects could in principle be observed in LGT Alvarez, Pisarski, Olesen... Finite T version of universal string corrections Deconfinement driven by the quantum fluctuations of the string. Effective string prediction for the ratio T c / σ. 3

5 1985 Olesen, Alvarez... The flux tube as an effective string It can only describe the large distance (R > 1/ σ) behaviour of the interquark potential First indications that the corrections due to the anomaly appear at high order in 1/R Dietz and Filk + Nordita group + Torino group + de Forcrand, et al. Functional form of the effective string corrections Extension of the universal quantum corrections to other geometries and to finite size shapes. Higher order corrections. Connection with CFT s 4

6 First attempts to test effective string predictions with MC simulations Main problem: too large statistical and systematic errors. Results: qualitative agreement with effective string predictions, but no clearcut evidence Torino group + M. Hasenbusch and K.Pinn Study of effective string corrections in the 3d gauge Ising model: test of the log increase of the flux tube width with the distance. test of the effective string predictions for Wilson loops, Polyakov loops and interfaces 5

7 Lucini and Teper, Necco and Sommer, Bali... First unambiguous evidences (with standard MC algorithms) in favour of a bosonic effective string theory in non-abelian LGT s Lüscher and Weisz, Teper and collaborators, Pepe, Lucini, Kuti, Drummond... A new multilevel algorithm is proposed for non-abelian LGT s. Exponential reduction of statistical errors: large interquark distances can be reached SU(N) LGT s both in d = 3 and d = 4 are well described by a bosonic effective string theory. Unexpected: the transition from perturbative to string behaviour takes place at surprisingly small distances. Main result: the Nambu-Goto effective string works impressively well both for the ground state and for the excited states 6

8 Torino group + M. Hasenbusch Using string inspired techniques the explicit expression of the corrections due to the Nambu-Goto string are obtained to all orders for Polyakov correlators, interfaces and Wilson loops. At the same time, high performance algorithms based on the duality transformation for 3d gauge Ising model allow to succesfully test these predictions up to the next to leading order Pepe, Wiese + Torino group First studies of the behaviour of the flux tube thickness at the next to leading order both in SU(N) and in Ising. The results agree remarkably well with the Nambu-Goto predictions 7

9 Lüscher and Weisz, Drummond, Aharony and collaborators Universality theorems for the effective string Effective string corrections are universal up the next to leading order in d > 3 and up to the third order in d = 3 8

10 Main contexts in which effective string contributions may appear Lüscher type corrections to the interquark potential (lot of papers...) Spectrum of flux tube excited states (Teper, Majumdar, Brandt...) Finite T dependence of the string tension and prediction of T c / σ Flux tube thickness Low lying states of glueball spectrum (Lucini, Teper...) Hagedorn transition (Meyer + Torino group) (see Feo s talk tomorrow) 9

11 String theory and Lattice Gauge Theories Conjecture: Two color sources in a confining gauge theory are bound together by a thin flux tube, which can fluctuate like a massless string. Main consequence: linearly rising potential quantum corrections to the interquark potential (Lüscher term) log increase of the flux tube thickness 10

12 Potential V (R) between two external, massive quark and anti-quark sources from Wilson loops W (L, R) e LV (R) (large L) V (R) = lim L 1 L log W (L, R) In the limit of infinite mass quarks (pure gauge theory) we find the famous area law for the Wilson loop Area law linear potential σ is the string tension W (L, R) e σrl ; V (R) = σr

13 Quantum corrections and effective models The area law is exact in the strong coupling limit (β 0). As the continuum limit (β ) is approached quantum corrections become important. Leading correction for large R: The Lüscher term V (R) = σ R π d 2 24 R + O ( ) 1 R 2. 12

14 It can be obtained summing up the quantum fluctuations of the transverse degrees of freedom of the string treated as d 2 massless modes [Lüscher, Symanzik and Weisz, (1980)]. This is the gaussian approximation of the effective string: W (L, R) e σrl DX i e σ 2 R dx 0 dx1 {( 0 X) 2 +( 1 X) 2 +int.s} Can also be derived assuming a SOS picture for the fluctuations of the surface bordered by the Wilson loop two-dimensional conformal field theory of d 2 free bosons Consequence: Using known results of CFT s we can predict the behaviour of the Wilson loop for finite values of L. 13

15 < W (R, L) >= e (σrl) Z q (R, L) Where Z q (R, L) is the partition function of d 2 free massless scalar fields living on the rectangle defined by the Wilson loop: R L Z q (R, L) [ ] d 2 η(τ) 2 R where η(τ) is the Dedekind η function and τ = il/r. The L R simmetry is ensured by this identity η ( 1 ) τ = iτ η(τ) known as the modular transformation of the η. 14

16 Defining: F (R, L) log < W (R, L) > and expanding the Dedekind function η(τ) = q 24 1 (1 q n ) ; q = e 2πiτ, one finds n=1 F (R, L) = σrl (d 2) From which we find as anticipated: [ πl 24R + 1 ] 4 log R V (R) = σ R π d 2 24 R

17 This result is universal: all confining gauge theories flow toward this gaussian effective action and thus share the same behaviour for the interquark potential (for large enough R) with no dependence on the gauge group and a trivial (linear) dependence on the space-time dimensions. Finding a string description able to select among different gauge groups (hence finding the QCD string theory ) requires going beyond the gaussian approximation The simplest proposal is the Nambu-Goto action 16

18 The Nambu-Goto string Action area of the surface spanned by the string in its motion: S = σ dξ 0 dξ 1 det g αβ (1) where g αβ is the metric induced on the w.s. by the embedding: g αβ = XM ξ α X N ξ β G MN (2) ξ α = world-sheet coords. (ξ 0 = proper time, ξ 1 spans the extension of the string) 17

19 Connection with the gaussian model One can use the world-sheet re-parametrization invariance of the NG action to choose the so called physical gauge : The w.s. coordinates ξ 0, ξ 1 are identified with two target space coordinates x 0, x 1 One can study the 2d QFT for the d 2 transverse bosonic fields with the gauge-fixed NG action Z = DX i e σ R dx 0 dx 1 1+( 0X) 2 +( 1X) 2 +( 0X 1 X) 2 = e σrl DX i e σ 2 R dx 0 dx1 {( 0 X) 2 +( 1 X) 2 +int.s} 18

20 The anomaly problem Problem: This gauge fixing is anomalous (unless we are in d = 26) Effective solution: It can be shown (Olesen, 1985) that the corrections due to the anomaly decay at least as 1/R 3 thus one can trust the perturbative expansion at least up to the order 1/R 2 i.e. the first order beyond the gaussian approximation. Stringy solution: The Nambu-Goto action can be rewritten (order by order in 1/L) so as to be anomaly-free in any dimension (Polchinski and Strominger, 1991): 19

21 Polchinski and Strominger action: S eff = 1 [ 1 dτ + dτ 4π a 2( +X X) ( ) D 26 ( X X)( + X 2 ] X) 12 ( + X X) 2 + O(L 3 ), (3) where τ ± are light-cone world-sheet coordinates, and a is a length scale related to the string tension. As a consistency check, it can be shown (Lüscher, Weisz, Drummond, Hari Dass, Mattlock, 2004) that the first perturbative correction beyond the gaussian contribution in both frameworks is the same. 20

22 Finite Temperature LGTs Finite temperature can be realized by imposing periodic boundary conditions in the (euclidean) time direction The (finite) temperature is given by the inverse of the lattice length in the compactified time direction T = 1/L A new set of observables can be constructed: the Polyakov loop which is the trace of the ordered product of timelike variable along a timelike axis of the lattice and behaves as an order parameter for the deconfinement phase transition The interquark potential is given by the correlator of two Polyakov loops. V (R, T ) = log(< P (x)p + (x + R) >) T = 1/L 21

23 22

24 Effective string corrections P (R)P (0) = Z(L, R) DXe S[X]. If we choose a Nambu-Goto action for the effective string (and set d = 2+1) then: S[X] = σ L 0 dτ R 0 dς 1 + ( τ X) 2 + ( ς X) 2. 23

25 Perturbative expansion in powers of 1/(σRL) Z(L, R) = e σrl Z 1 ( 1 + F 4 σr 2 + F ) 6 (σr 2 ) 2 + The leading order of this expansion: Z 1 corresponds to the partition function of a free boson on a cylinder There is no L R simmetry: the modular transformation of the Dedekind function τ 1 τ allows to relate the T and R dependences of the interquark potential, i.e. to predict the behaviour of σ as a function of the temperature T In string theory this modular transformation is known as open closed string duality 24

26 Z 1 can be easily evaluated where [ < P (x)p + (x + R) > e (σrl) η(i L ] (d 2) 2R ) η(τ) = q 1 24 (1 q n ) ; q = e 2πiτ, τ = i L 2R n=1 to be compared with analogous term in the Wilson case: [ η(i L < W (R, L) > e (σrl) R ) ] d 2 2 R In the large L/R limit (i.e. τ >> 1) the two expression give the same Lüscher term 25

27 Defining: F (R, L) log < P (x)p + (x + R) > and expanding the Dedekind function one finds [ ] πl F (R, L) = σrl (d 2) 24R +... From which we find as anticipated: V (R) = σ R π d 2 24 R + O ( ) 1 R 2. In the large R/L limit (i.e. τ << 1) instead we find V (R) = σ R (d 2) π 6L2R

28 F 4 was evaluated for the first time in (Dietz-Filk 1983): F 4 = π2 L 1152σR 3 ( [2E 4 i L ) 2R E 2 2 ( i L )] 2R, where E k are Eisentein functions of order k and are defined as: E 2 (τ) = 1 24 E 4 (τ) = σ 1 (n)q n (4) n=1 σ 3 (n)q n (5) n=1 q = e 2πiτ, (6) where σ 1 (n) and σ 3 (n) are, respectively, the sum of all divisors of n (including 1 and n), and the sum of their cubes. 27

29 We obtain in the large R limit V (R) = σ R (d 2) π 6L 2R (d 2) π 2 72σL4R

30 All orders calculations in the Nambu-Goto case In the Polyakov loop case the Nambu Goto partition function can be evaluated exactly if one neglects the anomaly F = 1 2 c k e LEk(R), k where the coefficient c k are the partitions of integers (they come form the expansion of the Dedekind function) and (Arvis 1982) E k (R) = σr 1 + 2π σr 2 ( k d 2 ) 24 29

31 Performing a L R transformation (i.e. going to the closed string channel) ( π 2 2 F = 2πL c k G (R; M(k)) σ)d where G (R; M) = propagator of a scalar field of mass M over the spatial distance R: k G(R; M) = 1 2π M 2πR «d 3 2 K d 3 (MR), 2 the mass M(k) is that of a closed string state with k representing the total oscillator number:» M 2 (k) = (σl) π k d 2 «σl

32 Two important remarks From the lowest mass (k = 0, m = 1) in the closed string channel: [ ] M 2 (1, 0) = (σl) 2 π(d 2) 1 3σL 2 one can obtain an estimate for the deconfinement temperature (recall that T = 1/L) which in this framework appears as a consequence of the tachionic state present in the NG string (Olesen, 1985): T c σ = 3 π(d 2) This estimate turns out to be in very good agreement with MC simulations for several LGT s. 31

33 SU(N) : continuum N T c / σ χ 2 /n df (36) (30) (42)[(81)] 3.7[1.0] (51) (107) NG Table 1: Data for SU(N) in d=4, (from B.Lucini, M.Teper and U. Wenger, JHEP0502:033,

34 In the large R and small L limit the NG partition function reduces to a single Bessel function. This means that the the NG effective string predicts the following behaviour for the Polyakov loop correlator: P (0)P (R) Kd 3(MR), 2 with M σl. This is exactly the limit in which dimensional reduction occurs ( Svetitsky-Yaffe conjecture). From the QFT approach to spin models we know that at high temperature and large distance whatever model we study the spin spin correlator will be dominated by the state of lowest mass whose propagator in d dimensions is given by G(R) K d 2 (mr) 2 Since d = d 1 this result exactly coincides with what we obtain with the NG string. 33

35 Comparison with MC simulations Duality and a new algorithm: the snake algorithm allow high precision simulations for very large values of R and L in the gauge Ising model. All the above predictions can be tested with precision δg G cases reaches which in some In the comparison there is no free parameter. The figures are not the result of a fitting procedure. The agreement at large distance with NG is impressive. distances deviations appear. At shorter 34

36 Ising, L= Gamma-Gamma_NG R Polyakov loop correlators in the (2+1) dimensional gauge Ising model at T = T c /10 (corresponding for this β at L = 80). 10 < R < 80 is the interquark distance. In the figure is plotted the deviation of Γ (the ratio G(R + 1)/G(R) of two correlators shifted by one lattice spacing) with respect to the Nambu-Goto string expectation Γ NG (which with this definition of observables corresponds to the straight line at zero). 35

37 0.02 R=32 Gamma-Gamma_LO L Polyakov loop correlators in the (2+1) dimensional gauge Ising model at a fixed interquark distance R = 32 varying inverse-temperature size (8 < L < 24). In the figure is plotted the deviation of Γ with respect to the asymptotic free string expectation Γ LO (which with this definition of observables corresponds to the straight line at zero). The curve is the Nambu-Goto prediction for this observable. 36

38 Universality of the effective string corrections The deep reason behind this impressive agreement are the universality theorems for the quartic (Lüscher and Weisz 2004) and sextic (Aharony and Karzbrun 2009) corrections to the interquark potential. In 3d the Nambu-Goto action is correct up to O(1/R 6 ). In particulr, contributions due to the anomaly appear only at order O(1/R 8 )! 37

39 The flux tube thickness. The flux density in presence of a pair of Polyakov loops is: < F µ,ν (x 0, x 1, h, R) >= < P (0, 0)P + (0, R)U µ,ν (x 0, x 1, h) > < P (0, 0)P + (0, R) > < U µ,ν > where x 0 denotes the timelike direction, x 1 is the direction of the axis joining the two Polyakov loops and h denotes the transverse direction. 38

40 h P x P U ΜΝ x,h 39

41 To evaluate the flux tube thickness we fix x 2 = R/2 to minimize boundary effects. (thanks to the periodic b.c. in the temperature direction we can instead choose any value of x 0 ) In the x 1 direction the flux density shows a gaussian like shape, the width of this gaussian is the flux tube thickness : w(r, L). w(r, L) only depends on the interquark distance R and on the lattice size in the compactified timelike direction L, i.e. on the inverse temperature of the model By tuning L we can thus study the flux tube thickness in the vicinity of the deconfinement transition 40

42 41

43 F ΜΝ x,h w 2 x x 42

44 Shape of the flux density generated by a Wilson loop in the Ising gauge model (at β = ). The dashed line is the gaussian fit. 43

45 Effective string predictions for the flux tube thickness. In the Nambu-Goto framework one should sum over all the surfaces bordered by the Polyakov loops and the plaquette with a weight proportional to the surface area. F µν (x, h) = surf e σarea[surf] w 2 (x) = dh h 2 F µν (x, h) dh Fµν (x, h) = dh h 2 dh surf e σarea[surf] surf e σarea[surf] 44

46 45

47 If we assume the size of the plaquette to be negligible with respect to the other scales, perform a physical gauge fixing and denote as h 0 the transverse coordinate of the plaquette then: dh0 h 2 0 w 2 h( x ( x 0 ) = 0 )=h 0 [Dh( x)] e σs[h] dh0 h( x 0 )=h 0 [Dh( x)] e σs[h] which can be rewritten as w 2 ( x 0 ) = [Dh( x)] h( x0 ) 2 e σs[h] [Dh( x)] e σs[h] h 2 ( x) with S[h] = σ d 2 x 1 + ( h) 2 46

48 This expectation value is singular and must be regularized using for instance a point splitting prescription. σw 2 ( x) = h( x + ɛ)h( x) G( x + ɛ; x) The U.V. cutoff ɛ has a natural physical meaning: ɛ plaquette size. Dealing with the whole NG action turns out to be too difficult. resort again to the free boson approximation ( SOS picture ) We S[h] σ d 2 x [1 + 1/2( h) 2 ] 47

49 Then G( x + ɛ; x) is the Green function of the Laplacian on the cylinder. which can be written (choosing the reference frame so as to have the two loops located in ±R/2) as: with q = e πl/2r G 2 (z; z 0 ) = 1 2π log θ 1 [π(z z 0 )/2R] θ 2 [π(z + z 0 )/2R] Setting z 0 = z + ɛ and performing an expansion in ɛ one finds: σw 2 (z) = 1 2π log π ɛ R + 1 2π log 2θ 2 (πrz/r) /θ 1 48

50 A similar calculation in the Wilson loop case gives σw 2 (z) = 1 2π with q = e πl/r. log π ɛ R + 1 2π log 2θ 2 (πrz/r)θ 4 (iπiz/r) θ 1 θ 3(πz/R) In both cases the dominant term diverges as 1 2π log R Both results assume L >> R 49

51 Performing a dual tranformation we can obtain the behaviour for R >> L i.e. in the high T regime. In this case the Green function can be written as: G(z; z 0 ) = 1 2π log and we have: σw 2 (z) = 1 2π Expanding in powers of R/L we find θ 1 [π(z z 0 )/L] θ 4 [π(z z 0 )/L] Imz Imz 0 LR q = e 2πR/L π ɛ log L + 1 2π log θ 4(2πiIz/L)/θ 1 (Iz)2 LR σw 2 (z) = 1 2π This time the R dependence is linear!! log 2π ɛ L + R 4L 50

52 Summary of the result for the Polyakov loop correlator Low temperature w 2 1 2πσ log( R R c ) +... (L >> R >> 0) High temperature (but below the deconfinement transition) w 2 1 2πσ (πr 2L + log( L ) +...) (R >> L) 2π ɛ Log increase of the flux tube width at zero temperature but Linear increase near the deconfinement transition! 51

53 Comparison with MC simulations. The 3d gauge Ising model is perfectly suited for studying the flux tube width. Thanks to duality one can create a vacuum containing the Wilson loop or the Polyakov loop correlators by simply frustrating the links in the dual lattice orthogonal to a surface bordered by the loops. Any choice of the surface is equivalent. 52

54 Measuring the energy operator (which is the dual of the plaquette operator) in this vacuum one can thus evaluate the ratio: < P (0, 0)P + (0, R)U µ,ν (x 0, x 1, h) > < P (0, 0)P + (0, R) > for any value of R and L with the same statistical uncertainty of the expectation value of the plaquette in the usual vacuum < U µ,ν >. 53

55 We used Wilson loops to test the low T predictions and Polyakov loop correlators for the high T ones. Low T we tested several values of β in the range β and several sizes of the Wilson loops (ranging from 12 2 to 64 2 ) so as to test a wide range of R σ values. The log fit of σw 2 as a function of R σ shows a very good χ 2 with an angular coef (5) to be compared with 1/2π High T we studied the model at β = which corresponds to at c = 1/8 and a 2 σ = (2) and tested lattice sizes 9 L 16 i.e. temperatures ranging from T = T c 2 to T = 8 9 T c 54

56 Squared width of the flux tube in units of sigma for the Z 2 gauge theory. The open symbols are Wilson loop data while the black circles refer to the (dual) Ising interface. 55

57 56

58 Σ L 57

59 Near the deconfinement transition the linear fit has very good χ 2, but the ang.coef. shows deviations with respect to the expected value 1/(4σL). This is likely to be the signature of higher order (universal) contributions in the effective string action. This conjecture can be tested using the 2 loop calculation recently obtained by Gliozzi, Pepe and Wiese. w 2 nlo = ( 1 + 4πf(τ) ) σr 2 wlo(r/2) 2 f(τ) = E 2(τ) 4E 2 (2τ), 48 ( E2 (τ) g(τ) = iπτ qd ) 12 dq f(τ) + g(τ) σ 2 r 2, (f(τ) + E 2(τ) 16 ) + E 2(τ) 96, The dominant term in the R >> L limit (i.e. τ 0) turns out to be 58

60 again linear in R: σwnlo 2 πr 24σ 2 L 3 + Combining together the leading and next to leading corrections we end up with the following expression: σw 2 = R 4σL ( 1 + π 6σL 2 + ) Extending these calculations to higher orders is too difficult. However dimensional reduction arguments suggest the following conjecture for the square width, resummed to all orders for the Nambu-Goto string: 59

61 σw 2 = R 4m = R 4σL 1 π 3σL 2 Assuming this expression and expanding up the third order (i.e. keeping only the universal terms) we find: σw 2 = R 4σL ( 1 + π 6σL 2 + π 2 ) 24(σL 2 ) 2 + The montecarlo data nicely agree with this conjecture 60

62 k T/T_c 61

63 Conclusions The effective string approach describes remarkably well the interquark potential, both at zero and at finite temperature, both in the 3d gauge Ising model and in non-abelian SU(N) LGTs. Recent MC results are so precise that also next to leading corrections (which have been proved to be universal) can be succesfully tested Assuming the Nambu-Goto action for the effective string a value for T c / σ can be obtained which is in good but not exact agreement with MC simulations The universality theorems explain why the Nambu-Goto action works so well, despite its anomaly. Corrections due to the anomaly arise only at very high orders in 1/σR 2 (or equivalently in 1 T/T c ) 62

64 The effective string predicts a logarithmic increase of the flux tube width at low temperature and a linear increase at high T (but still in the confining regime) This scenario is confirmed by MC simulations in the 3d gauge Ising model both at low and at high T, but an increasing discrepancy in the coefficient of the linear term appears as T c is approached if one truncates the Nambu-Goto action at the first order. Including the next to leading order contribution improves the agreement. Dimensional reduction also predicts a linear increase of the flux tube thickness at high temperature, with a T dependence of the coefficient which suggests how to resum to all orders the Nambu-Goto action. This conjecture agrees with the two loop calculation of Gliozzi Pepe and Wiese. 63

65 It is interesting to notice that at a difference with respect to the one loop approximation this expression for the amplitude of the linear growth divereges as the deconfinement temperature is approached. 64

66 Open issues Boundary terms Intrinsic thickness of the effective string Effective string versus dual superconductor models of confinement: Nambu-Goto string Abrikosov vortices Vortex configurations and the behaviour of space-like Wilson loops at high temperature 65

67 References M. Billo and M.C. Polyakov loop correlators from D0-brane interactions in bosonic string theory JHEP 0507 (2005) 038 M. Billo, M.C. and L. Ferro The partition function of interfaces from the Nambu-Goto effective string theory. JHEP 0602: (2006) 070 M.C., M. Hasenbusch and M.Panero Comparing the Nambu Goto string with LGT results. JHEP 0503 (2005) 026 High precision monte carlo simulations of interfaces in the three-dimensional Ising model: a comparison with the Nambu-Goto effective string model. JHEP 0603 (2006) 084 A. Allais and M.C Linear increase of the flux tube width at high temperature. JHEP 0901:073,

68 M. Billo, M.C. and R. Pellegrini New numerical results and novel effective string predictions for Wilson loops ArXiv:

69 The snake algorithm. Goal: compute the ratio G(R)/G(R + 1). Proposal: Use duality and factorize the ratio of partition functions in such a way that for each factor the partition functions differ just by the value of J ij at a single link Z L R Z L (R+1) = Z L R,0 Z L R,1... Z L R,M Z L R,M+1... Z L R,L 1 Z L R,L, where L R, M denotes a surface that consists of a L R rectangle with a M 1 column attached. 68

70 Figure 1: Sketch of the surface denoted by L R, M. In the example, L = 6, R = 8 and M = 2. The circles indicate the links that intersect the surface. 69

71 Each of the factors can be written as expectation value in one of the two ensembles: Z L R,M+1 Z L R,M = s i =±1 exp( βh L R,M (s)) exp( 2 βs k s l ) Z L R,M, where < k, l > is the link that is added going from L R, M to L R, M +1. Further improvement: hierarchical updates. Result: the error show no dependence on R!! 70

72 Dimensional reduction and the Svetitsky-Yaffe conjecture weak form of the S-Y conjecture: The high temperature behaviour of a (d+1) LGT with gauge group G can be effectively described by a spin model in d dimensions with (global) symmetry group C (the Center of G). strong form of the S-Y conjecture: If both the spin model and the LGT have continuous phase transitions then they share the same universality class. The mapping between the LGT and the effective spin model is based on the following identifications 71

73 For all the values L 10 the linear fit has very good χ 2, but the ang.coef. shows 72

74 These mappings should be intended in a renormalization group sense i.e. the Polyakov loop operator is mapped into a linear combination of all the (C-odd) operators in the spin model. For instance, in the 2d Ising case the whole conformal family of the spin operator. This combination will be dominated by the relavant(s) operator(s). In the Ising case only one: the spin operator. In the case of the plaquette operator the mapping will be in general a linear combination of the energy and the identity families. 73

75 Agreement between 2d spin model estimates and effective string predictions. With the Nambu-Goto effective string we obtain for the Polyakov loop correlator: P (0, 0)P (0, R) + = n=0 ) (Ẽn v n 2 K 0 (ẼnR). π This expression is expected to be reliable in the large distance limit. In this limit only the lowest state (n = 0) survives and we end up with a single K 0 function: 74

76 lim P (0, 0)P (0, R) + K 0 (Ẽ0R). R The spin-spin correlator of any 2d spin model in the symmetric phase is given by lim σ(0, 0)σ(0, R) + K 0 (mr). R The two expression coincide, they are universal (no dependence on the symmetry group) and allow to identify m with Ẽ0 = σl { } 1 π 1/2 3σL. 2 which, at the first order in 1/L becomes m σl = σ/t 75

77 Effective spin model description of the the flux tube thickness. Following the S-Y mapping the flux density in the Ising LGT in (2+1) dimensions becomes the ratio of connected correlators: σ(x 1 )ɛ(x 2 )σ(x 3 ) σ(x 1 )σ(x 3 ) in the high temperature phase of the 2d Ising model in zero magnetic field. The large distance behaviour of this correlator can be evaluated using the Form Factor approach. 76

78 Width of the flux tube The width of the flux tube evaluated at the midpoint between the two spins is given by w 2 r 2 (r) = dx x2 2K 0 (2mr) 1 + x 2e 2mr where R 2r is the distance between the two spins. 1+x 2. 77

79 In the large R limit we thus obtain w 2 R 4m +... to be compared with the effective string result: w 2 ( R 4σL + log( L ) 2π ) +... (R >> L) linear increase of the width in both cases but with a different T dependence of the coefficent m = Ẽ0 = σ T 1 πt 2 3σ = σl 1 π 3σL 2 This discrepancy is due to the free bosonic approx in the string calculation. 78

80 References M. Billo and M.C. Polyakov loop correlators from D0-brane interactions in bosonic string theory JHEP 0507 (2005) 038 M. Billo, M.C. and L. Ferro The partition function of interfaces from the Nambu-Goto effective string theory. JHEP 0602: (2006) 070 M.C., M. Hasenbusch and M.Panero Comparing the Nambu Goto string with LGT results. JHEP 0503 (2005) 026 High precision monte carlo simulations of interfaces in the three-dimensional Ising model: a comparison with the Nambu-Goto effective string model. JHEP 0603 (2006) 084 M.C and M. Zago A new simulation strategy for the study of the interquark potential in abelian LGTs. In preparation 79