Random Geometry meets Lilliputian strings Yuri Makeenko (ITEP, - - PowerPoint PPT Presentation
Random Geometry meets Lilliputian strings Yuri Makeenko (ITEP, Moscow) Based on: J. Ambjrn, Y. M. Phys. Lett. B 756, 142 (2016) [ arXiv:1601.00540 ] J. Ambjrn, Y. M. Phys. Rev. D 93, 066007 (2016) [ arXiv:1510.03390 ] Talk at the 2nd
Based on:
Talk at the 2nd French-Russian Conference “Random Geometry and Physics” Paris, October 17–21, 2016
———————————–
– (off-shell) Nambu-Goto versus Polyakov string – invariant cutoff Λ2√g
(mean field that becomes exact at large d) – the Alvarez-Arvis spectrum – the Lilliputian scaling limit
– Schwinger-Dyson equations versus path integral (d → d − 2) – 1/d correction via the path integral
———————————– inherited from 1980’s
in critical dimension (d=26) and on-shell
cutoff is Λ2√g where Λ is invariant cutoff (e.g. proper time) and g = det gab (of metric tensor)
Polyakov (1981)
Knizhnik-Polyakov-Zamolodchikov (1988), David (1988), Distler-Kawai (1989)
giving the string susceptibility index which is not real for 1 < d < 25 γ = d − 1 −
12
a continuum string for d ≤ 1 but does not for d > 1 (same for hypercubic latticization of Nambu-Goto string in d > 2)
———————————– String is formed by more fundamental constituents Effective or induced or emergent string makes sense when it is long Examples: – Abrikosov vertices in supercondactor – Nielsen-Olesen string in the Higgs model – Confining string in QCD In particular no tachyon for L > Ltachyon Pretty much like the view on Quantum Electrodynamics
———————————– Lagrange multiplier λab for independent metric tensor ρab K0
2
World-sheet parameters ω1, ω2 ∈ ωL × ωβ rectangle Closed bosonic string winding once around compactified dimension of length β, propagating (Euclidean) time L. No tachyon if β is large enough Classical solution X1
cl = L
ωL ω1, X2
cl = β
ωβ ω2, X⊥
cl = 0,
[ρab]cl = diag
L2
ω2
L
, β2 ω2
β
λab
cl = diag
Lωβ , Lωβ βωL
cl
√ρcl minimizes the Nambu-Goto action (a classical vacuum)
———————————– Gaussian path integral over Xµ
q by splitting Xµ = Xµ cl + Xµ q:
Seff = K0
2
+d − 2 2 tr log O, O = − 1 √ρ∂aλab∂b. in the static gauge modulo the ghost determinant Proper-time regularization of the trace tr log O = −
∞
a2
dτ τ tr e−τO, a2 ≡ 1 4πΛ2 with −O being the 2d Laplacian for λab = ρab√ρ The effective action governs λab and ρab which do not fluctuate at large d (exact mean field) like 2d O(N) sigma-model at large N. Then ghosts can be ignored Saddle-point values ¯ λab and ¯ ρab minimize the effective action
———————————– The minimum is reached for diagonal and constant ¯ λab and ¯ ρab, when Seff = K0 2
¯
λ11L2 ω2
L
+ ¯ λ22β2 ω2
β
+ 2
ρ11¯ ρ22 − ¯ λ11¯ ρ11 − ¯ λ22¯ ρ22
ωβωL
−d√¯ ρ11¯ ρ22 ωβωL 2 √¯ λ11¯ λ22 Λ2 − πd 6
λ22 ¯ λ11 ωL ωβ for L ≫ β The averaged induced metric ∂aX · ∂bX (that equals ¯ ρab at large d) depends on ω1 near the boundaries but this is not essential for L ≫ β
Ambjørn, Makeenko (2016)
———————————–
Ambjørn, Makeenko, Sedrakyan (2014)
Determinant for ω ∈ rectangle is given by the product over modes with the Dirichlet b.c. as the Dedekind η-function: det(−∆) =
∞
ω2
T
+ πn2 ω2
R
1 √2ωR η
ωR
uscher, Symanzik, Weisz (1980)
det(−∆) = 1 24π
∂z
∂z
ω(z) =
z
r
ds
, ωR ωT = K
√1 − r
√r
where K is the complete elliptic integral of the first kind. We have verified that indeed 1 2
√1 − r 1/2 η i
K
√1 − r
√r
=
1 25/6π1/2 [r(1 − r)]1/12
———————————–
Ambjørn, Makeenko (2016)
The minimum of the effective action (quantum vacuum) is given by ¯ ρ11 = L2 ω2
L
2C
C
2C − 1, ¯ ρ22 = 1 ω2
β
2C
2C − 1, β2
0 = πd
3K0 ¯ λab = C¯ ρab ¯ ρ, C = 1 2 +
4 − dΛ2 2K0 The conformal gauge ¯ ρ11 = ¯ ρ22 is realized for ωβ ωL = 1 L
C same as classical for β ≫ β0 alternatively to fix ρ11 = ρ22 and minimize with respect to ωβ
ωL.
These generalize the classical solution, the WKB (perturbative or loop) expansion about which goes in 1/K0 ∼ , recovering one loop. C takes for K0 > 2dΛ2 values between 1 (classical) and 1/2 (quan- tum), which plays a crucial role for existence of the continuum limit
———————————– Substituting the solution, we obtain the saddle-point value Seff = K0CL
0/C
which is L times the ground state mass The average area of typical surfaces essential in the path integral Area =
ρ11¯ ρ22 = L
0/2C
0/C
C (2C − 1). These results merely repeat Alvarez (1981) except he used the zeta- function regularization where formally our Λ = 0 and C = 1. We shall now see the scaling regime needs K0 → 2dΛ2 or C → 1/2
———————————– The ground state energy E(β) = K0C
0/C
does not scale generically because K0 > 2dΛ2 for C to be real (> 1/2). Let β2 > β2
min =
π 3dΛ2 for not to have a tachyon. Choosing the smallest possible value β = βmin, we find E(β) =
π
3 K0C Λ √ 2C − 1 which scales to m for C − 1 2 ∝ m2 Λ2 , K0 − 2dΛ2 ∝ m4 Λ2 This scaling does not exist for larger values of β and thus is particle-like similar to lattice regularizations of a string, where
Durhuus, Fr¨
———————————– Let us “renormalize” the units of length LR =
2C − 1 L, βR =
2C − 1 β, to obtain for the effective action Seff = KR LR
R −
πd 3KR , KR = K0(2C − 1) The renormalized string tension KR scales if K0 → 2dΛ2 + K2
R
2dΛ2 reproducing the Alvarez-Arvis spectrum of continuum string The average area is also finite Area = LR
R − πd 6KR
R − πd 3KR
It is simply the minimal area for β2
R ≫ πd/(3KR) and diverges when
β2
R → πd/(3KR) like for the zeta-function regularization
———————————– Everything is like for the zeta-function regularization, but length =
C lengthR ∼ √2dKR lengthR Λ in target space which is of order of the cutoff (Lilliputian) Nevertheless, the cutoff (in parameter space) ∆ω = 1/(Λ 4 √g) fixes maximal number of modes in the mode expansion Xq =
ωβ +bmn sin 2πmω2 ωβ
ωL , to be n(1)
max ∼ Λ 4
√gωL n(2)
max ∼ Λ 4
√gωβ Classically
4
√gωβ = β reproducing Brink-Nielsen (1973) Quantumly
4
√gωβ = β √2C − 1 = √ 2d Λβ √KR is much larger = ⇒ classical music can be played on the Lilliputian strings
———————————– Why it was not observed before? The Lilliputian scaling in nonperturbative in : Semiclassically (recalling that 1/K0 ∼ ) C = 1 2 +
4 − dΛ2 2K0 = 1 − dΛ2 2K0 + O(2) cannot approach 1/2 and thus the metric cannot become singular. The scaling regime of KR = K0(2C−1) cannot be seen semiclassically. We found a certain geometry (metric tensor) which self-consistently dominates the path integral over geometries and becomes singular as K0 → 2dΛ2 + K2
R
2dΛ2
———————————– Summary:
resolved (infinitely many stringy modes)
the zeta-function regularization
(this is why lattice string regularizations of 1980’s never reproduce canonical quantization)
———————————– Relativistic string can be quantized by the Schwinger-Dyson equations rather than a path integral Advantage: terms from ghosts and the gauge fixing mutually cancel Averages of non-invariant quantities may vanish, so we have to con- sider proper averages like
computed with fixed gauge (e.g. conformal gauge). Invariance of the measure under a shift of ρab gives
d 2K0 ω| e−a2O|ω
K0
δ
δρab(ω)
δ δρab(ω) →
δ δρab(ω′)
———————————– Computing the partial derivative by using the formula ∂√ρ ∂ρab = 1 2ρab√ρ we find
2K0 ω| e−a2O|ω
λab w.s. = Cρab√ρ, C = 1 2 +
4 − (d − 2)Λ2 2K0 It is valid for any d, accounting for the fluctuations
———————————– Integrating out Xµ
q we get
d − 2 2 tr ln
√ρ∂aλab∂b
1 n
✣✢ ✤✜
· · · · · · ··· resulting in the effective action Seff = K0
2
−d − 2 2 Λ2
√ρ
Expanding λab = (¯ λ + δλ)δab, ρab = (¯ ρ + δρ)δab we find (in conformal gauge) to quadratic order Seff = Ss.p.
eff − K0(2C − 1)
C
2C3 Λ2¯ ρ
with one positive and one negative eigenvalues = ⇒ stable fluctuations because δλ is imaginary and δρ is real (in particular justifying constant ¯ λ and ¯ ρ)
———————————– Polyakov string: 26 − 1 = 25 ghosts metric dc Nambu-Goto string: 26 − 1 + 1 = 26 ghosts metric Lagrange dc multiplier The signs are opposite because the metric and the Lagrange multiplier are real and imaginary, respectively
———————————– For Polyakov’s string in conformal gauge ρab = eϕδab (−∆reg)−(d−26)/2 ∝ e
(d−26) 96π
if ϕ is smooth (curvature R ≪ Λ2 is not large). Higher derivatives become essential otherwise. The expansion goes in R/Λ2. So the det may be finite in spite of the Liouville action diverges. Let ϕ is constant along ω2 and has a discontinuity from 0 to ϕ0 > 0 at certain ω1. Since the integral then diverges (opposite to Brownian trajectories), one might think this leads to an instability for d > 26. But det(−∆) for such a discontinuous ϕ is larger than one for constant ϕ = ϕ0, because all eigenvalues are larger. It cannot be zero as Liouville says. The one for constant ϕ0 is as above with ¯ λab = δab. det(−∆) for such a discontinuous ϕ can be explicitly computed using the Gel’fand-Yaglom technique and shown to be finite.
———————————– Pauli-Villars regularization of the Polyakov string [det(−∆)]reg = det(−∂2) det(−∂2 + M2ρ) To quadratic order, we expand ρ = ¯ ρ + δρ that generates the propa- gator of the field Xµ
M and the triple vertex:
M(k)Xν M(−k)
δab k2 + M2¯ ρ
M(k + p)Xν M(−p)
The effective action to quadratic order −d 2
(2π)2 δρ(p)δρ(−p) 2
(2π)2 M4 (k2 + M2¯ ρ)[(k + p)2 + M2¯ ρ] = − d 96π¯ ρ2
(2π)2
− d 96π
———————————–
in d > 2 only for the Lilliputian strings
this scaling regime
changes