Random Geometry meets Lilliputian strings Yuri Makeenko (ITEP, - PowerPoint PPT Presentation

Random Geometry meets Lilliputian strings Yuri Makeenko (ITEP, Moscow) Based on: • J. Ambjørn, Y. M. Phys. Lett. B 756, 142 (2016) [ arXiv:1601.00540 ] • J. Ambjørn, Y. M. Phys. Rev. D 93, 066007 (2016) [ arXiv:1510.03390 ] Talk at the 2nd French-Russian Conference “Random Geometry and Physics” Paris, October 17–21, 2016

Content of the talk ———————————– • effective string philosophy – (off-shell) Nambu-Goto versus Polyakov string – invariant cutoff Λ 2 √ g approximate solution of proper-time regularized bosonic string • (mean field that becomes exact at large d) – the Alvarez-Arvis spectrum – the Lilliputian scaling limit • accounting for semiclassical fluctuations about the saddle-point – Schwinger-Dyson equations versus path integral ( d → d − 2) – 1 /d correction via the path integral living in the Lilliputian world •

1. Introduction

Problems of string theory ———————————– inherited from 1980’s Path integral reproduces canonical quantization only • in critical dimension (d=26) and on-shell • Otherwise a non-linear problem emerges because the world-sheet cutoff is Λ 2 √ g where Λ is invariant cutoff (e.g. proper time) and g = det g ab (of metric tensor) Polyakov (1981) It can be solved for the (closed) Polyakov string • Knizhnik-Polyakov-Zamolodchikov (1988), David (1988), Distler-Kawai (1989) giving the string susceptibility index which is not real for 1 < d < 25 � d − 1 − ( d − 1)( d − 25) γ = 12 • Lattice regularization (by dynamical triangulation) scale to a continuum string for d ≤ 1 but does not for d > 1 (same for hypercubic latticization of Nambu-Goto string in d > 2)

Effective-string philosophy ———————————– 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 > L tachyon Pretty much like the view on Quantum Electrodynamics

2. Saddle-point solution to bosonic string

Nambu-Goto string via Lagrange multiplier ———————————– Lagrange multiplier λ ab for independent metric tensor ρ ab d 2 ω √ ρ + K 0 � � � d 2 ω λ ab ( ∂ a X · ∂ b X − ρ ab ) � d 2 ω det ∂ a X · ∂ b X = K 0 K 0 2 World-sheet parameters ω 1 , ω 2 ∈ ω L × ω β 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 cl = L cl = β X 1 X 2 X ⊥ ω 1 , ω 2 , cl = 0 , ω L ω β    L 2 , β 2 � � , Lω β βω L √ ρ cl λ ab = ρ ab [ ρ ab ] cl = diag cl = diag cl  ω 2 ω 2 Lω β βω L L β minimizes the Nambu-Goto action (a classical vacuum)

Effective action ———————————– Gaussian path integral over X µ q by splitting X µ = X µ cl + X µ q : d 2 ω √ ρ + K 0 � � d 2 ω λ ab ( ∂ a X cl · ∂ b X cl − ρ ab ) S eff = K 0 2 + d − 2 O = − 1 √ ρ∂ a λ ab ∂ b . tr log O , 2 in the static gauge modulo the ghost determinant Proper-time regularization of the trace � ∞ d τ 1 a 2 ≡ τ tr e − τ O , tr log O = − 4 π Λ 2 a 2 with −O being the 2d Laplacian for λ ab = ρ ab √ ρ The effective action governs λ ab and ρ ab which do not fluctuate at large d (exact mean field) like 2d O ( N ) sigma-model at large N . Then ghosts can be ignored λ ab and ¯ Saddle-point values ¯ ρ ab minimize the effective action

Effective action (cont.) ———————————– λ ab and ¯ The minimum is reached for diagonal and constant ¯ ρ ab , when   λ 11 L 2 λ 22 β 2 S eff = K 0 λ 11 ¯ λ 22 ¯  ω β ω L  ¯ + ¯ ρ 22 − ¯ ρ 11 − ¯ � + 2 ¯ ρ 11 ¯ ρ 22 ω 2 ω 2 2 L β − d √ ¯ � λ 22 ¯ ρ 11 ¯ ρ 22 ω β ω L Λ 2 − πd ω L √ ¯ λ 11 ¯ λ 11 ¯ λ 22 6 ω β 2 for L ≫ β The averaged induced metric � ∂ a X · ∂ b X � (that equals ¯ ρ ab at large d ) depends on ω 1 near the boundaries but this is not essential for L ≫ β Ambjørn, Makeenko (2016)

Side remark: a mathematical formula ———————————– 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: ∞ πm 2 + πn 2 � � � � 1 iω T � det( − ∆) = = √ 2 ω R η ω 2 ω 2 ω R m,n =1 T R Alternatively for z ∈ upper half-plane L¨ uscher, Symanzik, Weisz (1980) � � � � 1 ∂ω ( z ) ∂ω ( z ) � � � � � d 2 z ∂ a log det( − ∆) = � ∂ a log � � � � 24 π � ∂z � � ∂z � � � � with the Schwarz–Christoffel map (0 < r < 1) � √ 1 − r � � z K d s ω R ω ( z ) = = Gr¨ otzsch modulus , � √ r � � ω T K r (1 − s )( s − r ) s where K is the complete elliptic integral of the first kind. We have verified that indeed � √ 1 − r �   K 1 1 2 5 / 6 π 1 / 2 [ r (1 − r )] 1 / 12  = �� 1 / 2 η  i � √ r � √ 1 − r K � � 2 K

Saddle-point solution ———————————– Ambjørn, Makeenko (2016) The minimum of the effective action (quantum vacuum) is given by β 2 − β 2 � � 0 β 2 − β 2 ρ 11 = L 2 � � C ρ 22 = 1 C 0 = πd 2 C β 2 0 ¯ 2 C − 1 , ¯ 2 C − 1 , ω 2 β 2 − β 2 ω 2 � � 2 C 3 K 0 0 L β C � 4 − d Λ 2 � C = 1 � 1 λ ab = C ¯ ρ ab � � ¯ ρ, ¯ 2 + 2 K 0 The conformal gauge ¯ ρ 11 = ¯ ρ 22 is realized for � β 2 − β 2 ω β = 1 0 same as classical for β ≫ β 0 ω L L C alternatively to fix ρ 11 = ρ 22 and minimize with respect to ω β ω L . These generalize the classical solution, the WKB (perturbative or loop) expansion about which goes in 1 /K 0 ∼ � , recovering one loop. C takes for K 0 > 2 d Λ 2 values between 1 (classical) and 1/2 (quan- tum), which plays a crucial role for existence of the continuum limit

Saddle-point solution (cont.) ———————————– Substituting the solution, we obtain the saddle-point value � β 2 − β 2 S eff = K 0 CL 0 /C which is L times the ground state mass The average area of typical surfaces essential in the path integral β 2 − β 2 � � 0 / 2 C C � d 2 ω � � Area � = ρ 11 ¯ ¯ ρ 22 = L (2 C − 1) . � β 2 − β 2 0 /C 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 K 0 → 2 d Λ 2 or C → 1 / 2

3. Two scaling regimes: Gulliver’s vs. Lilliputian

Lattice-like scaling limit (Gulliver’s) ———————————– The ground state energy � β 2 − β 2 E ( β ) = K 0 C 0 /C does not scale generically because K 0 > 2 d Λ 2 for C to be real ( > 1 / 2). Let π β 2 > β 2 min = 3 d Λ 2 for not to have a tachyon. Choosing the smallest possible value β = β min , we find √ � π K 0 C E ( β ) = 2 C − 1 3 Λ which scales to m for 2 ∝ m 2 K 0 − 2 d Λ 2 ∝ m 4 C − 1 Λ 2 , Λ 2 This scaling does not exist for larger values of β and thus is particle-like similar to lattice regularizations of a string, where only the lowest mass scales to finite, excitations scale to infinity Durhuus, Fr¨ ohlich, Jonsson (1984), Ambjørn, Durhuus (1987)

Lilliputian string-like scaling limit ———————————– Let us “renormalize” the units of length � � C C L R = 2 C − 1 L, β R = 2 C − 1 β, to obtain for the effective action � πd β 2 S eff = K R L R R − , K R = K 0 (2 C − 1) 3 K R The renormalized string tension K R scales if K 0 → 2 d Λ 2 + K 2 R 2 d Λ 2 reproducing the Alvarez-Arvis spectrum of continuum string The average area is also finite � πd � β 2 R − 6 K R � Area � = L R � πd β 2 R − 3 K R It is simply the minimal area for β 2 R ≫ πd/ (3 K R ) and diverges when β 2 R → πd/ (3 K R ) like for the zeta-function regularization

Lilliputian string-like scaling limit (cont. 1) ———————————– Everything is like for the zeta-function regularization, but √ 2 dK R length R � 2 C − 1 length = length R ∼ Λ C in target space which is of order of the cutoff (Lilliputian) √ g ) fixes Nevertheless, the cutoff (in parameter space) ∆ ω = 1 / (Λ 4 maximal number of modes in the mode expansion � � a mn cos 2 πmω 2 + b mn sin 2 πmω 2 sin πnω 1 � X q = , ω β ω β ω L m,n ≥ 0 to be √ gω L √ gω β n (1) n (2) max ∼ Λ 4 max ∼ Λ 4 √ gω β = β Classically 4 reproducing Brink-Nielsen (1973) √ β 2 d Λ β √ gω β = 4 √ 2 C − 1 = Quantumly is much larger √ K R = ⇒ classical music can be played on the Lilliputian strings