Correlation functions and spectral curves in models of minimal gravity
aa r X i v : . [ h e p - t h ] O c t Correlation functions and spectral curvesin models of minimal gravity
O.KruglinskayaLebedev Physics Institute [email protected]
Abstract
The correlation functions for models of minimal gravity are discussed. An algorithmis proposed for calculations of invariant ratios from formulas of residues that can becompared with the coefficients of expansion of the partition function in Liouville theory.For (2,2K-1) models transition coefficient from basis of quasiclassical hierarchy to basisLiouville theory is obtained evidently, and the hypothesis about exact form of spectralcurve has been verified.
Keywords : generating functions, Liouville theory, integrable systems, correlation func-tions.
It is well known that matrix models describe 2-dimensional quantum gravity with central charge c ≤
1. In general case we should consider string loop expansion – over the world sheets withgenus more than zero, for describing theory of 2-dimensional gravity using 1 /N -expansion [1].Here we will consider only the limit N → ∞ , responsible for contribution of surfaces with zerogenus.The partition function of one-matix model can be presented as integral over the eigenvalues(see, for instance, [2] and references therein): Z = Z d Φ exp − ~ T rW (Φ) = 1 N ! Z N Y i =1 (cid:16) dφ i e − ~ W ( φ i ) (cid:17) ∆ ( φ ) . (1)In this expression Φ is matrix of size N × N , ~ = t N – parameter of quasiclassical expansion(term t remains fixed and finite at N → ∞ , ~ → W (Φ) is general determined by theformula: W (Φ) = X k> t k Φ k , (2)where t k are the sources, and tr Φ k – some basis in the operator space.The integration measure in the r.h.s. of (1) includes the Vandermonde determinant:∆( φ ) = Y i
Considering matrix models as discrete analog of continuous 2-dimensional gravity, the partitionfunction of the latter one in this approach is obtained from the partition function of matrixmodel as a result of non-trivial and singular double-scaling limit (see [8]). Details of this limitwere studied in [9], we present here only result.Minimal conformal theories [10], interacting with Liouville gravity [11] are called (p,q)-models of minimal gravity. To calculate the partition function of (p,q)-models for zero genuswe use the same method as for calculation of prepotential of matrix model. We can write thevariables t k , also known as times, and free energy F using formulas where (7),(8), where now: dS = Y dX, ξ = X /p . (11)For any (p,q)-point a couple of polynomials are taken: X = λ p + x p − λ p − + ... , Y = λ q + y q − λ q − + ... , (12)that satisfy the equation of a plane curve of the form: Y p = X q + X C ij X i Y j . (13)If p = 2 , q = 2 K −
1, then: Y = P K − ( X ) . (14)which describes generally Riemann surface of genus g = K −
1, in one case this curvebecomes singular with 2 K branch points: roots of polynomial P K − ( X ) = 0 plus X = Y = ∞ .According to the hypothesis proposed in [12], equation of curve (14) for minimal gravitycoincides with the equation on Chebyshev polynomials (in rescaled variables). Below thisassumption is precised and tested for a number of models by direct calculation.For comparison with Liouville theory ambiguity in the choice of basis needs to be takeninto account, due to the fact that times have scaling dimension, that can coincide with sum ofdimensions of other times in the same model t k → t k + At α t β , A = const, (15)∆ k = ∆ α + ∆ β , α, β < k. (16)In such case a resonance occurs (according to the terminology of [13], [14]). In formula (16)∆ k = p + q − k is scaling dimension of the variable t k , ∆ α , ∆ β – dimensions of t α t β respectively.In (2 , K −
1) minimal models of gravity cosmological constant µ = t K − always has thedimension ∆ µ = 4. Then resonances with cosmological constant are: t K − − i → t K − − i + A i · µ i +1 , i = 1 , , .... (17)Coefficients A i can be found from vanishing of one-point function ∂ F ∂t i = 0 and the stringequation.Below we describe the scheme of calculations of transition coefficients { A i } for (2 ,
9) model.In appendix calculated conversion factors are represented for cases K = 6 , , , , , , Example:(2,9) model
For polynomials (12) in this case we have: X = λ + x , Y = λ + y λ + y λ + y λ + y λ. (18)We calculate times by formulas (7): t = − x y + 34 x y − x y + 3564 x y − x ,t = 23 y − x y + 54 x y − x y + 10564 x ,t = 25 y − x y + 74 x y − x , (19) t = 27 y − x y + 94 x , t = 29 y − x = 0 , t = 211 . Solving latest five linear equations for y and substituting solution in equation for t , weobtain string equation: t + 32 x t + 63128 x + 158 x t + 3516 x t = 0 . (20)In this model only one resonance is possible: t → t + A µ , µ = t . From equality to zero one-point function ∂ F ∂t and string equation we find µ and constant A : µ = − x , A = 4936 . (21)Free energy F of models of minimal gravity (2 , K −
1) can be written as: F ( t i ) = µ K +1 / f ( τ j ) , (22)where f – scale invariant function, τ j – dimensionless ratios of times.In this example: F ( t , t , t , µ ) = µ f (cid:18) t µ / , t µ , t µ / (cid:19) , ∂ F ∂t = µ f (1) , ∂ F ∂t = µ f (11) = x , .... (23)and dimensionless ratios are: τ = t µ / , τ = t µ , τ = t µ / . (24)Rewriting string equation in terms of u = f (11) : τ + 3 uτ + 634 u + 152 u τ + 352 u = 0 . (25)We expand the function f in a series: f = f + f τ + f τ + f τ + 12 f τ + 12 f τ + 12 f τ + f τ τ + f τ τ + f τ τ + 16 f τ + 16 f τ + 16 f τ + 12 f τ τ τ + 12 f τ τ + 12 f τ τ + ... f and f in this expansion are uninteresting, because they are not universal.From equation (25) one can find leading coefficients, when τ = τ = 0: f = − f (3 f + 2) , f = − f + 35 f , .... (26)As a result we obtain dimensionless ratios, using (8) and (23),(26), for example: f f f = − , f f f = − . (27)In the same way, ratios can be easily calculated for other (2,2K-1). For instance, for model(2,11) with generating function F = µ / ˜ f (cid:16) ˜ t µ , ˜ t µ / , ˜ t µ , ˜ t µ / (cid:17) :˜ f ˜ f ˜ f = − , ˜ f ˜ f ˜ f = − . (28)It is dimensionless ratios (27),(28)that make sense to compare with results of calculationsin Liouville theory, which were obtained in recent years, using higher Liouville equations [15]. Now let us turn to the equation of spectral curve, that can be obtained from formulas (12),where y i expressed in terms of t k using (7). One should remember that resonances of times t k with cosmological constant µ (formula (17)) in a basis of Liouville theory are considered,whereupon t k assumed to be zero.For instance, we write equation of curve for (2,9) model, contained only µ and t . To dothis we express from the first equation (18) λ and substitute it in equation for Y . In turns all y i to be expressed through the times (19). Making replacement we find: Y = 1126 ( X − x ) (cid:0) x − Xx − X x + 8 X x + 16 X (cid:1) = 1126 x ( X ′ + 1) (2 X ′ − (cid:0) X ′ − X ′ − (cid:1) . Note that r.h.s. of this equation is expressed in terms of the Chebyshev polynomial T ( a )after substitution a = q X ′ +12 : T = (256 a − a + 432 a − a + 9 a ) = 12 ( X ′ + 1)(2 X ′ − (8 X ′ − X ′ − . Last expression coincides with an equation of curve rescaling of variables : Y ′ = T ( a ) , X ′ = T ( a ) = 2 a − . (29)and justifies Seiberg-Shih hypothesis [12].Similarly, hypothesis can be tested for all (2 , K −
1) models, some of which are presentedin Appendix. 5
Conclusion
In this paper algorithm of calculation of correlators was proposed, where correlators are coeffi-cients of expansion of generating functions in the models (2,2K-1) in the natural basis from thepoint of view continuous 2-dimensional gravity [13]. Dimensionless ratios of this coefficientscan be linked with results in Liouville theory, in which calculation are far more difficult (forexample, see [15]).Also exact equations of spectral curves in Liouville basis were obtained. The hypothesis [12]was tested by direct calculation, and the equations of curves were found indeed to coincide withequations of Chebyshev polynomials in rescaled variables.Note that for general case (p,q) models of minimal gravity with arbitrary p and q analgorithm of transition to Liouville basis is still unknown. Some some steps in this directionare, for instance, in [14], [16].I am grateful to A.Marshakov for clarifying discussions, and A.Anokhina for valuable re-marks. This work is partly supported by the Ministry of Education and Science of the RussianFederation under the contract 8498 and by RFBR grant 12-02-33095.6 ppendix (2,2K-1) Resonanceswith cosmological String equation and equation of curveconstant (2,11) t = µ , t = µ , t + x t + x t + x + x t + x t = 0 µ = − x Y = ( X − x )( x + 6 Xx − X x −− X x + 16 X x + 32 X ) (2,13) t = µ , t = µ , t + x t + x t + x + µ = − x + x t + x t + x t = 0 Y = ( X − x )(32 X x − X x − X x ++24 X x + 6 Xx + 64 X − x ) (2,15) t = µ , t = µ , t + x + x t + x t + t = µ , µ = − x + x t + x t + x t + x t = 0 Y = ( X − x )(128 X − x + 64 X x − X x −− X x + 80 X x + 24 X x − Xx ) (2,17) t = µ , t = µ , t + x t + x t + x t + x t + t = µ , µ = − x + x t + x t + x t + x = 0 Y = ( X − x )(256 X + x + 128 X x − X x −− X x + 240 X x + 80 X x − X x − Xx ) (2,19) t = µ , t + x + x t + x t + x t + t = µ , + x t + x t + x t + x t + x t = 0 t = µ , Y = ( X − x )(256 X x − X x − X x + t = µ , +672 X x + 240 X x − X x − µ = − x − X x + 10 Xx + 512 X + x ) t = µ , t + x + x t + x t + t = µ , + x t + x t + x t + t = µ , + x t + x t + x t + x t = 0 t = µ , Y = ( X − x )(512 X x − X x − µ = − x − X x + 1792 X x + 672 X x − X x −− X x + 60 X x + 10 Xx + 1024 X − x ) (2,23) t = µ , t + x + x t + x t + t = µ , + x t + x t + x t + t = µ , + x t + x t + x t + x t + x t = 0 t = µ , Y = ( X − x )( − x + 2048 X + 1024 x X − t = µ , − x X − x X + 4608 x X + 1792 x X − µ = − x − x X − x X + 280 x X + 60 x X − x X ) eferences [1] G. t’Hooft, Nucl.Phys.B , :3 (1974), 461-473.[2] A. V. Marshakov, TMPh , :2 (2006), 583-636.[3] Krichever I.M., Commun. Pure and Appl. Math. , :4 (1994), 437-475, arXiv:hep-th/920511.[4] A.Gorsky, I.Krichever, A.Marshakov, A.Mironov and A.Morozov, Phys. Lett. B , :3-4(1995), 466-474, arXiv:hep-th/9505035.[5] A. Marshakov, Seiberg-Witten Theory and Integrable Systems , World Scientific, Singapore,1999; ”Seiberg-Witten curves and integrable systems”,
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