September 29, 202 Gaussian integrals Happy birthday Costas
Of course only non-gaussian problems are of interest! Matrix models of 2D-gravity Z = dme NTrV (M) in which M = M is an N N matrix and V (M) = t n M n Kontsevich Airy matrix model Z = dm exp ( 2 TrΛM 2 + i 6 M 3 ) who proved that F = log Z satisfies Witten s conjectures. Namely define t n (Λ) = (2n )!!TrΛ (2n )
then F (t 0,, t n, ) = τ k 0 0 τ k n n 0 t k n n k n! in which the coefficients τ d τ dn are the intersection numbers of the moduli space of curves on a Riemann surface M g,n, space of inequivalent complex structures on a Riemann surface of genus g with n marked points with 3g = n 3 + n d i. It follows that F satisfies the KdV hierarchy : U = 2 F/ t 2 0 U t = U U t 0 + 2 3 U t 3 0
Penner model Z = dm exp NtTr(log( M) + M) F = log Z is the generating function of the Euler character of M g,n F = χg,nn 2 2g t 2 2g n from which he obtained χ g,n = ( ) n(n + 2g 3)!(2g ) n!(2g)! B 2g is a Bernoulli number. B 2g
SO WHY BE INTERESTED IN GAUSSIAN INTEGRALS? 2
Consider Gaussian matrix models with an external matrix source P A (M) = Z A e N 2 trm 2 NtrMA A = A a given matrix with eigenvalues a, a N. One can tune A to various situations.
2 2 Density of states (the dots are the eigenvalues of the source) 3
Two basic ingredients. Explicit results for K-point functions (at finite N). 2. An N-K duality which exchanges the size N of the matrix, with K the number of operators. Joint work with Shinobu Hikami 4
. Correlation functions in an external source: P A (M) = Z A e N 2 trm 2 NtrMA U A (s,, s K ) = N K trens M tre Ns KM is a function of the eigenvalues a α of the Hermitian source matrix A. 5
One point function Using HarishChandra, Itzyson-Zuber integral over the unitary group due TrUAU B = det ea ib j in which (A) = (a i a j ), one finds (A) (B) ens2 U A (s) = N trensm 2 = Ns du 2iπ ensu N ( u a α + s u a α ) For instance for A = 0 ens2 2 U 0 (s) = Ns du 2iπ ensu ( + s u )N 6
K-point functions U A (s,, s K ) = tre Ns M tre Ns km K Ns 2 i = e 2 K i= du i e Ns iu i det 2iπ u i u j + s i K N i= α= ( + s i ) u i a α
2. Duality : We consider the average of products of K characteristic polynomials det(λ M). Tr λ M = λ det(λ M) det(µ M) µ=λ The K-point function is defined as F K (λ,..., λ K ) = Z N < K α= det(λ α M) > A,M = Z N dm K α= det(λ i I M)e 2 Tr(M A)2 7
Define the K K diagonal matrix The duality reads = Z N [dm] N ( i) Nk Z k Λ = diag(λ,..., λ K ). K α= [db] K det(λ α I M)e 2 Tr(M ia)2 N j= det(a j I B) e 2 tr(b+iλ)2 where B is a K K Hermitian matrix. The K-point function with N N Gaussian random matrices (in a source) is equal to an N-point function with K K Gaussian matrices in a different source (like color exchanged with flavor ). 8
The proof is simple det(λ α M ) = d θ i αdθ i αe θ i α(λ α δ ij M ij )θ j α i = N and repeat this K times : Then Kα= det(λ α M ).. The Gaussian integral over M gives the exponential of a quartic form in the Grassmann variables : θ i αθ j α θ j βθ i β = α,β α,β ij i θ i αθ i β j θ j βθ j α 2. Disentangle the quartic by a Gaussian integration over a K K matrix B dbe [TrB2 + α,β B α,β i θ i αθ i β] 9
with B = B is K K. 3. integrate out the Grassmannian variables θ i, θ i. d θ i αdθ i αe θ i α(λ α δ α,β B α,β )θ i β = [ det K K (λ αδ α,β B α,β )] N
Simplest application : recovering Kontsevich model Consider the correlation function of the characteristic polynomials det(λ M) and take a trivial unit matrix for the source A = P A (M) = Z exp N 2 Tr(M )2 a trivial shift which brings one edge of Wigner s semi-circle at the origin. 0
= Z N [dm] N K α= det(λ α I M)e N 2 Tr(M )2 ( i) Nk [db] K [det( ib)] N e N 2 tr(b+iλ)2 Z k det( ib) N = e Ntr[ ib+ 2 B2 + i 3 B3 + ] The linear term in B shifts Λ, the quadratic term cancels. In a scale in which the initials λ k are close to one, or more precisely N 2/3 (λ k ) is finite, the large N asymptotics is given by K K matrices B of order N /3. Higher terms
are negligible and we are left with terms linear and cubic in the exponent, namely F k (λ,..., λ k ) e N 2 trλ2 dbe in 3 trb3 +intrb(λ ). The original Kontsevich partition function was defined as Z = dbe trλb2 + i 3 trb3 The shift B B + iλ, eliminates the B 2 term and one recovers up to a trivial rescaling the same integral.
The fact that Kontsevich s model is dual of a Gaussian model makes it easy to compute the intersection numbers. Duality and replicas If Λ is a multiple of the identity Λ = λ, the coefficients of /λ in the expansion of log Z are proportional to the intersection numbers of the moduli of curves with one marked point on a Riemann surface of genus g. To recover these numbers one can use replicas, i.e. the simple relation and lim n 0 n λ [det(λ M)]n = tr λ M. < [det(λ M)] n > A,M =< [det( ix)] N > Λ,X
where X is an n n random Hermitian matrix. Similarly with two marked points = lim n,n 2 0 U(s, s 2 ) = Tre s M Tre s 2M dλ dλ 2 e s λ +s 2 λ 2 2 λ λ 2 [det( ix)] N Λ with Λ = (λ,, λ, λ 2,, λ 2 ) degenerate n and n 2 times.
Dealing with 0 0 matrices The Gaussian average over the matrix X is given by U(s,, s k ) = n tres X tre s kx = ( ) k(k )/2 k s 2 i e 2 k du i 2iπ e k (u i s i ) k ( + s i u i ) n det u i + s i u j 2
The continuation to n 0 is straightforward lim U(s,, s k ) = ( ) k(k )/2 k s 2 i e 2λ n 0 k du i 2iπ e k (u i s i /λ) k log ( + s i u i ) det u i + s i u j 3
The calculation of the contour integrals is more cumbersome, but all the integration can be done explicitly to the end and give with σ = s + + s k. lim n 0 U(s,, s k ) = λ σ 2 k 2 sinh σs i 2λ, This function is the generating function for the n = 0 limit of lim n 0 n trxp trx p k. 4
For instance it gives lim n 0 n < (trx3 ) 4g 2 >= 3 3g 2 2 2g (6g 4)! ( 4g 2 g ) This gives the intersection numbers of the moduli of curves with one marked point. < τ 3g 2 > g = (24) g g! (g = 0,, 2,...). 5
Generalizations : spin curves It is clear that can obtain results for curves with several marked points by the same technique (although it is difficult to go beyong low genera). One can also tune the external source to obtain a generalized Kontsevich model. For instance if the source matrix A has N/2 eigenvalues equal to +, and N/2 equal to in the large N appropriate scaling limit < N i= det(a i ib) >=< [det( + B 2 )] N 2 > = dbe N 4 trb4 intrbλ This case is nothing but the critical gap closing situation. 6
Again an appropriate tuning of the source matrix can yield the p-th generalization of Kontsevich s Airy matrix-model, a model introduced by Marshakov-Mironov-Morozov, defined as Z = dbexp[ Z 0 p + tr(bp+ Λ p+ ) tr(b Λ)Λ p ] It can be used to recover the (p, q) models coupled to gravity. The free energy is the generating function of the generalized intersection numbers τ m,j for moduli of curves with spin j F = d m,j < m,j τ d m,j m,j > m,j t d m,j m,j d m,j!
where j p m(p+2) t m,j = ( p) 2(p+) m l=0 (lp + j + )tr Λ mp+j+ For instance for the quartic generalized model (p=3) < τ 8g 5 j 3,j > g= Γ( g+ 3 ) (2) g g! Γ( 2 j 3 ) where j = 0 for g = 3m + and j = for g = 3m. (For g = 3m + 2, the intersection numbers are zero). Witten conjecture : the free energy which generates intersection numbers for spin curves satisfies a Gelfand-Dikii hierarchy and indeed this is true for the correlators U(s,, s K ) providing thereby an alternative definition.
Generalization : higher Kontsevich-Penner models Z = dme T r[m p+ +k log M+ΛM] p=2 Airy p=- Penner p=-2 unitary matrix model
Conclusion Topological matrix models such as generalized Kontsevich and Penner models,describe special noncritical string theories. (Maybe not surprising since integration over M g,n is what one does in perturbative string theories.) It is still surprizing (to me) that they may be generated by simple Gaussian models... in appropriate external sources.