Random matrix models and applications S. Hikami Mathematical and - - PowerPoint PPT Presentation
Random matrix models and applications S. Hikami Mathematical and Theoretical Physics Unit OIST OIST-iTHES-CTSR July 9, 2016 Talk is based on With E. Brezin (1) Random Matrix Theory With An External Source, a preparation for book
Mathematical and Theoretical Physics Unit OIST OIST-iTHES-CTSR July 9, 2016
(1) Random Matrix Theory With An External Source, a preparation for book (2016) (2) Random matrix, singularities and open/close intersection numbers,
* Riemann surface with boundary ---- open/close intersection numbers
GAUSSIAN INTEGRALS INTERESTING MODELS ARE NON-GAUSSIAN
Z =
Z
dX exp (−1 2TrΛX2 + i 6X3) who proved that F = log Z satisfies Witten’s conjectures. Namely define tn(Λ) = −(2n − 1)!!TrΛ−(2n−1)
7
(Triangulation of random surface by matrix model )
then F(t0, · · · , tn, · · ·) =
X
hτk0
0 · · · τkn n · · ·i 1
Y
tkn
n
kn! in which the coefficients hτd1 · · · τdni are the ”intersection numbers of the moduli space of curves” on a Riemann sur- face Mg,n , space of inequivalent complex structures on a Riemann surface of genus g with n marked points with 3g = n 3 + Pn
1 ki.
It follows that F satisfies the KdV hierarchy : U = ∂2F/∂t2 ∂U ∂t1 = U ∂U ∂t0 + 1 12 ∂3U ∂t3
Z =
Z
dX exp −NtTr(log(1 − X) + X) F = log Z is the generating function of the Euler character
F =
X
χg,nN2−2gt2−2g−n from which he obtained χg,n = (−1)n(n + 2g − 3)!(2g − 1) n!(2g)! B2g B2g is a Bernoulli number.
Gaussian matrix model with an external source
Consider Gaussian matrix models with an external matrix source PA(X) = 1 ZA e−N
2 trX2−NtrXA
A = A† a given matrix with eigenvalues a1, · · · aN. ZA is of course trivial (ZA = e
N 2 trA2) but one can tune A to
various situations.
t t t t t y y
Density of states (the dots are the eigenvalues of the source)
8
PA(X) = 1 ZA eN
2 trX2NtrXA
The partition function is trivial, the correlation functions are not, but they are known explicitly (S.Hikami , EB) UA(s1, · · · , sK) = 1 NKhtreNs1X · · · treNsKXi is a function of the eigenvalues aα of the Hermitian source matrix A.
Consider the average of the product of K characteristic polynomials det(λ − M). The K-point function is defined as FK(λ1, ..., λK) = 1 ZN <
K
Y
α=1
det(λα − X) >A,X = 1 ZN
Z
dX
K
Y
α=1
det(λα · I − X)e−1
2Tr(X−A)2
Tr 1 λ − X = ∂ ∂λ det(λ − X) det(µ − X)|µ=λ Alternatively tr 1 λ − X = lim
n→0
1 n ∂ ∂λ[det(λ − X)]n.
Define the K × K diagonal matrix Λ =
B @
λ1 · · · λ2 · · · · · ·
1 C A
The duality reads 1 ZN
Z
[dX]N
K
Y
α=1
det(λα · I − X)e−1
2Tr(X−iA)2
= (−i)Nk 1 Zk
Z
[dY ]K
N
Y
j=1
det(aj · I − Y ) e−1
2tr(Y+iΛ)2
where Y is a K × K Hermitian matrix. The K-point function with N × N Gaussian random ma- trices (in a source) is equal to an N-point function with
K × K Gaussian matrices in a different source (like ”color” exchanged with ”flavor”). Trivial example : one-point function 1 ZN
Z
[dX]det(λ − X)e−1
2Tr(X−iA)2
= (−i)N 1 Z
Z
dy
N
Y
j=1
(aj − y) e−1
2(y+iλ)2
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×1, the coefficients of 1/λ in the expansion of log Z are proportional to the inter- section 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 lim
n→0
1 n ∂ ∂λ[det(λ − X)]n = tr 1 λ − X. and < [det(λ − X)]n >A,X=< [det(1 − iY )]N >Λ,Y
11
where Y is an n ⇥ n random Hermitian matrix. Similarly with two marked points U(s1, s2) = hTres1XTres2Xi = lim
n1,n2!0
Z
dλ1dλ2es1λ1+s2λ2 ∂2 ∂λ1∂λ2 h[det(1 iY )]NiΛ with Λ = (λ1, · · · , λ1, λ2, · · · , λ2) degenerate n1 and n2 times.
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 gener- alized Kontsevich model. For instance if the source matrix A has N/2 eigenvalues equal to +1, and N/2 equal to −1 in the large N appropriate scaling limit <
N
Y
i=1
det(ai − iY ) >=< [det(1 + Y 2)]
N 2 >
=
Z
dY e−N
4 trY 4−iNtrY Λ
This case is nothing but the critical gap closing situation.
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 et al. defined by Z = 1 Z0
Z
dY exp[ 1 p + 1tr(Yp+1 Λp+1) tr(Y Λ)Λp] It can be used to recover the (p, q) models coupled to grav- ity. The ’free energy’ is the generating function of the general- ized intersection numbers hQ τm,ji for moduli of curves with ’spin’ j F =
X
dm,j
<
Y
m,j
τdm,j
m,j >
Y
m,j
tdm,j
m,j
dm,j! where
tm,j = (−p)
j−p−m(p+2) 2(p+1)
m−1
Y
l=0
(lp + j + 1)tr 1 Λmp+j+1 For instance for the quartic generalized model (p=3) < τ8g−5−j
3
,j >g=
1 (12)gg! Γ(g+1
3 )
Γ(2−j
3 )
where j = 0 for g = 3m + 1 and j = 1 for g = 3m. (For g = 3m + 2, the intersection numbers are zero). Witten conjecture : the free energy which generates inter- section numbers for spin curves satisfies a Gelfand-Dikii hi- erarchy and indeed this is true for the correlators U(s1, · · · , sK) providing thereby an alternative definition.
Generalization : higher Kontsevich-Penner models Z =
Z
dXeTr[Xp+1+k log X+ΛX]
[22]
6 etc.
Intersection numbers for non-orientable surfaces There is no equvalent of the Itzykson-Zuber for real sym- metric matrices which generate non-orientable surfaces. How- ever the HarishChandra formula applies to Lie algebras such as o(N) (or sp(N)) which also produce non-orientable sur-
lation functions and duality.
Consider the Lie algebra of o(2N), namely real antisymmet- ric matrices. For given real antisymmetric matrices X and Λ the HarishChandra localization formula, for the integral
Z
SO(2N) dgetr(gXg−1Λ) = C
P
w∈W
(detw)exp[2
N
P
j=1
w(xj)λj]
Q
1≤j<k≤N
(x2
j − x2 k)(λ2 j − λ2 k)
where C = (2N 1)! Q2N1
j=1
(2j 1)!, w are elements of the Weyl group, (permutations followed by reflections (xi ! ±xi ; i = 1, · · · , N) with an even number of sign changes). From this, one derives again the correlation functions in closed form hO(X)iA = 1 ZA
Z
dX O(X) exp (1 2trX2 + trAX) A = a1v · · · aNv, v = iσ2 = 1 1
!
. 1 2N < treσX >A= 1 Nσ
I
du 2πi((u + σ
2)2 a2 γ
u2 a2
γ
) u u + σ
4
eσu+σ2
4
and similarly for the K-point functions.
2.Duality <
k
Y
α=1
det(λα · I − X) >A=<
N
Y
n=1
det(an · I − Y ) >Λ where X is a 2N × 2N real antisymmetric matrix, Y is a 2k × 2k real antisymmetric matrix. The matrix source A is also a 2N × 2N antisymmetric matrix. The matrix Λ is a 2k × 2k antisymmetric matrix, coupled to Y . We assume, without loss of generality, that A and Λ take the canonical form Λ = λ1v ⊕ · · · ⊕ λkv.
ces By an appropriate tuning of the an’s, and a corresponding rescaling of Y and Λ, one may generate similarly higher models Z =
Z
dY e−
1 p+1trY p+1+trY Λ
(1) where p is an odd integer. U(σ) = 1 Nσ
I
du 2iπe−
c p+1[(u+σ 4)p+1−(u−σ 4)p+1](1 − σ
4u)
p = −1 virtual Euler characteristics U(σ)OR = 1 2N
Z
dye−Ny e−y (1 − e−y)2 U(σ)NO = 1 4N
Z
dye−Ny[ 1 1 − e−y − 1 1 + e−y]
Why matrix model is so powerful ? “ M-theory “, 11 dimensions, unification of superstring theories, type I, type IIB, type IIA 11D supergravity, E8xE8 heterotic, SO(32)
(Witten 1995) “ M “ is matrix theory (Banks, Fischler, Shenker, Susskind). 1997. gravity/gauge duality -----AdS/CFT
Simple Gaussian random matrix with an external source generates the interesting topological matrix models such as generalized Kontsevich model, Penner model, and open/closed intersection numbers, non-
The duality and replica method is very powerfullfor the evaluation of topological invariants.