RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-6. Particle Physics

Duality and integrability of supermatrix model with external source†
T. Kimura∗1,∗2In quantum field theory, in order to compute correlation functions, it is convenient to introduce the generating function by adding an extra source term. Such agenerating function is defined in the sense of path integral, and thus it is quite difficult to compute in general.
However, in the matrix model, just a zero dimensional
theory, a number of methods for computation are established, which are also applicable to the model withthe external source. In this report we generalize theduality of the matrix model with the external sourcewith a characteristic polynomial, which was originallyfound in the Gaussian matrix model,1) to the supermatrix model with an arbitrary matrix potential.
The correlation function of the characteristic polynomial in the supermatrix model, which we study here,is given bypqMΨN,M ;p,q {ai }N,{b}
;
{λ}
,{µ}
j j=1
α α=1β β=1i=1p
Sdet(λα − Z)
1= dZ e− h̄ Str W (Z)+Str ZC α=1(1)qSdet(µβ − Z)
β=1

where Z is a size N + M Hermitian supermatrix, andthe external source is C = diag(a1 , . . . , aN , b1 , . . . , bM ).
This formula includes several useful situations, e.g., theordinary characteristic polynomial average (M = q =0), the average of inverses (M = p = 0), and the ratio average (M = 0). Therefore it provides a master
formula for the characteristic polynomial average invarious matrix models.
The matrix measure in the integral is invariant under
the supergroup transformation, dZ = d(U ZU −1 ) withU ∈ U(N |M ), which is expressed in terms of eigenvalues, dZ = ∆N,M (x; y)2 dN x dM y dU . Here the Jacobian is given by the Cauchy determinant,∆N,M (x; y) =NM

i<j (xi − xj )i<j (yi − yj ).
N,Mi,j (xi − yj )(2)Then, to compute the intgral, we now introduce theHarish-Chandra–Itzykson–Zuber formula for the supergroup U(N |M )2–4)
dU eU(N |M )Str ZU CU −1det exi ai det e−yi bj. (3)=∆N,M (x; y)∆N,M (a; b)

Applying this formula, we obtain the following expression for the matrix integral in terms of eigenvalues†∗1∗2

Condensed from the article in PTEP 2014 123A01 (2014)
RIKEN Nishina Center
Institut de Physique Théorique, CEA Saclay

ΨN,M ;p,q = N,M1

1

dxi dyj e− h̄ W (xi )+ h̄ W (yj )+xi ai −yj bji,j×∆N +p,M +q (x, λ; y, µ). (4)∆N,M (a; b)∆p,q (λ; µ)Since the Cauchy determinant can be written as a determinant
xk−1i(5)∆N,M (x; y) = det(xi − yj )−1
with i = 1, . . . , N , j = 1, . . . , M , k = 1, . . . , N − M , ifN ≥ M , we obtain the determinanal formula for thecharacteristic polynomial averagepqMΨN,M ;p,q {ai }N
i=1 , {bj }j=1 ; {λα }α=1 , {µβ }β=1
Qk−1 (ai ) Pk−1 (λα )1=det  R(ai ; bj ) SR (λα ; bj )  ,∆N,M (a; b)∆p,q (λ; µ)R̃λα ;µβSL (ai ; µβ )(6)where we have introduced auxiliary functions:
1Pi−1 (x) = xi−1 , R̃(x; y) =,x−y
1
Qi−1 (a) = dx Pi−1 (x) e− h̄ W (x)+xa ,R(a; b) =

(7)
(8)

1dxdy R̃(x; y) e− h̄ (W (x)−W (y))+xa−yb , (9)11e− h̄ W (x)+xa ,x−µ11SL (λ; b) = dye h̄ W (y)−yb .
λ−ySL (a; µ) =dx(10)(11)

The formula (6) actually shows a duality between
the external source and the characteristic polynomial,
which is just given by Laplace (Fourier) transforms,
pqM,{b};{λ},{µ}ΨN,M ;p,q {ai }N
jαβ
i=1
j=1
α=1
β=1

F.T.
pqN
= Ψp,q;N,M {λα }α=1 , {µβ }β=1 ; {ai }i=1 , {bj }M
j=1 .
(12)

This is because the auxiliary functions transformto each other through the Fourier transformation:
Pi−1 (x) ↔ Qi−1 (a), R(x; y) ↔ R̃(a; b), SL (λ; b) ↔
SR (a; µ).
References1) E. Brézin, S. Hikami, Comm. Math. Phys. 214 (2000) 11.
2) T. Guhr, J. Math. Phys. 32 (1991) 32.
3) J. Alfaro et al., J. Math. Phys. 36 (1995) 3085.
4) T. Guhr, Comm. Math. Phys. 176 (1996) 555.

- 164 -

完全版2014_本文.indd 164

15/10/16 17:50