Journal of Nonlinear Mathematical Physics Volume 8, Number 2 (2001), ??-?? ARTICLE Matrix Integrals and the Geometry of Spinors Johan VAN DE LEUR Mathematical Institute, Utrecht University P.O. Box 80010, 3508 TA Utrecht, The Netherlands E-mail: vdleur@math.uu.nl Received October 9, 2000; Accepted January 10, 2001 Abstract We obtain the collection of symmetric and symplectic matrix integrals and the collec- tion of Pfaffian tau-functions, recently described by Peng and Adler and van Moerbeke, as specific elements in the Spin-group orbit of the vacuum vector of a fermionic Fock space. This fermionic Fock space is the same space as one constructs to obtain the KP and 1-Toda lattice hierarchy. In memory of F D Veldkamp Introduction 1 Recently H Peng [?] showed that the symmetric matrix model and the statistics of the spectrum of symmetric matrix ensembles is governed by a strange reduction of the 2- Toda lattice hierarchy [?]. Adler and van Moerbeke [?, ?] (see also [?]) show that this reduction leads to vectors of Pfaffian tau-functions. However these Pfaffian tau-functions do not satisfy the Toda lattice hierarchy, but rather another system of PDE's, which can be identified as the BKP hierarchy in the form described by V Kac and the author in [?]. To be more explicit, let

\mathcal{S}_m(E)

be the set of

m \times m

symmetric matrices with spectrum in

E \subset \mathbb{R}

(a union of intervals),

dZ

the Haar measure on symmetric matrices and

V(t,z) = V(z) + \sum_{i=1}^{\infty} t_i z^i

, where

V(Z)

is a potential. Applying the spectral theorem to the symmetric matrix

Z = O^T \operatorname{diag}(z^0, ..., z_{m-1}) O

, with

O \in SO(m)

, we find upon integrating over the special orthogonal group the following formula for the symmetric matrix integral (see also [?]).

\hat{\tau}_m^E(t) = \int_{\mathcal{S}_m(E)} e^{TrV(t,Z)} dZ = c_m \int_{\mathbb{R}^m} |\Delta_m(z)| \prod_{i=1}^{m-1} (e^{V(t,z_i)} I_E(z_i) dz_i),

(1.1) where

\Delta_m(z) = \prod_{0 \le i < j \le m-1} (z_j - z_i)

is the Vandermonde determinant. Adder and van Moerbeke [?, ?] show (see also Section 2) that these integrals for

m = 2n

can be expressed in certain Pfaffians:

\hat{\tau}_{2n}^{E} = (2n)!c_{2n}\operatorname{Pf}((\mu_{k,\ell}(t))_{0 \le k,\ell \le 2n-1}).

A = (A_{ij})_{0 \le i,j \le 2n-1}

defined by

\operatorname{Pf}(A) = \frac{1}{2^n n!} \sum_{\sigma \in S_{2n}} \operatorname{sg}(\sigma) \prod_{i=0}^{n-1} A_{\sigma(2j), \sigma(2j+1)}.

(if

n=0

we assume

Pf(A)=1

) and

\mu_{ij}(t)=\langle y^i,z^j\rangle_t

are the moments of the time- dependent skew-symmetric inner product

\langle f, g \rangle_t = \iint_{\mathbb{R}^n} f(y)g(z)e^{V(t,y)+V(t,z)}\operatorname{sg}(z-y)dydz.

(1.3) The formula for

\hat{\tau}_m^E

will be obtained Section 2. If m is odd, it is also a Pfaffian, but the expression is more complicated. In this paper we will show, using the Clifford algebra techniques of [?], that replacing

\hat{\tau}_m^E

\tau_m^E = \frac{\hat{\tau}_m^E}{mlc_m}

, the generating series,

\tau^E(t,q) = \sum_{m=0}^{\infty} \tau_m^E(t) q^m,

of these Pfaffians is a specific element in the Spin group orbit of the vacuum vector in a fermionic Fock space. The fermionic Fock space which is constructed is more or less the same as the one one uses to obtain the KP hierarchy and 1-dim Toda lattice hierarchy. However, these tau-functions do not satisfy the KP or Toda lattice hierarchy, but we will show in section 4 that they satisfy the (charged) BKP hierarchy of [?] (see also Section 3):

\mathrm{Res} \; \Big( z^{n-1} e^{\xi(t',z)} e^{-\eta(t',z)} \tau_{n-1}(t') z^{-m-1} e^{-\xi(t'',z)} e^{\eta(t'',z)} \tau_{m+1}(t'') \\

+ \ (z)^{-n-1} e^{-\xi(t',z)} e^{\eta(t',z)} \tau_{n+1}(t')(z)^{m-1} e^{\xi(t'',z)} e^{-\eta(t'',z)} \tau_{m-1}(t'') \Big)

(1.4)

= \frac{1}{2} \left( 1 - (-1)^{n+m} \right) \tau_n(t') \tau_m(t''),

for all

n, m \in \mathbb{Z}

, where Res

\sum f_i z^i = f_{-1}

and

\xi(t,z) = \sum_{k=0}^{\infty} t_k z^k

and

\eta(t,z) = \sum_{k=0}^{\infty} \frac{1}{k} \frac{\partial}{\partial t_k} z^{-k}

. (1.5) Note that for all

n, m \in 2\mathbb{Z}

these are exactly the equations for the Pfaffian tau-functions obtained by Adler and van Moerbeke in [?]. The hierarchy of equations for all

n, m \in 2\mathbb{Z}

is called the DKP hierarchy in [?]. In fact using the KP boson-fermion correspondence one shows that

\tau^{E}(t,q) = \exp\left(\frac{1}{2} \iint_{\mathbb{R}^{2}} X(t,y,z) e^{V(y)+V(z)} \operatorname{sg}(z-y) dy dz\right)

\times \left(1 + \int_{E} X(t, w) e^{V(w)} dw\right) \cdot 1,

290 J van de Leur where

X(t,z)

and

X(t,y,z)

are the vertex operators

X(t,z) = az^{q\frac{\partial}{\partial q}} \exp(\xi(t,z)) \exp(-\eta(t,z)).

X(t, y, z) = X(t, z)X(t, y)

(1.6)

=q^{2}(z-y)(yz)^{q\frac{\partial}{\partial q}}\exp(\xi(t,z)+\xi(t,y))\exp(-n(t,z)-n(t,y)).

The symplectic matrix integrals, which can be treated in a similar way, are described in Section 8.

\mathbf{2}

Symmetric matrix integrals Let from now on

m = 2n

if m is even and

m = 2n + 1

if m is odd and recall formula (??):

\hat{\tau}_m^E(t) = c_m \int_{\mathbb{R}^m} |\Delta_m(z)| \prod^{m-1} \left( e^{V(t,z_i)} I_E(z_i) dz_i \right).

Denote by sg the sign-function and by

S_m

the permutation group of m letters. Using the identity

\sum_{\sigma \in S_n} \operatorname{sg}(\sigma) \prod_{j=0}^n \operatorname{sg}(\sigma(2j+1) - \sigma(2j)) = 2^n n!,

we find

\hat{\tau}_{m}^{E}(t) = m! c_{m} \int_{-\infty < z_{0} < z_{1} < \dots < z_{m-1} < \infty} \Delta_{m}(z) \prod_{i=0}^{m-1} \left( e^{V(t, z_{i})} I_{E}(z_{i}) dz_{i} \right)

= \frac{m!c_m}{2^n n!} \sum_{\sigma \in S} \operatorname{sg}(\sigma) \prod_{j=0}^{n-1} \operatorname{sg}\left(\sigma^{-1}(2j+1) - \sigma^{-1}(2j)\right)

\times \int_{-\infty < z_0 < z_1 < \dots < z_{m-1} < \infty} \Delta_m(z) \prod_{i=0}^{m-1} \left( e^{V(t, z_i)} I_E(z_i) dz_i \right)

= \frac{m!c_m}{2^n n!} \sum_{\sigma \in S} \operatorname{sg}(\sigma) \int_{-\infty < z_0 < z_1 < \dots < z_{m-1} < \infty} \Delta_m(z)

(2.1)

\times \prod_{i=0}^{m-1} \operatorname{sg} \left( z_{\sigma^{-1}(2j+1)} - z_{\sigma^{-1}(2j)} \right) \prod_{i=0}^{m-1} \left( e^{V(t,z_i)} I_E(z_i) dz_i \right)

= \frac{m!c_m}{2^n n!} \sum_{\sigma \in S_m} \int_{-\infty < z_{\sigma(0)} < z_{\sigma(1)} < \dots < z_{\sigma(m-1)} < \infty} \Delta_m(z)

\times \prod_{i=0}^{n-1} \operatorname{sg}(z_{2j+1} - z_{2j}) \prod_{i=0}^{m-1} \left( e^{V(t,z_i)} I_E(z_i) dz_i \right)

= \frac{(m)!c_m}{2^n n!} \int_{\mathbb{R}^m} \Delta_m(z) \prod_{i=0}^{m-1} \operatorname{sg}(z_{2j+1} - z_{2j}) \prod_{i=0}^{m-1} \left( e^{V(t,z_i)} I_E(z_i) dz_i \right).

291 Matrix Integrals and the Geometry of Spinors If

m = 2n

is even we continue as follows:

\hat{\tau}_{2n}^{E} = \frac{(2n)!c_{2n}}{2^{n}n!} \sum_{\sigma,\sigma} \operatorname{sg}(\sigma) \prod_{i=0}^{n-1} \int_{\mathbb{R}^{2}} \left( z_{2j}^{\sigma(2j)} z_{2j+1}^{\sigma(2j+1)} e^{V(t,z_{2j}) + V(t,z_{2j+1})} \right)

\times I_E(z_{2j})I_E(z_{2j+1})\operatorname{sg}(z_{2j+1}-z_{2j})dz_{2j}dz_{2j+1}

(2.2)

= \frac{(2n)!c_{2n}}{2^n n!} \sum_{s=0}^{n-1} \operatorname{sg}(\sigma) \prod_{s=0}^{n-1} \mu_{\sigma(2j),\sigma(2j+1)}(t) = (2n)!c_{2n} \operatorname{Pf}((\mu_{k,\ell}(t))_{0 \le k,\ell \le 2n-1}).

This is formula (??) of the introduction, with the moments

\mu_{ij}

defined by (??). If

m = 2n+1

is odd, we use the following lemma, which can be found in Adler, Horozov and van Moerbeke [?], to continue the calculation. Lemma 2.1. Let

A = (a_{ij})_{0 \le i,j \le 2\ell+1}

be a skew symmetric matrix, then

Pf(A) = \sum_{i=0}^{2\ell} (-)^k A_{k,2\ell+1}(Pf)((A_{ij})_{i,j\neq k,2\ell+1}).

Now,

\hat{\tau}_{2n+1}^{E}(t) = \frac{(2n+1)!c_{2n+1}}{2^{n}n!} \sum_{\sigma \in \mathcal{S}} \operatorname{sg}(\sigma) \int_{\mathbb{R}^{2n+1}} \prod_{i=0}^{n-1} \operatorname{sg}(z_{2j+1} - z_{2j})

\times \prod_{i=1}^{2n} z_i^{\sigma(i)} \left( e^{V(t,z_i)} I_E(z_i) dz_i \right)

= \frac{(2n+1)!c_{2n+1}}{2^n n!} \sum_{k=0}^{2n} \sum_{\sigma \in S_{2n+1}, \sigma(2n)=k} \operatorname{sg}(\sigma) \int_{\mathbb{R}^{2n+1}} z_{2n}^k e^{V(t, z_{2n})} I_E(z_{2n}) dz_{2n}

(2.3)

\times \prod_{i=1}^{n-1} \operatorname{sg}(z_{2j+1} - z_{2j}) \prod_{i=1}^{2n-1} z_i^{\sigma(i)} \left( e^{V(t,z_i)} I_E(z_i) dz_i \right)

= \frac{(2n+1)!c_{2n+1}}{2^n n!} \sum_{k=0}^{2n} (-)^k \sum_{\rho \in S_n^{(k)}} \operatorname{sg}(\rho) \int_{\mathbb{R}^{2n+1}} z_{2n}^k e^{V(t,z_{2n})} I_E(z_{2n}) dz_{2n}

\times \prod_{i=1}^{n-1} \operatorname{sg}(z_{2j+1} - z_{2j}) \prod_{i=1}^{2n-1} z_{i}^{\rho(i+\epsilon_{k}(i))} \left( e^{V(t,z_{i})} I_{E}(z_{i}) dz_{i} \right),

where

\rho \in S_{2n}^{(k)}

, the set of permutations of the numbers

1, 2, \ldots, k-1, k+1, \ldots, 2n

, is such that

\sigma(j) = \rho(j + \epsilon_k(j))

, with

\epsilon_k(j) = 0

j < k

and

j \geq k

. To give an idea of what we are doing, we give an example. If

n = 2

k = 1

and

\sigma = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 \\ 4 & 3 & 0 & 2 & 1 \end{pmatrix}

, then

\rho = \begin{pmatrix} 0 & 2 & 3 & 4 \\ 4 & 3 & 0 & 2 \end{pmatrix}

and

sg(\sigma) = -sg(\rho)

. Hence,