The Inst. of Natural Scielrces Nihon Univ. Proc. of The Inst. of Natural Sciences Vol. 25 (1990) pp. 49-65 HURWITZ PAIRS, SUPERCOMPLEX SOLITON EQUATIONS, AND QUASICOMFORMAL MAPP Osamu Suzuki, Julian Lawrynowicz and J (Received October 31. 1989) Hurwitz pairs are discussed in connection with algebra theory. The following results are obtained : ( i ) A Hurwitz pair determines a special Clifford algebr (ii) A field operator of the Dirac type, which is called use of a Hurwitz pair and its characterization is giv of the Neveu-Schwarz model of the superstring theory (E4, E3). (iil) Isospectral deformations of the Hurwitz operator various soliton equations (Theorem 111). (iv) A special complex structure, which is called sup (Definition (2. 3)) and there exists a correspondence of a Hilbert space and reduction solutions 0L the Sat ( v ) A quasicomformal mapping is obtained from a gener By these results we may conclude that Hurwitz pairs giv only in mathematics but also in physics. Introduction This paper is the second part of our study on Hurwi have given an outline of a field theory defined by field equation of the Dirac type and soliton equatio shall give a systematic treatment on Hurwitz pairs analysis and field theory. Our main concern is not to suggest relationships between Hurwitz pairs and othe considerations mainly to the simplest Hurwitz pair (E2 manner. Our main results are stated in Theorems I sections 4, 5, 6 and 7. The first three sections sup Although the preliminary part is overlapped with the do not hesitate to repeat them. During the prepara * The third narned author was supported by the Polish Academ during the preparation of this work.
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The Inst. of Natural Scielrces Nihon Univ. Proc. of The Inst. of Natural Sciences Vol. 25 (1990) pp. 49-65
where m is a non-negative constant and I is the identity operator, is called a mass pro-
ducing quantization, je;nn,p)= 7(?7~n,p)(f) is called the Hul-witz operator of the Hurwitz
pair (En, EP) ~)ith mass m. Especially whell ~)e choose m=0, we call it the Hurwitz
operator and is denoted by ye(n,p). The equation j~e(n,p)u! =0 is called the Hurwitz equation.
The Hurwitz equation is written as
ye(?e,p)c =0. X(?e,p) = ~ iTkalax~ k*1
Here we notice the selL-adjointness of the Hurwitz operator. We choose the linear space
Fo(RP-1. O~) of n-component functions of C=-class with compact supports. Then we see
that
(3. 18) ye(n,p) : ro(RP-1, cn) _=~ Fo(RP-1, cn).
We introduce the inner product by
(3. 19) J (~,c)= RP_1 ~*~)! dv for p,9)f~Fo(RP-1, cn).
Then the formal abjoint operator je(n,p)* of X(n,p) is given by
p-1 (3. 20) X(?e,p) = ~ ir~alaxk.
~=1
Hence the operator ~e(1e,p) is self-adjoint if and only if r~=rk(k=1,2, ...,p-1)
In terms of Hurwitz equations, we can give a characterization of pre-Hurwitz pairs :
Theorem 11 (L1l]). The foZlo~)ing statelnents af~e equivale'rt:
( i ) (En, EP) is a p7-e-Hul~-z()itz pair.
( ii ) There e,xist pure imaginal~y n-sized Inatl~ices rl'T2, "',rp-1 which satisfy the con-
dition (1. 12) and (1. 13).
(ili) ;e is a self-adjoilrt opel~ator of the form (3. 17) with pure imaginary matrices
rl' r2, "', rp-1 satisfying
(3. 21) = Ap_1 In' ;e2 -
* The introduction of mass by use of the spontanous symmetry breaking of the gauge structure 0L Hurwitz
operator is suggested by Dr. S. Kanemaki [also see [5~).
HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPl~~:'GS
Proof. (i) ~> (ii) is a direct consequence of the discussions in S 1. (ii) ~~ (iii). By
rl'T2, "',rp-1' we consider the operator ~e of the L0rm (3. 17). Then we see that its self-
adjointness follows from the conditon (1. 13). Also we see that (3. 21) follows from (1. 12).
(iii) ~> (ii). Since ye is a self-adjoint operator, we see that T~=rk(k=1,2, ・・・,p-1).
By the condition that rk is pure imaginary, we see that (1. 12) holds. Also we see that
(3. 21) implies (1. 12).
Remark. The Hurwitz condition (1. 1) can be stated in terms of the Hurwitz operator
X(n,p) as follows :
(-Ap_1 Inc, c) = I I je(?e,p)c1 12 for c~Fo(Rp-1, C"), (3. 22)
where I I I is defined by the inner product (3. 19).
4. The Neveu-Schwarz model
In this section we shall be concerned with the complex version of the Hurwitz pairs
(E4, E4) and (E4, E3) and obtain the equations of Neveu-Schwarz model of the superstring
thoery ( [24]).
At flrst we treat the Hurwitz condition in the complex form. We identify the flrst
element E4 of (E4, E4) with the hermitian space E~ m a natural manner and consider
the Hurwitz mapping as fc : E~XE4-->E~・ This mapping is called the complex form of the Hurwitz mapping fr. We write down its explicit form. At first we notice that the
Hurwitz mapping of (E4, E4) can be witten as follows :
(4. 2) = i O and r3=t O (13 ( O~ a a2 ( . rl - i O ' r2=~0 (T2/ (F3 O Here ol'(T2,(T3 are the well known Pauli matrices
O -i ~l O)' 2=(i O) and (;3= (Fl (;
Then we see that the complex form fc : E~XE4---~E2 becomes of the form c
(4. 4) fc= (iylerl + iy2a2 + iy3(73 +y412 ) x.
By Theorem I we see that (E4, E3) is also a Hurwitz pair. The Hurwitz mapping and its
complex form are
(4. 5) f (x, y) = ( - iylrl ~ iy2r2 +y314)x
and
(4. 6) fc(x, y) = (iyl(TI + iy2(T2 +y312)x.
* The Hurwitz condition in the complex form introduces the concept of Hermitian Hurwitz pairs. Detaile
studies are given in [9] and [5].
O. Suzuki, J. Lawryno~vicz J. Kalina
Here we consider the following mass producing quantization :
(4. 7) Xo : M2(C)OC[yl'y2,y3]'->M2(C)OC[ala(T, a/ a T]
defined by
(4. 8) Xo : yl~=~-ialaa, y2--~'a/a T, y3-~0 '1.
This may be regarded as the complex version of the mass producing quantization of S 3 (see
(3. 16)). Then the X(c4,3) =xo(fc) becomes
(4. 9) X(c4, 3) = (Tla / ac + i,T2 a / a lr ,
The corresponding Klein-Gordon operator is
(4. 10) a2/ aa2 - a2/ a -, 2.
Hence- the correponding equations are
(4. 11) (Tla~laa+ia2a~)r/aT=0 and a2p/aa2-a2~/a-,2=0.
We shall show that these equations are nothing but the equations which appear in the
Neveu-Schwarz model ([24]). We recall the Lagrangian of the Neveu-Schwar~ model
L = - (1/2a')JdadT {a.Xka"Xk+i~-kp"a.~k} (4. 12)
where
(O -i~ O ' 2 = ( i f) (4 13) , pl = ~ i O) p
and ~k(k 1 2) rs a two component splnor (see (2 32) n p 240 m E24]). As for the
notations we refer to [24]. The Euler equations of the Lagrangian are written as follows :
(4. 14) allklaa+a~lk/aT =0, al2kla(T- a~2k/af =0, a2Xk/acF2-a2Xk/ a ~2=0
We see easily that these equations are equivalent to the ones in (4. 11). In fact, these two
equations are transformed each o.ther by the transformation '
O l , ( , = ) ~ =A~'/ A I O
where we set ),, one of ;,k(k=1,2) by
~1 2=/'~ * ~),2)
5. Isospectral deformations of the Hurwitz operator of (E2, E2)
In [1l], we have given isospectral deformations of the Hurwitz operator in the case of
n =p=2 and optain the K-dV and the modifled K~lV equations. Unfortunately other
important equations are out 0L our donsiderations. In this section we shall give a full
description of isospectral deformations of the Hurwitz operator ;e(=X(z,2)). Isospectral
deformations of Hurwitz operators of more general cases are important in connection with
Gauge fields and will be discussed in the forthcoming paper. Firstly we show that isospectral
deformations of Je(= ije) are equivalent to those which have been treated by Abrowitz et
(14) - 58 =
HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPINGS
al. [1]. Hence we may say that known soliton equations are obtained by use of isospectral
deformations of our Hurwitz operator. Secondly we apply the 2-component K.-P, theory
due to M. and Y. Sato [23] to je and discuss soliton equations for our case.
We set
(O -i~ (5. 1) ^ and D=d/dx ;e=(TD, where (T= ¥ i O)
and consider an isospectral deformation on je with a deL0rmation parameter t whose generator
is given by a differential operator B :
(5. 2) Lc = ~c, L=;e + U {
ct =Bc
where ~ is a spectral parameter and U(=U(x, t)) is a matrix valued function. We show
that (5. 2) can be transformed to the isospectral deformation which has been introduced by
Abrowitz et al. ([14]) : Put
(5 3) _(- i -1~ ~l i )
and make A-lye A. Then we we see that this is equal to
1 O ( , = ) KD, where K o _1
which is nothing but the operator which is introduced by Abrowitz. Hence we can trans-
forrn the results to our case.
Proposition (5. 4). By isospectral deformations (5. 2), we can obtain the K.-dV, mo-
dlfied K.-dV. , sine-Gordon, ard nonlinear Schrddinger equations. More exactly, if we set
U=1(~(r+q)i -r+q) 2 ~_r +q (r+q) i
where r and q are functions of x and t, then we obtain
( i ) the modified K.-dV equation : rt+6r2rx+r*xx=0, when we choose r=q, real and
B as a differentiaj operatot~ of the third order,
( ii ) the nonliner Schredinger equation : irt+rxx+2jrl2r=0, when we choose r=q*, the
complex conjucate and B as a differential operator of the second order,
(ili) the sine-Gordon equation : rtx=sin r if we choose r=q, real and B as a pseudo-
differential operator of the order - 1.
As for proofs, see [14].
Next we deflne the K.-P. system for the operator je. Following [23], we introduce a
pseudo-differential operator P :
(5. 5) P= Po + plD-1 + . .. + p~D-n + . . .,
where P~=P~(x) is a matrix valued function and Po is a constant invertible matrix. We
set
O. Suzuki, J. LaT~;~rynowicz J. Kalina
(5. 6) A L=Pj~P-1 and L,e=P(~Dnp-1.
Then we have the following decomposition :
Ln = (L~) + + (L*) _,
(L~)+=B(n)Dn+B(n"-)1Dn-1+...+B(n) and (L ) B(")D 1+B(")D +
We notice that B(I~) = PoaD?ep~l. Here we introduce an infinite number of time parameters
t = (tl' t2, "') and consider P(=P(t)) depending on t. We introduce
Definition (5. 7). The equation
(5. 8) aL/at~= [(L~)+' L] n =1, 2, ...
is called the K.-P. system of the Hurwitz pair (E2, E2).
Following the discussions in [19], we can show that every solution of (5. 9) can be
obtained from that of the linear equation
(5. 10) aU/at~=aD7eU, n=1, 2, ....
We get special solutions of (5. 9) :
Proposition (5. 11). The K.-P. system can be reduced to the K.-P, system for the
KD when Po is chosen as A-1, ~e'hel'e A is given in (5. 3), Hence nonlinear Schredinger
equation and the 1'rodlfied K.-dV, equation can be obtained in the cases of n=2 and n=3
respectively.
Proof We choose P A-1. For a solution P of (5. 8), we set
(5. Il) P =P'A-1, P'=12 +p~D-1 +p2"D-2+ ...
L'=P'KDP'-1 and L~=P'KDnp'-1.
Then we see that L' satisfies
(5. 12) aL'/atn=[(L~)+' L'] (n=1 2 ...) ,, ,
which is nothing but the K.-P. system for KD. The latter part of the assertions are well
known*.
By summarizing the discussions we can obtain the foll07ving theorem :
Theorem 111. Isospectral deformations of the Hurwitz operator ~e oj (E2, E2) give
rise to the K.-dV, the modtfied K.-dV. , sine-Gordon and nonlinear Schrbdinger equations.
The K.-P. system for ;e can be defined by (5. 8) and linearization is given by (5. 10). In
the cases of n = 2, 3, this equation gives the modtfied K.-dV. and nonlinear Schrbdinger
equations respectively.
* Calculating the integrability conditions in the case of n=1, 2, we can obtain the so called nonlinear Schr6dinger equation ...etc (this is a private communication by Dr. Jimbo).
HURWITZ PAIRS, SUSERCOMPLEX STRUCTURES, SOLITON EQUATIONS, AND QUASICOMPORMAL MAPPlNGS
6. Supercomplex structure of a Hilbert space and reduction solutions of the K.-P.
system
In this section we generalize a concept of supercomplex structure to a Hilbert space
and state a correspondence between supercomplex structures and reduction solutions of the
original K.-P. system. We know that reduction solutions of the K.-P, system give those
of the K.-dV and other soliton equations ([3]). On the other hand, we know that the
Virassoro algebra is used to describe soliton equations ([22]). Taking into account the
fact that the Virassoro algebra can be obtained from the group of biholomorphic mappings
of C* we may say that our result will give one of the understandings of a reason why
the Virassoro algebra is used in the theory of soliton equations.
We begin with generalizing the concept of pre-Hurwitz pair to a Hilbert space. By H
we denote the separable Hilbert space over R. We make the following definition :
Definition (6. 1). A pair (H, EP) is called a pre-Hurwitz pair, if there exists a
bilinear mapping f: HX EP --~ H satisfying
(6. 2) Ilf(x,y)ll=1lxll llyll for any xeH and y~EP,
where I x I , I I yll denote the norms of H and EP respectively. A pre-Hurwitz pair is called
a decomposable pre-Hurwitz pair, if H has the decolmposition
(6. 3) H= (T"~ Ek where (E~) is a Hurwitz pair. k=1
In the following we treat only decomposable pre-Hurwitz pairs. Choosing supercomplex
structures' Jk on E~, we define a supercomplex structure J on H by
(6. 4) J= ~) Jk. k=1
The complex structure J is called a decomposabZe supercowplex structure on H, or simply,
a supercomplex structure on H. Then we can prove the following theorem :
Tneorem IV. There exists a correspondence between a set of decomposable supercom-
plex structures and a set of ~30 (p-1)-reduction solutions of the original Sato's K.-P.
system .
Before going to the proof of Theorem IV, we recall basic facts on reduction solutions
of the K.-P. system ([3]). The original K.-P. system is given in the following manner :
We set
(6. 5) L= WDW-1 where D=d/dx and W=1+ulD-1+u2D-2+ ... Here ul'u2, "' are functions of x and t, t=(tl' t2, "')' We call the equation
(6. 6) aL/at~= [(L")+' L] (n= 1, 2, 3, ・・・) the (original) K.-P, system ([23]). A solution of the K.-P, system is called an l-reduction
solution if
O. Suz*ki. J. La+ry~o~vic' J. Kalina
(6. 7) (Lt) + = Lz
with some integer l. We know that 2-reduction (or 5-reduction) solutions give rise to
those of the K.~lV. (resp. Boussinesq) equation. It is well known that l-reduction solutions
can be characterized in terms of the so called Kac-Moody Lie algebra, more exactly the
Lie algebra A~1)1 ([3]). An element ~~A(1) can be written as l-1
(6. 8) ~= ~ X(k);,k k=-* '
where X(k)e~~iC(1, R) and ), is a parameter. In a similar manner we can difine ~r-reduction
solutions. We choose a Lie subalgebra ~r of ~~~C(1, R) and set ~ by (6. 8) with X(k)e:~r
(keZ). The corresponding solutions of the K.-P. system are called ~r-reduction solutions.
Proof of Theorem IV. Let ~ be an ~~O (p-1)-reduction solution. Then ~ has the
form of (6. 8). Hence we have a sequence of infinite elements {X(k)}ke2; X(k)~J~50(p-1,R).
We set gk=exp(X(k))(e~SO(p-1,R)). By use of the identification (see (2. 4))