New infinite-dimensional multiple-symmetry groups for the Einstein–Maxwell-dilaton–axion theory

Journal of Geometry and Physics 58 (2008) 1030–1042

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New infinite-dimensional multiple-symmetry groups for the Einstein–Maxwell-dilaton–axion theory Ya-Jun Gao ∗ Department of Physics, Bohai University, Jinzhou 121013, Liaoning, People’s Republic of China

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Article history: Received 17 October 2007 Received in revised form 10 March 2008 Accepted 10 March 2008 Available online 15 March 2008 JGP SC: Strings and superstrings Lie groups and Lie (super)algebras Classical integrable systems MSC: 83E30 22E65 22E70 17B80

a b s t r a c t The symmetry structures of stationary axisymmetric Einstein–Maxwell-Dilaton–Axion (EMDA) theory are further studied. By using the so-called extended double (ED)-complex function method, the usual Riemann–Hilbert (RH) problem is extended to an ED-complex formulation. Two pairs of ED RH transformations are constructed and they are verified to give infinite-dimensional multiple-symmetry groups of the EMDA theory; each of these symmetry groups has the structure of a semidirect product of a Kac–Moody group Sp\ (4, R) and a Virasoro group. Moreover, the infinitesimal forms of these RH transformations are calculated and they are found to give exactly the same results as previous work; this demonstrates that the two pairs of ED RH transformations in this paper provide exponentiations of all the infinitesimal symmetries in our previous paper. The finite forms of symmetry transformations given in the present paper are more important and useful for theoretical studies and new solution generation, etc. © 2008 Elsevier B.V. All rights reserved.

Keywords: Einstein–Maxwell-dilaton–axion theory Extended double-complex method Riemann–Hilbert transformation Infinite-dimensional multiple-symmetry group

1. Introduction Recently, much attention has been attracted to the studies of symmetries for the dimensionally reduced low energy effective (super)string theories (e.g. [1–24]) owing to their importance in theoretical and mathematical physics. Such effective string theories describe various interacting matter fields coupled to gravity; the Einstein–Maxwell-dilaton–axion (EMDA) theory (see e.g. [3,11,12,14,17,18,23,24]) is a typical and important model of this kind. Some analogies between the EMDA theory and the reduced vacuum Einstein theory have been noted. However, the mathematical structures of the EMDA theory are much more complicated. For example, many scalar functions in pure gravity correspond, formally, to matrix ones in the effective string theory; thus the non-commuting property of the matrices gives rise to essential complications for the further study of the latter. Moreover, some important and useful formulas in some studies of the reduced vacuum gravity (e.g. [25–29]) will have no general analogues in the EMDA theory, so deeper researches and further extended studying methods are needed. The present paper is a continuation of our previous papers [23,24]. In [23,24], we found the doubleness symmetry of the stationary axisymmetric (SAS) EMDA theory. Further, by using the so-called extended double (ED)-complex function ∗ Tel.: +86 416 3400148; fax: +86 416 3400149. E-mail address: [email protected]. 0393-0440/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2008.03.009

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method [29], we constructed the ED-complex 4 × 4 matrix H, F -potentials and established two pairs of ED-complex Hauser–Ernst (HE)-type linear systems. On the basis of these linear systems, we explicitly constructed new infinitesimal multiple-symmetry transformations for the SAS EMDA theory and verified that they constitute quadruple infinite-dim (‘-dim’ stands for ‘-dimensional’ or ‘ dimensions’ here and hereafter) Lie algebras, each of which is a semidirect product of the Kac–Moody sp\ (4, R) and Virasoro algebras. However, for theoretical studies and new solution generation, etc., it is more important and useful to find finite symmetry transformations of the theory considered; this is the main aim of the present paper. In Section 2, some related concepts and notation for the ED-complex functions [29], the ED-complex H, F -potentials and two pairs of HE-type linear systems for EMDA theory [24] are briefly recalled. In Section 3, we construct a pair of ED Riemann–Hilbert (RH) transformations relating to the first pair of HE-type linear systems and then prove that they are indeed double-symmetry transformations of the EMDA theory. In Section 4, the equivalent integral equation formulations are given and the infinite-dim group structures of the ED RH transformations are verified. In Section 5, we find that a doubleduality mapping can be introduced and a second pair of ED RH transformations relating to the second pair of HE-type linear systems can be given. In Section 6, infinitesimal forms of the given ED RH transformations are calculated, which give exactly the same results as our previous paper [24]. These demonstrate that the two pairs of ED RH transformations in the present paper provide exponentiations of all the infinitesimal symmetry transformations given in [24]. Finally, Section 7 gives a summary and discussion. 2. ED-complex H , F -potentials and HE-type linear systems for EMDA theory For later use, here we briefly recall some related concepts and notation for the ED-complex function [29] as well as the ED-complex H, F -potentials and HE-type linear systems for EMDA theory [24]. 2.1. ED-complex function [29] Let i and J denote, respectively, the ordinary and the ED imaginary unit. We shall concern ourselves mainly with some P special values of J, i.e. J = j (j2 = −1, j 6= ±i) or J = ε (ε2 = +1, ε 6= ±1). If a series ∞ n=0 |an |, an ∈ C (ordinary P∞ 2n complex number), is convergent, then a(J) = n=0 an J is called an ED ordinary complex number, which can correspond to a pair (aC , aH ) of ordinary complex numbers, where aC := a(J = j), aH := a(J = ε). When a(J) and b(J) are both ED ordinary complex numbers, c(J) = a(J) + Jb(J) is called an ED-complex number; it can correspond to a pair (cC , cH ), where cC := c(J = j) = aC + jbC , cH := c(J = ε) = aH + εbH . We define a(J) := ReED (c(J)), b(J) := ImED (c(J)). If a(J) and b(J) are real, we call them double-real and call the corresponding c(J) simply a double-complex number. We would like to point out that, from the above definitions, J should be taken as an indeterminate rather than a discrete variable. The ED-complex method can be regarded as some “deformation” theory, in which J plays the role of a “deformation parameter” (or analytical link; cf. [30] for the non-extended case). By doubleness symmetry we in fact mean the symmetry property of the considered theory under this “deformation”. We call it an ED-complex method only because in most of its applications (e.g. in the present paper) we are mainly interested in the cases of J = j and J = ε. All ED-complex numbers with usual addition and multiplication constitute a commutative ring. Corresponding to the two imaginary units J and i in this ring, we have two complex conjugations: ED-complex conjugation “?” and ordinary complex conjugation “−” c(J)? := a(J) − Jb(J),

c(J) := a(J) + Jb(J).

(2.1)

These imply that J? = −J, J = J, i? = i, i = −i. If a(J) and b(J) are ED ordinary complex functions of some ordinary complex variables z1 , . . . , zn , then c(z1 , . . . , zn ; J) = a(z1 , . . . , zn ; J) + Jb(z1 , . . . , zn ; J) is called an ED-complex function. We describe c(z1 , . . . , zn ; J) as continuous, analytical, etc. if a(z1 , . . . , zn ; J) and b(z1 , . . . , zn ; J) both, as ordinary complex functions, have the same properties. For an ED-complex matrix W (J), we define W (J)+ := [W (J)? ]> ,

(2.2)

where “>” denotes transposition. The ED imaginary unit commutation operator “◦” is defined by ◦

◦ : J −→ J ,

◦

j=

,

◦

 = j.

(2.3)

◦

Obviously, J is the ED imaginary unit, too. 2.2. ED-complex H, F -potentials and HE-type linear systems [24] The EMDA action, which describes the bosonic sector of the heterotic string in 4-dim, and contains a metric gµν (signature

+ − − −, µ, ν = 0, 1, 2, 3), a U (1) vector field Aµ , a Kalb–Ramond antisymmetric tensor field Bµν and a dilaton field φ, can be written as [11,12]

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Z  S=

−R +

1 3

e−4φ Hµνλ Hµνλ + 2∂µ φ∂µ φ − e−2φ Fµν F µν



√ − g d4 x ,

(2.4)

where R is the Ricci scalar, g = det(gµν ) and Fµν = ∂µ Aν − ∂ν Aµ ,

Hµνσ = ∂µ Bνσ − Aµ Fνσ + cyclic.

(2.5)

In the SAS case, the 4-dim space-time line element can be written as [31] ds2 = fAB dxA dxB − e2γ δLN dxL dxN ,

(A, B = 0, 1, L, N = 2, 3),

(2.6)

and fAB can be parametrized as fAB =

f −f ω

−f ω , f ω2 − ρ2 f −1 !

(2.7)

where the set of non-trivial EMDA dynamical quantities also contains fields (A0 , A1 ), B01 and φ, and all of them are assumed to depend only on x2 , x3 . Denoting x2 , x3 as x, y for simplicity and introducing 2 × 2 symmetric real matrices P and Q by ! ! √ √ f − 2e−2φ A20 − 2e−2φ A0 ω − 2(A1 + ωA0 ) √ −2φ √ P= , Q = , (2.8) − 2e A0 −e−2φ − 2(A1 + ωA0 ) 2A0 (A1 + ωA0 ) − B01 the essential dynamical equations of the SAS EMDA theory can be written as [11,12] d(ρ−1 P ∗ d QP) = 0,

d(ρ ∗ d PP−1 + ρ−1 P ∗ d QPQ ) = 0,

(2.9)

and ρ = ρ(x, y) > 0 is a harmonic function in 2-dim {x, y}. As pointed out in Refs. [23,24], the SAS EMDA theory possesses so-called doubleness symmetry such that we can introduce 2 × 2 double-real symmetric matrices P(J) and Q (J) and define a double-real 4 × 4 matrix function M(J) = M(x, y; J) as ! M(J) =

P (J)

−Q (J)P(J)

−P(J)Q (J) Q (J)P (J)Q (J) + J2 ρ2 P (J)−1

,

(2.10)

and the motion Eq. (2.9) can be extended to a double formulation d(ρ−1 M(J)η ∗ d M(J)) = 0

(2.11)

with conditions M(J)> = M(J),

(2.12a)

M(J)ηM(J) = J ρ η,   0 I2 η := , −I2 0 2

2

(2.12b) (2.12c)

where I2 is the 2-dim unit matrix. If a solution of Eqs. (2.11) and (2.12) is known, then by the decomposition (2.10), we can obtain real solutions of the EMDA theory in pairs as follows:

(P, Q ) = (PC , QC ),

(2.13a)

(Pˆ , Qˆ ) = (T (PH ), VPH (QH )),

(2.13b)

where the transformations T , V are defined by T : P −→ T (P ) =

ρP−1 ,

V : P, Q −→ VP (Q ) =

Z

ρ−1 P(∂y Q )Pdx − ρ−1 P(∂x Q )Pdy,

(2.14)

and the existence of VPH (QH ) is ensured by the J =  case of Eq. (2.11). Eq. (2.11) implies that we can introduce a double-real 4 × 4 matrix twist potential N(x, y; J) by dN(J) = −ρ−1 M(J)η ∗ d M(J); then from (2.12) and the harmonicity of ρ(x, y) we can obtain N(J) − N(J)> = −2J2 zη with the real field z = z(x, y) introduced by ∗ dρ = dz. Thus, if we define a double-complex H-potential H(J) := M(J) + JN(J)

(2.15)

and define Ω := Jη, then the equations for N(J) and M(J) can be written together as 2(z + ρ∗ )dH(J) = (H(J) + H(J)+ )Ω dH(J).

(2.16)

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By introducing an ordinary complex parameter t and defining A(t; J) := I − t[H(J) + H(J)+ ]Ω ,

Γ (t; J) := tΛ(t)

−1

(I is the 4-dim unit matrix),

dH(J),

(2.18)

Λ(t) := 1 − 2t(z + ρ∗ ),

Λ(t)−1 = λ(t)−2 [1 − 2t(z − ρ∗ )], 2 1/2

λ(t) := [(1 − 2zt) + (2ρt) ] 2

(2.17)

,

(2.19) (2.20)

then Eq. (2.16) can be rewritten as tdH(J) = A(t; J)Γ (t; J),

(2.21)

and the associated ED-complex HE-type linear system can be established as dF (t; J) = Γ (t; J)Ω F (t; J),

(2.22)

F (0; J) = I,

(2.23a)

F˙ (0; J) = H(J)Ω ,

(2.23b)

λ(t)F (t; J) Ω F (t; J) = Ω ,

(2.24a)

F (t; J) Ω A(t; J)F (t; J) = Ω .

(2.24b)

>

+

Where F (t; J) = F (x, y, t; J) is a 4 × 4 ED-complex matrix function of x, y and t, and is holomorphic in a neighborhood of t = 0, F˙ (t; J) := ∂F (t; J)/∂t. Besides, we introduce another ordinary complex parameter w and define A˜ (w; J) := w − (H(J) + H(J)+ )Ω ,

˜ (w) Γ˜ (w; J) := Λ

−1

(2.25)

dH(J),

(2.26)

˜ (w) := w − 2(z + ρ ), Λ

˜ (w) Λ

−1

∗

˜ w) [w − 2(z − ρ )], = λ( −2

∗

˜ w) := [(w − 2z)2 + (2ρ)2 ]1/2 . λ(

(2.27) (2.28)

Then Eq. (2.16) can be rewritten as dH = A˜ (w; J)Γ˜ (w; J),

(2.29)

and the associated HE-type ED-complex linear system is dF˜ (w; J) = Γ˜ (w; J)Ω F˜ (w; J),

(2.30)

˜ w)F˜ (w; J)> Ω F˜ (w; J) = Ω , λ(

(2.31a)

F˜ (w; J)+ Ω A˜ (w; J)F˜ (w; J) = Ω ,

(2.31b)

where F˜ (w; J) = F˜ (x, y, w; J) is another ED-complex 4 × 4 matrix function of x, y and w, and is holomorphic around w = 0. 3. ED Riemann–Hilbert transformations At first, we generalize the usual RH problem to an ED-complex formulation. Let L denote a smooth contour surrounding the origin in the ordinary complex plane and symmetric with respect to the real axis, L+ and L− be the inside and outside (including ∞) of L, respectively. For an ordinary complex variable s, if a given ED-complex matrix function G(s; J) is holomorphic and invertible on L, then there exist a pair of ED-complex matrix functions X± (s; J) which are (respectively for ±) holomorphic in L± , continuous and invertible on L ∪ L± such that X− (s; J) = X+ (s; J)G(s; J),

s ∈ L.

(3.1)

We call (3.1) an ED-structural RH problem. For a fixed kernel G(s; J), the fundamental solution X± (s; J) of the RH problem is unique up to a non-singular constant matrix factor. A suitable boundary condition can cancel this indefiniteness. By using the above ED-structural RH problem formulation and solutions F (t; J), F˜ (w; J) of linear systems (2.22)– (2.24), (2.30)–(2.31), we can construct double-symmetry transformations for the EMDA theory. From definitions (2.15), (2.17)–(2.20) and (2.25)–(2.28), we may consistently choose the ED-complex matrix functions F (t; J) and F˜ (w; J) as F (t; J) = F (t; J),

F˜ (w; J) = F˜ (w; J)

in order to ensure the reality of M(J) and N(J) in the transformed H(J). We shall take this choice in the following.

(3.2)

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3.1. ED RH transformation for linear system (2.22)–(2.24) Following the spirit of [27], we introduce a scalar function v(t) which is independent of x, y and holomorphic on L ∪ L− except infinity where it tends to linear divergence such that v(t) is a linear function of t or has singularities in L+ . In addition, v(t) is real when t is real. Furthermore, we introduce two new real functions ρ0 = ρ0 (x, y), z0 = z0 (x, y) such that (for fixed x, y) λ(v(t)) = [(1 − 2zv(t))2 + (2ρv(t))2 ]1/2 and λ0 (t) := [(1 − 2z0 t)2 + (2ρ0 t)2 ]1/2 have the same zeros in t, and v˙ (t) 6= 0 at these zeros. Thus we have   1 1 = v−1 , (3.3a) 0 0 2(z ± iρ ) 2(z ± iρ) ∗

d ρ0 = dz0 .

(3.3b)

Eq. (3.3a) can be interpreted as a variable transformation under which Eqs. (2.16)–(2.24) are transformed. We shall use the notation with primes to denote the transformed functions, e.g. F (t; J) 7→ F 0 (t; J). Motivated by [25,27], for a given solution F (t; J) of (2.22)–(2.24), we select the contour L such that F (t; J) is holomorphic on L ∪ L+ and take the kernel G(t; J) of (3.1) as G(t; J) = F 0 (t; J)u(t)F (v(t); J)−1 ,

(3.4)

where the ordinary complex 4 × 4 matrix function u(t) (independent of x, y) is holomorphic in L ∪ L− and satisfies u(t)> ηu(t) =

η,

u(t) = u(t),

(3.5)

i.e. u(t) ∈ Sp(4, R) when t is real. By virtue of the above ED-structural RH problem, we have the following: Theorem 1. If X± (t; J) = X± (x, y, t; J) is a fundamental solution of the ED-structural RH problem (3.1), (3.4) and (3.5) with boundary condition X+ (0; J) = I,

(3.6)

then the ED-complex matrix function given by F (t; J) = X+ (t; J)F 0 (t; J)

in L ∪ L+

= X− (t; J)F (v(t); J)u−1 (t) in L ∪ L−

(3.7)

is holomorphic on L ∪ L+ and satisfies dF (t; J) = Γ 0 (t; J)Ω F (t; J),

(3.8a)

A(t; J)dF (t; J) = tdH(J)Ω F (t; J),

(3.8b)

F (0; J) = I,

(3.9a)

F˙ (0; J) = H(J)Ω ,

(3.9b)

λ (t)F (t; J) Ω F (t; J) = Ω ,

(3.9c)

F (t; J)+ Ω A(t; J)F (t; J) = Ω

(3.9d)

H(J) := H0 (J) − J2 X˙ + (0; J)Ω ,

(3.10)

0

>

with

Γ (t; J) := tΛ (t) 0

−1

0

dH(J),

(3.11a)

A(t; J) := I − t(H(J) + H(J)

+

)Ω .

(3.11b)

Theorem 2. The new ED-complex function H(J) given by (3.10) is an ED H-potential of the EMDA theory considered. Explicitly, H(J) satisfies 2(z0 + ρ0 )dH(J) = (H(J) + H(J)+ )Ω dH(J),

(3.12a)

H(J) − H(J)

(3.12b)

∗

>

= −2J Ω z , 2

M(J) := ReED (H(J)), M(J)Ω M(J) = J

2

02

ρ Ω.

0

M(J)

>

= M(J),

(3.12c) (3.12d)

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Before proving the Theorems 1 and 2, we first prove the following: Lemma 1. Let X± (t; J) be a fundamental solution of the ED-structural RH problem (3.1), (3.4)–(3.6); we can consistently define the following ED-complex functions of t as > W1 (t; J) = X+ (t; J)−1 Ω X+ (t; J)−1

= (λ(v(t))/λ W2 (t; J) =

0

on L ∪ L+

(t))X−> (t; J)−1 Ω X− (t; J)−1

+ X+ (t; J)−1 Ω A0 (t; J)X+ (t; J)−1

on L ∪ L− ,

(3.13a)

on L ∪ L+

= X−+ (t; J)−1 Ω A(v(t); J)X− (t; J)−1 on L ∪ L− , W3 (t; J) = dX+ (t; J)X+ (t; J)

−1

= dX− (t; J)X− (t; J)

−1

(3.14a)

+ X+ (t; J)Γ (t; J)Ω X+ (t; J) 0

−1

on L ∪ L+

+ X− (t; J)Γ (v(t); J)Ω X− (t; J)

−1

W4 (t; J) = Ω A(t; J)dX+ (t; J)X+ (t; J)

−1

= Ω A(t; J)dX− (t; J)X− (t; J)

−1

on L ∪ L− ,

+

+ tX+ (t; J)−1 Ω dH0 (J)Ω X+ (t; J)−1

+

+ v(t)X− (t; J)−1 Ω dH(J)Ω X− (t; J)−1

(3.15a)

on L ∪ L+ on L ∪ L−

(3.16a)

and have W1 (t; J) = Ω ,

(3.13b)

W2 (t; J) = Ω A(t; J),

(3.14b)

W3 (t; J) = Γ 0 (t; J)Ω ,

(3.15b)

W4 (t; J) = tΩ dH(J)Ω .

(3.16b)

02

(3.17)

A

>

(t; J)Ω A(t; J) = λ (t)Ω .

Proof. From (2.24a), (3.5), (3.4) and (3.1) we obtain > X+ (s; J)−1 Ω X+ (s; J)−1 =

(λ(v(s))/λ0 (s))X−> (s; J)−1 Ω X− (s; J)−1 ,

s ∈ L.

Noticing the properties of the functions X± (t; J), v(t), λ(t), since [λ(v(t))/λ0 (t)] is non-singular in L ∪ L− , the above equation implies that W1 (t; J) in (3.13a) is consistently defined and gives an ED entire function of t. Note that (λ(v(t))/λ0 (t))X−> (t; J)−1 Ω X− (t; J)−1 is regular at t = ∞, so W1 (t; J) is equal to a constant matrix. From the boundary condition (3.6), we get Eq. (3.13b). To prove (3.14b), we use (2.24b), (3.5), (3.4) and (3.1) to get + X+ (s; J)−1 Ω A0 (s; J)X+ (s; J)−1 = X−+ (s; J)−1 Ω A(v(s); J)X− (s; J)−1 ,

s ∈ L.

This implies that W2 (t; J) is consistently defined and gives an ED entire function of t. From the expression for A(v(t); J) and the property of v(t) at t = ∞, we conclude that W2 (t; J) is linear in t. By using (3.6) we can obtain the coefficients in this linear function such that W2 (t; J) turns to being Ω A(t; J) by definition (3.11b). The (3.15b) is proven as follows. From (3.1), (3.4) and (2.22) we have dX+ (s; J)X+ (s; J)−1 + X+ (s; J)Γ 0 (s; J)Ω X+ (s; J)−1

= dX− (s; J)X− (s; J)−1 + X− (s; J)Γ (v(s); J)Ω X− (s; J)−1 ,

s ∈ L.

(3.18)

Thus the function W3 (t; J) defined in (3.15a) is a meromorphic function of t and has simple singularity at the zeros of λ (t) (or, equivalently, λ(v(t))). According to the theory of the meromorphic function, we can express W3 (t; J) as U (J) + tΛ0 (t)−1 V (J). By using (3.6) and (3.10) we obtain 0

U (J) = 0,

V (J) = dX˙ + (0; J) + dH0 (J)Ω = dH(J)Ω ,

and thus W3 (t; J) = tΛ0 (t)−1 dH(J)Ω and this gives (3.15b) by (3.11a). To prove (3.16b), we note that from (3.18), (2.18), (3.14a), (3.14b) and (2.21) we have + Ω A(s; J)dX+ (s; J)X+ (s; J)−1 + sX+ (s; J)−1 Ω dH0 (J)Ω X+ (s; J)−1

= Ω A(s; J)dX− (s; J)X− (s; J)−1 + v(s)X−+ (s; J)−1 Ω dH(J)Ω X− (s; J)−1 ,

s ∈ L.

Similarly to above, W4 (t; J) in (3.16a) is a linear function of t and by using (3.6) we get (3.16b). As for (3.17), we first note that from (2.24) we have A> (t; J)Ω A(t; J) = λ2 (t)Ω . Then from (3.13a), (3.13b), (3.14a) and (3.14b) it follows that ( > X+ (t; J)−1 A0 > (t; J)Ω X+? (t; J)−1 Ω −1 X++ (t; J)−1 Ω A0 (t; J)X+ (t; J)−1 on L ∪ L+ > A (t; J)Ω A(t; J) = > ? X− (t; J)−1 A> (v(t); J)Ω X− (t; J)−1 Ω −1 X−+ (t; J)−1 Ω A(v(t); J)X− (t; J)−1 on L ∪ L−

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(

=

> X+ (t; J)−1 A0 (t; J)Ω A0 (t; J)X+ (t; J)−1 0 (λ (t)/λ(v(t)))X−> (t; J)−1 A> (v(t); J)Ω A(v(t); J)X− (t; J)−1 >

on L ∪ L+ on L ∪ L−

2

= λ0 (t)Ω .  Proof of Theorem 1. Eq. (3.8a) is derived from (3.15a), (3.15b), (3.7) and (2.22); (3.8b) is deduced from (3.16a), (3.16b), (3.7), (2.21), (2.22), (3.14a) and (3.14b); (3.9a) follows simply from (3.7), (2.23a) and (3.6); and (3.9b) from Eqs. (3.7), (3.6) and (2.23) and definition (3.10); (3.9c) follows from (3.13a), (3.13b), (3.7) and (2.24a); (3.9d) is derived from (3.14a), (3.14b), (3.7) and (2.24b).  Proof of Theorem 2. To prove (3.12a), note that Eqs. (3.11a), (3.8a) and (3.8b) imply Λ0 (t)dF (t; J) = A(t; J)dF (t; J). Thus from (2.19) and (3.11b) we have 2(z0 + ρ0 )dF (t; J) = (H(J) + H(J)+ )Ω dF (t; J). ∗

Taking the t-derivative of the above equation and then setting t = 0, we obtain (3.12a) by using (3.9b). Eq. (3.12b) is derived by taking the t-derivative of Eq. (3.9c) and then setting t = 0, and noting (3.9a) and (3.9b). (3.12c) is a trivial implication of (3.12b). To prove (3.12d), note that (3.12b) and (3.12c) imply A(t; J) = (1 − 2tz0 )I − 2tM(J)Ω , and thus from (3.17) we have (1 − 2tz0 )2 Ω − 4t2 Ω M(J)Ω M(J)Ω = λ0 (t)2 Ω ; this gives (3.12d).  3.2. ED RH transformation for linear system (2.30)–(2.31) Here we need another scalar function v˜ (w), which has the same properties as v(t) but the variable t is replaced by w, and according to the properties of v˜ (w) we may write v˜ (w)|w→∞ = aw,

a

> 0 (real number).

(3.19)

˜ v˜ (w)) = Relating to v˜ (w) we introduce two real functions ρ00 = ρ00 (x, y), z00 = z00 (x, y) such that (for fixed x, y) λ( [(v˜ (w) − 2z)2 + (2ρ)2 ]1/2 and λ˜ 00 (t) := [(w − 2z00 )2 + (2ρ00 )2 ]1/2 have the same zeros in w, and v˙˜ (w) 6= 0 at these zeros. Thus we have 2(z00 ± iρ00 ) = v˜ −1 (2(z ± iρ)), ∗

(3.20a)

d ρ00 = dz00 .

(3.20b)

Eq. (3.20a) can be interpreted as another variable transformation; the corresponding transformed functions will be denoted with double primes “00 ”, e.g. F˜ (w; J) 7→ F˜ 00 (w; J). Consider an ED-structural RH problem relating to (2.30), (2.31) as follows. We use L˜ and L˜ ± in the w-plane. For a given solution F˜ (w; J) of Eqs. (2.30) and (2.31), we select the contour L˜ such that F˜ (w; J) is holomorphic on L˜ ∪ L˜ + and take the kernel of (3.1) as

˜ (w; J) = a−1/2 F˜ 00 (w; J)u˜ (w)F˜ (v˜ (w); J)−1 , G

(3.21)

where the positive real number a is the same as in (3.19), and u˜ (w) has the same properties as u(t) except that t is replaced by w. By virtue of ED-structural RH problem (3.1) with kernel (3.21), we can obtain another ED RH transformation for the EMDA theory. Firstly, we have the following: Lemma 2. Let X˜ ± (w; J) be a fundamental solution of the ED-structural RH problem (3.1), (3.21) with boundary condition X˜ − (∞; J) = I.

(3.22)

Then we can consistently define the following functions of w as

˜ 1 (w; J) = X˜ +> (w; J)−1 Ω X˜ + (w; J)−1 W ˜ v˜ (w))/λ = a (λ( −1

˜ 00

on L˜ ∪ L˜ +

(w))X˜ −> (w; J)−1 Ω X˜ − (w; J)−1

˜ 2 (w; J) = X˜ ++ (w; J)−1 Ω A˜ 00 (w; J)X˜ + (w; J)−1 W

on L˜ ∪ L˜ − ,

(3.23a)

on L˜ ∪ L˜ +

= a−1 X˜ −+ (w; J)−1 Ω A˜ (v˜ (w); J)X˜ − (w; J)−1 on L˜ ∪ L˜ − ,

(3.24a)

˜ 3 (w; J) = dX˜ + (w; J)X˜ + (w; J)−1 + X˜ + (w; J)Γ˜ 00 (w; J)Ω X˜ + (w; J)−1 on L˜ ∪ L˜ + W = dX˜ − (w; J)X˜ − (w; J)−1 + X˜ − (w; J)Γ˜ (v˜ (w); J)Ω X˜ − (w; J)−1 on L˜ ∪ L˜ − ,

(3.25a)

˜ 4 (w; J) = Ω A˜ (w; J)dX˜ + (w; J)X˜ + (w; J)−1 + X˜ ++ (w; J)−1 Ω dH00 (J)Ω X˜ + (w; J)−1 on L˜ ∪ L˜ + W = Ω A˜ (w; J)dX˜ − (w; J)X˜ − (w; J)−1 + a−1 X˜ −+ (w; J)−1 Ω dH(J)Ω X˜ − (w; J)−1 on L˜ ∪ L˜ − ,

(3.26a)

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and have

˜ 1 (w; J) = Ω , W

(3.23b)

˜ 2 (w; J) = Ω A˜ (w; J), W

(3.24b)

˜ 3 (w; J) = Γ˜ 00 (w; J)Ω , W

(3.25b)

˜ 4 (w; J) = Ω dH˜ (J)Ω . W

(3.26b)

A˜

>

˜ 002

(w; J)Ω A˜ (w; J) = λ (w)Ω .

(3.27)

Here

˜ (J) := a−1 H00 (J) − J2 ∂τ X˜ − (w; J)|τ=0 Ω , H

τ := w−1 ,

00

(3.28)

˜ 00 (w)−1 dH˜ (J), Γ˜ (w; J) := Λ

(3.29a)

˜ (J) + H˜ (J)+ )Ω . A˜ (w; J) := w − (H

(3.29b)

Proof. The proof is similar to that of Lemma 1. However, here boundary conditions at w = ∞ such as (3.19) and (3.22) are used.  Theorem 3. The ED-complex matrix function given by F˜ (w; J) = a−1/2 X˜ + (w; J)F˜ 00 (w; J)

in L˜ ∪ L˜ +

= X˜ − (w; J)F˜ (v˜ (w); J)u˜ (w)

−1

in L˜ ∪ L˜ −

(3.30)

is holomorphic on L ∪ L+ and satisfies 00

dF˜ (w; J) = Γ˜ (w; J)Ω F˜ (w; J),

(3.31a)

˜ (J)Ω F˜ (w; J), A˜ (w; J)dF˜ (w; J) = dH

(3.31b)

λ˜ 00 (w)F˜ (w; J)> Ω F˜ (w; J) = Ω ,

(3.32a)

F˜ (w; J)+ Ω A˜ (w; J)F˜ (w; J) = Ω .

(3.32b)

Proof. The proof is similar to that of Theorem 1.



˜ (J) given by (3.28) is an ED H-potential of the EMDA theory considered. Explicitly, Theorem 4. The new ED-complex function H

˜ (J) satisfies H

˜ (J) = (H˜ (J) + H˜ (J)+ )Ω dH˜ (J), 2(z00 + ρ00 )dH

(3.33a)

˜ (J) − H˜ (J)> = −2J2 Ω z00 , H

(3.33b)

∗

˜ (J) := ReED (H˜ (J)), M

˜ (J) M

>

˜ (J), =M

002

˜ (J)Ω M ˜ (J) = J2 ρ Ω . M

(3.33c) (3.33d)

Proof. Firstly we note that Eqs. (3.29a) and (3.31a) imply

˜ 00 (w)dF˜ (w; J)F˜ (w; J)−1 = dH˜ (J)Ω . Λ

(3.34)

˜ 00 (w)dF˜ (w; J). Multiplying this equation from the left by To prove (3.33a), use (3.31a) and (3.31b) to get A˜ (w; J)dF˜ (w; J) = Λ ˜ 00 (w) and from the right by F˜ (w; J)−1 , then we obtain (3.33a) by using (3.34) and the definitions of A˜ (w; J) and Λ˜ 00 (w). Λ To prove (3.33b), note that from (3.32a), (2.28) and (2.27) we have ˜ 00 (w)dF˜ (w; J)F˜ (w; J)−1 Ω − [Λ˜ 00 (w)dF˜ (w; J)F˜ (w; J)−1 Ω ]> , 2Ω dz00 = Λ ˜ (J) − dH˜ (J)> = −2J2 Ω dz00 and then gives (3.33b) on selecting some suitable integral and this, by using (3.34), is followed by dH constant. The proofs of (3.33c) and (3.33d) are similar to those of (3.12c) and (3.12d). 

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4. Equivalent integral equations and group properties of the ED RH transformations Firstly, noting the analytic property of X− (t; J) in (3.7) on L ∪ L− (including ∞), we have 1 2πi

Z L

X− (s; J) s(s − t)

ds = 0 ,

t ∈ L+ .

(4.1)

Substituting Eq. (3.7) into (4.1), it follows that 1 2πi

Z

F (s; J)u(s)F (v(s); J)−1 s(s − t)

L

ds = 0,

t ∈ L+ ,

(4.2)

subject to the condition F (0; J) = I. As for ED RH transformation (3.30), by condition (3.22) we have 1 2πi

Z ˜ X− (s; J) L˜

(s − w)

ds = I ,

w ∈ L˜ + .

(4.3)

Now from (3.30) we obtain 1 2πi

Z ˜ F (s; J)u˜ (s)F˜ (v˜ (s); J)−1

(s − w)

L˜

ds = I,

w ∈ L˜ + .

(4.4)

In order to show the group structure of the above ED RH transformations explicitly, from the properties of v(t), we introduce ξ(t) := v−1 (t) on the contour L and define the action of (u, ξ) on any ED function Ψ (t; J) as

(u, ξ)Ψ (t; J) := u(t)Ψ (ξ−1 (t); J) = u(t)Ψ (v(t); J).

(4.5)

Then the integral Eq. (4.2) can be rewritten as 1 2πi

Z

F (s; J)(u, ξ)F (s; J)−1

L

s(s − t)

ds = 0,

t ∈ L+ .

(4.6)

If we carry out the ED RH transformation two times successively and define

(u, ξ) : F (t; J) −→ F (t; J),

(u1 , ξ1 ) : F (t; J) −→ F (t; J),

(4.7)

then from (3.7) (or equivalently (4.6)) we have 1 2πi

Z F (s; J)(u , ξ )(u, ξ)F (s; J)−1 1 1 L

s(s − t)

ds =

=

1 2πi 1 2πi

Z F (s; J)[u (s)u(v (s))]F (v(v (s)); J)−1 1 1 1 L

s(s − t)

Z F (s; J)(u γ (u), ξ ξ)F (s; J)−1 1 ξ1 1 L

s(s − t)

ds

ds = 0 ,

t ∈ L+ ,

where we have used the homomorphism γ : {ξ} → Aut{u} defined by

γ : ξ −→ γξ ,

γξ : u(t) −→ γξ (u)(t) = u(ξ−1 (t)).

(4.8)

Thus, we have an ED RH transformation (u2 , ξ2 ) : F (t; J) → F (t; J) such that

(u2 , ξ2 ) = (u1 , ξ1 )(u, ξ) = (u1 γξ1 (u), ξ1 ξ).

(4.9)

˜ w) := v˜ −1 (w) on the contour L˜ and defining Similarly, by introducing ξ( ˜ Ψ˜ (w; J) := u˜ (w)Ψ˜ (ξ˜ −1 (w); J) = u˜ (w)Ψ˜ (v˜ (w); J), (u˜ , ξ)

(4.10)

we obtain an ED RH transformation (u˜ 2 , ξ˜ 2 ) : F˜ (w; J) → F˜ (w; J) such that

˜ = (u˜ 1 γ˜ ˜ (u˜ ), ξ˜ 1 ξ). ˜ (u˜ 2 , ξ˜ 2 ) = (u˜ 1 , ξ˜ 1 )(u˜ , ξ) ξ1

(4.11)

According to the loop group theory [32], the composition laws (4.9) and (4.11) show that the ED RH transformations (3.7) (or (4.2)) and (3.30) (or (4.4)), joined together, provide an ED representation of the semidirect product of the affine Kac–Moody group Sp\ (4, R) and the Virasoro group. As special cases, when v(t) = t, v˜ (w) = w, we obtain an ED representation of the Kac–Moody group Sp\ (4, R); when u(t) = I, u˜ (w) = I, we get an ED representation of the Virasoro group.

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5. Double-duality symmetry and quadruple RH transformations The EMDA theory under consideration has even more symmetries. To see this, we note that the Eqs. (2.11) and (2.12) with Gauss decomposition (2.10) imply that there exists a double-duality mapping D(J) as follows: ! Pˆ (J) −Qˆ (J)Pˆ (J)

ˆ (J) = D(J) : M(J) 7→ M  ◦  Pˆ (J) = T P ( J ) ,

◦2

Qˆ (J) = J V

◦ P( J )

−Pˆ (J)Qˆ (J) Qˆ (J)Pˆ (J)Qˆ (J) + J2 ρ2 Pˆ (J)−1  ◦  Q (J ) ,

,

(5.1a) (5.1b)

where the transformations T , V and overcircle operation “◦” are defined by (2.14) and (2.3) respectively. It can be directly ˆ (J) satisfies verified that M

ˆ (J)η ∗ d M ˆ (J)) = 0, d(ρ−1 M ˆ (J) M

>

(5.2)

ˆ (J), =M

ˆ (J)ηM ˆ (J) = J M

2

(5.3a)

ρ η, 2

(5.3b)

which are the same in form as (2.11) and (2.12a) and (2.12b). Thus, paralleling the above discussions, we can have the

ˆ (J), Γˆ (J), Aˆ (J), Γˆ˜ (J), Aˆ˜ (J), Fˆ (t; J), Fˆ˜ (w; J), etc. and obtain hatted partners of the Eq. (2.16) and the linear system corresponding H pair (2.22)–(2.24), (2.30)–(2.31). ˆ (J) and M(J) give the same pair of real solutions of the EMDA theory, but M ˆ (J) 6= M(J) and It should be pointed out that M ˆ (J) 6= H(J), Γˆ (J) 6= Γ (J), Γˆ˜ (J) 6= Γ˜ (J), etc. Thus, starting from a single solution M(J) of Eqs. (2.11) and (2.12), we we have H obtain two different pairs of ED-complex linear systems and two different pairs of ED RH transformations. The two pairs of ED RH transformations constitute quadruple infinite-dim symmetry groups of the EMDA theory: each of these symmetry groups is a semidirect product of the Kac–Moody Sp\ (4, R) and Virasoro groups. These results demonstrate that the EMDA theory under consideration possesses very rich symmetry structures. 6. Infinitesimal quadruple RH transformations In order to find the relationship between the results in the present paper and that in [24], we discuss the infinitesimal forms of the above RH transformations. Setting v(t) = t and considering the following infinitesimal transformation: u(t) = I + δu(t),

F (t; J) = F (t; J) + δF (t; J),

then by (2.23a), Eq. (4.2) becomes

δF (t; J)F (t; J)−1 = −

t

F (s; J)δu(s)F (s; J)−1

Z

2πi

s(s − t)

L

ds,

t ∈ L+ .

(6.1)

Noticing the properties of u(t), without loss of generality, we can select δu(s) = δ(αk) u(s) = Tα s−k (k ≥ 0), where Tα = Ta αa , Ta are generators of sp(4, R) (the Lie algebra of Sp(4, R)), αa are infinitesimal real constants. Substituting these into (6.1), we have Z −k s F (s; J)Tα F (s; J)−1 t δ(αk) F (t; J)F (t; J)−1 = − ds, t ∈ L+ , 2πi L s(s − t) P 0 k (k) 0 and then the parameterized transformation δα (t0 )F (t; J) = ∞ k=0 t δα F (t ; J ) (t ∈ L+ ) is given by

δα (t0 )F (t; J)F (t; J)−1 = − =− =

t

2πi t

Z X ∞ 0k t F (s; J)Tα F (s; J)−1 L k=0

Z

2πi t

t − t0

L

sk+1 (s − t)

F (s; J)Tα F (s; J)−1

(s − t0 )(s − t)

ds

ds

[F (t0 ; J)Tα F (t0 ; J)−1 − F (t; J)Tα F (t; J)−1 ].

(6.2)

Similarly, taking v˜ (w) = w and considering the infinitesimal transformation u˜ (w) = I + δ˜ u˜ (w), F˜ (w; J) = F˜ (w; J) + δ˜ F˜ (w; J), then (4.4) becomes

δ˜ F˜ (w; J)F˜ (w; J)−1 = −

1 2π i

Z ˜ F (s; J)δ˜ u˜ (s)F˜ (s; J)−1 L˜

(s − w)

ds,

w ∈ L˜ + .

(6.3)

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Selecting δ˜ u˜ (s) = δ˜ (αk) u˜ (s) = Tα s−k (k ≥ 1) and denoting the corresponding δ˜ F˜ (w; J) by δ˜ (αk) F˜ (w; J), then from (6.3), the P 0 0 k ˜ (k) ˜ ˜ parameterized infinitesimal transformation δ˜ α (w0 )F˜ (w; J) = ∞ k=1 w δα F (w; J ) (w ∈ L+ ) is given by

δ˜ α (w0 )F˜ (w; J)F˜ (w; J)−1 =

w0

[F˜ (w0 ; J)Tα F˜ (w0 ; J)−1 − F˜ (w; J)Tα F˜ (w; J)−1 ].

w − w0

(6.4)

The ED RH transformations (4.2) and (4.4), in fact, contain more symmetries of the EMDA theory. To show this, we need “cross” infinitesimal variations δ˜ F (t; J) and δF˜ (w; J) brought about by u(t) = I + δ˜ u(t) and u˜ (w) = I + δu˜ (w), respectively. Considering the relation between t and w in (2.22) and (2.30), we can select δ˜ (αk) u(t) = Tα tk (k ≥ 1) and δ(αk) u˜ (w) = Tα wk (k ≥ 0). Correspondingly, Eqs. (6.1) and (6.3) give, respectively, Z k t s F (s; J)Tα F (s; J)−1 δ˜ (αk) F (t; J)F (t; J)−1 = − ds; 2πi L s(s − t) Z k˜ 1 s F (s; J)Tα F˜ (s; J)−1 δ(αk) F˜ (w; J)F˜ (w; J)−1 = − ds. 2πi L˜ (s − w) To obtain explicit expressions for the corresponding parameterized “cross” infinitesimal transformations δ˜ α (w)F (t; J) = P∞ k (k) P∞ k ˜ (k) ˜ ˜ ˜ k=1 w δα F (t ; J ) and δα (t )F (w; J ) = k=0 t δα F (w; J ), we note that since F (t ; J ), F (w; J ) have different analytic properties, from (2.22)–(2.24), (2.30)–(2.31) we can set F˜ (w; J)|w= 1 = t1/2 F (t; J),

F (t; J)|t= 1 = w1/2 F˜ (w; J).

t

w

(6.5)

Thus we have

δ˜ α (w)F (t; J)F (t; J)−1 = −

tw

2πi

1 − tw

δα (t)F˜ (w; J)F˜ (w; J)−1 = − =

ds

(1 − sw)(s − t) ˜ [F (w; J)Tα F˜ (w; J)−1 − F (t; J)Tα F (t; J)−1 ]; L

tw

=

F (s; J)Tα F (s; J)−1

Z

Z ˜ F (s; J)Tα F˜ (s; J)−1

1 2πi 1

L˜

1 − tw

(1 − ts)(s − w)

(6.6)

ds

[F (t; J)Tα F (t; J)−1 − F˜ (w; J)Tα F˜ (w; J)−1 ].

(6.7)

Next we consider the cases u(t) = I and u˜ (w) = I of the transformations (4.2) and (4.4). We first calculate infinitesimal transformations brought about by v(t) = t + ∆(t),

(6.8a)

˜ (w), v˜ (w) = w + ∆

(6.8b)

˜ (w) are infinitesimal functions of t and w, respectively. For (6.8a), we have where ∆(t) and ∆ F (t; J) = F (t; J) + ∆F (t; J),

F (v(t); J)−1 = F (t; J)−1 + ∂t [F (t; J)−1 ]∆(t).

Substituting u(t) = I and (6.9) into (4.2), we obtain Z ˙ t F (s; J)F (s; J)−1 ∆F (t; J)F (t; J)−1 = ∆(s)ds, 2πi L s(s − t)

t ∈ L+ .

(6.9)

(6.10)

Noticing the properties of v(t), without loss of generality, we select ∆(s) = ∆(σk) (s) = σ s1−k (k ≥ 0) and denote the corresponding ∆F (t; J) by ∆σ(k) F (t; J), where σ is an infinitesimal real constant; then from (6.10), the parameterized P 0 k (k) 0 transformation ∆σ (t0 )F (t; J) = ∞ k=0 t ∆σ F (t ; J )(t ∈ L+ ) is given by Z ˙ sF (s; J)F (s; J)−1 σt ∆σ (t0 )F (t; J)F (t; J)−1 = ds 2πi L (s − t0 )(s − t)

=

σt t − t0

[tF˙ (t; J)F (t; J)−1 − t0 F˙ (t0 ; J)F (t0 ; J)−1 ].

(6.11)

Similarly, for (6.8b) and (4.4), we obtain

˜ F˜ (w; J)F˜ (w; J)−1 = ∆

1 2πi

Z ˙˜ F (s; J)F˜ (s; J)−1

(s − w)

L˜

˜ (s)ds, ∆

w ∈ L˜ + .

(6.12)

˜ (s) = ∆ ˜ (k) (s) = s1−k (k ≥ 1) and denoting the corresponding ∆ ˜ F˜ (w; J) by ∆ ˜ (k) F˜ (w; J) ( is an infinitesimal real Selecting ∆ P∞ 0 ˜ 0 k ˜ (k) ˜ ˜ constant), then from (6.12), the parameterized transformation ∆ (w )F (w; J) = k=1 w ∆ F (w; J)(w0 ∈ L˜ + ) is given by ˜  (w0 )F˜ (w; J)F˜ (w; J)−1 = ∆

w0 w − w0

[wF˙˜ (w; J)F˜ (w; J)−1 − w0 F˜˙ (w0 ; J)F˜ (w0 ; J)−1 ].

(6.13)

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˜ F (t; J) and ∆F˜ (w; J) brought about by variations v(t) = As before, we also need “cross” infinitesimal transformations ∆ ˜ (t), v˜ (w) = w + ∆(w). In these cases, the ED RH transformations (4.2) and (4.4) give t+∆ ˜ F (t; J)F (t; J)−1 = ∆

Z ˙ F (s; J)F (s; J)−1

t

2πi

∆F˜ (w; J)F˜ (w; J)−1 =

1

s(s − t)

L

˜ (s)ds; ∆

Z ˙˜ F (s; J)F˜ (s; J)−1

2πi

(s − w)

L˜

(6.14)

∆(s)ds.

(6.15)

˜ (t) = ∆ ˜ (k) (t) = −t1+k (k ≥ 1); ∆(w) = ∆(σk) (w) = −σ w1+k (k ≥ 0) and denote the corresponding We select ∆ ˜ (k) F (t; J), ∆(σk) F˜ (w; J), respectively. In addition, from relations (6.5), we have transformations by ∆ 

[F˙˜ (w; J)F˜ (w; J)−1 ]|w= 1 = −t tF˙ (t; J)F (t; J)−1 +

1



;

2 (6.16)   1 ˙ − 1 − 1 [F˙ (t; J)F (t; J) ]|t= 1 = −w wF˜ (w; J)F˜ (w; J) + . w 2 P P∞ k (k) k ˜ (k) ˜ ˜ ˜  (w)F (t; J) = ∞ Thus the parameterized transformations ∆ k=1 w ∆ F (t ; J ) and ∆σ (t )F (w; J ) = k=0 t ∆σ F (w; J ) are given, respectively, by t

˜  (w)F (t; J)F (t; J)−1 = − ∆ =

tw sF˙ (s; J)F (s; J)−1 ds 2πi L (1 − sw)(s − t)  tw 1 ˙ tF˙ (t; J)F (t; J)−1 + wF˜ (w; J)F˜ (w; J)−1 + ; Z

tw − 1

∆σ (t)F˜ (w; J)F˜ (w; J)−1 = −

=

(6.17)

2

˙ σ sF˜ (s; J)F˜ (s; J)−1 ds 2πi L˜ (1 − st)(s − w)  σ 1 ˙ wF˜ (w; J)F˜ (w; J)−1 + tF˙ (t; J)F (t; J)−1 + . Z

tw − 1

(6.18)

2

Similarly, we can obtain the infinitesimal ED RH transformations of the hatted potentials. Finally, we give the corresponding infinitesimal transformations of ρ, z. Write

η± := z ± iρ,

η0± := z0 ± iρ0 ,

η00± := z00 ± iρ00 ,

t± := (2η± )−1 ,

w± := 2η± .

(6.19)

Then from (3.3a) and (6.8a), we have (k)

∆(σk) η± = (η0± − η± )(σ) = ∆σ (t)η± =

∞ X k=0

tk ∆(σk) η± =

1 2 2t±

∆σ(k) (t± ) = ση± (2η± )k ,

ση±

∞ X

tk (2η± )k =

k=0

σ

k ≥ 0,

η± 1 − 2tη±

,

and therefore ∆σ (t)z =

σ [z(1 − 2tz) − 2tρ2 ], λ(t)2

∆σ (t)ρ =

σ ρ. λ(t)2

(6.20)

Similarly, from (3.20a), (6.19) and (6.8b), we obtain 1 (k)  1−k (k) ˜ (k) η± = (η00± − η± )() ˜ (w± ) = − w± ∆ =− ∆ = −η± (2η± )−k , 2  2 ∞ ∞ X X η± ˜  (w)η± = ˜ (k) η± = −η± , ∆ wk ∆ wk (2η± )−k = w w − 2η± k=1 k=1

k ≥ 1,

and thus

˜  (w)z = ∆

w [z(w − 2z) − 2ρ2 ], ˜ w)2 λ(

˜  (w)ρ = ∆

w2 ρ. ˜ w)2 λ(

(6.21)

Eqs. (6.2), (6.4), (6.6), (6.7), (6.11), (6.13), (6.17) and (6.18) and their hatted partners give exactly the same infinitesimal transformations of F (t; J), F˜ (w; J), Fˆ (t; J), Fˆ˜ (w; J) and the associated H-potentials (by Eq. (2.23) and their hatted partners) as constructed in [24], while (6.20) and (6.21) give the same infinitesimal transforms of ρ, z as in [24]. These results demonstrate that the two pairs of ED RH transformations in this paper provide exponentiations of all the infinitesimal symmetry transformations given in [24].

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Y.-J. Gao / Journal of Geometry and Physics 58 (2008) 1030–1042

7. Summary and discussion The symmetry structures of the SAS EMDA theory are further studied. By using the so-called ED-complex function method, the usual RH problem is extended to an ED-complex formulation. Associating with the pair of ED-complex HEtype linear systems (2.22)–(2.24), (2.30)–(2.31), we construct a pair of ED RH transformations (3.7), (3.30) (or equivalently (4.2), (4.4)). In addition, owing to the double-dual symmetry stated in Section 5, we can construct another pair of (socalled hatted) ED RH transformations. These ED RH transformations generate new F , H-potentials from old ones, and give quadruple infinite-dim symmetry transformation groups of the EMDA theory considered. Each of these symmetry groups is verified to have the structure of a semidirect product of the complete Kac–Moody group Sp\ (4, R) and the Virasoro group. 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