Noncommutative Solitons and Quasideterminants

Noncomm utativ e Soliton s and Quasid eterminan ts Masashi Hamana k a 1 Nagoya University, Dep artment of Mathematics, Chikusa-ku, Nagoya, 464-8602, JAP AN Abstract W e discuss an extension of soliton theory and in tegrable sys tems to noncomm utativ e spaces, fo cusing on in tegrable asp ects of noncomm utativ e a nti-self-dual Y ang-Mills equa- tions. W e giv e a wide class of exact solutions by solving a Riemann-Hilb ert pro blem for the A tiy ah-W ard ansatz and prese nt B¨ ac klund transformatio ns for the G = U (2) noncom- m utativ e anti-self-dual Y ang-Mills equations. W e find that one kind of noncomm utative determinan t, quasideterminan ts, play crucial roles in the construction of noncomm utativ e solutions. W e also discuss reduction of a noncomm utative anti-self-dual Y ang-Mills equa- tion to noncomm utativ e in tegrable equations. This is partia lly base d on a collab oration with C. R. Gilson and J. J. C. Nimmo (Gla sgo w). 1 E-mail: hamamnak a@math.nag oy a-u.ac.jp 1 In tro duction The extension of integrable systems and solito n theories to non-comm utativ e (NC) space- times 2 has b een studied by man y a ut ho r s ov er the past couple of y ears and v arious kinds of in tegrable-lik e prop erties hav e b een reve aled. (F or reviews , see [1]- [7].) This is partially motiv ated b y the recent dev elopmen ts of noncomm utativ e g auge theories on D -branes. In the noncommutativ e ga uge theories, the noncommutativ e extension corresp onds to the presence of bac kground flux, and in the effectiv e theory of D-branes, noncomm utative solitons can b e iden tified with the lo w er-dimensional D-branes. (F or reviews, see e.g. [8]- [11 ].) This makes it po ssible to rev eal some aspects of D -brane dynamics, suc h as tac h y on condensations, by constructing exact noncomm utative solitons and studying their prop erties. Most of noncomm utativ e in tegrable eq uations suc h as noncomm utativ e Kortew eg-de V ries (KdV) equations belong, apparen tly , not to gauge t heories, but to s calar theories . Ho w ev er, it has no w b een pro v ed that they can b e deriv ed from noncomm utativ e an ti- self-dual (ASD) Y ang-Mills (YM) eq uations b y reduction (e.g. [12, 13]), whic h w as first conjectured explicitly b y the author and K. T oda [14]. (The original comm utative one w as prop osed b y R. W ard [15 ] and hence this conjecture is sometimes called the non- c omm utative War d’ s c onje ctur e .) As noncommutativ e an ti-self-dual Y ang-Mills equations b elong to gauge theories, lo we r-dimensional man y integrable equations m ust ha v e phy sical corresp ondence (in the background flux), and therefore analysis of exact soliton solutions of noncomm utative in t egrable equ atio ns could b e applied to the c orresp o nding phys ical situations in the framew ork of N = 2 string theories [16] - [19]. (In this con text, the signature is not Euclidean (+ + + + ) but ultrahyperb olic (+ + − − ) and the N = 2 string theory liv es in this signa t ure.) F urthermore, in tegrable aspects of anti-self-dual Y ang-Mills equation can b e under- sto o d fro m the geometrical framew ork of the twistor the ory . Via the W ard’s conjecture, the t wistor theory gives a new g eometrical viewp oin t in to the lo w er-dimensional in te- grable equations and some classification can b e carried out in such a w ay . These results are summarized in the b o ok of Mason and W o o dhouse elegantly [20]. (See also [21, 22].) In this pap er, w e discuss in tegrable asp ects of the noncomm utativ e anti-se lf- dual Y ang- Mills equations from the vie wp oint of the noncomm utativ e twistor theory . W e giv e a series of no ncomm utativ e Atiy ah- W a rd ansatz solutions b y solving a noncomm utative vers ion of the Riemann-Hilb ert problem. The solutions include not only noncomm utative instan tons (with finite action) but also noncommu tative non-linear pla ne w av es and so on (with infinite action). W e also find that noncommutativ e determinan ts of a particular kind, the quasideterminan ts, pla y crucial roles in construction of exact solutions and presen t a direct pro of of the results of the B¨ ac klund transformation and the solutions generated. These are due to collab oration with C. Gilson and J. Nimmo (G lasgo w) [23, 24]. Finally we giv e an example of noncomm utativ e W ard’s conjecture, r eduction of the noncomm utativ e an ti-self-dual Y ang-Mills equation in to the noncomm utative KdV equa- tion via the noncomm utativ e toroidal K dV equation. The reduce d equation actually has 2 In the presen t pape r, the w ord “noncommutativ e” alwa ys refer s to generaliza tion to nonc ommut ative spaces, not to non-a be lian a nd so on. 1 in tegrable-lik e prop erties suc h as infinite conserv ed quan tities, exact N-soliton solutions and so on. These results would lead to a kin d of classification of no ncommutativ e inte- grable equations from a geometrical vie wp oint and to applications to the corresp o nding ph ysical situations and geometry also. 2 Noncomm utativ e an ti-self-dual Y ang-Mills equation s In this section, we review some asp ects of the noncomm utativ e an ti-self-dual Y ang- Mills equation and establish notations. 2.1 Noncomm utativ e gauge theories Noncomm utativ e spaces are defined b y the noncomm utativit y of the co ordinates: [ x µ , x ν ] = iθ µν , (2.1) where θ µν are real constan ts called the non c ommutative p ar am eters . The noncommutativ e parameter is a n ti-symmetric with resp ect to µ, ν : θ ν µ = − θ µν and the rank is ev en. This relation lo oks like the canonical comm uta tion relation in quan tum mec hanics and leads to “space-space uncertaint y relation.” Hence singularities whic h exist on comm utativ e spaces could resolve on noncommutativ e spaces. This is one of t he prominen t features of noncomm uta tiv e field theories and yields v a rious new phy sical ob jects suc h as U (1) instan tons. Noncomm utativ e field theories are given b y the exc hange of ordinary pro ducts in the comm utativ e field theories for the star-pro ducts a nd realized as deformed theories from the comm utativ e ones . The ordering of non-linear terms a r e determined b y some additiona l conditions. The star-pro duct is defined for ordinary fields on c ommutativ e space s. F or Euclidean spaces, it is explicitly giv en b y f ⋆ g ( x ) := exp  i 2 θ µν ∂ ( x ′ ) µ ∂ ( x ′′ ) ν  f ( x ′ ) g ( x ′′ )    x ′ = x ′′ = x = f ( x ) g ( x ) + i 2 θ µν ∂ µ f ( x ) ∂ ν g ( x ) + O ( θ 2 ) , (2.2) where ∂ ( x ′ ) µ := ∂ /∂ x ′ µ and so on. This explicit represen tation is known as the Moyal pr o duct [25]. The star-pro duct has asso ciativit y: f ⋆ ( g ⋆ h ) = ( f ⋆ g ) ⋆ h , and returns bac k to the o rdinary pro duct in the comm utativ e limit: θ µν → 0. The mo dification of the pro duct mak es the ordinar y spatial co ordinate “noncomm utativ e,” that is, [ x µ , x ν ] ⋆ := x µ ⋆ x ν − x ν ⋆ x µ = iθ µν . Here are the noncomm utativ e Kadom tsev-P etviash vili (KP) and KdV equations: • NC K P equation in (2 + 1)-dimension (typic ally [ x, y ] ⋆ = iθ or [ t, x ] ⋆ = iθ ) ∂ u ∂ t = 1 4 ∂ 3 u ∂ x 3 + 3 4  ∂ u ∂ x ⋆ u + u ⋆ ∂ u ∂ x  + 3 4 ∂ − 1 x ∂ 2 u ∂ y 2 − 3 4  u, ∂ − 1 x ∂ u ∂ y  ⋆ , (2.3) 2 where t a nd x, y are time and spatial co ordinates, resp ectiv ely , and ∂ − 1 x f ( x ) = R x dx ′ f ( x ′ ). • NC K dV equation in (1 + 1)-dimension ([ t, x ] ⋆ = iθ ) ∂ u ∂ t = 1 4 ∂ 3 u ∂ x 3 + 3 4  ∂ u ∂ x ⋆ u + u ⋆ ∂ u ∂ x  . (2.4) The ordering of v ariables in non-linear terms is crucial to preserv e some sp ecial integrable prop erties and determined in the Lax formalism. (F or a review, see [3].) W e note that the fields themselv es tak e c-num b er v alues as usual and the differen tiation and the inte gra t ion for them a r e w ell-defined as usual, for example, ∂ µ ⋆ ∂ ν = ∂ µ ∂ ν , and the w edge pro duct of λ = λ µ ( x ) dx µ and ρ = ρ ν ( x ) dx ν is λ µ ⋆ ρ ν dx µ ∧ dx ν . Noncomm utativ e ga uge theories a r e defined in this wa y b y imp osing noncomm utativ e v ersion o f the gauge symmetry , where t he gauge transformation is defined as follo ws: A µ → g − 1 ⋆ A µ ⋆ g + g − 1 ⋆ ∂ µ g , (2.5) where g is an eleme nt of the ga uge gr o up G (The inv erse is assumed to exist in the sense of t he star pro duct in this pap er.) This is sometimes called the star gauge tr ans formation . W e note that b ecause of the noncomm utativit y , the comm utator terms in field strength are a lw a ys needed ev en when the gaug e gr o up is ab elian in order to preserv e the star gauge symmetry . This U (1) par t of t he gauge group actually plays crucial r o les in general. W e note that b ecause of the noncomm utativity of matrix elemen ts, cyclic symmetry of traces is brok en in general: T r A ⋆ B 6 = T r B ⋆ A. (2.6) Therefore, gauge in v aria n t quan tities b ecomes hard to define on no ncommutativ e spaces. 2.2 Noncomm utativ e an ti-self-dual Y ang-Mills equations Let us consider Y ang-Mills theories in 4-dimens ional noncomm utative spaces whose real co ordinates of the space are denoted by ( x 0 , x 1 , x 2 , x 3 ), where the gauge group is GL ( N , C ). Here, w e follo w the con v en tion in [20]. First, w e in tro duce double n ull co o rdinates of 4- dimensional space a s follo ws ds 2 = 2( dz d ˜ z − dw d ˜ w ) , (2.7) W e can reco v er v ario us kind of real spaces by putt ing the corresp onding realit y conditio ns on the double n ull co ordinates z , ˜ z , w , ˜ w as follo ws: • Euclidean Space ( ¯ w = − ˜ w ; ¯ z = ˜ z ): An example is  ˜ z w ˜ w z  = 1 √ 2  x 0 + ix 1 − ( x 2 − ix 3 ) x 2 + ix 3 x 0 − ix 1  . (2.8) 3 • Mink ows ki Space ( ¯ w = ˜ w ; z and ˜ z are real.): An example is  ˜ z w ˜ w z  = 1 √ 2  x 0 + x 1 x 2 − ix 3 x 2 + ix 3 x 0 − x 1  . (2.9) • Ultrahyperb olic Space ( ¯ w = ˜ w ; ¯ z = ˜ z ): Example are  ˜ z w ˜ w z  = 1 √ 2  x 0 + ix 1 x 2 − ix 3 x 2 + ix 3 x 0 − ix 1  , or z , ˜ z , w , ˜ w ∈ R . (2.10) The co or dinate v ectors ∂ z , ∂ z .∂ ˜ w , ∂ ˜ z form a null tetrad and are repre sen ted explicitly as: ∂ z = 1 √ 2  ∂ ∂ x 0 + i ∂ ∂ x 1  , ∂ ˜ z = 1 √ 2  ∂ ∂ x 0 − i ∂ ∂ x 1  , ∂ w = 1 √ 2  ∂ ∂ x 2 + i ∂ ∂ x 3  , ∂ ˜ w = 1 √ 2  ∂ ∂ x 2 − i ∂ ∂ x 3  . (2.11) F or the Euclidean and ultrah yp erb o lic signatures, the Ho dge dual op erator ∗ satisfies ∗ 2 = 1 and hence the s pace of 2-forms β decomposes in to the direct sum of eigen v alues of ∗ with eigen v alues ± 1, that is, s elf-dual (SD ) part ( ∗ β = β ) a nd an ti-self-dual (ASD) part ( ∗ β = − β ). F rom now on, w e treat these tw o signatures. T ypical examples of self-dual f orms are α = dw ∧ dz , ˜ α = d ˜ w ∧ d ˜ z , ω = dw ∧ d ˜ w − dz ∧ d ˜ z, (2.12) and those o f an ti-self-dual forms are dw ∧ d ˜ z , d ˜ w ∧ dz , dw ∧ d ˜ w + d z ∧ d ˜ z . (2.13) The noncomm ut a tiv e an ti-self-dual Y ang- Mills equation is derive d from the compati- bilit y condition of the f ollo wing linear system: L ⋆ ψ := ( D w − ζ D ˜ z ) ⋆ ψ = ( ∂ w + A w − ζ ( ∂ ˜ z + A ˜ z )) ⋆ ψ ( x ; ζ ) = 0 , M ⋆ ψ := ( D z − ζ D ˜ w ) ⋆ ψ = ( ∂ z + A z − ζ ( ∂ ˜ w + A ˜ w )) ⋆ ψ ( x ; ζ ) = 0 , (2.14) where A z , A w , A ˜ z , A ˜ w and D z , D w , D ˜ z , D ˜ w denote gauge fields a nd cov arian t deriv atives in the Y ang-Mills theory , resp ectiv ely . The constan t ζ ∈ C P 1 is called the sp e ctr al p ar ameter . The compatible condition [ L, M ] ⋆ = 0, giv es rise to a quadratic p o lynomial o f ζ and eac h co efficien t yields the follo wing equations: F w z = ∂ w A z − ∂ z A w + [ A w , A z ] ⋆ = 0 , F ˜ w ˜ z = ∂ ˜ w A ˜ z − ∂ ˜ z A ˜ w + [ A ˜ w , A ˜ z ] ⋆ = 0 , F z ˜ z − F w ˜ w = ∂ z A ˜ z − ∂ ˜ z A z + ∂ ˜ w A w − ∂ w A ˜ w + [ A z , A ˜ z ] ⋆ − [ A w , A ˜ w ] ⋆ = 0 , (2.15) whic h are equiv alen t to the noncommutativ e a nti-self-dual Y ang-Mills equations F µν = − ∗ F µν in the r eal represe ntation. 4 Gauge transformations act on the linear system (2.14) as L 7→ g − 1 ⋆ L ⋆ g , M 7→ g − 1 ⋆ M ⋆ g , ψ 7→ g − 1 ⋆ ψ , g ∈ G. (2.16) W e note that the solution ψ ( N × N matrix) of the linear system (2.14) is not regular at ζ = ∞ because of Liouville’s theorem. (If it is regular, then the gauge fie lds become flat.) Hence we hav e to consider another linear system on a nother lo cal patc h in ζ ∈ C P 1 whose co o r dinate is ˜ ζ = 1 / ζ as ˜ L ⋆ ˜ ψ := ˜ ζ D w ⋆ ˜ ψ − D ˜ z ⋆ ˜ ψ = 0 , ˜ M ⋆ ˜ ψ := ˜ ζ D z ⋆ ˜ ψ − D ˜ w ⋆ ˜ ψ = 0 . (2.17) The compatibility condition of this system also giv es rise to the anti-se lf- dual Y ang-Mills equation. 2.3 Noncomm utativ e Y ang’s equations and J , K -matrices Here w e discuss the p oten tial forms of the noncommutativ e an ti- self-dual Y ang-Mills equa- tions suc h as noncomm utativ e J, K -matrix formalisms and t he noncommu tative Y a ng’s equation, whic h is already presen ted b y e.g. K. T ak asaki [26]. Let us first discuss the J -matrix formalism of the noncommutativ e a nti-self-dual Y ang- Mills equation. The first equation of noncomm utative anti-se lf- dual Y ang-Mills equation (2.15) is the compatible condition of the linear system D z ⋆ h = 0 , D w ⋆ h = 0, where h is a N × N matrix. Hence w e get A z = − ( ∂ z h ) ⋆ h − 1 , A w = − ( ∂ w h ) ⋆ h − 1 . (2.18) Similarly , the second equation of noncomm utativ e an ti-self-dual Y ang-Mills equation (2.15) leads to A ˜ z = − ( ∂ ˜ z ˜ h ) ⋆ ˜ h − 1 , A ˜ w = − ( ∂ ˜ w ˜ h ) ⋆ ˜ h − 1 , (2.19) where ˜ h is a lso a N × N matrix. W e note that h ( x ) = ψ ( x, ζ = 0) , ˜ h ( x ) = ˜ ψ ( x, ζ = ∞ ). By defining a new matrix J = ˜ h − 1 ⋆ h , the third equation o f the noncommutativ e an ti-self-dual Y ang- Mills equation (2.1 5) b ecomes the noncommutativ e Y ang’s equation ∂ z ( J − 1 ⋆ ∂ ˜ z J ) − ∂ w ( J − 1 ⋆ ∂ ˜ w J ) = 0 , (2.20) or equiv alen tly , ∂  J − 1 ⋆ ˜ ∂ J  ∧ ω = 0 . (2.21) where ∂ = dw ∂ w + dz ∂ z , ˜ ∂ = d ˜ w ∂ ˜ w + d ˜ z ∂ ˜ z ω is the same one as in (2.12). Gauge transformations act on h a nd ˜ h as h 7→ g − 1 h, ˜ h 7→ g − 1 ˜ h, g ∈ G . (2.22) 5 Hence the Y ang’s J -matrix is ga uge inv arian t while the matrices h and ˜ h are gauge dep enden t. In this pap er, w e sometimes use the f o llo wing g auge for G = GL (2): h MW =  f 0 e 1  , ˜ h MW =  1 g 0 b  , (2.23) then J = ˜ h − 1 MW ⋆ h MW =  f − g ⋆ b − 1 ⋆ e − g ⋆ b − 1 b − 1 ⋆ e b − 1  . whic h is called the Maso n -Wo o dhouse g auge . There is another p ot en tial form of the no ncomm utativ e anti-self-dual Y ang-Mills equa- tion, kno wn as the K -matrix formalism . In the g auge A w = A z = 0, the third equation of (2.15) b ecomes ∂ z A ˜ z − ∂ w A ˜ w = 0 . This implies the existence of a p otential K suc h that A ˜ z = ∂ w K, A ˜ w = ∂ z K . Then the second equation of (2.15 ) b ecomes ∂ z ∂ ˜ z K − ∂ w ∂ ˜ w K + [ ∂ w K, ∂ z K ] ⋆ = 0 . (2.24) Then, w e get ψ = 1 + ζ K + O ( ζ 2 ) , ˜ ψ = J − 1 + O ( ˜ ζ ) , (2.25) and A ˜ w = J − 1 ⋆ ∂ ˜ w J = ∂ z K, A ˜ z = J − 1 ⋆ ∂ ˜ z J = ∂ w K . This g a uge is suitable for the discussion of the ( binar y) Darb oux transformations for the (nonc ommutativ e) an ti-self- dual Y ang-Mills equations [27, 28, 29]. 3 Twistor des cription of noncomm utativ e an ti-s elf- dual Y ang-Mil ls equations In this section, we construct wide class of exact solutions of the noncomm utative anti-self- dual Y ang-Mills equations from the geometrical viewp oin t of the noncommutativ e t wistor theory . The noncomm utat ive twis tor theory has b een dev elop ed b y sev eral authors and mathematical foundations are established [26, 30, 31, 32]. The tw istor t heory is based on a corresp ondence b et w een (complexifie d) space-time co ordinates ( z , ˜ z , w , ˜ w ) and t wistor c o ordinates ( λ, µ, ζ ) whic h are lo cal co ordinates of a 3-dimensional complex pro jective space (t wistor space). The explicit relation is called the incidenc e r elation , and represen ted as fo llo ws: λ = ζ w + ˜ z , µ = ζ z + ˜ w , (3.1) whic h implies that for an y t wistor f unction f ( λ, µ , ζ ), lf ( λ, µ, ζ ) := ( ∂ w − ζ ∂ ˜ z ) f ( λ, µ, ζ ) = 0 , mf ( λ, µ, ζ ) := ( ∂ z − ζ ∂ ˜ w ) f ( λ, µ, ζ ) = 0 . (3.2) 6 3.1 Noncomm utativ e P enrose-W ard transformation F or the a n ti-self-dual Y ang-Mills theory , t here is a one-to-one corresp o ndence b et w een solutions o f the an ti-self-dual Y ang-Mills equation and holo morphic v ector bundles on the twistor space. The for mer is give n by solutions ψ , ˜ ψ of the linear systems (2.14) and (2.17). The la t t er is giv en b y pat ching matrices P of the holomorphic v ector bundles. The explicit correspondence is called the Penr ose-War d c orr esp ond e n c e . Here w e just need the Moy al- deformed P enrose-W ard correspo ndence b etw een t he an ti-self-dual Y ang- Mills solution ψ , ˜ ψ and the patc hing matrix P . F rom given ψ a nd ˜ ψ , the patc hing matrix P is constructed as P ( ζ w + ˜ z , ζ z + ˜ w , ζ ) = ˜ ψ − 1 ( x ; ζ ) ⋆ ψ ( x ; ζ ) . (3.3) (Here w e note that ψ ( x ; ζ ) is regular w.r.t. ζ around ζ = 0 a nd ˜ ψ ( x ; ζ ) is regular w.r.t. ˜ ζ around ˜ ζ = 0 or equiv alently ζ = ∞ .) Con v ersely , if there exists the factor izat io n (3.3) in to ψ and ˜ ψ for a given P where ψ ( x ; ζ ) is regular w.r.t. ζ around ζ = 0 and ˜ ψ ( x ; ζ ) is regular w.r.t. ˜ ζ around ˜ ζ = 0, then the ψ a nd ˜ ψ are solutions of linear systems (2.14) and (2.17) for the noncomm utative anti-se lf- dual Y ang-Mills equations. (This f a ctorization problem is called the Riemann-Hilb ert pr oblem and solv ed formally [2 6 ]. Noncommutativit y can b e in tro duced into only t w o v a r iables ζ w + ˜ z and ζ z + ˜ w . Then ζ is a comm utativ e v ariable and the w ay s of solving the Riemann-Hilb ert problem b ecome similar to comm utative ones. ) 3.2 Noncomm utativ e A tiy ah-W ard ansatz solutions for G = GL (2) F rom no w on, w e restrict ourselv es to G = GL (2). F or this gaug e group, w e can tak e a simple ansatz for the Patc hing matrix P , whic h is called the A tiyah-War d an s atz in the comm utativ e c ase [3 3]. Noncomm utativ e g eneralization of this ansatz is stra ig h tforward and actually leads to a solution of the factorization problem. The l -th order noncomm u- tativ e Atiy ah-W ard ansatz is specified b y the following form of the patc hing matrix up to constan t matrix actions fro m b oth sides ( l = 0 , 1 , 2 , · · · ): P l ( x ; ζ ) =  0 ζ − l ζ l ∆( x ; ζ )  . (3.4) W e note that P satisfies eq. ( 3.2) and hence, the Lauren t expansion of ∆ w.r.t. ζ ∆( x ; ζ ) = ∞ X i = −∞ ∆ i ( x ) ζ − i , (3.5) giv es rise to the fo llowing recurrence relations in the co efficien ts as follo ws ∂ ∆ i ∂ z = ∂ ∆ i +1 ∂ ˜ w , ∂ ∆ i ∂ w = ∂ ∆ i +1 ∂ ˜ z . (3.6) 7 The w av e functions ψ and ˜ ψ can b e expanded by ζ and ˜ ζ = 1 / ζ , resp ectiv ely: ψ = h + O ( ζ ) =  h 11 + P ∞ i =1 a i ζ i h 12 + P ∞ i =1 b i ζ i h 21 + P ∞ i =1 c i ζ i h 22 + P ∞ i =1 d i ζ i .  , ˜ ψ = ˜ h + O ( ˜ ζ ) =  ˜ h 11 + P ∞ i =1 ˜ a i ˜ ζ i ˜ h 12 + P ∞ i =1 ˜ b i ˜ ζ i ˜ h 21 + P ∞ i =1 ˜ c i ˜ ζ i ˜ h 22 + P ∞ i =1 ˜ d i ˜ ζ i .  . (3.7) No w let us solv e the factorization problem ˜ ψ ⋆ P = ψ for the noncomm utative A tiy ah- W a r d ansatz. This is concretely written down as  ˜ ψ 11 ˜ ψ 12 ˜ ψ 21 ˜ ψ 22  ⋆  0 ζ − l ζ l ∆( x ; ζ )  =  ψ 11 ψ 12 ψ 21 ψ 22  , (3.8) that is, ˜ ψ 12 ζ l = ψ 11 , ˜ ψ 22 ζ l = ψ 21 , (3.9) ˜ ψ 11 ζ − l + ˜ ψ 12 ⋆ ∆ = ψ 12 , ˜ ψ 21 ζ − l + ˜ ψ 22 ⋆ ∆ = ψ 22 . (3.10) F rom Eqs. (3.7) and (3.9) w e find that some entries b ecome p olynomials w.r.t. ζ : ψ 11 = h 11 + a 1 ζ + a 2 ζ 2 + · · · a l − 1 ζ l − 1 + ˜ h 12 ζ l , ψ 21 = h 21 + b 1 ζ + b 2 ζ 2 + · · · b l − 1 ζ l − 1 + ˜ h 22 ζ l , ˜ ψ 12 = ˜ h 12 + a l − 1 ζ − 1 + a l − 2 ζ − 2 + · · · + a 1 ζ 1 − l + h 11 ζ − l , ˜ ψ 22 = ˜ h 22 + b l − 1 ζ − 1 + b l − 2 ζ − 2 + · · · + b 1 ζ 1 − l + h 21 ζ − l , (3.11) and so o n. By substituting these relations in to eq. (3.10), w e get sets of equations for h and ˜ h in the co efficien ts of ζ 0 , ζ − 1 , · · · , ζ − l : ( h 11 , a 1 , · · · , a l − 1 , ˜ h 12 ) ⋆ D l +1 = ( − ˜ h 11 , 0 , · · · , 0 , h 12 ) , ( h 21 , c 1 , · · · , c l − 1 , ˜ h 22 ) ⋆ D l +1 = ( − ˜ h 21 , 0 , · · · , 0 , h 22 ) , (3.12) where D l :=      ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ 1 ∆ 0 · · · ∆ 2 − l . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0      . (3.13) These linear equations can b e solved by taking inv erse matrix of D l +1 from righ t side and can b e rewritten in terms of quasideterminan ts (F or a brief review, see App endix A.): h 11 = h 12 ⋆ | D l +1 | − 1 1 ,l +1 − ˜ h 11 ⋆ | D l +1 | − 1 1 , 1 , h 21 = h 22 ⋆ | D l +1 | − 1 1 ,l +1 − ˜ h 21 ⋆ | D l +1 | − 1 1 , 1 , ˜ h 12 = h 12 ⋆ | D l +1 | − 1 l +1 ,l +1 − ˜ h 1 , 1 ⋆ | D l +1 | − 1 l +1 , 1 , ˜ h 22 = h 22 ⋆ | D l +1 | − 1 l +1 ,l +1 − ˜ h 21 ⋆ | D l +1 | − 1 l +1 , 1 . (3.14) 8 If w e tak e the Mason-W o o dhouse gauge (2.24), Eq. (3.1 4) c an b e solv ed for h and ˜ h in terms of quaside terminants of D l +1 : f = h 11 = − ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , e = h 21 = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , g = ˜ h 12 = − ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 2 · · · ∆ 0 − 1 , b = ˜ h 22 = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 . (3.15) This is the l -th order noncomm utativ e Atiy ah- W a rd ansatz solution. F or l = 0, the non- comm utativ e anti-self-dual Y ang-Mills equation b ecomes a no ncommutativ e line ar equa- tion ( ∂ z ∂ ˜ z − ∂ w ∂ ˜ w )∆ 0 = 0. (W e note that for the Euc lidean space, this is the n oncom- m utativ e Lapla ce equation b ecause of the reality condition ¯ w = − ˜ w . The fundamen tal solutions leads to noncommu tative instan to n solutions [34].) The plane w av e solutions yields a noncomm utativ e v ersion of non-linear plane wa ve solutions [35]. Other scalar functions ∆ i ( x ) is determined explicitly by the recurrence relation (3.6) from the solu- tion ∆ 0 ( x ) of this linear equation up to in tegral constan ts. Hence the noncomm utative A tiy ah-W ard ansatz solutions are exact. 3.3 B¨ ac klun d tra nsformation for the noncomm utativ e A tiy ah- W ard ansatz solutions Finally let us discuss an adjoint action for the patc hing matrices α : P l 7→ P l +1 = A − 1 P l A in the t wistor side, whic h leads to a B¨ acklun d tra nsformation for the no ncommutativ e an ti-self-dual Y ang-Mills equation in the Y ang-Mills side. This is a noncomm utative generalization of the Corrigan-F airlie-Y ates-Go ddard (CFYG) tra nsfor ma t ion [36 , 37, 20]. The adjoint a ctio n is defined by the following t w o kinds of adjoint actions: α = β ◦ γ 0 , β : P 7→ P new = B − 1 P B , γ 0 : P 7→ P new = C − 1 0 P C 0 , (3.16) where A = B C , B =  0 1 ζ − 1 0  , C 0 =  0 1 1 0  . (3.17) In order to find the corresponding transformations in the Y ang-Mills side, we hav e to observ e how the a dj o in t actions act on the matrices h and ˜ h , or ψ and ˜ ψ . Here w e tak e the Mason-W o o dhouse ga uge (2.24). W e can easily find that the γ 0 -transformation is just h 7→ hC 0 , ˜ h 7→ ˜ hC 0 and hence J = ˜ h − 1 ⋆ h 7→ C − 1 0 J C 0 . Then, w e can read the explicit form of the transformations f o r the v a r iables b, e, f , g in the Mason-W oo dhouse gauge (2.2 4 ). 9 As for the β -t ransformation for ψ and ˜ ψ , w e hav e to tak e a singular gauge transfor- mation due to regularit y w.r.t. ζ in the Birkhoff facto r ization as follows: ψ new = s ⋆ ψ B , ˜ ψ new = s ⋆ ˜ ψ B , (3.18) where the singular gauge transformation is s =  0 ζ b − 1 − f − 1 0  . (3.19) The explicit calculation giv es ψ new =  b − 1 ψ 22 − ζ b − 1 ⋆ ψ 21 − ζ − 1 f − 1 ⋆ ψ 12 f − 1 ⋆ ψ 11  , (3.20) where ψ ij is the ( i, j )-th elemen t of ψ . In the ζ → 0 limit, this reduces to the Mason- W o o dhouse g a uge: h new =  f new 0 e new 1  =  b − 1 0 − f − 1 ⋆ j 12 1  , (3.21) where ψ = h + j ζ + O ( ζ 2 ). Here we note that the linear sys tems can b e represen ted in terms of b, f , e, g as L ⋆ ψ = ( ∂ w − ζ ∂ ˜ z ) ⋆ ψ +  − f w ⋆ f − 1 ζ g ˜ z ⋆ b − 1 − e w ⋆ f − 1 ζ b ˜ z ⋆ b − 1  ⋆ ψ = 0 , M ⋆ ψ = ( ∂ z − ζ ∂ ˜ w ) ⋆ ψ +  − f z ⋆ f − 1 ζ g ˜ w ⋆ b − 1 − e z ⋆ f − 1 ζ b ˜ w ⋆ b − 1  ⋆ ψ = 0 . (3.22) By pic king the first order term of ζ in the 1-2 comp onen t of the first equation, w e get ∂ w ( f − 1 ⋆ j 12 ) = − f − 1 ⋆ g ˜ z ⋆ b − 1 . (3.23) Hence from the 1-1 and 2-1 comp onen ts of Eq. (3.21), we ha v e f new = b − 1 , ∂ w e new = ∂ w ( f − 1 ⋆ j 12 ) = − f − 1 ⋆ g ˜ z ⋆ b − 1 . (3.24 ) In similar wa y , w e can g et the o t her ones. 3.4 Summary and commen ts Here we can r econsider that the noncomm utative Atiy ah-W ard ansatz solutions are g en- erated b y the t w o kind of B¨ ac klund transformation from the seed solutions b = e = f = g = ∆ − 1 without solving the Riemann-Hilb ert problem. (The difference of signs in f , g is not essen tial b ecause they can b e absorb ed in to the reflection symmetry f 7→ − f , g 7→ − g of the noncommutativ e Y ang equation.) Firstly , let us summarize the previous results: 10 • β -transformation [20, 12]: e new w = − f − 1 ⋆ g ˜ z ⋆ b − 1 , e new z = − f − 1 ⋆ g ˜ w ⋆ b − 1 , g new ˜ z = − b − 1 ⋆ e w ⋆ f − 1 , g new ˜ w = − b − 1 ⋆ e z ⋆ f − 1 , f new = b − 1 , b new = f − 1 . (3.25) • γ 0 -transformation [23]:  f new g new e new b new  =  b e g f  − 1 =  ( b − e ⋆ f − 1 ⋆ g ) − 1 ( g − f ⋆ e − 1 ⋆ b ) − 1 ( e − b ⋆ g − 1 ⋆ f ) − 1 ( f − g ⋆ b − 1 ⋆ e ) − 1  . (3.26) W e not e that b o th tra nsfor ma t io ns are involutive , that is, β ◦ β and γ 0 ◦ γ 0 are the iden tity transformations. No w le t us consider the tw o series of noncommu tative Atiy ah-W ard ansatz solutions R l or R ′ l generated b y the β ◦ β and γ 0 ◦ γ 0 transformations as fo llo ws: R 0 α → R 1 α → R 2 α → R 3 → · · · β ց γ 0 l β ց γ 0 l β ց γ 0 l ց · · · R ′ 1 α ′ → R ′ 2 α ′ → R ′ 3 → · · · (3.27) where α = γ 0 ◦ β : R l → R l +1 and α ′ = β ◦ γ 0 : R ′ l → R ′ l +1 . In ev ery solution, the comm utativ e limit leads to b = f . The simplest ansatz R 0 and R ′ 1 lead to the so called the Corrigan-F ai rl i e -’t Ho oft-Wilczek (CFtHW) ansatz [38] - [41]. V a rious v a riables are repres ente d in terms of quasideterminan ts as follows: • Noncommutativ e Atiy ah-W ard ansatz solutions R l Noncomm utativ e Atiy ah-W ard ansatz solutions R l are repres ente d b y the explicit form of elemen ts b l , e l , f l , g l as quasideterminan ts of ( l + 1) × ( l + 1) matrices: b l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , f l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , e l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , g l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 . 11 J l =                 0 − 1 0 · · · 0 0 1 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0                 , J − 1 l =                 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 1 0 0 · · · 0 − 1 ∆ 0                 . In the Mason-W o o dhouse gauge, h l =                   0 1 0 · · · 0 0 0 0 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 0 0 0 0 · · · 0 1 1                   , h − 1 l =                 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 0 0 0 · · · 0 − 1 1                 . ˜ h l =                   1 1 0 · · · 0 0 0 0 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 0 0 0 0 · · · 0 1 0                   , ˜ h − 1 l =                 1 − 1 0 · · · 0 0 0 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0                 . • Noncommutativ e Atiy ah-W ard ansatz solutions R ′ l Noncomm utativ e Atiy ah-W ard ansatz solutions R ′ l are repres ente d b y the explicit form of elemen ts b ′ l , e ′ l , f ′ l , g ′ l as quasideterminan ts of l × l matrices: b ′ l = ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ 1 ∆ 0 · · · ∆ 2 − l . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 , f ′ l = ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ 1 ∆ 0 · · · ∆ 2 − l . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 , e ′ l = ∆ − 1 ∆ − 2 · · · ∆ − l ∆ 0 ∆ − 1 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l − 2 ∆ l − 3 · · · ∆ − 1 , g ′ l = ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 2 ∆ 1 · · · ∆ 3 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 1 . 12 J ′ l =                 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l − 1 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 0 1 0 · · · 0 0 0                 , J ′− 1 l =                 0 0 0 · · · 0 1 0 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 − 1 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0                 . In the Mason-W o o dhouse gauge, h ′ l =               ∆ l − 1 ∆ l − 2 · · · ∆ 0 0 ∆ − 1 ∆ − 2 · · · ∆ − l 1 ∆ 0 ∆ − 1 · · · ∆ 1 − l 0 . . . . . . . . . . . . . . . ∆ l − 2 ∆ l − 3 · · · ∆ − 1 0               , h ′− 1 l =                0 0 · · · 0 − 1 0 0 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . 0 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 1 ∆ l − 1 · · · ∆ 1 ∆ 0 0 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 1                , ˜ h ′ l =               0 ∆ 1 · · · ∆ 3 − l ∆ 2 − l . . . . . . . . . . . . . . . 0 ∆ l − 1 · · · ∆ 1 ∆ 0 1 ∆ l · · · ∆ 2 ∆ 1 0 ∆ 0 · · · ∆ 2 − l ∆ 1 − l               , ˜ h ′− 1 l =                1 ∆ l · · · ∆ 2 ∆ 1 0 0 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 0 ∆ 1 · · · ∆ 3 − l ∆ 2 − l 0 . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 · · · ∆ 1 ∆ 0 1 0 0 · · · 0 − 1 0                , where         a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44         :=               a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33             a 11 a 12 a 14 a 21 a 22 a 24 a 31 a 32 a 34             a 11 a 12 a 13 a 21 a 22 a 23 a 41 a 42 a 43             a 11 a 12 a 14 a 21 a 22 a 24 a 41 a 42 a 44               . Because J is gauge inv arian t, this sho ws that the presen t B¨ acklund transformation is not just a gauge transformation but a non-trivial o ne. The pro of of these results can b e made directly by using iden tities of quasideterminan ts only , suc h as, noncommutativ e Jacobi iden tity , homological relations, and Gilson-Nimmo’s deriv ativ e for mula [23, 24]. (F or the compact represen ta tions of J , see esp ecially App endix A in [23].) This implies t ha t nonc ommutative B¨ acklund tr ansformations ar e identities of quasideterminants . 13 4 Noncomm utativ e W ard’s Conjec ture Here w e briefly discuss reductions of the no ncommutativ e anti-self-dual Y ang-Mills equa- tion into lo w er-dimensional noncommutativ e in tegrable equations suc h as the noncomm u- tativ e KdV equation. The reductions ar e sp ecified b y a choice of gauge group, symmetry , gauge fixing and so on. G auge groups are in general GL ( N ). W e ha v e t o tak e t he U (1) part of t he ga uge gr o up into accoun t in noncommutativ e case. The noncomm utativit y in the reduced directions is assumed to b e eliminated b ecause of compatibilit y with the sym- metry . (Hence within the reduced directions, the symmetry is the same as comm utativ e one.) The residual gauge symmetry sometime s sho ws equiv alence of a few reductions. Here, w e presen t non-trivial reductions of the noncomm utativ e an ti-self-dual Y ang- Mills equation with G = GL (2) to the noncomm utativ e KdV equation via a (2 + 1)- dimensional integrable equation. Let us start with the standard an ti-self-dual Y ang-Mills equation (2.1 5) with G = GL (2 , C )) and imp ose the following translational in v ariance: Y = ∂ ˜ z , (4.1) and put the follo wing non-t r ivial reduction conditions for the gauge fields: A ˜ w = O , A ˜ z =  0 0 1 0  , A w =  q − 1 q w + q ⋆ q − q  , A z =  (1 / 2) q w ˜ w + q ˜ w ⋆ q + α − q ˜ w φ − (1 / 2) q w ˜ w − q ⋆ q ˜ w + α  , where α = ∂ − 1 w [ q w , q ˜ w ] ⋆ , ∂ − 1 w f ( w ) := Z w dw ′ f ( w ′ ) , { A, B } ⋆ := A ⋆ B + B ⋆ A, φ = − q z + 1 2 q w w ˜ w + 1 2 { q , q w ˜ w } ⋆ + 1 2 { q w , q ˜ w } ⋆ + q ⋆ q ˜ w ⋆ q + [ q , ∂ − 1 w [ q w , q ˜ w ] ⋆ ] ⋆ . Then w e get a noncomm utativ e v ersion of the toroidal KdV equation [14] b y iden t if ying 2 q w = u : u z = 1 4 u w w ˜ w + 1 2 { u, u ˜ w } ⋆ + 1 4  u ˜ w , ∂ − 1 w u ˜ w  ⋆ + 1 4 ∂ − 1 w [ u, ∂ − 1 w [ u, ∂ − 1 w u ˜ w ] ⋆ ] ⋆ . (4.2) This equation has hierarc h y and N-soliton solutions in terms of quasideterminan ts of the W ro nskian [42]. W e note that under the ultr a h yp erb olic signature (+ + −− ), all remaining co ordinates amo ng z , w , ˜ w can b e set to b e real [2 0]. If we take further r eduction ∂ w = ∂ ˜ w , that is, dimensional reduction to the X = ∂ w − ∂ ˜ w direction, then the reduced equation coincides with the noncomm utativ e KdV equation: ˙ u = 1 4 u ′′′ + 3 4 ( u ′ ⋆ u + u ⋆ u ′ ) . (4.3) 14 where ( t, x ) ≡ ( z , w + ˜ w ) and ˙ f := ∂ f /∂ t, f ′ := ∂ f /∂ x . W e note that the ga uge gr o up is not S L (2) but GL (2) b ecause A z is not traceless . This implies that the U (1) part of the gauge group pla ys a crucial role in the reduction pro cess also. This noncomm uta tiv e KdV equation has b een studied b y sev eral author s and prov ed to p ossess infinite conserv ed quan tities [43] in terms of Strachan’s pro ducts [44] and exact m ulti-soliton solutions in terms of quasideterminan ts [45, 42 ]. (See a lso [46].) 5 Conclus ion and Discus sion In this pap er, w e ha v e prese nted the B¨ a cklund transformations for t he noncomm utative an ti-self-dual Y ang-Mills equation with G = GL (2) a nd constructed a series of the exact noncomm utativ e A tiy ah-W ard ansatz solutions in terms of quasideterminan ts. The quasideterminan ts pla y imp o rtan t r o les in the construction of noncomm utativ e soliton solutions not only for the noncomm utativ e an ti-self-dual Y ang-Mills equation, but also v arious lo w er-dimensional noncomm utativ e in tegrable equations [47]-[6 2]. Suc h common prop erties ha v e b een rev ealed in the study of the noncomm utativ e ex- tension; ho w ev er, ev en within the commutativ e limit, it giv es us a new in sight. V arious prop erties and iden tities of the quasideterminan ts are actually v ery useful and suitable for the noncommutativ e soliton theory . Surprisingly , o bta ining a pro of by using the quaside- terminan ts is sometime s easier than achie ving the same end by using the comm utativ e determinan ts! This suggests that the quasideterminan ts migh t b e more essen tial than the usual determinan ts in the con text of soliton theories. (ev en within the comm utative limit!) In Sato’s theory of solitons, the Pl¨ uc k er relations of the W ronskian play crucial roles. The presen t results w ould suggest the p ossibilit y of b oth noncomm utativ e extension and higher-dimensional extension of his the ory . It migh t b e time to reconsider a fo rm u- lation of Sato’s theory of (noncommutativ e) an ti-self-dual Y ang-Mills equations from the viewpoint of quasideterminan ts. (F or comm utative discussions, see e.g. [63, 64]) Ac kno wledg emen ts The author would lik e to thank the o r g anizers of t he RIMS In ternational Conference on Geome try related to I ntegrable Systems, 25 - 28 Septem b er, 200 7 in Ky oto, and the COE w orkshop on Noncomm utative Geometry and Ph ysics, 2 6 F ebruary - 3 March, 2008 at Shona n Village Cen ter, Japan for the in vitation to presen t this w ork and for their hospitalit y . He is grateful to C. Gilson and J. Nimmo for a fruitful colla b oration leading to the completion of this work, and t o T. Asak aw a A. Dimakis, I. Kishimoto, O. Lec htenfe ld, L. Mason, F. M ¨ uller-Hoisse n, Y. Ohta and K. T ak asaki for useful commen ts. Thanks ar e due t o the org anizers and audiences during the worksh ops YITP-W-09-0 4 on “QFT 2009” for their hospitalit y and discus sion, resp ectiv ely . This w ork w as partially supp o rted b y the Daiko F oundation, the Sho w a Public-Rew ard F oundation and the T o y oaki Sc holarship F oundation. 15 A Brief Revi ew of Quasi determinan ts In this section, w e make a brief in tro duction of quasideterminan ts introduced b y Gelfand and R eta kh in 1991 [65] and presen t a few prop erties of them whic h play imp ortan t roles in section 4. A go o d surv ey is e.g. [66] and relatio n b et w een quasideterminan ts and noncomm utativ e symmetric functions is summarized in e.g. [67]. (See also, [68, 6 9]) Quasideterminan ts are not just a noncomm utative generalization of usual comm utative determinan ts but rather related to in v erse matrices. Let A = ( a ij ) b e a n × n matrix and B = ( b ij ) b e the in ve rse matrix of A . Here a ll matrix elemen t s are supp osed t o b elong to a (noncomm utativ e) ring with an asso ciativ e pro duct. This general noncommutativ e s ituatio n include s the Mo y al or noncomm utative deformation which w e discuss in the main sec tions. Quasideterminan ts of A are defined formally as the inv erse of the elemen ts of B = A − 1 : | A | ij := b − 1 j i . (A.1) In the commutativ e limit, this is reduced to | A | ij − → ( − 1) i + j det A det ˜ A ij , (A.2) where ˜ A ij is the matrix obtained from A deleting the i -t h ro w a nd the j - t h column. W e can write do wn more explicit form of quasidete rminants. In order to see it, let us recall the fo llo wing f o rm ula f or a square matrix:  A B C D  − 1 =  ( A − B D − 1 C ) − 1 − A − 1 B ( D − C A − 1 B ) − 1 − ( D − C A − 1 B ) − 1 C A − 1 ( D − C A − 1 B ) − 1  , (A.3) where A and D a r e square matrices, and all in v erses are supp osed to exist. W e note that any matrix can b e decomp o sed as a 2 × 2 matrix by blo c k decomp osition where the diagonal par t s are square matrices, and the ab o ve formula can b e applied to the decomp osed 2 × 2 matrix. So the explicit forms of quasideterminan t s are giv en iteratively b y the follo wing formula: | A | ij = a ij − X i ′ ( 6 = i ) ,j ′ ( 6 = j ) a ii ′ (( ˜ A ij ) − 1 ) i ′ j ′ a j ′ j = a ij − X i ′ ( 6 = i ) ,j ′ ( 6 = j ) a ii ′ ( | ˜ A ij | j ′ i ′ ) − 1 a j ′ j . (A.4) It is sometimes con v enien t to represen t the quasideterminan t as fo llo ws: | A | ij = a 11 · · · a 1 j · · · a 1 n . . . . . . . . . a i 1 a ij a in . . . . . . . . . a n 1 · · · a nj · · · a nn . (A.5) 16 Examples of quasidete rminants are, fo r a 1 × 1 matrix A = a | A | = a, (A.6) and for a 2 × 2 matrix A = ( a ij ) | A | 11 = a 11 a 12 a 21 a 22 = a 11 − a 12 a − 1 22 a 21 , | A | 12 = a 11 a 12 a 21 a 22 = a 12 − a 11 a − 1 21 a 22 , | A | 21 = a 11 a 12 a 21 a 22 = a 21 − a 22 a − 1 12 a 11 , | A | 22 = a 11 a 12 a 21 a 22 = a 22 − a 21 a − 1 11 a 12 , (A.7 ) and for a 3 × 3 matrix A = ( a ij ) | A | 11 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 − ( a 12 , a 13 )  a 22 a 23 a 32 a 33  − 1  a 21 a 31  = a 11 − a 12 a 22 a 23 a 32 a 33 − 1 a 21 − a 12 a 22 a 23 a 32 a 33 − 1 a 31 − a 13 a 22 a 23 a 32 a 33 − 1 a 21 − a 13 a 22 a 23 a 32 a 33 − 1 a 31 , (A.8) and so on. Quasideterminan ts hav e v ario us in teresting prop erties similar to those of determinants. Among them, the follo wing ones play imp o rtan t roles in this pap er. In the blo c k matrices giv en in these r esults, lo we r case letters denote single en tries and upp er case letters denote matrices of compatible dimensions so that the ov erall matrix is square. (By using b o xes, it b ecomes easier to calculate v arious iden tities. Suc h calculations are fully presen ted in e.g. [49, 59].) • Noncommutativ e Jacobi iden tity [65, 49]       A B C D f g E h i       =     A C E i     −     A B E h         A B D f     − 1     A C D g     . • Homolog ical relations [65]       A B C D f g E h i       =       A B C D f g E h i             A B C D f g 0 0 1       ,       A B C D f g E h i       =       A B 0 D f 0 E h 1             A B C D f g E h i       • Gilson-Nimm o’s derivative formula [49]     A B C d     ′ =     A B ′ C d ′     + n X k =1      A ( A k ) ′ C ( C k ) ′          A B e t k 0     , where A k is the k th column of a matrix A and e k is the column n -v ector ( δ ik ) (i.e. 1 in the k th row and 0 elsewhere). 17 References [1] A. Dimakis and F. M ¨ uller-Ho issen, “Extension of Moy a l-deformed hierarc hies of soli- ton equations,” nlin/0408023. [2] M. 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