Lax pairs for the discrete reduced Nahm systems
aa r X i v : . [ n li n . S I] S e p LAX PAIRS FOR THE DISCRETE REDUCED NAHM SYSTEMS
G. GUBBIOTTIA
BSTRACT . We discretise the Lax pair for the reduced Nahm systems andprove its equivalence with the Kahan–Hirota–Kimura discretisation proce-dure. We show that these Lax pairs guarantee the integrability of the discretereduced Nahm systems providing an invariant. Also, we show with an exam-ple that Nahm systems cannot solve the general problem of characterisationof the integrability for Kahan–Hirota–Kimura discretisations.
1. I
NTRODUCTION
W. Nahm in 1982 [20] introduced a model for self-dual multimonopoles interms of three coupled matrix differential equations:˙ T i = [ T j , T k ], T i = T i ( t ) ∈ M N , N ( C ) , (1.1)where the indices i , j , k are cyclic permutations of the set {1, 2, 3}, and N is apositive integer. The system of three equations (1.1) nowadays called Nahm’sequations.In [13] some special cases of Nahm’s equations with particular symmetrieswere studied in connection with the theory of monopoles. The obtained sys-tems of coupled two-dimensional differential equations are known as the re-duced Nahm systems : ˙ x = x + y y = − x y , (1.2a)˙ x = x − y , ˙ y = − x y − y , (1.2b)˙ x = x − y , ˙ y = − x y + y . (1.2c)Due to the symmetry of the associated Nahm matrices the systems (1.2) arecalled the tetrahedral Nahm system , octahedral Nahm system , and icosahedralNahm system respectively. The peculiarity of these systems is the fact that theyare algebraically integrable , in the sense that they possess an invariant ellipticcurve, i.e. a genus one curve, of degree three, four and six respectively. Formore information on the general Nahm equations in the context of the moderntheory of integrable systems we refer to [1].In recent years arose the interest in the problem of finding good discretisa-tion of continuous systems. By good discretisation, here we mean a discreti-sation, which preserves as much as possible the properties of its continuous Date : September 4, 2020.2010
Mathematics Subject Classification. G. GUBBIOTTI counterpart. Within this framework a procedure called
Kahan-Hirota-Kimura(KHK) discretisation became popular as a way of producing integrable discreteequations from systems of integrables ODEs. Specifically, given a systems offirst-order ordinary differential equations:˙ x = F ( x ) (1.3)its KHK is given by the following formula: x n + − x n h = F ³ x n + + x n ´ − F ( x n + ) + F ( x n )2 , x n = x ( nh ), h → + . (1.4)This formula was presented first by W. Kahan in a series of unpublished lec-ture notes [14], and applied by K. Kimura and R. Hirota to produced an inte-grable discretisation of the Lagrange top [17], This result attracted the interestof many scientists working in the field of geometric discretisation theory [18],from the Berlin school [21–23]. In particular in [21] it was noticed that whenthe function F in (1.3) is quadratic the discretisation rule (1.4) give raise to abirational map. Later, some general integrability properties of the KHK dis-cretisation were unveiled through the work of G. R.W. Quispel and his collabo-rators [4, 6].In particular in [21–23], Petrera, Pfadler and Suris developed an algebraicapproach for the search of invariants for KHK discretisations, called the Hirota–Kimura bases . With this approach they produced lots of examples, yet besidesinvariants and preserved measures, little was know about additional structuresof the discrete integrable systems they found. For instance, in the conclusionsof [22] the authors write:“Of course, it would be highly desirable to find some structures,like Lax representation, bi-Hamiltonian structure, etc., whichwould allow one to check the conservation of integrals in a moreclever way, but up to now no such structures have been foundfor any of the [K]HK type discretizations.”In this paper, we give an answer to the above comment made by Petrera,Pfadler and Suris in [22]. That is, using a technique presented in [15, 25], webuild the discrete analog of the reduced Nahm system from their Lax represen-tation. Then, we show that this discretisation is equivalent to the KHK discreti-sation discussed in [22]. These Lax pairs are used to produce invariants, andproving integrability of the discrete Nahm systems.The plan of the paper is the following: in section 2 we give a review of theliterature on the Lax pair for the continuous and discrete Nahm systems. Insection 3 we use such construction to produce the Lax pairs for the reducedNahm systems (1.2) and prove integrability. In the final section 4 we give someconclusions and an outlook for further researches. Moreover, we show with anexample that there exists Nahm systems whose Lax pair does not provide in-tegrability, yet the system is KHK discretisable, and both the continuous anddiscrete systems are algebraically integrable. This shows, that despite the suc-cess obtained in explaining the integrability of the Euler top [15, 16, 25], and of
AX PAIRS FOR THE DISCRETE REDUCED NAHM SYSTEMS 3 the reduced Nahm systems (1.2), the Nahm equation approach cannot solvethe general problem of characterisation of the integrability of KHK discretisa-tions. 2. L
AX PAIR FOR THE CONTINUOUS AND DISCRETE N AHM SYSTEMS
In the literature several different forms of the Lax pair for the Nahm equa-tions have been proposed. For instance, recently in [15] it was proposed thefollowing form: A ( λ ) = µ T − T T T ¶ + λ µ T − T ¶ + λ µ T T − T T ¶ , (2.1a) B ( λ ) = µ T − T ¶ + λ µ T T − T T ¶ (2.1b)The above matrices are such that the system (1.1) is equivalent to the followingcompatibility condition: ˙ A = [ A , B ] . (2.2)The Lax pair (2.1) consists of 2 N × N matrices. In this paper to avoid toocumbersome formulas we consider the inverse matrix complexifcation of thematrices in (2.1): M = µ M − M M M ¶ =⇒ M = M + i M . (2.3)That is, we consider the following Lax pair for the Nahm system (1.1): A ( λ ) = T + i T − λ T + λ ( T − i T ) , (2.4a) B ( λ ) = − i T + λ ( T − i T ) . (2.4b)The compatibility condition (2.2) gives again the Nahm equations (1.1) takingits real and imaginary part.Associated to the matrix B ( λ ) there exists a unique family of unitary matri-ces resolving the differential equation ˙ V = V B , with initial condition V (0) = I N .As proven in [19] this implies that the spectrum of the matrix A ( λ ) does notdepend on the independent variable t . So, the coefficients of the characteristicpolynomial of A ( λ ): p A ¡ µ ¢ = det ¡ A ( λ ) − µ I N ¢ , (2.5)do not depend on t . That is, the coefficients of the characteristic polynomial(2.5) are first integrals of the Nahm system (1.1). Moreover, since the system(1.1) does not depend on the variable λ too, for each coefficient we can havemultiple first integrals. Remark . Equation (2.5) yields at least N first integrals, but there is no apriori guarantee that these first integrals are functionally independent and/ornon-trivial. This implies that the integrability of the system (1.1) must be provedcase by case using the appropriate form of the matrices T i . G. GUBBIOTTI
In [16], followed by [15, 25], was introduced a method to discretise the com-patibility condition (2.2). Consider the discrete time interval: t n = nh , h → + , (2.6)hence we define f n ≡ f ( t n ). Then, the compatibility condition (2.2) can be dis-cretised as: A n + ( λ ) − A n ( λ ) h = A n + ( λ ) B n ( λ ) − B n + ( λ ) A n ( λ ) . (2.7)The corresponding system of difference equations is given by: T i , n + − T i , n h = T j , n + T k , n − T k , n + T j , n , (2.8)where the indices i , j , k are cyclic permutations of the set {1, 2, 3}. Remark . In principle a different discretisation of the compatibility condi-tion (2.2) can be given: A n + ( λ ) − A n ( λ ) h = A n ( λ ) B n + ( λ ) − B n ( λ ) A n + ( λ ) , (2.9)yield the following system of difference equations: T i , n + − T i , n h = T j , n T k , n + − T k , n T j , n + , (2.10)where the indices i , j , k are cyclic permutations of the set {1, 2, 3}. However, bydirect computation it is possible to show that, in the cases considered in thispaper, condition (2.10) is equivalent to (2.8) up to the transformation: x n + i ←→ x n − i , (2.11)where x n is the vector of the dynamical variables. That is, the evolution definedfrom (2.8) is the opposite of the evolution defined by (2.10). We notice that thisis a general fact when dealing with KHK discretisation as pointed out in [21].Equation (2.7) can be rearranged as: A n + ( λ ) ( I N − hB n ( λ )) = ( I N − hB n + ( λ )) A n ( λ ) . (2.12)Introducing the matrices: L n ( λ ) = A n ( λ ) , M n ( λ ) = I N − hB n ( λ ) , (2.13)which allows us to rewrite (2.12) as: L n + ( λ ) M n ( λ ) = M n + ( λ ) L n ( λ ) . (2.14)Following [26, 27] we have that equation (2.14) implies that the spectral data ofthe matrix L n ( λ ) M − n ( λ ) are constant along the evolution. Indeed, from (2.14),the matrices L n + ( λ ) M − n + ( λ ) and L n ( λ ) M − n ( λ ) are conjugate, so that theyhave the same characteristic polynomial. This implies that the that the coef-ficients of the characteristic polynomial of L n ( λ ) M − n ( λ ) are conserved quan-tities (invariants) for the system (2.8). Alternatively, using Binet’s rule, we have AX PAIRS FOR THE DISCRETE REDUCED NAHM SYSTEMS 5 that the coefficients of the characteristic polynomial of L n ( λ ) with respect to M n ( λ ): p L , M ¡ µ ¢ = det ¡ L n ( λ ) − µ M n ( λ ) ¢ , (2.15)divided by det M n ( λ ) are constants of motions. That is, writing such character-istic polynomial in the following way: p L , M ¡ µ ¢ = ( − N det M n ( λ ) µ N + c N − ( λ ) µ N − + · · · + c ( λ ) , (2.16)we can write these invariants in the following way: H = c ( λ )det M n ( λ ) , H = c ( λ )det M n ( λ ) , . . . , H N − = c N − ( λ )det M n ( λ ) . (2.17)Finally, we note that the same consideration on functional independence ofthe invariants (2.17) given in Remark 2.1 apply.3. D ISCRETE REDUCED N AHM SYSTEMS
In [13] where considered three special cases of Nahm’s equations (1.1) corre-sponding to symmetry groups of regular solids, namely tetrahedral , octahedral ,and icosahedral symmetry.Assume we are a given G ⊂ SO (3), a symmetry group of regular solid. Then,the G -invariant Nahm matrices T i have the following form: T i ( t ) = x ( t ) ρ i + y ( t ) S i . (3.1)Here ρ : R → su ( k ) is a representation of so (3) on C k , while ( S , S , S ) is a G -invariant vector in the symmetric power space S k V ⊂ R ⊗ su ( k ) where V isthe representation corresponding to G in SU (2).In the following we will consider the tetrahedral, octahedral, and icosahe-dral symmetry cases, with the definitions of the G -invariant Nahm matrices T i given in [13]. The discretisation of these continuous systems was obtainedin [22] using the KHK discretisation procedure and proved to be integrable byconstructing the invariant with the so-called Hirota–Kimura bases [21]. Herewe will prove that the KHK discretisation follows from the discretisation of theLax pairs and the invariant can be found using the associated characteristicpolynomial (2.15). Remark . We note that the results of [21] on the discrete Nahm systemswhere generalised simultaneously and independently in [5, 24]. Some com-ments on the geometry of these systems were given in [3]. Later, in [28] it wasproved how to construct the tetrahedral and the octahedral case discrete casesusing generalised Manin transform. Finally, in [10] it was pointed out that inthe octahedral case the geometric construction of [28] induces non-standardfeatures on the procedure of resolution of singularities, proving the existenceof families of particular solutions.
G. GUBBIOTTI
Tetrahedral symmetry.
Consider the reduced Nahm system with tetrahe-dral symmetry (1.2a). Its Lax pair is given by: A ( λ ) = λ ³ x + y ´ − i ¡ λ − ¢ ³ x − y ´ − λ ³ x − y ´ − ¡ λ + ¢ ³ x + y ´ i ¡ λ − ¢ ³ x + y ´ ¡ λ + ¢ ³ x − y ´ , (3.2a) B ( λ ) = ³ x + y ´ − i λ ³ x − y ´ − i ³ x − y ´ − λ ³ x + y ´ i λ ³ x + y ´ λ ³ x − y ´ . (3.2b)Considering the characteristic polynomial (2.5) we obtain the invariant givenin [13]: H = y ( y + x ). (3.3)Note that the level curves of the invariant H (3.3) are genus one (elliptic) curves.From (2.13) we obtain the following discrete Lax Pair: L n ( λ ) = λ ³ x n + y n ´ − i ¡ λ − ¢ ³ x n − y n ´ − λ ³ x n − y n ´ − ¡ λ + ¢ ³ x n + y n ´ i ¡ λ − ¢ ³ x n + y n ´ ¡ λ + ¢ ³ x n − y n ´ ,(3.4a) M n ( λ ) = − i h ³ x n + y n ´ i λ h ³ x n − y n ´ i h ³ x n − y n ´ λ h ³ x n + y n ´ − i λ h ³ x n + y n ´ − λ h ³ x n − y n ´ . (3.4b)The corresponding compatibility conditions are: x n + − x n h = x n + x n + y n + y n , (3.5a) y n + − y n h = − ¡ x n y n + + x n + y n ¢ , (3.5b)and coincide with the KHK discretisation of the reduced Nahm system withtetrahedral symmetry (1.2a), originally presented in [22]. The characteristicpolynomial of (3.4a) with respect to (3.4b): p L , A ¡ µ ¢ = µ − h x n + h y n + h x n λ y n + h y n λ ¶ µ − h λ y n (48 x n + y n ) µ + h y n x n + y n )(5 λ − µ − λ y n x n + y n )( λ − λ + λ + AX PAIRS FOR THE DISCRETE REDUCED NAHM SYSTEMS 7
From equation (2.17) we could get three different invariants, yet they will benecessarily dependent. So, we choose: H ( λ ) = − λ y n (48 x n + y n )( λ − λ + λ + − h x n + h y n + h x n λ y n + h y n λ . (3.7)This invariant is dependent on λ , therefore to find a non-trivial invariant wecan expand in Taylor series with respect to λ , that is H ( λ ) = P ∞ k = h k λ k , andtake the first non-constant element: h = − y n (48 x n + y n )1 − h x n + h y n Octahedral symmetry.
Consider the reduced Nahm system with tetrahe-dral symmetry (1.2b). Its Lax pair is given by: A ( λ ) = λ ¡ x + y ¢ λ ¡ x − y ¢ − y − x + y λ ¡ x − y ¢ λ ¡ x + y ¢ − x − y − λ ¡ x − y ¢ λ ¡ x − y ¢ λ y /3 0 − x + y − λ ¡ x + y ¢ , (3.9a) B ( λ ) = x + y λ ¡ x − y ¢ x − y λ ¡ x + y ¢
00 0 − ¡ x − y ¢ λ ¡ x − y ¢ λ y /3 0 0 − ¡ x + y ¢ . (3.9b)Considering the characteristic polynomial (2.5) we obtain the invariant givenin [13]: H = y ( x + y )( x − y ) . (3.10)Note that the level curves of the invariant H (3.10) are genus one (elliptic)curves.From (2.13) we obtain the following discrete Lax Pair: L n ( λ ) = λ ¡ x n + y n ¢ λ ¡ x n − y n ¢ − y n − x n + y n λ ¡ x n − y n ¢ λ ¡ x n + y n ¢ − x n − y n − λ ¡ x n − y n ¢ λ ¡ x n − y n ¢ λ y n /3 0 − x n + y n − λ ¡ x n + y n ¢ ,(3.11a) M n ( λ ) = − h ¡ x n + y n ¢ − λ h ¡ x n − y n ¢ − h ¡ x n − y n ¢ − λ h ¡ x n + y n ¢
00 0 1 + h ¡ x n − y n ¢ − h λ ¡ x n − y n ¢ − λ h y n /3 0 0 1 + h ¡ x n + y n ¢ .(3.11b) G. GUBBIOTTI
The corresponding compatibility conditions are: x n + − x n h = x n + x n − y n + y n , (3.12a) y n + − y n h = − ¡ x n + y n + x n y n + ¢ − y n y n + . (3.12b)and coincide with the KHK discretisation of the reduced Nahm system withoctahedral symmetry (1.2b), originally presented in [22]. Proceeding in analo-gous ways as in section 3.1, that is taking the first non-constant element of theTaylor series of H ( λ ), we obtain the following invariant: h = − y n ¡ x n + y n ¢ ¡ x n − y n ¢ h − h ¡ x n − y n ¢ i h − h ¡ x n + y n ¢ i . (3.13)3.3. Icosahedral symmetry.
Consider the reduced Nahm system with tetrahe-dral symmetry (1.2c). Its Lax pair is given by: L ( λ ) = λ ¡ x + y ¢ λ ¡ x − y ¢ − y λ y − x + y λ ¡ x − y ¢ λ ¡ x + y ¢ y − x − y λ ¡ x + y ¢ λ ¡ x − y ¢ − x + y − λ ¡ x + y ¢ λ ¡ x + y ¢ λ y
120 0 0 − x − y − λ ¡ x − y ¢ λ ¡ x − y ¢ λ y − λ y
120 0 0 − x + y − λ ¡ x + y ¢ ,(3.14a) M ( λ ) = x + y λ ¡ x − y ¢ y x − y λ ¡ x + y ¢ x + y λ ¡ x − y ¢ − x − y λ ¡ x + y ¢ λ y
120 0 0 0 − x + y λ ¡ x − y ¢ y − λ y
120 0 0 0 − x − y .(3.14b)Considering the characteristic polynomial (2.5) we obtain the invariant givenin [13]: H = y (3 x − y ) (4 x + y ) . (3.15)Note that the level curves of the invariant H (3.15) are genus one (elliptic)curves. A X P A I R S F O R T H E D I S C R E T E R E D U C E D N A H M S Y S T E M S From (2.13) we obtain the following discrete Lax Pair: L n ( λ ) = λ ¡ x n + y n ¢ λ ¡ x n − y n ¢ − y n λ y n − x n + y n λ ¡ x n − y n ¢ λ ¡ x n + y n ¢ y n − x n − y n λ ¡ x n + y n ¢ λ ¡ x n − y n ¢ − x n + y n − λ ¡ x n + y n ¢ λ ¡ x n + y n ¢ λ y n
120 0 0 − x n − y n − λ ¡ x n − y n ¢ λ ¡ x n − y n ¢ λ y n − λ y n
120 0 0 − x n + y n − λ ¡ x n + y n ¢ , (3.16a) M n ( λ ) = − h ³ x n + y n ´ − λ h ¡ x n − y n ¢ − h y n − h ¡ x n − y n ¢ − h λ ¡ x n + y n ¢ − h ¡ x n + y n ¢ − h λ ¡ x n − y n ¢ + h ¡ x n + y n ¢ − h λ ¡ x n + y n ¢ − h λ y n
120 0 0 0 1 + h ¡ x n − y n ¢ − λ h ¡ x n − y n ¢ − h y n
600 7 h λ y n
120 0 0 0 1 + h ³ x n + y n ´ . (3.16b) The corresponding compatibility conditions are: x n + − x n h = x n + x n − y n + y n , (3.17a) y n + − y n h = − ¡ x n y n + + x n + y n ¢ + y n y n + . (3.17b)and coincide with the KHK discretisation of the reduced Nahm system withicoshedral symmetry (1.2c), originally presented in [22]. Proceeding in analo-gous ways as in section 3.1, that is taking the first non-constant element of theTaylor series of H ( λ ), we obtain the following invariant: h = y n ¡ x n − y n ¢ ¡ x n + y n ¢ £ − h ¡ x n + x n y n + y n ¢¤ h − h ¡ x n − y n ¢ i h − h ¡ x n − y n ¢ i .(3.18)4. C ONCLUSIONS
In this paper we discretised the reduced Nahm systems [13] using the tech-nique employed for the Euler top presented in [15, 16, 25]. Our results showsthat the discretisation is analog to the so-called Kahan–Hirota–Kimura discreti-sation. We proved that such Lax pairs are “bona fide”. That is, they can be usedto produce (all) the invariants of the associated systems, and hence to proveintegrability. A Lax pair that cannot be used to produce (all) the invariants of agiven system is called a fake Lax pair [2, 11, 12].We conclude this paper noting that, unfortunately Nahm systems and theirgeneralisations are not enough to explain the integrability of all KHK discretis-able systems. To this end we consider the following system, which is a particu-lar case of the coupled Euler top introduced in [8]:˙ x = x x , (4.1a)˙ x = x x , (4.1b)˙ x = x x + x x , (4.1c)˙ x = x x , (4.1d)˙ x = x x . (4.1e)This system is naïvely integrable. We say that system of difference equationsis naïvely integrable when it possesses N − N is the number of degrees of freedom. In [22] itwas proved that the system (4.1) possesses five first integrals, four of which arefunctionally independent, proving naïve integrability. AX PAIRS FOR THE DISCRETE REDUCED NAHM SYSTEMS 11
It is easy to see that the system (4.1) arises from the following Nahm system: T = − x x x − x , (4.2a) T = x − x , (4.2b) T = x − x x − x . (4.2c)So, the system (4.1) has a Lax pair given by (2.4): A ( λ ) = (1 − i) λ − λ + + i − λ x − λ x − x − i λ x + i x λ x + λ x + x (1 − i) λ − λ + + i λ x − λ x + x i λ x − i x − λ x + λ x − x (1 − i) λ − λ + + i ,(4.3a) B ( λ ) = − i + (1 − i) λ − i x − λ x − i x i x + λ x − i + (1 − i) λ − i x + λ x i λ x i x − λ x − i + (1 − i) λ , (4.3b)and the characteristic polynomial (2.5) of L ( λ ) is: p L ¡ µ ¢ = µ + (3 − ¡ i λ − λ − i − λ ¢ µ + £ ( I − λ +
4i ( I − + λ − I λ +
4i ( I − − λ + I + ¤ µ − (1 − i) ( I − λ +
2i [2 (i − I + I − λ − − i5 [5 I + (12 + I − (8 − I − + λ +
4i (4 + I − I − I ) λ + + i5 (12 + − I − − i) I + + i) I ) λ −
2i [2 (1 + i) I − I − i ] λ − (1 + i ) ( I + i ) (4.4)where: I = x − x + x , I = x x − x x , I = x − x + x . (4.5)Taking the coefficients of (4.4) with respect to λ and µ we obtain that the threefunctions in (4.5) are the only independent invariants given by L ( λ ). This showsthat the Lax pair (4.3) does not prove the naïve integrability of the system (4.1).In this sense the Lax pair (4.1) is fake in the sense of [2, 11, 12]. Now consider the discrete Nahm system (2.9) corresponding to the Nahmmatrices (4.2): T n = − x n x n x n − x , (4.6a) T n = x n − x n , (4.6b) T n = x n − x n x n − x n . (4.6c)Unfortunately, we obtain the compatibility conditions are overdetermined, inthe sense that we have more compatibility conditions than independent vari-ables. For instance, let us consider the first discrete Nahm equation T n + − T n = h ¡ T n + T n − T n + T n ¢ writing down the coefficients explicitly: x n + − x n = h ¡ x n + x n − x n + x n + ¢ , (4.7a) x n + − x n =
0, (4.7b) x n + − x n = h ¡ x n + x n − x n + x n + ¢ , (4.7c) x n + − x n = h ¡ x n + x n + x n − x , n + ¢ , (4.7d) x n + − x n = h ¡ x n + x n + x n − x n + ¢ . (4.7e)It is clear that (4.7b) implies that the discrete time evolution of x n is trivial.So, the discrete Nahm system (4.7) cannot be a discretisation of the integrablesystem (4.1).On the other hand in [22] it was proven that the KHN discretisation of thesystem (4.1) exists and it is algebraically integrable. More precisely, using themethod of the Hirota–Kimura bases [21], the authors proved that such KHKdiscretisation preserves all five invariants of its continuous counterpart (4.1).We note that the functionally independent invariants of such KHK discretisa-tion can be found directly, that is without using the Hirota–Kimura bases, withthe method of [7]. See also [9] for an explanation of the method in the case ofdifference equations.So, to conclude, this example shows that, although very useful in severalcase, the Nahm’s equations approach is not enough to explain integrability ofquadratic vector equations in both the continuous and the discrete case.A CKNOWLEDGEMENTS
This research was supported by Dr. M. Radnoviˇc’s grant DP160101728 andby Prof. N. Joshi and Dr. Milena Radnoviˇc’s grant DP200100210 from the Aus-tralian Research Council.The author expresses his gratitude to Prof. N. Joshi, Prof. G. R. W. Quispeland Dr. D. T. Tran for their helpful discussions during the preparation of this
AX PAIRS FOR THE DISCRETE REDUCED NAHM SYSTEMS 13 paper. We thank the anonymous referee, whose comments led to a great im-provement of the paper. R
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