The Non-Compact Weyl Equation

The Non-Compact W eyl Equation Anastasia Doik ou ∗ and Theo dora Ioannidou † ∗ Department of Enginee r ing Sciences, University of Patras, GR-2650 0 Patras, Greec e E-mail : adoiko u@upat ras.gr † Department of Mathematics, Physics and Computatio na l Sciences, F aculty of Eng ineering, Aristotle University of Thessa loniki, GR-5412 4 Thessaloniki, Greece E-mail : ti3@auth .gr Abstract A non-c omp act v ersion of th e W eyl equation is pr op osed, b ased on the infi n ite di- mensional spin zero rep r esen tation of the sl 2 algebra. Solutions of the aforemen tioned equation are obtained in terms of the Kummer functions. In this con text, w e dis- cuss the ADHMN app r oac h in order to construct the corresp ondin g non-c omp act BPS monop oles. Con ten ts 1 In tro duction 1 2 The W eyl Equation 2 3 Conclusions 6 A App endix 7 1 In tro duction The Nahm equations pro vide a system o f non-linear ordinary differen tial equations dT i ds = 1 2 ε ij k [ T j , T k ] (1) for three n × n an t i- hermitian matrices T i (the so-called Nahm data) o f complex-v alued func- tions of the v ariable s , where n is the magnetic c harge of the BPS monop ole configurat io n. The tensor ε ij k is the totally an tisymmetric tensor. In the ADHMN approach, the construction of S U ( n + 1) monop ole solutions of the Bogomoln y equation with top ological charge n is translated to the following problem whic h is know n as the in v erse Nahm transform [1]. Giv en the Nahm dat a for a n -monop ole the one-dimensional W eyl equation  I 2 n d ds − I n ⊗ x j σ j + iT j ⊗ σ j  v ( x , s ) = 0 (2) for the complex 2 n - vector v ( x , s ), mu st b e solv ed. I n denotes the n × n iden tity matrix, x = ( x 1 , x 2 , x 3 ) is the p osition in space at which the monop ole fields are to b e calculated. In the minimal symmetry breaking case, the Nahm data T i ’s can b e cast as (see Reference [2], for a more detailed discussion) T i = − i 2 f i τ i , i = 1 , 2 , 3 (3) where τ i ’s form the n -dimensional represen tation of S U (2) and satisfy: [ τ i , τ j ] = 2 iε ij k τ k . (4) 1 Let us c ho ose an orthono r mal basis for these solutions, satisfying Z ˆ v † ˆ v ds = I . (5) Giv en ˆ v ( x , s ), the normalized v ector computed from (2) and (5), the Higg s field Φ and the gauge p otential A i are g iven b y Φ = − i Z s ˆ v † ˆ v ds, (6) A i = Z ˆ v † ∂ i ˆ v ds. (7) In [3, 4], we applied the ADHMN construction to obtain the S U ( n + 1) (for generic v alues of n ) BPS monop oles with minimal symmetry br eaking, by solving the W eyl equation. In this pap er, w e presen t a non-compact approac h of the ADHMN transform b y in tro ducing an infinite dimensional spin zero represen tation of the sl 2 algebra for the Nahm data. The aforemen tioned represen tation is expressed in terms of appropriate d i ff er ential op er ators ; hence, the W ey l equation is also written in terms o f the aforemen tioned differen tial op erato rs, and not in terms o f n × n matrices as in it s con v entional form (see, fo r example, R ef. [3 , 4]). In the App endix, we presen t the equiv alence b et w een t he t w o appro a c hes, i.e. matrix v ersus differen tial op erator description o f t he W ey l equation, whic h leads us to conjecture that the results of the presen t in v estigation should b y construction satisfy the Bogomolny equation. This is mainly due t o the structural similarit y b et wee n the equations arising in t he presen t case, a nd the ones emerging in the finite dimensi onal c ase describ ed in the App endix and in Ref. [3]. Nev ertheless, this is an in triguing issue, whic h merits further in v estigatio n, in particular when azimuthal dep endence is also implemen ted a long the lines describ ed in [4]. 2 The W eyl Equation In o rder to construct the non-compact BPS monop ole solutions of the W eyl equation, let us consider the sl 2 algebra, and fo cus on the non-trivial spin zero represen tation. Consider the general case: i.e. the spin S ∈ R represen tation o f sl 2 of the form τ 1 = −  ξ 2 − 1  d dξ + S  ξ + ξ − 1  , τ 2 = − i   1 + ξ 2  d dξ + S  ξ − 1 − ξ   , τ 3 = − 2 ξ d dξ . (8 ) 2 Also tak e the inner pro duct, in the basis of p olynomials of ξ on the unit circle ( ξ = e iθ ), to b e of the fo rm: h f , g i ≡ 1 2 iπ Z 1 ξ f ∗ g dξ (9) and immediately obtain the form ula h ξ m , ξ n i = δ nm . (10) Next consider the generic state v = ∞ X k = −∞ h k ξ k  b 1 √ η + b 2 √ η  , (11) where h k = h k ( r , s ) and b i = b i ( r , s ) for i = 1 , 2. Notice that using the represen ta t io n (8), f or S b eing an inte ger or half inte ger ; together with the inner pro duct (9) of an appro priate orthonormal basis { ˆ v 1 , . . . , ˆ v n +1 } where n = 2 S + 1 b eing t he dimension of the represen tation ( see also App endix for more details): Z n +1 0 h ˆ v i , ˆ v j i ds = δ ij (12) one may reco ver the Higgs field obta ined in [3] from the formula Φ ij = − i Z n +1 0 ( s − n ) h ˆ v i , ˆ v j i ds. (13) Next, w e fo cus on the the spi n zer o r epr esentation of sl 2 , a ssociated to the M¨ obius transformation and also relev ant in high energy QCD (see for example, Ref. [5, 6]) . Again w e consider the spherically symmetric case (that is, x i = r δ i 3 ) where the Nahm data are giv en by (3) for f i = f = − 1 s . Substituting t he Nahm data (3 ) where τ i ’s are defined b y (8) for S = 0 to the W eyl equation ( 2 ) and express ing σ i in terms of the spin 1 2 represen tation; that is equation (8) for S = 1 2 : σ 1 = −  η 2 − 1  d dη + ( η − 1 + η ) 2 , σ 2 = − i   1 + η 2  d dη + ( η − 1 − η ) 2  , σ 3 = − 2 η d dη (14) one gets  d ds + f ( ξ 2 − 1) 2 d dξ   η 2 − 1  d dη − ( η − 1 + η ) 2  − f (1 + ξ 2 ) 2 d dξ   1 + η 2  d dη + ( η − 1 − η ) 2  + 2 f ξ d dξ  η d dη  + 2 r η d dη  ∞ X k = −∞ h k ξ k  b 1 √ η + b 2 √ η  = 0 . (15) 3 Next, by setting w k = b 1 h k and u k = b 2 h k in (15), the follow ing set of linear differen tial equations is obtained ˙ w k − ( k + 1) s u k +1 −  k s − r  w k = 0 , ˙ u k +1 + k s w k +  ( k + 1) s − r  u k +1 = 0 , k ∈ ( −∞ , ∞ ) . (16) Here, ˙ w k and ˙ u k are the tot a l deriv ativ es of the functions w k and u k with respect to the argumen t s . Note that o ur results a re analog ous to the ones obtained in [3]. Let us now solv e these equations. The coupled equations f or u k +1 and w k are equiv alen t b y expressing u k +1 in terms of w k : u k +1 = s ( k + 1) ˙ w k − ( k − r s ) ( k + 1) w k , (17) to the single second-order equation s ¨ w k + 2 ˙ w k −  r 2 s − 2 r ( k + 1)  w k = 0 . (18) Then, the solution of (18) is given in a closed form, in terms o f the K ummer functions as w k = e − r s h c 1 ( r ) M ( − k , 2 , 2 r s ) + c 2 ( r ) U ( − k , 2 , 2 r s ) i (19) where c i ( r ) for i = 1 , 2 are constan ts. M ( − k , 2 , 2 r s ) is the regular c onfluent h yp e r ge ometric Kummer function and U ( − k , 2 , 2 r s ) is the T ric omi c onfluent hyp er ge ometric function defined in T able 1 1 . These functions are widely kno wn a s the Kummer functions of first and second kind, respective ly , and are linearly independent solutio ns of t he Kummer equation [7]. 1 Γ( a, z ) is the c omplementary o r upp er inc omplete Gamma function defined b y Γ( a, x ) = Z ∞ x t a − 1 e − t dt, ℜ ( a ) > 0 . 4 M ( − k, 2 , 2 rs ) U ( − k, 2 , 2 rs ) k = − 2 e 2 rs Γ ( − 1 , 2 rs ) e 2 rs k = − 3 (1 + rs ) e 2 rs 1+2 rs 4 rs − ( 1 + r s ) Γ (0 , 2 r s ) e 2 rs k = − 4 1 3  3 + 6 r s + 2 r 2 s 2  e 2 rs 1+5 rs +2 r 2 s 2 12 rs − 1 6  3 + 6 rs + 2 r 2 s 2  Γ (0 , 2 r s ) e 2 rs k = − 5 1 3  3 + 9 rs + 6 r 2 s 2 + r 3 s 3  e 2 rs (3+2 r s ) ( 1+8 rs +2 r 2 s 2 ) 144 rs − 1 18  3 + 9 r s + 6 r 2 s 2 + r 3 s 3  Γ (0 , 2 r s ) e 2 rs T able 1: Explicit expressions of the Kummer fu nctions M ( − k , 2 , 2 r s ) and U ( − k , 2 , 2 r s ) for k = − 2 , . . . , − 5. Finally , the corresp o nding function u k +1 giv en b y (17) tak es the simple f o rm u k +1 = k ( k + 1) e − r s h − c 1 ( r ) M ( − k + 1 , 2 , 2 r s ) + c 2 ( r )( k + 1) U ( − k + 1 , 2 , 2 r s ) i . ( 2 0) The next step is to c ho ose a n orthogonal basis of the infinite dimensional space. Consider the following functions: v k = ξ k √ η w k + ξ k √ η u k +1 , (21) whic h are orthogona l by construction. Then the norm of suc h a function is giv en by Z 1 −∞ < v k , v k > d s = Z 1 −∞  w 2 k + u 2 k +1  ds = N k . (22) As it can b e observ ed from T able 1 the arbitrar y constant c 2 ( r ) at (19) and (20) should b e set equal to zero in order to av oid the div ergencies of (22) at s → −∞ . Also, the norm (22) is w ell-defined only for k ∈ ( −∞ , − 2]. Some particular examples of t he v alues of the norm N k are N − 2 = c 2 1 ( r ) 2 r  3 + 4 r + 4 r 2  e 2 r , N − 3 = c 2 1 ( r ) 8 r  5 + 16 r + 28 r 2 + 16 r 3 + 4 r 4  e 2 r , N − 4 = c 2 1 ( r ) 162 r  63 + 324 r + 864 r 2 + 960 r 3 + 540 r 4 + 144 r 5 + 16 r 6  e 2 r , N − 5 = c 2 1 ( r ) 288 r  81 + 576 r + 2088 r 2 + 3456 r 3 + 3084 r 4 + 1536 r 5 + 432 r 6 + 64 r 7 + 4 r 8  e 2 r . (23) 5 Similarly to the finite case the associated Higgs field ma y b e then o btained via the generic expression: Φ k k = − i N k Z 1 −∞ s  w 2 k + u 2 k +1  ds. (24) 3 Conclus ions In this pap er, w e discuss the ADHMN construction in t he case of the non-compact sl 2 algebra. More precise ly , w e pro p ose a g eneralized v ersion of the W ey l equation in terms of differen tial op erators. The aforemen tioned (non- compact) W eyl equation is solv ed explicitly for the infinite dimensional spin zero represen t a tion of sl 2 , and the asso ciated solutions are expressed in terms of the so-called Kummer functions. Also, a suitable infinite set of orthogonal functions is chos en, and in ana logy to the finite case (see, f or example, [3] and References therein), expressions of the relev an t Higgs fields are prop osed. These expressions ha v e a simple and elegan t form, and should correspond to a kind o f infinite BPS monop ole configurations. The next natural step is to verify that o ur results satisfy the Bogomolny equation. Ho w- ev er in order to do so w e need to implemen t azim uthal dep endence to the spherically sym- metric solution presen ted here along the lines presen ted in [4] and thus , obtain the solution of the full non-compact W eyl equation. This requires the iden tification of a suitable transfor- mation [4] that reduces the full problem to the “diago na l” one treated here. This is a rguably a highly non-trivial t ask, and will b e pursued in full detail elsewhere together with the ph ysi- cal des cription of the full solution. In an y case, the results presen ted here are already of great significance give n that t hey provide solutions of the non-c omp act W eyl equation, op ening also the path to the study of no v el infinite type monop ole configurations. T o conclude, it w ould b e in t eresting to inv estigate an y p ossible relev ance of our find- ings with previous results of the classical vers ion of the Nahm equations related to infinite monop oles [8, 10 ] and S U ( ∞ ) Y ang-Mills theories [11, 1 2]. Note that in [13] t he Nahm equa- tions ar e asso ciated to the classic al sl 2 algebra ( Poiss o n br a c ket structure) and are line ar , whereas in our study w e consider the quantum sl 2 algebra and we deal with the cor r esp ond- 6 ing spin zero infinite dimensional (non-compact) represen ta t ion. O ne final commen t is in order: one should not confuse the spin zero infinite dimensional represen tation utilized here with the n → ∞ limit of the represen tatio n used, fo r example, in [3, 4]. The represen tation emplo yed in the presen t inv estigation is qualita tiv ely differen t from the n → ∞ case; hav ing, for instance, a zero Casimir (i.e. C = 0) as opp osed to the n → ∞ case where C → ∞ . A App e ndix In what fo llo ws w e briefly describe the equiv alence b etw een the matrix description of the W eyl equation presen ted in full g eneralit y in Ref. [3] and the differen tial op erator descrip- tion attempted here. More precisely , w e express the W eyl equation in terms of differen tial op erators via the spin S represen tation o f su 2 , for S b eing inte ger o r half inte ger . That is, w e fo cus on the finite represen ta tion of dimension n = 2 S + 1 and explicitly show the equiv alence with the generic results obtained in [3], via t he matrix description. First, consider the Nahm data (3) with f i = − 1 s ; substitute the represen t a tions τ i via the generic express ions (8) fo r S in teger or half in teger, and the matrices σ i via (14). Then the W eyl equation tak es the form:  d ds − 1 2 s  ( ξ 2 − 1) d dξ − S  ξ + ξ − 1     η 2 − 1  d dη − ( η − 1 + η ) 2  + 1 2 s   ξ 2 + 1  d dξ + S  ξ − 1 − ξ     η 2 + 1  d dη + ( η − 1 − η ) 2  − 2 s ξ d dξ  η d dη  + 2 r η d dη  v = 0 . (25) Finally , assume the generic form f or the function v : v = n X k =1 h k ξ k − 1 − S  b 1 √ η + b 2 √ η  , (26) where h k = h k ( r , s ) and b i = b i ( r , s ) for i = 1 , 2. Setting w k = b 1 h k and u k = b 2 h k in (25), the following set o f linear differen tial equations is obtained ˙ u 1 −  n − 1 2 s + r  u 1 = 0 , (27) ˙ u k +1 + k − n s w k −  n − 1 − 2 k 2 s + r  u k +1 = 0 , (28) ˙ w k − k s u k +1 +  n + 1 − 2 k 2 s + r  w k = 0 , (2 9 ) 7 ˙ w n +  1 − n 2 s + r  w n = 0 , (30) where ˙ u i and ˙ w i are the total deriv ativ es of u i ( r , s ) a nd u i ( r , s ) with resp ect to the arg ument s . Equations (27) and (3 0) can b e immediately in tegrated and their solutions are equal to: u 1 = κ 1 ( r ) s n − 1 2 e r s , w n = κ 2 ( r ) s n − 1 2 e − r s . (31) Note that the a foremen tioned solutions coincide with t he ones f o und in [3]. The coupled equations (28) and (29) are equiv alen t b y expressing u k +1 in terms of w k : u k +1 = 1 k  s ˙ w k +  n + 1 − 2 k 2 + r s  w k  , (32) to the single second-order equation s 2 ¨ w k + 2 s ˙ w k −  r 2 s 2 + ( n − 1 − 2 k ) r s + n 2 − 1 4  w k = 0 , (33) whic h may b e solv ed b y substituting w k = W k s and z = 2 r s . The la tter equation is then reduced to the fa miliar Whittaker equation: d 2 W k dz 2 +  − 1 4 + 2 k − n + 1 2 z + 1 − n 2 4 z 2  W k = 0 , (34) and coincides with the solution W k found in [3]. The next step is to c ho ose an orthogonal basis of the n -dimensional space. Consider the follo wing functions v 1 = ξ − S √ η u 1 , v k = ξ k − 1 − S  √ η w k + 1 √ η u k +1  , v n = ξ n − 1 − S √ η w n , k ∈ { 2 , . . . , n − 1 } (35) whic h are orthogona l by construction. Then the norm of suc h a function is giv en by Z n +1 0 < v k , v k > d s = Z n +1 0  w 2 k + u 2 k +1  ds = N k . (36) And one ma y r eadily recov er the Higg s field o btained in [3] from the form ula Φ k k = − i N k Z n +1 0 ( s − n )  w 2 k + w 2 k +1  ds. (37) Remark : It is clear that the presen t description is equiv alen t to the one discussed in [3]. 8 References [1] W. Nahm, Th e c onstruction o f al l self-dual multimonop oles by the ADHM me tho d , in Monop oles in Quantum Field T h e ory , eds N.S. Craigie, P . Go ddard and W. Nahm (W orld Scien tific, Singap o re, 1982). [2] N.S. Man to n and P .M. Sutcliffe, T op olo gic al Solitons , Cam bridge Monographs on Math- ematical Phy sics, Cam bridge Univ ersit y Press (2004). [3] A. Doik ou and T. Ioannidou, JHEP 1008, (2010) 105 . [4] A. Doik ou and T. Ioannidou, arXiv:1010 .5076 . [5] L. N. Lipat o v, Sov . Ph ys. JETP 63, 904 (198 6). [6] L.D. F addeev and G.P . Korc hemsky , Phy s. Lett. B342, 311 (1995). [7] M. Abramow itz and I. 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