Hitchins equations and integrability of BPS Z(N) strings in Yang-Mills theories

Hit hin's equations and in tegrabilit y of BPS Z N strings in Y ang-Mills theories Maro A. C. Kneipp Universidade F e der al de Santa Catarina (UFSC) 1 , Dep artamento de Físi a, CFM, Campus Universitário, T rindade, 88040-900, Florianóp ols, Br azil. Abstrat W e sho w that Z N string's BPS equations are equiv alen t to the Hit hin's equations (or self-dualit y equation) and also to the zero urv ature ondition. W e onstrut a general form for BPS Z N string solutions for arbitrary simple gauge groups with non-trivial en ter. Dep ending on the v auum solutions onsidered, the Z N string's BPS equations redue to dieren t t w o dimensional in tegrable eld equations. F or a partiular v auum w e obtain the equation of ane T o da eld theory . Keyw ords: In tegrable Equations in Ph ysis, In tegrable Hierar hies. 1 E-mail address: kneippfs.ufs.br. 1 In tro dution In S U ( N ) QCD, it is b eliev ed that partile onnemen t in the strong oupling regime happ ens due to  hromo eletri strings (QCD strings). Man y prop erties of the QCD strings ha v e b een studied in tensely in the last y ears using lattie alulation. On the other hand, it is b eliev ed that QCD strings in the strong oupling ma y b e dual to  hromomagneti strings in the Higgs phase in w eak oupling, whi h are easier to study analytially . Sine QCD strings in onning phase should b e formed only b y S U (3) gauge elds and not U (1) gauge elds, in reen t y ears w e are analyzing some prop erties of  hromomagneti Z N strings solutions whi h app ear in a theory with non-Ab elian simple gauge group G (without U (1) fators) brok en to its en ter. The Z N string solutions ha v e man y features similar to the QCD strings. In partiular they are asso iated to o w eigh ts of represen tations of G (or equiv alen tly to w eigh ts of the dual group 2 G ∨ ) and their top ologial setors are asso iated to the en ter elemen ts of the gauge group G . More preisely , the o w eigh ts of G an b e separated in osets asso iated to no des of the extended Dynkin diagram of G . All Z N string solutions asso iated to o w eigh ts in a giv en oset b elong to the same top ologial setor [ 1℄. The Z N strings asso iated to the fundamen tal w eigh ts of dieren t represen tations an ha v e dieren t tensions and for dieren t v auum solutions, the BPS b ounds for the tensions an satisfying either the sine la w saling or the Casimir saling [1℄[2 ℄, dieren tly from the non-Ab elian semi-lo al BPS string solutions with gauge group S U ( N ) × U (1) where the tension is only due magneti ux in the U (1) diretion and it dep ends on the U (1) winding n um b er[3 ℄. It is imp ortan t to note that the Casimir saling and the sine la w saling onsidered in [1 ℄[2℄ are lo w er b ounds for the non-BPS Z N string tensions. Previously w e analyzed the Z N string in soft brok en N = 2 [4℄[5 ℄ and N = 4 [1 ℄ Sup er Y ang- Mills theories, but in [2℄ and here w e do not onstrain t the p oten tial to b e sup ersymmetri sine w e are in terested in studying some general prop erties at the lassial lev el of the Z N strings whi h ma y b e useful for QCD and not neessarily onnemen t in sup ersymmetri theories. The Z N strings do es not neessarily p oin t in a diretion in the Cartan subalgebra (CSA). Ho w ev er, sine the monop oles' magneti ux is in the CSA [ 6℄, w e only onsider Z N string solutions with ux in the CSA whi h are the relev an t for onnemen t of these monop oles [ 5℄[1 ℄ whi h ma y b e dual to partile onnemen t. This result is analogous to the Ab elian dominane observ ed in QCD. In the presen t w ork w e sho w that Z N string's BPS equations [4℄[1 ℄ for Y ang-Mills theo- ries with salars in the adjoin t are equiv alen t to the Hit hin's equations [ 7℄ and onsequen tly to the four dimensional self-dualit y equation [7℄ and also equiv alen t to the zero urv ature ondition[11 ℄, implying that this set of BPS Z N string solutions in non-Ab elian Y ang-Mills theories is (quasi-)in tegrable. In tegrabilit y of BPS v orties in Ab elian-Higgs theory w as re- en tly onsidered in [8℄. In tegrabilit y of other soliton solutions for theories in dimensions higher than t w o are analyzed in [ 9℄. In reen t y ears, in tegrabilit y also had a renew ed in terest in gauge and string theories [10 ℄. On the other hand, Hit hin's equations app eared in man y distint problems as for example in Matrix string theory [11℄[12 ℄ and more reen tly in on- netion with the geometri Langlands program [13 ℄. In tegrabilit y of Hit hin's equations and self-dualit y equations are also disussed in [14℄[15 ℄. The equiv alene of BPS Z N string equations with Hit hin's equations, self-dualit y equa- tions and zero urv ature ondition is in teresting b eause allo ws us to apply metho ds and 2 W e shall onsider the dual group G ∨ as the o v ering group asso iated to the dual algebra g ∨ . 1 results of these systems to Z N string solutions and vie-v ersa. In the U (1) Ab elian-Higgs theory , the elds of rotationally symmetri BPS string solutions an b e written as funtions of a eld whi h satises Liouville's equation plus a onstan t and a singularit y at the origin [16 ℄[17℄. In this w ork w e generalize this result for the Z N strings sho wing that, for a partiular v auum resp onsible for the gauge symmetry breaking, the elds of the rotationally symmetri BPS Z(N) string solutions are funtions of a eld whi h satises ane T o da eld equation with a singularit y at the origin. In this pap er w e in tro due, in setion 2, some general results for BPS Z N strings and sho w the equiv alene of the Z N string BPS onditions with the Hit hin's equations, and onsequen tly to self-dualit y equation and the zero urv ature ondition. In setion 3 w e onstrut an Ansatz for the Z N strings and sho w that the Z N string's BPS equations redue to t w o dimensional in tegrable theories equations. In setion 4 w e sho w that for a partiular v auum, the BPS Z N string solutions redues to the equation of ane T o da theory whi h is a deformation of onformal T o da theory . In setion 5 w e analyze the sp eial ase of rotationally symmetri solutions. These solutions resem ble the Riemannian or stringy instan tons of Matrix string theories[11 ℄[12 ℄. 2 BPS Z N strings equations and the Hit hin's equations Let us onsider Y ang-Mills-Higgs theories with arbitrary gauge group G whi h is simple, onneted and simply onneted. In order to exist strings and onned monop oles w e shall onsider theories with t w o omplex salars elds φ s , s = 1 , 2 , in the adjoin t represen tation of G . W e also onsider that the v auum solutions φ v a 1 , φ v a 2 pro due the symmetry breaking pattern G φ v a 1 → U (1) r φ v a 2 → C G , (1) where r is the rank of G and C G its en ter, whi h w e onsider to b e non trivial. The Lagrangian of the theory w e are to study is L = − 1 4 G aµν G µν a + 2 X s =1 1 2 ( D µ φ s ) ∗ a ( D µ φ s ) a − V ( φ, φ ∗ ) (2) where a is a Lie algebra index, D µ = ∂ µ + ie [ W µ , ℄ and G µν = ∂ µ W ν − ∂ ν W µ + ie [ W µ , W ν ] . Let z = x 1 + ix 2 , ∂ z = 1 2 ( ∂ 1 − i∂ 2 ) , (3) W z = 1 2 ( W 1 − iW 2 ) , 2 and B i = − ǫ ij k G j k / 2 . F or a stati string solution with ylindrial symmetry in the x 3 dire- tion, the string BPS equations for a theory with gauge group G without U (1) fators are[ 4 ℄[1℄ 3 B 3 a = − d a , (4) D z φ s = 0 , (5) D ¯ z φ † s = 0 , (6) V ( φ, φ ∗ ) − 1 2 ( d a ) 2 = 0 , (7) with d = e 2 ( 2 X s =1 h φ † s , φ s i − mRe ( φ 1 ) ) , where m is a non-negativ e mass parameter. The string tension satises the b ound T ≥ me 2 | φ v a 1 | | Φ st | (8) where Φ st = 1 | φ v a 1 | Z d 2 x T r [ Re ( φ 1 ) B 3 ] (9) is the string ux in the φ 1 diretion, with the in tegral b eing taking in the plane orthogonal to the string. The equalit y in Eq. (8) happ ens only for the strings satisfying the BPS equations. W e shall onsider V ( φ, φ ∗ ) = 1 2 ( d a ) 2 , (10) whi h guaran tee that equation (7) is automatially fullled. Note that the BPS equation ( 7 ) do es not restrit the p oten tial to ha v e this form. In [4℄[5℄[1 ℄ w e onsidered soft brok en N = 2 and N = 4 p oten tials. Similarly to the Prasad-Sommereld limit [ 18 ℄ for BPS monop oles, w e tak e the limit m → 0 [4℄ in order for the BPS string equations to b e onsisten t with the equations of motion and retain the p oten tial terms resp onsible for the breaking of G in to its en ter C G . Note that in this limit, the tensions T → 0 , although the ratio of the string tensions are nite and an satisfy the Casimir saling la w or the sine la w as disussed in[ 2 ℄ and in setion 4. Let g b e the Lie algebra asso iated to the gauge group G and let the generators H i , i = 1 , 2 , ..., r , form a basis for the Cartan subalgebra (CSA) h with rank r . Let us adopt the Cartan-W eyl basis in whi h T r ( H i H j ) = δ ij , T r ( E α E β ) = 2 α 2 δ α + β , where the trae is tak en in the adjoin t represen tation. In this basis, the omm utation relations read [ H i , E α ] = ( α ) i E α , (11) [ E α , E − α ] = 2 α α 2 · H , 3 F or the sak e of simpliit y w e shall only onsider the string solutions whi h ha v e p ositiv e ux Φ st . F or the an tistrings, one m ust use opp osite signs in some of these equations as disussed in our previous w orks. 3 where α are ro ots and the upp er index in ( α ) i means the omp onen t i of α . W e denote b y α i and λ i , i = 1 , 2 , ..., r , the simple ro ots and fundamen tal w eigh ts of g resp etiv ely and α ∨ i = 2 α i α 2 i , λ ∨ i = 2 λ i α 2 i (12) are the simple o-ro ots and fundamen tal o-w eigh ts of g , and are also the simple ro ots and fundamen tal w eigh ts of the dual algebra g ∨ . They satisfy the relations α i · λ ∨ j = α ∨ i · λ j = δ ij . Moreo v er, α i = K ij λ j (13) where K ij = 2 α i · α j α 2 j (14) is the Cartan matrix asso iated to g . W e denote b y ψ the highest ro ot of g . Considering the on v en tion that ψ 2 = 2 , the highest ro ot an b e written as ψ = r X i =1 m i α ∨ i (15) where m i are in tegers whi h are the lev els (or marks) of the fundamen tal represen tations whi h ha v e λ i as highest w eigh ts. F or S U ( n ) , m i = 1 for i = 1 , 2 , ..., n − 1 . In order to pro due the symmetry breaking (1) w e an onsider a general v auum solution φ v a 1 = v · H , v = v i λ ∨ i , (16) φ v a 2 = r X l =0 b l E − α l , (17) where α 0 = − ψ is the negativ e of the highest w eigh t, v i are non-v anishing real onstan ts, b l are real onstan ts and for l = 1 , 2 , ..., r they an not v anish in order to G to b e brok en in to C G . Comparing with the general v auum solutions onsidered in [1 ℄[2℄, in (17) w e add a term asso iated to the ro ot − α 0 whi h do es not  hange the symmetry breaking but  hange some prop erties of the v auum solutions as w e shall explain b ello w. W e usually onsider Z N strings solutions with the gauge elds in the CSA with H i , i = 1 , 2 , ..., r as basis generators whi h are the relev an t for onnemen t of the standard monop ole solution, sine the monop oles ha v e magneti ux in diretion of the CSA. Then, as disussed in our previous w orks [4 ℄[1℄, sine the gauge elds are ev erywhere in the Cartan subalgebra, from the BPS equations D z φ 1 = 0 and D ¯ z φ † 1 = 0 results that the eld φ 1 ( x ) is onstan t and equal to its asymptoti form (16), i.e., φ 1 ( x ) = v · H . (18) Then, the BPS equations (4)-(6 ) in the limit m → 0 redue to G ¯ z z → − ie 4 h φ † 2 , φ 2 i , D z φ 2 = 0 , (19) D ¯ z φ † 2 = 0 , 4 whi h are exatly the Hit hin's equations [7 ℄. As it is kno wn, these equations are equal to a redution to t w o dimensions of the self-dualit y equation in Eulidean four dimensions [7℄, G µν = 1 2 ǫ µν ρσ G ρσ , with gauge elds W i = W i for i = 1 , 2 and W 3 = φ 2 r , (20) W 4 = φ 2 i , where φ 2 r and φ 2 i are resp etiv ely the real and imaginary parts of φ 2 and imp osing that the elds do es not dep end on the extra dimensions with o ordinates x 3 and x 4 . The Equations (19) an also b e written in the form of a zero urv ature ondition in t w o dimensions onsidering the onnetion [11 ℄ A z = W z + λ 2 φ † 2 , (21) A ¯ z = W ¯ z − 1 2 λ φ 2 , where λ is a sp etral parameter. Then, F ¯ z z = ∂ ¯ z A z − ∂ z A ¯ z + ie [ A ¯ z , A z ] =  G ¯ z z + ie 4 h φ † 2 , φ 2 i  + λ 2 D ¯ z φ † 2 + 1 2 λ D z φ 2 . Therefore, the system of equations (19 ) is equiv alen t to a zero urv ature ondition whi h im- plies the lassial in tegrabilit y of the set of BPS Z N string solutions. More preisely , due to the limit m → 0 , w e ha v e the ondition F ¯ z z → 0 , whi h w e ould all quasi-in tegrabilit y ondition. Ho w ev er, for simpliit y w e will use the equal sign in the follo wing in tegrable equations. The at onnetion A z , A ¯ z an b e written in terms of the self-dual elds W i (20 ) as A z = 1 2 ( W 1 − i W 2 ) + λ 2 ( W 3 − i W 4 ) , (22) A ¯ z = 1 2 ( W 1 + i W 2 ) − 1 2 λ ( W 3 + i W 4 ) . The equiv alene of BPS Z N string equations, with the Hit hin's equations, self-dualit y equations and zero urv ature ondition is in teresting b eause it allo ws to apply metho ds and results of these systems to the Z N string solutions and vie-v ersa. In the next setion w e sho w that for Z N string solutions onstruted from dieren t v auum are asso iated to dieren t in tegrable eld equations. 3 BPS Z N string solutions In the Higgs phase of the theory , when G is brok en to its en ter C G whi h w e onsider to b e non-trivial, there exist Z N string solutions and the monop oles are onned b y these strings. In order to ha v e nite string tension, asymptotially these solutions ha v e the form φ s ( ϕ, ρ → ∞ ) = g ( ϕ ) φ v a s g ( ϕ ) − 1 , s = 1 , 2 , (23) W i ( ϕ, ρ → ∞ ) = i e ( ∂ i g ( ϕ ) ) g ( ϕ ) − 1 , i = 1 , 2 , 5 where φ v a s are the v auum solutions (16 ), (17), ρ and ϕ are the radial and angular o ordinates. In order for the onguration to b e single v alued, g ( ϕ + 2 π ) g ( ϕ ) − 1 ∈ C G . Considering g ( ϕ ) = exp iϕ M , where M = ω · H it implies that exp(2 π iω · H ) ∈ C G , whi h results that ω ∈ Λ w ( G ∨ ) , where Λ w ( G ∨ ) = ( ω = r X i =1 n i λ ∨ i , n i ∈ Z ) (24) is the o w eigh t lattie of G or equiv alen tly the w eigh t lattie of the dual group G ∨ . Then, using the v auum solutions (16 ), (17 ), the asymptoti form of the Z N string solution (23 ) an b e written as φ 1 ( ϕ, ρ → ∞ ) = v · H , φ 2 ( ϕ, ρ → ∞ ) = r X i =0 b i { exp ( − iϕω · α i ) } E − α i , (25) W i ( ϕ, ρ → ∞ ) = ǫ ij x j eρ 2 ω · H, i = 1 , 2 . Therefore, for ea h w eigh t ω of the dual group G ∨ w e an onstrut a string solution. In [1℄ is sho wn ho w these strings are separated in dieren t top ologial setors. As men tioned b efore, w e an tak e φ 1 ( ϕ, ρ ) = v · H for the whole spae. In order to determine the other elds for the whole spae w e onsider the Ansatz φ 2 ( ρ, ϕ ) = r X i =0 f i ( ρ, ϕ ) b i E − α i exp( − iY ( ρ, ϕ ) · α i ) . (26) Similarly to the string solution in the Ab elian Higgs mo del, if ω is su h that for a giv en α i , the salar pro dut ω · α i 6 = 0 , then the orresp onding funtion f i ( ρ, ϕ ) m ust ha v e some zeros sine from the asymptoti form (25 ) w e see that the terms with ω · α i 6 = 0 ha v e non-v anishing winding n um b er. W e an rewrite this Ansatz, similarly to the string solutions in the Ab elian Higgs mo del [16℄[19 ℄, as φ 2 ( ρ, ϕ ) = G ( ρ, ϕ ) φ v a 2 G − 1 ( ρ, ϕ ) (27) where φ v a 2 is the v auum solution (17 ) and G ( ρ, ϕ ) = exp [ Z ( ρ, ϕ ) · H ] , Z ( ρ, ϕ ) = − e 2 X ( ρ, ϕ ) + iY ( ρ, ϕ ) , with Z ( ρ, ϕ ) , X ( ρ, ϕ ) and Y ( ρ, ϕ ) b eing r omp onen t real funtions with 2 ln f i = eX · α i . The p oin ts where f i ( ρ, ϕ ) v anishes, X · α i has a logarithmi singularit y . F rom the asymptoti form (25 ) w e an onlude that for ρ → ∞ , X ( ρ → ∞ , ϕ ) = 0 , Y ( ρ → ∞ , ϕ ) = ϕω . 6 F or the sp eial ase of rotationally symmetri solutions, w e an onsider Y ( ρ, ϕ ) = ϕω and X ( ρ, ϕ ) = X ( ρ ) is a radial funtion. This ansatz an b e used for an y Z N string solution, not only BPS. F rom Eq. (27) results that ∂ z φ 2 =  ( ∂ z G ) G − 1 , φ 2  . Therefore, from the BPS equation D z φ 2 = 0 , w e an onlude that W z = i e ( ∂ z G ) G − 1 + F z = i e ∂ z ( Z · H ) + F z (28) where F z ( x ) is a Lie algebra v alued funtion whi h omm utes with φ 2 . On the other hand, from the BPS equation D z φ 1 = 0 and the fat that φ 1 ( x ) = v · H , implies F z ( x ) should b elong to the CSA. Sine φ 2 is not in the CSA, it implies that F z ( x ) = 0 . Similarly , b y omputing ∂ ¯ z φ † 2 and rep eating the ab o v e argumen t w e an onlude that W ¯ z = i e ∂ ¯ z  G − 1  † G † = − i e ∂ ¯ z  Z † · H  . (29) Note that G † 6 = G − 1 . Therefore, from the rst BPS equation in (19) results that X satises 4 ∂ ¯ z ∂ z ( X · H ) − e 4 h e eX · H ( φ v a 2 ) † e − eX · H , φ v a 2 i = 0 . (30) This is the equation of motion of an Eulide an t w o dimensional in tegrable system sine it equiv alen t to the zero urv ature ondition with the onnetion (21 ) using the elds ongu- rations (27), (28) and (29 ) (whi h are solutions of D z φ 2 = 0 and D ¯ z φ † 2 = 0 ), that is A z = i e ∂ z ( Z · H ) + λ 2 exp  − Z † · H  ( φ v a 2 ) † exp  Z † · H  , (31) A ¯ z = − i e ∂ ¯ z ( Z † · H ) − 1 2 λ exp ( Z · H ) φ v a 2 exp ( − Z · H ) . Using the fat that φ v a 2 = P r l =0 b l E − α l , w e an write Eq. (30 ) as ∂ ¯ z ∂ z X − e 4 r X j =0 b 2 j α ∨ j e eα j · X = 0 , (32) remem b ering that X is an r omp onen t salar eld. F or the v auum solutions with b 0 = 0 , w e dene X α i = α i · X, i = 1 , 2 , ..., r. Then, Eq. (32) an b e written as ∂ ¯ z ∂ z X α i − e 4 r X j =1 K ij b 2 j exp  eX α j  = 0 , (33) 4 In order to arriv e to this equation w e are not onsidering the p oin ts where X has a singularit y . W e disuss more on this issue in the last setion. 7 where K ij is the Cartan matrix (14). On the other hand, for v auum solutions with b 0 6 = 0 , w e dene X α i = α i · X , i = 0 , 1 , 2 , ..., r . (34) In this ase, Eq. (32 ) an b e written as ∂ ¯ z ∂ z X α i − e 4 r X j =0 b K ij b 2 j exp  eX α j  = 0 , i = 0 , 1 , 2 , ..., r (35) where b K ij is the extended Cartan matrix whi h has the same form as the Cartan matrix Eq. (14 ), but with i, j = 0 , 1 , ..., r , and are asso iated to the un t wisted ane Lie algebras. Ho w ev er, from (15) and the fat that α 0 = − ψ w e an onlude that r X i =0 m i α ∨ i = 0 , where w e onsider m 0 = 1 . Therefore, the elds X α i in (34 ) are not indep enden t but satisfy the onstrain t r X i =0 2 m i α 2 i X α i = 0 . Therefore, equation (35) m ust b e sub jet to this onstrain t [20℄. As w e men tioned b efore, for a Z N string solution asso iated to a v etor ω , for the terms in Eq. (26 ) where ω · α i 6 = 0 , the orresp onding funtion f i m ust ha v e some zeros and hene X α i = α i · X has logarithmi singularities. Therefore equations (33 ) and (35 ) are v alid exept at the singularities of X α i . Similarly to the Ab elian ase [16℄[17 ℄, w e an allo w for these singularities b y inluding delta-funtions on the righ t hand side of the ab o v e equations. 4 A v auum solution and ane T o da eld theories Let us no w onsider a onrete v auum solution. In order to b e a v auum solution of the p oten tial (10), the onstan ts in (16 ), (17 ), m ust satisfy the relation m  K − 1  ij v j = b 2 i − m i b 2 0 . (36) In [1 ℄[2℄ w e analyzed t w o v auum solutions with b 0 = 0 , whi h are v alid for an y gauge group G : a) The rst v auum solution w e onsidered w as v i = a, (37) b i = v u u t ma r X j =1 ( K − 1 ) ij = p maδ · λ i , i = 1 , 2 , ..., r, (38) where a is a p ositiv e real onstan t and δ = P r i =1 λ ∨ i is the dual W eyl v etor. With this v auum, the Z N strings tensions satisfy the Casimir saling [1℄. 8 b) The seond v auum solution w as v i = ay (1) i , (39) b i = 1 2 sin π 2 h q may (1) , i = 1 , 2 , ..., r, (40) where a is a p ositiv e real onstan t and y (1) i are the omp onen ts of the P erron-F rob enius eigen- v etors of K ij asso iated to the eigen v alue 4 sin 2 π 2 h . With this v auum, the Z N strings tensions satisfy the sine la w saling [ 2 ℄. As men tioned b efore, in order for the Z N string's BPS equations to b e onsisten t with the equations of motion w e m ust tak e the limit m → 0 . Therefore, for the onstan ts b i or equiv alen tly φ v a 2 to b e nite, w e m ust tak e a → ∞ (and therefore v i → ∞ ) k eeping the pro dut ma nite [2℄. Another p ossibilit y w e use here is to onsider the v auum solutions (16 ), (17) with b 0 6 = 0 in whi h ase w e an k eep a nite. One an see this from Eq. ( 36 ) whi h implies that b i = q m ( K − 1 ) ij v j + m i b 2 0 . (41) F rom this equation, onsidering that v j is nite, when w e tak e m → 0 , it implies that b i → √ m i b 0 whi h is nite with b 0 b eing an arbitrary non-v anishing onstan t. That result hold when the omp onen ts v j satisfy either (37) or (39 ). In ea h ase, the ratio of tensions of the BPS Z N strings will on tin ue to satisfy the Casimir saling or the sine la w saling resp etiv ely . The v auum solution (17 ) will sta y φ v a 2 = b 0 r X l =0 √ m i E − α l = b 0 E , (42) where E = r X i =0 √ m i E α i and therefore asymptotially φ 2 ( ϕ, ρ → ∞ ) = b 0 r X i =0 √ m i { exp ( − iϕω · α i ) } E − α i . (43) The generator E satisfy [ E , E † ] = 0 . Hene it is diagonalizable and an b e em b edded in a new Cartan subalgebra. This generator w as originally in tro dued b y K onstan t [ 21℄. F or this v auum w e an write ( 30 ) as ∂ ¯ z ∂ z ( X · H ) − eb 2 0 4 h e eX · H E † e − eX · H , E i = 0 , (44) It an also b e written as ∂ ¯ z ∂ z X − eb 2 0 4 r X j =0 m j α ∨ j exp ( eα j · X ) = 0 (45) 9 or ∂ ¯ z ∂ z X α i − eb 2 0 4 r X j =0 b K ij m j exp  eX α j  = 0 . (46) Eq. (44 ) (or (45 ), (46 ) ) is the equation of motion of Eulidean t w o dimensional in tegrable ane T o da eld theory asso iated to the un t wisted ane Lie algebra b g obtained from g , with oupling onstan t e , equal to the oupling onstan t of the gauge theory , and mass parameter equal to eb 0 . In [2 ℄ w as sho wn that the sp etrum of BPS string tensions in the seond v auum solution (39) oinide with the solitons mass sp etrum of the orresp onding ane T o da theory . It is in teresting to note that the monop ole's BPS equations with spherial symmetry re- dues to the equation of onformal T o da theory [22 ℄. The equation of Ane T o da theory w as also obtained from Hit hin's equations for U ( N ) matrix theory in [11 ℄, but with some dier- enes as w e disuss in the next setion. A relation b et w een Hit hin equations and in tegrable systems w as also onsider in [14 ℄. F or g = su (2) , onsidering α 1 = 1 = − α 0 , equation (45 ) redues to the sinh-Gordon equation ∂ ¯ z ∂ z X − eb 2 0 2 sinh ( eX ) = 0 . 5 Rotationally symmetri solutions Let us no w onsider the sp eial ase of rotationally symmetri solutions. In this ase for a Z N string asso iated with the v etor ω of the w eigh t lattie of dual group G ∨ , Y ( ρ, ϕ ) = ϕω and X ( ρ, ϕ ) = X ( ρ ) is a radial funtion, and hene Z ( ρ, ϕ ) = − e 2 X ( ρ ) + iϕω . (47) Therefore, in this ase the salar elds has the form φ 1 ( ϕ, ρ ) = v · H , (48) φ 2 ( ϕ, ρ ) = b 0 r X i =0 √ m i n exp  e 2 X · α i − iϕω · α i o E − α i . (49) Generalizing some results from stringy instan tons of matrix string theories [11 ℄[12 ℄, w e an determine the form of the gauge elds: sine 2 ϕ = − i ln( z / ¯ z ) , the gauge elds (28) and (29 ) are giv en b y W z = i 2  − ∂ z ( X · H ) + 1 ez ω · H  , (50) W ¯ z = − i 2  − ∂ ¯ z ( X · H ) + 1 e ¯ z ω · H  . Using that ∂ z (1 / ¯ z ) = π δ (2) ( x ) , w e obtain that the magneti eld of the Z N string is B 3 = − 2 iG ¯ z z = − 2 h ∂ ¯ z ∂ z ( X · H ) − π e ω · H δ (2) ( x ) i . (51) F rom the requiremen t of regularit y at z = 0 implies that near the origin ∂ ¯ z ∂ z ( X · H ) ∼ π e ω · H δ (2) ( x ) + onst . 10 Therefore, near the origin X ( ρ → 0) ∼ − 2 e ω ln | z | + onst. (52) On the other hand, for ρ → ∞ , X ( ρ → ∞ ) → 0 , as w e men tioned b efore. Eq. ( 52) is onsisten t with the fat that in the general ase (not neessarily rotationally symmetri), X α i = α i · X has a logarithmi singularit y if α i · ω 6 = 0 . F rom (48) and (51 ) w e onlude that the ux (9) of a string asso iated to the o w eigh t w is Φ st = 2 π e ω · v | v | , whi h is onsisten t with the result in [ 1℄ using just the asymptoti form of the Z N string solution (25 ). F rom Eq. (46 ) and the b eha vior of the solution near the origin (52) w e an onlude that for the sp eial ase of rotationally symmetri solutions, for a Z N string asso iated with a o w eigh t w , the radial funtion X ( ρ ) m ust satisfy ∂ 2 X ∂ ρ 2 + 1 ρ ∂ X ∂ ρ − eb 2 0 r X j =0 α ∨ j m j exp ( eα j · X ) = 4 π e ω δ (2) ( x ) (53) or ∂ 2 X α i ∂ ρ 2 + 1 ρ ∂ X α i ∂ ρ − eb 2 0 r X j =0 b K ij m j exp  eX α j  = 4 π e ω · α i δ (2) ( x ) . (54) One ould arriv e diretly to this equation with the singularit y using (47) and (50) in the BPS equation (4) or in the onnetion (31). That result is similar to the string solution in the Ab elian-Higgs theory where for a rotationally symmetri onguration Ansatz the radial funtion, satises a rotationally symmetri form of a Liouville's equation plus a onstan t with a δ -funtion at the origin[16 ℄[17 ℄. Note that our ondition m → 0 orresp onds to tak e the limit a → 0 for the onstan t app earing in the p oten tial in Ab elian-Higgs theory , in whi h ase the radial funtion of the string solution w ould satisfy Liouville's equation with a δ -funtion at the origin. The Z N string solutions ha v e great similarit y with the Riemannian or stringy instan tons of matrix string theories [11℄[12 ℄, but also some dierenes: in the matrix stringy theories there is no gauge symmetry breaking. F or the stringy instan tons, the salar eld has a bran h ut in the origin instead of a zero for φ 2 ( ρ, ϕ ) for the rotationally symmetri Z N strings. They also ha v e dieren t angular dep endene sine the Z N string solutions are asso iated to lasses of Π 1 ( G/C G ) . Moreo v er, the ane T o da equation asso iated to stringy instan tons [11 ℄ has a singularit y struture dieren t from (53). Eqs. 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