Vortices as Instantons in Noncommutative Discrete Space: Use of Z 2 Coordinates

Transcription

1 Vortices as Instantons in Noncommutative Discrete Space: Use of Z Coordinates arxiv: v [hep-th] 14 Jul 009 Hideharu Otsu Faculty of Economics, Aichi University, Toyohashi, Aichi , Japan Toshiro Sato Faculty of Law Economics, Mie Chukyo University, Matsusaka, Mie , Japan Hitoshi Ikemori Faculty of Economics, Shiga University, Hikone, Shiga 5-85, Japan Shinsaku Kitakado Department of Physics, Faculty of Science Technology, Meijo University, Tempaku, Nagoya , Japan Abstract We show that vortices of Yang-Mills-Higgs model in R space can be regarded as instantons of Yang-Mills model in R Z space. For this, we construct the noncommutative Z space by explicitly fixing the Z coordinates then show, by using the Z coordinates, that BPS equation for the vortices can be considered as a self-dual equation. We also propose the possibility to rewrite the BPS equations for vortices as ADHM equations through the use of self-dual equation. otsu@vega.aichi-u.ac.jp tsato@mie-chukyo-u.ac.jp ikemori@biwako.shiga-u.ac.jp kitakado@ccmfs.meijo-u.ac.jp 1

2 1 Introduction Topological solitons play an important role in various field theories. These are kink, vortex, baby-skyrmion, monopole, skyrmion, instanton so on [1]. Some of the soliton equations are solved analytically, others are solved only numerically. It is interesting to look for the relations among the topological solitons. We consider a static soliton in Yang-Mills-Higgs YMH model in +1 dimensions. The static soliton is a vortex in -dimensional R space. Some properties of Abelian vortex non-abelian vortex in YMH model have been studied [, 3, 4]. The vortex configurations are solved numerically. The BPS Bogomol nyi-prasad- Sommerfield equations [5] for the vortex can be rewritten in terms of master equation plus half-adhmatiyah-drinfeld-hitchin-manin equation [6]. The solution of half-adhm equation contains information on the moduli space of the vortex, while instanton in 4-dimensional space are solved analytically by the ADHM method [7]. On the other h, Higgs fields can be treated as gauge fields [8]. Note that, in these works discrete spaces are treated in terms of differential forms without the explicit use of the coordinates. We have been investigating the possibility of describing a vortex in -dimensional space as an instanton in 4-dimensional space, which is R Z space in this paper. In the previous paper [9], from a viewpoint of the noncommutative differential geometry gauge theory in discrete space, we have shown that the instanton in R Z space is nothing but the vortex in R space. This means that difference of vortex instanton can be considered as that of the spaces R Z R 4. In ref. [9], we did not explicitly discuss the relations between the Yang eq. the master eq., due to lack of representation of the Z coordinates. The ADHM method for vortices also requires the coordinate representation. By introducing the explicit form of the Z coordinates, we can approach the problem from the new point of view. An attempt of this paper is the analysis using the explicit form of the noncommutative Z coordinates. On the other h, the arguments with the differential forms can not be cast straightforwardly into the coordinate picture. The purpose of this paper is to clarify the relation between instanton vortex using the noncommutative Z coordinates. We first define the coordinates for noncommutative Z space then investigate the relation between the instantons in R Z space the vortices in R space. In addition, we consider the relations among different descriptions of the vortices. In section, we summarize properties of YMH model fix the notations. In section 3, we construct a noncommutative Z space. In section 4, we discuss relation between the instanton in R Z space the vortex in YMH model in R space. In section 5, we discuss relations among BPS, master half-adhm equations. The final section is devoted to summary discussion. Some properties of Yang Mills Higgs model Let us summarize here some properties of the YMH model which has non-abelian gauge symmetry [9]. The model contains a Higgs field, represented by N L N R matrix, two gauge fields corresponding to U N L U N R gauge group. In this paper, we consider the models with N L N R N, where the solitons are local vortices. The Lagrangian in +1

3 dimensions is 1 L Tr +Tr F L µν F L g + 1 µν F R µν F R g µν D µ H D µ H g HH c1 N. 1 Where, we define a covariant derivative D µ field strength F L µν, FR µν as D µ H µ H +L µ H HR µ, F L µν µl ν ν L µ +[L µ,l ν ], 3 F R µν µr ν ν R µ +[R µ,r ν ], 4 Tr is a trace over the adjoint representation of U N. Two U N gauge fields L µ, R µ the Higgs field H are represented by N N matrices. In the following we take g c 1 for simplicity. The energy integral is of the form 1 E dx 1 dx Tr F L F R 1 + D1 H + D H + HH 1 N. 5 The BPS equations minimizing the energy are D 1 ±id H 0, 6 if L 1 ± HH 1 N 0, 7 if R 1 H H 1 N 0, 8 where we use the anti-hermitian gauge fields L µ L µ R µ R µ [4]. The solutions of the equations 6, 7, 8 are topologically stable solitons, called non-abelian vortices. Where, sets of equations are those for vortex for anti-vortex. It is obvious that only pure gauge configurations are allowed at the spacial infinity x. This means that the topological property of the non-abelian vortices is classified by the mapping index for S 1 U N U N. On account of the fact that U N is equal to U 1 SU N, the corresponding homotopy group is π 1 U N U N π 1 U 1 U 1 Z Z. 9 We can take the topological charges corresponding to 9 as Q L R i dx 1 dx Tr F1 L π 1 FR 0,±1,±, 10 Q L+R i π dx 1 dx Tr F L 1 +FR 1 0,±1,±,. 11 Here, Q L R is identified with the vortex number. On the other h, topological charge Q L+R is irrelevant to the vortex configuration, since gauge field TrL µ +R µ does not interact with 3

4 other fields. Although the general configurations are classified by two topological charges Q L R Q L+R, the vortex configurations are essentially classified by Q L R. Because the BPS equations 7 8 mean that the U1 part of F L 1 + FR 1 0, thus Q L+R 0 for the vortex solutions. Note that, although our YMH models have the U N U N gauge group with 1 N, vortex solutions have some relations to those of the model with U N gauge group. Particularly, the model of U 1 L U 1 R gauge group is equivalent to the model of U 1 L R gauge group, since one of the combinations of gauge field, i.e. L+R, decouples from other fields. For N, the relations among vortex solutions of the models with U N U N L U N R gauge groups are shown in section 5. Let us describe the notations for 4-dimensional space, since we construct the vortex in -dimensional space from a model in 4-dimensional space. The relation between Cartesian coordinates x 1,x,x 3,x 4 complex coordinates z, z,w, w in 4-dimentional space are z 1 x 1 +ix, z 1 x 1 ix, w 1 x 3 +ix 4, w 1 x 3 ix 4, z 1 1 i, z 1 1 +i, w 1 3 i 4, w 1 3 +i 4. 1 For R 4 space, z, z,w, w are usual complex coordinates. While, for R Z space used in this paper, w w are noncommutative coordinates to be defined in the next section. Gauge fields are defined by a z 1 a 1 ia, a z 1 a 1 +ia, 13 a w 1 a 3 ia 4, a w 1 a 3 +ia Finally, relations between the gauge field strength in the complex Cartesian coordinates are F z z if 1, F w w if 34, F z w 1 F 13 +if 14 +F 4 if 3, F zw 1 F 13 if 14 +F 4 +if 3, F zw 1 F 13 if 14 F 4 if 3, F z w 1 F 13 +if 14 F 4 +if

5 3 Noncommutative Z space In this section, we construct a -dimensional noncommutative discrete Z space, referring to the construction of noncommutative RNC space. In the case of R NC space, the complex coordinates are represented by the creation annihilation operators on the Fock space { n } with n 0,1,,3, [10]. Then the commutation relation of the complex coordinates is proportional to the noncommutative parameter. Now, we consider the coordinates w w of noncommutative discrete Z space as the operators on the Fock space with -states 0 1. Our definition of Z space is w 0 0, w 0 θ 1, w 1 θ 0, w 1 0, 16 ww n nθ n, n 0,1, 17 where θ is the noncommutative parameter. Then Z coordinates can be represented by matrices as w 0 1 θ, w 0 0 θ, where the Fock space is described by the vectors 1 0 0, From 18, coordinates w w are characterized by anti-commutation relations {w,w} { w, w} 0 0 {w, w} θ, 1 where {A,B} AB + BA. Note that, the noncommutative coordinates of Z space satisfy the anti-commutation relations 0 1, in contrast to the case of RNC space, where the commutation relations [w,w] [ w, w] 0 [ w,w] θ are satisfied. It means that the coordinates of Z space are fermionic, while those of RNC space are bosonic. Next, we define the differentiation by w w as right-differential, namely differentiation of a function f w, w by w or w is defined by the following procedure. Move w or w to the right for each term in f w, w with the help of 0 1, then differentiate by w or w on the right-h side. This definition of the differentiation can also be described by use of the commutator as w θ 1 [ w, ]σ 3, w θ 1 [w, ]σ 3. Because of the nilpotency of w w 0, arbitrary function of Z space can be exped in five terms, 1, w, w, w w, ww. 3 Here, four terms are linearly independent under the relation 1. Explicit form of the differentials are given by w 1 w w 0, w w 1, w w w w, w ww w 4 5

6 w 1 w w 0, w w 1, w w w w, w ww w. 5 These can also be represented by matrix form, corresponding to 18, as A B w 1 B 0 θ C D A+D B 6 where we used the fact that A B C D w A B C D 1 θ C A D 0 C, 7 A w w θ +B w θ +C w θ +D ww θ. 8 Furthermore, the integral in w space is defined by the super trace on the Fock space { 0, 1 } as Z Od w stro θ{ 0 O 0 1 O 1 }, 9 because of the anti-commutation relations 0 1. In the following, we take θ 1 for simplicity. 4 Vortices in R space as instantons in R Z space In this section, we discuss the YMH model in R space which descends from the Yang-Mills YM model in R Z space, where Z is the noncommutative discrete space. The following is the discussion on the self-dual equations in 4-dimensional R Z space BPS equations for the vortex in -dimensional R space. First, we sketch the argument in ref. [11] on the self-dual equations in pure UN YM model in commutative R 4 space. As we shall see later, applying this discussion to the R Z space, BPS equations in YMH model can be obtained. Their argument goes as follows. They consider the self-dual equation for pure UN YM model in commutative R 4 space. From the relation 15, the self-dual equation can be rewritten as F µν 1 ǫ µνρσf ρσ 30 F zw 0, 31 F z w 0, 3 F z z F w w. 33 6

7 in commutative z w coordinates. In this model, two UN matrix functions h h are introduced with the definition of gauge field as a z h 1 z h, a z h 1 z h, a w h 1 w h, a w h 1 w h. 34 Then, a part of self-dual equations 31 3 are satisfied automatically. And from F z z h 1 z g 1 z g h, F w w h 1 w g 1 w g h, 35 where another equation 33 takes the form g hh 1, 36 z g 1 z g + w g 1 w g Equation 37 is called Yang equation [1]. To apply the above argument to the case of R Z space, where Z space is noncommutative defined by 16, 17, we have to replace the coordinates w, w in the previous argument by noncommutative discrete ones for the equations from 1 to 15 from 30 to 37. Especially, the self-dual equations are F zw 0, F z w 0, F z z F w w, 38 where z, z are the commutative R coordinates w, w are the noncommutative Z ones. Futhermore, h h are expressed by the N N matrices 0 b 0 c h, h 1 1 c 0 b 1, b h, c 0 h 0 c 1 1 b Namely, h h are expressed as matrices 39, 40, each matrix elements b,c, b, c are UN matrices. Replacing R 4 space by R Z space, the gauge field corresponding to Z space can be considered as the Higgs field. In the following, we show the equivalence between the instanton of YM model in R Z space the vortex of YMH model in R space. Now, we shall consider the gauge fields strengths. We use the differential rules or 6 7 for the Z coordinates. The gauge fields are given by a z h 1 z h c 1 z c 0 0 b 1 z b a z h 1 z h c 1 z c 0 0 b 1 z b, 41, 4 7

8 a w h 1 w h 0 c 1 b c 1 b 1 0 w 0 b c 0 43 a w h 1 w h 0 c 1 b b 1 c 0 w 0 b c 0 Then we define the gauge fields L, R the Higgs field H as respectively. Here, h h are related as the gauge fields are anti-hermitan L z c 1 z c, L z c 1 z c,. 44 R z b 1 z b, R z b 1 z b 45 H c 1 b, H b 1 c, 46 h h 1 or b b 1, c c 1, 47 L z L z, R z R z. 48 The field strengths are calculated as follows. First, F zw F z w are calculated as F zw z a w w a z +[a z,a w ] 0 c 1 b c z 1 z c w 0 b 1 z b [ ] c + 1 z c 0 0 c 1 b 0 b 1, z b z c 1 b + c 1 z c c 1 b c 1 b b 1 z b D z H

9 where F z w z a w w a z +[a z,a w ] 0 1 c 1 z c 0 z b 1 w c 0 0 b 1 z b [ ] c + 1 z c b 1, z b b 1 c z b 1 c+b 1 z bb 1 c b 1 cc 1 z c D z H, 50 0 D z H z H +L z H HR z 51 D z H D z H z H H L z +R z H. 5 Note that the commutator term [a,a] is needed even for the U1 case because of the noncommutativity of Z space. As in the case of R 4, F zw 0 53 F z w 0 54 are satisfied automatically with the definition of gauge fields by h h. Equations mean D z H 0 55 D z H D z H 0 56 respectively, are nothing but a part of BPS equations for YMH model in R. Similarly, 0 Dz H 0 0 F zw, F z 0 0 w D z H 0 are derived, where 57 D z H z H +L z H HR z 58 D z H D z H z H H L z +R z H. 59 9

10 Finally, F w w becomes using F z z is calculated as F w w w a w w a w +[a w,a w ] c 1 b w b 1 w c [ ] 0 c 1 b 0 1 +, 1 0 b 1 c 0 HH H H 1, 60 F z z z a z z a z +[a z,a z ] c 1 z c 0 c 1 z c 0 z 0 b 1 z z b 0 b 1 z b [c + 1 z c, c 1 z c] [ 0 0 b 1 z b, b 1 z b] F L i F1 R. 61 From 60 61, the self-dual equation 33 reduces to the BPS equations These are also expressed by Yang equation where F z z F w w 6 if L 1 1 HH, 63 if R 1 H H z g 1 z g + w g 1 w g 0, 65 g hh 1, 66 h, h are given by The above argument shows that the vortex in R space can be regarded as an instanton in R Z space, since the self-dual equations 38 of YM model in R Z space is equivalent to the BPS equations of YMH model in R space. Furthermore, we can see that the YM model in R Z space also reduces to the YMH model in R space at the level of static part of the Lagrangian. For the static configurations, square of field strength becomes 1 4 Fµυ 1 F z z + 1 F w w + 1 F zwf z w + 1 F zwf z w L1 0, 67 0 L 10

11 where L 1 1 F L D zhd z H HH, L 1 F R D zh D z H H H. 68 Then, in the case of YM model for the UN UN gauge fields 1-Higgs field, the Lagrangian density of YM model in R Z space is given by TrL1 0 L σ 0 TrL 3, 69 where Tr means the trace of UN matrix σ 3 comes from the volume element derived from the metric of the Z space. Then the action S is obtained as S strld xd w R Z strld x { } TrL1 0 str σ 0 TrL 3 d x { 1 Tr F L F R D zhd z H + 1 D zhd z H + } 1 HH d x. 70 This gives the action of the YMH model in R space 1 with g c 1 for the static configurations. It can also be verified that the instanton number, denoted as Q I, in R Z space is just the vortex number in R space as follows. Q I 1 8π 1 8π 1 π { str TrF F } σ 3 d x str {Tr 1 } ǫ µναβf µν F αβ σ 3 d x str{tr F z z F w w F z w F zw +F zw F z w σ 3 }d x 1 [ Tr { if L π 1 HH 1 +if1 R 1 H H } +Tr { D z HD z H +D z HD z H } ]d x i Tr F1 L π 1 FR d x Q L R. 5 BPS, master half-adhm equations In the first part of this section, we study the BPS equation, master equation half-adhm equation for YMH models with UN UN UN gauge groups. In the latter part 11 71

12 of this section, we study the relation between formulations for soliton equation discussed in section 4 that of master equation plus half-adhm equation. We show that, for these two models, the BPS equation for the vortex with certain topological number can be expressed by the master equation plus half-adhm equation. Furthermore, we see that the vortex solution in two models satisfies the common half-adhm equation. In addition, we comment on some Abelian non-abelian vortices in both YMH models. Finally, we obtain the relation between the variables in the two formulations. First, we summarize the YMH model with UN gauge group [6]. The Lagrangian is L Tr 1 g F L µν F L µν +D µh D µ H g 4 HH c1 N Where, we define a covariant derivative D µ field strength F L µν as. 7 D µ H µ H +L µ H 73 F L µν µ L ν ν L µ +[L µ,l ν ]. 74 We take g c 1 in the following. BPS equations are D z H z H +L z H 0 75 if 1 1 HH. 76 LetusintroduceaN N invertible matrixsz, z GLN,Cconsider agaugeinvariant quantity defined by Ωz, z Sz, zs z, z. 77 Then the Higgs field gauge field can be written as H S 1 H 0, 78 L z S 1 z S. 79 Here, H 0 z is the N N matrix has elements consisting of holomorphic functions of z. The first BPS equation 75 could be solved for arbitrary S on account of these relations. And the second BPS equation 76 is written in the form of z Ω 1 z Ω 1 Ω 1 H 0 H This equation is called master equation [6] for the vortices. The vortex number is given by Q i dx 1 dx TrF 1 0,±1,±,. 81 π From the master equation, at z Ω H 0 H 0 8 1

13 for vortex configurations, since the left side of 80 is z Ω 1 z Ω S 1 F L 1 S 0 at z. 83 Then, the vortex number 81 can be rewritten as Q k 1 4π Im dz z log deth 0 H 0 1 π Im dz z logdeth This representation for the topological charge makes it clear that H 0 behaves like deth 0 z k at the spacial infinity z. Moreover, H 0 z can be considered as a solution of the half-adhm equation [6] L Here, L H 0 z,j z, Ψ, 86 z Z H 0, J, Ψ Z are N N, k N, k N k k matrices, respectively. Ψ Z are constant matrices have a meaning of moduli parameters. As a result, BPS equations reduce to the master equation plus half-adhm equation by introducing variables S H 0. Here, H 0 is given as a solution of the half-adhm equation. And, for given H 0, S is solved as a solution of the master equation. Next, we extend the above argument to the case of UN UN gauge fields L µ R µ 1. The BPS equations are D z H z H +L z H HR z 0, 87 if1 L 1 HH, 88 if1 R H H Expressing the Higgs field gauge field as H S 1 z, zh 0 zt z, z, 90 L z S 1 z S, 91 R z T 1 z T, 9 BPS equation 87 is satisfied automatically BPS equations are reduced to the two master equations z Ω 1 S zω S 1 Ω 1 S H 0Ω T H 0, z Ω 1 T zω T 1+H 0 Ω 1 S H 0Ω T, 93 13

14 where Ω S Sz, zsz, z, Ω T T z, zt z, z. 94 At z, we can see the following. Finite energy of the static energy 5 means that UN UN gauge fields L µ R µ go to pure gauge configurations. It is possible to send the SUN SUN part of gauge fields to zero, because of the homotopy Then S T can be expressed by elements of U1 as π 1 SU N Sz, z sz, z 1 N, T z, z tz, z 1 N, 96 where sz, z tz, z are scalar functions. Defining S sz, ztz, z 1, 97 Higgs field 90 U1 part of the gauge field are expressed as H S H 0 98 TrL z R z S 1 z S. 99 Then, by the replacement S S, the topological charge 81 in UN YMH model reduces to that in UN UN model. As a result, vortex number Q L R, given by 10, can be expressed by Q L R 1 4π Im dz z log deth 0 H 0, 100 which is same as 84. Therefore, H 0 satisfies the common half-adhm equation in each case of YMH model with UN U N UN gauge groups. On the other h, the master equation turns to the coupled equations for Ω S Ω T in the YMH model with U N UN gauge group. Here, we comment on the vortex solutions of U1 U1 UN UN YMH models. It is known that when F 1 H are a numerical vortex solution of U1 YMH model, a vortex solution of UN YMH model can be constructed by embedding this vortex solution as F 1 UdiagF 1,0,,0U 1, H UdiagH,1,,1U 1, 101 where U takes a value in CP N 1 [6]. On the other h, we can show that a vortex solution of U1 U1 YMH model is expressed by that of U1 model by comparing both BPS equations. That is, denoting a vortex configuration with topological number m of U1 model 14

15 7 with g 4 c 1 as F 1 H, a vortex of the U1 U1 YMH model 1 with g c 1 is given by F L 1 F R 1 1 F 1, H H. 10 And a non-abelian vortex of the UN UN YMH model is constructed as F L 1 Udiag 1 F 1,0,,0U 1, F R 1 Udiag 1 F 1,0,,0U 1, H Udiag H,1,,1U As mentioned in section, it is obvious that the topological charge Q L+R 0 for the Abelian vortex 10 non-abelian vortex 103, since Tr F1 1 L +FR 0. And charge Q L R i dx 1 dx Tr F1 L i π FR 1 dx 1 dx F π 1 m counts the vortex number. Finally, we consider the relation between variables h h variables S, T H 0 in YMH model with U N UN gauge group. A relation for the variables can take the form h 0 b c 0 0 H0 T zt H0 SzS We can check that two formulations lead the same Higgs field gauge fields. As the formulation given above in this section, taking the variables S,T,H 0, Higgs field gauge fields are given by H S 1 z, zh 0 zt z, z, 105 L z S 1 z S, R z T 1 z T, 106 respectively. On the other h, for the formulation discussed in section 4, taking the variable h as 104, Higgs field gauge fields are given by H c 1 b S 1 H0 S 1 H0 T T, 107 L z c 1 z c S 1 z S, 108 R z b 1 z b T 1 z T. 109 Then, the condition that the two formulations give the same fields is H S 0 z 1 H T 0 z H 0 z. 110 There exists some ambiguity in the relations of variables. A simple relation is given by 0 b 0 H h 0 zt z, z. 111 c 0 Sz, z 0 15

16 6 Summary Discussion In this paper, we have defined the coordinates for noncommutative Z space have investigated the relation between the instantons in R Z space the vortices in R space. We have shown that the vortices of YMH model in R space can be regarded as the instantons of YM model in R Z space. The BPS equation for the vortices can be considered as a self-dual Yang-Mills equation is related to the ADHM equation. We also have obtained the relations between the master equation for the vortices the Yang equation for the instantons. It may be expected that the ADHM method can also be applied to the construction of the vortex solutions. However, extension of ADHM equation into the R Z space is not straightforward. The reason can be traced to noncommutativity of w w or 3 4. Writing the Dirac operators as where are quaternions. Square of Dirac operators are written as where D x e µ D µ e µ µ +A µ, D x ē µ D µ D x, 11 e µ iσ i,1,ē µ iσ i,1 113 D x D x 1 [] D µ D µ +iη i+ µν σ i D µ D ν, 114 η i± µν ǫ iµν4 ±δ iµ δ ν4 δ iν δ µ4 115 are t Hooft symbols. The last term of equation 114 can be written for R 4 space as iη i+ µν σ i D µ D ν iσ 1 {F 3 +F 14 }+iσ { F 13 +F 4 } +iσ 3 {F 1 +F 34 }, 116 the condition [ D x D x,σ i ] leads to the anti-self-dual equation For R Z space, however, we have F µν 1 ǫ µναβf αβ. 118 iη i+ µν σ i D µ D ν iσ 1 {F 3 +F 14 }+iσ { F 13 +F 4 } +iσ 3 {F 1 +F 34 +[ 3, 4 ]}, 119 because of noncommutativity of Z space 117 does not lead to the self-duality equation we have to find a different constraint. Furthermore, unlike the case of noncommutative ADHM, [ 3, 4 ] is not a constant, thus we have to find a different modification. Consequently, 16

17 it is possible that ADHM equations are not pure algebraic equations but include differential equations in R space. And this could be related to the fact that it is impossible to obtain the vortex solutions analytically. We have compared our YMH model that contains two gauge fields with YMH model with only one gauge field. In the latter model, we can rewrite the BPS equations into the master equation plushalf-adhmequation. Wecothesame intheformermodel, thebps equations also reduce to the master plus half-adhm equations the half-adhm equations in both models coincide exactly with each other. Furthermore we have studied both Abelian non-abelian vortices the interrelations among them. Although we have defined our Z through equations 16, 17, there exist other possibilities they are probably worthwhile to be considered. Furthermore, it has been proposed that there exists similar relation in the case of the model on compact Riemann surface [13]. It would be interesting to examine the relations between our work their approach. Acknowledgments We would like to thank Akihiro Nakayama for his support hospitality. We also thank Yoshimitsu Matsui for discussions. 17

18 References [1] A. Actor, Classical solutions of SU Yang-Mills theories, Rev. Mod. Phys ; M.F. Atiyah N.S. Manton, Skyrmions from instantons, Phys. Lett. B ; P.M. Sutcliffe, Instanton moduli topological soliton dynamics, Nucl. Phys. B , [arxiv:hep-th/ ]. [] H.B. Nielsen P. Olesen, Vortex-Line models for dual strings, Nucl. Phys. B [3] E.J. Weinberg, Multivortex solutions of the Ginzburg-Lau equations, Phys. Rev. D ; N.S. Manton S.M. Nasir, Volume of vortex moduli spaces, Commun. Math. Phys , [arxiv:hep-th/ ]; F.A. Schaposnik, Vortices, [arxiv:hep-th/061108]. [4] D. Tong, The moduli space of noncommutative vortices, J. Math. Phys , [arxiv:hep-th/010010]; A. Hanany D. Tong, Vortices, instantons branes, JHEP , [arxiv:hep-th/ ]; D. Tong, TASI lectures on solitons, [arxiv:hep-th/050916]; N. Sakai D. Tong, Monopoles, vortices, domain walls D-branes: The rules of interaction, JHEP , [arxiv:hep-th/050107]. [5] E.B. Bogomol nyi, Stability of classical solutions, Sov. J. Nucl. Phys ; M.K. Prasad C.M. Sommerfield, An exact classical solution for the t Hooft monopole the Julia-Zee dyon, Phys. Rev. Lett [6] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi N. Sakai, Moduli space of non-abelian vortices, Phys. Rev. Lett , [arxiv:hep-th/ ]; M. Eto, Y. Isozumi, M. Nitta, K. Ohashi N. Sakai, Solitons in the Higgs phase: The moduli matrix approach, J. Phys. A R315 R39, [arxiv:hep-th/060170]; M. Eto et al., Non-Abelian vortices on cylinder: Duality between vortices walls, Phys. Rev. D , [arxiv:hep-th/ ]; M. Eto et al., Non-Abelian vortices of higher winding numbers, Phys. Rev. D , [arxiv:hep-th/ ]; M. Eto et al., Constructing Non-Abelian vortices with arbitrary gauge groups, [arxiv: [hep-th]]; M. Eto et al., Multiple layer structure of non-abelian vortex, [arxiv: [hep-th]]. [7] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld Y.I. Manin, Construction of instantons, Phys. Lett. A

19 [8] R. Coquereaux, G. Esposito-Farese G. Vaillant, Higgs fields as Yang-Mills fields discrete symmetries, Nucl. Phys. B ; R. Coquereaux, G. Esposito-Farese F. Scheck, Noncommutative geometry graded algebras in electroweak interactions, Int. J. Mod. Phys. A ; K. Morita Y. Okumura, Weinberg-Salam theory in noncommutative geometry, Prog. Theor. Phys ; K. Morita Y. Okumura, Reconstruction of Weinberg-Salam theory in noncommutative geometry on M 4 Z, Phys. Rev. D ; J. C. Varilly J. M. Gracia-Bondia, Connes noncommutative differential geometry the Stard Model, J. Geom. Phys ; A. Sitarz, Higgs mass noncommutative geometry, Phys. Lett. B , [arxiv:hep-th/ ]. [9] H. Ikemori, S. Kitakado, H. Otsu T. Sato, Non abelian vortices as instantons on noncommutative discrete space, JHEP , [arxiv: [hep-th]]. [10] J.A. Harvey, Komaba lectures on noncommutative solitons D-branes, [hep-th/010076]; N. Nekrasov A. Schwarz, Instantons on noncommutative R,0 superconformal six dimensional theory, Commun. Math. Phys , [arxiv:hep-th/980068]; R. Gopakumar, S. Minwalla A. Strominger, Noncommutative solitons, JHEP , [arxiv:hep-th/ ]. [11] A.N. Leznov A.V. Razumov, The canonical symmetry for integrable systems, J. Math. Phys , [arxiv:hep-th/ ]. [1] C.N. Yang, Condition of selfduality for SU gauge fields on Euclidean four-dimensional space, Phys. Rev. Lett [13] O. Lechtenfeld, A.D. Popov R.J. Szabo, Quiver gauge theory noncommutative vortices, Prog. Theor. Phys. Suppl , [arxiv: [hep-th]]; A.D. Popov, Integrability of vortex equations on Riemann surfaces, [arxiv: [hep-th]]; O. Lechtenfeld, A.D. Popov R.J. Szabo, SU3-equivariant quiver gauge theories nonabelian vortices, JHEP , [arxiv: [hep-th]]; A.D. Popov, Non-Abelian vortices on Riemann surfaces: an integrable case, Lett. Math. Phys , [arxiv: [hep-th]]; A.D. Popov, Explicit non-abelian monopoles instantons in SUN pure Yang-Mills theory, Phys. Rev. D , [arxiv: [hep-th]]. 19