Noncommutative Vortices and Instantons from Generalized Bose Operators
aa r X i v : . [ h e p - t h ] S e p Noncommutative Vortices and Instantons fromGeneralized Bose Operators
Nirmalendu Acharyya a ∗ , Nitin Chandra a † and Sachindeo Vaidya a,b ‡ a Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India b Department of Physics, McGill University, Montr´eal, QC, Canada H3A 2T8
Abstract
Generalized Bose operators correspond to reducible representations of the harmonicoscillator algebra. We demonstrate their relevance in the construction of topologicallynon-trivial solutions in noncommutative gauge theories, focusing our attention to fluxtubes, vortices, and instantons. Our method provides a simple new relation betweenthe topological charge and the number of times the basic irreducible representationoccurs in the reducible representation underlying the generalized Bose operator. Whenused in conjunction with the noncommutative ADHM construction, we find that thesenew instantons are in general not unitarily equivalent to the ones currently known inliterature.
Contents ∗ [email protected] † [email protected] ‡ [email protected] Instantons 11 U (1) Instanton . . . . . . . . . . . . . . . . . . . . . . . 15 ¯ A ∞
18B Large Distance Behavior 19C The New Solution Satisfies The Equation Of Motion 20D The New Solution Satisfies The ASD Condition 20E The Topological Charge Of Instantons 21
E.1 Usual Single ASD solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21E.2 The New Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Quantum field theories on noncommutative space-times are of considerable interest for avariety of reasons: they can provide a self-consistent deformation of ordinary quantumfield theories at small distances, yielding non-locality [1–3], or create a framework for finitetruncation of quantum field theories while preserving symmetries [4–9]. Arguments com-bining quantum uncertainties with classical gravity also provide an alternative motivationto study these theories [10], and the emergence of noncommutative field theories in stringtheory [11–13] have provided a considerable impetus to their investigation.Detailed investigations of noncommutative gauge theories have led to the discovery oflocalized static classical solutions in noncommutative spaces [14–22]. Among the models ofgauge theories in noncommutative spaces, one of the simplest is the Abelian-Higgs modelpossessing vortex–like solutions [23–26]. Another interesting class of solutions in noncommu-tative euclidean space are instantons in U ( N ) Yang-Mills theories. Nekrasov and Schwarzdeveloped a generalization of ADHM construction as given in [27] to find these noncom-mutative instantons [28]. The U ( N ) Yang-Mills instantons in R NC and R NC × R C werefurther studied in [29–35]. Pedagogical reviews can be found in [36, 37]. Apart from these,there are multitudes of other solutions in noncommutative gauge fields like merons [34], fluxtubes [38], monopoles [39], dyons [40], skyrmions [41], false vacuum bubbles [24], to namejust a few.In this article we present a new construction of such topological objects, based on ananalysis of the reducible representations of the standard harmonic oscillator algebra. Ourmethod gives rise to new instanton solutions (i.e. not gauge equivalent to the known ones),and in the process provides a simple interpretation for the instanton number: it simply“counts” the number of copies of the basic irreducible representation.2ur construction relies on operators called generalized Bose operators [42, 43] whichprovide an explicit realization of the reducible representations of the oscillator algebra, andare well-known in the quantum optics literature (see for examples [44,45]). As a warm-up, wefirst study the significance of generalized Bose operators in constructing fluxes and vorticeswith higher winding numbers, and then discuss instanton solutions in noncommutative YMtheories.The article is organized as follows. We start with a brief review of generalized Bose op-erators and its representations in section 2. In section 3, we discuss the flux tube solutionsof [38] in the language of generalized Bose operators and then we go on to show the rele-vance of these operators in noncommutative Nielsen-Olesen vortices. Section 4 discusses thenoncommutative instantons. using the generalized Bose operator in conjunction with theADHM construction, We construct a class of new instantons and compute their topologicalcharges. Our conclusions are presented in section 5. Brandt and Greenberg [42] give a construction of generalized Bose operators that changethe number of quanta of the standard Bose operator a by 2 (or more generally by a positiveinteger k ). We briefly recall their construction in this section.Consider the infinite-dimensional Hilbert space H spanned by a complete orthonormalbasis {| n i , n = 0 , , · · · , ∞} labeled by a non-negative integer n . Vectors in H are of theform | ψ i = P n c n | n i , c n ∈ C ∀ n such that P n | c n | < ∞ .The standard bosonic annihilation operator a acts on this basis as a | n i = n | n − i , ∀ n ≥ a | i = 0 (2.1)The annihilation operator is unbounded, and hence comes with a domain of definition: D a = { X n c n | n i| X n n | c n | < ∞} (2.2)Its adjoint a † satisfies a † | n i = ( n + 1) | n + 1 i , ∀ n ≥ D a .The number operator N ≡ a † a has as its domain of definition D N , where D N = { X n c n | n i , | X n n | c n | < ∞} . (2.4)The basis vectors {| n i} are eigenstates of N : N | n i = n | n i . (2.5)On D N , the operators a and a † satisfy [ a, a † ] = 1 . (2.6)3hile N counts the number of quanta in a state, the a and a † destroy and create respectivelya single quantum. Thus ( a, H ) is a representation of the oscillator algebra (2.6). It is alsothe unique (up to unitary equivalence) irreducible representation of this algebra (see forexample [46]).The Hilbert space H can split into two disjoint subspaces H + = { P c n | n i ∈ H} and H − = { P c n +1 | n + 1 i ∈ H} : H = H + ⊕ H − . The (projection) operatorsΛ + = ∞ X n =0 | n ih n | , Λ − = ∞ X n =0 | n + 1 ih n + 1 | (2.7)project onto the subspaces H ± . On the subspaces H ± , the operators b ± and its adjoint b †± can be defined as b + | n i = n | n − i , b † + | n i = ( n + 1) | n + 2 i , b + | i = 0 ,b − | n + 1 i = n | n − i , b †− | n + 1 i = ( n + 1) | n + 3 i , b − | i = 0 (2.8)with domain of closure D a ∩ H ± .On the domain D N ∩ H ± we have [ b ± , b †± ] = 1. Thus ( b − , H − ), ( b + , H + ) and ( a, H )are isomorphic to each other. In other words, there exist unitary operators U ± such that U ± b ± U †± = a .Using the projection operators Λ ± , one can define an operator bb = b + Λ + + b − Λ − (2.9)on H whose action on the basis vectors | n i is b | n i = n | n − i , b | n + 1 i = n | n − i . (2.10)Notice that both | i and | i are annihilated by b .The operator b satisfies the commutation relation [ N, b ] = − b and a new numberoperator can be defined as M = b † b = ( N − Λ − ) which has the states | n i as eigenstates buteach eigenvalue is two-fold degenerate. We can denote these eigenvalues by m n = ( n − λ − )where λ − (= h n | Λ − | n i ) takes values 0 and 1 for even and odd n ’s respectively. Then (2.10)can be rewritten as b | n i = m n | n − i and b † | n i = ( m n + 1) | n + 2 i . (2.11)The operator b has domain of closure D a and satisfy [ b, b † ] = 1 in the domain D N andthus ( b, H ) forms a reducible representation of the algebra [ a, a † ] = 1 having (2.9) as itsirreducible decomposition.The above can be generalized to construct an operator b ( k ) which lowers a state | n i by k − steps. We start by defining projection operators Λ i byΛ i = ∞ X n =0 | kn + i ih kn + i | , i = 0 , , · · · k − . (2.12)4hat project onto subspaces H i = { P n c kn + i | kn + i i} . In each subspace H i , one can defineoperators b i and their adjoints b † i that satisfy [ b i , b † i ] = 1 and hence correspond to the UIRof the oscillator algebra. A reducible representation is given by b ( k ) = k − X i =0 b i Λ i , b i | kn + i i = √ n | kn + i − k i , H = k − X i =0 H i (2.13)with [ b ( k ) , b ( k ) † ] = 1. Again,( b i , H i ) is isomorphic to ( a, H ) and ( b ( k ) , H ) forms a reduciblerepresentation of the algebra [ a, a † ] = 1.The equations (2.9)–(2.11) represent the case k = 2, the simplest non-trivial example ofthis construction. Henceforth we will use b for b (2) . An explicit expression for b is [43] b = 1 √ (cid:18) a √ N a Λ + + a √ N + 1 a Λ − (cid:19) (2.14)Before we end this section, let us point out a minor generalization of the Brandt-Greenberg construction. Under any unitary transformation U that acts on b + as b + → U b + U † , the fundamental relation [ b + , b † + ] = 1 is unchanged. In particular, if we choose U ≡ U + ( z + ) = e z + b † + − ¯ z + b + , then we find that b + ( z + ) ≡ U + ( z + ) b + U † + ( z + ) = b + − z + , (2.15)i.e. we get the “translated” annihilation operator. We can construct a reducible represen-tation using b + ( z + ) and b − ( z − ) (defined similarly) as b ( z + , z − ) = b + ( z + )Λ + + b − ( z − )Λ − = b + Λ + + b − Λ − − z + Λ + − z − Λ − . (2.16)Here b gets translated by different amounts in different subspaces H ± and the “translated”operator b ( z + , z − ) is unitarily related to b as b ( z + , z − ) = U ( z + , z − ) bU † ( z + , z − ) , U ( z + , z − ) = U + ( z + )Λ + + U − ( z − )Λ − . (2.17)More generally, using (2.13) we can write b ( k ) ( z , z , · · · , z k − ) = b ( k ) − k − X i =0 z i Λ i (2.18)and the unitary operator is U ( z , z , · · · , z k − ) = P k − i =0 U i ( z i )Λ i . Though minor, this gen-eralization will play a role in the construction of noncommutative multi-instantons.There exist other possibilities as well. For example, choosing U ( λ + ) = e λ +2 ( b − b † ) gives b + ( λ + ) = b + cosh λ + + b † + sinh λ + . (2.19)The above is the well known squeezed annihilation operator. Using b + ( λ + ) and b − ( λ − ), areducible representation may be constructed: b ( λ + , λ − ) = b + ( λ + )Λ + + b − ( λ − )Λ − (2.20)and more generally, b ( λ , λ , · · · , λ k − ) = k − X i =0 b i ( λ i )Λ i (2.21)5 Static Solutions In Noncommutative Gauge Theories
We are interested in exploring the relevance of the generalized Bose operators b ( k ) in non-commutative field theories. Let us start by considering two simple situations: • the flux tube solution in (3 + 1) − dimensional pure gauge theory • the vortex solution in (2 + 1) − dimensional abelian Higgs model. Consider pure U (1) gauge theory in (3 + 1)-dimensional spacetime with only spatial non-commutativity. This theory incorporates magnetic flux tube solutions which are importantin the context of monopoles and strings.We are interested in the non-trivial solutions of the static equation of motion. These so-lutions do not possess a smooth θ → x , ˆ x ] = iθ, [ˆ x , ˆ x ] = 0 , [ˆ x , ˆ x ] = 0 , (3.1)so that only the ˆ x − ˆ x plane is noncommutative.In noncommutative space, “functions” are elements of the noncommutative algebra (i.e.operators) generated by the operators ˆ x i . We will work directly with such operators. Deriva-tives in the ˆ x and ˆ x directions are defined via the adjoint action ∂ x f = iθ (cid:2) ˆ x , f (cid:3) , ∂ x f = − iθ (cid:2) ˆ x , f (cid:3) . (3.2)while the derivatives in the ˆ x and t directions are the same as in the commutative case.We can define a set of complex (noncommuting) variables z and ¯ z and a set of creation-annihilation operators as z = 1 √ x + i ˆ x ) , ¯ z = 1 √ x − i ˆ x ) , a = 1 √ θ z, a † = 1 √ θ ¯ z (3.3)which satisfy [ a, a † ] = 1. With this convention, the derivatives with respect to the complexcoordinates are given as ∂ z f = − √ θ h a † , f i , ∂ ¯ z f = 1 √ θ [ a, f ] . (3.4)Ordinary products become operator products and the integration on x − x plane isreplaced by trace over Fock space H : R dx dx f ( x , x ) → πθ Tr H ˆ f (ˆ x , ˆ x ).Taking A µ as anti-hermitian operator valued functions, the action of U (1) pure gaugetheory in the above noncommutative space is S = − πθ g Z dx dt Tr { F µν } (3.5)6here the field strength is F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ].We can define noncommutative covariant derivative as D µ = iθ µν x ν + A µ , ( A µ ) † = − A µ . (3.6)Noncommutative gauge theory has an alternative (and possibly more natural) formulationin terms of D µ rather than A µ . In terms of D µ , one works with the actionˆ S = − πθ g Z dx dt Tr { ˆ F µν } , where ˆ F µν = [ D µ , D ν ] , (3.7)which is (classically) equivalent to the S : ˆ S = S + πg θ R dtdx (Tr I − iθF ) = S +Constantterm + boundary term. Thus both ˆ S and S give the same equations of motion.For static, magnetic configurations ( ∂ t = 0 = A ) and with the choice ∂ A i = 0 , D = 0,we have [ D , [ ¯ D , D ]] = 0 , with D = 1 √ D + i D ) , ¯ D = 1 √ D − i D ) . (3.8)It is easy to check that D = a √ θ is a solution of (3.8), which is the vacuum configurationand has field strength F µν = 0. We can construct solutions about this vacuum by taking arotationally invariant ansatz D = af ( N ) and it can be shown that there exists a solutionof (3.8) D = a √ θ r N − n N ∞ X n = n | n ih n | , N = a † a, n = 0 , , .... (3.9)with a localized flux representing classical static circular magnetic flux tubes in x directioncentered about origin of the ( x , x ) plane with n related to its radius. The choice n = 0corresponds to the vacuum configuration and has zero magnetic flux.By virtue of the relation [ b, b † ] = 1, D = b √ θ is also a solution of (3.8). Again, we canstart with the ansatz D = G ( N ) b to construct the following solution of (3.8) D = b √ θ r N − n + Λ + − n − Λ − M ∞ X n = n + | n ih n | Λ + + ∞ X n = n − | n ih n | Λ − , n ± = 0 , , ... (3.10)This solution can be re-written in the form D = P g ( n ) | n ih n + 2 | : D = 1 √ θ ∞ X n = n + λ + r n − n + + 22 | n ih n + 2 | + ∞ X n = n − λ − r n − n − + 22 | n ih n + 2 | (3.11)which gives g (2 n ) = p n − n ′ + + 1 for n ≥ n ′ + and g (2 n + 1) = p n − n ′− + 1 for n ≥ n ′− .This solutuion is same as the higher moment solution obtained in [38] starting with theansatz D = F ( N ) a . Here λ + = cos (cid:0) nπ (cid:1) and λ − = sin (cid:0) nπ (cid:1) are eigen functions of Λ ± and n ′± is defined as n ′ + = n + , n ′− = n − − . Again, for the choice n + = 0 and n − = 1, thissolution reduces to D = b √ θ . 7urthermore, D = b ( k ) √ θ for k ≥ D = f ( n ) b ( k ) , thesolution D = b ( k ) √ θ s N − P ki =1 n i Λ i M ( k ) k X i =1 ∞ X n = n i | n ih n | Λ i , M ( k ) = b ( k ) † b ( k ) , n i = 0 , , ... (3.12)can be constructed, which is also same as the higher moment solution obtained from theansatz D = g ( n ) a k in [38]. These solutions with the generalized Bose operator representmagnetic flux tubes, nonperturbative in θ , with localised flux and varying radial profile de-termined by the set { n i } . This shows that there already exist certain solutions of noncom-mutative gauge theory which can be re-written in terms of the generalized Bose operators.This fact shows the importance of generalized Bose operators and motivates us to seek suchsolutions in other noncommutative gauge theories. The abelian Higgs model in noncommutative spaces is of some interest because of its sim-plicity as a noncommutative gauge theory and the existence of vortex solutions. Varioustopologically non-trivial vortex solutions in this context have been studied in detail.An interesting class of vortex solutions in this theory is studied in [23], which areanalogous to the Nielsen-Olesen Vortices in the commutative space. The model is in(2 + 1) − dimensions, and consists of a complex Higgs filed Φ which is a left gauge module(the gauge fields multiply the complex Higgs field Φ from left and ¯Φ from right). Minimizingthe static noncommutative energy functional, Bogomolnyi equations are generalized to thenoncommutative space and 1 /θ -expansion is done in large θ limit. The equations are thensolved order by order and the corrections to the leading order equation converge rapidly. Inthe large distance limit (which is the commutative limit in this case), the solution reducesto the Nielsen-Olesen vortex solution in ordinary (commutative) abelian Higgs model.The noncommutativity is same as that in section (3.1), with the only difference thatnow the space is 2 − dimensional, so the direction x is absent. The energy functional in thestatic configuration is given by [23] E = Tr (cid:20)
12 ( B + Φ ¯Φ − + D ¯ z Φ D z ¯Φ + T (cid:21) (3.13)where D is a covariant derivative with the gauge field A . The magnetic field B is definedas B = − i ( ∂ z A ¯ z − ∂ ¯ z A z ) − [ A z , A ¯ z ] and T is the topological term defined as T = D m S m + B where D m S m = ∂ m S m − i [ A m , S m ] , m = z or ¯ z. (3.14)with S m = i ǫ mn (Φ D n ¯Φ − D n Φ ¯Φ) . It can be shown that Tr T corresponds to the topologicalcharge.Our prime interest is to study the Bogomolnyi equations. Minimizing (3.13) we get thefollowing operator equations: D ¯ z Φ = 0 , D z ¯Φ = 0 , B = 1 − Φ ¯Φ . (3.15)8hich are the noncommutative Bogomolnyi equations. Now we can do a 1 /θ expansion ofthe Higgs field, gauge field and the magnetic field in the large θ limit asΦ = Φ ∞ + θ Φ − + .....A = √ θ ( A ∞ + θ A − + ..... ) B = θ ( B ∞ + θ B − + ..... ) . (3.16)The factor of √ θ and θ is used for scaling the variables A and B as they are are 1-form and2-form respectively. With this expansion, we can get the leading order O ( θ ) Bogomolnyiequation as Φ ∞ ¯Φ ∞ = 1 . (3.17)This equation admits a solution [47]Φ ∞ = 1 √ a n a † n a n , ¯Φ ∞ = (Φ ∞ ) † (3.18)which represents a n -vortex at origin. A more general solution is discussed in [23] whichrepresents n single vortices at n different points in the noncommutative plane and (3.18) isa special case of that general solution. But for our discussion, (3.18) is sufficient and dueto its simple form, computation and understanding becomes easier.Next we take the O (1) Bogomolnyi equation[ a, Φ ∞ ] = i ¯ A ∞ Φ ∞ (3.19)which can be solved to get (for details see appendix A)¯ A ∞ = − i √ N + 1 a (cid:16) √ N − √ N + n (cid:17) , A ∞ = ( ¯ A ∞ ) † (3.20)where N is the number operator.In the coherent state | ω i ( a | ω i = ω | ω i ), the expectation of the field Φ ∞ is h ω | Φ ∞ | ω i = ω n h ω | √ a n a † n | ω i with ω = | ω | e inϕ . (3.21)Here the phase dependence is e inϕ which comes solely from ω n as the other factor h ω | √ a n a † n | ω i is purely real, signifying a vortex in the noncommutative plane. The large distancebehavior is given by the large ω limit or equivalently large h N i limit. The coherent stateexpectations in this limit becomes (for details see appendix B) h ω | Φ ∞ | ω i ≈ e inϕ , h ω | ¯ A ∞ | ω i ≈ i n ω , h ω | A ∞ | ω i ≈ − i n ω , (3.22)which is exactly like the commutative n -Nielsen-Olesen Vortex.It is interesting to note that the leading order magnetic field is B ∞ = n | ih | , whichmeans the magnetic field of the solution is localized and the magnetic fluxes are confined.This problem is qualitatively similar to flux tube problem in Section 3.1, where we saw thatthe theory has solutions involving the generalized Bose operator b . This fact stimulates9s to seek vortex solutions with operator b in the Higgs model, i.e., a solution to (3.17)involving the b ’s and a corresponding new gauge field A ∞ . With this motivation in mindone can easily check Φ new ∞ = 1 √ b n b † n b n , ¯Φ ∞ = (Φ ∞ ) † (3.23)satisfies (3.17) and using this, as before, we can find (details in appendix A)¯ A new ∞ = − i (cid:18) a − √ b n b † n b n ab † n √ b n b † n (cid:19) , A new ∞ = ( ¯ A ∞ ) † . (3.24)The expectation value in the coherent state | ω i (eigenstate of a ) gives a phase dependenceof e i nϕ (as Φ new ∞ can always be reduced to the form F ( N ) a n ), a characteristic feature of2 n vortex in noncommutative plane. In the large ω limit it gives the large distance behavior: h ω | Φ new ∞ | ω i ≈ e i nϕ , h ω | ¯ A new ∞ | ω i ≈ i n ¯ ω , h ω | A new ∞ | ω i ≈ − i nω (3.25)which is exactly the commutative 2 n Nielsen-Olesen vortex.Now we need to compare the new solution (3.23) with the Witten’s solution (3.18). Forthe simplicity of expression and better understanding of the underlying algebra, we take n = 1 in (3.23). Using the explicit expression of the generalized Bose operator b , the newvortex solution can be written asΦ new ∞ = √ bb † b = 1 √ M + 1 1 √ a √ N a Λ + + a √ N + 1 a Λ − ) (3.26)where we know, M = b † b = N − Λ − . Further simplification can be done and the expression(3.26) for the new vortex reduces toΦ new ∞ = ( 1 √ N − Λ − + 2 1 √ N + 1 Λ + + 1 √ N − Λ − + 2 1 √ N + 2 Λ − ) a . (3.27)The eigenvalues of the projection operators Λ ± are 0 and 1 and they never contributesimultaneously. Owing to this fact the expression (3.27) simplifies toΦ new ∞ = 1 √ N + 2 1 √ N + 1 (Λ + + Λ − ) a = 1 p ( N + 1)( N + 2) a = 1 √ a a † a (3.28)which is same as the n = 2 Witten’s vortex. This calculation can be generalized for any n and it can be always shown that n -new vortex solution is same as the 2 n -Witten vortexfor all n . It is also easy to show that Φ ∞ = √ ( b ( k ) ) n ( b ( k ) † ) n ( b ( k ) ) n is also a solution of (3.17)and this solution is same as the kn -Witten vortex.This shows that the vortex solutions with the generalized Bose operators are alreadypresent in the old solutions but are subtly hidden. It is also very interesting to see how theprojectors intelligently conspire to make it happen.10 Instantons
Instantons are localized finite action solutions of the classical Euclidean field equations of atheory (for a review see [48]). The finite action condition is satisfied only if the Lagrangiandensity of the theory vanishes at boundary. This in turn can lead to different topologicalconfigurations of the field characterized by its “topological charge”. For Yang-Mills theories,the instantons are further classified as Self-Dual (SD) or Anti-Self-Dual (ASD) with theirtopological charges having opposite signs. A simple prescription to construct (anti-) self-dual instantons in the Yang-Mills theory is given in [27]. Let us first review this construction.In order to describe charge k instantons with gauge group U ( N ) on R one starts withthe following data:1. A pair of complex hermitian vector spaces V = C k and W = C N .2. The operators B , B ∈ Hom(V , V), I ∈ Hom(W , V), J ∈ Hom(V , W), which mustobey the equations[ B , B † ] + [ B , B † ] + II † − J † J = 0 , [ B , B ] + IJ = 0 (4.1)For z = ( z , z ) ∈ C ≈ R , define an operator D : V ⊕ V ⊕ W → V ⊕ V as D † = (cid:18) τσ † (cid:19) , τ = (cid:0) B − z B − z I (cid:1) , σ = − B + z B − z J (4.2)for anti-self-dual instantons and by D † = (cid:18) τσ † (cid:19) , τ = (cid:0) B − ¯ z B + z I (cid:1) , σ = − B − z B − ¯ z J (4.3)for self-dual instantons. Given the matrices B , B , I and J obeying all the conditionsabove, the actual instanton solution is determined by the following rather explicit formulae: A α = Ψ † ∂ α Ψ , (4.4)( α = 1 , , ,
4) where Ψ : W → V ⊕ V ⊕ W is the normalized solution of D † Ψ = 0 , Ψ † Ψ = 1 . (4.5)Here ∂ α is derivative with respect to the spacetime coordinates x α which are related to the z -coordinates as z = x + ix √ , ¯ z = x − ix √ , z = x + ix √ , ¯ z = x − ix √ . (4.6)For given ADHM data and the zero mode condition (4.5), the following completeness relationhas to be satisfied D D † D D † + ΨΨ † = 1 . (4.7)11t has been shown in [30] that (4.7) can be satisfied even for noncommutative spaces. Notethat the fields A α are anti-hermitian, consistent with Ψ † Ψ = 1. The field strength F αβ andits dual ˜ F αβ are given as F αβ = ∂ α A β − ∂ β A α + [ A α , A β ] , ˜ F αβ = 12 ǫ αβγδ F γδ . (4.8)The instantons found by the ADHM construction satisfy both the Yang-Mills equation ofmotion and the (anti-) self-duality condition D α F αβ = 0 , ˜ F αβ = ± F αβ (4.9)and has topological charge given by Q = − π Z d xT r ( ˜ F αβ F αβ ) . (4.10) To study instantons on a noncommutative R , we will use the notation outlined below. The4-dimensional noncommutative euclidean space is defined by the following noncommutativecoordinates: [ˆ x α , ˆ x β ] = iθ αβ , α, β = 1 , , , , (4.11)and θ αβ is a constant anti-symmetric 4 × x α ’s by A θ . There are three distinct cases one may consider:1. θ has rank 0 ( θ αβ = 0 ∀ α, β ). In this case A θ is isomorphic to the algebra of functionson the ordinary R . This space may be denoted by R C .2. θ has rank 2. In this case A θ is the algebra of functions on the ordinary R times thenoncommutative R , which may be denoted by R NC × R C . Without loss of generality,we can choose θ αβ = θ − θ . Let us define a system of complex coordinates and a set of operators as z = √ ( x + ix ) ¯ z = √ ( x − ix ) z = √ ( x + ix ) ¯ z = √ ( x − ix ) a = ¯ z √ θ a † = z √ θ a = ¯ z √ θ a † = z √ θ (4.12)which reduces the algebra (4.11) to[¯ z , z ] = θ, [¯ z , z ] = 0 , [ a , a † ] = 1 , [ a , a † ] = 0 . (4.13)Here a and a † are like annihilation and creation operators respectively while a † , a are ordinary complex numbers. We can define a number operator by N = a † a . TheFock space on which the elements of A θ act, consists of states denoted by | n , z i . Here n denotes the eigenvalues of the number operator N and can take only non-negativeintegral values, while z can be any complex number and denotes the eigenvalues of z . 12. θ has rank 4. In this case A θ is the noncommutative R . We choose θ to be of theform given by θ αβ = θ − θ θ − θ = θ − θ θ − θ , where we have assumed θ = θ = θ . Again, we can define a system of complexcoordinates and a set of operators as in (4.12) but now the algebra (4.11) becomes[¯ z , z ] = [¯ z , z ] = θ, [ a , a † ] = [ a , a † ] = 1 . (4.14)The Fock space on which the elements of A θ act consists of states denoted by | n , n i .Here n and n denote the eigenvalues of the number operators N = a † a and N = a † a respectively which can take only non-negative integral values.As already mentioned in Section 3.1, differentiation in the noncommutative space is impli-mented as an adjoint as in (3.2), while the integration is implimented by a suitable trace. ADHM construction for instantons has been generalized to a noncommutative space in [28].The construction effectively remains same as in the commutative case, only change being thereplacement of 0 in the right hand side of the first equation of (4.1) by the noncommutativeparameter θ for the case of R NC × R C and by 2 θ for the case of R NC × R NC respectively:[ B , B † ] + [ B , B † ] + II † − J † J = θ, [ B , B ] + IJ = 0 (4.15)for R NC × R C as in [32] and[ B , B † ] + [ B , B † ] + II † − J † J = 2 θ, [ B , B ] + IJ = 0 (4.16)for R NC × R NC . In noncommutative space, the gauge field is defined as ˆ D x α = − iθ αβ x β + A x α where A x α is the Yang-Mills gauge connection. Then the ansatz for gauge field isˆ D x α = − iθ αβ Ψ † x β Ψ (4.17)where θ αβ satisfies θ αβ θ βγ = δ αγ . The fields A x α are again anti-hermitian. In R NC × R C ,the components of the gauge field along the commutative directions will be given by (4.4),while those along the noncommutative axes by ˆ D x α .Let us first discuss the usual single anti-self-dual U (1) instanton solutions ( k = 1 , N = 1)in R NC × R C [31, 32] and in R NC × R NC [30, 31, 35, 37]. For k = N = 1, B , B , I and J areall complex numbers. As the noncommutative space (described by the coordinates z ) hastranslational invariance, we can always choose the origin in such a way that B and B in τ can be taken to be zero. Thus (4.15) or (4.16) ensures that either I or J is zero. Withoutthe loss of any generality we can choose J = 0. Now let us discuss the two cases seperately.13 NC × R C : Here we get I = √ θ from the equation (4.15). The phase in I does not effect the solutionfor the gauge field and hence has been taken to be zero. The operator (4.2) for anti-self-dualinstantons becomes D † = (cid:18) − z − z √ θ ¯ z − ¯ z (cid:19) = √ θ (cid:18) − a † − a † a − a (cid:19) (4.18)and its normalized zero mode solution is given byΨ = ψ ψ ξ ; ψ = a √ δ ∆ , ψ = a √ δ ∆ , ξ = r δ ∆ (4.19)with δ = a † a + a † a and ∆ = δ + 1 . Note that the inverse of the operator ∆ is well–defined,but that of δ is not since | n = 0, z = 0 i is a zero–mode of δ . R NC × R NC : In this case, (4.16) gives I = √ θ and the operator in (4.2) becomes D † = (cid:18) − z − z √ θ ¯ z − ¯ z (cid:19) = √ θ (cid:18) − a † − a † √ a − a (cid:19) . (4.20)The zero mode solution is again a 3-element column matrix Ψ where we write ψ = √ θa v and ψ = √ θa v . Then (4.5) becomesˆ∆ v = √ θξ, v † ˆ∆ v + ξ † ξ = 1 (4.21)where ˆ∆ = θ ˆ N = θ ( a † a + a † a ). But this operator does not have an inverse since it has azero–mode ˆ∆ | , i = 0 (4.22)and hence finding v and ξ is a bit tricky. We define a shift operator S such that SS † = 1 , S † S = 1 − P, P = | , ih , | . (4.23)Note that although the inverse of ˆ∆ is not defined otherwise, it is well–defined when sand-wiched between S and S † . Now we can solve for ξ and v : ξ = ˆΦ − S † , v = √ θ − S † , where ˆΦ = 1 + 2 θ ˆ∆ = 1 + 2ˆ N (4.24)which satisfy (4.21).Thus we get ψ = √ θa √ θ − S † , ψ = √ θa √ θ − S † , ξ = ˆΦ − S † . (4.25)14e can define the components of the gauge field in terms of the complex coordinates asˆ D = ˆ D x + i ˆ D x , ˆ D = ˆ D x + i ˆ D x , ˆ D ¯1 = ˆ D x − i ˆ D x , ˆ D ¯2 = ˆ D x − i ˆ D x . (4.26)Then the ansatz (4.17) translates intoˆ D a = 1 θ Ψ † ¯ z a Ψ , ˆ D ¯ a = − θ Ψ † z a Ψ = − ˆ D † a . (4.27)The solution for k = 1 U (1) ASD instanton becomesˆ D a = 1 √ θ S ˆΦ − a a ˆΦ S † , ˆ D ¯ a = − √ θ S ˆΦ a † a ˆΦ − S † (4.28)where the shift operator S , written explicitly, is S = 1 − ∞ X n ′ =0 | n ′ , ih n ′ , | (cid:18) − √ N + 1 a (cid:19) S † = 1 − ∞ X n ′ =0 (cid:18) − a † √ N + 1 (cid:19) | n ′ , ih n ′ , | . (4.29)In our notation of complex coordinates, the field strength is given by F a ¯ a = 1 θ + h ˆ D a , ˆ D ¯ a i , F a ¯ b = h ˆ D a , ˆ D ¯ b i ; a, b = 1 , a = b (4.30)and the ASD condition translates to F = − F , F = F ¯1¯2 = 0 . (4.31)The equations of motion in the R NC × R NC in terms of the fields ˆ D x α are[ ˆ D x α , [ ˆ D x α , ˆ D x β ]] = 0 . (4.32)The solution (4.28) satisfies both the ASD condition and the equations of motion. U (1) Instanton
We can try to get new solutions for noncommutative instantons by using the the generalizedBose operators. R NC × R C : Let us first define two operators b and cb = √ a √ N a Λ + + √ a √ N +1 a Λ − ,c = √ a √ N a Λ + + √ a √ N +1 a Λ − (4.33)15here N = a † a , Λ + = cos ( πN ) , Λ − = sin ( πN ) . Here b is the generalized operatordefined in (2.14). We can easily show that b † b = 12 ( N − Λ − ) , c † c = 12 N N − Λ − N − , where N = a † a . (4.34)The zero-mode solution Ψ is given by ψ = √ c, ψ = √ b, ξ = δ (cid:18) √ N a Λ + + 1 √ N + 1 a Λ − (cid:19) . (4.35)But this solution is not normalized i.e. Ψ † Ψ = 1. Usually the single instanton solution in R NC × R C with U (1) gauge group is normalized as [31]Ψ = Ψ q Ψ † Ψ . (4.36)But in our solution this technique cannot be used because in this case Ψ † Ψ = N − Λ − N − δ ( δ − | i and the inverse of Ψ † Ψ does not exist. We fixthis problem by definingΨ new = Ψ (1 − p ) 1 q Ψ † Ψ (1 − p ) u † , uu † = 1 , u † u = 1 − p (4.37)where p = | ih | is a projection operator, and u † is a shift operator projecting out thevacuum. The operator u can be written as u = ∞ X n =0 | n , z ih n + 1 , z | , u † = ∞ X n =0 | n + 1 , z ih n , z | . (4.38)It should be noted that the new solution in R NC × R C is completely non-singular. R NC × R NC : We claim the new solution to beˆ D a = 1 √ θ S new ˆΦ − new b a ˆΦ new S † new , ˆ D ¯ a = − √ θ S new ˆΦ new b † a ˆΦ − new S † new (4.39)where ˆΦ new = 1 + M and M = M + M = P a =1 b † a b a . Here again the operator M is notwell-defined otherwise (as M | , i = M | , i = M | , i = M | , i = 0), but is well-definedwhen sandwiched between S new and S † new (defined below) which projects out the states | , i , | , i , | , i and | , i : S new S † new = 1 , S † new S new = 1 − P new , P new = | , ih , | + | , ih , | + | , ih , | + | , ih , | . (4.40)16he explicit form for the new shift operator is as follows S new = 1 − ∞ X n ′ =0 | n ′ , ih n ′ , | (cid:18) − √ M + 1 b (cid:19) − ∞ X n ′ =0 | n ′ , ih n ′ , | (cid:18) − √ M + 1 b (cid:19) S † new = 1 − ∞ X n ′ =0 (cid:18) − b † √ M + 1 (cid:19) | n ′ , ih n ′ , | − ∞ X n ′ =0 (cid:18) − b † √ M + 1 (cid:19) | n ′ , ih n ′ , | (4.41)The operator b a (corresponding to a a ) is defined as in (2.14). We can check that this newsolution satisfies the ASD condition (4.31) and the equations of motion (4.32) (for detailssee Appendices C and D). The topological charge of this new solution can be shown to be4 times the charge of the usual single ASD instanton: Q new = − k = − U (1) gauge group despitethe topological charge being the same. We can understand this by observing that thedifference q between the numbers of a ’s and a † ’s in the two solutions is not same: q = 1 forthe usual ADHM solutions irrespective of its charge and the gauge group, whereas q = 2for the new solution (coming solely because of the operator b in the expression of ˆ D a givenby (4.39)). (The number of a ’s in Ψ is equal to that of a † ’s in Ψ † and vice-versa.)The new solution cannot be reduced to the usual instanton by a unitary transformationand hence represents gauge inequivalent configuration.If we use the operator b a ( z a + , z a − ) defined in (2.17), we can construct an instantonsolution by repeating the steps we have outlined above. This instanton also has charge − z , z , z − , z − can be thought as characterizing the “locations” offour instantons with charge −
1. It is easy to see that in the coincident limit z = z = z − = z − = 0, we recover (4.39).We can use the above technique to find a new solution in terms of the generalized Boseoperator b ( p a ) a : ˆ D a = 1 √ θ S new ( ˆΦ new ) − b ( p a ) a ( ˆΦ new ) S † new , ˆΦ new = 1 + 2 M , M = M ( p )1 + M ( p )2 , M ( p a ) a = b ( p a ) † a b ( p a ) a S new = 1 − p − X i =0 ∞ X n ′ =0 | n ′ , i ih n ′ , i | − q M ( p )1 + 1 b ( p )1 , (4.42)with S new satisfying S new S † new = 1 and S † new S new = 1 − i = p − ,j = p − X i =0 ,j =0 | i, j ih i, j | . This solutionrepresents an ASD instanton with charge Q = − p p . Again, it is gauge inequivalent to the k = − p p instanton known in the literature. We could as well have used a different shiftoperator given by S ′ new = 1 − p − X i =0 ∞ X n ′ =0 | i, n ′ ih i, n ′ | − q M ( p )2 + 1 b ( p )2 . (4.43)17heir actions are given by S † new | n , n i = (cid:26) | n + p , n i if 0 ≤ n ≤ p − | n , n i if n ≥ p , (4.44) S ′† new | n , n i = (cid:26) | n , n + p i if 0 ≤ n ≤ p − | n , n i if n ≥ p . (4.45)Other multi–instanton solutions can be constructed using the reducible representationsinvolving squeezed operators (2.21). Our construction of multi–instantonic solutions usingreducible representations of the standard harmonic oscillator algebra may also be generalizedfor 4k-dimensional instatons as discussed in [49, 50]. This exercise will be left as a futurework . We have described static classical solutions of noncommutative gauge theories in variousspacetime dimensions and showed that the generalized Bose operators are significant inconstructing solutions with higher topological numbers. While the flux tubes and vorticeswith higher winding numbers correspond to known solutions, the case of multi–instantonsis different. The multi–instantons with charge − p p ( p , p non-negative integers) are notgauge equivalent to known solutions. Another significant result of this article is an explicitrelation between the instanton number and the representation theory labels p and p .Using the “translated” b operators (2.17) we have constructed multi-instantons that de-pend explicitly on p p complex parameters. While the full moduli space of noncommutativemulti–instantons is still not well understood, we hope that this identification contributespartially to this question.Though we have only considered a few cases, there is actually a large variety of situ-ations in noncommutative gauge theories where generalized Bose operators may be used.In particular we expect that this procedure may shade new light on merons, monopoles,dyons, skyrmions etc. We plan to revisit some of these questions in future. Acknowledgement
SV would like to thank the High Energy Theory Group at McGill University for financialsupport.
AppendicesA Determination Of ¯ A ∞ ¯ A ∞ can be determined by multiplying (3.19) with ¯Φ ∞ from right and using (3.17)¯ A ∞ = − i ( a − Φ ∞ a ¯Φ ∞ ) . (A.1) We thank T. A. Ivanova for pionting out these references to us. A ∞ . We know that a n a † n = ( N + n )( N + n − ..... ( N + 1) . (A.2)Using this, we can show that¯ A ∞ = − i (cid:0) a − Φ ∞ a ¯Φ ∞ (cid:1) (A.3)= − i a − a p ( N + n )( N + n − ..... ( N + 1) p ( N + n − N + n − .....N ! (A.4)= − ia √ N − √ N + n √ N (A.5)= − i √ N + 1 a (cid:16) √ N − √ N + n (cid:17) . (A.6)Similar technique gives us ¯ A new ∞ as in (3.24). B Large Distance Behavior
Let | ω i be the coherent states of the operator a , where ω is a complex number labeling thestates. In the large ω limit, the expectation value h ω | Φ ∞ | ω i and h ω | ¯ A ∞ | ω i gives the largedistance behavior of the solution: h ω | Φ ∞ | ω i = h ω | √ a n a † n a n | ω i (B.1)= ω n h ω | p ( N + n )( N + n − .... ( N + 1) | ω i (B.2) ≈ ω n h ω | √ N ) n | ω i (B.3) ≈ ω n ( h ω | N | ω i ) n (B.4) ≈ ω n ωω ) n (B.5) ≈ e inϕ . (B.6)For the gauge field, h ω | ¯ A ∞ | ω i = h ω | − i √ N + 1 a ( √ N − √ N + n ) | ω i (B.7) ≈ − i h ω | N + 1 √ N + 1 a (1 − (1 + n N ) | ω i (B.8) ≈ i n h ω | a N | ω i (B.9) ≈ i n h ω | N a | ω i (B.10) ≈ i n ω . (B.11)Similar thing can done for Φ new ∞ and ¯ A new ∞ to get (3.25).19 The New Solution Satisfies The Equation Of Motion
The equations of motion are[ ˆ D x β , [ ˆ D x β , ˆ D x α ]] = 0 , α, β = 1 , , , . (C.1)There are four equations for different values of α . We will show only for α = 1 and otherfollow similarly. For α = 1 the equation becomes[ ˆ D x , [ ˆ D x , ˆ D x ]] + [ ˆ D x , [ ˆ D x , ˆ D x ]] + [ ˆ D x , [ ˆ D x , ˆ D x ]] = 0 . (C.2)In terms of the complex coordinates, (cid:16) [ ˆ D , [ ˆ D ¯1 , ˆ D ]] + [ ˆ D , [ ˆ D ¯2 , ˆ D ]] (cid:17) + (cid:16) [ ˆ D ¯1 , [ ˆ D ¯1 , ˆ D ]] − [ ˆ D ¯2 , [ ˆ D , ˆ D ¯1 ]] (cid:17) = 0 . (C.3)Now using the explicit expression for the solutions (4.39) we get[ ˆ D ¯1 , [ ˆ D ¯1 , ˆ D ]] = S new b † r M + 3 M + 1 r MM + 2 (cid:18) M + 2 M M − M + 1 M − MM + 2 M + 3 M + 1 ( M + 1) − MM + 2 M + 3 M + 1 ( M + 1) + M + 1 M + 3 M + 4 M + 2 ( M + 2) (cid:19) S † new , (C.4)[ ˆ D ¯2 , [ ˆ D , ˆ D ¯1 ]] = − S new b † r M + 3 M + 1 r MM + 2 (cid:18) M + 2 M M − M + 1 M − MM + 2 M + 3 M + 1 M − MM + 2 M + 3 M + 1 ( M + 1) + M + 1 M + 3 M + 4 M + 2 ( M + 2) (cid:19) S † new . (C.5)Adding above two we get [ ˆ D ¯1 , [ ˆ D ¯1 , ˆ D ]] + [ ˆ D ¯2 , [ ˆ D , ˆ D ¯1 ]] = 0 . (C.6)Similarly we can show [ ˆ D ¯1 , [ ˆ D ¯1 , ˆ D ]] − [ ˆ D ¯2 , [ ˆ D , ˆ D ¯1 ]] = 0 and hence the equation of motionis satisfied. D The New Solution Satisfies The ASD Condition
The commutator between ˆ D a and ˆ D ¯ a is[ ˆ D a , ˆ D ¯ a ] = − θ S new (cid:18) MM + 2 M + 3 M + 1 ( M a + 1) − M + 2 M M − M + 1 M a (cid:19) S † new . (D.1)Summing over a = 1 , D , ˆ D ¯1 ] + [ ˆ D , ˆ D ¯2 ] = − θ S new (cid:18) M M + 3 M + 1 − ( M + 2) M − M + 1 (cid:19) S † new = − θ . (D.2)Therefore, F + F = θ + [ ˆ D , ˆ D ¯1 ] + [ ˆ D , ˆ D ¯2 ] = 0. Similarly one can show F = F ¯1¯2 = 0.20 The Topological Charge Of Instantons
The topological charge of the instanton is defined as Q ∝ θ T r ( F F − F F ) = ⇒ Q = cθ ∞ X n ,n =0 h n , n | ( F F − F F ) | n , n i (E.1)where we denote the constant of proportionality by c .Further, the ASD condition gives F F = − F = − (cid:18) θ + 2 θ h ˆ D , ˆ D ¯1 i + h ˆ D , ˆ D ¯1 i (cid:19) . (E.2) E.1 Usual Single ASD solution
The solutions (4.28) can be simplified toˆ D a = 1 √ θ S s N ( N + 3)( N + 1)( N + 2) a a S † , ˆ D ¯ a = − √ θ Sa † a s N ( N + 3)( N + 1)( N + 2) S † . (E.3)An arbitrary commutator becomes h ˆ D a , ˆ D ¯ b i = − θ Sa † b a a N ( N + 1)( N + 2) S † − δ ab θ S N ( N + 3)( N + 1)( N + 2) S † (E.4)and the components of the field strength (and their products) can be calculated to be F F = − θ S ( N − N ) N ( N + 1) ( N + 2) S † . (E.5)Again using (E.4) we get F = − θ Sa † a N ( N + 1)( N + 2) S † , F = − θ Sa † a N ( N + 1)( N + 2) S † . (E.6)Multiplying the above two and simplifying gives F F = 16 θ S ( N + 1) N N ( N + 1) ( N + 2) S † . (E.7)Thus we get Q = − c ∞ X n =0 ∞ X n =1 ( n + n ) + 4 n ( n + n ) ( n + n + 1) ( n + n + 2) − c ∞ X n =0 n + 2) ( n + 3) (E.8)21 .2 The New Solution Again the solutions (4.39) can be written asˆ D a = 1 √ θ S new s M ( M + 3)( M + 1)( M + 2) b a S † new , ˆ D ¯ a = − √ θ S new b † a s M ( M + 3)( M + 1)( M + 2) S † new . (E.9)Then the commutators evaluate to h ˆ D a , ˆ D ¯ b i = − θ S new b † b b a M ( M + 1)( M + 2) S † new − δ ab θ S new M ( M + 3)( M + 1)( M + 2) S † new . (E.10)The product of the field strength becomes F F = − θ S new ( M − M ) M ( M + 1) ( M + 2) S † new , F F = 16 θ S new ( M + 1) M M ( M + 1) ( M + 2) S † new . (E.11)Hence the charge is Q new = − c ∞ X n =0 ∞ X n =2 ( m n + m n ) + 4 m n ( m n + m n ) ( m n + m n + 1) ( m n + m n + 2) + 2( m n + 2) ( m n + 3) (cid:19) (E.12)with m n = 12 ( n − λ − n ) , λ − n = (cid:26) n = even1 ; n = odd (E.13)Now for all even n ’s we have m n = m n +1 . Hence any absolutely convergent series over n (or n ) whose terms depend only on m n and m n can be broken into equal sums to give ∞ X n =0 G ( m n , m n ) = 2 X n =0 , , ,... G ( m n , m n ) . (E.14)Hence ∞ X n =0 ∞ X n =2 F ( m n , m n ) = 2 X n =0 , , ,... X n =2 , , ,... F ( m n , m n )= 4 X n =0 , , ,... X n =2 , , ,... F ( m n , m n ) . (E.15)Again m n = n for all even n ’s. 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