Monopoles, Dirac operator and index theory for fuzzy SU(3)/(U(1)×U(1))
aa r X i v : . [ h e p - t h ] N ov Monopoles, Dirac operator and index theoryfor fuzzy
S U (3) / ( U (1) × U (1)) Nirmalendu Acharyya ∗ a and Verónica Errasti Díez † a,ba Centre for High Energy Physics, Indian Institute of Science, Bangalore-560012, India b Physics Department, McGill University, 3600 University St., Montreal, QC H3A 2T8,Canada
Abstract
The intersection of the 10-dimensional fuzzy conifold Y F with S F × S F isthe compact 8-dimensional fuzzy space X F . We show that X F is (the analogueof) a principal U (1) × U (1) bundle over fuzzy SU (3) / ( U (1) × U (1)) (cid:0) ≡ M F (cid:1) . Weconstruct M F using the Gell-Mann matrices by adapting Schwinger’s construction.The space M F is of relevance in higher dimensional quantum Hall effect and matrixmodels of D -branes.Further we show that the sections of the monopole bundle can be expressed inthe basis of SU (3) eigenvectors. We construct the Dirac operator on M F fromthe Ginsparg-Wilson algebra on this space. Finally, we show that the index of theDirac operator correctly reproduces the known results in the continuum. ∗ [email protected] † [email protected] Introduction
Fuzzy spaces emerge naturally in the discussion of various theories like quantumHall effect (QHE) and matrix models of D -branes in the presence of certain back-ground fields. In the context of QHE in 2-dimensions, the Hilbert space of thelowest Landau level corresponds to symmetric representations of SU (2) . The ob-servables for the lowest Landau level then correspond to observables of S F (forexample, see [1–3]).Higher dimensional ( d > generalizations of QHE are interesting for vari-ous reasons. For example, they extend the notion of incompressibility to higherdimensions. In QHE in d > , the Landau problem is replaced by a particlemoving on a compact coset space in the presence of background gauge fields (say,monopoles). One such coset space is SU (3) / ( U (1) × U (1)) . The intersection of the10-dimensional conifold Y ( ≡ { z α , w β : z α w α = 0 , α, β = 1 , , } ) with S × S is a U (1) × U (1) monopole bundle on this coset space. For a particle moving onthis coset space in the presence of background U (1) monopoles, the Hilbert spaceof the lowest Landau level has an exact correspondence with the representations of SU (3) (for instance, see [4]). Consequently, the observables for the lowest Landaulevel are observables of fuzzy SU (3) / ( U (1) × U (1))( ≡ M F ) . Thus in the presenceof background fields, the natural description of the space becomes fuzzy and theemergent compact fuzzy space like M F and monopoles on it are relevant in theunderstanding of such QHE. M F is described by matrix algebras on the carrier space of the representationsof SU (3) . This space appears in the study of matrix models describing branes.The low energy effective action of a N (coincident) D -brane system is that ofa U ( N ) Yang-Mills theory. It is well known that the corresponding transversegeometry is inherently noncommutative [5]. Owing to the non-Abelian nature ofsuch theories, the
N Dp -brane system couples to Ramond-Ramond field strengthsof degree ≥ p + 4 [6, 7]. In particular, when the RR background is chosen to beproportional to the SU (3) structure constants, an 8-matrix model has the action S = T T r (cid:20)
12 ˙ φ i + 14 [ φ i , φ j ] − i κf ijk φ i [ φ j , φ k ] (cid:21) , i = 1 , , . . . , . (1.1)Here, κ is a coupling constant, φ i ’s are N × N matrices and f ijk ’s are the SU (3) structure constants. This describes N coincident Dp -branes (with p ≤ ). One ofthe ground state configurations of the above action is M F [8]. SU (3) / ( U (1) × U (1)) is of particular relevance in string theory. For instance, a7-dimensional space with G holonomy can have a conical singularity on this space[9]. Understanding how this holonomy appears in the fuzzy case is an interestingquestion by itself. We leave it for a seperate investigation in future. Here we focusprimarily on the construction of M F .To discuss higher dimensional QHE , we should introduce fermions (electrons)on M F . To this end, we construct the Dirac operator on M F . The Dirac operatoris also necessary to discuss supersymmetry on this fuzzy space, which is of interestto many. In a seperate context, since this is a finite dimensional model, it is mportant to study the fermion-doubling problem with the Dirac operator. Theconstruction of the Dirac operator on a fuzzy coset space like M F is nontrivial andintrinsically interesting. A beautiful (expected) relation exists between the indexof the Dirac operator and the topological objects on M F , which we make explicit.Our construction of M F derives from a Schwinger-like construction using theGell-Mann matrices and six independent oscillators. With these six oscillators,the 10-dimensional fuzzy conifold Y F can be constructed as in [10, 11]. X F = Y F ∩ (cid:0) S F × S F (cid:1) describes a subspace of this 6-dimensional oscillator’s Hilbertspace, which is the carrier space of all the representations of SU (3) . In section 2,we show that there exists a Hopf-like map X F → M F . X F is a U (1) × U (1) principal bundle over M F . The monopoles can be charac-terized by linear maps from one representation space of SU (3) to another [10–14].Such sections are rectangular matrices that map a M F of a given size to anotherof different size. In section 3, we construct these matrices in the basis of the SU (3) D -matrices.In section 4, we show that a Ginsparg-Wilson (GW) algebra is associated with M F . The Dirac operator can be constructed using the elements of this GW algebra,which are functions of the generators of SU (3) . We compute the index of the Diracoperator using the representations of SU (3) and their quadratic Casimir values asin [15]. The index is equal to Tr ( F ∧ F ∧ F ) . SU (3) / ( U (1) × U (1)) C F is described by six independent oscillators ˆ a α , ˆ b α ( α = 1 , , ): h ˆ a α , ˆ a β i = 0 , h ˆ a α , ˆ a † β i = δ αβ , h ˆ b α , ˆ b β i = 0 , h ˆ b α , ˆ b † β i = δ αβ , h ˆ a α , ˆ b β i = 0 , h ˆ a α , ˆ b † β i = 0 . (2.1)These oscillators act on the Hilbert space F spanned by the eigenstates of thenumber operators ˆ N a ( ≡ ˆ a † α ˆ a α ) and ˆ N b ( ≡ ˆ b † α ˆ b α ) : F ≡ span (cid:8) | n a , n a , n a ; n b n b , n b i : n αa , n αb = 0 , , , . . . (cid:9) . (2.2)A fuzzy conifold is described by a subalgebra in C F [10, 11]. We define theoperator ˆ O ≡ X α =1 ˆ b α ˆ a α , (2.3)which has as its kernel ker( ˆ O ) = span n |·i ∈ F : ˆ O|·i = 0 o ⊂ F . (2.4) We use the symbol |·i to denote the state | n a , n a , n a ; n b n b , n b i .) The algebra of (ˆ a α , ˆ b β ) ’s restricted to ker( ˆ O ) describes a 10-dimensional fuzzy conifold Y F .For convenience of normalization, we will also work with the set of operators ˆ χ α ≡ ˆ a α √ N a , ˆ ξ α ≡ ˆ b α √ N b , (2.5)which satisfy ˆ χ † α ˆ χ α = 1 , ˆ ξ † α ˆ ξ α = 1 . (2.6)The ˆ χ α ’s (or ˆ ξ α ’s) are well-defined if we exclude the states for which ˆ N a = 0 (or ˆ N b = 0 ). Then the algebra generated by ˆ χ α ’s and ˆ ξ α ’s describes S F × S F .The operator ˆ O ′ ≡ ˆ ξ α ˆ χ α also vanishes in ker( ˆ O ) . Therefore the algebra ofthe ˆ χ α ’s and ˆ ξ α ’s restricted to ker( ˆ O ) describes an 8-dimensional fuzzy space X F .Informally, we can think of X F as Y F ∩ (cid:0) S F × S F (cid:1) . X F → M F ≡ fuzzy SU (3) / ( U (1) × U (1)) Using the matrices T i = λ i ( λ i = Gell-Mann matrices, i = 1 , , . . . , ) satisfying [ T i , T j ] = if ijk T k , { T i , T j } = 13 δ ij + d ijk T k , (2.7)we can define a Schwinger-like construction (similar to [16]): ˆ y i = ˆ a † α ( T i ) αβ ˆ a β − ˆ b α ( T i ) αβ ˆ b † β , (2.8) ˆ s i = ˆ χ † α ( T i ) αβ ˆ χ β − ˆ ξ α ( T i ) αβ ˆ ξ † β . (2.9)The ˆ y i ’s obey ˆ y † i = ˆ y i , [ˆ y i , ˆ y j ] = if ijk ˆ y k (2.10)and the Casimirs are ˆ C ≡ ˆ y i ˆ y i and ˆ C ≡ d ijk ˆ y i ˆ y j ˆ y k : ˆ C = 13 h ˆ N a + ˆ N b + ˆ N a ˆ N b + 3 (cid:16) ˆ N a + ˆ N b (cid:17)i − ˆ O † ˆ O , (2.11) ˆ C = 118 (cid:16) ˆ N a − ˆ N b (cid:17) (cid:16) N a + ˆ N b + 3 (cid:17) (cid:16) ˆ N a + 2 ˆ N b + 3 (cid:17) + ˆ N a − ˆ N b O † ˆ O . (2.12)The Hilbert space F can be split into the subspaces F n a ,n b : F n a ,n b ≡ span ( |· i : X α n αa = n a , X α n αb = n b ) , F = ⊕F n a ,n b , (2.13)where the direct sum ⊕ is over n a and n b . In ˜ F n a ,n b ≡ F n a ,n b ∩ ker( ˆ O ) , the Casimirstake fixed values: ˆ C (cid:12)(cid:12)(cid:12) ˜ F na,nb = c I , ˆ C (cid:12)(cid:12)(cid:12) ˜ F na,nb = c I , (2.14) ith c = 13 (cid:2) n b + n a + n b n a + 3( n a + n b ) (cid:3) = fixed , (2.15) c = 118 ( n a − n b ) (2 n a + n b + 3) ( n a + 2 n b + 3) = fixed . (2.16)Also in ˜ F n a ,n b , ˆ s i ˆ s i = fixed , d ijk ˆ s i ˆ s j ˆ s k = fixed . (2.17)Therefore, the algebra of ˆ s i ’s restricted to ˜ F n a ,n b describes a 6-dimensional fuzzyspace M F .Each ˜ F n a ,n b is a carrier space of a finite dimensional irrep of SU (3) . Thisrepresentation is characterized by a pair of positive integers ( p, q ) = ( n a , n b ) andis of dimension dim SU (3) = 12 ( n b + n a + 2) ( n b + 1) ( n a + 1) . (2.18)The ˆ y i ’s (and ˆ s i ’s) are square matrices in ˜ F n a ,n b . Thus M F is the fuzzy version of SU (3) / ( U (1) × U (1)) and (2.8) is a noncommutative U (1) × U (1) fibration.Note that in the above construction neither n a nor n b can be chosen to be zero.In case the construction is done with only one set of oscillators (either ˆ a α ’s or ˆ b α ’s),one would get fuzzy C P , as in [7, 17]. Nevertheless, for n a >> n b (or n b >> n a ), M F looks like fuzzy C P in some sense [7]. Let H n a ,n b → l a ,l b be the space of linear operators Φ , which map ˜ F n a ,n b to ˜ F l a ,l b : Φ : ˜ F n a ,n b → ˜ F l a ,l b , Φ ∈ H n a ,n b → l a ,l b . (3.1)In general, these Φ ’s are rectangular matrices. H n a ,n b → n a ,n b is a noncommutative algebra which maps ˜ F n a ,n b → ˜ F n a ,n b and any Φ ∈ H n a ,n b → n a ,n b is a square matrix. In this algebra, the rotations are generatedby the adjoint action of ˆ y ( n a ,n b ) i ( ˆ y ( n a ,n b ) i is the restriction of ˆ y i in ˜ F n a ,n b ): Ad (cid:16) ˆ y ( n a ,n b ) i (cid:17) Φ ≡ ˆ L i Φ ≡ [ˆ y ( n a ,n b ) i , Φ] , Φ ∈ H n a ,n b → n a ,n b . (3.2) ˆ L i ’s generate a SU (3) : [ ˆ L i , ˆ L j ] = if ijk ˆ L k . (3.3) The Casimirs of a ( p, q ) representation of SU (3) are ˆ C = 13 [ p + q + pq + 3( p + q )] I , ˆ C = 118 ( p − q )(2 p + q + 3)( p + 2 q + 3) I . hen n a = l a , n b = l b , the spaces H n a ,n b → l a ,l b are noncommutative bimodulesand any Φ ∈ H n a ,n b → l a ,l b is a rectangular matrix. For the bimodules, the generatorsof the SU (3) in (3.3) act by a left- and a right-multiplication: ˆ L i Φ = ˆ y ( l a ,l b ) i Φ − Φˆ y ( n a ,n b ) i . (3.4)This SU (3) action is reducible and we will give its explicit decomposition shortly.Any element Φ ∈ H n a ,n b → l a ,l b can be expanded in the basis of the eigenvectors of ˆ L , ˆ L , ˆ L i ˆ L i and d ijk ˆ L i ˆ L j ˆ L k .To construct the basis vectors, let us start as follows. The operator ˆ f = (ˆ a † ) ˜ l a (ˆ a ) ˜ n a (ˆ b † ) ˜ l b (ˆ b ) ˜ n b (3.5)is an element of H n a ,n b → l a ,l b if (˜ l a , ˜ n a , ˜ l b , ˜ n b ) are positive integers satisfying κ a ≡ l a − n a = ˜ l a − ˜ n a , κ b ≡ l b − n b = ˜ l b − ˜ n b . (3.6)It is easy to see that ˆ U + ˆ f ≡ (cid:16) ˆ L + i ˆ L (cid:17) ˆ f = 0 , ˆ V + ˆ f ≡ (cid:16) ˆ L − i ˆ L (cid:17) ˆ f = 0 , ˆ W + ˆ f ≡ (cid:16) ˆ L − i ˆ L (cid:17) ˆ f = 0 , ˆ L ˆ f = (cid:16) ˜ n a + ˜ l b (cid:17) ˆ f , ˆ L ˆ f = − √ (cid:16) l a + 2˜ n b + ˜ n a + ˜ l b (cid:17) ˆ f . (3.7)So ˆ f is the highest weight vector of the SU (3) representation characterized by ( p, q ) , with p = ˜ l a + ˜ l b + ˜ n a + ˜ n b , q = ˜ l a + ˜ n b , p ≥ q ≥ . (3.8)The quadratic and the cubic Casimirs for this representation take values C = 13 (cid:0) p + q − pq + 3 p (cid:1) , (3.9) C = 118 ( p − q )(2 p − q + 3)( q + p + 3) , (3.10)while the dimension is d = 12 ( p − q + 1)( p + 2)( q + 1) . (3.11)The lower weight vectors belonging to the same irrep ( p, q ) are generatedby the action of the lowering operators ˆ U − (cid:16) ≡ ˆ L − i ˆ L (cid:17) , ˆ V − (cid:16) ≡ ˆ L + i ˆ L (cid:17) and ˆ W − (cid:16) ≡ ˆ L + i ˆ L (cid:17) on ˆ f . generic vector belonging to the ( p, q ) irrep is labelled by m and m – the ˆ L and ˆ L values respectively: ˆ L Ψ m ,m p,q = m Ψ m ,m p,q , ˆ L Ψ m ,m p,q = m Ψ m ,m p,q , ˆ L i ˆ L i Ψ m ,m p,q = C Ψ m ,m p,q , d ijk ˆ L i ˆ L j ˆ L k Ψ m ,m p,q = C Ψ m ,m p,q . (3.12)In the following, we specify the allowed values of ( p, q ) (i.e. which irreps occur). In(3.6), κ a and κ b can be both positive and negative. The ranges of the pairs (˜ l a , ˜ n a ) and (˜ l b , ˜ n b ) are different for each choice of the sign of κ a and κ b . Consequently, theirreps appearing in such maps also differ. We find the irreps for each case. Case 1: l a ≥ n a and l b ≥ n b In this case, both κ a , κ b ≥ . The ranges of ˜ n a and ˜ n b are ≤ ˜ n a ≤ n a , ≤ ˜ n b ≤ n b . (3.13)Therefore the allowed values of p and q are ˜ n a ˜ l a ˜ n b ˜ l b p q κ a κ b κ a + κ b κ a κ a + 1 κ b κ a + κ b + 2 κ a + 1 κ a κ b + 1 κ a + κ b + 2 κ a + 1 · · · · · · · · · · · · · · · · · · n a l a n b l b J a + J b l a + n b where J a ≡ l a + n a and J b ≡ l b + n b . Case 2: l a ≤ n a and l b ≥ n b Here, κ a ≤ and κ b ≥ and ≤ ˜ l a ≤ l a , ≤ ˜ n b ≤ n b . (3.14)Hence p and q take the following values: ˜ l a ˜ n a ˜ n b ˜ l b p q − κ a κ b − κ a + κ b − κ a + 1 κ b − κ a + κ b + 2 1 − κ a κ b + 1 − κ a + κ b + 2 1 · · · · · · · · · · · · · · · · · · l a n a n b l b J a + J b l a + n b ase 3: l a ≥ n a and l b ≤ n b When κ a ≥ and κ b ≤ , the ranges of ˜ n a and ˜ l b are ≤ ˜ n a ≤ n a , ≤ ˜ l b ≤ l b . (3.15)Then the allowed values of p and q are ˜ n a ˜ l a ˜ l b ˜ n b p q κ a − κ b κ a − κ b κ a − κ b κ a + 1 − κ b κ a − κ b + 2 κ a − κ b + 1 κ a − κ b + 1 κ a − κ b + 2 κ a − κ b + 1 · · · · · · · · · · · · · · · · · · n a l a l b n b J a + J b l a + n b Case 4: l a ≤ n a and l b ≤ n b In this case, κ a ≤ and κ b ≤ and ≤ ˜ l a ≤ l a , ≤ ˜ l b ≤ l b . (3.16)Thus the irreps have ( p, q ) values ˜ l a ˜ n a ˜ l b ˜ n b p q − κ a − κ b − ( κ a + κ b ) − κ b − κ a + 1 − κ b − ( κ a + κ b ) + 2 − κ b + 1 − κ a − κ b + 1 − ( κ a + κ b ) + 2 − κ b + 1 · · · · · · · · · · · · · · · · · · l a n a l b n b J a + J b l a + n b Any arbitrary element Φ ∈ H n a ,n b → l a ,l b can be expressed in terms of the SU (3) harmonics as Φ = X m ,m ,p,q c m ,m p,q Ψ m ,m p,q , c m ,m p,q ∈ C . (3.17)These Φ ’s are identified as the noncommutative analogue of the sections of theassociated line bundle. opological charge The sections of the associated line bundle carry two topological charges, corre-sponding to each U (1) fibre. In H n a ,n b → l a ,l b , we can define two topological chargeoperators: ˆ K a ≡ h ˆ N a , i , ˆ K b ≡ h ˆ N b , i . (3.18)It is easy to see that Φ in (3.17) has topological charges ( κ a , κ b ) given by ˆ K a Φ = ˜ l a − ˜ n a κ a , ˆ K b Φ = ˜ l b − ˜ n b κ b , κ a , κ b ∈ Z . (3.19)Therefore Φ is a section of a complex line bundle with topological charges ( κ a , κ b ) . SU (3) / ( U (1) × U (1)) is a 6-dimensional space embedded in R . This space iscurved and non-symmetric. Also, as U (1) × U (1) ⊂ Spin (6) ∼ = SU (4) , this spaceadmits a spin structure. In the commutative case, the Dirac operator contains threeterms: the kinetic term, the spin-connection term and the background monopoleterm (if any) [15].There are many possible ways to obtain the Dirac operator on this space (forexample, [18]). In the fuzzy case, we do so by constructing a Ginsparg-Wilsonalgebra on M F .We look at the zero charge sector first. On H n a ,n b → n a ,n b , ˆ y i has left and rightactions ˆ y Li f ≡ ˆ y i f, ˆ y Ri f ≡ f ˆ y i . (4.1)These satisfy (2.10)-(2.12): [ˆ y Li , ˆ y Lj ] = if ijk ˆ y Lk , [ˆ y Ri , ˆ y Rj ] = − if ijk ˆ y Rk , [ˆ y Li , ˆ y Rj ] = 0 , ˆ y i ˆ y i ≡ ˆ y Li ˆ y Li = ˆ y Ri ˆ y Ri = c I , d ijk ˆ y Li ˆ y Lj ˆ y Lk = d ijk ˆ y Ri ˆ y Rj ˆ y Rk = c I . (4.2)The γ -matrices on R are × matrices which generate the Clifford algebra { γ i , γ i } = 2 δ ij , γ † i = γ i , i = 1 , , . . . , . (4.3)Using γ i ’s we can construct t i ≡ i f ijk γ j γ k , [ t i , γ j ] = if ijk γ k , (4.4)which generate a SU (3) : [ t i , t j ] = if ijk t k . (4.5)We can define Γ ≡ a γ i (ˆ y Li + 13 t i ) , ˜Γ ≡ − a γ i (ˆ y Ri − t i ) , (4.6) here the normalization a is given by a I = ˆ y i ˆ y i + 13 t i t i , t i t i = 3 I , ˆ y i ˆ y i = c I . (4.7) Γ and ˜Γ generate of a Ginsparg-Wilson algebra: A GW = { Γ , ˜Γ : Γ = I = ˜Γ , Γ † = Γ , ˜Γ † = ˜Γ } . (4.8)From this algebra, one can construct a Dirac operator DD = a (cid:16) Γ + ˜Γ (cid:17) = γ i ˆ L i + 23 γ i t i , where ˆ L i = ˆ y Li + ˆ y Ri , (4.9)and a chirality operator Γ ch Γ ch = a (cid:16) Γ − ˜Γ (cid:17) = γ i (cid:0) ˆ y Li + ˆ y Ri (cid:1) . (4.10)It is easy to check that {D , Γ ch } = 0 , (4.11) ν ≡ index D = Tr (Γ ch ) (index theorem) . (4.12)It is interesting to note that in (4.9), the second term ( ∼ γ i t i ) is the spin-connectionterm in [15].The Dirac operator is of the form D = (cid:18) AA † (cid:19) . (4.13)When the action of D is restricted to the algebra ( H n a ,n b → n a ,n b × M at ) , it is theDirac operator on M F . In this case, there is no monopole background, and hencethe gauge field contribution to the Dirac operator is zero.On the bimodule H n a ,n b → l a ,l b with n a = l a , n b = l b or either, there is abackground monopole. On this bimodule, ˆ L i is the covariant derivative whichincludes the monopole contribution. Hence, if we restrict D to the bimodule ( H n a ,n b → l a ,l b × M at ) , we automatically incorporate the background monopoleinformation. There is no need to add the monopole term in the Dirac operator.Rather, restricting the algebra of D to the proper subspace accounts for monopoles. SU (3) / ( U (1) × U (1)) is a non-symmetric space with positive curvature. On thisspace, there is an additional connection due to the torsion, which appears in thesquare of the Dirac operator [15]: D = −∇ + curvature + torsion + possible gauge field contribution . (4.14) he Dirac Laplacian ∇ on this coset space is a positive operator. For the Diracoperator to have zero modes, we would need the cancellation of the lowest eigen-value of the Laplacian with the lowest eigenvalue of the sum of the curvature,torsion and gauge field contributions. These considerations require that the num-ber of zero modes of the Dirac operator on SU (3) / ( U (1) × U (1)) is given by thedimension of the SU (3) irrep with the minimum value of the quadratic Casimir C [15]. We adopt the same requirement to compute the number of zero modes inthe fuzzy case. Case 1: κ a ≥ and κ b ≥ In this case C takes the minimum value in the representation with ( p, q ) =( κ a + κ b , κ a ) : C min = 13 (cid:0) κ a + κ b + κ a κ b (cid:1) + κ a + κ b . (4.15)The dimension of this representation is the index of D : ν = d min = 12 ( κ a + κ b + 2)( κ a + 1)( κ b + 1) . (4.16)The discussion for the other cases is similar: Case 2: κ a ≤ and κ b ≥ ( p, q ) = ( − κ a + κ b , , C min = ( − κ a + κ b ) − κ a + κ b ,ν = d min = ( − κ a + κ b + 2)( − κ a + κ b + 1) . (4.17) Case 3: κ a ≥ and κ b ≤ ( p, q ) = ( κ a − κ b , κ a − κ b ) , C min = ( κ a − κ b ) + κ a − κ b ,ν = d min = ( κ a − κ b + 2)( κ a − κ b + 1) . (4.18) Case 4: κ a ≤ and κ b ≤ ( p, q ) = ( − κ a − κ b , − κ b ) , C min = (cid:0) κ a + κ b + κ a κ b (cid:1) − κ a − κ b ,ν = d min = ( − κ a − κ b + 2)( − κ a + 1)( − κ b + 1) . (4.19) hen there is no monopole, κ a = 0 = κ b and the index is ν = 1 , (4.20)which is consistent with [15, 17]. Also, as in the commutative case, the index ν gives Tr ( F ∧ F ∧ F ) for the monopole fields. Our realization of M F can be used to study large N limits of matrix models of D -branes. Among other things, M F as the vacua of the matrix model (1.1) can bein an irreducible or reducible representation. It is easy to generalize the Schwingerconstruction of M F by using Brandt-Greenberg oscillators and obtain reduciblealgebras of M F , as in [14]. We can obtain the quantum states for those reducible M F ’s using the prescription of Gelfand-Naimark-Segal (GNS). Those quantumstates will be inherently mixed and will carry entropy, which is typically large.These informations will play a vital role in the understanding of the vacua andtachyon condensations in the matrix model.Using the Dirac operator and the index theory, one may try to construct thespinor bundle on M F and thus find the supersymmetric analogue of X F → M F .The super conifold in the continuum has various interesting features [19] whichshould be manifest in the fuzzy version as well. We leave this for a future investi-gation.The fermion-doubling problem on this finite dimensional space can also bestudied. It has been shown that there is no fermion-doubling on the fuzzy sphere[20, 21]. There might be such dramatic consequences on M F as well. Acknowledgements
V.E.D. thanks I.I.Sc. for hospitality during her stay from September 2014 to Oc-tober 2015. We would like to thank A. P. Balachandran, Sachindeo Vaidya andKeshav Dasgupta for illuminating discussions and suggestions. We are indebted toDiptiman Sen, who pointed out an error in an earlier version of the draft.
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