Bose-Fermi Equivalence in Three Dimensional Non-commutative Space-Time
aa r X i v : . [ h e p - t h ] F e b Bose-Fermi Equivalence in Three DimensionalNon-commutative Space-Time
Ajith K M ∗ , E. Harikumar † and M. Sivakumar ‡ School of Physics, University of Hyderabad,Hyderabad, 500046, India.November 19, 2018
Abstract
We study the Fermionisation of Seiberg-Witten mapped action (toorder θ ) of the λφ theory coupled minimally with U(1) gauge fieldgoverned by Chern-Simon action. Starting from the correspondingpartition function we derive non-perturbatively (in coupling constant)the partition function of the spin theory following Polyakov spinfactor formalism. We find the dual interacting fermionic theory is nonlocal. This feature persist also in the limit of vanishing self coupling.In θ → ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Non commutative (NC) space-time[1] has gained a lot of interest in recenttime due to (i) relevance to quantum aspects of gravity (ii) as a regularizationin field theory (iii) certain limit of the string theory. Well studied among themis the Moyal space-time whose co-ordinates obey[ x µ , x ν ] ∗ = iθ µν (1)In Moyal space-time, the usual product is replaced by the ∗ product definedas f ( x ) ∗ g ( x ) = e i θ ij ∂ xi ∂ yj f ( x ) g ( y ) | x = y (2)and θ µν is a constant parameter characterizing the noncommutativity. Fieldtheories on such space-time have several interesting features which are dis-tinct from the models on the commutative space-time. These includes UV/IRmixing, novel topological soliton solutions, twisted symmetries etc. Gaugetheories in NC space time can be mapped using Seiberg-Witten (SW) map[2] to gauge theories in commutative space time. In this note we study Bose-Fermi equivalence of scalar field theory in 2+1 dimensional NC space-time.Bose-Fermi transmutation is studied for the NC theories after re expressedin the commutative space-time using SW map(keeping up to order θ terms.)Bosonisation of Fermionic theories (and vice versa)have been well studied incommuative space-time. Polyakov proposed a study of Bose-Fermi equiva-lence in 2+1 dimension using Chern-Simon gauge action[3]. It was shown thatthe expectation value of the Wilson loop averaged over Chern-Simon(CS)gauge action ( and for a suitable coefficient) is given by < e i H c Adx > CS = e iπW ( C ) (3)where W ( C ) is the writhe of the space curve. It was also shown [3, 4] that W ( C ) =Ω( C ) + (2 k + 1) , k ∈ Z . Here Ω( C ), known as Polyakov spin factor,represent the solid angle subtended by the tangent to the curve C on aunit sphere. This is also related to the overlap of spin coherent states andforms the symplectic 2-form of SU(2) group manifold which is S . The oddinteger (2k+1) has been shown to be related to statistics. The expressionin the Eqn.(3) has been applied, to derive fermionic theory from scalar fieldcoupled with U (1) gauge field governed by Chern-Simon action by one of us[5, 6]. Interesting feature of this approach is that, it is non-perturbative incoupling constant. In this work we apply this approach to NC Space-time.2ose-Fermi equivalence can be seen as duality equivalence. Duality as-pects of NC field theories have been well studied [8 , , , ,
12] in recenttimes. There were many studies using different approaches to generalise theknown duality equivalence between Maxwell-Chern-Simon theory(MCS) toSelf-dual model(SD) in NC spaces. This was studied in [9 ,
10] using masteraction method but with different conclusions. By applying a dual projectionprocedure to NCSD model, a dual model was constructed in [11 ,
12] andwas shown to be different from NCMCS theory. In [13], using a differentapproach dual of SW mapped NCMCS was obtained and shown to be dif-ferent from SW mapped NC (St¨uckelberg compensated) SD model. Theseinvestigations showed that the duality relation present in the commutativespace time need not carry forward to NC space-time. Bosonization in twoand three dimensional NC space-time has been studied in [14, 15, 16]. Hence,it is interesting to investigate bosonisation in NC space time following thePolyakov approach.In this work we study Fermionisation of λφ theory coupled to U ∗ (1) gaugefield governed by Chern-Simon action in NC space-time. We apply SW mapto re-express the theory interms of fields in commutative space-time keepingterms up to the first order in θ , and apply the methods devolepd in [5, 6].We derive the Fermionic partition function, exact in self coupling. The dualFermionic theory obtained is nonlocal, interacting theory. We see that theFermionic mass term does not get θ correction. U ∗ (1) In this section, we consider Fermionisation of self interacting scalar fieldtheory in the fundamental representation of U ∗ (1). We start with the massivecomplex scalar field in fundamental representation, coupled to a Chern-Simonterm in noncommutative Euclidean space described byˆ S φ = Z d x [( ˆ D µ ˆ φ ) ∗ ( ˆ D µ ˆ φ ) † + m ˆ φ ∗ ˆ φ † − λ ( ˆ φ † ∗ ˆ φ ) ∗ ( ˆ φ † ∗ ˆ φ ) − i π ǫ µνλ ( ˆ A µ ∂ ν ˆ A λ + 2 i A µ ˆ A ν ˆ A λ )] (4)3n the above action hated fields are functions of non commutative (NC) co-ordinates. The covariant derivative is defined byˆ D µ ˆ φ = ∂ µ ˆ φ − i ˆ A µ ∗ ˆ φ. Using SW map we rewrite the action in Eqn.(4) in terms of commutativefields and θ . For this we use the SW solution for ˆ φ and ˆ A µ ˆ A µ = A µ − θ αβ A α ( ∂ β A µ + F βµ ) (5)ˆ φ = φ − θ αβ A α ∂ β φ. (6)to order θ [17]. Using this, from Eqn.(4) we get (to order θ ) S φ = Z d x [ D µ φ ( D µ φ ) † − y µν D µ φ ( D ν φ ) † + m (1 + y µµ ) φφ † − λ ( φφ † ) (1 − θ αβ F αβ ) − i π ǫ µνλ A µ ∂ ν A λ ] (7)where y µν = 12 ( θ µα F να + θ να F µα + 12 η µν θ αβ F αβ ) . Note that we have used the fact that NC Chern-Simon term goes to commu-tative Chern-Simon term under SW map [18]. The coefficient of Chern-Simonterms is chosen sothat dual theory is that of spin- fermion. Before integrat-ing the scalar fields we linearise λ term in the above and re-express this termusing Hubbard-Stratnovich field χ , λ ( φφ † ) (1 − θ αβ F αβ ) = − χ ( x ) + 2 √ λχ ( x )( φφ † )(1 − θ αβ F αβ ) (8) The Euclidean path integral with the above action is given by Z = Z DφDφ † DADBDCDχ e − S e − ( R d x [ i π ǫ µνλ A µ ∂ ν A λ ]+ iC µν ( y µν + B µν )+ χ ( x ) ) (9) S = Z d x h D µ φ ( D µ φ ) † + B µν D µ φ ( D ν φ ) † + [ ˜ m (1 − B µµ )] φφ † i µν and B µν were introduced to linearize the θ depended coupling of A fieldto scalar field and and we use ˜ m ( x ) = m − √ λ χ ( x ).After integrating the φ and φ † fields we get partition function as Z = Z DADBDC e − ln det R e − i ( R C µν ( y µνθ + B µν )+ χ )+ iλ π ǫ µνλ A µ ∂ ν A λ ) (10)where the operator R is given by R = ( − ( δ µν + B µν ) D µ D ν − ( D µ B µν ) D ν + V )and V ( x ) = [ ˜ m (1 − B µµ )] . (11)We can use the heat kernel representation of the logarithm of determinant[19],treating R as the Hamiltonian, i.e, ln det R = Z ∞ dαα T r e − α R (12)Applying the standard path integral method to this gauge invariant “Hamil-tonian”, R we obtain lnDet R = Z ∞ dαα Z x ( α )= x (0) Dx ( τ ) e − R α dτ [ H + ˙ x µ ∂ ρ B ρµ − i ˙ x µ A µ ] (13)In the above the measure Dx ( τ ) = (4 πǫ ) − N Q N − i =0 d x , and H = ( M µν ) − ˙ x µ ˙ x ν + V ( x ( τ )) with M µν = ( δ µν + B µν ). Here we take ǫ → . Also we omittedterms quadratic in B µν as they are of order θ which can be seen by inte-grating C-field. After expanding the e − ln det R in power series, the partitionfunction becomes Z = Z D A ∞ X n =0 n ! n Y i =1 h Z ∞ dα i α i Z x (0)= x ( α i ) Dx e R αi dτ [ N ( τ i )] i e − R G ( x ) d x (14)Here N ( τ i ) and G(x) are N ( τ i ) = H ( τ i ) + 12 ˙ x µi ∂ ρ B ρµ ( τ i ) − i ˙ x µi A µ ( τ i ) and G ( x ) = iC µν ( y µν + B µν ) + χ + i π ǫ µνλ A µ ∂ ν A λ respectively and the measure D A = DADBDCDχ .5 Gauge field integration
We first rewrite R x C µν y µνθ as R x Γ νθ A ν (after omitting surface terms) where Γ νθ is Γ νθ = [ − θ µα ∂ α C νµ + θ µν ∂ σ C µσ − θ αν ∂ α C γγ ] . (15)For gauge field integration we collect all the A µ terms in the above partitionfunction and write them as e − i R d x [ π A µ d µν A ν ] − A µ ( x )( − J µ +Γ µθ )] where we have used the definition for the particle current, the current asso-ciated with the particle moving along the Wilson loop, as J µ = Z dτ ˙ x µ dτ δ ( x − x c ( τ )) (16)and d µλ = ǫ µνλ ∂ ν . Note that unlike in the commutative case, in the absenceof Chern-Simon term, particle current is non-vanishing. After the gauge fieldintegration (omitting θ terms) the partition function become Z = Z D Ω ∞ X n =0 n ! n Y i =1 h Z ∞ dα i α i Z x (0)= x ( α i ) Dxe − S e − R αi dτ [ ω i ] i . (17)In the above ω i = H i − π Γ µθ ( d − ) µρ J ρi + L i and the measure D Ω =
DBDCDχ.
Where we have used L i = iπ ( J µi ( d µν ) − ) J νi and S = Z d x [ iC µν B µν + χ ]In the above partition function the integral e − R dτL i is of the form e − iπ R d xJ µ ( d µν ) − J ν = e iπ ( W ( C n )+ P i = j n ij ) (18)where W ( C n ) is the writhe of the curve C n (= S C i ) and n ij is the linkingnumber of the curves C i and C j [3]. The linking number term does not con-tribute. The writhe W ( C n ) has the expression W ( C n ) = Ω( C n ) + 2 k + 1,where Ω( C n )is the Polyakov factor[3], and 2k+1 is an odd integer. Thus e − πiW ( C n ) = ( − e − i Ω( C n ) . (19)6he coefficient of W ( C n ) is dictated by the choice of the coefficient of Chern-Simon term. Interestingly this encodes both spin and statistics of the trans-formed field. The (-1) in the above equation is responsible for the expressionappearing as determinent rather its inverse (see Eqn.(27) below ), whichleads to Grassmanian nature of the transmuted field. The coefficient ofPolyakov’s spin factor is responsible for the spin nature of the transmutedfield through the well known properties of spin coherent states [20]. UsingEqn.(19), the partition function becomes. Z = Z D Ω e − R d x ( iC µν B µν + χ ) e − R dαα R Dx ( τ ) e − R α dτ [ ˜ M ]+( − i
12 Ω − iVµJµ (20)where ˜ M = 14 ( δ µν − B µν ) ˙ x µ ˙ x ν + V ( τ i )and V µ is given by V µ = [2Γ σθ ( d σµ ) − + i ∂ ρ B ρµ ] (21)The addition of Polyakov spin factor to the path integral for spinless particleboth in free and in the presence of background scalar and vector fields havebeen studied in [7]. Following this procedure, we obtain − Z ∞ Λ − dαα Z Dx ( τ ) e − R α dτ [ ( δ µν − B µν ) ˙ x µ ˙ x ν + V )]+( − i Ω − i H V µ dx µ = ( − Z ∞ Λ − dαα T re − α [ DA + ˜ V + M F ] (22)where Λ is cut-off. This makes use of the well known result Z ˆ n (0)=ˆ n ( l ) D ˆ n e i R l dτ ( H (ˆ n )+ Ω(ˆ n )) = T r D ˆ n | e iH ( τ µ ) | ˆ n E (23)where ˆ n are the SU (2) coherent states and τ µ are the Pauli matrices. Herewe define D = ( i∂ µ − V µ ) τ µ , A = q det ( δ µν − B µν ) , (24)˜ V = − √ π h ( m B µµ + 2 √ λχ ( x )(1 − B µµ ))] (25)and M F = √ π
4Λ ( m + Λ ln
2) (26)7here M F is mass of the Fermion. Using the above result in (20) we get Z = Z DBDCDχ e − R d x ( C µν B µν + χ ) det h DA + ˜ V + M F i (27)Note that -1 in Eqn.(22)is responsible for the determenant to appear in thenumerator. This can be written as functional integral over fermionic fieldsand then integrating over χ , we get the partition function as Z = Z DBDCD Ψ D ¯Ψ e − R d x ( C µν B µν ) e − R d x ¯Ψ [ DA + ˜ V + M F ] Ψ − g (Ψ ¯Ψ) (28)where ˜ V = − √ π
4Λ [ m B µµ ] and g ( x ) = πλ (1 − B µµ ( x )) . This result is non perturbative in λ and the interacting fermionic theory isnon-local. When λ → ∗ product to first orderin θ . Such theory will not have non-locality. In the limit θ →
0, Lagrangianin Eqn.(28) becomes L = Z d x ( C µν B µν ) + 2 ¯Ψ 1 A [ i∂ µ + i ∂ ρ B ρµ ] τ µ Ψ − ¯Ψ √ π
4Λ [ m B µµ ] + M F ¯ΨΨ − πλ (1 − B µµ ( x )) ( ¯ΨΨ) . (29)Now integration over the field C µν (In the partition function) set B µν tovanish. Hence in the case of θ → λ = 0 the commutative result in [6],which is a local (Ψ ¯Ψ) fermionic theory is retrived. When θ → λ = 0we get the commutative result, i.e free fermion. Thus the commutative limitis smooth. In this paper we have studied Fermionization in 3 dimensional NC space,where NC λφ theory coupled to U ∗ (1) gauge field governed by Chern-Simonaction. The dual Fermionic partition function derived, (non-perturbative in λ ) is non-local for both λ = 0 and λ = 0 cases. As it is clear Fermionicmass term does not get θ correction. In θ → ,
6] is recoverd. Note also when m →
0, Dirac paricle has a non zeromass (dependent on cut off Λ). This is expected as Chern-Simon term in theBosonic theory is parity violating, which is reflected in the non zero mass ofthe fermionic theory. In the NC case it is possible for a real scalar to couplewith gauge field (unlike in the commutative case). Thus it is natural to seekFermionisation of real scalar coupled to Chern-Simon term. The SW mappedaction for real scalars is (up to ordrt θ ) S ϕ = 12 Z d x (cid:20) ∂ µ ϕ∂ µ ϕ + 2 θ µα F αν (cid:18) − ∂ µ ϕ∂ ν ϕ + 14 η µν ∂ ρ ϕ∂ ρ ϕ (cid:19)(cid:21) . (30)Here the coupling to gauge field is through non-minimal coupling only. Forthe application of Poliyakov’s approach it is necessary to have Wilson loopterm (i.e, minimal coupling), which is absent here. Hence, straight forwardextension of this procedure to real scalar is not possible. It is an interesingproblem to see how the real scalar can be fermionised. Acknowledgement
KMA and MS acknowledges DST for support througha project.
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