On the Hopf Structure of W - Algebra and N=1 Superconformal 2 Algebra in the OPE Language 4 0 0 2 H. T.O¨zer∗ n a Physics Department, Faculty of Science and Letters, J 9 Istanbul Technical University, 2 80626, Maslak, Istanbul, Turkey 3 v 0 February 1, 2008 8 1 0 1 Abstract 2 0 HopfstructureoftheprototyperealizationsoftheW2-algebraandalso h/ N = 1 superconformal algebra are obtained using the bosonic and also t fermionic Feigin-Fuchs type of free massless scalar fields in the operator - p product expansion (OPE) language. e h : v 1 Introduction i X r Conformal symmetry has played an important role in the developments of the a physics contents of such models: i.e. (super)string theory [1],statistical physics [2], as well as in mathematics [3]. Its underlying symmetry algebra is Virasoro algebra which is a Lie algebra. It is well-known that a universal enveloping algebra of any simple Lie algebra is always a Hopf algebra [4]. Let g a simple Lie algebra and U(g) its universal enveloping algebra.Then define the comalti- plication △ ,the counit ǫ and the antipode S for U(g) as follows : △(x) = x⊗1+1⊗x, (1. 1) ǫ(x) = 0, (x∈g) (1. 2) S(x) = −x. (1. 3) ∗e-mail : ozert @ itu.edu.tr 1 In particular one can show that the coproduct rule define a Lie algebra homo- morphizm: △([x,y]) = △(x)△(y)−△(y)△(x) (1. 4) = [△(x),△(y)] (1. 5) For example, if one take a Virasoroalgebra(or also similar infinite dimensional algebras) and feed it into this machinery one obtains a Hopf algebra.Although nobody in general taken care of this structure ,the purpose of this paper is to confirm this structure in these kind of algebras by using the operator product expansion (OPE) language.In this context we note that the universal envelop- ing algebra of the underlying symmetry algebra in the two dimensional (su- per)conformal field theory is essentially the same as that of the corresponding universal enveloping algebra of any simple Lie algebra. We must also empha- size here that one reason for the importance of the Hopf algebra is that the Hopf structure of analgebrafacilities the constructionof representationsof the sample algebra. It is known that W2 - algebra is equivalent to the Virasoro algebra in the W-algebra framework [5]. The W2-algebra, involving the modes of a spin-two field T(z)≡ L z−m−2, is described by the OPE m m P 1c 2T(w) ∂T(w) T(z)T(w)= 2 + + (1. 6) (z−w)4 (z−w)2 z−w wherecisthecentralcharge. Accordinglythe VirasorogeneratorsL ’s,which m are the Laurent coefficients of T(z),satisfy the Virasoro algebra c [Ln,Lm]=(n−m)Ln+m+ (n3−n)δn,−m, [Lm,c]=0 (1. 7) 12 Onemustemphasizehere thatthe Virasoraalgebrais a Liealgebra. Therefore The Hopf Structure of this algebra is defined by: △(L ) = L ⊗1+1⊗L , (1. 8) m m m △(c) = c⊗1+1⊗c, (1. 9) ǫ(L ) = 0, (1. 10) m ǫ(c) = 0, (1. 11) S(L ) = −L , (1. 12) m m S(c) = −c, (1. 13) In this work we will concentrate over conformal field concept, so we will only recapitulate the above Hopf structure for the energy-momentum tensor T(z) instead of L modes for that reason : m △(T(z)) = T(z)⊗1+1⊗T(z), (1. 14) ǫ(T(z)) = 0, (1. 15) S(T(z)) = −T(z) (1. 16) 2 It can be verified that this comultiplication rule is an algebraic homomrphism for W2-algebra (1.6): △(1c) △(2T(w)) △(∂T(w)) △(T(z))△(T(w))= 2 + + (1. 17) (z−w)4 (z−w)2 z−w Theorganizationofthis paperisasfollows. Thenextsectioncontainsthe Hopf structureoftheFeigin-Fuchstypeoffreemasslessscalarfieldquantizationinthe commutatorandOPElanguage,respectively.Insection3. theHopfstructureof W2-Algebra is realized.Insection 4.Boththe Hopf structure ofthe Feigin-Fuchs type of free massless fermionic scalar field quantization and the Hopf structure of N =1 superconformal algebra are realized, respectively. 2 The Hopf Structure of Free Massless Bosonic Scalar Fields A Feigin-Fuchs type of free massless bosonic scalar field ϕ(z) is a single-valued function on the complex plane and its mode expansion is given by h(z) ≡ i∂ϕ(z)= a z−n−1. (2. 1) n nX∈Z Canonical quantization gives the commutator relations [a ,a ] ≡ a a − a a = κmδ , (2. 2) m n m n n m m+n,0 where κ is a central element commuting with all the modes {a }, [a ,κ] = n n 0 ,and the aim of the central element κ is to provide to Hopf algebra structure forthefreefieldmodealgebra(2.2). ThisassocoativealgebraisaHopfalgebra with △(a ) = a ⊗1+1⊗a , (2. 3) m m m △(κ) = κ⊗1+1⊗κ, (2. 4) ǫ(a ) = 0, (2. 5) m ǫ(κ) = 0, (2. 6) S(a ) = −a , (2. 7) m m S(κ) = −κ (2. 8) From the consistency of this Hopf structure with (2.2),i.e. the coproduct oper- ation is given by △[a ,a ] = △(a )△(a )−△(a )△(a ) (2. 9) m n m n n m = 1⊗[a ,a ]+[a ,a ]⊗1 (2. 10) m n m n = △(κ)mδ (2. 11) m+n,0 3 Ontheotherhand,thecommutatorrelations(2.2)areequivalenttothecontrac- tion κ h(z)h(w) = +:h(z)h(w): (2. 12) (z−w)2 So,weshallnowdemonstratethatthiscontractionrelationhasaHopfstructure in the OPE language. △(h(z)) = h(z)⊗1+1⊗h(z), (2. 13) ǫ(h(z)) = 0, (2. 14) S(h(z)) = −h(z) (2. 15) One can check that (2.13-15) satisfy the Hopf algebra axioms and that the defining relation (2.12) is consistent with them. i.e.the coproduct △ △(h(z))△(h(w)) = 1⊗h(z)h(w)+h(z)h(w)⊗1 + h(z)⊗h(w) + h(w)⊗h(z) (2. 16) △(κ) = +:△(h(z))△(h(w)): (2. 17) (z−w)2 In the above,it is seen that the last two terms in the (2.16)does not contribute to the OPEs as the singular terms. 3 The Hopf Structure of W -Algebra 2 One can says that the W2-algebra is realized by the Feigin-Fuchs type of free masslessscalarfields{h(z)},ofconformalspin-1. Letusdefineaconformalfield T(z) having conformal spin-2 as follows: 1 T(z)= :h(z)h(z): (3. 1) 2 Using the contraction (2.12), we want to construct W2-algebra . So the non- trivial OPE of T(z) with itself takes the form 1κ2 2κT(w) κ∂T(w) T(z)T(w)= 2 + + +:T(z)T(w): (3. 2) (z−w)4 (z−w)2 z−w For this construction,we used the following OPEs: κh(w) κ∂h(w) T(z)h(w) = + +:T(z)h(w):, (3. 3) (z−w)2 z−w κh(w) h(z)T(w) = +:h(z)T(w): (3. 4) (z−w)2 4 The corresponding Hopf structure is given by 1 △(T(z)) = :△(h(z))△(h(z)): (3. 5) 2 = 1⊗T(z)+T(z)⊗1+h(z)⊗h(z) (3. 6) one can say here that this coproduct is not as in (1.6),but this additional term h(z)⊗h(z)will give us a relationbetween △(c) and△(κ2) in the following cal- culations.Finally,itcanbeverifiedthatthiscomultiplicationruleisanalgebraic homomorphism for the W2-algebra (3.2). △(T(z))△(T(w)) = 1⊗T(z)T(w)+T(z)T(w)⊗1 + h(w)⊗T(z)h(w)+T(z)h(w)⊗h(w) + h(z)⊗h(z)T(w)+h(z)T(w)⊗h(z) + h(z)h(w)⊗h(z)h(w) (3. 7) by using the explicit operator product expansions (3.2-4) 1κ2 2κT(w) κ∂T(w) = 1⊗ 2 + + +:T(z)T(w): (cid:26)(z−w)4 (z−w)2 z−w (cid:27) 1κ2 2κT(w) κ∂T(w) + 2 + + +:T(z)T(w): ⊗1 (cid:26)(z−w)4 (z−w)2 z−w (cid:27) κh(w) κ∂h(w) + h(w)⊗ + +:T(z)h(w): (cid:26)(z−w)2 z−w (cid:27) κh(w) κ∂h(w) + + +:T(z)h(w): ⊗h(w) (cid:26)(z−w)2 z−w (cid:27) κh(w) κh(w) + h(z)⊗ +:h(z)T(w): + +:h(z)T(w): ⊗h(z) (cid:26)(z−w)2 (cid:27) (cid:26)(z−w)2 (cid:27) κ κ + +:h(z)h(w): ⊗ +:h(z)h(w): (cid:26)(z−w)2 (cid:27) (cid:26)(z−w)2 (cid:27) (3. 8) and also Taylor expansion for h(z) at w 1 h(z) = h(w)+(z−w)∂h(w)+ (z−w)2∂2h(w)+··· (3. 9) 2 we obtain △(1κ2) △(2κT(w)) △(κ∂T(w)) △(T(z))△(T(w)) = 2 + + (z−w)4 (z−w)2 z−w + :△(T(z))△(T(w)): (3. 10) 5 we must emphasize here that the aims of the element κ is to provide to Hopf algebra structure for the free field algebra (2.12),then the coproduct of central element c for the abstract W2-algebra (1.6) must be △(c)=c⊗1+1⊗c as in equation (1.9), but in the present realization c˜= 1 since there is one free field and one can says that the contribution of the one free field to the central term isonly one,andthenitis seenthatthe centraltermandallstructureconstants of the W2-Algebra depend only generator κ , so there is a relation between the centralelement△(c)and△(κ2)ingeneral,butthisappearsinthepresentwork as △(c) = △(c˜.κ2) =△(c˜)△(κ2)=△(c˜)△(κ)△(κ) (3. 11) = κ2⊗1+1⊗κ2+2κ⊗κ (3. 12) where △(c˜= 1) = 1⊗1 . Therefore, this relations prevent a doubling of the centralchargecandκ2. ThispointsofviewisalsovalidforN=1superconformal algebra. 4 The Hopf Structure of Superconformal Alge- bra N = 1 superconformal algebra [5] is generated by a fermionic spin-3/2 chiral field G(z) and stres-energy tensor T(z) ,which are satisfy the following OPEs: 2c 2T(w) G(z)G(w) = 3 + (4. 1) (z−w)3 (z−w) 3G(w) ∂G(w) T(z)G(w) = 2 + (4. 2) (z−w)2 z−w 1c 2T(w) ∂T(w) T(z)T(w) = 2 + + (4. 3) (z−w)4 (z−w)2 z−w This algebra can be realized a Feigin-Fuchs Type of free massless scalar field h(z) ≡ i∂ϕ(z) (2.1) and a real fermion field ψ(z)= ψnz−n−12 (4. 4) nX∈Z InadditiontotheHopfstructureoftheFeigin-FuchsTypeoffreemasslessscalar boson field quantization as in section 2, we have to give the Hopf structure of the Feigin-Fuchs Type of free massless scalarfermionfield quantization. So the canonical quantization gives the following anti-commutator statement, {ψ ,ψ } ≡ ψ ψ + ψ ψ = κδ (4. 5) m n m n n m m+n,0 6 and [ψ ,κ]=0 . This assocoative algebra is a Hopf algebra with n △(ψ ) = ψ ⊗1+1⊗ψ , (4. 6) m m m ǫ(ψ ) = 0, (4. 7) m S(ψ ) = −ψ (4. 8) m m From the consistency of this Hopf structure with eqn. (4.5),i.e. the coproduct operation is given by △{ψ ,ψ } = △(ψ )△(ψ )+△(ψ )△(ψ ) (4. 9) m n m n n m = 1⊗{ψ ,ψ }+{ψ ,ψ }⊗1 (4. 10) m n n m = △(κ)δ (4. 11) m+n,0 whereweusedaparitycondition(a1⊗b1)(a2⊗b2)=−a1a2⊗b1b2 (ifb1 anda2 areodd). Onthe other hand,the anti-commutatorrelations(4.5)areequivalent to the contraction statement κ ψ(z)ψ(w)= +:ψ(z)ψ(w): (4. 12) z−w So,weshallnowdemonstratethatthiscontractionrelationhasaHopfstructure in the OPE language. △(ψ(z)) = ψ(z)⊗1+1⊗ψ(z) (4. 13) ǫ(ψ(z)) = 0 (4. 14) S(ψ(z)) = −ψ(z) (4. 15) Onecancheckthatequations(4.13-15)satisfytheHopfalgebraaxiomsandthat the defining relation equation (4.12) is consistent with them. i.e.the coproduct △ △(ψ(z))△(ψ(w)) = 1⊗ψ(z)ψ(w)+ψ(z)ψ(w)⊗1 +ψ(z)⊗ψ(w)+ψ(w)⊗ψ(z) (4. 16) △(κ) = +:△(ψ(z))△(ψ(w)): (4. 17) z−w In order to realize the Hopf structure of N = 1 Superconformal algebra which aregivenasineqn.(4.1-3). LetusdefineaconformalfieldG(z)havingconformal spin-3, as follows : 2 G(z) = :ψ(z)h(z): (4. 18) By using the statements (2.12) and (4.12), the OPE of G(z) with itself takes the form, κ2 2κT(w) G(z)G(w)= + +:G(z)G(w): (4. 19) (z−w)3 (z−w) 7 where T(z) is stress-energy tensor, which is in the form of : 1 1 T(z) = :h(z)h(z): + :ψ(z)∂ψ(z): (4. 20) 2 2 and the OPEs with itself and also with G(z)are 1κ2 2κT(w) κ∂T(w) T(z)T(w) = 2 + + +:T(z)T(w): (4. 21) (z−w)4 (z−w)2 z−w 3κG(w) κ∂G(w) T(z)G(w) = 2 + +:T(z)G(w): (4. 22) (z−w)2 z−w respectively.For this realization ,we emphasize here that we used the OPEs in the equations (3.3-4),and also the following OPEs: 1κψ(w) κ∂ψ(w) T(z)ψ(w) = 2 + +:T(z)ψ(w): (4. 23) (z−w)2 z−w κψ(w) κ∂ψ(w) G(z)h(w) = + +:G(z)h(w): (4. 24) (z−w)2 z−w κh(w) G(z)ψ(w) = +:G(z)ψ(w): (4. 25) z−w The Hopf structure of N =1 superconformal algebra is given by △(G(z)) = :△(ψ(z))△(h(z)): (4. 26) = 1⊗G(z)+G(z)⊗1+h(z)⊗ψ(z) +ψ(z)⊗h(z)(4. 27) and 1 1 △(T(z)) = :△(h(z))△(h(z)): + :△(ψ(z))△(∂ψ(z))(:4. 28) 2 2 = 1⊗T(z)+T(z)⊗1+h(z)⊗h(z) + ψ(z)⊗∂ψ(z)+∂ψ(z)⊗ψ(z) (4. 29) Finally, One can easily verify that these comultiplication rules are an algebraic homomorphism for the N =1 Superconformal algebra (4.1-3). 2 △(κ ) △(2κT(w)) △(G(z))△(G(w)) = + (z−w)3 (z−w) + :△(G(z))△(G(w)): (4. 30) △(1κ2) △(2κT(w)) △(κ∂T(w)) △(T(z))△(T(w)) = 2 + + (z−w)4 (z−w)2 z−w + :△(T(z))△(T(w)): (4. 31) and △(3κG(w)) △(κ∂G(w)) △(T(z))△(G(w)) = 2 + (z−w)2 z−w + :△(T(z))△(G(w)) : (4. 32) 8 5 Conclusions Inthis latterwehavepresentedthattheuniversalenvelopingalgebraofthe un- derlyingsymmetryalgebrainthetwodimensional(super)conformalfieldtheory is essentially the same as that of the corresponding universal enveloping alge- bra of any simple Lie algebra, with examples only for the W2-algebra and also N =1 superconformal algebra. We will try to extand these studies beyond the W2-algebraand also at least N =2 superconformalalgebra. The investications in this directions are under study.At this point this paper does nothave a com- posedsystem,but,besidessomepreviousarticles[6]adetailedcalculationswill be also given for the connection between the two products in the subsequent works which will be the complementary to this one. 6 Acknowledgement I would like to thank M. Chaichian ,M.Arik and E.Hizel for their valuable discussions and excellent guidance throughout this research. References [1] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory , Vols. 1, 2 (Cambridge Univ. Press. Cambridge. 1987). [2] C. Itzykson, H. Saluer and J.B. Zuber, eds, Conformal Invariance and Ap- plications to Statistical Mechanics (World Scientific, Singapore, 1988). [3] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1984) 333. [4] E. Abe, Hopf Algebras (Cambridge University Press,1980). [5] P. Bouwknegt and K. Schoutens, Phys. Rep. 223 (1993) 183-276. [6] H.T.O¨zer,”OnTheConstructionofWn -AlgebrasInTheFormofWAn−1 -Casimir Algebras”, Mod. Phys. Lett. A, Vol.11, No.14 (1996) 1139-1149, H.T.O¨zer,”Miura-LikeFreeFieldRealizationofFermionicCasimirWB3- Algebras”, Mod. Phys. Lett. A , Vol.14,No.7, (1999) 469-477, H.T. O¨zer, ” On the Super Field Realization of Super Casimir WA - Algebras”, Int. J. n Mod. Phys. A, Vol. 17, No. 3 (2002) 317-325. 9