ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaA Treatment of the Schwinger Modelwithin Noncommutative GeometryH. GrosseP. Pre�snajder

Vienna, Preprint ESI 554 (1998) May 18, 1998Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

UWThPh-22-1998hep-th/9805085A Treatment of the Schwinger Modelwithin Noncommutative GeometryH. GrosseInstitut for Theoretical Physics, University of Vienna,Boltzmanngasse 5, A-1090 Vienna, AustriaP. Pre�snajder1Department of Theoretical Physics, Comenius UniversityMlynsk�a dolina, SK-84215 Bratislava, SlovakiaAbstractWe describe a free spinor �eld on a noncommutative sphere startingfrom a canonical realization of the algebra U(u(2j1)) and a sequenceof su(2)-invariant embeddings of su(2) representations. The gaugeextension of the model - the Schwinger model on a noncommutativesphere is de�ned and the model is quantized. Due to the noncom-mutativity of the sphere, the model contains only a �nite numberof modes, and consequently is non-perturbatively UV-regular. Thefermionic determinants and the e�ective actions are calculated. Theorigin of chiral anomaly is clari�ed. In the commutative limit standardformulas are recovered.1Supported by the "Fond zur F�orderung der Wissenschaftlichen Forschung in�Osterreich" project P11783-PHY and by the Slovak Grant Agency VEGA project1/4305/97.

1 IntroductionThe basic notions of the noncommutative geometry were developed in [1],[2], [3], and in the form of the matrix geometry in [4], [5]. The essence of thisapproach consists in reformulating �rst the geometry in terms of commutativealgebras and modules of smooth functions, and then generalizing them totheir noncommutative analogs. The notion of the space as a continuum ofpoints is lost, and this is expected to lead to an UV-regular quantum �eldtheory.One of the simplest models for a noncommutativemanifold is the noncom-mutative (fuzzy) sphere. It was introduced by many authors using varioustechniques, [6], [7], [8], [9]. In general, these are related to �nite dimensionalrepresentations of the SU(2) group. Thus, the models in question are basi-cally matrix models. Quantum �eld theoretical models with a self-interactingscalar �elds on a truncated sphere were described in [10], [11]. Since, the �eldsposses only �nite number of modes the models are UV-regular.The basic ingredient of the noncommutative geometry developed in [1], [2]is the spectral triple (A;D;H), together with a chirality operator (grading �)and a charge conjugation (antilinear isometryJ ). Besides a noncommutativealgebra A, the spectral consists of a Dirac operator D acting in the Hilbertspace of spinor �elds H. The free spinor �elds on a noncommutative spherein the spirit of noncommutative geometry were introduced in [11] and [12]in the framework of supersymmetric approach. The spectrum of the freeDirac operator was found identical but truncated to the standard one on acommutative sphere. 1

Our aim is to demonstrate, that the axioms of the noncommutative geom-etry can be implemented to a nontrivial �eld theoretical model. The modelwe wish to describe is the noncommutative analog of the Schwinger model ona sphere. The commutative quantum version was analyzed in detail in [14].Its noncommutative matrix version was proposed in [15], an approach goingbehind matrix models was sketched in [16]. Recently a systematic classicalnoncommutative formulation was found in [17], where a di�erential calculuson a supersphere in the commutative and noncommutative cases is describedin detail. Gauge models (for topologically trivial �eld con�gurations) aredescribed in both commutative and noncommutative versions.Here we shall �rst modify the supersymmetric descpription of topologi-cally nontrivial spinor �eld con�gurations on a noncommutative sphere foundin [11]. Instead of using in the formulation of the model �nite dimensionalrepresentations of the superalgebra osp(2j1) (thus working in fact within thematrix model approach), we shall start from an in�nite dimensional canonicalrealization of the enveloping algebra U(u(2j1)). We introduce linear su(2)-invariant embeddings of various sequences of �nite dimensional representa-tions of the even su(2) subalgebra (similar embeddings are important withina general approach to the quantization of vector bundles proposed in [18];this is natural since, our interpretation of �elds on a nocommutative sphereis technically close to the mentioned quantization). Only the evaluation ofthe action is performed within particular �nite dimensional representations.A free Dirac operator is introduced and the model is gauged. After speci�-cation of the gauge degrees of freedom the model is quantized. The resultingmodel is well de�ned nonperturbatively and UV-regular. This allows an exact2

analysis of various non-perturbative objects like fermionic determinants ande�ective actions. The non-perturbative origin of chiral symmetry breakingis clari�ed. In the commutative limit the standard formulas are reproduced(see e.g. [14]).The paper is organized as follows. In Section 2 we describe the free spinor�eld on a noncommutative sphere within the u(2j1) supersymmetric formal-ism and we discuss linear embeddings of �nite dimensional representationsof the even su(2) subalgebra. In Section 3 we introduce gauge degrees offreedom, and we present a �eld action for noncommutative Schwinger model.In Section 4 we quantize nonperturbatively the model, and then we calculatefermionic determinants and e�ective �eld actions. Last Section 5 containsconcluding remarks.2 Free spinor �eldFirst we summarize the commutative version of the model in question in theSU(2)-invariant supersymmetric formulation (see [14] for SU(2)-invariantformalism, and [11], [12], [17] for its supersymmetric reformulation). In thisapproach the �elds are functions of two pairs ��, ���, � = 1; 2, of complexvariables and of one anticommuting (Grassmannian) pair a; a�:� = X a��mn��m�na��a� ; (1)where n = (n1; n2) is a two component index and �; � = 0; 1 (we are using amultiindex notations: �n = �n1; �n2 , jnj = n1 + n2, n! = n1!n2!, etc.). Thespace sHk of �elds with a topological winding number 2k is de�ned as the3

space of �elds (1) with 2k = jnj � jmj+ �� � �xed. Any �eld from sHk canbe expanded as� = �0(�;��) + f(�;��)a + g(�;��)a� + F (�;��) ; (2)where = a�a�aa�, �0; F 2 Hk, f 2 Hk+ 12 and g 2 Hk� 12 . HereHk0 denotesthe subspace of �elds from sHk0 with � = � = 0.The subspace of odd elements from sHk is identi�ed with the space Sk ofspinor �elds with a given topological winding number 2k, i.e. any �eld fromSk can be expanded as = f(�;��)a + g(�;��)a� ; (3)where f 2 Hk+ 12 and g 2 Hk� 12 . The chirality operator � in Sk is given as� = P+ � P�, where P� are projectors Sk ! S�k given by:P+ = a@a = fa ; P� = a�@a� = ga� : (4)The chirality operator takes in S�k the value �1. The charge conjugation Jis de�ned as followsJ = J (fa + ga�) := g�a � f�a� = �� = � : (5)Obviously, J : Sk ! S�k, and J 2 = �1. The inner product in Sk we de�neas (1;2) = Z d��12 = Z d�(f�1 f2 + g�1g2) ; (6)where d� = 116�2rd2�d2��dada��(����� + a�a� r) and d� = 12�rd3x�(x2i � r2)are normalized measures on a supersphere sS3 and sphere S2 respectively.4

In the superspace C2j1 we have a natural action of the superalgebra u(2j1)of matrices [19]0B@ X 00 B 1CA + 1� ip2 0B@ 0 V�V � 0 1CA ; X = �X� ; B = �B� ; (7)written in a (2j1)-block diagonal form. In the space C2j1 a super-Poissonbracket algebra can be introduced by postulating elementary bracket rela-tions among C2j1 coordinate functions[��; ���] = ��� ; [a; a�] = 1 (8)(with all other elementary brackets vanishing). The u(2j1) superalgebra isthen realized in terms of super-Poisson brackets by choosing its basis in thefollowing way: xi = i2�+�i� ; b = �+� + 2a�a ;v� = ��a� ; �v� = "�����a ; r = �+� + a�a : (9)The operators xi; b are even generators and v�; �v� the odd ones of the superal-gebra su(2j1); r is the central element extending it to the u(2j1) superalgebra.The u(2j1) superalgebra is realized in the space of sHk as the adjointPoisson-bracket superalgebra fXi; B; V�; �V�; Rg:Xi = [xi;] ; B = [b;] ; R = [r;]V� = [v�;] ; �V� = [�v�;] : (10)Obviously, the space sHk is an invariant space with respect to (11). Thisaction can be extended to the superalgebra B = U(u(2j1)) = B0 � B1 (the5

enveloping algebra of the superalgebra U(u(2j1)). Since the function r =�+�+ a�a is an invariant function with respect to this action, all functions(�elds) � can be factorized by the relation r = const.The spinor space Sk is invariant with respect to the action of the evenPoisson bracket subalgebra corresponding to A := B0. The free Dirac op-erator is an operator in Sk which has the following form in terms of u(2j1)generatorsD0 = 12"��(V�V� + �V� �V�) = "��(��@���a�@a + ���@��a@a�) : (11)We stress that this Dirac operator already contains a topological gauge �eldterm (the k-monopole magnetic �eld). Similarly, the chirality operator canbe expressed as � = B � R = a@a � a�@a� : (12)The chirality operator anticommutes with the Dirac operator: D0� = ��D0,i.e. the Dirac operator is a chiral odd operator mapping S�k ! S�k . The freespinor �eld action is de�ned byS[;�] = Z d��D0 : (13)Now we shall describe a noncommutative version of the free spinor model.We shall go beyond matrix models starting from the in�nite dimensionalsuperalgebra B = U(u(2j1)) = B0 � B1 and its even subalgebra A := B0.Only the evaluation of the action will be performed at some �nite level.We shall work within the canonical �B realization of the superalgebra B:we insert graded commuting variables a; a�, ��, ���, � = 1; 2 by annihilation6

and creation operators satisfying graded commutation relations[��; ���] = ��;� ; [a; a�] = 1 (14)(all other elementary commutators vanish). They act in the Fock (super)-space sF = fjn; �i = 1pn!��na��j0ig = sF0 � sF1 ; (15)possesses a natural grading with respect to the fermion occupation number� (in eq. (16) n is a two-component multiindex, � = 0; 1, and j0i is thecorresponding normalized vacuum state: ��j0i = aj0i = 0). In what followswe shall use the notation F = sF0, and we shall simply write B and Ainstead of �B and �A.The generators satisfying in sF the U(u(2j1)) graded commutation rela-tions (10) are given by eqs. (9) (commuting parameters are replaceded byannihilation and creation operators and graded Poisson brackets by corre-sponding graded commutators). The Fock subspacesFN = fjn; �i ; jnj+ � = Ng = sF0N � sF1N ; (16)is the carrier space of the unitary irreducible representation �0N of u(2j1)superalgebra. We shall use the notation FN = sF0N .We de�ne the space sHk of super�elds with the winding number 2k asthe set of operators in the Fock space of the form (1) with 2k = jnj �jmj+ � � � �xed. Obviously, � 2 sHk maps any space sFN into the spacesFM , M = N + 2k. The u(2j1) superalgebra is realized in sHk as theadjoint graded commutator superalgebra (10). Any super�eld from sHk can7

be expressed as in (2) with coe�cients from Hk (however now expressed interms of anihilation and creation operators).By restricting the action of super�elds � 2 sHk from sF to sFN we obtaina space of mappings sFN ! sFM , M = N + 2k. We shall denote them bysHJ 0k , k = 12(M � N), J 0 = 12(M + N � 1) = J � 12 (these relations amongM;N; J; J 0 and k are assumed in what follows). From the de�nition of sHJ 0kit follows that on sHJ 0k acts the direct product ��0M �0N of the irreduciblerepresentations ��M and �N of the superalgebra u(2j1).Note: Representations �0N of u(2j1) are well known. They correspondsto the atypical (q;�q) representations of su(2j1), in the notation of [20]:quadratic and cubic Casimir operators vanish and they are classi�ed by thevalue R = N of the central element in u(2j1) superalgebra. However, theirproducts are indecomposable and have quite pathalogical properties, [20].This causes problems with the identi�cation of auxiliary �elds in the non-commutative case.Such problems do not occur for the coe�cient spaces Hk of operators act-ing in FN := sF0N . The subspaces HJk are de�ned as the spaces of mappingsFN ! FM . In HJk we introduce the scalar product(�1;�2)Jk = 1J + 1Tr(��1�2) : (17)InHJk acts the direct product representation ��M�N of the irreducibleSU(2)representations ��N and �N in FM and FN respectively. The direct product��M�N possesses a decomposition to irreducible SU(2) representations, and8

consequently HJk = JMj=jkj V Jjk ; (18)where V Jjk denotes a SU(2) representation space corresponding to a spin j.For each J and k we constructed in Appendix A an orthormal basisfDjkm, j = 0; 1; : : : ; jmj � jg in Hk. The orthonormal basis fDJjkm = cJkjDjkm,j = 0; : : : ; J , jmj � jg in HJk with respect to the scalar product (17) isdetermined by rescaling coe�cients cJkj determined in Appendix A.We de�ne the isometrical projection operator CJk : Hk ! HJk byCJkDjkm =DJjkm, and by C 0Jk : HJk ! Hk we denote the reversed isometrical embedding.We can extend this construction by taking K � J and de�ning the isomet-rical projections CKJk : HKk ! HJk , together with the reversed isometricalembeddings C 0JKk : HJk ,!HKk . It holdsCKJk = CJkC 0Kk ; C 0JKk = CKk C 0Jk : (19)These are the key objects of our approach which give the prescription howare related to each other for various values of J the (matrix) realizations inHJk of a given �eld from Hk.Similarly as in the graded commutative case, we de�ne the space Sk ofspinor �elds with a topological winding number 2k as the odd subspace ofthe space sHk. Such spinor �elds can be expanded as = f(�;��)a + g(�;��)a� ; (20)where f 2 Hk+ 12 and g 2 Hk� 12 . The chirality operator � : Sk ! Sk andthe charge conjugation J : Sk ! S�k are de�ned in the same way as in thecommutative case. 9

The subspaces SJ 0k are again given by the restriction: they are odd map-pings from HJ 0k , i.e. odd mappings sFN ! sFM with M = J 0 + k + 12,N = J 0 � k + 12 . In the space SJk we choose the inner product(1;2)J 0k = 12J 0 + 1sTrN ( �12) ; (21)where sTrN denotes the supertrace in the space of mappings sFN ! sFN .Now we shall concentrate our attention to the space of mappings Sk ! Sk.Such mappings have the formXAiBoi := XAiB�i ; (22)where Ai,Bi are elements from sH0 such that all products AiBi are even(Bo means the right multiplication of by B�). In the space of mappingsSk ! Sk we have a natural grading induced by the chirality operator: themapping PAiBoi is even (odd) if all Ai and Bi are even (odd). Obviously,S�k ! S�k for the even mappings and S�k ! S�k for the odd ones.In the noncommutative case the free Dirac operator is given by the �rstequation in (11) but with graded Poisson brackets inserted by graded com-mutators. The resulting expression can be rewritten in the formD0 = v��vo� � �v�vo� ; (23)which was proposed earlier in [11]. This is the simplest odd mapping com-muting with the su(2) generators Xi = xi � xoi , i = 1; 2; 3.The spectrum of the free Dirac operator contains:(i) non-zero modesEJ 0�jk = �s(j + k + 12)(j � k + 12) ; j = jkj+ 12 ; jkj+ 32 ; : : : ; J ; (24)10

(ii) and 2jkj zero-modes corresponding to j = jkj � 12 (if k 6= 0).The corresponding eigenfuctions fJ 0j�km g, j = jkj � 12 ; jkj+ 12; : : : ; J 0, jmj � jare presented in Appendix A. They form an orthonormal basis in SJ 0k . Anyspinor �eld from SJ 0k can be expanded as = J 0Xj=jkj+1=2 Xjmj<j(aj+kmJ 0j+km + aj�kmJ 0j�km ) : (25)For a given k and J = J 0 + 12 the free spinor �eld action we take asSJk [;�] = (;D0)Jk : (26)The spinor �eld space Sk is obviously anA-bimodule. The algebra AAoacts in Sk as left-multiplication by A 2 A and right-multiplication by B� 2 A(see (22)). The Dirac operator acts in AAo as a commutator:D0(ABo) = [D0; ABo] + ABoD0 : (27)Thus, D0 : ABo ! [D0; ABo]. Consequently, the graded Leibniz rule inA Ao is satis�ed. Obviously, this changes the grading of the mapping.The charge conjugation J induces in the space Sk ! Sk an automorphismABo ! J �1ABoJ . It is easy to see that the Dirac operator corresponds toan J -odd mapping: J �1D0J = �D0.3 Gauge symmetryIn the commutative case the gauge transformations of the spinor �eld aregiven in the following way ! ! ; � ! !�� ; (28)11

where ! is an arbitrary unitary element from H0. To guarantee the gaugeinvariance one should replace the free Dirac operator D0 by the full Diracoperator D = D0 + qoA : (29)Here qo is a dimensionless coupling constant related to the usual couplingconstant q and the radius r of the sphere in question as qo = qr. Thecompensating gauge �eld A transforms inhomogenously under gauge trans-formations A ! A + ![D0; !�] : (30)Since, D0 already contains the topological (k-monopole) gauge �eld, the �eldA is a globally de�ned gauge �eld. Such gauge �elds can be expressed interms of the real prepotentials � 2 H0 corresponding to the pure gauge�eld A (exact 1-form) and � 2 H0 corresponding to the dynamical gauge�eld A (coexact 1-form). The full Dirac operator can be written in terms of~ = UQP+ + UQ�1P� with U = exp(i�) and Q = exp(qo�). The formulafor D can be rewritten as D = ~D0~� : (31)The square of the Dirac operator has the formD2 = [�() + qoF ()]P+ + [�()� qoF ()]P� ; (32)where �() is the Laplace operator on a sphere with the gauge �eld included,and the �eld strenght operator F () isF () = q�1o (���@��� � ��@��) + �� : (33)12

The di�erential term takes in Sk the constant value q�1o k and it correspondsto the contribution of the k-monopole, the last term is the �eld strenghtgenerated by the dynamical �eld �.The pure gauge prepotential � can be gauged away, and there are norestrictions on it. There are limitations on the dynamical prepotential, since� enters the Schwinger model �eld actionS[;�;] = 14 Z d�F 2() + Z d��D ; (34)which is obviously gauge invariant. Expanding � into spherical functions(denoted as Dj0m):� = 1Xj=0 Xjmj�j bjmDj0m ; bj�m = (�1)mbj�m ; (35)we obtain a contribution to the action proportional to1Xj=0 Xjmj�j j2(j + 1)2jbjmj2 :This is �nite provided that bjm = o(j�5=2) : (36)In the noncommutative case we identify the group of gauge transforma-tions with the group of unitary operators ! 2 H0, and we postulate that thespinor �elds 2 Sk transform under gauge transformations as follows ! ~! = !!o = !!� : (37)This is an unitary J -invariant mapping in Sk. In order to obtain a gaugeinvariant action we introduce compensating gauge degrees of freedom. We13

shall describe them by the operators of the form = UQ(+)P+ + UQ(�)P�,where U 2 H0 is an unitary operator describing pure gauge degrees of free-dom and Q(�) 2 H0 are positive operators corresponding to the dynamicalgauge degrees (they will be speci�ed below). To any we assign the mapping~ in Sk: ! ~ = o = � : (38)Explicitely, acting by ~ and ~� on = fa+ ga� 2 Sk we obtain~ = UQ(+)fQ(�)U�a + UQ(�)gQ(+)U�a� ;~� = Q(+)U�fUQ(�)a + Q(�)U�gUQ(+)a� : (39)We take the interacting Dirac operator in the formD = ~D0~� : (40)Postulating that under gauge transformation ! ! ; (41)we enjoy the transformation property: D! !D!�.Formula (40) for the full Dirac operator has the important property thatit preserves the index of the Dirac operator. Any zero-mode 0 of D0 yieldsa zero-mode ��10 is of D and vice versa. Nonzero-modes appear in pairs,if (+)E is an eigenstate of the full Dirac operator to the eigenvalue E;E > 0,then (�)E = �(+)E is an eigenstate to the eigenvalue �E.The square of the Dirac operator is an even operator in Sk. Similarly asin the commutative case, the square of the Dirac operator can be written asD2 = [�() + qoF ()]P+ + [�()� qoF ()]P� ; (42)14

where �() is the noncommutative analog of a Laplace operator on a spherewith gauge degrees of freedom included, F () is the operator coorespond-ing to the �eld strenght and qo is the coupling constant. The topological(monopole) contribution to the �eld strenght is obtained by putting = 1in F (). In Sk it takes the same value q�1o k as in the commutative case.For a given J and k the spinor �eld action interacting with gauge �eldhas the form SJk [;�;] = 12J 0 + 1sTrN [ �D]+ 1J2 + J � k2 sTrJ 0k [F 2()P+ � F 2()P�]g ; (43)where sTrN in the �rst term denotes the supertrace in the space of mappingssFN ! sFN , and sTrJk in the second one denotes the supertrace in the spaceof mappings SJ 0k ! SJ 0k . The factor in the second term guarantees that thepure topological contribution obtained for = 1 is properly normalized toq�2o k2. The action (46) is gauge invariant. We can gauge U away and �x thegauge by putting = Q(+)P+ +Q(�)P�.In the noncommutative case the dynamical gauge degrees of freedom aredescribed the hermitean operator � 2 H0 possessing an expansion like in (35),however now in terms of a noncommutative analogs of spherical functions:Dj0m 2 H0, j = 0; 1; : : : ; jmj � j, (see Appendix A). Since, the noncommuta-tive gauge �eld action approaches in the commutative limit its commutativeform we shall assume the asymptotic behaviour (36) for the expansion coef-�cients too.We de�ne the operators Q(�) in the Fock space F by their restrictionsQ(�)N 0 to subspaces FN 0. For �xed J and k in the action appear restrictions15

with N 0 = J 0 � k � 12 . The maximal value of N 0 for �xed J and arbitraryjkj � J is 2J . Therefore, it is enough to de�ne the the operators Q(�)N 0 ,N 0 � K, for some �xed K > 2J . Namely, we putQ(�)N 0 = exp�(�)KN 0 ; N 0 � K ; (44)where the operators�(�)KN 0 = (ln �C 0KN 00 � exp �CKN 00 )(�eo�) ; (45)are de�ned with the help of isometrical mappings CKN 00 : HKk ! HN 0k andC 0KN 00 : HN 0jk ,! HKjk introduced in Appendix A (eo is a dimensionless pa-rameter speci�ed later).This de�nition guarantees that �KN 0 = CKN 00 � posessess an expansion inFN 0 �KN 0 = CKN 00 � = N 0Xj=0 Xjmj�j bjmDN 0j0m ; (46)with the truncated subset of coe�cients fbjm, j = 0; 1; : : : ;K, jmj � jg.Thus, the untruncated modes have the intensities independent on N 0. The�eld � is for us a primary object, whereas Q(�) are secondary ones.4 Quantization and fermionic determinantWe quantize the Schwinger model within the path integral approach. Firstwe describe the quantization in the commutative case (for details see [14]),and then its noncommutative version.The quantum expectation values of the �eld functionals P [�] (where � =f;�; �; �g denotes the collection of all �elds in question) are de�ned by16

the formula hP [�]i = Z�1 Xk Z (D�)k P [�] e�S[�] ; (47)with the summation over all topological winding numbers � = 2k and thenormalization Z = Z (D�)0 e�S[�] : (48)The action S[�] is given in (43). The symbol (D�)k = (DD�D�D�)k de-notes a formal integration over all �eld con�gurations with a given topologicalnumber. The �eld can be expanded, e.g. into corresponding eigenfuctionsof the free Dirac operator = 1Xj=jkj�1=2 Xjmj�j (aj+kmj+km + aj�kmj�km) (49)with Grassmannian coe�cients aj�km (for j = jkj�1=2 the expansion containsonly one component corresponding to the zero modes). Then Dk � Q daj�kmmeans a formal in�nite dimensional Berezin integration. The symbol D�k isde�ned analogously.If the �eld functional P [�] is gauge invariant, then the integrand in (47)does not depends on a pure gauge prepotential �. A gauge �xing condition in(47) should be imposed, or alternatively, one can factorize out a pure gaugeprepotential �, and take in (47) only the relevant �elds � = f;�; �g.Then (D�)k = (DD�D�)k. Expanding � into spherical fuctions (as in(35)) we introduce a formal in�nite dimensional integration D� � Q dbjm(with bj�m = (�1)mbjm).Thus, the expectation values are de�ned only formally, and some regu-larization procedure is needed. In [14] were calculated in this way various17

non-perturbative quantities. If the functional P [�] in question is a polyno-mial in spinor �elds then the calculation over (DD�)k can be performedexplicitely. This leads to the fermionic determinants and e�ective actionswhich are needed for the calculation of fermionic condensates.The gauge �eld e�ective action Sregk [�] is obtained by integration overDD� in (47) and is given asSeffk [�] = Sk[�] + �regk [�] ; (50)where Sk[�] is the classical gauge �eld action and the second term representsthe quantum correction de�ned by the equationCregk detkD = exp(��regk [�]) : (51)Here Cregk is a �eld independent regularizating factor and detkD denotes theproduct of non-zero eigenvalues of D in the sector with topological windingnumber � = 2k. The formula derived in [14] reads�regk [�] = 2 Z d�(r�)2 � �ij�j � 12 j�j2 � 2j�j ln(j�j!) + 2 j�jXn=1 n lnn : (52)The e�ective action appears in formulas for many important quantities,e.g. the fermionic condensate - the mean value of the �eld functional (;) =R d��. The simplest way how to calculate it consists in adding a "mass"term m(;) to the action and using the formulah(;)i = � @@mhe�m(;)ijm=0 : (53)There are two equal contributions h(;)i�1=2 appearing in the numeratorin (47) for k = �1=2. 18

In the noncommutative case we shall start from the general formula (47),however there are important di�erences. We �x J and restrict the admissiblerange of k to jkj � J . The symbol (D�)Jk = (DD�D�)Jk denotes anintegration over all �eld con�gurations with a given topological number:(i) The spinor �elds 2 SJ 0k and � 2 SJ 0�k can be expanded into cor-responding eigenfuctions of the free Dirac operator with independent Grass-mann coe�cients a�jm , a��jm . The fermionic part of the integration measure(D�)Jk denotes the �nite Berezin integral(DD�)Jk = Yjmj�jkj�1=2 da0mpJ J 0Yj=jkj+1=2 Yjmj�j da�jmpJ da��jmpJ ; (54)(the �rst product comes from the contributions of over zero-modes and ispresent only for k 6= 0).(ii) We restrict � to FK with some �xed K > 2J . Expanding � as in (46)with expansion coe�cients bjm = (�1)mbj�m we put(D�)Jk = KYj=0 dbj0 jYm=1 dbjm : (55)There are no problems with the gauge �xing. The gauge group is isomor-phic to SU(K + 1) possessing a �nite volume which can be factorized out.The number of all modes, the dimension of the measure (D�)Jk , is �nite, andconsequently there are no ultraviolet divergencies. This allows to calculatestraightforwardly various non-perturbative quantities.Let the observable P [�] in (47) be independent of fermion �elds. Weperform a bosonization of the model by integrating out the fermionic degreesof freedom and obtaining an e�ective action SeffJk [�] = SJk [�] + �Jk [�]. Here19

SJk [�] is the classical noncommutative gauge action (the second term in (43))and �Jk [�] is the quantum correction given asexp(��Jk [�]) = detkDdetD0 = R (D0D0�)Jke�( �;D)JkR (DD�)J0 e�( �;D0)J0 ; (56)where (D0D0�)Jk denotes the integration over non-zero modes. It is moreconvenient to take instead of D = ~D0~� the shifted Dirac operator ~(D0+m)~�, possessing 2jkj modes with E = m instead of zero-modes. Thenexp(��Jk [�]) = limm!0 1m2jkj R (DD�)Jke�( �;~(D0+m)~�)JkR (DD�)J0 e�( �;D0)J0 : (57)We rewritte it as a product of two factors. The �rst onelimm!0 R (DD�)Jk e�( �;~(D0+m)~)JkR (DD�)Jke�( �;(D0+m))Jk = det~ det~� (58)do not depend on m.In Appendix B we derived the formula for the rescaling mappingC 0KN 00 = expf� 1Xk=0[(N 0 + 1)�2k�1 � (K + 1)�2k�1]Sk(�)g ; (59)with a known polynomials Sk(�) in Laplace operator � = X2i , and an anal-ogous formula for CKN 00 di�ering just by the sign in the exponent. Usingthem we calculated in Appendix B the commutative limit J !1 of the �rstfactor. Taking K = O(J "), " > 1, and normalizing the constant eo properlyby e2o = q2o(2J + 1)�1 we obtained the leading term in the asymptotic formof (58): det~ det~� = expf�2q2o Z d�(Xi�)2 + o(J�1)g : (60)The second factor gives with � = 2klimm!0 1m2jkj R (DD�)Jke�( �;(D0+m))JkR (DD�)J0e�( �;D0)J0 = J2jkj2 detkD0det0D0 =20

exp[�ij�j+ 12 j�j2 + 2j�j ln(j�j!)� 2 j�jXn=1 n lnn + o(J�1)] : (61)Eqs. (60) and (61) yields in the limit J !1 the well-known expression forthe e�ective action (see e.g. [14]).5 Concluding remarksIn this article we described a noncommutative regularization of the Schwingermodel on a noncommutative sphere. We constructed Connes real spectraltriple (A;H;D0) supplemented by the the grading � and the antilinear isom-etry J (identi�ed with the chiralitity operator and charge conjugation). Animportant new aspect with respect to our previous work [10]-[12] lies in thefact that as the spectral algebra A we take an in�nitedimensional associativealgebra - the even part of the superalgebra B = U(u(2j1)). We have workedwithin its canonical realization.The elements ABo 2 A Ao act in the Hilbert space H of spinor �eldsas the left multiplications by A and the right one by B�. Introducing thedi�erentials dA = [D0; A] and dBo = [D0; Bo] we obtain that in our case thecondition [A; dBo] = [dA;Bo] = 0, usually required in the noncommutativegeometry, is violated.We gauged the model in a standard way: ! !!�, with ! unitary.The compensating gauge �elds A enter the full Dirac operator which can beexpressed as D = D� �D�+D�� �D�� with D� = V�+ qoA� and �D� = �V�+ qo �A�.Such a form of D with general A� and �A� was recently proposed in [17]within supersymmetric extension of the Schwinger model. This formulation21

contains besides physical �elds auxiliary �elds. We eliminated them by ex-pressing A� = UQ[V�; U�1Q�1U�1] and �A� = UQ�1[ �V�; QU�1] in terms ofprepotentials. Moreover, our choice preserves the index of the Dirac operator.The physical model is determined if we choose the action for the �elds inquestion. The evaluation of the action we performed by restricting the �eldsto a particular �nite dimensional mappings in the Fock space. This requiredsequences of linear SU(2)-invariant embeddings of various representations ofA. We evaluated the rescaling coe�cients of these embeddings which arenecessary for the evaluation of the action.The action of the resulting noncommutative Schwinger model contains�nite number of modes for each �eld in question. This allows nonpertur-batively to quantize the model. As an application we calculated the chiralanomaly and the e�ective actions. Although we obtained standard results inthe commutative limit our interpretation is very di�erent:- in the commutative case the nontrivial value of the fermionic determi-nants is due to the singularity of the operator exp(�"D2) for "! 0, and thisleads to the chiral symmetry breaking,- in our approach the chiral anomaly follows straightforwardly from thebasic postulates - its appearance is a direct consequence of the fact that theisometric embeddings and exp-log mappings do not commute.The noncommutative formulation of the Schwinger model leads not onlyto the expected ultraviolet regularization, but straightforwardly to highlynonperturbative results. It provides a better understanding of the procedureof regularization and renormalization, and the mechanisms of bosonizationand chiral symmetry breaking. It would be interesting to extend our investi-22

gations in various directions, e.g. to include gravity along lines presented in[3].Appendix AHere we shall describe various SU(2) representations in the space Hk. Anyoperator � from this space can be expanded as � = P bjmDjkm with m =�j;�j+1; : : : ;+j, j = jkj; jkj+1; : : :. The operators Djkm can be constructedas follows:(i) The operatorDjk;�j = [(2j + 1)!=(j + k)!(j � k)!]1=2 �̂�j+k2 �̂j�k1 (62)is the lowest weight with respect to the adjoint action of the operator X0because X0Djk;�j = [x0;Djk;�j ] = �jDjk;�j and X�Djk;�j = [x�;Djk;�j ] = 0(here �̂� = ��(�+ �)�1=2 and �̂�� = (�+ �)�1=2���).(ii) For a given j all other Djkm can be obtained by a repeated action ofX+, Djkm = [(j +m)!=(j �m)!(2j)!]1=2 Xj�m+ Djk;�j : (63)The operators Djkm, jmj � j (for a given j and k) span a representation spaceV jk of the SU(2) representation with spin j.By de�nition we take fDjkm; j = jkj; jk+1; : : : ; jmj � jg as an orthonormalbasis in Hk. Thus, Hk = 1Mj=jkj V jk : (64)Restricting the action of super�elds � 2 Hk from sF to sFN we obtaina space of mappings sFN ! sFM , M = N + 2k. We shall denote it by HJk ,23

k = 12(M �N), J = 12(M +N). The expansion (64) is now truncated:HJk = JMj=jkj V Jjk ; (65)where V Jjk denotes the space spanned by the operators (63) restricted tosFN . If we introduce in HJk the scalar product(�1;�2)Jk = 1J + 1Tr(��1�2) ; (66)then the orthonormal basis in V Jjk is given asDJjkm = cJkjDjkm ; m = �j; : : : ;+j ; (67)with cJkj = [(J + 1)(J + k)!(J � k)!=(J + j + 1)!(J � j)!]1=2 : (68)Summing up over j = jkj; : : : ; J one obtains aan orthonormal basis in HJk .Taking K � J we de�ne the isometrical projection operator CKJk : HKk !HJk by DKjkm = CKJk DJjkm, and by C 0KJk : HJjk ! HKjk we denote the reversedisometrical embedding. The operators CKJk and C 0KJk are both diagonal withthe eigenvalues cKkj=cJkj and cJkj=cKkj respectively, as follows from the equationcJkjDKjkm = cKkjDJjkm ; j � J � K : (69)Now we determine the spectrum of the free Dirac operator. Since, [X0;D0]= 0, we classify the eigenstates of D0 according to the eigenvalues values ofX0. The spinor �eld = ���m1 �n�12 a + ���m�11 �n2a� (70)24

is the lowest weight of X0 because, X0 = [x0;] = (m + n � 1) andX� = [x�;] = 0. Moreover,D0 = �m��m�11 �n2a� + �n��m1 �n�12 a : (71)This is an eigenstate to the value E = �pmn provided that �pm = �pn.The spinor �eld lowest weight eigenstates arej�k;�j = c(pn��m1 �n�12 a � pm��m�11 �n2a�) ; (72)where j = 12(m+n�1), k = 12(m�n) (if k 6= 0 we obtain for m = 0 or n = 0zero modes corresponding to the �rst or second term in (74) respectively).The remaining eigenstates of D0 to the same eigenvalue are given by thestandard formulaj�km = [(j +m)!=(j �m)!(2j)!]1=2 Xj�m+ j�k;�j : (73)The admissible values of j then are j = jkj; jkj+ 1; : : : J . The normalizationconstant c in (72) is speci�ed if we restrict the space of spinor �elds Sk tothe space SJk with the inner product:(1;2)Jk = 12J + 1 sTr(�12) : (74)Appendix BFrom the formula (68) for the rescaling coe�citients it follows thatcN0j = exp2412 jXn=1 ln 1� n=(N + 1)1 + n=(N + 1)35 =exp "� 1Xk=0(N + 1)�2k�1Sk(j(j + 1))# ; (75)25

where Sk(t) = [(2k + 1)(2k + 2)]�1tk + : : : are polynomials de�ned bySk(j(j + 1)) = 12k + 1 jXn=1 n2k+1 :The eigenvalues of the operator C 0KN 00 can be expressed ascN 00j =cK0j = expf� 1Xk=0[(N 0 + 1)�2k�1 � (K + 1)�2k�1]Sk(j(j + 1))g : (76)The expansion in (75) is convergent provided j � N 0 � K. The eigenvaluesof the operator CKN 00 di�ers just by the sign in front of the sum in exponent.Replacing j(j+1) by the (noncommutative) Laplace operator on a sphere� = X2i one obtains directly from (76) the formulas for the operators C 0KN0and CKN 00 . This allows to derive the leading term formulaQ(�)N 0 = C 0KN 00 expfCKN 00 (�eo�)g =[1� K �N 02(K + 1)(N 0 + 1)� + : : :] exp[�eo(1 + K �N 02(K + 1)(N 0 + 1)� + : : :)�]= exp[�eo� � e2o K �N 02(K + 1)(N 0 + 1)(Xi�)2 + o(K�2; N 0�2)] : (77)Using the well known formula detA B = (detA)M(detB)N (valid formatrices A 2 Mat(N N) and B 2 Mat(M M)) and the de�nition of theoperators ~ : SJ 0k ! SJ 0k it can be easily seen that in the gauge U = 1 wehave det ~ = det[o] =(detM�1Q(+))N+1(detNQ(�))M(detMQ(�))N (detN�1Q(+))M+1 ; (78)where N = J 0� k+ 12 and M = J 0+ k+ 12 as usually. Inserting here (77) weobtain 26

ln det ~ = eo[(N + 1)TrM�1� �MTrN� �NTrM�(M + 1)TrN�1�]� e2o2(K + 1) "(N + 1)(K �M + 1)M TrM�1(�)2 + M(K �N)N + 1 TrN (�)2 +N(K �M)M + 1 TrM (�)2 + (M + 1)(K �N + 1)N TrN�1(�)2+# (79)Taking K = O(J "), " > 1, normalizing the constant eo by e2o = q2o(2J + 1)�1and using relations1N 0TrN 0� ! Z d�� ; 1N 0TrN 0(r�)2 ! Z d�(r�)2 ;(which are valid for N 0 ! 1 due to the asymptotic behaviour (36)) weobtain in the limit J !1 the asymptotic formulaln det ~ = �q2o Z d� (Xi�)2 + o(J�1) ; (80)leading sraightforwardly to eq. (60).For a given J and k 6= 0 the free Dirac operator possesses non-zero modesEJ�jk = �q(j + 12)2 � k2, j = jkj + 12 ; jkj + 32 ; : : : ; J , with the multiplicity2j + 1. Putting j 0 = j + 12 and J 0 = J + 12 we obtainJ2jkj2 detkD0det0D0 = (�1)2kJ2jkj2 QJ 0j0=jkj+1(j2 � jkj2)2j0QJ 0j0=1 j04j0 =(�1)2k jkjYm=1 1 +m=J 01 + (m� jkj)=J 0!2J+2m 2jkjYn=1 n2jkj�2n J2jkj2Qjkjn=1(J 0 + n)2jkj : (81)Denoting � = 2k we obtain in the limit J !1 desired eq. (61).27

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