Axial Symmetry, Anti-BRST Invariance and Modified Anomalies

Dedicated to Ahmad Shafiei Deh Abad for his 66th birthday Axial Symmetry, Anti-BRST Invariance and Modified Anomalies Amir Abbass Varshovi [email protected] School of Physics, Institute for Research in Fundamental Science (IPM) P. O. Box 19395-5531, Tehran, IRAN. Department of Physics, Sharif University of Technology P.O. Box 11365-9161, Tehran, IRAN. Abstract; It is shown that anti-BRST symmetry is the quantized counterpart of local axial symmetry in gauge theories. An extended form of descent equations is worked out which yields a set of modified consistent anomalies. Introduction Soon after the pioneering articles of Adler, Bell and Jackiw [1, 2] on the anomalous axial Ward identities for -gauge theory, Bardeen [3] showed that for more general theories a generalized version of anomalous terms exists. More precisely, Bardeen proved that the root of the anomalous behavior is not entirely confined in the axial currents but is basically in the renormalization of the theory. In the other words, the anomalous terms come from the Feynman diagrams with few enough vertices which diverge linearly or more. On the other hand, Bardeen showed that the scalar and the pseudo1 scalar fields do not appear in the anomalous terms after renormalization while the whole anomalous term is never completely removed by using counter terms. Soon after, Wess and Zumino [4] found a consistency condition as an alternative criterion for gauge invariance of the anomalous terms. The anomalous terms which satisfy the consistency condition are referred to as consistent anomalies. In [4] it was shown that the Bardeen’s results were also confirmed by the consistency condition. Later on Fujikawa [5-8] showed that the root of the anomalies lies in the path integral measure of partition functions. Indeed he rigorously showed that anomalies are intimately related to the celebrated Atiyah-Singer index theorem [9, 10]. By this a fascinating feature for anomalous behaviors of gauge theories was worked out via the spectacular achievements of differential geometry. On the other hand, after Faddeev and Popov [11] discovered their path integral quantization method for non-Abelian gauge theories, an alternative formulation of classical gauge symmetry was found [12, 13]. This discovered quantized setting of gauge invariance is usually referred to as BRST symmetry which stands for the names of its founders. Indeed, the Faddeev-Popov quantization method replaced the classical gauge symmetry by its quantized counterpart, BRST. In this formulation two kind of unphysical fields, ghost and anti-ghost, emerged in the quantum Lagrangian in the path integral setting with fermion statistics and scalar behaviors. Soon after, the anti-BRST, another quantized symmetry of the quantum Lagrangian was found [14, 15]. It was then shown that a deep duality exists between BRST and anti-BRST symmetries and respectively for ghost and anti-ghost fields [16]. Then Stora and Zumino independently [1719] discovered a sequence of descent equations which could relate the consistent anomalies and Schwinger terms [20] through a set of deRhamBRST differential equations. Faddeev showed that this new point of view can be explained in the setting of cohomology algebras [21] and finally Zumino related these concepts to the cohomology of the gauge group [22]. Soon after some rigorous geometric models for Faddeev-Popov quantization method, ghosts, (anti-) BRST transformations and BRST cohomology were constructed to explain the path integral quantization of gauge theories in the setting of differential geometry [23-31] (for more collected and recent works 2 see [32-35]). These geometric formulations led to elegant topological descriptions of consistent anomalies and consistent Schwinger terms [36-38]. In this geometric point of view, ghosts are the left invariant operator valued 1forms over the Lie group of gauge transformations and can be described by principal connections over the principal bundle of Affine space of gauge fields modulo gauge transformations. The geometric procedure produces an alternative setting for Faddeev-Popov path integral quantization method. Indeed, this formulation can be considered as the geometric version of BRST quantization of gauge theories [39-43]. More precisely, although the structure of BRST invariance made its first appearance as an accidental symmetry of the quantum action, but it became apparent that it takes its most sophisticated setting in the Hamiltonian formulation of gauge theories, where it is used in the homological reduction of Poisson algebra of smooth functions on a symplectic manifold on which a set of first class constraints is defined. This constructive formulation lead to more drastic and deeper understanding of ghost fields, gauge fixing, and eventually BRST transformations in terms of symplectic geometry. Such geometric method of quantization of gauge theories based on symplectic data is conventionally referred to as geometric quantization [44-46]. In this article the main ideas of geometric quantization is applied for studying some general version of gauge theories which admit the local axial symmetry. It is shown that because of the enlargement of the gauge transformation group, the algebra of ghosts and BRST transformations get larger. Also it is seen that the behavior of the new geometric objects are in complete agreement with anti-ghosts and anti-BRST transformations [14, 15]. Indeed, a generalized version of BRST and anti-BRST transformations is found by considering the local axial symmetry. In fact, this consideration leads to an enlargement of the algebra generated by gauge fields, ghosts and anti-ghosts by introducing the axial gauge fields. It is shown that despite of the ordinary algebra generated by gauge, ghost and anti-ghost fields, this enlarged algebra is closed under BRST and anti-BRST transformations [16]. The necessity of Nakanishi-Lautrup fields [47, 48] is removed here and these auxiliary fields are replaced by a set of functionals of ghosts and anti-ghosts. This functional 3 can be considered as a colored scalar field with ghost-number zero. On the other hand, the extra degrees of freedom carried by auxiliary fields are compensated by axial gauge fields in the extended Lagrangian. In fact, this development gives a conceptual framework and a geometric formalism to algebraic anti-BRST transformations and anti-ghosts. This also leads to a geometric description of extended BRST quantization of gauge theories [49, 50]. Using these extended BRST transformations via the approach of [17-19], an extended sequence of descent equations and consequently a modified version of consistent anomalies and Schwinger terms is found. According to [3], none of vector and axial currents is the only reason of anomalies. Indeed the anomalous behavior of the theory should be studied by considering the contribution of vector and axial currents simultaneously. Thus, it seems more natural to consider the modified consistent anomalies and Schwinger terms as the anomalous behaviors of the theory. In section 1, Axial Extension of a Gauge Theory, the idea of axially extended gauge theories is studied in the setting of differential geometry. In second 2, Extended BRST Transformations, a summarized review over geometric ghosts and BRST transformations is given and then the geometrical aspects of antighosts and anti-BRST transformations are worked out. Finally, in section 3, Extended Descent Equations, the modified version of anomalous terms is derived and studied properly. In fact, in this article it is shown that the following diagram commutes for gauge theories; lassi al auge etr → al etr → ial B nti etr B etr Although it was thought that the quantization approach causes the classical gauge symmetry to produce two different quantum symmetries [16], BRST 4 and anti-BRST, in this article it is shown that the anti-BRST invariance is the quantized version of local axial symmetry, which is broken in standard YangMills theories. This broken symmetry revives through quantization and produces the anti-BRST invariance. 1. Axial Extension of a Gauge Theory Extending a renormalizable field theory is the process of adding a number of new renormalizable terms to the Lagrangian density in order to enlarge its symmetry group. This process may or may not add a number of new fields to the theory. It can be seen that this can replace a global symmetry by a local one. From the perturbation theory point of view this may result in appearance of a number of new Feynman diagrams which will affect the renormalization process and consequently the anomalous behaviors of the theory. In this section the axial extension of gauge theories is studied in a geometric framework. By axially extension we mean enlarging a Yang-Mills theory to produce an axial gauge theory which admits local axial symmetry. 1-1. Axially Extended Gauge Theory The main goal of axial extension of a gauge theory is to add a number of new renormalizable terms to the Lagrangian density in order to make the axial symmetry take the local form. Thus, consider a gauge theory over by; ̅ with gauge group an element (1-1-1) and the gauge transformation group . Under the action of with; 5 d (1-1-2) on remains invariant. But the action of axial transformation ̅ , which yields a nontrivial variation; cannot be compensated by variations of gauge fields . Essentially, this is the reason that axial transformations are global symmetries. To overcome this difficulty one should introduce a number of new axial gauge fields to capture the variations of under axial transformations. To this end, a set of new terms, ̅ , each of which coupled to an axial gauge field, say , should be added to the Lagrangian. The resulting theory takes the form of; ̅ ̅ (1-1-3) where is a functional of and which will be determined in the following. It can be easily checked that the invariance of under gauge transformation requires (1-1-2) together with; d where (1-1-4) . On the other hand, the invariance of under axial transformation requires; d 6 (1-1-5) Notice that the axial transformation mixes the vector and axial gauge fields. This is a crucial fact which is discussed in the following. To find a gauge/axial invariant Lagrangian density , one should look for a gauge/axial invariant . The second part of (1-1-5) forces to be a functional of a mixed . Moreover, according to (1-1-2), (1-1-4) and (1-1-5), and form of the gauge and axial transformations are respectively given by; d d (1-1-6) Both of these transformations are completely similar to the gauge field part of (1-1-2) which keeps the Maxwell Lagrangian invariant. Thus, if one defines; ) ( (1-1-7) would be a the Maxwell Lagrangian density compatible candidate for . Note that although the trace of vanishes the Lagrangian cannot be split into two different parts, each one a functional of or . Take the infinitesimal gauge transformation of , ; d | d d | d d d (1-1-8) , and the infinitesimal axial transformation of ( ) d | d d 7 ; (1-1-9) Indeed, it can be seen that the infinitesimal axial transformations of gauge and axial gauge fields are given by; (1-1-10) If one defines the vector and axial gauge fields as operator valued 1-forms, then the gauge and axial transformations of and are respectively given by the following forms; d (1-1-11) and; d (1-1-12) Therefore, an extended Lagrangian density, , is found properly. Moreover as we expected before, not only contains (the original theory) but it admits the axial symmetry in the local form. Indeed, up to anomalous behaviors, is a renormalizable theory by power counting. Note that from the perturbation theory point of view the fact of which contains , results in the appearance of all Feynman diagrams of the original theory in the perturbative calculations of the extended theory. Thus may be considered as a sub-theory for . In the next subsection, it is shown that the extended Lagrangian, , can be considered as a gauge theory with an enlarged gauge group. 8 1-2. Extended Gauge Transformation Group As it was stated above, an axially extended gauge theory admits the local axial symmetry, but it has not shown yet that the extended Lagrangian density is also a gauge theory by itself. To show this fact the geometric theory of gauge fields is revisited now. Let be an even dimensional spin manifold and a (semi-simple) Lie group with Lie algebra . Consider a principal -bundle over ; (1-2-1) and set a connection over with the Cartan connection form [51]. Now suppose that is a complex irreducible representing space of . Also consider a Hermitian inner product for , say , and assume that the action of is unitary. The representation of on defines a complex vector bundle over with; (1-2-2) where; | and (1-2-3) It can easily be checked that is a vector bundle and any one of its sections can be represented by an equivalence class, say , for some and . Clearly, by definition (1-2-3), the choices of and , are not unique at all. As it will be discussed in the following, these different choices produce the gauge symmetry. On the other hand, the connection of induces a connection over with covariant derivative; 9 d where any d and d (1-2-4) is the action of , induced by the representation of , define; | (for ). The definition (1-2-4) is independent of the choices and . Moreover, the pull back of through , , defines an operator valued 1-form over . Conventionally is called the connection form. Basically these operator valued 1-forms play the role of the gauge fields in the context of gauge theories. To see this note that; d d which is similar to the gauge field part of (1-1-2) with replacing d ). ) with – d (resp. – (1-2-5) (resp. Now consider the tensor product bundle; (1-2-6) for the spin bundle and its standard fiber . The definition (1-2-4) and the connection of the spin bundle naturally induce a covariant derivative, , over the tensor product bundle (1-2-6). Thus if is the Dirac operator of , then the Lagrangian density of matter is given by; (1-2-7) for . According to definitions (1-2-3) and (1-2-4), it can be seen that is a gauge invariant functional. The other gauge invariant functional on 4-dimensional is the Maxwell's lagrangian density; 10 ( with ) (1-2-8) the curvature of and the Hodge star operator. Obviously, when , the spin bundle is trivial and thus it is seen from (1-2-4) that; . Usually is defined by referring to . Thus d ( one may conventionally define; ) . Now the axially extended gauge theories can be studied by these geometric structures. First, note that for any given -dimensional Lie algebra there exists a natural way to define a -dimensional Lie algebra which contains as a Lie sub-algebra. To see this, chose a basis for , say , and define a set of new elements together with following commutation relations; (1-2-9) It can be checked easily that the Jacobi identity is satisfied by the relations of (1-2-9) and thus they form a Lie algebra, . Indeed can be considered as an extension of . On the other hand, it is clear that for a given representation of , say , and for a nontrivial involutive matrix which commutes with all s, one can produce a representation of , say . It is enough to set . This can be done with for any representation of the Lie algebra , over an even dimensional spin manifold. Thus, the extending procedure of (1-2-9) is referred to as axial extension of a Lie algebra. As it was stated above, is itself a Lie algebra and thus there exists a simply connected Lie group ̃ with ie ̃ . Since contains as a Lie sub-algebra, then there is a closed Lie subgroup of ̃ , with Lie algebra . Actually ̃ contains a simply connected Lie subgroup, say , with ie . Indeed, ̃ . To see this note that , forms a basis for the following Lie brackets; 11 with ] [ ] [ ] [ (1-2-10) This shows that can be decomposed into the direct sum of two copies of , , it follows . On the other hand, since that ̃ , and thus the axial extension of any Lie algebra induces an extension of its simply connected Lie group. Therefore, the axial extension of will induce an extension of . Hence the axial extension of a -gauge theory leads to a -gauge theory. Generally, for simply connected and ̃ the inclusion of d defines a ̃ [52]. Thus, if natural inclusion of into ̃ denoted by and ̃ are principal bundles over for simply connected Lie groups and ̃ ̃ is a smooth map which makes the following respectively, and if diagram commutative ̃ ̃ ̃ (1-2-11) then a natural reduction of ̃ -gauge theories to -gauge theories over is defined through . It is enough to set the Cartan connection form over , the pull back of that over ̃ , say , through , i.e. . Such a reduction procedure is always possible for simply connected , e.g. . The reduction procedure can be considered as the inverse project of extension the gauge theories. As it was stated above, one can consider as a representing basis of the extended Lie algebra . With this notation, (1-2-9) also confirms the infinitesimal gauge and axial transformations of (1-1-11) and (1-1-12). 12 More precisely, (1-1-3) defines an ordinary ̃ -gauge theory. Consequently, the gauge and axial transformations are considered as the gauge transformations for the extended Lagrangian density . These transformations are usually called extended gauge transformations. The relations of (1-2-10) assert that one can consider a mixed form of and to define a number of new gauge fields which do not mix under extended gauge transformations. To this end set; (1-2-12) where and are respectively the projections onto the right and left handed Weyl spinors. Thus the infinitesimal transformations take the following form; d d (1-2-13) for (resp. ) the right- (resp. left-) handed infinitesimal chiral transformation. This splits the gauge fields into right- and left-handed components. 2. Extended BRST Transformations It is known that the BRST invariance is the quantized version of the classical gauge symmetry in the context of gauge theories [11-13]. Actually, the BRST invariance can reproduce all the information of gauge symmetries within the 13 quantized formalism. In this section the geometric structure of anti-BRST transformations and consequently of anti-ghost is given by using the ideas of geometric quantization. In the first subsection, a review over the geometrical theory of ghosts and BRST transformations is given. 2-1. Ghosts and BRST transformations, a Geometric Approach The geometric description of Fddeev-Popov quantization, not only illustrates the canonical structures of ghost fields and BRST transformations, but it results in simple proofs for their substantial behaviors [36-38]. To see this assume that , the space-time, is an even dimensional spin manifold. Consider the principal -bundle of (1-2-1) and the Cartan connection form . The space of all connection forms is an Affine space, denoted by . The gauge transformation defines a right action of the infinite dimensional Lie group on . For example the action of on is given by; d Consider Then the free action of d (2-1-1) to be the set of base point preserving elements [36]. on defines a principal -bundle; ⁄ (2-1-2) Fix a connection over this principal bundle and denote its Cartan connection form by . For a fixed define the fiber map; (2-1-3) 14 Define as a -valued one form over with; (2-1-4) , where and are the . Then set for pull backs of the first and the second parts of respectively. Conventionally is called ghost. Indeed, is a -valued left invariant 1-form over and then its color components anti-commute with each other as 1-forms. Thus its color components may be considered as Grassmannian numbers like ghosts. Denote the exterior derivative operator over by d pull back through . Then a direct calculation gives; and define to be its d (2-1-5) , we find that; and d d , Moreover, since d d d similar to ordinary BRST derivation. This is the reason for and to be respectively considered as ghost and BRST transformation [36]. In fact, for better understanding (2-1-5), it should be compared with the vector gauge field part of (1-1-11). Indeed, any 1-parameter group gauge transformation, say , , induces a fiber-wise vector field over , shown by , given by ∫ ∑ for . Moreover, di ( (2-1-6) di and ) the coordinate functions of in (2-1-5), is a differential 2-form on d d 15 with; (2-1-7) for the internal multiplier operator. More generally, infinitesimal gauge transformations are infinitesimal moves through the fibers of [36] and therefore, they are smooth sections of tangent bundle . Basically infinitesimal gauge transformations are fiber-wise vector fields on . On the other hand, ghosts are fiber-wise 1-forms over and thus, they can evaluate the infinitesimal gauge transformations as their dual objects. In fact, the equivalence of gauge symmetry and BRST invariance can be illustrated by noting that the former is defined with the elements of but the later one is given in terms of their dual objects in . It is the same idea of Legendre transformation in classical mechanics which translates the Lagrangian formalism into the Hamiltonian formulation over a symplectic manifold. This intuition is one of the cornerstones of geometric quantization. 2-2. Anti-Ghosts and Anti-BRST Transformations As stated above, the extension of gauge group extends the algebra of ghosts. In fact, in this subsection it is shown that how anti-ghosts emerge through the axial extension of a gauge theory. Consider and as the set of all vector and axial connection forms respectively and take the space of extended connection forms as the Cartesian product space . Then consider , ̃ and ̃ as the axially extended forms of Lie algebra , gauge group , and gauge transformation group , respectively. Finally define and similar to (2-1-4) and let and be projections. The fiber map for a fixed point is given by; ̃ (2-2-1) Now define . Thus if and 16 , with and are the pull backs of the exterior derivative operators on be shown that; , d , and d ,d , respectively, then it can similarly (2-2-2) which is in complete agreement with BRST transformations, replacing the Nakanishi-Lautrup field [47, 48, 53] by . Note that is a colored scalar field with ghost number zero similar to . On the other hand, as it will be shown in the following vanishes and this implies that annihilates both and similarly. Thus, can be considered as the Nakanishi-Lautrup field which appears in the path integral quantization formalism. Indeed, it is seen that the ordinary BRST transformation is not closed for classical fields, ghosts and anti-ghosts [12, 13, 16], but this new fashion of BRST algebra is closed by itself. Actually, the auxiliary field is replaced by in (2-2-2) which closes the BRST algebra. Moreover, the degrees of freedom carried by the auxiliary field are compensated with those due to axial gauge fields. However, is also called the BRST operator while and are referred to as ghost and anti-ghost fields, respectively. This gives an elegant geometric description to anti-ghosts in terms of 1-forms dual to infinitesimal axial trnsformations. On the other hand, one can similarly show that; d (2-2-3) Actually, since d d . According to (2-2-3), , then gives hand a set of transformations which are similar to BRST ones and keep 17 the quantum Lagrangian invariant. Clearly is in complete agreement with the algebraic anti-BRST transformation [14-16]. Thus, conventionally is called the anti-BRST operator. On the other hand, it can be seen that from d d d d d d , one finds that; d d d d (2-2-4) These anti-commutation relations will result in a generalized form of descent equations [17-19] by using d, and alternatively. Note that, is given in term of the auxiliary field in the context of the ordinary anti-BRST transformation [14-16]. Thus, (2-2-3) equalizes the Nakanishi-Lautrup field and , which confirms the last part of (2-2-2). The union of (2-2-2) and (2-2-3) is referred to as extended BRST transformation. Consequently, it was shown by (2-2-2)-(2-2-4) that the anti-BRST invariance is the quantized counterpart of the local axial symmetry. 3. Extended Descent Equations In this section the extended BRST transformation is used to generalize the descent equations [17-19] in a concrete manner. Indeed this process leads to a modification of consistent anomalous terms in gauge theories. In the other words, the enlargement of gauge group yields a set of generalized form of anomalous terms including consistent anomalies and consistent Schwinger terms via extended BRST. The result will be delivered in a lattice diagram of differential forms which commutes up to deRham exact forms. 3-1. Analytic BRST and Anti-BRST Transformations The smooth action of a Lie group on a given smooth manifold will induce a set of vector fields over . Indeed, if is a left invariant vector field over , is a and , , is its integral curve, then for any , 18 smooth curve in . Actually the collection of all vectors ̃ d | defines a smooth vector field over with their integral curves , . Thus, the action of ̃ on can be considered as a collection of fiberwise smooth vector fields over . To formulate this idea analytically, consider a coordinate system over , say , with di and di vector field, , over . Thus, the gauge transformation with; induces a ∑ ∫ (3-1-1) A generalized form of in (2-1-6). Moreover, the axial transformation defines another vector field, , over with; ∑ ∫ (3-1-2) where it is supposed that; (3-1-3) 19 since form a basis for tangent spaces of these notations the infinitesimal gauge transformations of and and axial gauge fields with respect to , i.e. (1-1-8), are given by; . Using as vector (3-1-4) While the infinitesimal axial transformations of and as vector and axial , i.e. (1-1-10), are given by; gauge fields with respect to (3-1-5) Now to extract the analytic formulation for BRST and anti-BRST operators one should go from the tangent bundle to its dual and use the differential form alternatives. In fact, the Cartan connection form , makes it possible to translate the fiber-wise tangential formalisms to the cotangential ones. According to (3-1-1) and (3-1-2) one finds; ∑ ∫ (d d ) (3-1-6) where s an and the pull back of (3-1-6) via the fiber map 20 . Consequently taking leads to; d | d ∑ ∫ for | d | d the formal left invariant vector field on ̃ due to (3-1-7) . But since any fiber of is precisely a copy of ̃ , then any vertical vector of the tangent | | space is an element of s an . Thus it can be seen that, ∑ ∫ (3-1-8) is the fiber-wise exterior derivative operator over the total space equivalently is the extended BRST derivative operator. , and Note that using the same procedure, it is seen that for the ordinary geometric feature of path integral quantization, anti-ghosts never appear in (3-1-8) to form the exterior derivative operator. Therefore, for such cases, (3-1-8) is replaced by; ∑ ∫ (3-1-9) 21 while the anti-ghost part is exclusively related to the axial extension; ∑ ∫ (3-1-10) Considering that each fiber of is canonically diffeomorphic with ̃ , it is seen that the ghost part of (3-1-7) defines the analytic BRST derivative. Specially, for any one has, ∑ ∫ (3-1-11) Similarly it is seen that (3-1-10) defines the analytic anti-BRST derivative. Thus, for any we have; ∑ ∫ (3-1-12) 3-2. Modified Anomalies and Schwinger Terms Obviously, the form of consistent anomalies and Schwinger terms are completely dependent on the formulations of BRST operators via the descent equations. Thus it is expected that when the algebra of ghost fields and eventually the BRST transformations get larger, then the consistent anomalies and Schwinger terms should be modified properly. 22 Gauge anomaly is the deviation of a second quantized Yang-Mills theory from its classical gauge symmetry. Indeed, anomalies cannot be canceled by renormalization counter terms while they ruin the renormalizability of the theory [1, 33, 53]. To see this note that, classically the equations of motion of the Lagrangian density (1-1-3) show that the vector and axial currents, should obey the following conservation laws; (3-2-1) On the other hand, the variation of the quantum action with respect to (resp. ) is (resp. ). Thus from (3-1-11) and (3-1-12) we have; ∑∫ ∑∫ Thus, the properties of (∑ ∫ and (3-2-2) asserts that; ) (∑ ∫ ) (∑ ∫ ) (∑ ∫ 23 ) (3-2-3) where the first equation is called the consistency condition [4, 21, 22] (with ). The second equation is somehow unfamiliar despite of its relation to axial currents. It is known that if a theory is anomalous, then both and participate in anomalous behaviors simultaneously [3]. Thus, it is more convenient to consider as the anomalous term. Actually, the counter terms affect this description of anomalies such that a suitable choice of counter terms may lead to as a desired result. But, cohomologically the extended BRST class of is unaffected with respect to counter terms. Indeed, the cohomology class of is independent of renormalization methods; in the other words reveals the pathology of renormalizing the theory [22, 53]. Thus BRST transformation produces a framework to study the anomalous behaviors of gauge theories. Geometrically, using the Quillen supper connection [54] and the family index theory [55-56], it can be seen that ∫ is the Cartan connection form of the principal -bundle due to the determinant line bundle constructed from perturbed Dirac operators over [36, 55, 57]. Any -gauge transformation causes the anomalous term to be added by a local term over , say . Moreover, this local term is an exact extended BRST form, i.e. , which never changes the cohomology class of . Specially, such s are exactly the possible counter terms appear in renormalizing the theory. Actually, (3-2-3) shows that the connection of Quillen , determinant bundle, restricted to the fibers of , is flat, which is a geometric description of consistency conditions [36]. In noncommutative geometric point of view it is known that this anomalous behavior arises because , for Dirac operator , is not a trace class operator in general [58, 59]. Actually, belongs to the Schatten class for di which causes the quantum action not to be well defined over . To see this recall that; ∫ ̅ ∫ ̅ ̅ ̅ e (∑ {( 24 ) }) (3-2-4) But for small enough integers , di ,( ) s are not trace class operators. Then one needs to modify the trace. This can be considered as the regularization method in the setting of operator theory. Such fascinating feature of regularization takes place by using the Dixmier trace and zeta Riemann function [58-60]. This causes the quantum action to vary through the fibers. Indeed, is not even a single valued function over . Therefore, belongs to a nontrivial deRham (extended BRST) cohomology class of ̃ [22, 58, 59]. As an intuitive example for this situation one can consider the angle function over . is not a continuous function but its exterior derivation d , forms a nontrivial cohomology class of . Thus, integrating the pull back of through a smooth ̃ over map measures the anomalous behavior of the theory [58, 59, 61]. In this way, one computes an element of the holonomy group of the Quillen determinant bundle over ̃ . As it was stated above, this bundle is equipped with a flat connection which leads to a projective group l . Actually, since this homomorphism of ( ̃) ( ̃) connection preserves the metric of , then l [55-57]. But since is single valued over , then l( . Therefore, ) one concludes that ∫ for the winding number of the phase of ̃ [61]. This can be computed in terms of det around the loop periodic cyclic cohomology, using the Chern-Connes character and the local index formula [58-60]. To extract the modified consistent anomaly according to the Stora-Zumino procedure [17, 18], one should use d, and alternatively to provide a generalized formulation of descent equations. Initially, the Bianchi identity asserts that; d (3-2-5) 25 where d is the curvature. Thus, (3-2-5) implies that a deRham and (anti-) BRST closed form. Set and consider as a -form over . Thus, the Poincare lemma leads to; d d d d d is d (3-2-6) where is a deRham differential -form with ghost number , while . Actually, is simultaneously a differential -form over and a differential -form over . On the other hand, it can easily be shown that d . Indeed, and differ in an exact differential form. This fact plays an important role in calculating the consistent Schwinger term. From (3-2-6) it is seen that; ( ) d (3-2-7) and hence; ∫( Therefore, up to a factor ( ) ) (3-2-8) can be considered as the modified nonintegrated consistent anomaly; in the other words, ∫ ( candidate for . A direct calculation shows that when ) is a then; (3-2-9) 26 which is the modified consistent anomaly up to a factor of . Indeed, if one sets the resulting form is equal to the well-known consistent anomaly [22, 53] (the anti-ghost will be killed automatically after taking the trace). On the other hand, ghost number counting leads to; d d (3-2-10) which is called ghost consistent anomaly. Moreover, the remained term; d d (3-2-11) is called anti-ghost consistent anomaly. =∫ , for a non-local form implies that; If , then ∫ ( ) d d (3-2-12) for a BRST (resp. anti-BRST) closed form (resp. ). Thus, the added term (or ) must be simultaneously a BRST and an anti-BRST closed form. This undetermined BRST/anti-BRST closed form can be considered as the gauge fixing term in the Faddeev-Popov quantization method [11, 53]. It asserts that the quantum Lagrangian density is simultaneously invariant under BRST and anti-BRST transformations, the fact that was pointed out in [16]. Moreover, continuing the sequence of equations (3-2-6), results in; 27 d d , . (3-2-13) Indeed, fallowing the standard descent equations for d and following equality, d ( yields the ) (3-2-14) Ghost number counting implies that, d d (3-2-15) and, d d (3-2-16) Therefore, (3-2-17) is a candidate for the modified consistent Schwinger term up to a factor For , the modified consistent Schwinger term is given by; d . (3-2-18) where; 28 d d d d d d (3-2-19) are respectively called ghost/ghost, anti-ghost/anti-ghost and ghost/antighost consistent Schwinger term. Note that the ghost consistent anomaly and the ghost/ghost consistent Schwinger term are related by an ordinary descent equation of d and . It is also the case for anti-ghost consistent anomaly and anti-ghost/anti-ghost consistent Schwinger term with replacing by . More generally, the extended descent equation will give rise to a lattice diagram which commutes up to exact deRham forms; d d d d ↔ ↔ ↔ ↔ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ ⁄ (3-2-20) 29 In the diagram of (3-2-20) the differential forms inner vertices are considered as (resp. ⁄ at the ) for the range (resp. domain) of the relevant incoming (resp. outgoing) arrows. Finally it can be seen that the diagonal elements of (3-2-20) (the elements with the same number of ) produce modified consistent anomalies, Schwinger terms and so on. More precisely, the extended formalism of descent equations is given for the diagonals of (3-2-20). Note that the top row and the left column of (3-2-20) show respectively the ordinary deRham/BRST and deRham/anti-BRST descent equations as was pointed out above. Conclusion In this article it was shown that in gauge theories, anti-BRST invariance is the quantized version of local axial symmetry. Basically, the axially extension of a gauge theory enlarges the gauge group and consequently the algebra of ghost fields. Initially, it was shown that the anti-BRST derivation and infinitesimal axial transformations are mutually dual in the sense of differential geometric objects. Moreover, an elaborate geometric description for anti-ghosts and anti-BRST transformation was given. Finally a collection of extended descent equations was formulated by using BRST and anti-BRST derivatives alternatively. This resulted in a modified version for consistent anomalies and consistent Schwinger terms. Acknowledgement The author acknowledges F. Ardalan, for discussions. Also my special thanks go to Ahmad Shafiei Deh Abad for his useful comments and discussions in geometry and for his ever warm welcom. Finally the author expresses his 30 gratitude to R. Bertlmann, A. Davody, D. Freed, M. Khalkhali and D. Perrot for usefull comments. Also this article owes most of its appearance to S. 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