You are an expert at summarizing long articles. Proceed to summarize the following text: 
a long - standing problem in quantum field theory is a nonperturbative formulation of chiral fermions . our only general nonperturbative formulation of quantum field theory is the renormalization - group limit of functional integrals defined on a lattice . but when chiral symmetry is an issue , there are notorious problems @xcite . briefly , one must either sacrifice locality or positivity , tolerate additional states ( doubling or mirror states ) , or break chiral symmetry explicitly . when the coupling of fermions to gauge fields is vector - like , the standard formulations @xcite are adequate , if imperfect . on the other hand , when fermions couple to chiral gauge fields , it has been difficult to prove a conceptually clean theory ; see ref . @xcite for a review . an encouraging proposal replaces the functional integral over fermions with an auxiliary quantum - mechanical system @xcite , inspired by domain - wall @xcite and lattice pauli - villars @xcite methods . this paper offers constructions of vector - like and chiral gauge theories , coupling continuum fermions to lattice gauge fields by introducing an interpolation of the latter . ideas of this type were first discussed by flume and wyler for the schwinger model @xcite , and recently t  hooft advocated a similar approach for four - dimensional gauge theories @xcite . the appeal stems from the nontrivial ( instanton ) topology of continuum gauge fields , because the atiyah - singer index theorem @xcite implies an intimate relation between chiral properties of fermions and the topology of the gauge  field .    in 19871988 there was some discussion about topology and fermions in lattice gauge theory @xcite . except for a conference reports @xcite , however , none of the applications to chiral gauge theories have been published . spurred by ref . @xcite , i  would like to present my variation on the theme . @xcite regulates the gauge field with the lattice and the fermions with a standard pauli - villars scheme . the number of fermionic degrees of freedom ( per unit volume ) remains infinite  in the words of smit , the method is desperate @xcite . in particular , a numerical evaluation of the effective action would require infinite computation , even at fixed cutoff . this paper , on the other hand , examines a sharp cutoff on determinants , which was first studied in ref . the number of fermionic degrees of freedom is now finite , so the numerical computation of the effective action is finite  too . another difference between this paper s proposal and the one in ref . @xcite is the strategy for removing the cutoffs . let @xmath0 denote the lattice spacing and @xmath1 the ultraviolet cutoff on fermions . in the formulation of ref . @xcite one takes @xmath2 for @xmath0 fixed , and afterwards @xmath3 . as stressed in sect . [ gth cutoff ] the cutoff in ref . @xcite maintains the gauge symmetry of the chiral theory only in the @xmath2 limit . with the sharp cutoff formulated below , however , it is permissible and natural to take @xmath3 , @xmath2 with @xmath4 constant . the latter approach is far superior in a numerical computation . if @xmath1 must vary at fixed @xmath0 , it will be extremely difficult to generate useful ensembles of lattice gauge fields , because to obtain a renormalized theory the bare gauge coupling must depend on @xmath1 . [ vector - like ] begins with a discussion of vector - like theories . the analysis starts with anomalies , because they are such a stumbling block for lattice formulations @xcite . the functional integral over fermion fields is defined in the manner of fujikawa @xcite , and a specific cutoff procedure is formulated and justified in sect .  [ regulator ] . these sections define an effective action for a continuous background gauge field . [ link ] summarizes the essential features of an interpolation from the lattice field of parallel transporters to a connection . [ metric ] derives a heuristic relation between the fermion measure of this proposal and the usual measure of lattice fermions . while inessential to the main line of argument , the derivation suggests a rationale for relating the cutoff on determinants to the lattice spacing . [ mc ] briefly considers numerical aspects . as usual , the generalization to chiral gauge theories is not immediate , but sect . [ chiral ] produces a satisfactory definition of the chiral effective action , including fermion nonconservation . finally , sect . [ conclusions ] remarks on some of the loose ends , and compares the status of this formulation with ref .  @xcite . the formal expression for the euclidean functional integral for fermions is @xmath5 let us assume a background gauge potential ( or connection ) @xmath6 . in the application to lattice field theory , this connection is a determined from the lattice gauge field , cf . [ link ] . staying momentarily `` in the continuum , '' the action is @xmath7 formal integration over the fermion fields yields the boltzmann factor @xmath8 the objective is to give a rigorous meaning to the measure @xmath9 , and/or to the determinant . consider the eigenfunctions and eigenvalues of the dirac operator @xmath10 @xmath11 since @xmath10 is anti - hermitian , the @xmath12 are real . the dirac operator transforms covariantly under the gauge group , so the @xmath12 are gauge invariant . the dirac operator anti - commutes with @xmath13 , i.e.  @xmath14 . hence , if @xmath15 is an eigenfunction with eigenvalue @xmath12 , then @xmath16 is an eigenfunction with eigenvalue @xmath17 . as usual , one imagines that the theory is defined on a compact space - time , and the infinite volume limit is taken at the end . then the spectrum of @xmath10 is discrete . the dirac operator can possess zero modes , @xmath18 . in this subspace it is convenient to sort the eigenfunctions according to chirality , i.e.  the eigenvalue of @xmath13 . let @xmath19 be the number of modes with @xmath20 and @xmath21 , and let @xmath22 . the determinant should then be @xmath23 except that the infinite product still requires an ultraviolet regulator ; this is postponed to sect .  [ regulator ] . ( the second equality follows because nonzero eigenvalues come in pairs @xmath24 . ) it is easy to see that the eigenvectors are orthonormal and form a complete set : @xmath25 @xmath26 where @xmath27 , @xmath28 denote spinor and color indices . these properties of the eigenfunctions permit the expansions @xmath29 \bar{\psi}(x ) & { \displaystyle \sum_n } \bar{a}_n \varphi^\dagger_n(x ) , \end{array}\ ] ] where the coefficients @xmath30 and @xmath31 are grassman numbers . to obtain eq . ( [ unregulated ] ) from eqs . ( [ basic integral ] ) and ( [ vlike action ] ) one takes the functional integral over fields given by eqs . ( [ eigenbasis ] ) , i.e.  the fermion measure is _ defined _ to be @xmath32 fujikawa @xcite makes a formal argument pursues a similar , yet complementary , line of thought . ] to relate the right - hand side of eq . ( [ measure ] ) to @xmath33 . since the product over a continuous index is formal , it is logically cleaner to assert eq . ( [ measure ] ) as a definition . the fujikawa measure is analogous to one based on fourier modes , @xmath34 the momenta correspond to eigenvalues of @xmath35 , and are also discrete ( in a box ) . with a coupling to gauge fields , however , the momenta are not gauge invariant . hence , the definition based on eigenfunctions of @xmath10 is preferable . one must check that the formalism reproduces the axial anomaly . consider space - time dependent chiral transformations @xmath36 \bar{\psi}(x ) & \bar{\psi}'(x ) e^{-\alpha^a(x)t^a\gamma_5 } , \end{array}\ ] ] and @xmath37 is a generator of a global symmetry group . ( for u(1 ) take @xmath38 . ) under such transformations the measure of eq . ( [ measure ] ) is not invariant , but @xmath39 and similarly for @xmath40 , with sums on @xmath41 implied . the rules of berezin integration imply @xmath42 the jacobian determinants are then responsible for the anomaly . using an eigenfunction expansion of @xmath43 one can write @xmath44 with @xmath45 then @xmath46 where the anomaly @xmath47 collecting the anomaly from @xmath48 and @xmath49 , as well as terms from the chiral transformation of the action , yields the ( anomalous ) ward - takahashi identity @xmath50 the preceding discussion skirts the need to regulate the determinants in the ultraviolet . let us start with a kind of pauli - villars regulator . ( [ anomaly ] ) becomes @xmath51 let the index @xmath52 run over @xmath53 with the convention that @xmath54 denotes nonzero modes ; let us order the nonzero modes by @xmath55 and take @xmath52 odd ( even ) if @xmath12 is negative ( positive ) . the regulating function @xmath56 is chosen to look like the sketch in fig . [ regulator - function ] . =    the cutoff @xmath57 and smearing parameter @xmath58 should satisfy @xmath59 @xmath60 where @xmath61 is even ( so @xmath62 ) , and one takes @xmath63 with @xmath57 fixed . the fermi function @xmath64 changes rapidly from 1 to 0 as @xmath65 passes through 1 . since @xmath66 and @xmath67 and all its derivatives vanish at @xmath68 , it supports the standard analysis @xcite that leads to @xmath69\ ] ] in four dimensions , where the two traces are over global and gauge - group indices . details in appendix  [ convergence ] show that all higher dimension terms , e.g.  @xmath70 , are proportional to @xmath71 as @xmath63 .    to check that the regulator does not spoil the derivation of eq . ( [ awti ] ) , one restricts the chiral transformations @xmath72 to those for which @xmath73 if one of @xmath74 . if both @xmath52 and @xmath75 , then the left - hand side of eq . ( [ block c ] ) need not vanish . the jacobian matrix @xmath76 takes a block form , and the regulator decouples the block with large eigenvalues . nevertheless , the transformation function @xmath72 is sufficiently arbitrary to derive the ward - takahashi identity . the @xmath63 limit corresponds to keeping the first @xmath61 modes of @xmath10 completely and eliminating the rest . in a loose analogy with the lattice cutoff , in which the fourier measure of eq . ( [ fourier measure ] ) is truncated , one can carry out the truncation of the modes throughout . thus , the regulated measure is @xmath77 with eigenvalues ordered as above . the functional integral is now over fields given by @xmath78 \bar{\psi}(x ) & { \displaystyle \sum_{n=1-n_0}^n } \bar{a}_n\varphi_n^\dagger(x ) . \end{array}\ ] ] the effective boltzmann factor becomes @xmath79 similarly , the regulated fermion propagator is @xmath80 the finite - mode cutoff has been examined before @xcite , with emphasis on anomalies . those papers did not notice the disappearance of higher - dimensional terms as the fermi function becomes infinitely sharp , a feature that is especially important in the chiral gauge theory , cf . [ chiral ] .    in perturbation theory , one must be careful to impose the sharp cutoff with @xmath10 , rather than with loop momenta . for example , the dependence of the regulator on @xmath10 , and hence on @xmath6 , induces additional terms in the gauge current , to maintain gauge invariance . how should one choose @xmath61 ? one ought to take nonzero modes in pairs : it would be silly to take @xmath12 and not @xmath17 . one also ought to retain all zero modes , because they are the most infrared of all ! but from the atiyah - singer index theorem @xcite the number of zero modes is even ( odd ) if the topological charge @xmath81 is even ( odd ) . thus , @xmath61 must depend on the gauge field and be even ( odd ) if @xmath81 is even ( odd ) .    the order of magnitude of @xmath61 can be specified only vaguely . an examination of cutoff effects , appendices  [ convergence ] and  [ cutoff ] , introduces expansions . for a generic cutoff , one would need to maintain @xmath82 . the absence of non - universal terms with the finite - mode cutoff , however , permits a lower cutoff . if , as in the next section , @xmath6 is obtained from a lattice gauge field , @xmath83 , where @xmath76 is a gauge - group dependent constant . with the finite - mode cutoff it is thus natural to take @xmath84 . indeed , on an @xmath85 lattice one expects @xmath86 fermionic degrees of freedom in the lattice fermion field ( per flavor ) , where @xmath87 is the dimension of the fermion s gauge - group representation . an obvious choice would be to augment the usual number with the zero modes , i.e.  @xmath88 . to extend the formulation to lattice gauge theory , one must provide an interpolation determining a connection @xmath89 from the lattice gauge field @xmath90 . @xcite gives a prescription adequate for defining traces of products of @xmath91 , but does not prescribe @xmath6 . other possibilities , based on definitions of the topological ( instanton ) charge of a lattice gauge field @xcite , do prescribe @xmath6 explicitly @xcite . the latter have two crucial properties :    1 . the only singularities in @xmath6 are instanton - like the interpolation @xmath6 transforms as a connection under _ lattice _ gauge transformations .    ref . @xcite does prove a useful theorem on the spectrum of the dirac operator for a bounded gauge field . the ( manifestly ) topological interpolations obey the hypothesis of the theorem .    the great insight of ref . @xcite was to recognize that the lattice gauge field can be used to define a fiber bundle . then the topological charge is the second chern number of the bundle . lscher originally provided expressions for transition functions , which encode changes of gauge @xmath92 from a patch @xmath93 of space - time to a neighboring one @xmath94 . for consistency , @xmath95 . in the fiber - bundle formalism , the gauge is fixed separately within each patch , such that @xmath96 has no singularities . the winding responsible for topological charge then resides in the transition functions @xcite . the more familiar patch - independent connection is @xmath97 where the section @xmath98 is related to the transition functions by @xmath99 the first step of refs . @xcite is a patch - wise continuous , bounded interpolation for the nonsingular @xmath96 . ( [ connection ] ) produces a continuous , bounded connection @xmath6 , but when the transition functions have nontrivial winding , the sections , and thus @xmath6 , have directional singularities . by construction , therefore , the _ only _ singularities in the globally defined @xmath6 are those induced by the instanton - winding of the section .    the other crucial property of the reconstructed gauge potentials of refs . @xcite is the response to a lattice gauge transformation . the transformation law of the lattice gauge field is @xmath100 where @xmath101 , @xmath102 denote lattice sites . for @xmath103 in the patch @xmath104 associated with @xmath101 , the reconstructed section obeys @xcite @xmath105 the interpolation @xmath106 is independent , eq . ( [ v = ww ] ) requires that the interpolation obey @xmath107 . ] of @xmath104 , but , of course , @xmath108 is independent of @xmath103 . the interpolated connection transforms as @xmath109 with the _ same _ function @xmath106 . if @xmath110 at each site , the three interpolated gauge transformation fields obey the composition law @xmath111 note that the interpolation depends on the underlying lattice gauge field , which is emphasized here by the second argument . consequently , every interpolated @xmath112 can be built up from infinitesimal , site - by - site steps . for the present discussion the complicated expressions for the interpolated @xmath106 and @xmath6 are not illuminating . interested readers can consult refs . @xcite for lscher s bundle . ( for phillips and stone s bundle , analogous results can be obtained @xcite . )    our prescription for coupling fermions to lattice gauge fields is to start with the lattice gauge field @xmath113 , interpolate to obtain the connection @xmath6 , use the associated dirac operator to define the measure , and regulate the determinant using the sharp cutoff . because the dirac operator is constructed using fiber bundles , it automatically satisfies the atiyah - singer index theorem @xmath114 where @xmath81 is the topological charge ( as defined by ref . @xcite ) of the lattice gauge field . moreover , from eq . ( [ transform a ] ) the eigenvalues of the dirac operator , and hence the effective action , are invariant under lattice gauge transformations . from eq . ( [ regulated effective action ] ) the boltzmann factor @xmath115 is also positive and finite . it is intriguing to contrast the regulated measure of eq . ( [ regulated measure ] ) with the usual one @xmath116 where @xmath117 and @xmath118 denote the lattice fermion field and its conjugate at site @xmath101 . there is an interpolation of the fermion field analogous to @xmath6 . in particular , under a lattice gauge transformation the interpolated fermion field transforms as @xmath119 with the same interpolated @xmath106 mentioned in sect . [ link ] . let us expand the interpolated field @xmath120 as in eq . ( [ eigenbasis ] ) , use the orthonormality to solve for the @xmath30 and take differentials . then @xmath121 d\bar{a}_n & { \displaystyle\sum_s\int}d^4x\,d\bar{\psi}(s ) \dfrac{\partial\bar{\psi}(x)}{\partial\bar{\psi}(s ) } \varphi_n(x ) = : { \displaystyle\sum_s } d\bar{\psi}(s)\bar{m}_{sn}. \end{array}\ ] ] if the number of modes @xmath61 equals the number of lattice sites , @xmath41 and @xmath122 are @xmath123 matrices and one can write @xmath124 the determinants here are akin to ones appearing in ref . formally , one can combine the determinants into a `` metric '' @xmath125 and rewrite the measure as @xmath126 the metric depends covariantly on the gauge field , by construction of the interpolation . one can develop a geometric picture by imagining a grassman line element @xmath127 that is gauge invariant . in this language the customary lattice fields are curvilinear coordinates , and the expansion coefficients are rectilinear . according to this picture , the usual lattice measure mistakenly neglects the curvature . note that without the gauge interaction , the metric becomes flat : the two bases are then fourier transforms of one another . the metric also becomes trivial in the naive continuum limit . then the interpolation is unnecessary and @xmath128 . the factor @xmath129 is essential , however , for obtaining the correct anomaly , index theorem , etc . in particular , the anomaly arises because one may remove the regulators only after calculating with the functional integral , not ( as in the naive continuum limit ) before .    of course , eq . ( [ metric measure ] ) is merely heuristic . to make the manipulations rigorous , the interpolated fields must be smooth enough that the dirac eigenmode expansions leading to eq . ( [ a - psi ] ) stop at the @xmath61th term . here @xmath61 is both the number of modes kept and the number of lattice sites . this condition presumably puts constraints on the smoothness of the lattice gauge field similar to , but perhaps more stringent than , those imposed by uniqueness considerations in the fiber bundle constructions @xcite . one should emphasize , however , that such constraints are on the connection between the present measure and the usual lattice one . they do not detract from eq . ( [ measure ] ) as a definition of the functional integral , or @xmath115 as a definition of the effective action . since the effective action @xmath130 is real and positive , the functional integral over the gauge bosons is amenable to the monte carlo method with importance sampling . the dirac eigenvalues , and hence the effective action , depend on the lattice gauge field @xmath131 so they must be recomputed for every change of  @xmath131 . if the effective action is defined with a gaussian @xmath132 @xcite or standard pauli - villars @xcite regulator , one would need to compute all eigenvalues of @xmath10 and weight them appropriately  an infinite computation . with eq . ( [ regulated effective action ] ) , however , only the lowest @xmath61 eigenvalues are needed  a finite computation . times more costly , where @xmath133 characterizes machine precision . ]    to compute the eigenvalues one can introduce an auxiliary lattice much finer than the original one . parallel transporters for the fine lattice are constructed from the interpolated gauge field . now consider any discretization of the dirac operator , and denote its eigenvalues by @xmath134 . the discretization and auxiliary lattice spacing must be chosen such that @xmath135 ( up to tolerable floating - point precision ) for @xmath136 . on a fine enough auxiliary lattice , `` any '' discretization becomes precise enough . to avoid problems sorting the eigenvalues , however , a discretized operator without a doubled spectrum is preferable . suitable examples would be the wilson discretization @xcite , or one derived by gauging the fixed - point action of a free fermion @xcite . the discretization may break chiral symmetry , provided the breaking is numerically significant only for modes above the cutoff  @xmath61 . similar remarks apply to the construction for chiral gauge theories , sect . [ chiral ] . an important difference is that the boltzmann factor @xmath115 can be complex ( for fermions in `` complex '' representations ) , and in any case not positive definite . monte carlo integration is then much more difficult , because fluctuations in sign reduce the effectiveness of importance sampling , cf . [ how to ] . nevertheless , @xmath115 can be evaluated with finite computation . the preceding sections provide a definition of the functional integral for vector - like fermions . it is nonperturbative , has the correct axial anomaly , and by construction provides a natural association between a dirac operator and a topological charge , so that the index theorem is obeyed . the ideas are now extended to chiral fermions , which appear in the standard model and in grand unified theories . the essential feature of chiral gauge theories is that positive and negative chirality fermions transform under different representations of the gauge group . in a basis of the dirac matrices with @xmath13 diagonal it is useful to split the four - component dirac spinor into two two - component weyl spinors . without loss of generality one can charge - conjugate the negative chirality part and assemble everything into one positive chirality field . this weyl spinor is henceforth denoted @xmath137 , and the representation of the gauge group under which it transforms is denoted @xmath138 , with generators  @xmath139 . the kinetic term of the action is @xmath140 as before one wants to define the boltzmann factor @xmath141 but @xmath142 maps positive chirality weyl spinors into negative chirality weyl spinors . the underlying difficulty in constructing a chiral gauge theory is that the right - hand side of eq . ( [ chiral gamma ] ) is _ not _ a ( functional ) determinant . a well - formulated chiral gauge theory should also exhibit fermion nonconservation @xcite . if the vector - like operator @xmath10 has zero modes , the right - hand side of eq . ( [ chiral gamma ] ) should vanish : recall that the zero modes have definite chirality . thus the zero modes possess natural projections from dirac spinors @xmath143 onto weyl spinors @xmath144 . for positive chirality the projection implies @xmath145 , and for negative chirality @xmath146 ( integration by parts implied ) . in eq . ( [ chiral gamma ] ) integration over the @xmath147 component of @xmath137 or over the @xmath148 component of @xmath149 yields zero . on the other hand , if there are @xmath19 zero modes of each chirality ( counting individual species appropriately ) the integral @xmath150 does not vanish if @xmath151 . ( the notation @xmath152 is schematic for a product of @xmath153 suitable components or positions of @xmath137 . ) amplitudes for fermion nonconservation are proportional to integrals like @xmath154 .    before discussing how to regulate the integrals in eqs . ( [ chiral gamma ] ) and ( [ b violation ] ) , one should list the properties of eq . ( [ chiral gamma ] ) that the regulator should respect . in addition to the connection between zero modes and fermion nonconservation , one wants    1 . [ invariant ] @xmath155 , under all circumstances . [ anomaly free ? ] @xmath156 , only if @xmath138 is `` anomaly - free . '' [ complex ] @xmath157 , if @xmath138 is complex ; such that @xmath158 , then @xmath138 is real ; otherwise it is complex . the representation generated by @xmath159 is denoted @xmath160 . if @xmath138 is real , then @xmath161 is real . ] indeed @xmath162 . here @xmath163 is a gauge transform of @xmath6 , and the notation @xmath164 stresses the fermion representation . condition  [ anomaly free ? ] is imposed so that the nonperturbative definition reproduces perturbation theory . condition  [ complex ] identifies a diagnostic feature of the effective action of chiral fermions . even though the right - hand side of eq . ( [ chiral gamma ] ) is by nature not a determinant , ultraviolet regulators almost always turn it into one . for example , in the most naive ( and unsuccessful ) lattice formulation , @xmath142 becomes a large , square , numerical matrix , and @xmath165 is the matrix determinant . so let us provisionally assume @xmath166 in anticipation of a cutoff analogous to the one for vector - like theories ( sect . [ regulator ] ) , the product runs only over the first @xmath61 eigenvalues . the most transparent way to realize conditions  [ invariant ] and  [ anomaly free ? ] is as follows : write @xmath167 , with @xmath168 and @xmath169 real , and note that @xmath170 if the moduli @xmath171 of the eigenvalues are gauge invariant and one orders the eigenvalues by @xmath172 , condition  [ invariant ] is satisfied . by condition  [ anomaly free ? ] , in anomalous ( sub)representations the phases of the eigenvalues would be gauge variant , but the variation from one species could cancel that of another . unfortunately , it seems that eigenvalue problems with these simple gauge - transformation properties leave the total phase @xmath173 unspecified , flouting condition  [ complex ] . is related to the vector - like theory and @xmath174 to the spectral asymmetry of a certain operator @xcite . this method , however , requires three regulators : one for the gauge fields , one for the vector - like fermions , and one for @xmath175 . ]      a standard way to cast the effective action as a determinant is to introduce a new negative chirality partner @xmath176 with no dynamics . the action is now @xmath177 where the four - component spinor @xmath178 and @xmath179 the matrix forms of eqs . ( [ dirac spinor ] ) and ( [ d hat ] ) presumes a @xmath180-matrix basis with @xmath13 diagonal . formally , the functional integral over @xmath181 is trivial , but the combined integral @xmath182 can be expressed as a determinant , as in the vector - like theory . the operator @xmath183 is not self - adjoint , but it is elliptic @xcite , so on a compact space - time it still has a discrete spectrum . the eigenvalue problem now has different right and left eigenfunctions . there are zero modes @xmath184 where @xmath185 are zero - mode eigenfunctions of the vector - like operator @xmath10 , with chirality @xmath186 . nonzero modes come in pairs . this operator is again not self - adjoint , though elliptic , so there are left and right eigenfunctions @xmath187 . most papers either ignore this subtlety , or try to circumvent it . for example , some tricks turn @xmath183 into a hermitian operator , thus leaving @xmath173 unspecified . they can not be adopted here . ] @xmath188 ( i\hat{d})^\dagger\chi_n & \lambda_n^ * \chi_n \end{array}\ ] ] that are mutually orthonormal : @xmath189 in addition @xmath190 . furthermore , since @xmath191 , if @xmath192 has eigenvalue @xmath12 , then @xmath193 has eigenvalue @xmath17 , and similarly for @xmath194 , @xmath195 . the functional integral is now defined to be over fields @xmath196 \bar{\psi}(x ) & { \displaystyle \sum_{n=1}^{n_-}}\bar{z}_n \varphi_{-,n}(x ) & { \displaystyle \sum_{n=1}^n }     \bar{a}_n \chi^\dagger_n(x ) , \end{array}\ ] ] i.e.  the measure is @xmath197 as in sect . [ regulator ] the functional integral is cut off by retaining only the lowest @xmath61 nonzero modes . the principle for ordering the eigenvalues is revealed below , but clearly the two modes with the same @xmath55 should be adjacent . analogously to sect . [ regulator ] , it is convenient to take @xmath52 odd ( even ) if @xmath198 is negative ( positive ) .    let us first integrate over the @xmath30 and @xmath31 . ( [ chiral fields ] ) and ( [ chiral measure ] ) yield @xmath199 for the functional integral . in a real representation these eigenvalues come in pairs @xmath200 ; @xmath201 satisfies condition  [ complex ] . but one should not expect @xmath202 to be a suitable definition of the effective action  hence the tilde  because neither the moduli nor the phases of the nonzero eigenvalues are gauge invariant .    under the gauge transformation @xmath203 the chiral dirac operator transforms as @xmath204 , where @xmath205 . the zero modes are gauge invariant . to first order in @xmath206 the nonzero eigenvalues vary by @xmath207 which immediately yields the variation of @xmath201 : @xmath208 where @xmath209 the last expression applies if the eigenvalues are ordered by increasing @xmath210 , which is justified because it reproduces the consistent gauge anomaly .    from eq . ( [ chiral anomaly ] ) and appendix  [ convergence ] , the imaginary part of @xmath211 is @xmath212,\ ] ] the familiar consistent anomaly @xcite . it vanishes if @xmath213 in representation @xmath138 @xcite ; @xmath201 satisfies condition  [ anomaly free ? ] . even if the anomaly cancels , however , the real part @xmath211 is not gauge invariant . with @xmath214 and @xmath215 from appendix  [ convergence ] @xmath216 after taking @xmath63 , all higher - dimension terms drop out because they are proportional to @xmath217 . following bardeen , the gauge variation of the real part can be compensated by counter - terms @xcite . let @xmath218 s_4 & { \displaystyle \frac{1}{48\pi^2}\int}d^4x\,\tr_\rho\left [ { \mbox{\small $ \frac{1}{2}$}}a_\mu\partial^2a_\mu     + ( \partial\cdot a)^2 + { \mbox{\small $ \frac{1}{4}$}}a_\mu a_\nu a_\mu a_\nu - { \mbox{\small $ \frac{1}{2}$}}(a^2)^2 \right ] , \end{array}\ ] ] and @xmath219 . if @xmath220 is computed using the _ interpolated _ gauge field , its gauge variation cancels that of @xmath221 exactly . the choice of @xmath222 is not unique , because one could also add a gauge invariant term proportional to @xmath223 . but the ambiguity corresponds to a shift in the ( inverse ) bare gauge coupling , so it should make no difference once all cutoffs are removed .    the counter - term @xmath224 is supposed to remove a quadratic divergence , but the number of modes @xmath61not the mass @xmath57 that appears in eq . ( [ counter - terms])defines the cutoff . the number of fermion modes with momentum below a cutoff @xmath1 is @xmath225 approximating the sum by an integral yields @xmath226 . but @xmath61 is an integer , so the error in this approximation can be eliminated by taking @xmath227 this value of @xmath57 is the one needed to cancel the quadratic divergence . if one chooses @xmath228 , then @xmath229 . the combination @xmath230 is thus gauge invariant under infinitesimal gauge transformations . by eq . ( [ composition ] ) this is enough to show that @xmath130 is invariant under all lattice gauge transformations . hence , @xmath130 satisfies all three conditions . this is the main result . finally , let us integrate over the zero modes . unless there are enough factors of the fermion field in the amplitude , the integral vanishes . with the minimal number of fields in eq . ( [ b violation ] ) @xmath231 where @xmath232 , @xmath233 and @xmath234 , @xmath235 denote discrete indices and positions of the fields in eq . ( [ b violation ] ) . functional integration alone would lead to eq . ( [ amplitudes ] ) with @xmath201 instead of @xmath236 . the zero modes present no substantive changes in the computation of the gauge variation of @xmath201 , so the same counter - terms restore gauge symmetry .      although ref . @xcite focuses primarily on the vector - like theory with pauli - villars cutoff , it does prove its important convergence theorem for theories with vector and axial - vector couplings . this suggests that the chiral coupling is also intended as an application . with the cutoff proposed there , the analysis of cutoff effects leads to somewhat different conclusions . appendix  [ pv ] recasts t  hooft s cutoff in a way that makes appendix  [ convergence ] directly applicable . one finds that the quadratic divergence drops out , but the universal term @xmath215 still spoils gauge invariance of the real part . moreover , the coefficients of the higher - dimension terms , @xmath237 and so forth , no longer vanish , though they are suppressed by powers of @xmath238 . to eliminate these violations of gauge symmetry , one must take @xmath1 to infinity , on each gauge field separately . this is cumbersome , and perhaps logically inconsistent , because to obtain a renormalized theory , the bare gauge coupling must be adjusted to keep physical masses  properties of the ensemble average  cutoff independent .      to summarize the results of this section it is worth sketching how to compute correlation functions . because the integrals @xmath239 sometimes vanish trivially , let us denote the sector of lattice gauge fields with @xmath19 zero modes @xmath240 . nonzero integrals with then be over one or so sectors . consider first fermion - conserving observables . one would like to compute a ratio of the form @xmath241 where @xmath242 is the lattice - gauge - field action . here the gauge - invariant observable @xmath243 is constructed from the gauge field and fermion propagators . ( [ observable ] ) one requires an ensemble of fields in @xmath244 , distributed with weight @xmath245 for amplitudes of this type , all other sectors carry weight 0 , because of the zero modes , so in the monte carlo they are simply omitted . consider next a fermion - violating amplitude . for simplicity , suppose that @xmath243 does not contain fermion species that are being created or annihilated . now one would like to compute a ratio of the form @xmath246 the numerator and denominator are averages over different sectors , is typically the sector with topological charge @xmath247 , which would have one zero mode for each species . ] and recall that in sectors with zero modes @xmath248 is defined to be the product of the first @xmath61 nonzero eigenvalues , ordered by @xmath210 . one can re - write eq . ( [ violation ] ) as @xmath249 where the average @xmath250 is over @xmath251 . in addition to the weight @xmath252 , one must compute the zero - mode eigenfunctions @xmath253 for created and annihilated species . the other factor in the fermion - nonconserving amplitude is the ratio of two partition functions . to compute it accurately some special numerical techniques @xcite are available , which keep track of the system s preference for one sector or the other . various versions of these `` histogram methods '' @xcite may also prove useful in obtaining the nontrivial phase of @xmath252 , inherent to a complex representation . some theories , the simplest of which is su(2 ) with one weyl doublet , are afflicted by a global anomaly @xcite . the representations in question are real , and therefore @xmath115 , as defined by eqs . ( [ gamma tilde ] ) and  ( [ main result ] ) , is real and positive . thus @xmath130 is  real . let us focus on su(2 ) . because @xmath254 , there are nontrivial gauge transformations , for which the proof of gauge invariance of @xmath130 breaks down . ) shows that the interpolations inherit this property , the two classes can be separated by lattice gauge transformations for which the interpolation is ill - defined . ] let @xmath255 be in the nontrivial class . the variation @xmath256 is real , but does it vanish ? following ref . @xcite one can compute the difference by embedding su(2 ) into su(3 ) and taking a trajectory from @xmath257 to @xmath258 in su(3 ) . @xmath259 does not change for any infinitesimal step , and since @xmath260 , the trajectory can be constructed from infinitesimal steps . thus @xmath261 for the embedded field , and hence likewise for the su(2 ) fields themselves .    on the other hand , witten argued that the two configurations @xmath131 and @xmath262 should have boltzmann weights equal in magnitude but _ opposite _ in sign @xcite . indeed the gauge variation of @xmath263 integrated along the su(3 ) trajectory supports his conclusion @xcite . but , given a lattice gauge field @xmath264 , the algorithm for @xmath265 can not determine whether @xmath266 or @xmath267 . and thanks to the pains taken ensure gauge invariance , the computed weight is the same in either case . this is not a serious drawback , however . if the global anomaly applies , one can replace numerator and denominator of eq . ( [ observable ] ) by witten s original result , @xmath268 , eliminating all computation . the seeming incompatibility of lattice fermions and chiral symmetry has inspired the recurring idea @xcite of treating the fermions in the continuum , even if the underlying gauge field is on the lattice . smit calls the idea desperate @xcite . how desperate are the main results presented here ? the colorful terminology refers to the tacit assumption that a continuum requires an infinite number of degrees of freedom ( per unit volume ) . then the arithmetic needed to evaluate functional integrals is infinite : we are desperate because a computer can not do the job . but with a finite - mode cutoff there are @xmath61 eigenvalues ; the computation is finite . alas , even in the vector - like theory the effort needed to obtain higher eigenvalues will be high ( cf . the construction for chiral fermions is yet more computationally intensive , because , first , the counter - terms must now be computed accurately , and , second , the phase , which stems not from the regulator but from the chiral coupling itself , requires extra care . there is a potential shortcoming to the finite - mode regulator . one would like to verify that the spectrum remains chiral , in perturbation theory and beyond . appendix  [ cutoff ] demonstrates perturbative universality for fermion loops . @xcite and appendix  [ convergence ] do the same for anomalous diagrams only . ) but there is ( as yet ) no comparable proof for diagrams with external fermion lines , which are needed to examine the fermion spectrum . the resolution is not straightforward , because it depends on details of the gauge - boson propagator , i.e.  on the lattice gauge action . and even if the perturbative test is a success , one should be cautious until the nonperturbative spectrum has been checked . the counter - terms required in the chiral gauge theory may be unsettling at first sight . but their necessity arises from the unassailable observation @xcite that a _ regulated _ functional integral either respects a symmetry or it does not . to obtain the anomaly ( and hence the physically powerful requirement of anomaly cancellation ) the functional integral @xmath269 can not be gauge symmetric . consequently , the final result for the chiral boltzmann factor @xmath270 is the product of a functional integral and a symmetry - restoring factor . a related peculiarity is the fate of witten s global anomaly @xcite . again , an ansatz for the effective action is either gauge invariant or not . the effective action @xmath130 _ is _ gauge invariant , even when the gauge transformation is in the nontrivial class . without asserting that this paper s formulation succeeds at defining the globally anomalous theories , one might suggest that it could shed light on the dynamical puzzles that originally motivated ref . an optimistic possibility is that the dynamics of @xmath130 fail in globally anomalous theories , but not otherwise . a serious complication is that the formulation is in euclidean field theory . at the nonperturbative level the wick rotation does no good , and instead one constructs the minkowski theory via the imaginary - time evolution operator on the hilbert space of states . this procedure defines a hamiltonian that can then be used to propagate the states in real time . the eigenfunctions used to define the functional integral are fundamentally four - dimensional , so the constructive approach @xcite does not seem helpful . perhaps the axiomatic approach @xcite will prove more promising . finally , let us compare the present chiral construction with the overlap formalism @xcite . both define an effective action with a gauge - invariant real part . both generate an imaginary part in a complex fermion representation , but not in a real representation . with the interpolation and @xmath271 the gauge variation of the imaginary part is the consistent anomaly and with the finite - mode regulator ( or in the limit @xmath2 @xcite ) nothing else . with the overlap the gauge variation of the imaginary part contains the anomaly and higher - order gauge breaking terms , analogous to @xmath272 ( notation of appendix  [ convergence ] ) , as well . @xcite argues they should be tolerably small , and tests in two dimensions @xcite indicate that this conclusion may be correct . both methods provide fermion nonconserving amplitudes : here the gauge - field topology @xcite drives fermion nonconservation , and in ref . @xcite the fermion nonconserving amplitudes define the gauge - field topology . ( for smooth lattice gauge fields the two topologies coincide . ) whereas appendix  [ cutoff ] includes an explicit verification of fermion - loop coupling - constant renormalization , it does not seem that an explicit calculation starting from the lattice overlap is available yet @xcite . on the other hand , ref . @xcite includes several numerical cross checks that have not been done here . an important , dynamical test is whether the fermion spectrum remains chiral after integrating over gauge fields . neither construction has been subjected to this test yet , because it requires a full - fledged monte carlo calculation . sects .  [ vector - like][metric ] are based on an unpublished draft ( 1988 ) . that work was aided by discussions with m. gckeler , g. schierholz , and u .- j . i also thank h. joos for discussions on this topic then and since . while formulating sect . [ chiral ] , i had helpful conversations with w.a . bardeen , m.f.l . golterman , and t. onogi . fermilab is operated by universities research association , inc . , under contract de - ac02 - 76ch03000 with the u.s . department of energy .    while this paper was being written , a preprint appeared by s.d.h . hsu ( yale university report yctp - p5 - 95 , hep - th/9503058 ) , expanding on the idea of gckeler and schierholz @xcite to exploit the @xmath175-invariant formulated in ref . this appendix fills in the steps from eq . ( [ regulated anomaly ] ) to eq . ( [ fdualf ] ) . the modifications needed to obtain eq . ( [ consistent anomaly ] ) are provided in sect . [ chiral convergence ] . the analysis is standard , but it is provided to demonstrate the form of higher - dimension terms suppressed by powers of @xmath273 . the sharp limit @xmath63 turns out to be special , because it eliminates these terms _ before _ taking @xmath274 . first some preliminaries on fourier transforms . for definiteness the space - time is a box with sides @xmath275 , and volume @xmath276 . the fourier transform @xmath277 allowed values of @xmath278 depend on the boundary conditions . this appendix presents details for a wide class of almost periodic boundary conditions , such that @xmath279 , @xmath280 . strictly periodic directions have @xmath281 ; anti - periodic directions have @xmath282 ; @xmath76-periodic directions have a certain combination of the foregoing and @xmath283 . the inverse transform ( assuming convergence ) is @xmath284 the application needed here is the fourier transform of eq . ( [ complete ] ) @xmath285    consider any function @xmath286 obeying @xmath287 and @xmath288 @xcite , and let @xmath289_{ij},\ ] ] with the summation convention over the spin - flavor - color multi - indices @xmath290 . in sect . ( [ regulator ] ) the fermi function @xmath291 appears , with mass @xmath57 and the limit @xmath63 . for comparison with refs . @xcite , however , it is convenient to keep @xmath286 arbitrary . fourier transforming the eigenfunctions and using eq . ( [ k complete ] ) one obtains @xmath292_{ij}.\ ] ] from eq . ( [ detc ] ) one recalls that here @xmath293 . hence , @xmath294_{ij},\ ] ] expanding in @xmath295 @xmath296 where now @xmath297 , and @xmath298 . because the functions under consideration are smooth and vanish rapidly at infinity , the sums can be approximated by integrals @xmath299 with little error . odd powers vanish . note that the finite - size effects also vanish when the ultraviolet regulator is removed .    reorganizing the terms according to the power of @xmath1 @xmath300 + & \left . m^{d-4 } f^{(2-d/2)}(0)\,\alpha_4^a(x ) + m^{d-6 } f^{(3-d/2)}(0)\,\alpha_6^a(x ) + \cdots\right ] , \end{array}\ ] ] where @xmath301 , @xmath302,\ ] ] @xmath303 + { \mbox{\small $ \frac{1}{6}$}}[d_\mu , d_\nu][d_\mu , d_\nu ] \right)\right\},\ ] ] and @xmath304 contains terms with 6 @xmath305 s , combined to produce a function ( rather than a differential operator ) . the result depends on the cutoff function via the coefficients @xmath306 , defined by @xmath307 , @xmath308 the @xmath1 independent term has the universal coefficient @xmath287 ; the other coefficients differ for different cutoffs .    with the fermi function @xmath309 , @xmath54 , plus terms of order @xmath310 . hence , the @xmath311 , @xmath312 , are power - law divergences , unless the traces vanish . on the other hand , for @xmath54 then @xmath313 as @xmath314 . thus , the sharp cutoff has no `` scaling violations . ''    for the vector - like theory @xmath315f_{\mu\nu}$ ] . if @xmath316 , the only surviving term @xmath214 yields the well - known result . if @xmath317 , the dirac trace makes @xmath214 vanish as well as everything in @xmath318 except the term @xmath319 , yielding the familiar axial anomaly , eq . ( [ fdualf ] ) .      in the chiral gauge theory one wants @xmath320_{ij}.\ ] ] the left and right eigenfunctions of @xmath183 are not complete , but @xmath321 where @xmath322 projects onto zero modes . this nuisance is easiest to handle with anti - periodic boundary conditions . if the function @xmath286 drops to zero below the lowest momentum mode , then @xmath323 . thus , the projector can be dropped after fourier transforming . the hole in @xmath286 can not affect the momentum sums , so the correct approximation in eq . ( [ integrals ] ) uses a function without the hole on the right - hand side .    the algebraic manipulations still hold , but one must replace @xmath324 by @xmath139 , @xmath10 by @xmath271 _ and _ the symbol @xmath325 by @xmath326 . from eq . ( [ d hat ] ) @xmath327 then @xmath328 consequently , @xmath329 where the trace is now only over gauge indices . in @xmath316 this is the universal anomaly , plus a term that can be compensated by a local counter - term ; in @xmath317 it only the latter survives , and it is a power - law divergence . the function in @xmath318 is too lengthy to present explicitly , but after the trace @xmath330 , where @xmath331\quad(d=4)\ ] ] is the well - known consistent anomaly @xcite , and @xmath332 - 2 [ \partial_\mu \partial{\!\cdot\!}a , a_\mu ] + \{\partial{\!\cdot\!}a , a^2\ } \rule{0.0em}{0.9em}\right.\right . \\[1.0em ]    & \left.\left.\rule{0.0em}{0.9em } - a_\mu(\partial{\!\cdot\!}a)a_\mu - [ \partial_\mu a_\nu , [ a_\mu , a_\nu ] ] + a_\mu(\partial_\mu a_\nu + \partial_\nu a_\mu)a_\nu \right)\right ] \end{array}\ ] ] is the almost as well - known quantity that , like @xmath333 , can be compensated by local counter - terms @xcite . the analysis of the previous section can be applied directly to the effective action . it shows that the finite - mode cutoff is in the same ( perturbative ) universality class as pauli - villars regulators . as in appendix [ convergence ] , the analysis is performed for an arbitrary smooth function , but once again the sharp limit is special , because it has no power corrections .    for a good infrared behavior , this section considers only anti - periodic boundary conditions . one can write the regulated effective action as @xmath334 where the effective lagrangian @xmath335 and , for any function @xmath286 obeying @xmath287 and @xmath288 , @xmath336 and @xmath337 . the notation applies to the vector - like theory , but the substitutions needed for the chiral gauge theory are obvious . the effective lagrangian for arbitrary @xmath286 will be denoted @xmath338 . fourier transforming the eigenfunctions , using eq . ( [ k complete ] ) , and using @xmath293 one obtains @xmath339_{ij},\ ] ] where @xmath340 . expanding in @xmath295 @xmath341 where @xmath342 . the derivatives take the form @xmath343 the @xmath344 term can be called universal , because after summing over @xmath278 the universal @xmath287 remains .    in the vector - like theory , the fermion mass ( presumed nonzero ) regulates the infrared and one can use eq . ( [ integrals ] ) . in the chiral gauge theory ( or any massless case ) , infrared singularities make the integrals poor estimates of the sums . for the present purposes , however , the function @xmath286 can also be used as an infrared regulator . one simply chooses @xmath286 to drop to zero for momentum less than the smallest allowed by the anti - periodic boundary condition , say for @xmath345 . the delicate infrared behavior has consequences when reorganizing the effective lagrangian according to the dimension of the interactions . one finds @xmath346 + & \left . m^{d-4 } l_f^{(2-d/2)}(\delta^2)\,l_4(x ) + m^{d-6 } l_f^{(3-d/2)}(\delta^2)\,l_6(x ) + \cdots\right]+\delta{{\cal l } } , \end{array}\ ] ] where @xmath347 yields a constant of no dynamical significance , @xmath348 @xmath349 + { \mbox{\small $ \frac{1}{6}$}}[d_\mu , d_\nu][d_\mu , d_\nu]\right\},\ ] ] and @xmath350 contains same combination of 6 @xmath305 s as @xmath304 , which is a function rather than a differential operator . the interactions in @xmath351 are proportional to @xmath352^l$ ] ; hence they survive only when @xmath353 . the effective lagrangian depends on the cutoff via the coefficients @xmath354 . if @xmath355 ( i.e.  @xmath356 ) these become @xmath357 . for example , with the fermi function @xmath358 plus terms of order @xmath359 . the higher - dimension functions have coefficients @xmath360 , @xmath54 . from eq . ( [ lfn ] ) one sees that they consist of a universal , @xmath286- and @xmath1-independent term , plus terms non - universal proportional to @xmath361 . the remaining dimensions are balanced by infrared scales @xmath41 or @xmath362 . again , for the fermi function the non - universal terms are suppressed by @xmath363 and drop out in the sharp limit . since interactions of all dimension appear in eq . ( [ lagrangian expansion ] ) , it must be interpreted as a perturbative series .    in the vector - like theory , regulated as in sect .  [ regulator ] , @xmath364 vanishes . indeed , the gauge invariance of the _ regulated _ theory forbids any dimension 2 terms . in @xmath365 the dirac trace eliminates the nested commutator leaving , for @xmath317 , @xmath366 which renormalizes the gauge coupling .    in the chiral gauge theory of sect . [ chiral ] this analysis applies to @xmath201 . the dimension - two term @xmath364 does not vanish , because now the regulator breaks the gauge symmetry . in four dimensions one has @xmath367 to restore gauge symmetry , one must add the counter - term @xmath224 in eq . ( [ counter - terms ] ) . on the other hand , @xmath365 induces coupling constant renormalization . ( the gauge non - invariant pieces cancel . ) with some patience one can accumulate the nonvanishing @xmath180-matrix traces in @xmath365 to obtain @xmath368 up to a total derivative . notice that , as expected , the renormalization term is half that of the vector - like theory . the dimension - four interactions in @xmath351 , present because @xmath369 , include gauge - breaking terms ; they are cancelled by @xmath222 . @xcite defines the fermion functional integral with a time - honored @xcite pauli - villars regulator : @xmath370 in the vector - like case , and similarly @xmath371 with @xmath271 in the chiral case . the determinants are products over the infinitely numerous eigenvalues . the masses for vector - like gauge theories ; @xmath372 for chiral gauge theories . ] @xmath373 and signatures @xmath374 satisfy @xmath375 @xmath376 for @xmath377 , eq . ( [ logs ] ) defines the overall scale @xmath1 . ( [ powers ] ) and ( [ logs ] ) require infinite series only if one requires the identities for all @xmath52 . the function needed to describe the effective lagrangian , cf . ( [ gamma_f])([lagrangian expansion ] ) , is @xmath378 note that @xmath379 and all its derivatives vanish at infinity , by virtue of eqs . ( [ powers ] ) and  ( [ logs ] ) , so the manipulations of appendix  [ cutoff ] still hold . ( [ awti ] ) and ( [ gauge variation ] ) also hold as before , but with @xmath380 replaced by @xmath381 . here @xmath382 again @xmath383 and all its derivatives vanish at infinity , so the manipulations of appendix  [ convergence ] hold .    to apply eq . ( [ general result ] ) one notes the universal normalization @xmath384 . power - law divergences drop out , but `` scaling violations '' remain ( @xmath54 ) : @xmath385 and analogously for @xmath386 . one might remark that the absence of power - law divergences relies on integrating @xmath103 to infinity . they re - appear if one truncates @xmath387 when , say , @xmath388 becomes small . karsten and j. smit , nucl . * b183 * ( 1981 ) 103 . nielsen and h. ninomiya , nucl . phys . * b185 * ( 1981 ) 20 , ( e ) * b195 * ( 1982 ) 541 ; * b193 * ( 1981 ) 173 ; + d. friedan , commun . math * 85 * ( 1982 ) 481 . wilson , in _ new phenomena in subnuclear physics _ , edited by a. zichichi ( plenum , new york , 1977 ) . l. susskind , phys . rev . * d16 * ( 1977 ) 3031 ; + t. banks , j.b . kogut , and l. susskind , phys . rev . * d13 * ( 1976 ) 1043 . petcher , nucl . b ( proc . suppl . ) * 30 * ( 1993 ) 50 . r. narayanan and h. neuberger , phys . lett . * 71 * ( 1993 ) 3251 ; nucl . b412 * ( 1994 ) 574 ; institute for advanced study report iassns - hep-94/99 ( hep - th/9411108 ) . kaplan , phys . b288 * ( 1992 ) 342 . frolov and a.a . slavnov , phys . b309 * ( 1993 ) 344 ; nucl . b411 * ( 1994 ) 647 ; + r. narayanan and h. neuberger , phys . lett . * b302 * ( 1993 ) 62 . r. flume and d. wyler , phys . 108b * ( 1982 ) 317 . g. t  hooft , utrecht university report thu-94/18 ( hep - th/9411228 ) . m. atiyah and i.m . singer , ann . math . * 93 * ( 1971 ) 139 . kronfeld , nucl . b ( proc . suppl . ) * 4 * ( 1988 ) 329 . j. smit , nucl . b ( proc . suppl . ) * 4 * ( 1988 ) 451 . m. gckeler and g. schierholz , nucl . b ( proc . suppl . ) * 29 * ( 1992 ) 114 ; * 30 * ( 1993 ) 609 . a. andrianov and l. bonora , nucl . phys . * b233 * ( 1984 ) 232 , 247 . k. fujikawa , phys . lett . * 42 * ( 1979 ) 1195 ; phys . rev . * d21 * ( 1980 ) 2848 , ( e ) * d22 * ( 1980 ) 1499 . m. lscher , comm . * 85 * ( 1982 ) 29 . a. phillips and d. stone , comm . math . phys . * 103 * ( 1986 ) 599 . m. gckeler , a.s . kronfeld , g. schierholz , and u .- j . wiese , nucl . * b404 * ( 1993 ) 839 . kronfeld , unpublished ( 1988 ) . van baal , comm . * 85 * ( 1982 ) 529 . m. gckeler , a.s . kronfeld , m.l . laursen , g. schierholz , and u .- j . wiese , nucl . b292 * ( 1987 ) 349 . wiese , phys . * b315 * ( 1993 ) 417 . g. t  hooft , phys . * 37 * ( 1976 ) 8 ; phys . rev . * d14 * ( 1976 ) 3432 , ( e ) * d18 * ( 1978 ) 2199 . l. alvarez - gaum and s. della pietra , in _ recent developments in field theory and statistical mechanics _ , edited by j. ambjrn , b.j . durhuus , and j.l petersen ( elsevier , amsterdam , 1985 ) ; + l. alvarez - gaum , s. della pietra , and v. della pietra , phys . lett . * 166b * ( 1986 ) 177 ; + s. della pietra , v. della pietra , and l. alvarez - gaum , commun . * 109 * ( 1987 ) 691 . l. alvarez - gaum and p. ginsparg , nucl b243 * ( 1984 ) 449 . balachandran , g. marmo , v.p . nair , and c.g . trahern , phys . rev . * d25 * ( 1982 ) 2713 . m.b . einhorn and d.r.t . jones , phys . * d29 * ( 1984 ) 331 . w.a . bardeen , phys . rev . * 184 * ( 1969 ) 1848 . j. wess and b. zumino , phys . lett . * 37b * ( 1971 ) 95 . gross and r. jackiw , phys . rev . * d6 * ( 1972 ) 477 . a. borelli , l. maiani , g.c . rossi , r. sisto , and m. testa , nucl . * b333 * ( 1990 ) 335 . alonso , ph . boucaud , j.l . corts , and e. rivas , mod . a5 * ( 1990 ) 275 ; phys . d44 * ( 1991 ) 3258 . bodwin and e.v . kovcs , nucl . b ( proc . suppl . ) * 20 * ( 1991 ) 546 ; * 30 * ( 1993 ) 617 . g. bhanot , s. black , p. carter , and r. salvador , phys . lett . * b183 * ( 1987 ) 331 ; + g. bhanot , k. bitar , and r. salvador , phys . lett . * b187 * ( 1987 ) 381 ; * b188 * ( 1988 ) 246 ; + g. bhanot , a. gocksch , and p. rossi , phys . lett . * b199 * ( 1988 ) 101 . m. karliner , s. sharpe , and y.f . chang , nucl . b302 * ( 1988 ) 204 . a. gocksch , phys . lett . * b206 * ( 1988 ) 290 ; + u .- j . wiese , nucl . b318 * ( 1989 ) 153 . e. witten , phys . * 117b * ( 1982 ) 324 . f. klinkhamer , phys . lett . * b256 * ( 1991 ) 41 . wilson and j. kogut , phys . reports * 12c * ( 1974 ) 75 ; + m. lscher , commun . math . phys . * 54 * ( 1977 ) 283 . k. osterwalder and r. schrader , commun . * 31 * ( 1973 ) 83 , * 42 * ( 1975 ) 281 ; + k. osterwalder and e. seiler , ann . phys . * 110 * ( 1978 ) 440 . r. narayanan and h. neuberger , rutgers university report ru-94 - 99 ( hep- lat/9412104 ) ; + r. narayanan , h. neuberger , and p. vranas , institute for advanced study report iassns - hep-95 - 19 ( hep - lat/9503013 ) . s. randjbar - daemi and j. strathdee , ictp ( trieste ) report ic-95 - 6 ( hep - lat/ 9501027 ) . m.f.l . golterman and y. shamir , institute for theoretical physics report nsf - itp-95 - 60 ( hep - lat/9501035 ) . g. t  hooft , nucl . phys . * b33 * ( 1971 ) 173 .
lattice fermions have well - known difficulties with chiral symmetry . to evade them 
 it is possible to couple _ continuum _ fermions to _ lattice _ gauge fields , by introducing an interpolation of the latter . following this line of thinking , 
 this paper presents two euclidean formulations of the effective action that appears after functional integration over fermion fields , one for vector - like and the other for chiral couplings . with suitable finite - mode regulators both 
 effective actions can be evaluated in a finite computation . 
 the prescriptions provided here contain some details not found in previous work marrying continuum fermions to the lattice via an interpolation . 
 for example , the counter - terms needed to maintain chiral gauge invariance are explicitly given . by construction 
 coupling - constant renormalization , anomaly structure , and ( in the chiral gauge theory ) fermion nonconserving amplitudes all satisfy one s expectations from perturbative and semi - classical analyses .    addtoresetequationsection    # 1#1^ # 1#1 # 1 # 1#1    # 1#2 # 1#2