Recent results on zero modes of the Abelian Dirac operator in three
dimensions support to some degree the conjecture that the Chern-Simons action
admits only certain quantized values for gauge fields that lead to zero modes
of the corresponding Dirac operator. Here we show that this conjecture is wrong
by constructing an explicit counter-example.Comment: version as published in PRD, minor change

A.A. Balinsky

B. Muratori

C. Adam

C. Nash

D.M. Elton

J. Fröhlich

L. Erdos

M. Loss

M.P. Fry

English

arXiv

Crossref

Chern-Simons action for zero-mode supporting gauge fields in three dimensions

Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree the conjecture that the Chern-Simons action admits only certain quantized values for gauge fields that lead to zero modes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing an explicit counter-example.Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree the conjecture that the Chern-Simons action admits only certain quantized values for gauge fields that lead to zero modes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing an explicit counter-example

Adam, C.

Muratori, B.

Nash, C.

CERN Document Server

PHYSICAL REVIEW D 67, 087703 ~2003!Chern-Simons action for zero-mode supporting gauge fields in three dimensionsC. Adam*Wolfgang-Pauli Institute c/o Institute of Mathematics, Vienna University, 1090 Vienna, AustriaB. Muratori†CERN, SL-AP Division, Geneva 32, CH-1211, SwitzerlandC. Nash‡Department of Mathematical Physics, National University of Ireland, Maynooth, Ireland~Received 29 November 2002; published 28 April 2003!Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree theconjecture that the Chern-Simons action admits only certain quantized values for gauge fields that lead to zeromodes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing anexplicit counterexample.DOI: 10.1103/PhysRevD.67.087703 PACS number~s!: 11.15.TkIn the last few years a considerable amount of interest hasbeen devoted to the study of zero modes of the Abelian Diracoperator in three-dimensional Euclidean space, that is, tosquare-integrable solutions of the Dirac equationD C[sW @ i]W1AW ~xW !#C~xW !50 ~1!where xW5(x1 ,x2 ,x3), sW are the Pauli matrices, and C is atwo-component spinor. In addition, the gauge field AW is as-sumed to obey certain integrability conditions ~e.g., squareintegrability of the related magnetic field BW 5]W3AW ).On the one hand, such solutions are relevant for the quan-tum mechanical behavior of nonrelativistic electrons ~seee.g., Ref. @1#!, because solutions to the above equation are, atthe same time, solutions to the Pauli equation ~the Pauliequation is obtained by just squaring the Dirac operator inthe above equation, i.e., D 2C50). On the other hand, solu-tions to the Dirac equation are also relevant for ~Euclidean!quantum electrodynamics, as was discussed, e.g., in Refs.@2,3#.Some first examples of zero modes were constructed inRef. @4#. In Ref. @5# a class of Dirac operators and their zeromodes was constructed which depend on a function that isarbitrary up to certain boundary conditions, thereby relatingthe existence of these zero modes to some topological con-dition. Some further examples of zero modes were given inRef. @6# and in Ref. @7#. In Refs. @8,9# the first examples ofDirac operators with multiple zero modes were given,thereby demonstrating the existence of the phenomenon ofzero mode degeneracy. Further, a relation between the num-ber of zero modes and a certain topological linking number~the Hopf index! of the corresponding gauge field was estab-lished in Ref. @9#. A very detailed and more geometrical dis-*Email address: adam@mat.univie.ac.at†Email address: Bruno.Muratori@cern.ch‡ Email address: cnash@stokes2.thphys.may.ie0556-2821/2003/67~8!/087703~3!/$20.00 67 0877cussion of these Dirac operators with multiple zero modes,based on the concept of Riemannian submersions, was givenin Ref. @10#.In Ref. @11# the following two results were proved: ~i! Forthe one-parameter family of gauge potentials tAW zero modesmay exist for at most a finite set of values t i for any tP(t0 ,t1), and ~ii! The set of gauge potentials with no zeromodes is a dense subset of the set of all gauge potentials~with certain decay properties!. Recently, some results on thedimensionality of the space of gauge potential with zeromodes were obtained in Ref. @12#. There it was proven thatlocally the space of gauge potentials with ~at least! one zeromode is of co-dimension one within the space of all gaugepotentials ~with certain decay properties!. In addition, someresults on the dimensionalities of spaces of gauge potentialswith multiple zero modes were proven.The above-described results would suit well with the as-sumption that there exists a certain functional of the gaugepotential which may admit only fixed or quantized values forgauge potentials that support zero modes. The simplest func-tional one can imagine is the Chern-Simons action, whichhas the additional attractive feature of being a topologicalinvariant ~i.e., independent of the metric!. Therefore, if theexistence and degeneracy of zero modes is related to sometopological features, as was speculated, e.g., in Ref. @3#, theChern-Simons action would be an obvious candidate.In addition, the assumption of quantized Chern-Simonsaction for gauge potentials with zero modes is further sup-ported by the results of Ref. @9#, where a whole class ofgauge potentials with an arbitrary number of zero modes wasconstructed. For all these gauge potentials, which are char-acterized by an arbitrary function and an integer l ~the num-ber of zero modes for a given gauge potential!, the Chern-Simons action indeed admits only the quantized values116p2E d3xAW BW 5 14 S l1 12 D2~2!where l is the number of zero modes for the given gauge©2003 The American Physical Society03-1BRIEF REPORTS PHYSICAL REVIEW D 67, 087703 ~2003!potential ~the Chern-Simons action for all the gauge poten-tials of Ref. @9# was calculated explicitly in Ref. @13#!.Remark: In Ref. @9# a fixed universal background gaugepotential was added to all the gauge potentials, AW →AW˜ 5AW1AW b, in order to relate the resulting gauge potentials AW˜ toHopf maps. For these resulting gauge potentials the Chern-Simons action is automatically quantized, (1/16p2)*d3xAW˜BW˜ 5(1/4)(l11)2, because the integer Hopf index is givenby the Chern-Simons action. However, even the originalzero-mode supporting gauge potentials, without the back-ground field, lead to the quantized Chern-Simons action ~2!,although they cannot be directly related to Hopf maps.Another, more general argument in favor of quantizedChern-Simons action for zero-mode supporting gauge fieldsis related to the anomaly equation in Minkowski space,]mJm5116p2emnabFmnFab . ~3!Here, Jm is the current density of a chiral ~left-handed! WeylFermion which couples to the Abelian gauge field Am withfield strength Fmn . Choosing the gauge A050 and integrat-ing the above equation over all space and over the finite timeinterval @ t i ,t f # results in the equationQ~ t f !2Q~ t i!5214p @ICS~ t f !2ICS~ t i!# ~4!Q~ t ![E d3xJ0~ t ,xW !, ICS~ t ![E d3x~AW BW !~ t ,xW ! ~5!where the left-hand side of Eq. ~4! is the change in particlenumber between t i and t f @13#. If we further assume that thechange in particle number, Q(t f)2Q(t i), is two times thenumber of levels ~zero modes! L crossed in the ~adiabatic!change from AW (t i ,xW ) to AW (t f ,xW )—as is usually assumed—then the equation2L5214p @ICS~ t f !2ICS~ t i!# ~6!results. Consequently, L must be independent of the pathAW (t ,xW ) which connects AW (t i ,xW ) and AW (t f ,xW ) ~see Ref. @13#and the literature cited there for details!. Obviously, Eq. ~6!is automatically satisfied when level crossing may occur onlyfor fixed, quantized values of the Chern-Simons action ICS .Therefore, Eq. ~6! is compatible with the assumption thatzero modes only exist for certain quantized values of theChern-Simons action. The correctness of this assumptionwould, in fact, be the simplest way to realize Eq. ~6!.All in all, the existing results on zero modes provide someevidence for the assumption that the Chern-Simons action isquantized for gauge potentials with zero modes. Therefore, afurther investigation of this question is of some interest. To-pologically the Chern-Simons action corresponds to a coho-mology class in H3(S3;RÕZ) and this is not constrained to bediscrete. In the remainder of the paper we want to demon-08770strate that the above assumption is incorrect in the generalcase by constructing a simple counterexample.For this purpose, let us briefly review some results fromRef. @5#. There it was shown that the ansatz (r[uxuW )C5g~r !expS i f ~r ! xWrsW D S 10 D ~7!for the spinor leads to a zero mode for the gauge fieldAi5h~r !C†s iCC†C~8!provided that g(r) and h(r) are given in terms of the inde-pendent function f (r) as (8[d/dr)g8522rt211t2g . ~9!h5~11t2!21S t81 2rt D ~10!wheret~r !“tan f ~r !. ~11!Here, a sufficient condition on t(r) leading to smooth, non-singular and L2 spinors and smooth, nonsingular gauge po-tentials with finite energy @*(BW )2# and finite Chern-Simonsaction (*AW BW ) ist~0 !50, t~r !;c1r1O~r2! for r→0, ~12!t~‘!5‘ . ~13!Further, the Chern–Simons action for this class of gaugefields may be expressed likeE d3xAW BW 54p~2p24 !E0‘drrth211t214p~22p18 !E0‘drr2t8h211t254pE0‘drr~r f 81sin 2 f !2@~22p18 !r f 81~p22 !sin 2 f # ~14!where spherical coordinates (r ,u ,w) were introduced and theangular integrations have already been performed.Next, let us introduce the one-parameter family of func-tionsta~r !5r~a1r2! ~15!where a is an arbitrary real number. ta obeys the integrabilityconditions ~12! and ~13! for all values of a, therefore ta3-2BRIEF REPORTS PHYSICAL REVIEW D 67, 087703 ~2003!defines a zero-mode supporting gauge potential and a well-behaving zero mode for all values of a. We easily computeta85a13r2, ha53a15r211r2~a1r2!2andE d3xAW BW 516pE0‘drr2@a1~52p!r2#3~3a15r2!2@11r2~a1r2!2#3. ~16!This last integral may be easily evaluated numerically withthe help of Mathematica. Before presenting the result of thenumerical integration, we want to remark that, in the limita→1‘ , ta is equal to the simplest case t(r)5r up to an~infinite! rescaling, which does not change the value of theChern-Simons action. Further, the simplest gauge potentialwith t5r is, at the same time, the simplest gauge potential ofthe class of gauge potentials constructed in Ref. @9#, with onezero mode ~i.e., l51). The resulting Chern-Simons actionmay be evaluated with the help of Eq. ~2!, and leads toE d3xAW BW 516p2 14 S 32 D259p2. ~17!Therefore, the Chern-Simons action ~16! normalized accord-ing to (9p2)21*d3xAW BW should approach the value one inthe limit of large positive a.08770In Fig. 1 we plot the Chern-Simons action ~16! multipliedby (9p2)21, as a function of a. For a→1‘ it indeed ap-proaches the value 1. Further, it increases for decreasing val-ues of a and reaches arbitrarily large values for sufficientlynegative values of a. Obviously, the Chern-Simons action inFig. 1 may admit all values greater than 1. This result dem-onstrates that the Chern-Simons action for gauge potentialswith zero modes is, in general, not quantized, which was thepurpose of this brief report.The authors thank R. Jackiw for helpful comments. Fur-ther, C.A. acknowledges support from the Austrian STARTaward project FWF-Y-137-TEC of N.J. Mauser.FIG. 1. The appropriately normalized Chern-Simons action(1/9p2)*AW BW as a function of the parameter a.@1# J. Fro¨hlich, E. Lieb, and M. Loss, Commun. Math. Phys. 104,251 ~1986!.@2# M.P. Fry, Phys. Rev. D 54, 6444 ~1996!.@3# M.P. Fry, Phys. Rev. D 55, 968 ~1997!; 56, 6714~E! ~1997!.@4# M. Loss and H.-T. Yau, Commun. Math. Phys. 104, 283~1986!.@5# C. Adam, B. Muratori, and C. Nash, Phys. Rev. D 60, 125001~1999!.@6# D.M. Elton, J. Phys. A 33, 7297 ~2000!.@7# C. Adam, B. Muratori, and C. Nash, Mod. Phys. Lett. A 15,1577 ~2000!.@8# C. Adam, B. Muratori, and C. Nash, Phys. Lett. B 485, 314~2000!.@9# C. Adam, B. Muratori, and C. Nash, Phys. Rev. D 62, 085026~2000!.@10# L. Erdos and J.P. Solovej, Rev. Math. Phys. 13, 1247 ~2001!.@11# A.A. Balinsky and W.D. Evans, J. Funct. Anal. 179, 120~2001!.@12# D.M. Elton, Commun. Math. Phys. 229, 121 ~2002!.@13# C. Adam, B. Muratori, and C. Nash, Phys. Rev. D 62, 105027~2000!.3-3

Chern-Simons action for zero-mode supporting gauge fields in three
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Chern-Simons action for zero-mode supporting gauge fields in three dimensions

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