PLE:"ARY TALKS REVISTA MEXICANA DE FÍSICA 49 SUVLEME~"TO 2. 52-57

Towards nonassociativc gaugc thcorics

SEJYllE~BRE 2(()3

A.I. NesterovDepartamento de Ftsica, Centro Universitario de Ciencias /;,'xacla.\' e ingenier(a, Universidad de Guada/ajara

Guada/ajara, jalisco, M¿xicoe-maii: nesterov@udgserv.cencar.udg.mx

Recibido el 30 de mayo de 2002; aceptado el 16 de agosto de 2{)()1

A nonassociative gcncralil.ation of lhe principal fibrc hundlcs thcory with a sm(xllh loop mapping on lhe fibcr is presentcd. Our approach

allows to canstruct a ncw kind of gauge theones which involvc Ihe "nonassociativc syrnmclries".

Keywords: Smooth loops; quasigroups; gaugc lhcorics; fibrc bundlcs

Se presenta una gcncmlil..a.ción no asociativa de la lcona de haces fibrados principales con un mapco de law suave en la fibra. Nuestroenfoque permite construir una nueva clase de teorías de norma que involucrnn "simetrías no asociativas".

DescriplOres: Lazos suaves; cuasigrupos; teorías de nonna; haces fibrados

I'ACS: t4.80Ilv; 03.65.-w; 03.50.De; 05.30.l'r; t LI5.-e¡

1. Introduction

The concept of fibre bundles is one of the importnnt applica-tions of the theory of traosfonnation groups, Lie group in asmooth case. It is applied not only to various fields of mathc-matics, such as: differential geometry; differential topology;charactcristic cla"es, etc.; but to physics (gauge field the-ory in particular) as wcll. The theory of fibre hundles andLie group have becn cxtraordinarily successful in the deve-lopment of modero physics, in particular, in the descriptionof the strong. wcak. and eleclfomagnetic interactions. Howe-ver, current difficulties to unify all interaetions prompt to lookfoc a malhematical s1ructure bcyond groups. For instance, lhequantum group approach is a cons1ruction of lhis type wherethe Lie group symmetry is replaced by a quantum group sym-metry and the lalter reduced to the standard group symmetryin the sorne limil.

Another possibility is a nona<isociative gcneralization ofthe Lie group, which bcgin to play an important role inphysics. For instance. nona<isociative objects such a<ilhree-eocycles are linked with the ehiral anomalies in the fieldtheory, whieh appear as Sehwinger tenns io current alge-bra 11,2]. These terms have a eohomologieal nature and re-lated to the failure of the Jacobi ideotity, .l(A, B, C) l' 0,where for given threc operalors A, E, e we define

.l(A,B,C) = [[A,BJ,C) + [[B,C),AJ + [[C,AJ,BJ,

and J(A, B, C) = Owhenever lhe Jacobi idctltilY is preser-ved. The failure of the Jacobi identity yields. io general, vio-lation of al)sociativily in the corresponding group that forceslo make use of qua'iigroups and loops instead of groups.

It has beco fouod that eleetric fields in chiral gauge the-ories obcy the so-called Mareev algebra 13-71. satisfying

fourth-order Marcev identity

.l(E., Ej,[E •• E¡J) = [.J(E.,Ej,E.),E¡J,

instead of the Jacobi ideotity. Mareev algebra is a tnngentalgebra of the analytie Moufaog loop, and there exists a rcla-tioo betweeo local analytie Moufang loops and real Mal'cevalgebras 181. It generalizcs the elassical correspondeoee bct-ween local Lie groups and Lie algebras as follows: for thegiven tangent Mal'cev algebra there exiS1S an unique sim-ply coooccted analytical Moufang loop 19]. Qoe well knownexample of the aoalytical Moufang I(XlP associated with theoctonions is the sphere 57 and othcrs are lhe spaces 53 x R1

and 57 x R7.Breakdown of the Jacohi identity also appears in the

quark model (ID] and in the conlext of quantum mechanics ofa charge particle in a field of magnetic monopole 13, 11-1.11:

.l(v', v2, v3

) ~ 'V . B l' °where Vi are lhe components of the gauge.invariant veln-city operator, and B is a magnetic ficld of the lIlonopole.More recently the qua,igroups ami loops have becn em-ployed in general rclativity aod for thc description of theThnmas prccession. cohercnt slates and geometric phasesas we1l18, 9,14-22].

In this paper we present an introduction to the nonasso.ciative principal bundles. we say principal loop bundles, andrelevaot gauge theories. The principal pcculiarity of the no-nassicative gauge lheory is that a gauge group in the convcn-tional sense does not exist. It is replaced by a gauge loop andthe Lic algebra is replaced by a qua'iialgebra with slmclu-re funclions instcad of stmcture constant~of lhe Lie algebra.The definilion of principal loop fibre bundles is givcn in lennsof transfonnation loops ami in a smooth ca'ie it may considcrthe theory of loop bundles as an application of the theory ofslIlooth loops 118,23-25].

TOWARDS ~ONASSOCIA'"IlVE GAl1CiE TIIEORIES 53

where V is a lefl fundamental veclor field.

V, E T,(Q), Va E Ta(Q),

w(Va) = V" V, = L;;.' Va,

(5)(6)

(L,w) Va = I(O,a).w(Va),

(R,w) Va = Ad¡;l(a)w(Va).

"J1¡eorem1 123,24). The eanonical Ad-form w is a Idlfundamenlal form amI il is transformed under lefl (right) tran-slations as

Dejinition 2. A veclor-valued one-form w is said lo beeanonieal Ad-form if il is defincd Va E Q lhrough !he rela-tionGeneral resullS on lhe lhe()ry of qusigroups and loops may be

found in Refs. 25-30 and a good hislorieal review in Refs. 25and 31 (seo also referenees lherein).

Lel (Q,.) be a groupoid. i.e. a sel wi!h a binary opera-tion (a, b) >-+ a . b. A groupoid (Q,.) is ealled a quasigroupif eaeh of lhe equalions a.x = b, and y.a = b ha~ uniquesolulions: x = a\b. y = bla. A loop is a quasigroup wilha two-sided identity a.e :::;:e.a :::;:a, where e is a neutralelemenl. A loop lhal is also a differential manifold and anoperation a.b is a smoo!h map is eaJlcd a smooth loop. \Vedefine

2. Smooth loops: basic notations and examples

- -1l(a.O)= La.o o I(a,b)o La.O (1) 2.1. Examples

e• R" e. R" e'> R"ij,n k + jk,n i + ki,n j

+ e¡;Cft + e}.e¡; + ete;; = o. (3)

where Lo. : T,(Q) >-+ To(Q) denotes lhe differential of!helefllranslalion.

(9)1-(¡¡

I«,")E = 1 _ '1 "(E

and can be wrinen a!so as 1«",) = exp(ia), where a =2 arg(l - (¡¡).

Lel us eonsider a sel of lhe unilary malriees QSU(2)C SU(2).wbere an arbitrary cJemenl of lbe loop QSU(2) ha~ lbe fonn

U = 1 (1 '1) (10)" JI + 1'112 -;¡ 1 .

For arbilrary elemeOls of QSU(2) we define lbe nonasso-ciative binary opcrations as follows

2.1.2. i.t)OP Q C

2.1.3. l.DopQSU(2) 114)

x • y = X + Y + ¡(x) + ¡(y) - ¡(X + y), (7)

wbere ¡(X) = (1 - COS21tx)/4 is defincd on R/Z. Then(R/Z,') is an analytieal loop 132, 33J.

2.1.1. I.DOp (R/Z,')

Lel

Let e be a complex plane. The nona'isociative multiplicatiotl(*) is delincd by

(+'1(*'1=---, (,'lEC, (8)

1 - ('1whcrc a bar denotes complex conjugation. Thc a'isociator1«,") is given by

(2)

In view of noneommulativily of lbe rigbl and lefl transla-lions !here exiSl~ a problem in !he delinition of lbe 'adjoint'map. \Ve introduce a generalil.cd adjoiOl map of Q on il~elfas !he foJlows: A map

wherc La is a lefttranslation, Rb is a righttranslation, l(a,b)

is a left associator, T(b,c) is a right associator and i(a,b) is anadjoint associator.

Lel T,(Q) be !he langeOl spaee of Q atlhe neulral ele-meOl e. Then for eaeh X, E T,(Q). we eonslruel a smoo!hveclOr field on Q

where e~(a) are lhe strueture Junetions salisfying lhe modi-fied Jaeobi identily

Lel r¡ = R{DIDa;, i = 1,2, ... , r be a ba~is of lhe leflfundamenlal veelor lields. Then we have

Defmition l. A veclOr field X on Q whieh satisfies !he re-lation La.Xo = i(a,o).Xa.O for any a,b E Q is ealled a leJt

fundamental or left quasi-invariant vector field.

Ado(a) = L;;' o R¡;I o La.o: Q >-+ Q (4) (11 )

wbere on lhe righl-band side lbe matrices are multiplicd in anusual way amiis eaJlcd an Ad-map. The Ad-map leaves invarianllhe neutral

elemenl e and generales lhe map T,(Q) >-+ T,(Q):

(e¡~

A(r¡, () = O e~i~) , '{! = 2 arg (1 - ;¡().

Rev. Mex.. Ffs. 49 S2 (2(X)3) 52-57

54 1\.1. NESTEROV

2.1.5. Loop QflR [81

2.1.4. Loop Q//(2)

The qUalemionie eonjugalion (denoled by +) is defined by

Lel M be a manifold and (Q,., e) a smoo!h lwo-sided loop.A prineipalloop Q-bundle is a triple (P,", M), where Pis amanifolds and !he following eondilions hold:

O) q ael, freely on P by the right map: (p, a) E P x Q >->RaP=pa E P.

(2) M is !he quotienl spaee of P by !he equivalenee re-lation indueed by Q, M = P/ Q, and !he eanoniea!projection 1r: P -+ ),\1 is a smoolh map OOlO,

(3) Pis locally Irivial, that is, for eaeh x E M, Ihere exislsan open neighborh,xxl U and a diffeomorphism 4>:,,-1 (U) >-> U x Q, sueh Ihal for any poinl u E ".-1 (U)il has lhe form 4>(u) = (".(u), <p(u)), where <pis Ihemap from ".-1 (U) lo Q salisfying <p(RaP) = Ra<p(P).

3. Principal loop bundles and nonassociativegauge theories

Wc say P is a total space oc bundle space. Q is a strUClu-

re loop, M is a base oc base space and 1r Ís a projeclion. Forany x E M Ihe inverse image ".-1 (x) is lhe Fz jiber over x.Any fibcr is diffeomorphie lo Q and on eaeh fiber we have

Rb o RaP = Ra.bi'(a,b)P,

where i'(a,b) satisfies <p(i'(a,b)P) = r(a,b}<P(P)'Let us eonsider a eovering {Uo}oEJ of M with

UoEJ Uo = M, whieh can be ehosen in sueh a way !halthe rcstriction of the fibralion lo cach opcn sel Ua is lrivia/i-zable . This implies, lhal lhere exisls a diffeomorphism 4>0:".-1 (Uo) >-> Uo x Q. The seIS {Uo,4>o} areealleda familyof local trivializations.

Let {Uo, 4>o}oEJ he a lrivializing eover, i.e. a farnily ofloeai trivializations with lhe eovering {Uo}: UoEJ Uo = M.Then lhe farnily of maps {q¡¡o(p) = R;;.lq¡¡: ".-1 (Uo nU¡¡) >-> Q}, where qo : = <Po(p), q¡¡ : = <P¡¡(p), P E".-1 (Uo n U¡¡), is ealled lhe family of Iransi/ion fimc-lions of the bundle P(M, Q) eorresponding lo Ihe eovering(Uo,4>o}oEJ'

One can find lhal transition funelions ehange under lheright translations al)

(12)

(14)

( 13)

q+ = a - {3i - 'Yj - ók,

Lel us eonsider !he qualemionie algebra over eompIex field<C(l,i)

LlXlpQH(2) is a"ociated with lhe group 5UO,I) and ilS ae-lioo on Ihe Iwo-dimensional unit hyperboloid H'. Let D e <Cbelheopenunildisk,D = {( E <C: 1(1 < 1}. Wedefine!henon'lsociative binary operalion * as follows [14]

wilh !he mullipliealion operation defioed by lhe property ofbilinearily and following rules for i, j, k:

and can be wriuen also as 1«",) = exp(io:),o: = 2arg(1 +(1)). lnside D !he sel of eomplex numbers wilh Ihe opera-lion (*) fonns Iwo-sided loop QH(2), whieh is isomorphielo lhe geodesie ioop of Iwo-dimensional Lobaehevsky spaeerealized as lhe upper parl of lwo-sheeled unil hyperboloid.The isomorphism is established by ( = e'''' tanh(O /2), whe-re (O, <p) are inner eoordinales on H2

i' =j' =k' = -1, jk= -kj =i,

ki = -ik = j, ij = -ji = k.

He = {o:+ {3i +'Yj +ók I O:,{3,'Y,óEq

The 'lssocialOr 1«(,") on QH(2) is delermined by

for q = a + {3i+ 'Yj + ók.We reslriel ourselves lO !he sel of qualemions HR:

HR = {( = (0 + i((li + ('j + (3k) :

i' = -1, iE <C,(0,(\(',(3 E IR}

with !he norm 11(11' given by

IlIlroducing a binary opcration

( 15)

and obey !he so-ealled cocyc!e eondilion:

q¡¡o(x,p)'qo(p) = q¡¡,(x,p)' [q,o(x,p)'qo(P)]

vXEuanu¡¡nU,andVpE".-I(z), (17)

Dejinilion 3 Lel {Uo, 4>0} be !he iocal trivialization ande E Q a neutral clement. then Ua = ep~l (x, e), x E Ua iscalled the local section associatcd wit.h this triviali7.ation.

(16)

whcrc K is constant and / denotes lhe right division wc ob-rain a loop QHR.

Ir the point x lics in the inlcrsection of the neighborhoodsUo allll U¡¡, x E Uo n U¡¡,lhen lhe following formula holds:

( 18)

Rev. Mex. Frs. 4982 (2003) 52-57

TOWARDS ~ONASSOCIATIVEGAUC:ili TIIEORIES 55

(22)

Assume lhat tI.ere are two families of local sections. Uo'

and u~.These sections are linkcd by lhe right translations.a~ = Rq", aa. Using lhe sel {u~} one can introduce lhe newtmnsition functions q:tJ3' Then lhe following lransfonnationlaw holds:

3.1. I'rincipal QU(I) hundle uver S2

Lel us defiue a smoolh loop QU(l) as a loop of multipliealiouby unimodular complex numhers

ein * ei13 = ei(a+I3), O::; Q < 211", (19)

where a+,13 = a + ,13+ F(a, ,13),amI F(a,,I3) is a smo-olh fuueliou such lha! F(a,,B) = F(,I3,a), F(a,O)F(O,,I3) = ° aud F(a,,I3) + F(a+,I3,,) t- F(,I3,,) +F(a,,I3+t).

We eouslruellhe hundle hy l.1king

Base M = S2 with coordinates ° ~O < "-,° ~ c.p < 2,,-;

FiberQU(l) = S' with coordinates eÚ'.

We hre.1k S2 inlo lwo hemispheres H" wilh H+ n H_being a lhin slrip parametrized by lhe cquatorial angle c.p. Lo-eally the bundle looks like

H_ x QU(l) with coordinates (O, c.p, e;a,),

H+ x QU(l) with coordinates (O,cp,e'a+).

In ,,--1 (H _ nH+) lhe elemenls e'a+ and e'a, must be rela-led by

tangent vector X: 11".X = X. where X is lhe tangent vectorlo ,(t). A sel (X l is called a horizontal sub'pace Hu.

The main idea of connection is to compare the poims inlhe "neighboring" fibers in a way lhal is nol depcndent ona local trivialization. The connection allows liS 10 dccompo-se any veclor Z E Tp(P) in lhe form Z = X + Y, whereX = hor Z E Hp is lhe horizon~11 component of lhe veelorZ and Y = ver Z E Vp is lhe vertical one. The mal' c.p induces(he map of the vertical subspace Vp onto the tangent space 10

Q, Vp ~ 7~(Q), q = c.p(p):

1Ip = ddñU(t)Pi ~ 1Iq = ddRa(t)ql .t t=o t t=O

The veclor lield V is lhe left quasi-invarianl veelor field. In-deed, il can he wriuen as Vq = (Lq), V" where

(23)

Lel V, E T,(Q). The veclor field 11 eonneeling wilh V,by means of (22) ami (23) is caUed aJ¡mdamen/al veclor field.

lJefiniliofl 4 A connectionform 011 a principal Q-bundle is aveclor-valoed I-fonu laking values al T,(Q), whieh salisfies:

(i) w(Xp) = X" where Xp E Vp, X, E T,(Q) are de-lermined aceording lo (22) and (23).

(ji) (ñ~w)Xp = Ad;;-I (q)w(Xp), where q = c.p(P).

(iii) The horizontal suhspace 1ip is defined as a kemel of w:

This gives

a+ = a_ +, +Fh,a_).

Taking ioto accollnt thatthe reslllting stnlclure must he a ma-nifold we find lhal

, + Fh,Q-) = nc.p, n E Z. (20)

3.2. Connl'Ctioll. cun'aturc and Uianchi idcntitics

Lel P(M,Q) be lhe principal loop Q-bundle over lhe mani-fold M. For any u E P a langenl spaee al u we denole asTu(P) and !he langenl spaee lo lhe fiber passiug lhrough uas Vu• We eaH Vu a vertical subspace. It is generaled by lherighl transla!ions on lhe fiber: u >-+ R.u, a E Q, u E P:

Wilh tlle given connection fonn a local l-fonn taking va-lues in T,(Q) can be assoeialed. Lel a: U e M >-+ a(U) eP, ,,-0 a = id be a local see!ion of a Q-bundle Q >-+ P >-+ Mwhieh is equipped wilh a conneetion l-form w. Define lhelocal a-represen/atil'e of w lo be lhe veclor-valued I-form(taking values al T,(Q) wU on lhe opcn sel U e M givenbywU := a'w.

Lel us eoosider a principal loop Q-bundle (P,"-, M) overM ami fix a lrivializling covering {Ua,<I>alaEJ. Lelld :Ua >-+ Ua x Q hy x >-+ (x, e). A lrivia!izalion <l>adefinesa caTloniral section u O' by the equation

-1 -Ua = <PO' o Id,

and vice versa. We denote by Wc a canollieal Ad-form.

'l1/eore", 2 114,23). Lel {Un, <l>alaEJ be a family of localtrivializations for P with UnEJ Ua = M. then at Va n UJ3

Deflllilion 5. Let Wo = a~w. wherc w is the connectionfonn. The form Wu 011 Ua is callcd the conneetionform in(he local tridalization {VOl <pa}.

(21)Xu E Vu, a E Q, u E P.d - IXu = d-Ra(t)1.L 1

t t=O

For ,(t) E M heing a smoolh curve a horizon/allij/ of, isa curve ;¡(t) E P sueh lhal,,-(;¡(t)) = ,(t). Evidenlly, fordetennining .:y(t) it is sufHcient to define al any point of it a

Rev. Mex. Fís. 49 S2 (2(X)3) 52-57

56 A.l. NESTEROV

~t3.Cm!arian! dcrivativc. Curvaturc form

whcrc w is a conncction form, is caBed a C!lrvature formo

Definition 8. A veclor-valued 2-fonn rl(X, Y) defined as

(33)

From (30) one gets lhe Bianehi identity

Drl=O.

Choosing a nalUral coordinale syslem (x, q) on P, wherex E M and q E Q, we wrile lhe Lagrangian C. of lhe freegauge field as

C. = -~(F.v, F.V), (34)

where ( , ) denotes Ad-invarianl sca!ar praducl on Q.Physieal fields are considered as funetions on P in lhe fo-

llowing way. With w(x), x E M we relale a funetion q;(x, q)(lift of lhe function w(:;» on the principal Q-bundle as fo-llows: we assume lhal w(x, q) Iransfonns under a finite gau-ge transfonnalions U (q) of lhe some definile (nonassociative)representalion of lhe loop Q [18,14], according to lhe inverserule,

¡¡)(x,q'. q) = U(q-l)¡¡)(X,q'), q E Q.

Then lhe lift q;(x, q) of lhe funetion w(x, q) is defined as

q;(x,q) = U(q-I)q;(x,e), (35)

where q;(x,e) := w(x) and e E Q is a neulral element.The covariant derivative i)q; = D. q;dx. is given by

i)q; = dq; + wq; = (8. + A.(x,q») q;(x, q)dx. (36)

- -1where we set A. = Adq (e)A.(x).The nonassociative gauge invariant Lagrangian of malter

has lhe following strueture

Cm = C(q;,i).q;,x)

and describes a malter field w minimally coupling with thegauge field A•.

3.4. Nonassocialive gauge lhcories on principal loop hun-die

Let us compule n.v = 2rl(D., Dv). The eompulalionglvcs

n.v = -w([D.,Dv]) = F;vw(L;) = F;v(Ad;'(e»)~tj,

where {tj = w(Lj)} is lhe basis of lhe left quasi-invariantveclor fields al T,(Q), q = <p(P), pE P. Choosing lhe fa-mily oC the local scctions a ex associated with the trivialii".ationVa' <flo and taking ioto aceDunt, that <Po (uaJ = e whcrc <Pois rcstriction of <Poon 1r-1 (Va). wc obtain

1I"rll" = ~F.vdx. 1\ dxv. (32)

Sinee rl is a Ad-fonn, the following transformation law holds

(29)rl(X, Y) = Dw(X, Y),

Lel {x. ,y;} be a local eoordinale system in the neighborho-od 11'-' (U,,): x. being eoordinales in U" E M and y; coor-dinales in the fiber. Loeally 11'-1 (U,,) can be presented as adireel produetion U" x Q. The eonneclion fonn can be writ-len as w = wi Li, {L;} being lhe basis of left fundamentalfields and {wi} basis of l-fonns. In lhe eoordinates {x., yi}the eonneetion fonn reads

wi = (Adii'(e»)~A~(x)dx. +wij(y)dyj, (25)

where q = <p(y), A~(x)dx. = 1I"(w) and wij(y)dyj =q.we.

Theorem 3 [14, 23J. The eurvature fonn rl satisfies lhe strue-ture cquation

Defmilion 6. A cOI'arianl derivalive D. in lhe principal Q-bundle is defincd as follows:

D. = 8. - A~(x)Li,

where Li = (R.)18/8yj are generators of lhe left Iransla-t¡oos.

The eomputation of lhe eommutalor [D., Dv] yields

[D.,Dv] = (8vA~ - 8.A~)L; + A~At[Li,Lj]and inlradueing [Li, Lj] = Cf;(y)Lp we gel

[D., Dv] = - F;vL;, (26)

F;v :=8.A~ - 8vA~ - A~A~Cjp' (27)

Defmilion 7. Lel w be a veetor-va!ued r-focm inlhe principalQ-bundle. A (r + I)-fonn Dw defincd by

Dw(X1, X2, ••• , Xr+d = dw(horX" ... , horXr+.J (28)

is eallcd a covariant difJerential of lhe fonn W.

local eonneelion fonns w" and w{3, eorresponding to lhe sa-me conneclion W 00 P, are rclatcd by

w{3= Ad;.', (q{3")w,, + l(qp.,q.,).£J,,{3, (24)

whcrc qa¡3 are the transition functions, and Oal3 = q~{3WC

denOles lhe pullbaek on U" n U{3of the eanoniea! ¡-fonnWc on Q. Vice versa, for any set of lhe loca! fonns {w,,}(u E J), satisfying (24), lhere exists the unique eonnectionfonn w on P generating lhis family of the loca! fonns, na-mcly. Wa ::;::a~w, Va: E .J.

rl=dw+wl\w (30)

and is transfonncd undcr the right translations a"l

(ñ~rl)(X, Y) = Ad;'(q)rllp(X, Y),

where q = <p(P).

Acknowledgments

(31) I am gratefullo L.V. Sabinin for helpful discussions and com-mcnts.

Rev. Mex. Fls. 49 S2 (2003) 52-57

TOWARDS NONASSOCIATIVE GAUGE TBEORIES 57

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