Chia-Hsiung Tze- Linking the Gauss-Bonnet-Chern Theorem. Essential Hopf Maps and Membrane Solitons with Exotic Spin and Statistics

8/3/2019 Chia-Hsiung Tze- Linking the Gauss-Bonnet-Chern Theorem. Essential Hopf Maps and Membrane Solitons with Exoti

1/22

V P I - I H E P - 8 9 / 4LINKING TH E GAUSS-BONNET-CHERN THE OREM .

ESSENTIAL HOPF MAPS AND MEMBRANE SOLITONSWITH EXOTIC SPIN AND STATISTICS

C O N F - 8 9 0 7 1 4 0 - - 2Chia -HsiungTze** D E 9 0 0 0 1 4 5 1

Institute of High Energy PhysicsDepartment of Physics, Virginia Tech.

Blacksburg VA 24061

ABSTRACTBy way of the Gauss-Bonnet-Chem theorem , we present a higher dimensionalextension of Polyakov's regularization of W ilson loop s of point solitons. Spacetim e pathsof extended objects become hyper-ribbons with self-l inking, twist ing and writhingnum bers. Specifically w e dis cus s the exotic spin and statistical phase entang lem ents ofgeometric n-me mb rane soli tons of D-dimen sional K P j a-m odels with an added Hopf-Chern -Simon s term where ( n, D, K) = ( 0, 3, C ) , (2, 7, H) , ( 6, 15, Q ) . They areuniquely l inked to the com plex (C) and quaternion (H) and octon ion (Q ) d ivisionalgebras.

Tw o overlapping research areas have recently attracted much attention. They are thetopological quantum field theories (TQFT) in D > 3 spacetime dimensions [ 1 , 2 ] , inspiredby the wo rks of Donaldson and Jon es [3], and theories of particles bearing any spin andstatistics or anyons [4] .fuelled by the quest for a .nechanism underlying the fractionalquantum Hall effect or high temperature superconductivity.

By relating quantum D=2 conformal and D=3 Chern-Simons field theories, Witten[5] showed a correspondence between the expectation values of the W ilson loops traced bycolored' point sources in spa cetim e and Jone's polyno mia ls for kn ots. Notably , to soobtain the fundamental Skein relations, he had to regularize or frame the W ilson loops, inaddition to doing the standard regularization. It is here that an interesting o verlap occu rswith the theory of anyons. In fact, such a regulariza tion had been dis co ve red by Polyakov

* Talk given at the XVUJ1" International Conference on Differential Geometric Methods inTheoretical Physics: Physics and Geometry , U.C . Dav is, July 2-July 8, 1989.** Work supported by the U.S. DOE under Grant NO.DE-ASO5-8OER1O713.


2/22

.til in his proof of the D =3 fermi-bose transmutation of baby Skyrmions in their pomt-like'.unit. Our talk, takes off at this intersection, a meeting point of topology, geometry andphysics. It is based on work do ne w uh Soonkeon N am . 17] . We defer to this longerarticle tor greater details and a comprehensive list of references as time and spacelimitations only permit a quick review of some of the highligh ts here.

We stan from the well-studied C P, a- model with a Che m-S imo ns term [6]

A = d 3 x| | D^Z | 2 + - 3 - e ^ Ap. F v X + A ^ H . (1 )j . SK 2 i0 < 6 < 7T. The two com pone nt com ple x spinor field Z = [Z \, Z T ) with I Z T = 1 l ives

on S J . Th e unit norm ed field n is given by the Hopf projection m ap taki ng Z e S " to n= Z ^o Z e S~ . D^ is the co va na m den vaa ve w ith i ts holonomic U (l ) gauge f ieldA u = i Z ' 3 u Z , F ^ being the field strenght. This mo del admits exa ct S^ - soiitons.While the third term is the Aharonov-Bohm term, the second term also reads as

= ^ - j d 3 x A^ (for 8 = n ) i.e. as an interaction between the field A^ and theconserved topological current Ju = r^-e ^v xE abd i^i^ d n c . Th e latter is so normalized thatQ = Jd "x JQ ,Lhe soliton (electric) charge is integral and labels the ele m en ts of ^ ( S 2 ) = Z.The field boundary condition is such that spacctimc is R.3 u ( ) = S ; the Ch ern-Sim onsacnon is the Hopf invariant for the maps n : S^CPj = S^^ classified by the generators ofK^ (S^) = Z, the additive group of the integers.

Wilczek and Zee [4] show ed that interchanging two Q=l- soi iton s or equivalentlyrotating one of them around the other by 2K gives raise to a statistical, alias spin phase

10 afactor e to the wave function. H enc e the soliton is an anvon with ex oti c spin spin s = 2Kand intermediate statistics Th is pha se originates from th e gauge intera ction s among anyonsand corresponds to a mapping with Hopf invariant 1 . The latter feature is a key ingredientin our higher dimensional generalizations of the 9-spin and statistics connection.


3/22

L6] in his proof of the D =3 fenm -bosc transmutation of baby Skyrmions in their point-likelimit. Our talk takes o;T at this intersection, a meeting point of topolog y, geometry andphysics. It is based on work done with Soonkeon Nam.17] . We defe r to this longerarticle for greater details and a comprehensive list of references as time and spacelimitations only permit a quick review of some of the highlights here.

We stan from the well-studied CP j a- model with a Chern-Simons term [6]

IJ 2 + - 9 - e ^ A ^ F v X + A ^ l . (1)8rc2 J0 < 9 < it. The two component complex spinor field Z = (Z \, Zo) with I Z 1*"= 1 lives

on S J . The unit normed field n is given by the Hopf projection map taking Z e S J to n= Z + a Z e S . D^ is the covarian t derivative w ith its holonomic U( 1) gauge fieldA^= i Z ^ Z , F ^ v being the field strcnght. This model admits exact S 2 - solitons.While the third term is the Aharonov-Bohm term, the second term also reads asS H = 1 d 3x A^W (for 0 = n ) i.e. as an interaction between the field A ^ and the

onserved topological current Jn = ^ -e ^vx 6*** 1 1^"^ c The latter is so normalized thatQ = Id x J o ,the soliton (electric) charge is integral and labels die elements of ^ ( S ) = Z.The field boundary condition is such that spacetime is R^ u () ~ S ; the Chem -Simonsaction is the Hopf invariant for ihe maps n : S^-CPi S classified by the generators of713 (S2) = Z, the additive group of the integers.

Wilczek and Zee [4] showed that interchanging two Q=l -so lito ns or equivalentlyrotating one of them around the other by 2rc gives raise to a statistical, alias spin phase

i0 Qfactor e to ihe wave function. Hence die soliton is an anyon with exotic spin spin s = ^~and intermediate statistics Th is phase originates from the gauge interactions among anyonsand corresponds to a mapping w ith Hopf invariant 1 . Th e latter feature is a key ingredientin our higher dimensional generalizations of the 9-spin and statistics connection.


4/22

Polyakov opted for a tractable W ilson loop approach to the large distance behavior ofsolnon Green functions of system (1). To study the effects induced by the long rangeChem-Simons interactions, he approximated the parudon function Z by

ill closed paLhsZ = e-m L(P) (2)(P) p

P denotes a Feynman path of a soliton seen as a curve in spacetime R , L(P) is the totalpath length .

The first exponential factor in (2) is the Schwinger action of a free relativisuc point-like soliton of mass m. The other factor O(P) = ( exp(i (&> A^ dx^ ) \ where the functional

\ J? Iaveraging < ...> is w.r.t. the Hopf action embodies the Aharonov-Bohm e ffe ct, typical oftopologically massive gauge theories . The Chem-Simons-Hopf action induces magneticflux on electric charges and vice versa. Being Gaussian (and for 9 = K ), this phase isexacdy calculable, hence the analytic appeal of the Polyakov approximation. By directintegration of the equation of mo tion [8] , (P) = 1 - expj i S 0 - i d 3xlN j i S 0 - i[ J (3)

SQ is the free point particle action and N is a suitable normalization. The conserved current

of a Q= l point source is now J^ x ) = I d x8 (x -y (x )) - . From (1) the key equationfor J,, reads

(4)


5/22

L'pon substitution in i3) with 6 = K it y ields d>(P) = exp i f d ' - - A2 J -AAU

So tor ihe.ibove %iven point current and. in the Lorcntz gauge d aA a=4), A u can be solved to givePolvakov s result :

l (P) ) (5)Gwnere

dv V . (6)ix-yp

:n the limit where the two smooth closed 3-space cu rves C a and Cp coincide, namelyC a - Cj3 = P. the soliton worldline.

Were C i and C 2 in R^ ( or S^ )disjoint curves , (6) would just be their Gausslinking coefficient. If Q ( M->) is the solid angle subtended by C j at the point Nfo of C2,then Stok es theorem yields I Q = -7 I d Q( M 2) , which measures the variation otthis solid angle divided 4n as M2 runs along C2 ; it is the algebraic number of loops of onecurve around the other.

There is however an indeterminacy in the analytic result (6 ). W hile its integrand isthat of Gauss' invariant, the integration is over one and the same curve. I G ( P ) . whichrepresents the net action of the Aharonov-Bohm and the Chern-Simons terms in (1) on acharged particle, is thus undetermined. Clearly this artifact of the geometric point-limitapproximation must be corrected for by a proper definition or regularization of IQ(P)-


6/22

Upon substitution in (3) with 9 = K it yields (P) = exp . So tor theabove given po int current and, in the Lorentz gauge d aA a=O, A^ can be solved to givePolvakov s result :

where(P) (5)

C 0) =

m the limit where the rwo smooth closed 3-space curv es C a and C$ coincide, namelyC a = Cjj = P, the soliton worldline.

Were C i and C 2 in R^ ( or S^ )disjoint curves , (6) would just be their Gausslinking coefficient. If Q (M 2) is the solid angle subtended by C i at the point M2 of C2,then Stoke s theorem yields I Q = ~j~ I d ^ M 2) , which measures the variation ofthis solid angle divided 4ic as M2 runs along C2 ; it is the algebraic number of loops of onecurve around the other.

There is however an indeterminacy in the analytic result (6 ). W hile its integrand isthat of Gauss' invariant, the integration is over one and the same curve. 1Q(P), whichrepresents the net action of the Aharonov-Bohm and the Chern-Simons terms in (1) on acharged particle, is thus undetermined. Clearly this artifaa of the geometric point-limitapproximation must be corrected for by a proper definition or regularization of I Q ( P ) .


7/22

Po ly ak ov 's c un ng pre scn ptio n is to trade the delta function in the alternative ^S i x -y t o f 4 ; I 1 Q for the Gaussian 6fU-yi=(27 - x-v

to be - T (P ), the total torsion or tw ist of the cu rve P in spacenm eT I S ) ds. s and n de no te the arc length and the

principal normal vector to P at the poiiu X(s). What then are the geometnc underpinnings ofthis regulanzation procedure ?

By substituring the Gaussian die dominant contribution of the surface integral comesfrom an infinitesimal strip I p ; Polyako v [61 effectiv ely turns a spacetime curve into aribbon . Indeed, in 1959, Calugareanu [9] discovered that IQ { C a ^ Ca) is in fact aperfectly well defined ennry , a new topolog ical invariant SL , die self-lin kin g number fora simple closed ribbon[9]. SL is in fact the linking number of C3 with a twin curve C amoved an infinitesimallv small distance e along the principle normal vector field to Co.Being disjoint these two curves can be linked and unlinked, like the strands of a circularsupercoiled DN A mo lecule [1 0] . Now adays in knot dieory this construction is called theframing of a curve Cn . M ost significant to our consid eration s is the existen ce of the"conservation law": SL = T + W whereby SL, is the algebraic sum of two differentialgeometric characteristics of a clo sed ribbon, its total torsion or twisting number T and itswnthing number or writhe W . W hile both T and W , wh ich are metrical properties ofthe ribbon and its "axis" respective ly, can take a continu um of v alues, their sum SL mustbe an integer. The writhing num ber W, better kno wn in topolo gy [11 ] as the Gaussintegral for the map fy : S x S > S*", is the elem ent so lid angle or the pullback volume

dH, of S 2 under 0. So Calugareanu's formula reads

(7)


8/22

I'sing the inva na nc e of W un de r dilatations and let (D be the m ap, et s,u ) (e - = 1), al F r en e t - S e r re t f r am e v ec t o r a t t a ch ed t o t h e cu rv e , we g et

rds du Eabc c*dsebduz- . a.b.c = i 1,2.3) and 8 S = 9/ds, d u = d/du. AJ oconformally in va na ni action for the frame field e, W is man ifestly a Wess -Zu min o-

Nov ikov-W itten term and read ily identified as a Berry phase upon exp one ntiati on. It isprecisely Polyakov's [6] double integral representation (modulo an integer) for the torsionT(P) . From our vantage point, this equivalence.whose importance will soon be clear, isjust the equality W = - T (mo d Z) . A mondane illustration of the relation W + T = SL fora nbbon is a coiled phone cord. W he n unstressed with its axis curling like a helix in space,us w nth e is large wh ile its twist is sm all. When stretch ed with its axis alm os t straight, itstwist is large while its wnthe is small.

By now many people [12] have verified that the regularized Hopf (statistical) phasefactor is given by the writh e , OCP) = exp(ijcW(P)} . Its alterna te form O (P ) = exp( - inT(P)) exp(+iit n) by w,*y of the relation W = - T (Mod Z) is the "spin " phase factoressential to Polya kov 's proo f that the 1-solitons in (1) are fermions by obeying a Diracequation in their point-like limit. Had 0 been kept arbitrary, we would have the moregeneral theory of poindike anyons carrying fractional spin and intermediate statistics [4]. Inother words the relation W = -T + SL is the mathematical expression of the connectionbetween statistics and spin in the geometric point soliton limit

In the geom etry of 2-surfac es, a form of the celebrated G aus s-B onn et theorem is K= 2xi ^ telling prop erty , uniq ue in the wh ole of differential g eo m etr y, is thefollowing : Like the formula S L = T + W , it relates entities defined solel y in terms oftopology such as the Euler characteristics x of a closed surface M and metrical entitiesdefined purely in terms of distances and angles such as total Gaussian curvature K for M.It is perhaps no t too surprising th at Fuller [ 10] showed that the relation SL = T + W is infact a conse quen ce of the Ga uss -B on net formula. So we see an interesting role that thelatter theorem plays in a fundamental physics principle - the relation between spin andstatistics!


9/22

= - L | dsJ o Jo

Using the invariance of W under dilatations and let be the map, e(s,u) (e^ = 1), alocal Frenet-Serrct frame vector attached to the curve, we get

due.be e 4 d 5 e ba u e c , a,b,c = d.2 ,3) and B s = d/ds, dn = d/du. AJoconformally inva riant action for the frame field e, W is manifesdy a Wess-Zumino-Novikov-Witten term and readily identified as a Berry phase upon expo nentiation. I; isprecisely Polyakov's [6] double in tegral represen tation (modulo an integer) for the torsionT(P) . From our vantage point, this equivalence,whose importance will soon be clear, isjust the equality W = - T (mod Z ) . A mondane illustration of the relation W + T = SL fora ribbon is a coiled phone cord. When unstressed with its axis curling like a helix in space,its wnihe is large while its twist i s sm all. When stretched with its axis alm ost straight, itstwist is large while its writhe is sm all.

By now many people [12] have verified that the regularized Hopf (statis tical) phasefactor is given by the writhe ,


10/22

Furthermore s ince White 111] obtained the h igher d imensional vers ion ofCalugareanu s formula from his formulat ion of Gau ss-Bon net-Che rn theorem forRiemannian manifolds [13], it is natural for us to extend Polyakov s analysis to the D>3counterparts of the Wilczek-Zee mo del (1). We first define the generalized Ga uss linkingnumber for manifolds.

Gen eralizing the procedure for linking 3-space curve s, consider [11] two c ontinuousmaps f(M) and g(N) from two smooth, oriented, non intersecting manifolds M and N,dim(M) = m a n d dim(N) = n , in to Rm+n+1. Let S m + n be a unit (m+n)-sphere centeredat the origin of Rm+n+1 and d l ^ be the pull-back volume form of S m + n under the map e

M x N s n +m where to each pair of points (m ,n) e M x N w e associate the unitvector e in Rm+n+ : e(m ,n) = . The degree of this map is then the generalizedig (n ) - f (m) |Gauss linking number of M and N, namely

L(f(M),g(N)) s L(M VN) = f dln+ m (8 )

^ (=27t(n+ 1>''2/r((n+l)/2)) is the volum e of S n . W e first note that the linkin g num ber foreven dimensional subrnanifolds M and N is zero due to the non-commutativity propertyL(M, N) = ( _i/n*-Kn-l)

Next w e specialize W hite's ma in theorem : Let f: M n R D= 2 n + l ^ ^ s m o o t hembedding of a closed oriented differentable manifold into Euclidean (2n+l) space. Let vbe an unit vector along the mea n curva ture vector of M n . If n is gdji ( i. e . D = 3, 7, 11,15etc. .) then

SL(f,f6) = f dO ,n + - (9 )


11/22

ii tne seif-linking numbe r of a hy pe r-n bb on mad e up of M n and ihe same manifoldreformed a small distance e along v . and w here the terms on the RH S of (9) arerespectively the writhing and tw isnn g num bers, W and T , of the hy per -nb bon . For neven ( D=l,5,9, .~), both W and T are zero and hence also SL =0 , the situation isuninteresting for the physics of nonintegrabie phases.

The un ivers ality of the formula SL = W -+ T m irro rs that of Gauss-B onne t-Cherntheorem. It is only natural to exp ect that for higher dim en sio na l solitons in suitable modelsWhite's formula T = -W (mod Z) similarly links their spin and statistical phases . Asapplied to physics, it couid define and relate the twisting and writhing of odd dimensionalclosed S J - , S \ S 7-. . . hype r-nbb ons, the world vo lum es of topological S^ -, S 4 -, S^-mem branes soiitons in D=7, 11, 15. . . dimensional spac etim e respectively. How to cutdown thii infinity of choices ? What are the natural D >3 a-model counterparts of (1)which may admit such solitons with exotic spin and statistics ?

One approach suggests itself. These models should have the three essent ialingredients of the C P( 1) model [4] : 1) Th e existence o f topolog ical me mb rane solitons; 2)the presence in the action of an Abelian C her n-S im ons te rm , the Hopf invaria nt; 3) theassociated Hopf mappings S^""1 S n include ones with Hopf invariant 1. The first twoconditions are embodied in the time component of the key equation (4). Its integration overill of space yields the topological charge-magnetic flux coupling which is the very basis ofthe statistics phenom enon in (2+1) dim ensions. As to the third condition, a crucial elementin the Wilc zek -Ze e proof of the fractional spin and statis tics for one soliton [ 4], we have thefollowing pro per ty of Hopf m app ings in higheT dim en sio ns. W hile for any n evgn therealways exists a m ap f: S211"1 S n with only even inte ge r Hopf invariant y{T), but as forHopf maps of invariant 1. the celebrated Adams' theorem [14] tells us : If there exists aHopf m ap f: S D >S of Hop f invariant y{f) = 1, indeed one with an y integer -y(f),then D must equal 1,3, 7 and 15 i .e. ra = (D +l)/2 = 1 , 2 , 4 and 8.

From it follow other relevant theorems[ 15 ]. T he y are1) R n admit the structure of a real division algebra if and only if n= 1, 2,4 and 8 . The


12/22

is the self-linking num ber of a hypcr-ribbon made up of M n and the same manifolddeformed a sma ll distance e along v , a nd where the terms on the RHS of (9) arcrespec tively the writhing and twisting numbers, W and T , of the hypcr-ribbon. For neven ( D=l,5,9,.. .), both W and T are zero and hence also SL =0 , the situation isuninteresting for the physics of nonintegrable phases.

The universality of the formula SL = W + T mirrors that of G auss-Bonnet-Cherntheorem. It is on ly natural to expect that for highe r dim ension al so litons in suitable m od elsWhite's formula T = -W (mod Z) similarly links their spin and statistical phases . Asapplied to phy sics, it cou ld define and relate the twisting and writhing of odd dimen sionalclosed S 3- , S 5-, S 7 -.. . hyper-ribbons, the wo rld volum es of topological S 2 -, S 4 - , S 6 -membrancs solitons in D=7, 11, 15... dimensional spaccrime respectively. How to cutdow n this infinity of c ho ices ? What are the natural D > 3 o- m od el counterparts of (1)which may admit such solitons with exotic spin and statistics ?

One approach suggests itself. These models should have the three essentialingredients of the CP (1) mo del [4] : 1) Th e existen ce of topo logica l membrane solitons; 2)the presence in the action of an Abe lian Che m -Sim on s term, the Hopf invariant; 3) theassociated Hopf mappings S2""1 S n includ e on es with H op f invariant i. The first twoconditions are embo died in the time com ponen t of the key equation ( 4). Its integration o verail of space yields the topological charge-magnetic flux c oup ling w hich is the very b asis ofthe statistics pheno men on in (2+1) dimen sions. A s to the third condition, a crucial elem entin the Wilczek-Zee proof of the fractional spin and statistics for one soliton [4], we have thefollowing property of Ho pf mappings in higher dimension s. W hile for any n eve n therealway s exists a map f: S 20" 1 -* S n with only even integer Hopf invariant 7(f), but as forHo pf m aps of invariant 1. the celebrated Adam s' theorem [1 4] tells us : If there ex ists aHopf map f: S D -+ S of Hopf invariant yif) = 1, indeed one with an y integer 7(f),then D must equal 1 ,3 ,7 and 15 i .e. m = (D+ l)/2 = 1 , 2 , 4 and 8.

From it follow other relevant theorems[ 1 5 ] . They are1) R n admit the structure of a real division algebra if and only if n= 1, 2,4 and 8 . The


13/22

algebras are the real numbers (R), the complex numbers (C), the quaternions (H) and theoctonions (Q).2) The only parallelizable spheres a re S i , S-3 and S'3) There exists a vector product on R n if and only if n = 3 or 7 . Th ese products are thefamiliar Gibbs-Hamilton product of vectors or imaginary quaternions and the Cayleyproduct of imaginary octonions.4) Among the spheres S n only S~ or S^ admit an almost complex structure .

So Adams' theorem singles out 4 unique sets of spacetime and field topologies forthe sought for field theories. These unique four families of Hopf maps , the K = R, C,H, Q Hopf bundles and their various properties are made manifest by the tell-all diagram

= SO(2)II2* = 0(1) = S -> S 1 -> S l / Z 2 = RP(1) - S0(2)/Z o

IIS0(2) = U(l) = S l -> S 3 -> S 2 = CP(1) SU(2)/U(1)

IISU(2) = Sp(!) = S 3 -+ S 7 -> S 4 = HP(1) - Sp(2)/S p(l)xSp(l)II

Spin(8)/Spin(7) = S 7 -> S15 (=Spin(9)/Spint7))-> S 8= QP(l)-Spin(9)/Spim8).

The Hopf maps f : S2"" 1 S n, n= l, 2, 4, 8 ,with Hopf invar iant one have foundimportant applications in condensed matter physics and in quantum field theory.Particularly the connection with exotic spin and statistics had long been lurking in thebackground. Thus, in the D= 2 ty* field theory, the n= l real Hopf m ap realizes the 1-kinksoliton bearing intermediate spin and obeying exotic statistics [16] . Th e n=2 complexHopf map underlies the 0 spin and statistics of D=3 CP (1) model with Hopf term. Wehave shown this pattern to persist in models built on the remaining two Hopf fiberings,namely the D=7 quatemionic HP(1) (=S4) and the D=15 octonionic QP (1) (=S8) a-models augmented with their respective Hopf invariant term . Modulo possible Skyrme


14/22

terms needed for soliton stability, their action generalize (1) as

S (n ) = jn = 4. 8, M = S 7 , S 1 5

l V

where N. N 2=l, the (n+l)-vecnr parametrizing S n . If K = (K j, IC,) such that K +K=l~n'ith K , K , E H , O , is a K -va lue d 2-spinor parametrizing S* * , the Hopf map is

N= ScJK^K) with K M K , , K 2 ) , YH={ ^ , \i= 0, l,...m-l and Ym=( ! ) , m= 4 and 8. The Y"S are just the Dirac matrices of SL (2, K) with ym being the analog of y5oi the standard D=4 formalism.

The A n ,! are composite rank (n-1) Kalb-Ramond fields. They are nonlocal in N butlocal in K . The second term in ( ) is up a multiplicative constant 6 a"1 the Whitehead-Chem-Simons form of the Hopf invariant 7() classifying the mappings : S - S n.It also reads

i..4ii (11)

i.e. as an interaction of the field A n .! with the conserved top olo gica l currentJn-i = r~*Fn the * means Hodge duality. The sources of Jn_! are charged solitonic

4 7 C 2

2-membranes and 6-membrancs respectively, in accordance with the Kleman-Toulouseclassification of topological defects [17].

Since these membrane solitons can be readily shown to have a thin London limit,Polyakovs approximation translates into the Chem-Simons-Kalb-Ramond electrodynamicsof Nambu-Goto mem branes.To get the statistical phase, we consider [1 2] the propagationof two pairs of membranes-antimembrane and seek the phase upon adiaba'icallyexchanging the membranes. W e then get


15/22

terms needed forsoiiton stability, their action generalize (1) as

S(n) =| auN^Nd^-'x+ll AnJ M JMn = 4, 8, M = S 7 , S 1 5

where N, N 2 * ! . the (n+ l)-vecto r parametrizing Sn . If KT = (Ky KJ such that K +K=lwith Kj, K_ e H, Q , is a K -va lue d 2-spinor parametrizing S n" , the Hopf map isN= Sc (Rty K) with K1" =(K!,K2) , jJ ^ 1 , H=0, l,...m-l andym4 l ) , m= 4 and 8.The Ys are just theDirac matrices of SL(2, K) with ym being theanalog of Y5of the standard D =4 formalism.

The A n_i are composite rank (n-1) Kalb-Ramond fields. They arenonlocal in N butlocal in K . Thesecond term in ( ) is up a multiplicative constant 8 a'1 the Whitehead-Chem-Simons form of the Hopf invariant T ( O ) classifying the mappings O : S2"" 1 -Sn .it also reads

S l = ( n J n r j d 2n- li.e. as an interaction of the field An.i with the conserved topological currentJn-i = j ~ * F n *m c a n s Hodge duality. The sources of Jn_i arecharged solitonic2-membranes and6-membranes respectively, in accordance with the Kleman-Toulouseclassification of topological defects [ 1 7 ] .

Since these membrane solitons can bereadily shown to have a thin London limit,Polyakovs approximation translates into the Chem-Simons-Kalb-Ramond electrodynamicsof Nambu-Goto membranes.To get thestatistical phase, weconsider [ 12 ] thepropagationof two pairs of membranes-antimembrane and seek the phase upon adiabaticallyexchanging the membranes. Wethenget


16/22

(12)

Pj and PT are S^ hyper-cu rves. T he functional a verag e () is taken ever the Hopf acnon(9). As in D=3 case , the resulting phase here is the sum of three phases. The firstcontribution yields the phase factor exp{2i (7r/0) L) with L being the generalized Gauss'linking coefficieiit ( ) for two S^-loops. We get K~/Q *"or the statistical phase. The othertwo phase factors ^ (P j) arc given by the expectation va lue of one loop:

(13)( P ) = |

In the London-N ielsen-Olese n lim it we use the effective action

s = S o d ? x

S Q is the free N ambu-G oto action for a relativistic 2-mem brane, 0 < 8 < TC and theconstant a is yet to be chosen. By direct integration of the equation of motio n

with the given current F"Hy) = I d3x 5 (x-y) - , in the Lo rentz ga uge

x-y|

However just like in the D=3 case , the double integration is over one and the samehyperc urve S so the phase (16) is undetermined. W e thus apply White's result to get the


17/22

regularized phaseO ( P = S 3) = 46 '

^ y s 3 x s iwhere WfP) = -^ -1 dHft is the writhe of the Feynman path P of the Nam bu-Goto S~-membrane, a S hyper-nbon in 7-spacetimes. A similar computation gives the same resultfor the octonionic case of P = S ' in S* * -spacetime .

For the value of a = 47t:. O(P )=ex p(iriW /e). By setting 8 = it, we get the S3- (S 7-)counterpan of Po lyakov's phase factor


18/22

regularized phaseO ( P =S3) = expji-2-W(P)\ (17)I 46 /

where W(P) = -1 I d i ^ is the writhe of the Feynm an path P of the Nambu-Goto S~-membrane, a S hyper-nbon in 7-spacetimes. A similar computation g ives the same resultfor the octonionic case of P = S ' in S ^ -spaccame .

For the va lue of a = 4rc2, ^ (P Js ex p ^ W /e ) . By setting 8 = TC we get the S 3- (S 7-)coun terpart of Po lya kov's phase factor


19/22

topological obstrucaon s. Acco rding to G.W. Whitehead [18] all F a in F have the samehomotopy type i.e. K-(V ) = rc(Fn). Of releva nce to the question of exo tic spin andstatistics, for the 1-soliton sector are the key relations

= Z for (l . n) = ( l ,2 ) , (3 ,4 ), T , 8). (19)

by the W hitehe ad and H ure w icz [ 19] isomorp hisms, the latter stating rc n

The connections (IS) reflect the multi-valuednessof Fj and im ply the possibility ofadding to the KP(1) a-model action a Hopf invariant -ftN), the generator of the torsion freepan of * i + n , S

n) (n , (S

2)= Z , i c i S

4) = Z Z 1 2 and n 1 5 (S

7) = Z e Z , 2 0 > . Generalizing theCP(1) model {(i.n) = (1,2)), the nontnviality of these TC.(F' ) implies the possibilities of

Aharonov-Bohm effects of a multiply connected configuration space F and signals theexistence for the membrane solitons of a higher dimensional analog of an 8 spin andstatistics connection.

In the CP(1) case, the H opf term induces, upon a 2JI rotation P of the Skyrmion or aninterchange of two Sky rm ions, a projective spin phase factor O (P ) = ex p(i9 ) = exp(i2ns), s being the soliton spin .The equality 9 = 2its for this process of rotation is a physicalrealization of the homomorphism:

(20)

It establishes the equality of the kinematicaUy allowed exotic spin to the dynamicallyinduced 8-spin by way of the H opf term. Yet (20) is but a spec ial ca se of the Hopf-Whitehead J-homo morphism [201 7t.(SO(n) ) ~ \ . (S n)- Indeed we hav e the followingchain of homomorphisms :

= ;: I+ n(S n) = i(S O (n) ) = Z (21)


20/22

wh (i, n) = (1 ,2 ), (3, 4), (7, 8) . rt3(SO(4)) n 7(S 4) = Z , ^ ?(SO(8)) = *15(S 81 = ZIt suffices to say the physical interpretation of these topological relations implies a

dynamically induced exotic spin and statistics connection for the 2-and 6-membranes.

In conclusion, our work may cast new light on the unchaned problem of the spin andstatistics connection for higher dimensional extended objects. It is also a first step both inthe bosonic functional integral formulation for spinnir.g extended objects [6,21] and in thestudy of the 8-vacuum phenom enon in Kaluza-Klein compactification. As their historicaldevelopments [22] testify , division algebras stand at the crossroad of several frontier areasof mathematics and phy sics. Ove r the last deca de, ever more linkag es have beendiscovered between division algebras and basic aspects of unified the ories . Examples arerieid theories with rigid supersymmetnes, superstrings and supermembranes in criticaldimensions. Here we have uncovered their special relevance in yet another connection , ageneralized 9-spin and statistics connection.

Acknow ledgment: I wish to thank Soonkeon Nam for a very enjoyable collaboration.

ReferencesI.E. Witten, " Topological Quantum Field Theory", to appear in Comm. Math. Phys.I. G. T. Horo w itz," Exactly Soluble Diffeomorphism Invariant Theo ries" N SF-ITP-88-178; G.T. H orowitz and M . Srednicki,"A Quantum Field Theoretic Description of LinkingNumbers and Their Generalization", UCSB-TH-89-14.3. See M. A tiyah ," New Invariants of Three and Four D imensional M anifolds" to appearin Symposium on Mathematical Heritage of Hermann Weyl, ed. R. Wells et al. ( Univ.North Carolina, May 1987) and references therein.4. F. Wilczek, Phys. Rev. Lett. 48 (1982), 1144; 49 (1982), 957; F. WUczek and A. Zee,Phys. Rev. Lett. 51 (1983), 2250.5. E. W itten, "Quantum Field Theory and The Jones Polynom ial", IASSNS-H EP-88/33(1988), to appear in Proc. of the IAM P Congress,Swansea, 1988.6. A.M. Polyakov, Mod. Phy s. L ett. A3 (1988), 325.7. C-H. Tze and S. Nam, " Topological Phase Entanglements of Membrane Solitons inDivision Algebra a-models with a Hopf-Term", VPI-IHEP-88-10, to be published in


21/22

with (i, n) = (1 , 2) , (3 . 4), (7, 8) . 313(8 0(4)) - rc?(S 4) - Z , 7T7(SO(8)) = K^SS) = ZIt suffices to say the physical interpretation of these topological relations implies a

dynamically induced exotic spin and statistics connection for the 2-and 6-racmbrancs.

In conclusion, our work may cast new light on the uncharted problem of the spin andstatistics con nection for higher dim ension al extended objects. It is also a first step both inthe bosoruc functional integral formulation for spinning extended objects [6,21] and in thestudy of the 9-vacuu m p heno me non in Kaluza-Klein compactification. A s their historicaldevelopments [22] testify, division algebras stand at the crossroad of several frontier areasof mathematics and phy sics. O ver the last deca de, ever more linka ges have beendiscovered betw een division a lgebr as and basic aspects of unified the ories . Examples arefield theories with rigid supersymmctries, superstrings and supermembranes in criticaldimensions. Here we have un cov ered their special relevance in yet another connection , ageneralized 9-spin and statistics connection.

Ack now ledgm ent: I wish to thank Soonkeon Nam for a very enjoyable collaboration.

References

1. E. Witten," Top ological Q uantum Re id Theory", to appear in Co mm . M ath. Phys.2. G. T. Horowitz," Exactly Soluble Diffeomorphism Invariant Theories" NSF-ITP-88-178; G.T. Horow itz and M Srednicki,"A Quantum H eld Theoretic Description of Linkingand Their Generalization", U CSB -TH -89-14.3. See M At iyah ," New Invariants of Three and Four Dimensional M anifolds" to appearin Symposium on Mathematical Heritage of Hermann Weyl, ed. R. Wells et al. ( Univ.North Carolina, May 1987 ) and referen ces therein.4. F. Wilczek, Phys. Rev. Lett. 48 (1982), 1144; 49 (1982), 957; F. Wilczek and A. Zee,Phys. Rev. Lett. 51 (1983), 2250.5. E. W itten, "Quantum Field Theo ry and The Jones Polynomial", IAS SN S-H EP -88/33(1988), to appear in Proc. of the IAMP Congress,Swansea, 1988.6. A.M. Polyakov, Mod. Phys. Lett. A3 (1988), 325.7. C-H. T ze and S. Nam, " To po log ical Phase Entanglements of Mem brane Solitons inDivision Algebra o-models with a Hopf-Term", VPI-IHEP-88-10, to be published in


22/22

Annals of Physics.8. R.I. Nepomechie and A. Zee, in "Quantum Field Theory and Quantum Statistics(Fradkin Festschrift)" (LA. Batalin et al. Ed.), Vol. 2, Adam Hilger, Bristol, 1986.9. G. Calugareanu. Rev. Math. Pures Appl. 4 (1959), 5; Czech. Math, J. 11 (1961), 588.10. F. Brock-Fuller, Proc. Nat. Acad. Sci. USA 68 (1971), 815; Proc . Natl.Acad. Sci.USA 75 (1978) 3557; F.H.C. Crick, Proc. N at Acad. Sci. USA 73 (1976 ), 2639.11. J.H. W hite, Am. J. Math. 91 (1969), 693.12. J. Grundberg, T.H. Hansson, A. Karlhede and U. Lindstrbm, "Spin, Statistics ar.uLinked L oop s," Stockholm preprint (1988).13. S.S. Chern, Ann. Math. 43 (1942), 178; 46 (1945).14. J.F. Adams, Ann. of Math. 72 (lOfO). 2\j.15. M.W. Hirsrh. nuieren tial Topology ," Springer-Verlag, Berlin, 1976.16. Sec for instance J. Frohlich, "Statistics of Fields, the Yang-Baxter Equation, and theTheory of K nots and Links", in "Nonperturbative Quantum Field Theory, 1987 Cargese"(G. 't Hooft ct al. Ed.), Plenum, New York, 1988; "Statistics and Monodromy in Tw o andThree Dimen sional Quantum Field Theo ry", in "Proceedings of the 1987 ComoConference ," (K. Bleuler et al. Eds.).17. G. Tou louse and M.J. Kleman, J. de Physique Lett. 37 (1976), L-14 9.18. G.W. Whitehead, Ann. Math. 47 (1946), 460; J.H.C. Whitehead, Ann. Math. 58(1953), 418.19. W. H urewicz, Proc. Akad. Am sterdam 38 (1935), 112; 39 (1936), 117, 215.20. See for instance D. Ravenei and A. Zee, Comm. M ath. Phys. 98 (19 85), 239 .21 . T. Jacobson, Boston University Preprint BUHEP-88-12; P. Orland, VPI preprint,VPI-IHEP-88/6; K. Johnson, MIT Preprint CTP# 1656 (1988) and references therein.22. F. Giirsey, in "Symmetries in Physics (1600-1980)" ( M.G. Doncel, A. Herm ann, L.Michel and A. Pais Ed.) p. 557, Un iv. Autonoma de Barcelona Publications

DISCLAIMERThis report s pre par ed as . in account ul work sponso red by an ag en t) of the United StalesGovernment Neith er the I !nited Slates Government nor any agency there.il. nor an> of theiremployees , makes j m warranty , express or implied, or ass jme s any legal h ahihthility lor the accuracy. completeness, or usefulness of any information, apparaluprocess disclosed, or rep resen ts that Us use would not infringe p rivately ow nedem:e herein in an;, specific commercial product, process, or service by trade name. Irademark,

or responsi-, product, orights Refer-

Documents

Chia-Hsiung Tze- Linking the Gauss-Bonnet-Chern Theorem. Essential Hopf Maps and Membrane Solitons with Exotic Spin and Statistics