Topological tensor current of p̃-branes in the φ-mapping theory

Topological tensor current of p -branes in the -mapping theoryYishi Duan, Libin Fu, and Guang Jia Citation: Journal of Mathematical Physics 41, 4379 (2000); doi: 10.1063/1.533347 View online: http://dx.doi.org/10.1063/1.533347 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A hierarchy of topological tensor network states J. Math. Phys. 54, 012201 (2013); 10.1063/1.4773316 A note on the improvement ambiguity of the stress tensor and the critical limits of correlation functions J. Math. Phys. 43, 2965 (2002); 10.1063/1.1475766 Dynamical topology change in M theory J. Math. Phys. 42, 3171 (2001); 10.1063/1.1377038 Topology and perturbation theory J. Math. Phys. 41, 5710 (2000); 10.1063/1.533434 Stress-energy-momentum tensors in constraint field theories J. Math. Phys. 38, 847 (1997); 10.1063/1.531873

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Topological tensor current of p -branes in the f-mappingtheory

Yishi DuanInstitute of Theoretical Physics, Lanzhou University,Lanzhou, 730000 Gansu, Peoples Republic of China

Libin Fua)LCP, Institute of Applied Physics and Computational Mathematics,P.O. Box 8009 (26), 100088 Beijing, Peoples Republic of China

Guang JiaInstitute of Theoretical Physics, Lanzhou University,Lanzhou, 730000 Gansu, Peoples Republic of China

~Received 27 April 1999; accepted for publication 23 February 2000!

We present a new topological tensor current ofp-branes by making use of thef-mapping theory. It is shown that the current is identically conserved and behavesasd(fW ), and every isolated zero of the vector fieldfW (x) corresponds to a mag-netic p-brane. Using this topological current, the generalized Nambu action formulti p-branes is given, and the field strengthF corresponding to this topologicaltensor current is obtained. It is also shown that the magnetic charges carried byp-branes are topologically quantized and labeled by Hopf index and Brouwer de-gree, the winding number of thef mapping. 2000 American Institute of Phys-ics. @S0022-2488~00!01707-2#

I. INTRODUCTION

Extended objects withp spatial dimensional, known as branes, play an essential role inrevealing the nonperturbative structure of the superstring theories andM -theories.14 Antisym-metric tensor gauge fields have been widely studied in the theories ofp-branes.58 In the contextof the effectiveD510 or D511 supergravity theory ap-brane is ap-dimensional extendedsource for a (p12)-form gauge field strengthF. It is well-known that the (p12)-form strengthF satisfies the field equation

mFmm1mp115 j m1mp11,

wherej m1mp11 is a (p11)-form tensor current, corresponding to the electric source, and the dualfield strength* F satisfies

m* Fmm1m p115 j m1m p11

in which j m1m p11 is a (p11)-form tensor current, corresponding to the magnetic source.911

Thef-mapping theory proposed by Professor Duan12,13 is important in studying the topologi-cal invariant and topological structure of physics systems and has been used to study the topo-logical current of magnetic monopole,12 topological string theory,13 topological structure ofGaussBonnetChern theorem,14 topological structure of the SU~2! Chern density,15 and topo-logical structure of the London equation in superconductor.16 We must point out that thef-mapping theory is also a powerful tool to investigate the topological defects theory,1719 andhere the vector fieldfW is looked upon as the order parameters of the defects.

a!Author to whom all correspondence should be addressed; electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 7 JULY 2000

43790022-2488/2000/41(7)/4379/8/$17.00 2000 American Institute of Physics


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In this paper, we present a new topological tensor current of magneticp-branes by makinguse of thef-mapping theory. One shows that each isolated zero of thed-dimensional vector fieldfW (x) corresponds to ap-brane (p5D2d21), and this current is proved to be the general currentdensity of multi-p-branes. Using this current, the generalized Nambu action for multi-p-branes isobtained. This topological tensor current will give rise to the inner structure of the field strengthFincluding the contribution of the magneticp-branes. Finally, we show that the charges carriedby multi-p-branes are topologically quantized and labeled by the Hopf index and Brouwer degree,the winding number of thef mapping.

II. THE TOPOLOGICAL TENSOR CURRENT OF p -BRANES

Let X be a D-dimensional smooth manifold with metric tensorgmn and local coordinatesxm(m,n50,...,D21) with x05t as time, and letRd be an Euclidean space of dimensiond,D.We consider a smooth mapf:XRd, which gives ad-dimensional smooth vector field onX,

fa5fa~x!, a51,2,...,d. ~1!

The direction unit field offW (x) can be expressed as

na5fa

ifi, ifi5Afafa. ~2!

In thef-mapping theory, to extend the theory of magnetic monopoles12 and the topological stringtheory,13 we present a new topological tensor current, with the unit magnetic chargegm ,defined as

j m1mD2d5gm

A~Sd21!~d21!! S 1AgD em1mD2dmD2d11mD2d12mD3ea1a2ad]m(D2d11)n

a1]m(D2d22)na2]mDn

ad, ~3!

where g is the determinant of the metric tensorgmn and A(Sd21) is the area of

(d21)-dimensional unit sphereSd21. Obviously, this magnetic tensor current is identicallyconserved,

m1 jm1mD2d50, i 51,...,D2d. ~4!

From ~2! we have

]mna5

1

ifi]mf

a1fa]mS 1ifi D , ~5!]

]fa S 1ifi D52 fa

ifi3. ~6!

Using the above expressions, the general tensor current can be rewritten as

j m1mD2d5gmCdS 1AgD em1mD2dmD2d11mDea1ad]m(D2d11)f

a1]m(D2d22)fa2]mDf

ad(a

]

]fa]

]fa~Gd~ ifi !!, ~7!

4380 J. Math. Phys., Vol. 41, No. 7, July 2000 Duan, Fu, and Jia


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whereCd is a constant

Cd5H 2 1A~Sd21!d! ~d22! for d.214p

for d52,

andGd(ifi) is a generalized function

Gd~ ifi !5H 1ifid22 for d.2ln~ ifi ! for d52.

If we define a generalized Jacobian tensor as

ea1adJm1mD2dS fx D5em1mD2dmD2d11mD2d12mD]m(D2d11)fa1]m(D2d22)fa2]mDfad ~8!and make use of the generalized Laplacian Green function relation inf space

(a

]

]fa]

]fa~Gd~ ifi !!5H 2 4pd/2G~d/221! d~fW ! for d.2

2pd~fW ! for d52,

~9!

we obtain ad-function like tensor current13

j m1mD2d5gmd~fW !Jm1mD2dS fx D S 1AgD . ~10!

We find that j m1mD2d0 only whenf50. So, it is essential to discuss the solutions of theequations

fa~x!50, a51,...,d. ~11!

Suppose that the vector fieldfW (x) possessesl isolated zeroes, according to the deduction ofRef. 13 and the implicit function theorem,20,21 when the zeroes are regular points off-mapping,i.e., the rank of the Jacobian matrix@]mf

a# is d, the solution offW (x)50 can be parametrized by

xm5zim~u1,u2,...,uD2d!, i 51,...,l , ~12!

where the subscripti represents thei th solution and the parametersu5u(u1,...,uD2d) span a(D2d)-dimensional submanifold ofX, denoted byNi , which corresponds to ap-brane (p5D2d21) with spatialp-dimension andNi is its world volume. One sees that the tensor currentj m1mD2d is not vanished only on the world volume manifoldsNi ( i 51,...,l ), each of whichcorresponds to ap-brane. Therefore, every isolated zero offW (x) on X corresponds to magneticp-branes. These magneticp-branes had been formally discussed and were not studied based ontopology theory.8,22 Here, we must point out that thep-branes sometimes may be considered astopological defects,11,23 in this case for our theory the vector fieldfa(x) (a51,...,d) may belooked upon as the generalized order parameters19 for p-branes.

In the following, we will discuss the inner structure of the topological tensor currentj m1mD2d. It can be proved that there exists ad-dimensional submanifoldM in X with theparametric equation

4381J. Math. Phys., Vol. 41, No. 7, July 2000 Topological tensor current of p-branes


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xm5xm~v1,...,vd!, m51,...,D, ~13!

which is transversal to everyNi at the pointpi with

gmn]xm

]uI]xv

]vA Upi

50, I 51,...,D2d, A51,...,d. ~14!

This is to say that the equationsfW (x)50 have isolated zero points onM .As we have pointed in Refs. 14 and 15, the unit vector field defined in~2! gives a Gauss map

n:]MiSd21, and the generalized winding number can be given by this Gauss map

Wi51

A~Sd21!~d21!! E]Mi n* ~ea1adna1dna2``dnad!5

1

A~Sd21!~d21!! E]Miea1adna1]A2na2]AdnaddvA2``dvAd5

1

A~Sd21!~d21!! EMieA1Adea1ad]A1na1]A2na2]Adnadddv, ~15!where]Mi is the boundary of the neighborhoodMi of pi on M with pi]Mi , MiM j5B.Then, by duplicating the derivation of~3! from ~10!, we obtain

Wi5EMi

d~fW ~v !!JS fv Dddv, ~16!whereJ(f/v) is the usual Jacobian determinant offW with respect tov,

ea1adJS fv D5eA1Ad]A1na1]A2na2]Adnad. ~17!According to thed-function theory24 and thef-mapping theory, we know thatd(fW (v)) can beexpanded as

d~fW ~v !!5(i 51

l

b ih idd~vW 2vW ~pi !! ~18!

on M , where the positive integerb i5uWi u is called the Hopf index of the mapvfW (v) andh i5sgn(J(f/v))upi561 is the Brouwer degree.

14,16 One can find the relation between the Hopfindex b i , the Brouwer degreeh i , and the winding numberWi ,

Wi5b ih i , ~19!

One sees that Eq.~18! is only the expansion ofd(fW (x)) on M . In order to investigate theexpansion ofd(fW (x)) on the whole manifoldX, we must expand thed-dimensionald function ofthe singular point in terms of thed function on the singular submanifoldNi which had been givenin Ref. 24

d~Ni !5ENi

dD~x2zi~u!!Agud(D2d)u, i 51,...,l

in which



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gu5detS gmn ]xm]uI ]xn

]uJD , I ,J51,...,~D2d!. ~20!Then, from Eq.~18!, and by considering the property of thed function, one will obtain

d~fW ~x!!5(i 51

l

b ih iENi

dD~x2zi~u!!Agud(D2d)u. ~21!

Therefore, the general topological current of thep-branes can be expressed directly as

j m1mD2d5S 1AgD Jm1mD2dS fx D(i 51l

b ih iENi

dD~x2zi~u!!Agud(D2d)u, ~22!

which is a new topological current theory ofp-branes based on thef-mapping theory.If we define a Lagrangian as

L5A 1~D2d!!

gm1n1gm(D2d)n(D2d) jm1mD2d j n1nD2d, ~23!

which is just the generalization of Nielsens Lagrangian,25 from the above deductions, we canprove that

L5S 1AgD d~fW ~x!!. ~24!Then, the action takes the form

S5EXLAgdDx5E

Xd~fW ~x!!dDx. ~25!

By substituting the formula~21! into ~25!, we obtain an important result,

S5EX(i 51

l

b ih iENi

dD~x2zi~u!!Agud(D2d)udDx5(i 51

l

b ih iENi

Agud(D2d)u, ~26!

i.e.,

S5(i 51

l

h iSi , ~27!

whereSi5b i*NiAgud(D2d)u. This is just the generalized Nambu action for multi-p-branes (p

5D2d21), which is the straightforward generalization of Nambu action for the string world-sheet action.26 Here this action for multi-p-branes is obtained directly byf-mapping theory, andit is easy to see that this action is just Nambu action for multistrings whenD2d52.13

III. THE GAUGE FIELD CORRESPONDING TO THE TOPOLOGICAL CURRENT

In this section, we will study the antisymmetric tensor gauge field corresponding to thetopological tensor current presented in Sec. II. We know thatp-branes naturally act as the elec-tric source of a rankp12 field strength

F5dA, ~28!



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whereA is a (p11)-form as the tensor gauge potential and satisfies the gauge transformation

AA1dLp .

From Eq.~28!, one has the Bianchi identity

dF[0, ~29!

and the electric current density associated with the source can be expressed as

j m1mp115mFmm1mp11. ~30!

Just as the usual Maxwells equation, we know that Eqs.~28!~30! imply the presence of anelectric charge, i.e.,p-branes, but no magnetic source.11

Now, let us discuss the case when there exists the magnetic source. For this case, one mustintroduce another (p12)-form G for the magnetic source, and the field strengthF must bemodified to

F5dA1G, ~31!

which is the generalized field strength including the contribution of the magnetic source, i.e.,magnetic branes:p-branes withp5D2p24.

To obtain the explicit expression forG, let us consider that the current density corresponds tothe magnetic source which is given by

j m1m p115m* Fmm1m p11. ~32!

Using ~31! and ~32!, we obtain

j m1m( p11)51

Ag]mS Ag emm1m p11m p12mD21Ag Gm p12mD21D . ~33!

It has been pointed out in Sec. II that the current density of the magnetic branes is a topologicalcurrent given by Eq.~3!, which can be rewritten as

j m1mD2d5gm

A~Sd21!~d21!! S 1AgD ]m(D2d11)~em1mD2dmD2d11mD2d12mD3ea1a2adn

a1]m(D2d22)na2]mDn

ad!, ~34!

where (D2d)5 p11, i.e., p5D2d21. Comparing Eq.~33! to ~34!, we can obtain

Gm1md215~21!(D2d)gm

A~Sd21!~d21!!ea1a2adn

a1]m1na2]md21n

ad, ~35!

and

G5~21!(D2d)gm

A~Sd21!~d21!!ea1a2adn

a1dna2``dnad. ~36!

Of equal interest is the magnetic charge carried by the multip-branes, which is given by

QM5ES

j m1m p11Agdsm1m p11 ~37!



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where S is a d-dimension (d5p13) hypersurface inX, while dsm1m p11 is the convariantsurface element27 of S. From ~32! and ~37!, it is easy to prove that

QM5E]S

F,

where]S is the boundary ofS and a (p12)-dimension hypersurface. Substituting~22! into ~37!,we have

QM5gmESJm1m p11S fx D(i 51

l

b ih iENi

dD~x2zi~u!!Agud(D2d)udsm1m p11, ~38!

from ~8!, and the relation

1

~ p11!!em1m p11n1nddsm1m p115dx

n1``dxnd,

expression~38! can be rewritten as

QM5gmEf(S)

(i 51

l

b ih iENi

1

AgdD~x2zi~u!!Agud(D2d)ud(d)f. ~39!

Since on the singular submanifoldNi we have

fa~x!uNi5fa~zi

1~u!,...,ziD~u!![0, ~40!

this leads to

]mfa]xm

]uI UNi

50. ~41!

Using this expression, one can prove

Jm1mD2dS fx D UfW 50

5AgAgu

e I 1I (D2d)]xm1

]uI 1

]xm(D2d)

]uI (D2d). ~42!

Then we obtain a useful formula

d(d)fAgud(D2d)u5AgdDx. ~43!

By making use of the above formula and~39!, we finally get

QM5gm(i 51

l

b ih iEXdD~x2zi~u!!d

Dx5gm(i 51

l

b ih i . ~44!

Equation ~44! shows that thei th brane carries the magnetic chargeQiM5gmb ih i5gmWi ,

which is topologically quantized and characterized by Hopf indexb i and Brouwer degreeh i , thewinding numberWi of the f mapping.

IV. CONCLUSION

In this paper thef-mapping theory is introduced to study thep-branes theory, which is adevelopment of our former theories of magnetic monopoles and topological strings. We present anew topological tensor current of magnetic multi-p-branes and discuss the inner structure of this



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current in detail. It is shown that every isolated zero of the vector fieldfW ~i.e., order parameters!is just corresponding to a magnetic brane,p-brane (p5D2d21). The generalized Nambu actionfor multi-p-branes can be obtained directly in terms of this topological current. The topologicalstructure of the charges carried byp-branes shows that the magnetic charges are topologicallyquantized and labeled by the Hopf index and Brouwer degree, the winding number of thefmapping. The theory formulated in this paper is a new concept for topologicalp-branes based onthe f-mapping theory.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China and DoctoralScience Foundation of China.

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Topological tensor current of p̃-branes in the φ-mapping theory

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