CSBIon - a charged soliton of the 3-dimensional CS BI Abelian gauge theory Horatiu NastaseaandJacob Sonnenscheinbc

2025-04-24 0 0 588.57KB 36 页 10玖币

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CSBIon - a charged soliton of the

3-dimensional CS + BI Abelian gauge theory

Horatiu Nastasea∗and Jacob Sonnenscheinb,c†

aInstituto de F´ısica Te´orica, UNESP-Universidade Estadual Paulista

R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil

bSchool of Physics and Astronomy,

The Raymond and Beverly Sackler Faculty of Exact Sciences,

Tel Aviv University, Ramat Aviv 69978, Israel

cSimons Center for Geometry and Physics,

SUNY, Stony Brook, NY 11794, USA

Abstract

In this paper, we construct a charged soliton with a ﬁnite energy and no delta

function source in a pure Abelian gauge theory. Speciﬁcally, we ﬁrst consider

the 3-dimensional Abelian gauge theory, with a Maxwell term and a level NCS

term. We ﬁnd a static solution that carries charge N, angular momentum N

2and

whose radius is Nindependent. However, this solution has a divergent energy.

In analogy to the replacement of the 4 dimensional Maxwell action with the

BI action, which renders the classical energy of a point charge ﬁnite, for the 3

dimensional theory which includes a CS term such a replacement leads to a ﬁnite

energy for the solution of above. We refer to this soliton as a CSBIon solution,

representing a ﬁnite energy version of the fundamental (sourced) charged electron

of Maxwell theory in 4 dimensions. In 3 dimensions the BI+CS action has a

static charged solution with ﬁnite energy and no source, hence a soliton solution.

The CSBIon, similar to its Maxwellian predecessor, has a charge N, angular

momentum proportional to Nand an N-independent radius. We also present

other nonlinear modiﬁcations of Maxwell theory that admit similar solitons. The

CSBIon may be relevant in various holographic scenarios. In particular, it may

describe a D6-brane wrapping an S4in a compactiﬁed D4-brane background.

We believe that the CSBIon may play a role in condensed matter systems in

2+1 dimensions like graphene sheets.

∗E-mail address: horatiu.nastase@unesp.br

†E-mail address: cobi@tauex.tau.ac.il

arXiv:2210.06581v2 [hep-th] 22 May 2023

Contents

1 Introduction 1

2 Motivation and comparison with BIon solution in 4 dimensions 3

2.1 3 dimensional BIon solution to BI theory . . . . . . . . . . . . . . . . . . . 5

3 Solutions for Maxwell plus Chern-Simons in 3 dimensions 6

3.1 The basic static solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Regularization with a conducting circle around the origin . . . . . . . . . . 10

3.3 Time-dependent solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 The 3-dimensional BI action plus CS term 12

4.1 Equations of motion and constitutive relations . . . . . . . . . . . . . . . . 12

4.2 The analysis of possible solutions . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Charge, energy and angular momentum of the soliton solution . . . . . . . . 18

4.4 ModMax and ModMax precursor generalizations in 3 dimensions . . . . . . 20

5 Conclusions and discussion 22

A Nonrelativistic BI-type model 23

B Relativistic BI-type model 27

C Attempts of ﬁnding an analytic solution 30

C.1 Attempt1..................................... 31

C.2 Attempt2..................................... 32

C.3 Attempt3..................................... 32

C.4 Attempt4..................................... 33

1 Introduction

Classical solutions of quantum ﬁeld theories with ﬁnite energy are physically very important

and are rare. In gauge theories there are certain ﬁnite energy solutions with some ﬁnite

charge, usually topological in nature, though not only (for instance, consider the Q-ball

solution [1]). In the case of nonabelian gauge theories, one can have topological soliton

solutions involving the gauge ﬁelds only, for instance the BPST instanton solution [2],

though in that case the solution only exists in Euclidean signature. If one adds matter,

speciﬁcally scalars, there are more soliton solutions possible, like the ’t Hooft monopole in

the 3+1 dimensional nonabelian case [3], and the Nielsen-Olesen vortex in 2+1 dimensional

Abelian-Higgs theory [4]. One can also have ﬁnite energy solutions that are sourced by a

delta function, like the BIon solution, invented by Born and Infeld [5] in order to describe

the electron as a ﬁnite energy solution with a delta function source.

But until now, to our knowledge, there were no soliton solutions in pure abelian gauge

theory. In this paper, we ﬁrst derive a static solution of the Maxwell + level NCS theory.

This explicit solution has a charge N, angular momentum N/2 and a radius which is N

independent. However, it has a divergent energy and a delta function source. We cure

both problems by uplifting the system into a BI + CS one. We refer to the corresponding

soliton solution as the CSBIon. For that case were not able to derive an analytic explicit

solution, but we show that indeed it has ﬁnite energy, and charge, angular momentum

and radius similar to those of the predecessor Maxwell + CS theory, but no delta function

source. Moreover, the electric charge associated with the solution does not arise from a

topological number.

The Maxwell + CS electromagnetism in 2+1 dimensions has many applications to

condensed matter physics. These are described in the reviews [6–8] and in references

therein. Probably in a similar manner one can consider applications of the BI + CS action

to solid states systems. In particular a phenomenological description of the dynamics of

the graphene sheets in terms of a DBI action was proposed in [9]. The CSBIon may be a

source outside of the sheet.

Gauge ﬁeld theories, abelian and non-abelian, described by an action built of BI and

CS terms, are very common on the worldvolumes of D-branes. As such they show up in

various string and holographic models. An example of such an abelian gauge theory in three

dimensions is associated with a D6-brane that resides in the background of compactiﬁed

D4-branes and wraps an S4. This model has been suggested [10] as the holographic dual

of the proposal to describe an Nf= 1 baryon in terms of a quantum Hall droplet [11].

The paper is organized as follows. The next section is devoted to the motivation

for this work and to a comparison with BIon solution in 4 dimensions. In section 3 we

derive solutions of the Maxwell + CS action. First we derive the basic static solution and

compute its classical energy, angular momentum and radius. We then derive a solution

with ﬁnite energy for the case where the origin is encircled by a conducting circle and a

time dependent solution. In section 4 we uplift the Maxwell term to a BI one. We write

down the equations of motion and the constitutive relations. We analyze the structure

of the solution and conclude that it has to have ﬁnite energy and charge and angular

momentum that are linear with Nand radius which is independent of it. Next we describe

certain ModMax generalizations. In the next section we summarize, conclude and write

down several open questions. The paper includes also three appendices. In the ﬁrst we

describe a non-relativistic BI-type model, followed in the second by a relativistic one. We

then present 4 attempts of approximating the exact solution in the third one.

2 Motivation and comparison with BIon solution in 4 di-

mensions

As motivation for our work, we can take the point of view of the formal theoretical physicist,

and simply look for an answer to a mathematical physics question: can we ﬁnd in Abelian

gauge theory a ﬁnite energy soliton solution, which is not sourced by a delta function?

In 4 dimensions, the BIon solution to the BI action [5] (modiﬁcation of Maxwell elec-

tromagnetism) has a ﬁnite energy, which is why Born and Infeld constructed it. But it is

also sourced by a delta function, so as to be able to be identiﬁed with a ﬁnite ﬁeld energy

version of the electron. At r→ ∞, the BIon solution becomes the regular Maxwell electron,

so ~

E∝1/r2, which gives a ﬁnite energy at inﬁnity, since E ∼ 4πRr2dr ~

E2/2∼Rdr/r2,

while at r→0, the BIon modiﬁcation keeps ~

Eﬁnite.

But the BIon is necessarily sourced, since ~

∇ · ~

D≡4π˜ρf, with ˜ρfthe free, or external,

charge density, which is found to be qδ3(r). There are no static solutions that are ﬁnite

energy and not sourced, either in Maxwell or in BI theory.

In Maxwell theory (see [12–14]) and in its BI generalization [15,16], there are time-

dependent knotted solutions with non-trivial topological charges.

So it is natural to look to 3 dimensions, and see if we can ﬁnd something there. But

in 3 dimensions, even the regular Maxwell electron has ~

E∝1/r, so a diverging energy at

inﬁnity, since now E ∼ 2πRrdr ~

E2/2∼Rdr/r. So one needs to consider a modiﬁcation of

Maxwell theory at large distances, or small energies (in the IR). Luckily, in 3 dimensions

we have the CS term that we can add, and will dominate in the IR.

We can now ask: can we ﬁnd such an action, of Maxwell + CS, or BI + CS in a physical

system? The answer for BI+ CS is in the aﬃrmative, as follows.

Consider the D4-brane holographic system, or the doubly-Wick rotated nonextremal

D4-brane (Witten model) with a large Nnumber of D4-branes, and consider a D6-brane

wrapping the transverse S4in it, and the other 3 directions being parallel to the D4-brane.

The CS term on the D6-brane will contain a nontrivial term of the type RA∧dA ∧F(4),

and since on the transverse sphere F(4) ∼N(4), we obtain on the 3 directions common to

the D4- and D6-brane an Abelian gauge theory term

SCS+BI =SBI +N

2πZd3xµνρAµ∂νAρ.(2.1)

But, before we continue, we will review the 4-dimensional BIon solution.

The 4-dimensional BI action is

L(b;~

E, ~

B) = b2h1−p1 + F−G2i,(2.2)

where bis the dimensional parameter, of dimension 2, that deﬁnes the theory, and

F=1

b2(~

B2−~

E2) = 1

2b2Fµν Fµν , G =1

b2~

E·~

B=−1

4b2Fµν ˜

Fµν ,(2.3)

with ˜

Fµν =1

2µνρσ Fρσ.

As always in nonlinear electromagnetism theories, be it inside a material, or in vacuum,

we deﬁne the objects

H=−∂L

∂~

B=~

B−G~

√1 + F−G2,~

D=∂L

∂~

E=~

E+G~

√1 + F−G2,(2.4)

the above H(E, B) and D(E, B) being constitutive relations for the material, or the vacuum

theory.

In terms of ~

E, ~

D, ~

B, ~

H, the Maxwell equations without sources have form

∇ × ~

E=−1

c∂t~

B , ~

∇ · ~

B= 0 ,

∇ × ~

H=1

c∂t~

D , ~

∇ · ~

D= 0.(2.5)

In the presence of sources, one has

∇ · ~

D= ˜ρext ,(2.6)

which contains only the external (or free) charge density ˜ρext (or ˜ρf), which means delta

function sources, introduced as an extra term in the Lagrangian of the type R˜ρextA0,

whereas we also have

∇ · ~

E=˜ρ

0

,(2.7)

but here in ˜ρwe also have charges due to the polarization of the material, or in this case,

of the vacuum, leading as usual to the fact that this total charge density is spread out.

In 4 dimensions, the Hamiltonian is the Legendre transform of the Lagrangian over

E=F0i=−˙

Ain the A0= 0 gauge,

H=~

D− L =b2





1 + ~

r1 + ~

B2−~

b2−~

B·~

b22−1





,(2.8)

and since we can calculate that

2s≡~

D2+~

B2=

E2+~

B21 + ~

B2−~

b2+ 2(~

E·~

B)2

1 + ~

B2−~

b2−~

B·~

b22

p2≡~

D2~

B2−(~

B·~

D)2=

E2~

B2−(~

E·~

B)2

1 + ~

B2−~

b2−~

B·~

b22,(2.9)

we can re-express it in terms of its natural variables, ~

Dand ~

B, as

H(b;~

D, ~

B) = b2"r1 + 2s

b2+p2

b4−1#=b2

s1 + ~

D2+~

b2+~

D2~

B2−(~

D·~

B)2

b4−1

.

(2.10)

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CSBIon-achargedsolitonofthe3-dimensionalCS+BIAbeliangaugetheoryHoratiuNastasea*andJacobSonnenscheinb;caInstitutodeFsicaTeorica,UNESP-UniversidadeEstadualPaulistaR.Dr.BentoT.Ferraz271,Bl.II,SaoPaulo01140-070,SP,BrazilbSchoolofPhysicsandAstronomy,TheRaymondandBeverlySacklerFacultyofExactSciences,T...

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CSBIon - a charged soliton of the 3-dimensional CS BI Abelian gauge theory Horatiu NastaseaandJacob Sonnenscheinbc

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