Class of self-dual models in three dimensions

Class of self-dual models in three dimensions

A. de Souza Dutra*

UNESP/Campus de Guaratingueta´-DFQ, Av. Dr. Ariberto Pereira da Cunha, 333 Guaratingueta´, Sa˜o Paulo, Brazil

C. P. Natividade

UNESP/Campus de Guaratingueta´-DFQ, Av. Dr. Ariberto Pereira da Cunha, 333 Guaratingueta´, Sa˜o Paulo, Brazil and Instituto de Fı´sica–Universidade Federal Fluminense, Av. Litoraˆnea S/N, Boa Viagem, Nitero´i,

Rio de Janeiro, Rio de Janeiro, Brazil ~Received 26 June 1998; published 21 December 1999!

In the present paper we introduce a hierarchical class of self-dual models in three dimensions, inspired in the original self-dual theory of Towsend-Pilch-Nieuwenhuizen. The basic strategy is to explore the powerful property of the duality transformations in order to generate a new field. The generalized propagator can be written in terms of the primitive one ~first order!, and also the respective order and disorder correlation functions. Some conclusions about the ‘‘charge screening’’ and magnetic flux were established.

PACS number~s!: 11.10.Ef, 11.10.Kk, 11.15.2q

From the mathematical point of view, topological theories in three dimensions contain a rich variety of models which have received much attention in the last years. One of them is the self-dual model@1#, which presents a close connection with the well established Chern-Simons~CS!theory@2#. This fact could be confirmed in different ways: for instance, by comparing the Green functions of the Maxwell-Chern-Simons~MCS!theory and the self-dual~SD!model@2,3#, by inspecting the constraint structure of each model @3#, or through the bosonization of the massive Thirring model in three dimensions, which is related to the MCS theory in the large mass limit @4#. In this last case, the equivalence be-tween both models has been obtained starting from a careful analysis of the partition function and was improved later through the calculation of higher order derivative terms@5#. In the present work, we shall introduce a hierarchical fam-ily of dual models in three dimensions, related to the original SD model. The mathematical structure of the SD theory of-fers an alternative way of building up N families of dual models. In the final step, a master Lagrangian density corre-sponding to a higher order derivative model is generated. A very interesting aspect of this model is the existence of an isomorphism between its observables and those obtained in its first order form. This fact can be proved through different procedures: first, in the canonical analysis of the fields and their momenta, by using the treatment of order reduction@6#. In what follows, we will use a method developed in a series of papers @7# ~see also @8# for related works!, in order to describe the magnetic flux and charge on the plane (x1,x2) through two dual operators (m,s), called disorder and order operators, respectively.

In order to implement our alternative model, let us begin exploring the mathematical structure of the self-dual fields. In this sense, let us consider the duality transformation of the primary field A_m(N),

A_m(N)₅emab]aAb(N11), ~1!

where the index N is an integer which identifies the family of the respective self-dual field. The relation ~1! gives rise to the possibility of generating a class of Lagrangian densities indexed by N.

Let us start our study by considering the following La-grangian density:

L152

a

4~Fmn!

2

1b]mFml]nFnl1uemnr]sAm]n]sAr, ~2!

which has been examined recently @9#, with a, b, and u de-fined in it. Now, we are going to show that the Lagrangian density appearing in Eq.~2!is a higher order extension from the Proca-Chern-Simons one:

L₁(0)₅₂a

2Am

(0)_Am(0)

1b₂~F_mn(0)!2

1uemnrA_m(0)]nAr(0). ~3!

By using the transformation~1!, with N₅0, it is lengthy but straightforward to show that we arrive at the Lagrangian den-sity ~2!. The propagators can be related among them, since

^

A_m(0)A_n(0)

&

52e_mabe_na`<sub>b</sub>`

^

]aA_b(1)]a`

A_b`

(1)

&

.

From the above considerations it becomes clear that the results obtained here can be generalized from the N order to the (N₁1) one. Therefore, from the basic Lagrangian den-sity given by Eq. ~3!, we can build up the following generic higher-order Lagrangian density:

L(N)₅~21!

(N21)

4 Fmn

(N) h(N21)

~a_h1b!Fmn(N)

2~₂1!(N21)_uemnr_A m

(N)_]

nh(N)Ar

(N)

. ~4!

The above Lagrangian density belongs to a class such that the first one is related to the bosonization of the massive *Email address: dutra@feg.unesp.br

PHYSICAL REVIEW D, VOLUME 61, 027701

Thirring model@5#. In order to simplify the calculation of the canonical momenta, we are going to define the quantities

f_m[

A

_hN21_A

m, f˙m[

A

hN21A˙m, ~5! where_h2n

5*@d3k/(2p)3#(k2)2n_eikx_{. Now, if we take the} rescaling _h2n_→(_h2V)2n, and take V_→0 at the end of the calculations, the expansion in powers of the d’Alambertian can be employed by acting on the fields. Con-sequently, we can derive the canonical momenta associated to independent variables ( f_m, f˙_m) in a natural way,

dS(n)5

E

d3x

(

n50

`

d

dt~pn

(n)_d

fn1S_n(n)df˙n!,

where now the action S(n)₅*dtL(n) is the reduced form from those in Eq.~4!. Therefore, the momenta become

pn(N)

5~21!N21_{$b f}0n(N)

1a~]k_]l_f l

0(N)_d

k n

2]0_]l_f l n(N)

!

22uemln_] l]nfm%,

sn(N)5~21!N21_2a~]

mfmn(N)2d0

n_]

rf0r(N)!2ue0ln]0fl

(N)

, ~6!

which relates physical quantities from N theory with the first-order one. From the above equations we conclude that the basic commutators of the present theory in the Coulomb gauge are

@f_l~N!~¯x!,pk

(N)

~¯y!#x0_5y05

A

¹2(N21)@A_l~N!~¯x!,p_k~N!~¯y!#_x0_5y0

52i~₂1!N21_$bd

lkd2~¯x2¯y!

1~b1a¹2_!]

l]kG~¯x2¯y!%, ~7!

@f˙l~¯x!,sk

(N)

~¯y!#x0_5y052i~21!N21bd_lkd2~¯x2¯y!

5

A

¹2(N21)_@A˙

l ~N!_~¯x!,s

k

(N)

~¯y!#x0_5y0,

~8!

and also

@pl ~N!

~¯x!,pk

(N)

~¯y!#x05y052iu~21! N21

~b₁a¹2 !

3eik]l]id2~¯x2¯y!, ~9!

where G(x¯ ,y¯) obeys the equation

~b₁a¹2

!¹2G~¯ ,y¯x !5d2~¯x₂¯y!. ~10!

Here we remark that the application of the expansion of

A

_hN21

on the above brackets, extracts the temporal part of the d’Alambertian operator.

The Lagrangian density~4!permits us to infer the corre-sponding form of the photon propagator in momentum space

D_mn(N)~k!5 21

k2N~f2₂4u2_k2 !

HS

a1

b

k2

D

Pmn12ie

mln_k l

J

2j_f kmkn

k2N12, ~11!

where the last term corresponds to a gauge fixing. By using Fourier transform we can obtain the equivalent propagator in the coordinate space. Here, we adopt P_mn5k2g_mn2k_mknand

f[b₂ak2. Hence, if we fix some parameters in the original Lagrangian density given by Eq.~4!like a[4a2_{, b}

51 and

N₅1, we obtain the photon correlation function

^

A_m~x!An~y!

&

dual5@~124a2h!Pmn1i¯uemna]a#

3

S

1

F

¯u a

S

12

a2

u

¯2

DG

221

D

S

4a2

u

¯2

D

3e

2~ u¯ /a !(12a2/¯u2)R

4pR 2

1 4p

3j]m]n

S

2a2

R

4

D

~12!

with¯u_{[iu, 4a}2

,u¯2 _{and ‘‘dual’’ stands for the generalized}

model defined through the propagator of the Lagrangian den-sity. The above equation represents the photon correlation function of the problem mentioned in Ref.@9#.

At this point, we are able to extract a very interesting and useful result about the order-disorder correlation functions, starting from Eq. ~12!. We remember to the reader that the order-disorder formalism has been introduced first by Kadanoff and Ceva@10#in order to discuss the existence of a generalized statistics. Posteriorly this was extended to the continuum quantum field theory @11#. This procedure has been applied to some models in (211) dimensions by using a new interpretation of the operators that generate the statis-tics. Now, over the plane (x1,x2), the Maxwell theory has a nontrivial value for the topological charge associated with the identically conserved current Jm₅emnr_]

nAr. The mag-netic flux content correspondent to Jmis described by a non-local operator ~vortex operator! m(x) defined on a certain curve C. The correlation function

^

m(1)m(2)

&

of the disor-der operator is given as Euclidean functional integrals. In the same way, we can define the charge bearing operators(x) as being a dual version of m.

In order to give a better understanding of the role of order-disorder correlation functions, we will take as an ex-ample the case of the Maxwell-Chern-Simons theory, since its photon propagator in the coordinate space will be useful in the following.

The order correlation function for the MCS theory is de-fined in terms of the following Euclidean functional integral:

BRIEF REPORTS PHYSICAL REVIEW D 61 027701

^

s~x!s*~y!

&

5Z21

E

_DA

mexp

H

2

E

d3z

3

F

1₂A_m~Pmn₁uCmn₁Gmn!A_n1C_mAm

GJ

~13!

where Pmn[_2hdmn_1]m_]n_{, C}mn_[2ieman_]

a and Gmn is the usual gauge fixing term. Here we adopt an external field

C_m.

Integrating over A_m we readily obtain

^

s~x!s*~y!

&

5exp

H

1 2

E

d

3_{z d}3_z

8

C_m~z!

3@Pmn₁uCmn₁Gmn#21_C

n~z

8

!

J

~14! with @Pmn₁uCmn₁Gmn#21

5

^

A_m(x)A_n( y )

&

M CS being the Euclidean propagator of the A_mfield in MCS theory. Its ex-plicit expression in the coordinate space is given by

^

A_m~x!An~y!

&

M CS5@Pmn1iueman]a#

F

12e2uR

4pu2_R

G

2lim m→0

j]m_]n

F

1

m2

R

8p

G

. ~15!

Before going on, we should remark that

^

s(x)s*( y )

&

is not a gauge invariant quantity. The reason is that under a formal gauge transformation A_m→A_m1L, the charge operator changes tos

₈

₅exp„2piL(x)…s. In this way, going back to Eq. ~14!, we must extract the gauge independent part of

^

s(x)s*( y )

&

. This will be achieved by inserting the gauge independent part of

^

A_m(x)A_n*( y )

&

M CS, namely dmn and eman_{proportional terms. At the end of the calculations, it can} be shown that only the diagonal part of

^

A_m(x)A_n*( y )

&

M CS proportional to dmn

h contributes to the order correlation function. Therefore we obtain the following expression:

^

s~x!s

_*

~y!

&

M CS

5exp$

^

A_m~x!A_n*~y!

&

j50

diag %exp

S

e

2uR

4pR

D

~16!

5exp

H

pa

2

u

¯ @e 2¯ Ru

21#

J

⇒

^

sR~x!sR

*

~y!

&

5exp

H

pa

2

u

¯ e

2¯ Ru

J

_~17!

where the renormalizationsR[sepa 2_/2u¯

was adopted and a is a charge parameter. As a consequence, lim

R→`

^

sR(x)sR*( y )

&

51, which reflects the screening of

the charge associated with the mass generation for the gauge field induced by the CS term.

Now, going back to our model, we begin considering the limit which excludes the Podolsky term,u¯2

@a2_{, in Eq.~12!,}

^

A_m~x!A_n~y!

&

j50

dual_>

@P_mn12i¯u_emna]a_#

S

1 u

¯221

D

3

S

4

u

¯2

D

e2u¯ R

4pR, ~18!

in order to compare it with results of the MCS case. By examining the diagonal part of the above propagator we have

^

A_m~x!An~y!

&

dual>hdmn

S

1

u

¯221

DS

4

u

¯2

D

e2¯ Ru

4pR

>

S

1 u

¯221

D

e2u¯ R

4pR1extra terms

52

S

1

u

¯221

D

^

Am~x!An~y!

&

M CS ~19!

where ‘‘extra terms’’ are proportional to the d-functions. From Eqs.~15!and~17!we expect that

^

sR~x!sR

*

~y!

&

dual>exp

FS

1

u

¯221

D

21

G

3

^

sR~x!sR

*

~y!

&

M CS, ~20! for R_→`

^

sR~x!sR

*

~y!

&

dual→const. ~21!

Therefore the order correlation function, which is associated with charge screening, in our model has a similar behavior to that of the MCS theory. The result given by Eq. ~18! ex-presses the charge screening, which in this case is

Qdual5

E

d2z J05u

E

d2z¹_j2_e

i j]z i

Ai~z,j!, ~22!

which differs from the usual MCS charge by a second order derivative operator. Here J_m[]nJ_mn defined in Eq. ~23! be-low. Note that the presence of the differential operators in

Qdual does not alter the long range distance behavior of the order correlation function when compared with MCS theory. Now, in order to build the disorder correlation function

^

m(1)m(2)

&

in our model, we begin defining the vortex op-erator which is associated to the magnetic flux on the plane (x1,x2). This is obtained by coupling a certain external field

W_m to the dual current through

J_umn[Fmn2 uhe

mna

~1₂4a2

h!Aa, ~23!

which comes from the equation of motion. The generalized disorder operator can be written as

m_udual

~x!5exp

H

₂ib

E

d3z J_umnW_mn

J

, ~24!

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where Wmn is an external tensor field, W_mn[]_mWn

2]nWm, which would be coupled to the conserved current

J_umn in order to obtain the correct correlation function

^

mu~x!m*u~y!

&

dual5

E

DAmexp

H

2

E

d3z

F

1

2A m_D

mn dual_An

1AmD_mn(GCS)Wn1

1

4~Wmn!

2

GJ

_,

~25!

with GCS standing for generalized Chern-Simons, and D_mndual is given by

D_mndual[~1₂4a2

h!P_mn2j]m]n2ueman]ah,

D_mnGCS[P_mn2ue_an]a_h~124a2

h!21_{, ~26!}

where P_mn[_2hd_mn1]m]n. Now, if we consider the action of the operators P_mn andue_man]a_{h over W}

mn, this gives rise to

P_mnWm→Zn5

E

dlnd3~z2l !,

u_he_man]a_Wm

~1₂4a2

h! →Un5u

E

dl m_e

man]ah~124a2h!21

3d3~z₂l !. ~27!

Therefore, after integration over the field A_min Eq.~22!we get

^

mu~x!m*u~y!

&

N51

dual

5exp

H

1 2

E

d

3_{z d}3_z´„Z

m~z,x, y!

1U_m~z,x,y!…

^

A_m~x!A_*n~y!

&

dual

3„Zn~z´,x, y!1Un~z´,x, y!…

2

1

4~Wmn!

2

J

_{, ~28!}

where

^

A_m(x)A_*n( y )

&

dual is given by Eq.~12!. Now, if we now turn our attention to the fact that in the limit u@4a2_,

the photon correlation function is given by Eq. ~18!and the field Un will not depend on the factor 4a2, we will have

^

m_u~x!m_*u~y!

&

N51

dual

5exp

H

1 2

S

12

1

u

¯2

D

E

d

3_{z d}3_z´~Z

m1U˜m!

3

^

A_mA_*n

&

M CS_~Z n1U˜n!

21₄~W_mn!2

J

_{, ~29!}

with U˜_n5u*_x,Ldlme_man]ahd3_(z 2l).

Now, we note that up to _h term into U˜n field, the inte-grand of the above equation corresponds to the correlation function of MCS theory. However, since

^

A_m(x)A_*n( y )

&

M CS depends on 1/uz₂z´uthe contractions which involve the field

U˜n give rise to delta functionsd3(z2l) andd3(z2h) such that the line integral over dl_mand dh_mvanishes. This means that the integrand of the Eq.~30! corresponds to that of the MCS theory or,

^

m_u~x!m*u~y!

&

dual5exp(121/u 2₎

^

m_u~x!m

_*

_u~y!

&

M CS_. ~30!

Therefore, since the behavior of the vortex correlation func-tion operator in the MCS theory for very large distances ux2yu→` is a constant, indicating thatmu does not create genuine vortex excitations, we expect the same behavior for the dual theory.

For a future program, we intend to investigate the possible connection with the interesting formalism developed by Barci et al., where a mapping was made among some models in three dimensions@12#. This was done by using a nonlinear redefinition of the gauge field, in contrast to the linear self-dual transformation used in this work.

The authors are grateful to CNPq and FAPESP for partial financial support, and to D. Dalmazi for discussions.

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