Note on two-dimensional nonlinear gauge theories

12 Pages·0.12 MB·English

by  C. Bizdadea

Note on two-dimensional nonlinear gauge
theories
2
0
0
2 C. Bizdadea ∗

n Faculty of Physics, University of Craiova
a
J 13 A. I. Cuza Str., Craiova RO-1100, Romania

9

February 1, 2008
1
v
9
5
0
1 Abstract
0
2 A two-dimensional nonlinear gauge theory that can be proposed
0
for generalization to higher dimensions is derived by means of coho-
/
h mological arguments.
t
- PACS numbers: 11.10.Ef
p
e
h A big step in the progress of the BRST formalism was its cohomologi-
:
v cal understanding [1], which allowed, among others, a useful investigation of
i
X many interesting aspects related to the perturbative renormalization prob-
r lem [2]–[3], anomaly-tracking mechanism [3]–[4], simultaneous study of local
a
and rigid invariances of a given theory [5], as well as to the reformulation
of the construction of consistent interactions in gauge theories [6] in terms
of the deformation theory [7], or, actually, in terms of the deformation of
the solution to the master equation. Joint to these topics, the problem of
obtaining consistent deformations has naturally found its extension at the
Hamiltonian level by means of local BRST cohomology [8]. There is a large
varietyofmodelsofinterest intheoreticalphysics thathave beeninvestigated
in the light of the deformation of the master equation [9]–[10].
In this paper we investigate the consistent interactions that can be added
among a set ofscalar fields, two types of one-formsand asystem oftwo-forms
∗e-mail address: [e-mail]
1
in two dimensions, described in the free limit by anabelian BF theory [11], in
order to construct a two-dimensional nonlinear gaugetheory that can be pro-
posed for generalization to higher dimensions. Nonlinear gauge theories [12]
are important as they include pure two-dimensional gravitation theory [13],
which is expected to offer a conceptual mechanism for the study of quantum
gravity in higher dimensions from the perspective of gauge theories. More
precisely, when the nonlinear algebra is the Lorentz-covariant extension of
the Poincar´e algebra, the theory turns out to be the Yang-Mills-like formula-
tion of R2 gravity with dynamical torsion, or generic form of ‘dilaton’ gravity
[14].
Our strategy goes as follows. Initially, we determine the antifield-BRST
symmetry of the free model, that splits as the sum between the Koszul-Tate
differential and the exterior derivative along the gauge orbits, s = δ + γ.
Next, we deform the solution to the master equation of the free model. The
first-order deformation belongs to H0(s|d), where d is the exterior space-time
derivative. The computation of the cohomological space H0(s|d) proceeds
by expanding the co-cycles according to the antighost number, and by fur-
ther using the cohomological spaces H(γ) and H (δ|d). Subsequently, we
2
show that the consistency of the first-order deformation requires that all the
higher-order deformations vanish. With the help of the deformed solution to
the master equation we finally identify the interacting theory and its gauge
transformations, which form a nonlinear gauge algebra (open algebra) that
only closes on-shell.
We begin with a free model given by an abelian two-dimensional BF
theory involving a set of scalar fields, two types of one-forms and a system
of two-forms
1
S Aa,Ha,ϕ ,Bµν = d2x Ha∂µϕ + Bµν∂ Aa , (1)
0 µ µ a a µ a 2 a [µ ν]
h i Z (cid:18) (cid:19)
subject to the irreducible gauge invariances
δ Aa = ∂ ǫa, δ Ha = ∂νǫa , δ ϕ = 0, δ Bµν = 0, (2)
ǫ µ µ ǫ µ µν ǫ a ǫ a
where the notation [µν] means antisymmetry with respect to the indices
between brackets. A consistent deformation of the free action (1) and of
its gauge invariances (2) defines a deformation of the corresponding solu-
tion to the master equation that preserves both the master equation and
2
the field/antifield spectra. So, if SL Aa,Ha,ϕ ,Bµν + g d2xα + O(g2)
0 µ µ a a 0
stands for a consistent deformation ofhthe free action,iwithRdeformed gauge
transformations δ¯Aa = ∂ ǫa + gβa + O(g2), δ¯Ha = ∂νǫa + ρa + O(g2),
ǫ µ µ µ ǫ µ µν µ
δ¯ϕ = gβ +O(g2), δ¯Bµν = βµν+O(g2), then the deformed solution to the
ǫ a a ǫ a a
master equation
S¯ = S +g d2xα+O g2 , (3)
Z (cid:16) (cid:17)
satisfies S¯,S¯ = 0, where
(cid:16) (cid:17)
S = SL Aa,Ha,ϕ ,Bµν + d2x A∗µ∂ ηa +H∗µ∂νηa , (4)
0 µ µ a a a µ a µν
h i Z (cid:16) (cid:17)
and α = α +A∗µβ¯a +H∗µρ¯a +ϕ∗aβ¯ +B∗aβ¯µν +‘more’ (g is the so-called
0 a µ a µ a µν a
deformation parameter or coupling constant). The terms β¯a, ρ¯a, β¯ , β¯µν are
µ µ a a
obtained by replacing the gauge parameters ǫa and ǫa respectively with the
µν
fermionic ghosts ηa and ηa in the functions βa, ρa, β and βµν. The fields
µν µ µ a a
carrying a star denote the antifields of the corresponding fields or ghosts.
The Grassmann parity of an antifield is opposite to that of the corresponding
field/ghost. The pureghost number (pgh)andtheantighost number (antigh)
of the fields, ghosts and antifields are valued like
pghΦα0 = pgh Φ∗ = 0, pgh(ηα1) = 1, pgh η∗ = 0 (5)
α0 α1
(cid:16) (cid:17) (cid:16) (cid:17)
antighΦα0 = 0, antighΦ∗ = 1, antigh(ηα1) = 0, antigh η∗ = 2, (6)
α0 α1
(cid:16) (cid:17)
where we employed the notations
Φα0 = Aa,Ha,ϕ ,Bµν , Φ∗ = A∗µ,H∗µ,ϕ∗a,B∗a , (7)
µ µ a a α0 a a µν
(cid:16) (cid:17) (cid:16) (cid:17)
ηα1 = ηa,ηa , η∗ = (η∗,η∗µν). (8)
µν α1 a a
(cid:16) (cid:17)
The BRST symmetry of the free theory, s• = (•,S), simply decomposes as
the sum between the Koszul-Tate differential δ and the exterior derivative
along the gauge orbits γ, s = δ + γ, where the degree of δ is the antighost
number (antigh(δ) = −1, antigh(γ) = 0), and that of γ is the pure ghost
number (pgh(γ) = 1, pgh(δ) = 0). The grading of the BRST differential
is named ghost number (gh) and is defined in the usual manner like the
difference between the pure ghost number and the antighost number, such
3
that gh(s) = 1. The actions of δ and γ on the generators of the BRST
complex can be written as
δΦα0 = 0, δηα1 = 0, (9)
δA∗µ = ∂ Bνµ, δH∗µ = −∂µϕ , δϕ∗a = ∂µHa, (10)
a ν a a a µ
1 1
δB∗a = − ∂ Aa , δη∗ = −∂ A∗µ, δη∗µν = ∂[µH∗ν], (11)
µν 2 [µ ν] a µ a a 2 a
γAa = ∂ ηa, γHa = ∂νηa , γϕ = γBµν = 0, (12)
µ µ µ µν a a
γηα1 = γΦ∗ = γη∗ = 0. (13)
α0 α1
The master equation S¯,S¯ = 0 holds to order g if and only if
(cid:16) (cid:17)
sα = ∂ jµ, (14)
µ
for some local jµ. This means that the nontrivial first-order consistent in-
teractions belong to H0(s|d), where d is the exterior space-time derivative.
In the case where α is a coboundary modulo d (α = sλ + ∂ bµ), then the
µ
deformation is trivial (it can be eliminated by a redefinition of the fields).
In order to investigate the solution to (14), we develop α according to the
antighost number
α = α +α +...α , antigh(α ) = k, (15)
0 1 J k
where the last term can be assumed to be annihilated by γ, γα = 0. Thus,
J
we need to know the cohomology of γ, H(γ), in order to determine the terms
of highest antighost number in α. From (12-13) it is simple to see that the
cohomology of γ is generated by Fa = ∂ Aa , ∂µHa, ϕ , Bµν, the antifields
µν [µ ν] µ a a
together with their derivatives, as well as by the ghosts. If we denote by
eM (ηα1) a basis in the space of the polynomials in the ghosts, it follows that
the general solution to the equation γa = 0 takes the form
a = a Fa , ∂µHa [ϕ ],[Bµν], Φ∗ , η∗ eM (ηα1)+γb, (16)
M µν µ a a α0 α1
(cid:16)h i h i h i h i(cid:17)
where the notation f [q] signifies that f depends on q and its derivatives up
to a finite order. At this point we recall the cohomology of δ modulo the
exterior space-time derivative, H(δ|d). On account of the results inferred in
[10] it follows that it is vanishing for all antighost numbers strictly greater
4
than two, H (δ|d) = 0 for J > 2. Starting from the general form of an
J
object of antighost number two, a = Nα1η∗ + Mα0β0Φ∗ Φ∗ , where Nα1
2 α1 α0 β0
and Mα0β0 are functions of Φα0 and their derivatives, and requiring that a
2
belongs to H (δ|d), hence δa = ∂ mµ, we get that, up to a trivial term, the
2 2 µ
most general element from H (δ|d) can be represented as
2
δW δ2W δ2W
a = K η∗ − B η∗µν + A∗µH∗ −
2 δϕ c δϕ δϕ dµν c δϕ δϕ c dµ
c c d c d
1 δ3W δU
B H∗µH∗ν +K η∗µν+
2δϕcδϕdδϕe dµν c e ! µν δϕc c
1 δ2U
H∗µH∗ν = Ka˜ +K a¯µν, (17)
2δϕcδϕd c d ! 2 µν 2
where W and U are some functions involving only the scalar fields ϕ , while
a
K and K represent some constants, with K = −K . From (17) we find
µν µν νµ
in straightforward manner that δa˜ = ∂ m˜µ and δa¯µν = ∂ m¯βµν. Moreover,
2 µ 2 β
a is γ-closed, γa = 0. We have enough information as to solve the equation
2 2
(14). SinceH (δ|d)vanishesforJ > 2, onecanassume thatα = α +α +α .
J 0 1 2
As explained in the above, the general solution to the equation γα = 0 is
2
(up to a trivial term) α = α eM (ηα1), where pgh eM (ηα1) = 2 and
2 M
antigh(α ) = 2. Consequently, we have that (cid:16) (cid:17)
M
1 1
α = α ηaηb +αµν ηaηb + αµνλρηa ηb , (18)
2 2 ab ab µν 2 ab µν λρ
where α , αµν and αµνλρ are γ-invariant functions of antighost number two,
ab ab ab
that should satisfy in addition the symmetry properties
α = −α , αµν = −ανµ , αµνλρ = −αλρµν. (19)
ab ba ab ab ab ba
Here come in the results connected with H (δ|d) = 0 for J > 2. Equation
J
(14) projected on antighost number one is locally expressed by δα +γα =
2 1
∂ nµ. A necessary condition for the last equation to possess solution (or,
µ
equivalently, for α to exist) is that the functions α , αµν and αµνλρ belong
1 ab ab ab
to H (δ|d)
2
δα = ∂ kµ , δαµν = ∂ kβµν, δαµνλρ = ∂ kβµνλρ. (20)
ab µ ab ab β ab ab β ab
5
The existence of α demands in addition that the functions kβµν and kβµνλρ
1 ab ab
satisfy the equations
kβµν∂ ηb = µν ∂µηb , (21)
ab β µν ab µν
kβµνλρ∂ ηa ηb = σνλρ ∂µηa ηb +σµνρηa ∂ληb , (22)
ab β µν λρ ab µν λρ ab µν λρ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
for some µ and σ. In other words, only the objects from H (δ|d) that fulfill
2
the relations (21-22) are allowed to enter the solution (18). On the other
hand, the result (17) ensures that we can take
δW δ2W δ2W
α = K abη∗ − ab B η∗µν + ab A∗µH∗ −
ab δϕ c δϕ δϕ dµν c δϕ δϕ c dµ
c c d c d
1 δ3W
ab B H∗µH∗ν , (23)
2δϕcδϕdδϕe dµν c e !
δU 1 δ2U
αµν = K¯ abη∗µν + ab H∗µH∗ν , αµνλρ = 0, (24)
ab δϕc c 2δϕcδϕd c d ! ab
where W and U depend on the same fields respectively like W and U,
ab ab
W = −W , and K¯ is a constant. In this way, we can write down α like
ab ba 2
1 δW δ2W δ2W
α = K abη∗ − ab B η∗µν + ab A∗µH∗ −
2 2 δϕc c δϕcδϕd dµν c δϕcδϕd c dµ
1 δ3W δU
ab B H∗µH∗ν ηaηb +K¯ abη∗µν+
2δϕcδϕdδϕe dµν c e ! δϕc c
1 δ2U
ab H∗µH∗ν ηaηb . (25)
2δϕcδϕd c d ! µν
Rigorously speaking, we could have also added the term ε αµν (resulting
µν ab
from the admissible choice K = ε , as ε are the only covariant antisym-
µν µν µν
metric constants in two dimensions) to α . However, we avoided this term
ab
because we intend to construct only those deformations that are independent
of the space-time dimension. The presence of ε αµν would result, at the
µν ab
level of the deformed action and accompanying gauge structure, in quantities
proportional to ε , and will be therefore omitted. If we compute δα , we
µν 2
consequently deduce that the term of antighost number one in the first-order
6
deformation of the solution to the master equation is expressed by
δW δ2W
α = K ab A∗µAa −B∗aµνB + ab H∗νB Aaµ ηb +
1 δϕ c µ cµν δϕ δϕ d cµν
c c d !
(cid:16) (cid:17)
δU
K¯ U B∗aµνηb +ϕ∗bηa − abH∗ν Aaµηb +Hbηa . (26)
ab µν δϕ c µν ν
c !
(cid:16) (cid:17) (cid:16) (cid:17)
Withα athand, wedetermine α asthesolutiontotheequationδα +γα =
1 0 1 0
∂ lµ, which actually reads as
µ
K δW
α = abBµνAaAb −K¯U AaµHb. (27)
0 2 δϕ c µ ν ab µ
c
Thus, so far we have completely determined the deformation to order g,
S = d2xα.
1
If we denote by S = d2xβ the second-order deformation, the master
R 2
equation S¯,S¯ = 0 holds to order g2 if and only if ∆ = −2sβ+∂ θµ, where
R µ
(S ,S ) =(cid:16) d2(cid:17)x∆. This means that in order to have a deformation that is
1 1
consistent to order g2, the integrand of (S ,S ) should be s-exact modulo d.
R 1 1
This takes place if and only if
K = K¯, U = W , (28)
ab ab
and also
δW
t ≡ W cd] = 0. (29)
bcd a[b δϕ
a
Indeed, on account of (28) we get that
δt δ2t δ3t
∆ = K2 t ubcd + bcdv bcd + bcd z bcd + bcd w bcd , (30)
bcd δϕ e δϕ δϕ en δϕ δϕ δϕ enm
e e n e n m !
where
ubcd = AbµAcν −B∗bµνηc ηd − AbµHd +ϕ∗bηd ηc, (31)
µν µ
(cid:16) (cid:17) (cid:16) (cid:17)
v bcd = BµνAbAd +A∗µAbηd ηc + BµνB∗b −H∗µHb−
e e µ ν e µ e µν e µ
1 (cid:16) (cid:17) (cid:16)
η∗ηb ηcηd + H∗νAbµ −η∗µνηb ηcηd , (32)
3 e e e µν
(cid:19) (cid:16) (cid:17)
7
1
z bcd = η∗µνB ηb −B H∗νAbµ − H∗µH∗νηb −
en e nµν eµν n 2 e n µν
(cid:18)
A∗ H∗µηb ηcηd, (33)
eµ n
(cid:17)
1
w bcd = B H∗µH∗νηbηcηd. (34)
enm 6 eµν n m
From (31-34) one observes that ∆ given in (30) cannot be s-exact modulo
d, therefore it should vanish. This happens if and only if the functions
W satisfy (29), which is nothing but Jacobi’s identity for a nonlinear gauge
ab
algebra [12]. In conclusion, the consistency at order g2 implies S = 0. Then,
2
the higher-order deformation equations are identically satisfied if we choose
S = ··· = S = ··· = 0.
3 k
For concreteness, we work with K = 1, such that the deformed solution
tothe master equation, consistent to allorders inthedeformationparameter,
reads as
1 δW
S¯ = d2x HaDµϕ + BµνF¯a +A∗µ ∂ ηa −g bcAcηb +
Z µ a 2 a µν a µ δϕa µ !
δW δW δ2W
H∗µ ∂νηa +g bcAbνηc − bcHcηb + bc B Acνηb −
a µν δϕ µν δϕ µ δϕ δϕ dµν
a a a d !!
δW
gϕ∗aW ηb +gB∗aµν W ηb − abB ηb +
ab ab µν δϕ cµν
c !
g δW δ2W δ2W
abη∗ − ab B η∗µν + ab A∗µH∗ −
2 δϕc c δϕcδϕd dµν c δϕcδϕd c dµ
1 δ3W δW
ab B H∗µH∗ν ηaηb +g abη∗µν+
2δϕcδϕdδϕe dµν c e ! δϕc c
1 δ2W
ab H∗µH∗ν ηaηb , (35)
2δϕcδϕd c d ! µν!
where
δW
Dµϕ = ∂µϕ +gW Abµϕ , F¯a = ∂ Aa +g bcAbAc. (36)
a a ab a µν [µ ν] δϕ µ ν
a
At this stage, we have all the information for identifying the gauge theory
behind our deformation procedure. From the antighost number zero piece
8
in (35), it follows that the Lagrangian action that describes the deformed
model has the expression
1
S¯ Aa,Ha,ϕ ,Bµν = d2x HaDµϕ + BµνF¯a , (37)
0 µ µ a a µ a 2 a µν
h i Z (cid:18) (cid:19)
while from the antighost number one components we read the corresponding
deformed gauge transformations
δW
δ¯Aa = ∂ ǫa −g bcAcǫb, (38)
ǫ µ µ δϕ µ
a
δW δW δ2W
δ¯Ha = ∂νηa +g bcAbνǫc − bcHcǫb + bc B Acνǫb , (39)
ǫ µ µν δϕ µν δϕ µ δϕ δϕ dµν
a a a d !
δW
δ¯ϕ = −gW ǫb, δ¯Bµν = g W ǫbµν − abBµνǫb . (40)
ǫ a ab ǫ a ab δϕ c
c !
The form of the coefficients of the terms proportional with one antifield of
the ghosts and two ghosts indicate that the gauge algebra is nonlinear, and,
moreover, thepresenceofthequantitiesinvolvingtwoantifieldsoftheoriginal
fields and two ghosts shows that this algebra only closes on-shell. It is now
clear that the deformed Lagrangian action, as well as the resulting gauge
structure, does not contain in any way the two-dimensional antisymmetric
symbol, aswehavepreviously required. Inview ofthis, thereishopethatour
deformation mechanism can be properly generalized to higher dimensions.
To conclude with, in this paper we have investigated the consistent in-
teractions that can be introduced among a set of scalar fields, two types of
one-forms and a system of two-forms in two dimensions, described in the
free limit by an abelian BF theory. Starting with the BRST differential for
the free theory, s = δ + γ, we compute the consistent first-order deforma-
tion with the help of some cohomological arguments. Next, we prove that
the deformation is also second-order consistent, and, moreover, matches the
higher-order deformation equations. As a result, we are precisely led to a
two-dimensional nonlinear gauge theory, that can be in principle extended
to higher dimensions. Our deformation procedure modifies the gauge trans-
formations, as well as their algebra. Moreover, the deformed gauge algebra
is open and closes on-shell.
9
Acknowledgment
This workhasbeensupportedbyaRomanianNationalCouncilforAcademic
Scientific Research (CNCSIS) grant.
References
[1] E. S. Fradkin, G. A. Vilkovisky, Phys. Lett. B55 (1975) 224; I. A.
Batalin, G. A. Vilkovisky, Phys. Lett. B69 (1977) 309; E. S. Fradkin, T.
E.Fradkina,Phys. Lett.B72(1978)343; I.A.Batalin, G.A.Vilkovisky,
Phys. Lett. B102 (1981) 27; I. A. Batalin, E. S. Fradkin, Phys. Lett.
B122 (1983) 157; I. A. Batalin, G. A. Vilkovisky, Phys. Rev. D28
(1983) 2567; I. A. Batalin, G. A. Vilkovisky, J. Math. Phys. 26 (1985)
172; M. Henneaux, Phys. Rep. 126 (1985) 1; A. D. Browning, D. Mc
Mullan, J. Math. Phys. 28 (1987) 438; M. Dubois-Violette, Ann. Inst.
Fourier 37 (1987) 45; D. Mc Mullan, J. Math. Phys. 28 (1987) 428; M.
Henneaux, C. Teitelboim, Commun. Math. Phys. 115 (1988) 213; J. D.
Stasheff, Bull. Amer. Math. Soc. 19 (1988) 287; J.Fisch, M. Henneaux,
J. D.Stasheff, C. Teitelboim, Commun. Math. Phys. 120(1989)379; M.
Henneaux, Nucl. Phys. B (Proc. Suppl) 18A (1990) 47; M. Henneaux,
C. Teitelboim, Quantization of Gauge Systems (Princeton University
Press, Princeton, New Jersey) 1992
[2] B.Voronov,I.V.Tyutin, Theor.Math.Phys.50(1982)218;B.Voronov,
I. V. Tyutin, Theor. Math. Phys. 52 (1982) 628; J. Gomis, S. Weinberg,
Nucl. Phys. B469 (1996) 473; S. Weinberg, The Quantum Theory of
Fields (Cambridge University Press, Cambridge) 1996
[3] O. Piguet, S. P. Sorella, Algebraic Renormalization: Perturbative
Renormalization, Symmetries and Anomalies (Lecture Notes in Physics,
Springer Verlag, Berlin) Vol. 28 1995
[4] P. S. Howe, V. Lindstr˝om, P. White, Phys. Lett. B246 (1990) 130;
W. Troost, P. van Nieuwenhuizen, A. van Proeyen, Nucl. Phys. B333
(1990) 727; G. Barnich, M. Henneaux, Phys. Rev. Lett. 72 (1994) 1588;
G. Barnich, Mod. Phys. Lett.A9 (1994) 665; G. Barnich, Phys. Lett.
B419 (1998) 211
10