BRST gauge fixing and the algebra of global supersymmetry

18 Pages·0.17 MB·English

by  Kazuo Fujikawa

UT-785
BRST gauge fixing and
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9 the algebra of global supersymmetry
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Kazuo Fujikawa and Kazumi Okuyama
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Department of Physics,University of Tokyo
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v Bunkyo-ku,Tokyo 113,Japan
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[email protected] , [email protected]
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Xi Abstract
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a A global supersymmetry (SUSY) in supersymmetric gauge theory is generally
broken by gauge fixing. A prescription to extract physical information from such
SUSY algebra broken by gauge fixing is analyzed in path integral framework. If
δ δ Ψ = δ δ Ψ for a gauge fixing “fermion” Ψ, the SUSY charge
SUSY BRST BRST SUSY
density is written as a sum of the piece which is naively expected without gauge
fixing and a BRST exact piece. If δ δ δ Ψ = δ δ δ Ψ, the
SUSY SUSY BRST BRST SUSY SUSY
equal-time anti-commutator of SUSY charge is written as a sum of a physical piece
and a BRST exact piece. We illustrate these properties for N = 1 and N = 2
supersymmetric Yang-Mills theories and for a D = 10 massive superparticle (or
“D-particle”) where the κ-symmetry provides extra complications.
1
1 Introduction
Insomeoftheapplicationsofsupersymmetric (SUSY)gaugetheory[1],theSUSYalgebra,
in particular, its possible central extension provides important physical information[2].
On the other hand, the global SUSY in generic supersymmetric gauge theory is generally
broken by gauge fixing, at least in the component formulation in the Wess-Zumino gauge.
The SUSY charge thus generally becomes time-dependent after gauge fixing. In such a
situation, it is important to specify how to extract physical information from the broken
charge algebra. In the present note, we present a general prescription how to extract
physical information from the equal-time anti-commutator of such broken SUSY charge
densityintheframeworkofpathintegralandtheBjorken-Johnson-Low(BJL)prescription
[3]. The quantities in path integral , which are defined in terms of T∗-product , are
converted into those in the conventional T-product by BJL prescription, which in turn
readily provide information about the equal-time (anti-) commutator.
The essence of our analysis is summarized as follows: If one defines the gauge fixing
Lagrangian by = δ Ψ, where Ψ is a gauge fixing “fermionic” function, the SUSY
g BRST
L
charge density J0 is written as
SUSY
J0 = J0 + BRST exact piece (1.1)
SUSY SUSY(naive)
if
δ δ Ψ = δ δ Ψ (1.2)
SUSY BRST BRST SUSY
where δ is a localized (coordinate dependent) SUSY transformation, and J0
SUSY SUSY(naive)
is the naive charge density expected before gauge fixing.
If one has the relation with another SUSY transformation
δ′ δ δ Ψ = δ δ′ δ Ψ (1.3)
SUSY SUSY BRST BRST SUSY SUSY
one obtains the equal-time anti-commutation relation
Qα(x0),Qβ(y0) x0=y0 = physical piece+BRST exact piece (1.4)
{ }
where the physical piece as well as the BRST exact piece are obtained from a SUSY
transformation of the SUSY charge density in path integral formulation. These properties
holdwhen oneassigns alltheunphysical (ghost) particles tobe SUSY scalar, namely, they
are not transformed under SUSY transformation.
2
An advantage of this path integral formulation is that one can readily identify the
possible presence of generic “Schwinger terms” such as the physical central extension. It
turns out that the second property in (1.4) holds for a more general class of theories and
gauge fixing than the first property in (1.1). The properties (1.1) and (1.2) have been
analyzed in the past in Refs.[4] and [5]. To our knowledge, a systematic analysis of (1.3)
and (1.4) has not been performed before.
In the following, we illustrate these properties for N = 1 and N = 2 supersymmetric
Yang-Mills theories and for a D = 10 massive superparticle which may be regarded as
a “D-particle”. In the latter case, the so-called κ- symmetry gives rise to additional
technical complications.
= 1
2 N super Yang-Mills theory
As a first example, we consider supersymmetric Yang-Mills theory . The Lagrangian is
given by
1 1
= 2tr F Fmn +iλ¯σ¯mD λ (2.1)
L0 −g2 4 mn m
(cid:18) (cid:19)
where
D λ = ∂ λ i[A ,λ]
m m m
−
F = ∂ A ∂ A i[A ,A ] (2.2)
mn m n n m m n
− −
Here we normalized the generators Ta of gauge symmetry by trTaTb = 1δab. The Landau-
2
type gauge , for example, is defined by
= 2tr(B∂ Am ic¯∂ Dmc) = 2trδ (c¯∂ Am) (2.3)
g m m BRST m
L −
The SUSY transformation rules are
¯ ¯
δ A = i(ξσ¯ λ λσ¯ ξ)
SUSY m m m
−
δ λ = σmnξF
SUSY mn
δ λ¯ = ξ¯σ¯mnF (2.4)
SUSY mn
−
and the BRST transformation rules are defined by using a Grassmann parameter ǫ as
δA = iǫD c = iǫ(∂ c i[A ,c])
m m m m
−
3
¯ ¯
δλ = [ǫc,λ], δλ = [ǫc,λ]
− −
1
δc = [ǫc,c]
−2
δc¯= ǫB, δB = 0 (2.5)
Here c andc¯aretheFaddeev-Popov ghost andanti-ghost, andB is theNakanishi-Lautrup
field. We can confirm that δ and δ commute on A and λ by the above trans-
BRST SUSY µ
formation rules
δ δ A = δ δ A = [ǫc,δ A ]
BRST SUSY µ SUSY BRST µ SUSY µ
−
δ δ λ = δ δ λ = [ǫc,δ λ] (2.6)
BRST SUSY SUSY BRST SUSY
−
In other words, the gauge fixing fermion Ψ = c¯∂ Am in (2.3) satisfies
m
δ δ Ψ = δ c¯∂ (δ Am)
BRST SUSY BRST m SUSY
= B∂ (δ Am)+c¯∂ (δ δ Am)
m SUSY m BRST SUSY
= B∂ (δ Am)+c¯∂ (δ δ Am)
m SUSY m SUSY BRST
= δ (δ (c¯∂ Am))
SUSY BRST m
= δ δ Ψ (2.7)
SUSY BRST
and similarly
δ δ′ δ Ψ = δ′ δ δ Ψ (2.8)
BRST SUSY SUSY SUSY SUSY BRST
even for the localized (coordinate dependent) SUSY transformation. In fact these prop-
erties hold for a more general class of gauge fixing fermion Ψ = c¯F(A ,λ). Note that
m
all the un-physical particles are assigned to be SUSY scalar, namely, they are not trans-
formed under SUSY transformation. These properties suggest that the supercurrent and
superalgebra have the structure (1.1) and (1.4), which can be confirmed below.
The variation of the total Lagrangian = + under SUSY transformation with
0 g
L L L
position dependent parameters is
δ = ∂mξ¯S¯ +ξ¯J¯+S ∂mξ +Jξ (2.9)
SUSY m m
L
where the SUSY currents and SUSY violating source terms are given by
2i 2i
S = tr(λ¯σ¯ σ Fkl) 2itrδ (c¯λ¯σ¯ ) = tr(2λ¯σ¯nF+ ) 2itrδ (c¯λ¯σ¯ )
m −g2 m kl − BRST m −g2 nm − BRST m
4
2i 2i
S¯ = tr(Fklσ¯ σ¯ λ) 2itrδ (c¯σ¯ λ) = tr(2F− σ¯nλ) 2itrδ (c¯σ¯ λ)
m −g2 kl m − BRST m −g2 mn − BRST m
J = 2itrδ (c¯∂mλ¯σ¯ )
BRST m
−
J¯= 2itrδ (c¯σ¯ ∂mλ) (2.10)
BRST m
These SUSY currents S and S¯ have the structure indicated in (1.1). Here we defined
m m
1 1 1
F± = (F Fˆ ) = F i ε Fkl (2.11)
mn 2 mn ± mn 2 mn ± 2 mnkl
(cid:18) (cid:19)
where Fˆ = ˆF = i F. In Lorentz metric ˆ2F = F, and F± are a self-dual (or anti-self-
∗ ∗ ∗
dual) part of F ,i.e., ˆF± = F±.
∗ ±
A Ward-Takahashi identity for supersymmetry is obtained by starting with
S (y)η dµS (y)ηei Ld4x (2.12)
a a
h i ≡
Z R
with a global Grassmann parameter η and performing the change of field variables un-
¯
der the localized SUSY transformation parametrized by ξ(x) and ξ(x). By taking the
¯
variational derivative with respect to ξ(x) and ξ(x) , one obtains W-T identities. For
notational simplicity, we here write the identities obtained by multiplying global ξ and ξ¯
anew:
d3xξ[∂ T∗S0(x)S (y)η T∗J(x)S (y)η ]
0 a a
h i−h i
Z
+ d3xξ¯[∂ T∗S¯0(x)S (y)η T∗J¯(x)S (y)η ]
0 a a
h i−h i
Z
= δ(x0 y0) 2ξ¯σ¯mη T˜ (y) 2trδ (c¯F− )(y) +ξM (y)η
− ma − BRST ma a
∂ D (cid:16) (cid:17) E
δ(x0 y0) 4tr λ¯σ¯ σ0lηδ A (y) (2.13)
a SUSY l
−∂x0 −
D (cid:16) (cid:17) E
where we defined
3
ξM η = (iξσ η iη ξη)2tr[λ¯σ¯nσkD λ¯] (2.14)
a an an k
− 2
˜
and T is the energy-momentum tensor for supersymmetric Yang-Mills theory
ma
T˜ = TYM +Tλ
ma ma ma
1
g2TYM = 2tr F F k η F Fkl
ma mk a − 4 ma kl
(cid:18) (cid:19)
i 1
g2Tλ = 2tr(λ¯σ¯ D λ+3λ¯σ¯ D λ) η 2tr(iλ¯σ¯kD λ)
ma 4 m a a m − 4 ma k
1
ε 2tr(λ¯σ¯nDkλ) (2.15)
makn
− 4
5
We here recall the basic idea of the BJL prescription [3]. The correlation function
defined by T∗-product
T∗A(x)B(y) (2.16)
h i
can be replaced by the conventional T-product if
lim d4xeiq(x−y) T∗A(x)B(y) = 0 (2.17)
q0→∞ h i
Z
If the above quantity does not vanish, one defines the T-product by
d4xeiq(x−y) TA(x)B(y) d4xeiq(x−y) T∗A(x)B(y)
h i ≡ h i
Z Z
lim d4xeiq(x−y) T∗A(x)B(y) (2.18)
− q0→∞ h i
Z
In either case we have
lim d4xeiq(x−y) TA(x)B(y) = 0 (2.19)
q0→∞ h i
Z
which defines the T-product in general.
In the present context, the term proportional to ∂ δ(x0 y0) in the right-hand side
∂x0 −
is subtracted away if one uses the T-product in eq.(2.13). We then obtain
d3xξ[∂ TS0(x)S (y)η TJ(x)S (y)η ]
0 a a
h i−h i
Z
+ d3xξ¯[∂ TS¯0(x)S (y)η TJ¯(x)S (y)η ]
0 a a
h i−h i
Z
= δ(x0 y0) 2ξ¯σ¯mη T˜ (y) 2trδ (c¯F− )(y) +ξM (y)η (2.20)
− ma − BRST ma a
D (cid:16) (cid:17) E
If one performs the explicit time derivative operation in the left-hand side and if one uses
the current conservation condition following from (2.9)
d3x(∂ S0(x) J(x)) = d3x(∂ S¯0(x) J¯(x)) = 0 (2.21)
0 0
− −
Z Z
one obtains the following commutation relations
ξ¯Q¯(x0),S (y)η = 2ξ¯σ¯mη T˜ 2trδ (c¯F− ) (y) (2.22)
a x0=y0 ma − BRST ma
h i (cid:16) (cid:17)
ξQ(x0),S (y)η = ξM (y)η (2.23)
a x0=y0 a
h i
Eq.(2.22) shows that SUSY algebra at equal-time closes up to a BRST exact term. If
the right-hand side of Eq.(2.23) does not vanish, it would represent a possible “central
6
extension of N=1 SUSY algebra”. Note that nowhere in our calculation the equations of
motion have been used. The right-hand side of Eq.(2.23) vanishes if one uses the (safe)
equation of motion in Eq.(2.14). We thus recover the conventional SUSY algebra defined
in Poincare invariant vacuum , as is described in (1.4). A further comment on the central
extension will be given in Section 5.
In passing, we here note several useful relations in our path integral manipulation. To
obtain (2.10) and also identify the energy-momentum tensor (2.15), we use the following
identities,
σ¯lσmσ¯n = ηlmσ¯n ηmnσ¯l +ηlnσ¯m iεlmnkσ¯
k
− − −
F σ¯klσ¯m = 2Fmnσ¯
kl − n
σ¯mσklF = 2σ¯ Fnm (2.24)
kl n +
For the general 2-form A and B , we have the identities
mn mn
A+ B−mn = 0
mn
1
A± B±k +B± A±k = η A±B±kl
mk n mk n 2 mn kl
A+ B−k = B− A+k = A− B+k (2.25)
mk n mk n mk n
Using these identities, theenergy-momentum tensor ofgaugefieldTYM in(2.15)iswritten
mn
as
4 4
TYM = trF+ F−k = trF− F+k (2.26)
mn g2 mk n g2 mk n
In the evaluation of (2.14), we used the relation
ξσmkηtr[λ¯σ¯ D λ¯] = ξσ ηtr[λ¯σ¯mkD λ¯] (2.27)
ma k ma k
= 2
3 N super Yang-Mills theory
We next analyze the superalgebra of N = 2 super Yang-Mills theory. The Lagrangian of
N = 2 super Yang-Mills theory is given by Ref.[6].
2 1
= tr FmnF D φ†D φ iλ¯σ¯mD λ iψ¯σ¯mD ψ
L0 g2 −4 mn − m m − m − m
(cid:18)
1 θ
+i√2(ψ¯[λ¯,φ]+[λ,φ†]ψ) [φ,φ†]2 + trF F˜mn (3.1)
mn
− 2 16π2
(cid:19)
7
where F˜mn = 1εmnklF . This Lagrangianisinvariant under N = 2SUSY. The first SUSY
2 kl
transformation δ(1) is defined by
δ(1)φ = √2ξψ , δ(1)φ† = 0
ξ ξ
δ(1)ψ = 0 , δ(1)ψ¯ = i√2D φ†ξσm
ξ ξ − m (3.2)
δ(1)λ = (σklF +i[φ,φ†])ξ , δ(1)λ¯ = 0
ξ kl ξ
(1) ¯
δ A = iλσ¯ ξ
ξ m − m
¯
WewriteonlytheSUSYtransformationwiththeparameterξ;SUSYwiththeparameterξ
is given by its complex conjugation. The variables λ and ψ form an SU(2) doublet. The
R
second SUSY transformation δ(2) is obtained from δ(1) by the SU(2) rotation (λ,ψ)
R
→
(ψ, λ),
−
δ(2)φ = √2ξλ , δ(2)φ† = 0
ξ − ξ
δ(2)ψ = (σklF +i[φ,φ†])ξ , δ(2)ψ¯ = 0
ξ kl ξ (3.3)
δ(2)λ = 0 , δ(2)λ¯ = i√2D φ†ξσm
ξ ξ m
(2) ¯
δ A = iλσ¯ ξ
ξ m − m
The gauge fixing term for the Landau gauge is given by
= 2tr(B∂ Am ic¯∂ Dmc) = 2trδ (c¯∂ Am) (3.4)
g m m BRST m
L −
BRST transformation rules are defined by
δA = iǫD c
m m
δλ = [ǫc,λ] , δψ = [ǫc,ψ]
− −
δφ = [ǫc,φ]
−
1
δc = [ǫc,c]
−2
δc¯= ǫB , δB = 0 (3.5)
We can confirm that δ and δ thus defined commute on φ,ψ,λ andA . See (2.6).
BRST SUSY m
We thus conclude that the general analyses (1.1) - (1.4) apply to the present case, as is
explicitly demonstrated below.
The variation of the total Lagrangian = + under SUSY transformation with
0 g
L L L
position dependent parameters ξ (A = 1,2) is
δ(A) = S(A)∂mξ +J(A)ξ (3.6)
ξ L m
8
where the supercurrents and SUSY violating densities are given by
2i
S(1)η = tr λ¯σ¯ (σklF +i[φ,φ†])η i√2D φ†ησkσ¯ ψ 2itrδ (c¯λ¯σ¯ η)
m g2 − m kl − k m − BRST m
h i
2i
= trδ(1)( λ¯σ¯ λ+ψ¯σ¯ ψ) 2itrδ (c¯λ¯σ¯ η) (3.7)
g2 η − m m − BRST m
2i
S(2)η = tr ψ¯σ¯ (σklF +i[φ,φ†])η i√2D φ†ησkσ¯ λ 2itrδ (c¯ψ¯σ¯ η)
m g2 − m kl − k m − BRST m
h i
2i
= trδ(2)( ψ¯σ¯ ψ +λ¯σ¯ λ) 2itrδ (c¯ψ¯σ¯ η) (3.8)
g2 η − m m − BRST m
J(1) = 2itrδ (c¯∂ λ¯σ¯m) (3.9)
BRST m
−
J(2) = 2itrδ (c¯∂ ψ¯σ¯m) (3.10)
BRST m
−
We multiplied supercurrents by a global Grassmann parameter η to form Lorentz scalar
quantities.
An interesting property of N = 2 superalgebra is the general existence of the cen-
tral charge, which can appear in Q(1),Q(2) [1]. To calculate this anti-commutator, we
{ α β }
start with the Ward-Takahashi identity for supersymmetry in path integral formulation,
which is obtained from the expression S(2)(y)η and a change of integration variables
h m i
(1)
corresponding to localized supersymmetry δ ,
ξ
d3x ξ ∂ T∗S0(1)(x)S(2)(y)η T∗J(1)(x)S(2)(y)η
0h m i−h m i
Z h 2 i
= 2√2iδ(x0 y0) ξη ∂ntr (F +iF˜ )φ† (y) ξσ σ¯ η trδ (c¯Dkφ†)(y)
nm nm k m BRST
− − g2 −
(cid:28) h i (cid:29)
∂ 2
δ(x0 y0) tr ψ¯σ¯ σ0kηδ(1)A (y) (3.11)
−∂x0 − g2 m ξ k
(cid:28) (cid:16) (cid:17) (cid:29)
where, for notational simplicity, we wrote the Ward-Takahashi identity multiplied by a
global ξ. We also used the equations of motion
DnF = i[D φ†,φ] i[φ†,D φ] [λ¯,σ¯ λ] [ψ¯,σ¯ ψ] (3.12)
nm m m m m
− − −
σ Dmλ = √2[φ†,ψ] (3.13)
m
intheabovederivation, whichisexpectedtobeasafeoperation. Thetermproportionalto
∂ δ(x0 y0) in (3.11) can be dropped if one uses the T-product by the BJL prescription.
∂x0 −
We then obtain
2
ξQ(1)(x0),ηS(2)(y) = 2√2i ξη ∂ntr (F +iF˜ )φ† (y)
m x0=y0 − g2 nm nm
h i (cid:18) h i
ξσ σ¯ η trδ (c¯Dkφ†)(y) (3.14)
k m BRST
−
(cid:19)
9
after rewriting (3.11) in terms of the T-product and operating the time-derivative in the
left-hand side explicitly. The electric charge n and magnetic charge n are given by
e m
2 θ
d3x ∂ tr(Fn0φ†) = a∗ n + n (3.15)
n e m
g2 2π
Z (cid:16) (cid:17)
2 4π
d3x ∂ tr(F˜n0φ†) = a∗n (3.16)
n m
g2 −g2
Z
where a is the asymptotic value of φ at spatial infinity. Eq.(3.15) represents the so called
“Witten effect” [7]. Note that θ does not appear in the expressions of supercurrents, and
it enters in the superalgebra only through (3.15).
The equal-time commutator of supercharges is obtained from (3.14) by integrating
over d3y as
ξQ(1)(x0),ηQ(2)(x0) = 2√2i ξηZ∗ ξσ σ¯0η d3x trδ (c¯Dkφ†)(x) (3.17)
k BRST
− −
h i (cid:18) Z (cid:19)
where the central charge is given by
Z = an +τan (3.18)
e m
θ 4π
τ = +i (3.19)
2π g2
Eq.(3.17) has a general structure as in (1.4), and Z gives rise to the standard formula of
thecentralchargeofN = 2super Yang-Millstheory[2]. Weemphasize thatourderivation
of (3.17) (3.19) is fully quantum mechanical in the framework of path integral, which is
∼
based on the Lorentz covariant Landau gauge.
= 10
4 D massive superparticle (D-particle)
In this section we analyze the super algebra for a D = 10 massive superparticle in the
BRST framework. Due to the well-known complications arising from the off-shell non-
closure of κ-symmetry, a straightforward prescription does not work. Nevertheless, we can
devise a working prescription for this case also. The Lagrangianofa massive superparticle
is given by [8]
Π2 M2
= m e iMθ¯Γ θ˙ (4.1)
0 11
L 2e − 2 −
where e is the einbein , M is the mass of the superparticle and we defined
Π = X˙ iθ¯Γ θ˙ (4.2)
m m m
−
10