AXIAL Symmetry and Anti-BRST Invariance Amir Abbass...
Transcript of AXIAL Symmetry and Anti-BRST Invariance Amir Abbass...
arXiv:hep-th/1011.1095
1. Introduction to Geometric BRST
2. Axial Extension of Gauge Theories
3. Anti-Ghost and Anti-BRST Symmetry
4. Application and Conclusions
๐บ๐๐ข๐๐ ๐๐ฆ๐๐๐๐ก๐๐ฆ๐ธ๐๐๐๐๐๐๐๐๐๐๐
๐ต๐ ๐๐ ๐๐ฆ๐๐๐๐ก๐๐ฆ
๐ฌ๐๐๐๐๐๐๐๐ โ โ ๐ฌ๐๐๐๐๐๐๐๐๐ด๐ฅ๐๐๐ ๐๐ฆ๐๐๐๐ก๐๐ฆ
๐ธ๐๐๐๐๐๐๐๐๐๐๐๐ด๐๐ก๐ โ ๐ต๐ ๐๐ ๐๐ฆ๐๐๐๐ก๐๐ฆ
.
1. Introduction to Geometric BRST
Let ๐ be a 2๐-dimensional spin manifold with spin bundle ๐ โช ๐(๐) โ ๐ and ๐บ be a semi simple Lie group with a unitary irreducible representation on finite dimensional complex vector space ๐.
Suppose that ๐บ โช ๐ โ ๐ is a principal ๐บ -bundle equipped with a connection structure given by Cartan connection form ๐ and curvature ฮฉ = d๐ + ๐2. Then ๐ธ โ ๐ ร๐บ ๐, the equivalency classes of (๐, ๐ฃ) for;
๐, ๐ฃ ~ ๐ โฒ ๐, ๐โ1 โณ ๐ฃ ,
๐ โ ๐ and ๐ฃ โ ๐, produces a vector bundle over ๐ with standard fiber ๐;
๐ โช ๐ธ โ ๐.
Moreover the Cartan connection form ๐ induces a parallelism structure over ๐ โช ๐ธ โ ๐. This induced connection structure together with the pull back of Levi-Civita connection over โ2๐ โช ๐๐ โ ๐ via;
๐ต ๐ ๐ ร ๐๐๐๐ 2๐ ๐ด๐๐ก๐๐๐
๐ต(๐ ๐ ) โ ๐
โ ร โ โ = ๐ต ๐๐ ร ๐๐ 2๐
๐ด๐๐ก๐๐๐ ๐ต ๐๐ โ ๐
,
fixes a connection over the ๐ฎ-gauge theory bundle; ๐โจ๐ โช ๐ธโจ๐ ๐ โ ๐.
Therefore the ๐บ -gauge theory bundle is equipped with covariant derivative ๐ป locally given by;
๐ป๐๐ = ๐๐ โ ๐๐ด๐๐๐๐ โ ๐๐
๐๐ ๐พ๐ , ๐พ๐ ๐,
for anti-Hermitian matrices *โ๐๐๐+๐ as a representing basis of ๐ค โ Lie๐บ, and for Dirac matrices *๐พ๐+๐.
Let ๐ข โ ๐ถโ(๐, ๐บ) be the gauge transformation group, ๐บ โช ๐ โ ๐ be a trivial bundle (only for simplicity) and ๐: ๐ โ ๐ be considered as the global identity section. Also let ๐ be the Affine space of ๐โ(๐ ) for Cartan connection forms ๐ over ๐บ โช ๐ โ ๐. Indeed ๐ can be considered as the space of gauge fields ๐ด = โ๐๐ด๐
๐๐๐d๐ฅ๐. There is a free action of ๐ข on ๐
from right; ๐ด โฒ ๐ โ Ad๐โ1 ๐ด + ๐โ1d๐,
for ๐ด โ ๐, ๐ โ ๐ข. Therefore we have the following principal ๐ข-bundle; ๐ข โช ๐ โ ๐/๐ข.
Set a connection over this principal bundle with Cartan connection form ฮ , fix a connection over ๐บ โช ๐ โ ๐ with Cartan connection form ๐0, ๐ด0 = ๐โ(๐0), and define the fiber map ๐๐ด0
: ๐ ร ๐ข โ ๐ ร ๐, with;
๐๐ด0๐, ๐ โ ๐, ๐ด0 โฒ ๐ .
Let ฮ โ ฮฉ1(๐ ร ๐)โจ๐ค be defined by;
ฮ ๐ฃ, ๐ โ ๐ด ๐ฃ + ฮ ๐ ๐ ,
for ๐ฃ, ๐ โ ๐๐๐ ร ๐๐ด๐, and define; ๐๐ด0
โ d๐ = ๐ฟ (the exterior derivative
over ๐ข). Then it can be easily shown that; ๐๐ด0
โ ฮ (๐,๐)
= ๐ด + ๐, for
๐ด = ๐ด0 โฒ ๐ and for ๐ โ ๐๐ด0
โ ฮ a left invariant Lie๐ข-valued 1-form over
๐ข.
Moreover we have;
๐ฟ๐ด = d๐ + ๐ด, ๐ , ๐ฟ๐ = ๐2 =1
2,๐, ๐-
๐ฟd + d๐ฟ = ๐ฟ, d = 0 , ๐ฟ2 = 0.
It is seen that by considering ๐ด as the gauge field, ๐ and ๐ฟ are respectively in complete agreement with ghost field and BRST derivation.
2. Axial Extension of Gauge Theories
Let ๐ be โ2๐ equipped with the Minkowski (Euclidean) metric and ๐บ be a simply connected semi simple Lie group with Lie algebra ๐ค. Also suppose that ๐บ acts unitarily and irreducibly on finite dimensional complex vector space ๐ with *โ๐๐๐+๐ the anti-Hermitian represented basis for ๐ค.
Consider a Yang-Mills ๐บ-gauge theory over โ2๐ with Lagrangian density;
โ = โ1
4๐ก๐*๐น๐๐๐น๐๐+ + ๐๐ ๐พ๐๐๐๐ + ๐ด๐
๐๐ ๐พ๐๐๐๐,
for ๐น๐๐ = (๐๐๐ด๐ โ ๐๐๐ด๐) โ ๐,๐ด๐ , ๐ด๐-, ๐ด๐ = ๐ด๐๐๐๐.
โ is invariant under gauge transformations of ๐โ๐๐ผ with ๐ผ โ ๐ถโ(๐, ๐ค); ๐ โ ๐โ๐๐ผ๐,
๐ด๐ โ ๐โ๐๐ผ๐๐๐๐๐๐ผ + Ad๐โ๐๐ผ ๐ด๐ .
To impose the local axial symmetry to the theory, one should replace โ by;
โ๐๐ฅ = โ๐บ๐๐ข๐๐ + ๐๐ ๐พ๐๐๐๐ + ๐ด๐๐๐ ๐พ๐๐๐๐ + ๐ต๐
๐๐ ๐พ๐๐พ5๐๐๐,
with โ๐บ๐๐ข๐๐ โ โ1
4๐ก๐*๐น๐๐๐น๐๐+, for;
๐น๐๐ = ๐๐ ๐ด๐ + ๐ต๐ โ ๐๐ ๐ด๐ + ๐ต๐ โ ๐,(๐ด๐ + ๐ต๐) , (๐ด๐ + ๐ต๐)-,
๐ด๐ = ๐ด๐๐๐๐, and ๐ต๐ = ๐ต๐
๐๐๐๐พ5.
Therefore the Lagrangian density โ๐๐ฅ is invariant under the union of gauge and local axial transformations respectively given by;
๐ โ ๐โ๐๐ผ๐,
๐ด๐ + ๐ต๐ โ ๐โ๐๐ผ๐๐๐๐๐๐ผ + Ad๐โ๐๐ผ ๐ด๐ + ๐ต๐ ,
and; ๐ โ ๐โ๐๐ผ๐พ5๐,
๐ด๐ + ๐ต๐ โ ๐โ๐๐ผ๐พ5๐๐๐๐๐๐ผ๐พ5 + Ad๐โ๐๐ผ๐พ5 ๐ด๐ + ๐ต๐ ,
for ๐ผ โ ๐ถโ(๐, ๐ค).
1. 1. Does โ๐๐ฅ define a gauge theory with semi simple gauge group?
2. 2. Can this theory be quantized due to path-integral or geometrical versions of Faddeev-Popov quantization?
3. 3. If โ๐๐ฅ be a gauge theory, would it preserve nontrivial topological aspects of gauge theory โ such as instantonic and anomalous ones?
Let *๐ก๐+๐=1๐ be a basis for Lie algebra ๐ค. By definition the axial extention
of ๐ is a 2๐ -dimensional Lie algebra ๐ค generated with elements *๐ก๐, ๐ ๐+๐=1
๐ and commutation relations; ๐ ๐ , ๐ ๐ = ๐ก๐, ๐ก๐ , ๐ ๐ , ๐ก๐ = โ ๐ก๐, ๐ ๐ .
It can be easily seen that the process of axial extension produces a natural transformation in the small category of Lie algebras;
๐๐ค: ๐ค โ ๐ค .
It can also be seen that ๐ค is semi simple iff ๐ค is semi simple.
Let ๐บ be the simply connected semi simple Lie group with Lie algebra ๐ค .
Thus there is a smooth injective homomorphism ฮฃ๐ค: ๐บ โ ๐บ such that;
dฮฃ๐ค = ๐๐ค. In fact ๐บ is a closed Lie subgroup of ๐บ .
Using the principal ๐บ-bundle ๐บ โช ๐ โ ๐, the Cartan connection form ๐ on ๐ and the homomorphism ฮฃ๐ค, one can naturally define a principal ๐บ -
bundle over ๐, say ๐บ โช ๐ โ ๐, a principal bundle homomorphism;
๐ ร ๐บ ๐ด๐๐ก๐๐๐
๐ โ ๐ ๐ โ ร โ ๐ด๐ค ๐ โ = ,
๐ ร ๐บ ๐ด๐๐ก๐๐๐
๐ โ ๐
and a Cartan connection form over ๐ , say ๐ , such that; ๐โ ๐ = ๐.
Moreover the Lie algebra homomorphism; ๐ก๐ โฆ โ๐๐๐ , ๐ ๐ โฆ โ๐๐๐๐พ5,
leads to a unitary irreducible representation of Lie group ๐บ on ๐.
It would be clear that this representation together with principal ๐บ -bundle ๐บ โช ๐ โ ๐, lead to a Yang-Mills ๐บ -gauge theory which its Lagrangian density is given by โ๐๐ฅ.
More precisely ๐ดโจ๐ต = โ๐ ๐ด๐ + ๐ต๐ d๐ฅ๐ can be considered as extended
gauge field.
In fact โ๐๐ฅ is a gauge theory with semi simple gauge group ๐บ and consequently can be quantized by Faddev-Popov quantization.
On the other hand โ๐๐ฅ is the most natural extension of โ and thus it preserves all the nontrivial topological aspects of โ such as instantons and anomalies even when ๐บ โช ๐ โ ๐ is topologically nontrivial.
3. Anti-Ghost and Anti-BRST Symmetry
Consider the assumptions of last section.
Let ๐๐: ๐ค โ ๐ค be the projection map; ๐๐ ๐ก๐ = ๐ก๐ , ๐๐ ๐ ๐ = 0,
for ๐ = 1, โฆ , ๐, and set ๐๐5 = 1 โ ๐๐.
Suppose that ๐ and โฌ be respectively the spaces of ๐โ(๐๐ โ ๐ ) and ๐โ(๐๐5 โ ๐ ) for Cartan connection forms ๐ over ๐บ โช ๐ โ ๐ and for identity global section ๐: ๐ โ ๐ . Indeed ๐ and โฌ can respectively be considered as the spaces of vector and axial parts of extended gauge fields.
Finally denote the (extended) gauge transformation group ๐ถโ(๐, ๐บ ) by ๐ข .
There is a free action of ๐ข on ๐ ร โฌ from right; (๐ดโจ๐ต) โฒ ๐ โ Ad๐โ1 ๐ด + ๐ต + ๐โ1d๐,
for ๐ดโจ๐ต โก (๐ด, ๐ต) โ ๐ ร โฌ, ๐ โ ๐ข .
Therefore we have the following principal ๐ข -bundle; ๐ข โช ๐ ร โฌ โ (๐ ร โฌ)/๐ข .
Set a connection over this principal bundle with Cartan connection form ฮ , fix a connection over ๐บ โช ๐ โ ๐ with Cartan connection form ๐ 0, for ๐โ ๐ 0 = (๐ด0, ๐ต0), and define the fiber map ๐(๐ด0,๐ต0):๐ ร ๐ข โ ๐ ร (๐ รโฌ), with;
๐(๐ด0,๐ต0) ๐, ๐ โ ๐, (๐ด0, ๐ต0) โฒ ๐ .
๐ ร โฌ
(๐ ร โฌ)/๐ข
๐ข
Let ฮ โ ฮฉ1 ๐ ร ๐ ร โฌ โจ๐ค be given by;
ฮ ๐ฃ, ๐ โ (๐ดโจ๐ต) ๐ฃ + ฮ ๐ ๐ ,
for ๐ฃ, ๐ โ ๐๐๐ ร ๐(๐ด,๐ต)(๐ ร โฌ).
Then it can be shown that;
๐(๐ด0,๐ต0)โ ฮ
(๐,๐)= ๐ดโจ๐ต + ๐โจ๐โ ,
for gauge field ๐ดโจ๐ต โก (๐ด, ๐ต) = (๐ด0, ๐ต0) โฒ ๐ and for ๐โจ๐โ a left invariant Lie๐ข -valued 1-form over ๐ข . Indeed;
๐ = ๐๐ โ ๐(๐ด0,๐ต0)โ ฮ , ๐โ = ๐๐5 โ ๐(๐ด0,๐ต0)
โ ฮ .
Set ๐(๐ด0,๐ต0)โ d๐ = ๐ฟ, then we have;
๐ฟ๐ด = d๐ + ๐ด, ๐ , ๐ฟ๐ = ๐2 =1
2,๐, ๐-
๐ฟ๐ต = ๐ต, ๐ , ๐ฟ๐โ = ๐โ, ๐ ๐ฟd + d๐ฟ = ๐ฟ, d = 0 , ๐ฟ2 = 0.
On the other hand for ๐(๐ด0,๐ต0)โ dโฌ = ๐ฟโ we have;
๐ฟโ๐ด = ๐ต, ๐โ , ๐ฟโ๐ = 0
๐ฟโ๐ต = d๐โ + ๐ด, ๐โ , ๐ฟโ๐โ = ๐โ2 =1
2๐โ, ๐โ
๐ฟโd + d๐ฟโ = 0 , ๐ฟโ2 = 0.
And eventually; ๐ฟ๐ฟโ + ๐ฟโ๐ฟ = ๐ฟ, ๐ฟโ = 0.
It is seen that by considering ๐ด as the vector part of extended gauge field and noting that ๐ต is a pure gauge field, ๐โ and ๐ฟโ are respectively in complete agreement with anti-ghost field and anti-BRST derivation.
To see this more precisely it is enough to replace the Nakanishi-Lautrup (auxiliary) field in the standard formulation of BRST/anti-BRST derivation with ๐โ, ๐ .
More precisely in the standard formulation of Yang-Mills theories and Faddeev-Popov Quantization, when local axial symmetry is broken, the ghost field ๐ is a differential 1-form along the directions of local symmetry (gauge transformations), but the anti-ghost field ๐โ is a differential 1-form along the directions of global symmetry (axial transformations).
4. Application and Conclusions
Consider the assumptions of previous sections.
One of the most important application of BRST/anti-BRST correlation for axially extended gauge theories is to work out an extended counterpart of consistent anomaly called modified anomaly and to give a new classification of anomalous behaviors in the setting of an extended version of equivariant BRST cohomology called extended BRST cohomology.
To extract the modified anomaly according to the Stora-Zumino procedure, one should use d, ๐ฟ and ๐ฟโ alternatively to provide a generalized formulation of descent equations.
Initially, by Bianchi identity we have;
๐ฟ๐ = ๐๐ โ ๐ ๐ = ,๐, ๐ -, ๐ฟโ๐ = ๐โ๐ โ ๐ ๐โ = ,๐โ, ๐ -,
d๐ = ๐ ๐ดโจ๐ต โ ๐ดโจ๐ต ๐ = ,๐ , ๐ดโจ๐ต-,
for ๐ = d๐โ(๐ ) + ๐โ(๐ )2 = d(๐ดโจ๐ต) + (๐ดโจ๐ต)2 the pull back of the curvature. Indeed ๐น๐๐d๐ฅ๐ โง d๐ฅ๐ = 2๐๐ .
Thus ๐ก๐*๐ ๐+1+ , the (๐ + 1) th Chern character, is simultaneously a deRham and (anti-) BRST closed form.
Consider ๐ก๐*๐ ๐+1+ as a (2๐ + 2)-form over โ2๐+2. Thus, the Poincare lemma leads to;
๐ก๐ ๐ ๐+1 = dฮฉ2๐+10,0 ,
๐ฟฮฉ2๐+10,0 = dฮฉ2๐
1,0, ๐ฟโฮฉ2๐+10,0 = dฮฉ2๐
0,1,
๐ฟฮฉ2๐1,0 = dฮฉ2๐โ1
2,0 , ๐ฟโฮฉ2๐0,1 = dฮฉ2๐โ1
0,2 ,
๐ฟโ๐ฟฮฉ2๐+10,0 = dฮฉ2๐
1,1,
where ฮฉ๐๐,๐
is a deRham differential ๐-form with ghost number ๐ โ ๐,
while ๐ + ๐ + ๐ = 2๐ + 1. Actually, ฮฉ๐๐,๐
is simultaneously a differential ๐-form over ๐ and a differential ๐-form over โฌ.
It is seen that;
๐ฟ + ๐ฟโ ฮฉ2๐1,0 + ฮฉ2๐
0,1 = d(ฮฉ2๐โ12,0 + ฮฉ2๐โ1
0,2 + ฮฉ2๐โ11,1 ),
and hence;
๐ฟ + ๐ฟโ ฮฉ2๐1,0 + ฮฉ2๐
0,1
โ2๐
= 0.
Thus, up to a constant factor ฮฉ2๐1,0 + ฮฉ2๐
0,1 can be considered as the modified nonintegrated anomaly.
In the other words, ฮฉ2๐1,0 + ฮฉ2๐
0,1
โ2๐ is a candidate for (๐ฟ + ๐ฟโ)๐ for quantum action ๐. A direct calculation shows that when ๐ = 2 then;
ฮฉ41,0 + ฮฉ4
0,1 = ๐ก๐*d(๐โจ๐โ)( ๐ดโจ๐ต d ๐ดโจ๐ต โ1
2๐ดโจ๐ต 3)+,
which is the modified consistent anomaly up to a factor of ๐2 =1
24๐2.
Indeed, since ๐ต is a pure gauge then one can set ๐ต = 0 to achieve the well-known consistent anomaly (the anti-ghost will be killed automatically after taking the trace).
On the other hand, ghost number counting leads to;
๐2ฮฉ41,0 =
1
24๐2๐ก๐*d๐( ๐ดโจ๐ต d ๐ดโจ๐ต โ
1
2๐ดโจ๐ต 3)+,
which is called ghost consistent anomaly and is the anomaly of (vector) gauge current.
Moreover, the other term;
๐2ฮฉ40,1 =
1
24๐2๐๐*d๐โ( ๐ดโจ๐ต d ๐ดโจ๐ต โ
1
2๐ดโจ๐ต 3)+,
is called anti-ghost consistent anomaly and is the anomaly of axial current.
Moreover;
๐ฟ + ๐ฟโ ฮฉ2๐โ12,0 + ฮฉ2๐โ1
0,2 + ฮฉ2๐โ11,1 = d(ฮฉ2๐โ2
3,0 + ฮฉ2๐โ22,1 + ฮฉ2๐โ2
1,2 + ฮฉ2๐โ20,3 ).
Therefore,
ฮฉ2๐โ12,0 + ฮฉ2๐โ1
0,2 + ฮฉ2๐โ11,1 ,
is a candidate for the modified Schwinger term up to the factor ๐๐.
For ๐ = 2, the modified Schwinger term is given by;
๐2(ฮฉ32,0 + ฮฉ3
0,2 + ฮฉ31,1) =
1
24๐2๐ก๐*(d(๐โจ๐โ))2(๐ดโจ๐ต)+.
where;
๐2ฮฉ32,0 =
1
24๐2๐ก๐ d๐ 2๐ด ,
๐2ฮฉ30,2 =
1
24๐2๐ก๐ d๐โ 2๐ด ,
๐2ฮฉ31,1 =
1
24๐2๐ก๐ d๐d๐โ + d๐โd๐ ๐ต ,
are respectively called ghost/ghost, anti-ghost/anti-ghost and ghost/anti-ghost consistent Schwinger term and are respectively the anomalous terms of vector/vector, axial/axial and vector/axial currents commutation relations.
Consequently the extended descent equations give rise to a bi-complex which commutes up to exact deRham forms;
ฮฉ2๐+10,0
d,๐ฟ ฮฉ2๐
1,0 d,๐ฟ
ฮฉ2๐โ12,0
d,๐ฟ ฮฉ2๐โ2
3,0 d,๐ฟ
. . . .
โ d, ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ . . . .
ฮฉ2๐ 0,1 ๐ฟ
โฮฉ2๐ 2๐โ1 1,1 ๐ฟ
โฮฉ2๐โ1 2๐โ2 2,1 ๐ฟ
โฮฉ2๐โ2 2๐โ3 3,1 ๐ฟ
โ . . . .
โ d, ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ . . . .
ฮฉ2๐โ10,2 ๐ฟ
โฮฉ2๐โ1 2๐โ2 1,2 ๐ฟ
โฮฉ2๐โ2 2๐โ3 2,2 ๐ฟ
โฮฉ2๐โ3 2๐โ4 3,2 ๐ฟ
โ . . . .
โ d, ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ . . . .
ฮฉ2๐โ20,3 ๐ฟ
โฮฉ2๐โ2 2๐โ3 1,3 ๐ฟ
โฮฉ2๐โ3 2๐โ4 2,3 ๐ฟ
โฮฉ2๐โ4 2๐โ5 3,3 ๐ฟ
โ . . . .
โ d, ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ โ โ๐ฟโ . . . .
โฎ โฎ โฎ โฎ . . . .
In fact modified anomalies and extended descent equations produce a generalized formulation of BRST cohomology, called extended BRST cohomology, which is the cohomology of total complex of the given bi-complex.