R. P. Malik Physics Department, Banaras Hindu University, Varanasi, INDIA
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Transcript of R. P. Malik Physics Department, Banaras Hindu University, Varanasi, INDIA
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R. P. Malik Physics Department, Banaras Hindu University,
Varanasi, INDIA
31st July 2009, SQS’09, BLTP, JINR 1
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NOTOPH opposite of PHOTON
Nomenclature : Ogieveskty & Palubarinov
(1966-67)
Notoph gauge field =
Antisymmetric tensor gauge field 2
[Abelian 2-form gauge field]
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VICTOR I. OGIEVETSKY
(1928—1996)
&
I. V. PALUBARINOV
COINED THE WORD
``NOTOPH’’
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Why 2-form Why 2-form gauge gauge
theory?theory?
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QCD and hairs on
the Black hole
Celebrated B ^ F term
mass & gauge
invariance
Non-commutativity
in string theory
[ Xμ, Xv ] ≠ 0
Dual description
of a massless
scalar field
Spectrum of quantized
(super) string theory
Irrotational fluid
OgievetskyPalubarinov (’66-’67)
R. K. Kaul(1978)
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The Kalb-Ramond ( KR) Lagrangian density for the Abelian 2- form gauge theory is (late seventies)
3-form:
: Exterior Derivative
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Constraint Structure
KR Theory =
e.g. R. K. Kaul PRD (1978)
Momentum:
Gauge Theory
First-class constraints BRST formalism
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Earlier Works: (1) Harikumar, RPM, Sivakumar: J. Phys. A: Math.Gen.33 (2000)
(2) RPM: J. Phys. A: Math. Gen 36 (2003)
BRST (Becchi-Rouet-Stora-Tyutin) invariant
Lagrangian density:
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Notations:
: (anti-)ghost field [ghost no. (-1)+1]
: Nakanishi – Lautrup auxiliary field
: Massless scalar field
: Bosonic ghost & anti-ghost field with
ghost no. (± 2)
Auxiliary ghost fields
ghost no. (± 1)
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BRST symmetry transformations:
anti-BRST symmetry transformations:
Notice:
anticommutativity
gone!!
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Starting point for the superfield formalism!!
Why superfield formalism ??
Gauge Theory BRST formalism
BRST Symmetry (sb)
Local Gauge Symmetry
anti-BRST Symmetry (sab) 11
Bonora, Tonin, Pasti (81-82)Delbourgo, Jarvais, Thompson
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Key Properties:
1: Nilpotency ,
(fermionic nature)
2: Anticommutativity Linear independence of
BRST & anti-BRST
Superfield formalism providesi) Geometrical meaning of Nilpotency &
Anticommutativity
ii) Nilpotency and ABSOLUTE Anticommutativity are always present in this formalism. 12
(Bonora, Tonin)
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Outstanding problem: How to obtain absolute anticommutativity??
LAYOUT OF THE TALK
HORIZONTALITY CONDITION
CURCI-FERRARI TYPE RESTRICTION
COUPLED LAGRANGIAN DENSITIES
ABSOLUTE ANTICOMMUTATIVITY
(RPM, Eur. Phys. J. C (2009))
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Horizontality Condition
Gauge invariant quantity (Physical)
(N = 2 Generalization)
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: Grassmannian variables
(Gauge transformation)
Recall
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4D Minkowski space (4, 2)-dimensional Superspace
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The basic superfields, that constitute the super2-form , are the generalizations of the 4D local fields onto the (4, 2)-dimensional Supermanifold.
The superfields can be expanded along the Grassmannian directions, as
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The basic fields of the BRST invariant 4D 2-form theory are
the limiting case of the superfields when
r.h.s of the expansion = Basic fields + Secondary fields
Horizontality condition is the requirement that the SuperCurvature Tensor is independent of the Grassmannian variables.
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r.h.s of the H. C. =
(Soul-flatness/horizontality condition)
[Independent of ]
In other words, in the l.h.s.
all the Grassmannian components of the curvaturetensor are set equal to zero.
Consequence: All the secondary fields are expressed in terms of the basic and auxiliary fields.
(H. C.)
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The explicit expression for
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The horizontality condition requires that all the differentialforms with Grassmann differentials should be set equal tozero because the r.h.s.
is independent of them.
Thus, equating the coefficients of , ,
and equal to zero, we obtain
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Choosing
We have the following expansions
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Equating the rest of the coefficients of the Grassmannian differentials
We obtain the following relationships
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It is extremely interesting to note that equating the
coefficient of the differential equal to zero yields
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Where we have identified the following
The above equation is the analogue of the celebrated Curci-Ferrari restriction that we come across in the4D non-Abelian 1-form gauge theory
It can be noted that all the secondary fields of the superexpansion have been expressed in terms of the basic and auxiliary fields of the 2-form theory. For instance
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Which can also be expressed, in terms of the BRST and anti-BRST Symmetry transformations, as
In exactly similar fashion, all the superfields can be re-expressed in terms of the BRST and anti-BRST symmetry transformations.
(After H. C.)
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This shows that the following mapping is true
Any generic superfield can be expanded as
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Superfield approach : Abelian 2-form gauge theory :
Field Superfield
(4D) (4,2)-dimensional
Geometrical Interpretations
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One of the most crucial outcome of the superfield approach to 4D Abelian 2-form gauge theory is:
Emergence of a Curci-Ferrari type restriction
for the validity of the absolute anticommutativity
of the (anti-)BRST transformations
Nilpotency property is automatic.
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BRST and anti-BRST symmetry BRST and anti-BRST symmetry
transformations must anticommute transformations must anticommute becausebecause
- and directions are independent on - and directions are independent on
(4,2)-dimensional supermanifold.(4,2)-dimensional supermanifold.
This shows the linear independence This shows the linear independence of the BRST and anti-BRST of the BRST and anti-BRST symmetriessymmetries
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The following coupled Lagrangian densities:
and
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respect nilpotent and absolutely anticommuting (anti-)BRST and (anti-)co-BRST symmetry transformations [Saurabh Gupta & RPM Eur. Phys. J. C (2008)]
These are coupled Lagrangian densities because:
define the constrained surface [1-form non-Abelian theory]. Here and are the new Nakanishi-Lautrup typeauxiliary fields
Curci-Ferrari-Type restrictions [1-form non-Abelian theory]
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The BRST transformations are:The BRST transformations are:
The anti-BRST transformations are:The anti-BRST transformations are:
BRST and anti-BRST transformations imply:BRST and anti-BRST transformations imply:
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Anticommutativity check:
and
where
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Summary of results at BHU
arXiv: 0905.0934 [hep-th]
LB & RPM Phys. Lett. B (2007)
RPM Eur. Phys. J C (2008)
Hodge Theory
( Symmetries) [SG, RPM, HK,
SK]
Non-Abelian
Nature↔ Gerbes
[SG, RPM, LB]
Similarity with 2D
Anomalous Gauge Theory
[SG, RK, RPM]
New Constraint Structure
(Hamiltonian Analysis)
[BPM, SKR, RPM]
SG & RPM Eur. Phys. J C (2008)SG & RPM arXiv:0805.1102 [hep-th] RPM Europhys. Lett. (2008)
arXiv: 0901.1433 [hep-th]
Superfield formalism
[RPMEur.Phys. J. C (2009)]
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Acknolwedgements:
DST, Government of India, for funding
Collaborators:
Prof. L. Bonora (SISSA, ITALY)
Dr. B. P. Mandal (Faculty at BHU)
Mr. Saurabh Gupta, (Ph. D. Student)
Mr. S. K. Rai (Ph. D. Student)
Mr. Rohit Kumar (Ph. D. Student)
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