r. p. malik physics department, banaras hindu university, varanasi, india
DESCRIPTION
Notoph Gauge Theory : Superfield Formalism. R. P. Malik Physics Department, Banaras Hindu University, Varanasi, INDIA 31 st July 2009, SQS’09, BLTP, JINR. NOTOPH opposite of PHOTON Nomenclature : Ogieveskty & Palubarinov (1966-67) Notoph gauge field = - PowerPoint PPT PresentationTRANSCRIPT
R. P. Malik Physics Department, Banaras Hindu University,
Varanasi, INDIA
31st July 2009, SQS’09, BLTP, JINR 1
NOTOPH opposite of PHOTON
Nomenclature : Ogieveskty & Palubarinov
(1966-67)
Notoph gauge field =
Antisymmetric tensor gauge field 2
[Abelian 2-form gauge field]
3
VICTOR I. OGIEVETSKY
(1928—1996)
&
I. V. PALUBARINOV
COINED THE WORD
``NOTOPH’’
44
Why 2-form Why 2-form gauge gauge
theory?theory?
5
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)
6
The Kalb-Ramond ( KR) Lagrangian density for the Abelian 2- form gauge theory is (late seventies)
3-form:
: Exterior Derivative
7
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!!
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
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)
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))
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
It is extremely interesting to note that equating the
coefficient of the differential equal to zero yields
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
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.)
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)]
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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