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

8

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:

9

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)

10

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)

14

: Grassmannian variables

(Gauge transformation)

Recall

15

4D Minkowski space (4, 2)-dimensional Superspace

16

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

17

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.

18

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.)

19

The explicit expression for

20

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

21

Choosing

We have the following expansions

22

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

27

Superfield approach : Abelian 2-form gauge theory :

Field Superfield

(4D) (4,2)-dimensional

Geometrical Interpretations

28

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.

2929

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

30

The following coupled Lagrangian densities:

and

31

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]

3232

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:

33

Anticommutativity check:

and

where

34

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)

3636