R. P. Malik Physics Department, Banaras Hindu University, Varanasi, INDIA 31 st July 2009, SQS’09, BLTP, JINR 1

  NOTOPH opposite of PHOTON   Nomenclature : Ogieveskty & Palubarinov ( )   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’’

4 Why 2-form gauge 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 Ogievetsky Palubarinov (’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 TheoryBRST formalism BRST Symmetry (s b ) Local Gauge Symmetry anti-BRST Symmetry (s ab ) 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 provides i) 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 super 2-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 Super Curvature 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 curvature tensor 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 differential forms with Grassmann differentials should be set equal to zero 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 the 4D non-Abelian 1-form gauge theory  It can be noted that all the secondary fields of the super expansion 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.

29  BRST and anti-BRST symmetry transformations must anticommute because transformations must anticommute because - and directions are independent on - and directions are independent on (4,2)-dimensional supermanifold.  This shows the linear independence of the BRST and anti-BRST symmetries

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 type auxiliary fields Curci-Ferrari-Type restrictions [ 1-form non-Abelian theory]

32  The BRST transformations are: The anti-BRST transformations are: BRST and anti-BRST transformations imply:

33  Anticommutativity check: and where

34 Summary of results at BHU arXiv: [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: [hep-th] RPM Europhys. Lett. (2008) arXiv: [hep-th] Superfield formalism [RPM Eur.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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