FERMIONIC GENERALIZATION OF LINEAL GRAVITY

12 Ñ.À. Äóïëèé, Ä.Â. Ñîðîêà, Â.À. Ñîðîêà Î ôåðìèîííîì îáîáùåíèè ëèíåéíîé ãðàâèòàöèè ...

«Â³ñíèê Õàðê³âñüêîãî óí³âåðñèòåòó», ¹ 664, 2005 ñåð³ÿ ô³çè÷íà «ßäðà, ÷àñòèíêè, ïîëÿ», âèï. 2 /27/

УДК 539.12

FERMIONIC GENERALIZATION OF LINEAL GRAVITY IN CENTRALLY EXTENDED FORMULATION S.A. Duplij1 , D.V. Soroka2 , V.A. Soroka2 1

Theory Group, Nuclear Physics Laboratory, Kharkov National University Svoboda Sq. 4, 61001 Kharkov, Ukraine E-mail: [email protected]. Internet: http://www.math.uni-mannheim.de/˜duplij 2 Kharkov Institute of Physics and Technology Akademicheskaya St. 1, 61108 Kharkov, Ukraine E-mail:[email protected], [email protected] Received June 5, 2005 We generalize the central extension of the (1 + 1)-dimensional Poincar´e algebra by including fermionic charges which obey not supersymmetric algebra, but special graded algebra containing in the right hand side a central element only. We verify selfconsistency of Jacobi identities and derive the Casimir operator. Then we introduce the correspondent gauge fields and construct the classical gauge theory based on this graded algebra, present field transformations and derive the black hole mass in (1 + 1)-dimensions. KEY WORDS: Poincar´e algebra, lineal gravity, central extention, fermionic generator, Casimir operator

The Einstein-type theory in (1 + 1)-dimensions — lineal gravity — provides a great interest in connection e.g. with black hole solutions, while drastic simplification of group theoretical properties in lowest dimension allows a more deep understanding of gravitational effects. 1 In (1+1)-dimensions the Einstein tensor Gµν = Rµν − gµν R vanishes identically, therefore the standard equations 2 of motion vanish, and one needs to introduce additional fields to endow theory with sensible dynamics. The most studied models of lineal gravity are Liouville theory [1], a simplest non-trivial theory based on the scalar curvature R and additional scalar field, and so-called ’string-inspied’ model [2,3], where scalar (dilaton) field arises from string theory. Both the models can be also obtained by dimensional reduction from (2 + 1)-dimensions [4–6]. In [2] it was shown that the lineal gravity in (1 + 1)-dimensional space-time [7, 8] can be treated as a gauge theory with a central extension of the two-dimensional Poincar´e algebra taken as a gauge algebra. The quantization of lineal gravity was obtained in [9, 10]. Recently a possible fermionic generalization of the central extension of the Poincar´e algebra was proposed [11]. By gauging of this algebra, we give here a special fermionic generalization of the lineal gravity model which differs from the standard supersymmetry. CENTRAL EXTENSION OF POINCARE´ ALGEBRA The central extension of the (1 + 1)-dimensional Poincar´e algebra is [2, 11] [Pa , Pb ] = ab Y, b

[Pa , J] = a Pb , [Pa , Y ] = 0, [J, Y ] = 0,

(1) (2) (3) (4)

where Pa are translation generators, J is a Lorentz generator, Y is a central element Y =

i ΛI, 2

(5)

here Λ is some real constant, and ab = −ba , 01 = 1 is the totally antisymmetric two-dimensional Levi-Civita tensor, the tangent space indices a, b, c . . . = (0, 1) are lowered a b = hac cb by means of the flat metric hab = diag (−1, 1). The shape of central tensor charges Zab for the Poincar´e group in 1 + 1 dimensions Zab = ab Y,

(6)

was introduced in [2] while investigating the string-inspired lineal gravity, and for any dimensions similar tensor charges were proposed in [11]. Note that for the gauge theory of the de Sitter group Y = 12 ΛJ [6, 12]. If Zab = 0, the standard way of the supersymmetric extension of the Poincar´e algebra (1)-(2) is introducing some (0) fermionic generators Qα which satisfy anticommuting relations [13] (0)

a {Q(0) α , Qβ } = −2i (γ C)αβ Pa .

(7)

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But in case of nonvanishing tensor charges Zab = 0 the Jacobi identity and P -invariance forbid appearance of the standard terms with momenta Pa in the right hand side of (7). However we can introduce fermionic generators with nonvanishing anticommutator in another way [11] (using the central element Y ) which satisfy the graded algebra {Qα , Qβ } = −q(γ5 C)αβ Y, 1 [Qα , J] = (γ5 Q)α , 2 [Y, Qα ] = 0, [Pa , Qα ] = 0,

(8) (9) (10) (11)

where C is the charge conjugation matrix, q is some constant (which controls nonanticommutativity of the fermionic 1  a b γ , γ . As a real (Majorana) representation for the two-dimensional charges themselves) and γ5 = 12 ab γa γb , σ ab = 4 γ-matrices and charge conjugation matrix C we use γ 0 = C = −C T = −iσ2 , γ 1 = σ1 , γ5 = 12 ab γa γb = σ3 , {γa , γb } = 2hab , h11 = −h00 = 1, C −1 γa C = −γa T , where σi are Pauli matrices. The matrices γa satisfy γa γ5 = ab γ b ,

γa γb = hab − ab γ5 .

(12)

The Jacobi identities of the extended Poincar´e algebra (1)–(4),(8)-(11) are [Pa , [Pb , Pc ]] + [Pc , [Pa , Pb ]] + [Pb , [Pc , Pa ]] = 0, [Pa , [Pb , Qα ]] + [Qα , [Pa , Pb ]] + [Pb , [Qα , Pa ]] = 0,

(13) (14)

[Pa , {Qα , Qβ }] − {Qα , [Pa , Qβ ]} − {Qβ , [Pa , Qα ]} = 0,

(15)

[J, [Pa , Pb ]] + [Pa , [J, Pb ]] + [Pb , [J, Pa ]] = 0, [J, [Pa , J]] + [Pa , [J, J]] + [J, [J, Pa ]] = 0,

(16) (17)

[Qα , {Qβ , Qγ }] − [Qγ , {Qα , Qβ }] − [Qβ , {Qγ , Qα }] = 0, [J, {Qα , Qβ }] − {Qα , [J, Qβ ]} − {Qβ , [J, Qα ]} = 0,

(18) (19)

where (13)-(18) satisfy trivially, while (19) follows from antisymmetry of matrix C. We note that the full algebra (1)–(4),(8)-(11) is not semi-simple, because it is a semi-direct sum of the subalgebra generated by the Lorentz generator J and the graded ideal consisting of momenta Pa , central charge Y and fermionic generators Qα . It has the following Casimir operator K = Pa P a − 2Y J −

 αβ 1 Qα C −1 Qβ . 2q

(20)

GAUGE TRANSFORMATIONS The gauge 1-form A = dx Aµ (x) can be expanded in terms of the generators, and the gauge connection is µ

Aµ (x) = eaµ (x) Pa + ωµ (x) J + wµ (x) Y + ψµα (x) Qα ,

(21)

where eaµ (x) is the zweibein which determines the metric tensor of space-time gµν (x) = eaµ (x) ebν (x) hab , ωµ (x) is the spin connection, wµ (x) is the gauge field corresponding to the central element Y and ψµα (x) is the Grassmann Majorana “gravitino” field associated with Qα . Greek indices µ, ν . . . = 0, 1 refer to the world space. Infinitesimal gauge transformations corresponding to the full algebra (1)–(4),(8)-(11) are δA = dΩ + [A, Ω],

(22)

Ω = y a (x) Pa + ϕ (x) J + z (x) Y + εα (x) Qα ,

(23)

with the gauge generator where y a (x) are space-time translations, ϕ (x) is the Lorentz boost parameter, z (x) is the central element translation and εα (x) is the fermionic translation. From (1)–(4),(8)-(11) and (22) it follows   δeaµ (x) = a b −ϕ (x) ebµ (x) + y b (x) ωµ (x) + ∂µ y a (x) , (24) δωµ (x) = ∂µ ϕ (x) , δwµ (x) =

eaµ

b

(25) qψµα

β

(x) y (x) ab + (x) ε (x) (γ5 C)αβ + ∂µ z (x) ,   1 α ϕ (x) ψµβ (x) − ωµ (x) εβ (x) (γ5 )β + ∂µ εα (x) . δψµα (x) = 2

(26) (27)

14 Ñ.À. Äóïëèé, Ä.Â. Ñîðîêà, Â.À. Ñîðîêà

«Â³ñíèê Õàðê³âñüêîãî óí³âåðñèòåòó», ¹ 664, 2005

The finite gauge transformations

¯ = e−Ω deΩ + e−Ω AeΩ A→A

(28)

have the following form: e¯aµ (x) = eaµ (x) − N a b (x) (ebµ (x) − ωµ (x) sb (x) − b c ∂µ sc (x)) + sa (x) ∂µ ϕ (x) ,

(29)

ω ¯ µ (x) = ωµ (x) + ∂µ ϕ (x) ,

(30)

a

b

ebµ

b

c

 (x) − ωµ (x) s (x) −  c ∂µ s (x) w ¯µ (x) = wµ (x) − sa (x) N b (x)  α − 2q ψµα (x) − ωµ (x) ρα (x) + 2∂µ ργ (x) (γ5 )γ Tα β (x) ρβ 

+ ∂µ z (x) − ϕ (x) sa (x) ∂µ sa (x) − 2qϕ (x) ρα (x) ∂µ ρα (x) , ψ¯µα

(x) =

ψµα

(x) −

[ψµβ

β

β

γ

(31)

α

α

(x) − ωµ (x) ρ (x) + 2∂µ ρ (x) (γ5 )γ ]Tβ (x) + ∂µ ϕ (x) ρ (x) ,

(32)

where the notations N a b (x) = δ a b − (M −1 (x))a b , Tα β (x) = δα β − Sα β (x), ρα (x) =

α

(33)

a

ε (x) y (x) , sa (x) = , ϕ (x) ϕ (x)

(34)

with the finite Lorentz transformations for vectors and spinors respectively M a b (x) = δ a b coshϕ (x) + a b sinhϕ (x) , Sα β (x) = δα β cosh

(35)

ϕ (x) ϕ (x) β + (γ5 )α sinh 2 2

(36)

are introduced. The multiplet of the curvature 2-form is F=

  1 µ 1 dx ∧ dxv Fµν (x) = dA + A ∧ A = dxµ ∧ dxv ∂[µ Aν] (x) + A[µ (x) Aν] (x) , 2 2

(37)

where antisymmentrization [µν] is implied without the factor 1/2. The field strength F can be expanded in terms of the generators a α (x) Pa + rµν (x) J + vµν (x) Y + ξµν (x) Qα , (38) Fµν (x) = fµν with f a fµν (ω) eaν] (x) , (x) = ∂[µ eaν] (x) + a b ω[µ (x) ebν] (x) ≡ D[µ

(39)

rµν (x) = ∂[µ ων] (x) ,

(40)

1 q α β vµν (x) = ∂[µ wν] (x) + ea[µ (x) ebν] (x) ab + ψ[µ (x) ψν] (x) (γ5 C)αβ , 2 2 1 α ξ β α α α ξµν (ω) ψν] (x) , (x) (γ5 )β ≡ D[µ (x) = ∂[µ ψν] (x) − ω[µ ψν] 2

(41) (42)

where Dµf (ω) and Dµξ (ω) are covariant derivatives corresponding to boson and fermion translations, while rµν (x) is the strength tensor corresponding to the Lorentz boost, and vµν (x) is the strength tensor corresponding to the central element Y. The components of Fµν (x) (39)-(42) are transformed by the adjoint representation of the fermionic generalization (8)-(11) of the centrally extended Poincar´e group (1)-(4) as follows  b  a a f¯µν (x) = fµν (x) − N a b (x) fµν (x) − rµν (x) sb (x) , (43) r¯µν (x) = rµν (x) ,

(44) a



b fµν

b

 (x) − rµν (x) s (x)

v¯µν (x) = vµν (x) − sa (x) N b (x)  β  − 2q ξµν (x) − rµν (x) ρβ (x) Tβ α (x) ρα (x) ,   ξ¯α (x) = ξ α (x) − ξ β (x) − rµν (x) ρβ (x) Tβ α (x) . µν

µν

µν

(45) (46)

This transformation law can be equivalently expressed in the form F¯ A = (U −1 )A B F B

(47)

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with U AB

M a b (x) La c (x) sc (x)  0 1 =  −sc (x) Lc b (x) sc (x) Lc l (x) sl (x) + 2qργ (x) Rγ λ ρλ (x) 0 ρλ (x) Rλ α (x) 

where La b (x) = δ a b − M a b (x) , We observe that s det U =

0 0 1 0

 0  0 , 2qRβ γ (x) ργ (x)  α (S −1 (x))β

β

Rα β (x) = δα β − (S −1 )α (x) . det M = 1. det S −1

(48)

(49) (50)

LAGRANGIAN AND EQUATIONS OF MOTION An invariant Lagrangian density can be constructed using a multiplet of Lagrange multipliers   1 µν 1 A a α  ηA Fµν (x) = µν ηa fµν (x) + η2 rµν (x) + η3 vµν (x) + ηα ξµν (x) , 2 2 ηA = (ηa , η2 , η3 , ηα ), L=

(51)

which obey the coadjoint transformation law η¯A = ηB U B A , or in manifest form η¯a = (ηb + η3 sb (x))M b a (x) − η3 sa (x) ,

(52)

η¯2 = η2 + (ηa + η3 sa (x))La b (x) sb (x) − ρα (x) Rα β (x) (ηβ − 2qη3 ρβ (x)),

(53)

η¯3 = η3 ,

(54) β

η¯α = (S −1 )α (x) (ηβ − 2qη3 ρβ (x)) + 2qη3 ρα (x) .

(55)

The corresponding equations of motion are A Fµν (x) = 0,   a ∂µ η = ebµ (x) η3 + ωµ (x) η b b a , 1 α ∂µ η2 = eaµ (x) η b ab − ψµβ (x) (γ5 )β ηα , 2 ∂µ η3 = 0,

1 α β β α ωµ (x) η − qψµ (x) η3 (γ5 )β . ∂µ η = 2

(56) (57) (58) (59) (60)

The fermionic generalization of the centrally extended Poincar´e algebra (1)–(4),(8)-(11) in the coadjoint representation possesses a nonsingular invariant graded metric hAB = (−1)pB (pA +pC ) hCD U D B U C A ,

(61)

which has the form

hAB

s det hAB



hab  0 = (−1)pA pB hBA =   0 0 det hab 1 = 2, =− 2 4q det C 4q

0 0 u −1 −1 0 0 0

 0  0 ,  0 −2qCαβ

(62)

(63)

where pA ≡ p(A) is a Grassmann parity of the quantity A and u is an arbitrary constant. The inverse metric hAB hAB hBC = δC A

(64)

having the property hAB = (−1)pA +pB +pA pB hBA can be used to upper the indices η A = hAB ηB . In the compact notation XA = {Pa , J, Y, Qα } the full algebra (1)–(4),(8)-(11) can be presented as [XA , XB } ≡ XA XB − (−1)pA pB XB XA = fABC XC ,

(65)

16 Ñ.À. Äóïëèé, Ä.Â. Ñîðîêà, Â.À. Ñîðîêà

«Â³ñíèê Õàðê³âñüêîãî óí³âåðñèòåòó», ¹ 664, 2005

where fABC are the structure constants, and nonvanishing ones have the form fabY = ab , faJb = a b , fαβY = −q (γ5 C)αβ , fαJβ =

1 β (γ5 )α . 2

(66)

In terms of η = η A XA the equations of motion (57)–(60) can be written as dη + [A, η] = 0,

(67)

where A = AB XB . The invariant quantity

1 ηA hAB ηB (68) 2Λ can be interpreted as the black-hole mass [7], if it is constant (Λ is from (5)). Indeed, from the equations of motion (67) we have 1 1 dM = − ηA dη A = ηA η C AB fBCA . (69) Λ Λ As a consequence of the structure constant properties we obtain ηA η C AB fBCA = 0, and therefore M = const. Using the inverse metric hAB , the set of independent Casimir operators of the algebra (65) can be presented as M =−

K (u) = XA hAB XB = K − uY 2 ,

(70)

which is marked by the additional parameter u from (62). CONCLUSION Thus, we have constructed a special fermionic generalization of the lineal gravity which is not the standard supersymmetric two-dimensional gravity [14]. We presented the algebra of generators, the field transformations and found Lagrangian and equation of motion. We obtained the constant black hole mass and derived the Casimir operator which depends on an additional parameter. REFERENCES 1. Jackiw R. Lower dimensional gravity // Nucl. Phys. - 1985. - V. B252. - P. 343–356. 2. Cangemi D., Jackiw R. Gauge invariant formulations of lineal gravity // Phys. Rev. Lett. - 1992. - V. 69. - P. 233–236. 3. Cangemi D., Jackiw R. Poincare gauge theory for gravitational forces in (1+1)- dimensions // Ann. Phys. - 1993. - V. 225. - P. 229– 263. 4. Ach´ucarro A. Lineal gravity from planar gravity // Phys. Rev. Lett. - 1993. - V. 70. - P. 1037–1040. 5. Cangemi D. One formulation for both lineal gravities through a dimensional reduction // Phys. Lett. - 1992. - V. B297. - P. 261–267. 6. Grignani G., Nardelli G. Poincar´e gauge theories for lineal gravity // Nucl. Phys. - 1994. - V. B412. - P. 320–344. 7. Verlinde H. Black holes and strings in two dimensions // Sixth Marcel Grossmann Meeting on General Relativity. - Singapore. World Scientific, 1992. - P. 214–223. 8. Callan C. G., Giddings S. B., Harvey J. A., Strominger A. Evanescent black holes // Phys. Rev. - 1992. - V. D45. - P. 1005–1009. 9. Cangemi D., Jackiw R. Quantal analysis of string inspired lineal gravity with matter fields // Phys. Lett. - 1994. - V. B337. - P. 271– 278. 10. Cangemi D., Jackiw R. Quantum states of string inspired lineal gravity // Phys. Rev. - 1994. - V. D50. - P. 3913–3922. 11. Soroka D. V., Soroka V. A. Tensor extension of the Poincar´e algebra // Phys. Lett. - 2005. - V. B607. - P. 302–305. 12. Fukuyama T., Kamimura K. Gauge theory of two-dimensional gravity // Phys. Lett. - 1985. - V. B160. - P. 259–267. 13. Cangemi D., Leblanc M. Two dimensional gauge theoretic supergravities // Nucl. Phys. - 1994. - V. B420. - P. 363–386. 14. Howe P. Super Weil transformations in two dimensions // J. Phys. - 1979. - V. A12. - P. 393–401.

Харьковский национальный университет им. В.Н.Каразина, пл. Свободы 4, Харьков 61077 Национальный Научный Центр "Харьковский Физико-технический Институт", ул. Академическая 1, Харьков 61108 1

2

О ФЕРМИОННОМ ОБОБЩЕНИИ ЛИНЕЙНОЙ ГРАВИТАЦИИ В ФОРМУЛИРОВКЕ С ЦЕНТРАЛЬНЫМ РАСШИРЕНИЕМ С.А. Дуплий1 , Д.В. Сорока2 , В.А. Сорока2

В работе обобщается центральное расширение алгебры Пуанкаре в (1 + 1) измерениях путем включения фермионных зарядов, которые удовлетворяют не стандартной суперсимметричной алгебре, а специальной градуированной алгебре, которая в правой части содержит только центральные заряды. Проверена самосогласованность тождеств Якоби и получен оператор Казимира. Далее вводятся соответствующие калибровочные поля и построена классическая калибровочная теория, основанная на данной градуированной алгебре, представлены преобразования полей и получена масса черной дыры в (1 + 1) измерениях. КЛЮЧЕВЫЕ СЛОВА: алгебра Пуанкаре, линейная гравитация, центральное расширение, фермионный генератор, оператор Казимира