Perturbative Quantum Gravity on Complex Spacetime

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Physics Letters B 705 (2011) 120–123

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Physics Letters B

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Perturbative quantum gravity on complex spacetime

Mir Faizal

Department of Mathematics, Durham University, Durham, DH1 3LE, United Kingdom

article i nfo a bstract

Article history: In this Letter we will study non-anticommutative perturbative quantum gravity on spacetime with a Received 19 July 2011 complex metric. After analysing the BRST symmetry of this non-anticommutative perturbative quantum Received in revised form 13 September gravity, we will also analyse the effect of shifting all the ﬁelds. We will construct a Lagrangian density 2011 which is invariant under the original BRST transformations and the shift transformations in the Batalin– Accepted 15 September 2011 Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-ﬁxing term and the ghost Available online 1 October 2011 Editor: M. Cveticˇ term for this shift symmetry invariant Lagrangian density can be elegantly written down in superspace with a single Grassmann parameter. Keywords: © 2011 Elsevier B.V. Open access under CC BY license. BV formalism Non-anticommutative quantum gravity

1. Introduction invariance under the original BRST and shift transformations in the BV formalism. Consequently, we will express our results in super- Noncommutative field theory arises in string theory due to space formalism. presence of the NS antisymmetric tensor background [1–3].How- The BRST symmetry for Yang–Mills theories [22–25] and spon- ever, other backgrounds like the RR background can generate non- taneously broken gauge theories [26] has already been analysed in anticommutative field theory [4,5]. Quantum gravity on noncom- noncommutative spacetime. The invariance of a theory, under the mutative spacetime has been thoroughly studied [6–9].Infact,it original BRST transformations and shift transformations, that oc- is hoped that noncommutativity predicts the existence of the cos- curs naturally in background field method can be analysed in the mological constant which is of the same order as the square of the BV formalism [27–30]. Both the BRST formalism [31,32] and the Hubble’s constant [10]. Perturbative quantum gravity on noncom- BV formalism can be given geometric meaning by the use of su- mutative spacetime has been already analysed [11].Itwasfound perspace [33–36]. that the graviton propagator was the same as that in the commu- BV formalism has been used for quantising W3 gravity [37–39]. tative case. However, the noncommutative nature of spacetime was It has also been used in quantising metric-affine gravity in two experienced at the level of interactions. dimensions [40]. The BRST symmetry for perturbative quantum The idea of noncommutativity of spacetime has been gener- gravity in four-dimensional flat spacetime have been studied by alised to non-anticommutativity. In addition to this, quantum field a number of authors [41–43] and their work has been sum- theory has been studied on non-anticommutative spacetime [12]. marised by N. Nakanishi and I. Ojima [44]. The BRST symmetry Non-anticommutativity of spacetime occurs if the metric is com- in two-dimensional curved spacetime has been studied thoroughly plex. Spacetime with complex metric has been studied as an in- [45–47]. Similarly, the BRST symmetry for topological quantum teresting example of nonsymmetric gravity [13–15].Even though gravity in curved spacetime [48,49] and the BRST symmetry for nonsymmetric quantum gravity was initially studied in an attempt perturbative quantum gravity in both linear and nonlinear gauge’s to unify electromagnetism and gravity [16,17],itisnowmainly has also been analysed [50]. However, so far no work has been studied due to its relevance to string theory [18–20]. Quantum done in analysing the non-anticommutative perturbative quantum gravity on this non-anticommutative complex spacetime has also gravity in superspace BV formalism. This is what we aim to do in been discussed before [21]. this Letter. In this Letter we will discuss the perturbative quantum gravity on this non-anticommutative complex spacetime. We will first 2. Perturbative quantum gravity analyse the BRST symmetry of this theory and then study its the

We shall analyse perturbative quantum gravity with the follow- E-mail address: [email protected]. ing hyperbolic complex metric [21],

( f ) = ( f ) + ( f ) L = gbc bbc ωabc , (1) g sΨ, (11) where ω is the pure imaginary element of a hyperbolic complex where Clifford algebra with ω2 =+1. Here ω forms a ring of numbers 1 and not a field which the usual system of complex numbers do. ¯a b Ψ = c 3 ∂ hab − k∂ah + ba , (12) The advantage of using ω is that the negative energy states coming 2 from the purely imaginary part of the metric will be avoided. with k = 1. Now the sum of the ghost term, the gauge-fixing term Now we can introduce non-anticommutativity as follows, and the original classical Lagrangian density is invariant under the following BRST transformations xâ, xˆb = 2xâxˆb + iωτ ab, (2) = e 3 a − 3 b a where τ ab is a symmetric tensor. We use Weyl ordering and ex- shab Dab ce, sc cb ∂ c , press the Fourier transformation of this metric as, sc¯a =−ba, sba = 0. (13) ( f ) − ˆ ( f ) gˆ (xˆ) = d4k πe ikx g (k). (3) BV formalism is used to analyse the extended BRST symmetry. ab ab This extended BRST symmetry for perturbative quantum gravity can be obtained by first shifting all the original fields as, Nowwehaveaonetoonemapbetweenafunctionofxˆ to a func- tion of ordinary coordinates y via → − ˜ a → a − ã hab hab hab, c c c , ( f ) − ( f ) a a a a a ã = 4 ikx c¯ → c¯ − c¯˜ , b → b − b , (14) gab (x) d k πe gab (k). (4) and then requiring the resultant theory to be invariant under both So, the product of ordinary functions is given by the original BRST transformations and these shift transformations. ( f )ab 3 ( f ) This can be achieved by letting the original fields transform as, g (x) gab (x) ω ( f ) a a = ab 2 1 ( f )ab shab = ψab, sc = φ , exp τ ∂a ∂ g (x1)g (x2) = = . (5) 2 b ab x1 x2 x sc¯a = φ¯a, sba = ρa, (15) ( f )a Now R given as, bcd and the shifted fields transform as ( f )a R =−∂ Γ a + ∂ Γ a + Γ a 3 Γ e − Γ a 3 Γ e , (6) bcd d bc c bd ec bd ed bc ˜ = − e 3 − ˜ shab ψab Dab (ce ce), and we also get R = Rd . Thus finally R( f ) is given by bc bcd scã = φa + (c − c˜ ) 3 ∂b(c − c˜ ), b b a a ( f ) a a a ã a a ( f ) = ( f )ab 3 sc¯˜ = φ¯ + b − b , sb˜ = . (16) R g Rab . (7) ρ a ¯a The Lagrangian density for pure gravity with cosmological constant Here ψab, φ , φ , and ρa are ghosts associated with the shift sym- λ can now be written as, metry and their BRST transformations vanish, √ L = ( f ) 3 ( f ) − a ¯a a c g R 2λ , (8) sψab = sφ = sφ = sρ = 0. (17) where we have adopted units, such that 16π G = 1. In perturba- We define antifields with opposite parity corresponding to all the ( f ) tive gravity on flat spacetime one splits the full metric gab into original fields. These antifields have the following BRST transfor- ηab which is the metric for the background flat spacetime and hab mations, which is a small perturbation around the background spacetime, ∗ ¯ ∗a a sh = bab, sc = B , ( f ) = + ab g ηab hab. (9) ∗a a ∗a ¯a ab sc¯ = B , sb = b . (18) Here both η and h are complex. The covariant derivatives along ab ab ¯ a a ¯a with the lowering and raising of indices are compatible with the Here bab, B , B , and b are Nakanishi–Lautrup fields and their BRST transformations vanish too, metric for the background spacetime. The small perturbation hab is viewed as the field that is to be quantised. ¯ a a ¯a sbab = sB = sB = sb = 0. (19) 3. BV formalism It is useful to define

All the degrees of freedom in hab are not physical as the La- = − ˜ a = a − ã hab hab hab, c c c , grangian density for h is invariant under the following gauge ab ¯a ¯a ˜¯a a a ã transformations, c = c − c , b = b − b . (20) The physical requirement for the sum of the gauge-fixing term and δ h = De 3 Λ Λ ab ab e the ghost term is that all the fields associated with shift symmetry = e + e + ce 3 δb∂a δa ∂b g (∂chab) vanish. This can be achieved by choosing the following Lagrangian density, + gec 3 h ∂ + ηec 3 h ∂ 3 Λ . (10) ac b cb a e ˜ ¯ ãb ∗ab e Lg =−b 3 h − h 3 ψ − D 3 c In order to remove these unphysical degrees of freedom, we need ab ab ab e to fix a gauge by adding a gauge-fixing term along with a ghost a ∗ a b a − B 3 cã + c¯ 3 φ + c 3 ∂ c term. In the most general covariant gauge the sum of the gauge- a b + a 3 ¯˜ − ∗ 3 ¯a + a + a 3 ˜ + ∗a 3 fixing term and the ghost term can be expressed as, B ca ca φ b B ba b ρa. (21) 122 M. Faizal / Physics Letters B 705 (2011) 120–123

The integrating out the Nakanishi–Lautrup fields in this Lagrangian Furthermore, if we define Ψ as, density will make all the shifted fields vanish. = + If we choose a gauge-fixing fermion Ψ , such that it depends Φ Ψ θsΨ only on the original fields and furthermore define L = sΨ ,then g = + − δΨ 3 + δΨ 3 + δΨ 3 ¯ − δΨ 3 we have Ψ θ ψab φa ¯ ψa ρa , δhab δca δca δba δΨ δΨ δΨ δΨ L =− 3 + 3 + 3 ¯ − 3 (30) g ψab φa ¯ ψa ρa. (22) δhab δca δca δba then we can express Lg given by Eq. (22) as, The total Lagrangian density is given by ∂ L = ˜ ˜ g Φ. (31) L = Lc(h − h) + Lg + Lg . (23) ∂θ After integrating out the Nakanishi–Lautrup fields, this total La- Now the complete Lagrangian density in the superspace formalism grangian density can be written as, is given by, ∂ ∂ L = L − ˜ + ∗ 3 ab e + ¯∗a 3 b 3 L = + ˜ ∗ 3 ˜ ab + ˜¯∗ 3 ã − ˜¯a 3 ˜∗ − ˜¯a 3 ˜∗ c(h h) hab De c c c ∂bca Φ ωab ω ηa η η ηa η ηa ∂θ ∂θ ∗ ∗a ∗ δΨ ab ∗ δΨ a − ˜ 3 ˜ a + L − ˜ − c 3 ba − h + 3 ψ + c¯ + 3 φ fa f c(hab hab). (32) ab δhab a δca Upon elimination of the Nakanishi–Lautrup fields and the ghosts ∗ δΨ ∗ δΨ − c − 3 φ¯a + b − 3 ρa. (24) associated with shift symmetry, this Lagrangian density is man- a ¯a a a δc δb ifestly invariant under the BRST symmetry as well as the shift Now integrating out the ghosts associated with the shift symmetry, symmetry. we get the following expression for the antifields, 5. Conclusion ∗ δΨ ∗ δΨ h =− , c a = , ab δhab δc¯a In this Letter we analysed non-anticommutative perturbative δΨ δΨ gravity with a complex metric. As this theory contained unphys- ¯∗a =− ∗a = c , b . (25) ical degrees of freedom, we added a gauge-fixing term and a ghost δca δba term to it. We found that the sum of the original classical La- These equations along with Eq. (12) fix the exact expressions for grangian density, the gauge-fixing term and the ghost term was the antifields in terms of the original fields. invariant under the BRST transformations. As the shifting of fields occurs naturally in the background field method, we analysed the 4. Superspace formulation effect of the shift symmetry in the BV formalism. Finally, we expressed our results in the superspace formalism using a single In this section we will express the results of the previous sec- Grassmann parameter. tion in superspace formalism with one anti-commutating variable. It is well known that the sum of the original classical La- Let θ be an anti-commutating variable, then we can define the fol- grangian density, the gauge-fixing term and the ghost term for lowing superfields, most theories posing BRST symmetry is also invariant under an- other symmetry called the anti-BRST symmetry [44].Itwillbe ˜ e ω = h + θψ , ω˜ = h + θ ψ − D 3 c , ab ab ab ab ab ab ab e interesting to investigate the anti-BRST version of this theory. Fur- = + ˜ = ˜ + + 3 b thermore, the invariance of a gauge theory under the BRST and ηa ca θφa, ηa ca θ φa c ∂ ca , b the anti-BRST transformations along with the shift transformations ¯ = + ¯ ˜¯ = ¯˜ + ¯ + ηa ca θφa, ηa ca θ φa ba , has already been analysed in the superspace BV formalism [33]. ˜ ˜ Thus, after analysing the anti-BRST symmetry for this theory, the f = b + θρ , f = b + θρ , (26) a a a a a a invariance of this theory under the original BRST and the original and the following anti-superfields, anti-BRST transformations along with shift transformations can be studied in the superspace BV formalism. ˜ ∗ = ∗ − ¯ ˜∗ = ∗ − ωab hab θbab, ηa ca θ Ba, It will also be interesting to generalise the results of this Letter ˜¯∗ = ¯∗ − ˜ ∗ = ∗ − ¯ to general curved spacetime. The generalisation to arbitrary space- ηa ca θ Ba, fa ba θba. (27) time might not be so simple as it is still not completely clear how From these two equations, we have the BRST symmetry works for general curved spacetime. There will also be ambiguities due to the definition of vacuum state. ∂ ˜ ∗ab ˜ ¯ ãb ∗ab e We know it is possible to define a vacuum state called the Eu- ω 3 ωab =−bab 3 h − h 3 ψab − D 3 c , ∂θ ab e clidean vacuum in maximally symmetric spacetime [51].Wealso ∂ ∗ ∗ know the ghosts in anti-de Sitter spacetime do not contain any η˜¯ 3 ηã =−Ba 3 c˜ + c¯ a 3 ψ + c 3 ∂b c , ∂θ a a a b a infrared divergence. Therefore, the generalisation of this work to anti-de Sitter spacetime can be done easily [52].However,asthe ∂ − ˜¯a 3 ˜∗ = a 3 ¯˜ − ∗ 3 ¯a + a η ηa B ca ca φ b , ghosts in de Sitter spacetime contain infrared divergence, this work ∂θ cannot be directly extended to de Sitter spacetime [52].Inorderto ∂ − ˜ ∗ 3 ˜ a = ¯a 3 ˜ + ∗a 3 generalise this work to de Sitter spacetime we will have to modify fa f b ba b ρa. (28) ∂θ the BRST transformations accordingly. ˜ Now we can express Lg given by Eq. (21) as, References ∂ ∗ab ∗ a a ∗ ˜ ∗ ˜ a L˜ = ω˜ 3 ω˜ + η˜¯ 3 η˜ − η˜¯ 3 η˜ − f 3 f . (29) g ∂θ ab a a a [1] N. Seiberg, E. Witten, JHEP 9909 (1999) 032. M. Faizal / Physics Letters B 705 (2011) 120–123 123

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