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This content was downloaded from IP address 131.169.4.70 on 08/12/2017 at 01:42 X Workshop of the Gravitation and Mathematical Physics Division, Mexican Physical Society IOP Publishing Journal of Physics: Conference Series 545 (2014) 012016 doi:10.1088/1742-6596/545/1/012016 Supersymmetric closed string tachyon cosmology: a first approach V V´azquez-B´aez1;2 and C Ram´ırez2 1Benem´eritaUniversidad Aut´onoma de Puebla, Facultad de Ingenier´ıa,Blvd. Valsequillo esq. Av. San Claudio s/n, Ciudad Universitaria, Edificio 108-C, 72570 Puebla, M´exico 2Benem´eritaUniversidad Aut´onomade Puebla, Facultad de Ciencias F´ısicoMatem´aticas,P.O. Box 165, 72000 Puebla, M´exico E-mail: [email protected] Abstract. We give a worldline supersymmetric formulation for the effective action of closed string tachyon in a FRW background. This is done considering that, as shown by Vafa, the effective theory of closed string tachyons can have worldsheet supersymmetry. The Hamiltonian is constructed by means of the Dirac procedure and written in a quantum version. By using the supersymmetry algebra we are able to find solutions to the Wheeler-DeWitt equation via a more simple set of first order differential equations. 1. Introduction String theory has in its lowest mode tachyons which for the lack of knowledge on how to handle unstable configurations were ignored for many years. With the inclusion of supersymmetry they can be consistently eliminated by the GSO truncation. Some years ago it has been seen that the evolution of these tachyonic instabilities can be described by the condensation of the tachyonic modes. This was first perfomed in the somewhat simpler case of open strings, resumed by the well known Sen conjectures [1]. For closed strings the situation is more complicated, because it involves the structure of space-time. An interesting fact in this case, is that closed string tachyons, which are nonsupersymmetric in target space, can have worldsheet supersymmetry [2]. In [3] a Lagrangian of supersymmetric tachyons in the framework of a FRW background has been worked out, in a worldline superspace where the time variable is extended to the superspace of supersymmetry [6, 7]. This work is performed along such approach considering the covariant formulation of one-dimensional supergravity of the so called `new' Θ variables [8, 9], which allows in a straightforward way to write supergravity invariant actions. Supersymmetric cosmology was studied in a variety of different schemes, we refer the interested reader to the well known books by D'Eath and Moniz for a review [4, 5], 2. Closed String Tachyon Effective Action The bosonic closed string tachyon effective action is given according to [10] as Z 1 p −2φ h 2 2 i D S = 2 −ge R + 4 (@φ) − (@T ) − 2V (T ) d x; (1) 2κD Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 X Workshop of the Gravitation and Mathematical Physics Division, Mexican Physical Society IOP Publishing Journal of Physics: Conference Series 545 (2014) 012016 doi:10.1088/1742-6596/545/1/012016 where T is the closed string tachyon field, V (T ) is the tachyon potential and φ is the dilaton string φ Einstein field. This action can be written in the Einstein frame by means of gµν = e gµν , which is more suitable for our cosmological approach. For a four dimensional FRW metric and in the Einstein frame, the action takes the form Z 3_a2a 3Nka a3 a3 a3Ne2φV (T ) S = − + + φ_2 + T_ 2 − dt; (2) κ2N κ2 κ2N 2κ2N κ2 where N is the lapse function and a is the scale factor. This Lagrangian is invariant under time reparametrizations and this invariance is extended to supersymmetry introducing a Grassmann superspace associated to the bosonic time coordinate t (see Tkach et al. in [6, 7]). 3. Supersymetric Closed String Tachyon Model As stated in [11] superspace is the natural framework for a geometrical formulation of supersymmetry and supergravity, it extends spacetime by anticommuting Grassmann variables, xm ! (xm; θµ), and the field content of the superfields is given by the Grassmann power P µ1 µn expansion in the anticommuting variables φ(z) = n 1=n! θ ··· θ φµ1···µn (x). The supersymmetric cosmological model is obtained upon an extension of the time coordinate into a supermultiplet t ! (t; Θ; Θ).¯ Due to this generalization the fields of the theory are generalized as superfields, in this the expansion mentioned above is given by A t; Θ; Θ¯ = a (t) + iΘλ¯ (t) + iΘ¯ λ (t) + B (t)ΘΘ¯ ; T t; Θ; Θ¯ = T (t) + iΘ¯η (t) + iΘ¯ η(t) + G (t)ΘΘ¯ ; (3) Φ t; Θ; Θ¯ = φ (t) + iΘ¯χ (t) + iΘ¯ χ (t) + F (t)ΘΘ¯ ; where, A, T and Φ are the superfields of a, T and φ. The supersymmetric generalization of the action is given by S = SRsusy + SMsusy; (4) where, SRsusy is the cosmological supersymmetric generalization of the free FRW model p Z ! 3E 3 k 2 S = Ar¯Ar A − EA dΘdΘ¯ dt; (5) Rsusy κ2 θ θ κ2 and the supersymmetric matter term is Z 1 3 1 3 3 S = −EA r ¯ Φr Φ − EA r ¯ T r T + EA W (Φ; T ) dΘdΘ¯ dt; (6) Msusy κ2 Θ Θ 2 Θ Θ where W (Φ; T )p is the superpotential, and E is an invariant density playing the role of a supersymmetric −g in the action and given by i E = −e − (ΘΨ¯ + Θ¯ ); (7) 2 see [3, 8, 9] for details of its calculation. @W @W 1 @2W The superpotential expansion is W (Φ; T ) = W (φ, T ) + @φ (Φ − φ) + @T (T − T ) + 2 @T 2 (T − 2 1 @2W 2 @2W T ) + 2 @φ2 (Φ − φ) + @T @φ (T − T )(Φ − φ). This expansion is finite because the terms (T − T ) and (Φ − φ) are purely grassmannian and nilpotency holds for these kind or variables. 2 X Workshop of the Gravitation and Mathematical Physics Division, Mexican Physical Society IOP Publishing Journal of Physics: Conference Series 545 (2014) 012016 doi:10.1088/1742-6596/545/1/012016 Integrating over the Grassmann parameters, the total Lagrangian is 3aa_ 2 3aa_ a3T_ 2 a3T_ 3ia2T_ a3φ_2 L = − + λ − ¯λ¯ + − η − ¯η¯ + λη¯ + λη¯ + eκ2 eκ2 2eκ2 2eκ2 2κ2 eκ2 a3φ_ 3ia2φ_ 3ia ia3 ia3 ¯ ¯ ¯_ ¯ _ _ _ − 2 χ − χ¯ + 2 λχ¯ + λχ + 2 λλ + λλ − 2 (ηη¯ +η ¯η_) − 2 (χχ¯ +χ ¯χ_) eκp p κ κ 2κ κ 6e kλλ¯ 3ia k 6aeW λλ¯ 3ia2W ia3W + + λ + ¯λ¯ − − λ + ¯λ¯ − T η + ¯η¯ κ2 κ2 κ2 2κ2 2κ2 3a2eW ia3W 3a2eW a3eW + T λη¯ − λη¯ − φ χ + ¯χ¯ + φ λχ¯ − λχ¯ − TT ηη¯ κ2 2κ2 κ2 κ2 3 3 3 3 a eWT φ a eWφφ 3a ¯λλ¯ a ¯ηη¯ a χχψ¯ ¯ 3aeλλη¯ η¯ + 2 (¯χη − χη¯) − 2 χχ¯ − 2 + 2 + 2 − 2 κ κ p 2eκ 4eκ 2eκ κ 6aeλλχ¯ χ¯ 3aB2e 6aBe k 3Beλλ¯ 3a2Beηη¯ 3a2Beχχ¯ 3a2BeW − − + − − − − κ2 κ2 κ2 κ2 2κ2 κ2 κ2 a3eG2 3a2eG a3eGW a3eF 2 3a2eF a3eF W + + λη¯ − λη¯ − T + + λχ¯ − λχ¯ − φ ; 2κ2 2κ2 κ2 κ2 κ2 κ2 where the subscripts in W denote partial differentiation with respect to φ and T respectively. When we perform the variation of the Lagrangian with respect to the fields B, F and G, as usual algebraic constraints are obtained, that is B, F and G play the role of auxiliary fields, and they can be eliminated from the Lagrangian. Upon solving for the auxiliary fields and making the further rescalings for convinience λ ! κa−1=2λ, λ¯ ! κa−1=2λ¯, η ! κa−3=2η,η ¯ ! κa−3=2η¯, χ ! κa−3=2χ,χ ¯ ! κa−3=2χ¯, we find the Lagrangian p p 2 _ 2 3 3 _ _ 3aa_ 3 aa_ ¯¯ 3eka T a a T ¯ 3iT ¯ L = − 2 + λ − λ + 2 + 2 − η − η¯ + λη¯ + λη eκ p eκ κ 2eκ 2eκ 2 φ_2a3 a3φ_ i ¯ _ ¯ ¯_ ¯ _ _ + 2 − χ − χ¯ + 3iφ λχ¯ + λχ + 3i λλ + λλ − (ηη¯ +η ¯η_) eκ eκ p p p p 2 3e kλλ¯ 3e kηη¯ 3e kχχ¯ 3i ak − i (χχ¯_ +χ ¯χ_) + − − + λ + ¯λ¯ a 2a a κ p 2 3 2 2 3 eW 2a3 3eW a 3e ka W eWT a φ 9 ¯ 3 3 + 2 − 2 − 2 − 2 − eW λλ + eW ηη¯ + eW χχ¯ 4κ κ p 2κ 4κ 2 4 p2 3ia3=2W i a3W 3eW i a3W − λ + ¯λ¯ − T η + ¯η¯ + T λη¯ − λη¯ − φ χ + ¯χ¯ 2κ 2κ 2 2κ 3eW 3eκ2 3 + φ λχ¯ − λχ¯ − eW ηη¯ + eW (¯χη − χη¯) − eW χχ¯ + ηηχ¯ χ¯ − ¯λλ¯ 2 TT T φ φφ 4a3 2e 1 1 + ¯χχ¯ + ¯ηη:¯ 2e 4e 3 X Workshop of the Gravitation and Mathematical Physics Division, Mexican Physical Society IOP Publishing Journal of Physics: Conference Series 545 (2014) 012016 doi:10.1088/1742-6596/545/1/012016 4.