Extension to the superstring - Bi-invariant symplectic gravity

5.3 Bi-invariant symplectic gravity

5.3.9 Extension to the superstring

Having reformulated successfully the bosonic low energy effective action of string theory in a frame with dynamical variables (ˆg, β, φ), the question arises if this is extendable also to the type II superstring actions. As spinors are independent of gauge transformations of the B-field and are not affected by the Lie algebroid constructions so far, we do not expect a nontrivial transformation behavior of them under the field redefinitions (5.3.89). In contrast to this, the spin connection uses the tangent bundle and should therefore be generalized to Lie algebroids. Let us discuss the different sectors in turn.

The R-R sector

In the R-R sector we first consider the p-form fields C_a₁_...a_p. They are tensorial in the standard sense and they remain invariant under gauge transformations of the B-field. As discussed in section 5.3.5, especially equation (5.3.62) we can convert the p-form fields by use of the anchor form the (G, B, φ)-frame to the (ˆg, β, φ)-frame

Cˆ^a¹^...a^p = β^a¹^b¹· · ·β^a^p^b^bCb1...bp . (5.3.98)

From the theory established in section 5.3.5, we know that the totally antisym-metricp-vectors ˆC^a¹^...a^p are standard- andβ-tensors. In addition we know that the β-covariant derivative ˆ∇of these multi-vectors again gives a bi-covariant tensor:

Fˆ^a¹^...a^p+1 = ˆ∇^[a¹Cˆ^a²^...a^p+1^]. (5.3.99) However, to interpret them as proper physical field strengths, we have to show that they are invariant under the corresponding gauge transformations of the p-form fields ˆC^a¹^...a^p:

δΛCˆ^a¹^...a^p = ˆ∇^[a¹Λ^a²^...a^p^]. (5.3.100) Let us show the invariance explicitly:

δ_ΛFˆ^a¹^...a^p+1 = ˆ∇^[a¹∇ˆ^a²Λ^a³^...a^p+1^]

= ˆR_k^[a³^a¹^a²Λ^a⁴^...a^p+1^]k = 0, (5.3.101) where in the last line we first used vanishing torsion for the Levi-Civita connection and then the Bianchi identity (5.3.68) for the Riemann curvature tensor in (ˆg, β, φ)-frame. Therefore, identifying C_a and C_a₁_a₂_a₃ with the one- and three-form gauge potentials of type IIA supergravity, we have found corresponding expressions in the symplectic frame.

In analogy to the standard formulation, we then introduce generalized field strengths of the form:

Fˆ₂ = ˆF₂ , Fˆ₄ = ˆF₄ −Θ∧Cˆ₁, (5.3.102) and for the corresponding action we consider:

Sˆ_IIA^R-R = 1 2κ²₁₀

d¹⁰xp

−|ˆg|

β⁻¹

−¹₂|Fˆ2|²−¹₂|Fˆ4|²

, (5.3.103)

where we employ:

|Fˆp|² = 1 p!

Fˆa1...apFˆ^a¹^...a^p . (5.3.104) As was explained in detail, these expressions are bi-invariant scalars and combined with the corresponding measure in (5.3.103), we get a bi-invariant action for the R-R field strengths. It remains to reformulate the Chern-Simons part of the R-R sector of type IIA supergravity which is given by:

S_IIA^CS = 1 4κ²₁₀

H∧F₄∧C₃

= 1

4κ²₁₀ 1 3! 4! 3!

d¹⁰x ^a¹^...a¹⁰H_a₁_a₂_a₃F_(4)a₄_a₅_a₆_a₇C_(3)a₈_a₉_a₁₀ .

(5.3.105)

5.3 Bi-invariant symplectic gravity 111 Here, we denote by ^a¹^...a¹⁰ = ±1 the epsilon-tensor density. Noting the fact that ^a¹^...a¹⁰/p

−|G|transforms as a standard tensor field and is invariant underB-field gauge transformations, one can show in a straightforward way that_b₁_...b₁₀/p

−|ˆg|

is a β-tensor. Thus, we are able to write down the Chern-Simons part of the type IIA action in the symplectic frame:

Sˆ_IIA^CS = 1 4κ²₁₀

1 3! 4! 3!

d¹⁰x β⁻¹

_b₁_...b₁₀Θ^b¹^b²^b³Fˆ₍₄₎^b⁴^b⁵^b⁶^b⁷Cˆ₍₃₎^b⁸^b⁹^b¹⁰ . (5.3.106) Let us conclude our exposition of the R-R part by the following remark. Having already seen the invariance of the field strengths ˆF under gauge transformations of the multi-vectors ˆC, by using also the Bianchi identity of Θ, rewritten in the form

∇ˆ^[aΘ^bcd]= 0 , (5.3.107)

it is possible to show the invariance of the complete transformed type IIA action under the following set of gauge transformations:

δ_Λ₍₀₎Cˆ^a = ˆ∇^aΛ₍₀₎ , δ_Λ₍₂₎Cˆ^a¹^a²^a³ = ˆ∇^[a¹Λ^a₍₂₎²^a³^], δ_Λ₍₀₎Cˆ^a¹^a²^a³ =−Λ₍₀₎Θ^a¹^a²^a³ .

(5.3.108)

The NS-R and R-NS sectors

As was mentioned in the introduction to this section, the spin connection uses the tangent bundle and therefore should be transformed carefully to the Lie algebroid setting. Due to its structure, involving Christoffel symbols and derivatives of vielbein matrices, it is not clear a priori, if the transformation works. But it turns out, due to the inhomogeneous transformation behavior of the Christoffel symbols (5.3.58), that we get a spin connection in the (ˆg, β, φ)-frame having the same structure as the standard one with partial derivatives replaced by the D^a -derivative and standard Christoffel symbols replaced by the ones in the symplectic frame.

Let us first establish our notation and state that

α, β, γ, . . . denote Lorentz-frame indices, a, b, c, . . . denote space-time indices.

The vielbein matricese_α^arelating these two frames viae_a=e_a^αe_αande^a=e_α^ae^α are defined in the usual way by requiring that:

e_α^ae_β^bG_ab =η_αβ , (5.3.109)

with η_αβ = diag (−1,+1, . . . ,+1). In these conventions the spin connection can be expressed in terms of the Christoffel symbols Γ^c_ab as follows

ω_c^α_β =e_a^αe_β^bΓ^a_cb+e_a^α∂_ce_β^a. (5.3.110) Introducing spacetime-dependent gamma matrices by contracting with the vielbein γ^a = γ^αeαa and using standard notation γ^a¹^...a^p = γ^[a¹γ^a²· · ·γ^a^p^], we write the kinetic term for the dilatinoλ as:

L^λ_IIA =λγ^a ∂a− ₄ⁱωa αβγ^αβ

λ . (5.3.111)

Let us now transform this expression into the symplectic frame. By applying the anchor map to a basis in the Lorentz frame, we get:

e^α = e^α_adx^a →e^α_aβ^ab∂_b =:η^αγeˆ^m_γ∂_m . (5.3.112) Note, that this is compatible with the expected definition of the vielbein matrices in the symplectic frame:

e^α_aeˆ^β_bgˆ^ab=η^αβ . (5.3.113) We thus take the following definition for the new vielbein matrices in terms of the standard ones:

e^α_a =η^αβe_β^bβ_ba. (5.3.114) To finally state the result for the dilatino kinetic term and the corresponding form of the new spin connection let us recall that spinors are not affected by transformation to a Lie algebroid. Thus, ˆλ = λ and let us in addition define the new gamma matrices by:

γ_a = γ_αeˆ^α_a = γ^βe_β^bβ_ba . (5.3.115) Due to its importance let us state the result in a separate proposition and give a detailed proof of the transformation.

Proposition 5.3.5. With the definitions given above, the kinetic term for the dilatino in the symplectic frame is given by:

Lˆ^λ_IIA = ˆλγˆ_a D^a−₄ⁱωˆ^a_βδˆγ^βδˆλ , (5.3.116) where the transformed spin connection takes the form:

ˆ ω^aαβ

= ˆe^bαeˆ^βcΓˆbac

+ ˆe^bαD^aˆe^βb. (5.3.117)

5.3 Bi-invariant symplectic gravity 113 Proof. Writing out the definition and employing the relation (5.3.58) together with the transformation behavior (5.3.114) gives

L_IIA = ¯λγ^ah

∂_a− ₄ⁱ

e_k^α⁰e_β^bΓ^k_ab+e_k^α⁰∂_ae_β^k

η_αα⁰γ^αβi λ

= ¯λˆγ_a⁰β^a⁰^a h

∂_a− ₄ⁱ(e_k^α⁰e_β^b(−β^kk⁰β_aa⁰⁰β_bb⁰Γˆ_k⁰^a⁰⁰^b⁰ −β_b⁰_b∂_aβ^kb⁰) +β_kk⁰η^α⁰^α⁰⁰eˆ^k⁰_α⁰⁰∂_a(β^k⁰⁰^kη_ββ⁰eˆ^β⁰_k⁰⁰))η_αα⁰γ^αβi

=λˆ¯ˆγ_a⁰h

D^a⁰ − ₄ⁱ(ˆe^k⁰_αeˆ^β⁰_bΓˆ_k⁰^a⁰^b⁰ + ˆe^k⁰_α∂_aeˆ^β⁰_k⁰)η^αα⁰γ_α⁰_β⁰i λ ,ˆ

(5.3.118)

where we also used the transformation behavior of the inverse vielbein in the symplectic frame ˆe^aα = ηαββ^abebβ, which gives again the right sign in the second term.

The remaining part in this sector is the kinetic term for the gravitino Ψa, which also has a spacetime-vector index. It is given by the Rarita-Schwinger Lagrangian:

L^Ψ_IIA = Ψaγ^abc ∇b−₄ⁱωb αβγ^αβ

Ψc. (5.3.119)

In this form, the connection coefficients of the covariant derivative drop out due to the totally anti-symmetrized gamma matrices. Note that this is a special property of the Levi-Civita connection of Riemannian geometry and is different e.g. in theories with torsion. In the case of a general Lie algebroid, even the Levi-Civita connection can have an antisymmetric part and therefore we keep the general form (5.3.119). To transform the above expression into the symplectic frame, we first note the transformation of the vector index of the gravitino:

Ψˆ^a =β^abΨ_b . (5.3.120)

Performing an analogous transformation as done for the dilatino kinetic term, we observe that this factor of β^ab is crucial: It is needed to transform the covariant derivative in the right way to the symplectic frame. The result is then given by:

Lˆ^Ψ_IIA = ˆΨ^aγˆ_abc

∇ˆ^b −₄ⁱωˆ^b_αβγ^αβ

Ψˆ^c . (5.3.121) Note, that now the antisymmetric part of the symplectic Christoffel symbols enter in the new kinetic term: ˆΓ_c^[ab]= ¹₂Q_c^ab.

This completes our exposition of the fermionic terms in type IIA supergravity, since all the terms can be transformed to the symplectic frame in the ways des-cribed above. We therefore constructed a complete type IIA supergravity theory in the symplectic frame. A natural question to ask is, whether this theory can be obtained from first principles, i.e. by supersymmetrizingβ-diffeomorphims and then writing down an invariant theory. We will comment about this question in chapter 6 on future directions.

Im Dokument Lie algebroids, non-associative structures and non-geometric fluxes (Seite 121-126)