Fluxes

6.2 Fluxes

Compactifying string theory with fluxes leads, in the low energy limit, to gauged supergravity.

They correspond to deformation of abelian supergravities where a subgroup G₀ of the global symmetry group G of the supergravity theory is promoted to a local symmetry. The embedding of the gauge group G0 into the global symmetry group G can be described by an object called the embedding tensor, which corresponds exactly to the fluxes. Supersymmetry and gauge invariance of the embedding tensor then leads to a set of linear and quadratic constraint on the embedding tensor, which by extension should be verified by the fluxes of the corresponding low energy limit of string theory [75, 76].

Here we derive the expression of the generalised fluxes for the SL(3) × SL(2) × R⁺ ex-ceptional field theory. They will have to verify both linear and quadratic constraints so that the corresponding 8 dimensional gauged maximal supergravity we obtain in the low energy limit after compactification with fluxes is consistent. Considering the warp factor in the duality group will lead us to consider gauged supergravity with a gauged trombone symmetry. The gauging of the trombone symmetry forSL(3)×SL(2) exceptional field theory has never been done before due to the group product structure of this particular theory. We construct it here similarly to what is done in [67] where the trombone symmetry is gauged for simple groups.

6.2.1 Embedding tensor structure of D=8 gauged maximal Supergravity with trombone symmetry

A way to describe the gauging of a subgroup of a global symmetry group G in supergrav-ity theories is through the constant embedding tensor Θ_M^Γ [75, 76], where Γ is an index of Adj(G) = Adj(SL(3)×SL(2))in our case andM corresponds to the fundamental representa-tion (3,2). A consistent local gauging of the theory forces one to consider two constraints on

66 CHAPTER 6. GEOMETRY OF E₃₍₃₎×R⁺EFT AND F-THEORY this embedding tensor: a linear one and a quadratic one. Let us recall the results already known for the particular case of E₃₍₃₎ exceptional field theory, without the scale factor of the general extended group. A priori the embedding tensorΘ_M^Γ of the theory lives in

(3,2)×((8,1) + (1,3)) = [(3,2) + (6,2) + (15,2)] + [(3,2) + (2,4)]], (6.9)

but due to the linear and quadratic constraints, the embedding tensor only has(6,2) and(3,2) components. Using this linear constraint we can write the generators of the gauge group of the theory using the embedding tensor and the generators of the adjoint of the gauge group{tΓ}

(X_M)_N^P = Θ_M^Γ(t_Γ)_N^P = Θ_mγ,n^pδ_η^ρ+ Θ_mγ,η^ρδ^p_n (6.10)

with

Θ_mγ,η^ρ=ξ_mηδ_γ^ρ− 1

2δ_η^ρξ_mγ =P(3_SL(2))ρ ηδ

γξ_mδ Θmγ,np

=fγ(pb)

bmn− 3

4(ξnγδ_m^p − 1

3ξmγδ_n^p) =fγ(pb)

bmn −3

4P^(8)pnrmξrγ

(6.11)

withΘ_mγ,η^ρδ^p_n ∈ (3,2)and Θ_mγ,n^pδ^ρ_η ∈ (6,2)¹. To avoid confusion between the fundamental rep-resentation ofSL(3) and the adjoint ofSL(2) we write the later(3_SL(2)).

This is not the more general setting of supergravity gauging however, as one can gauge the trombone symmetry [66, 67]. In order to do that we have to consider a more general ansatz than the one used in [67], as the global symmetry group is not simple in our case but a product of simple groups. Considering the R⁺ factor in the duality group leads to an additional generator (t₀)_N^P = −δ_N^P in equation (6.10), and a corresponding additional component of the embedding tensorΘ_M⁰ ≡ K_M. This component lives in the(3,2)representation, and we expect it to appear in the same way as the other (3,2) parameter ξ_M. This leads to the following ansatz for the

1fγmnandξmα need to verify a set of quadratic constraints which can be found in [77].

6.2. FLUXES 67

where ζ₁ and ζ₂ are two real parameters. The symmetric part of the generators of the gauge group, the intertwining tensor, should be in the same representation whether or not we consider an R⁺ gauging. This is necessary in order to preserve the two-form field content of the theory [67]. This is verified forζ₁ =−ζ₂ = 6. The generators still have to verify a set of constraints which can be expressed in terms of the tensors introduced before as

0 =X_{M N}^PK_P + 6P(8)r

Now that we described the embedding tensor of maximal supergravity in 8 dimensions with a gauged trombone symmetry we look at the fluxes ofSL(3)×SL(2)×R⁺EFT. First let us consider the generalised metric of the extended space. We can define a generalised metric H living in the quotient ^SL(3)_SO(3) × _SO(2)^SL(2) ×R⁺ which transforms covariantly underSL(3)×SL(2)×R⁺ and is invariant under the maximal compact subgroup of E₃₍₃₎ i.e SO(3)×SO(2). Due to the product structure of the group, we define a generalised bein which splits as

EA¯M =e^−∆e_¯a^ml_α¯^γ (6.15)

68 CHAPTER 6. GEOMETRY OF E₃₍₃₎×R⁺EFT AND F-THEORY where∆is theR⁺component of the metric,e_¯a^mandl_α¯^γ theSL(3)andSL(2) beins respectively.

¯

aandα¯areSO(3)andSO(2)planar indices respectively. The metric of the internal space is then

H^{M N} =EA¯MEB¯Nδ^A¯B¯ =e^−2∆H^mng^γη (6.16)

where

H^mn =e_a¯^me¯bnδ^¯a¯b g^γη =l_α¯^γlβ¯ηδ^α¯β¯

(6.17)

correspond to anSL(3) anSL(2)metric respectively.

Having defined the generalised bein and a consistent generalised Lie derivative of the theory, one defines the generalised fluxes similarly to the fluxes in general relativity as²

LEA¯EB¯ =FA¯B¯ C¯

EC¯. (6.18)

In a coordinate frame, we find the fluxes to be

F_{M N}^P = Ω_{M N}^P −(2P(8,1)P NR

S+ 3P(1,3)P NR

S)Ω_RM^S+ 1

6Ω_RM^Rδ_N^P (6.19) where

Ω_{M N}^P = (E⁻¹)_N^A¯∂_MEA¯P (6.20) is the Weitzenböck connection³. Now, using the expressions of the bein (6.15) we obtain

Ω_{M N}^P =−∂_M∆δ_N^P +δ^ρ_η(e⁻¹)_n^¯a∂_mγ(e_¯a^p) +δ_n^p(l⁻¹)_η^α¯∂_mγ(l_α¯^ρ)

=−∂M∆δ_N^P +δ^ρ_ηΩmγ,np

+δ_n^pΩmγ,ηρ

(6.21)

2For more details see [74].

3We are abusing notation as the Weitzenböck connection should be globally defined, which is a priori not the case here.

6.2. FLUXES 69

After some manipulations we find the following generalised fluxes

FM NP

andζ is only used to write the fluxes in a similar form compared to the gauge generators (6.12).

Choosingζ =−6gives us the the same expressions we found after considering the intertwining tensor constraint in the context of D=8 gauged maximal supergravity with gauged trombone symmetry. We thus have to consider the quadratic constraints (6.13) and (6.14) on K and f. We present simplified expressions of these constraints for the type IIB supergravity solution of the section condition in section 6.4.3, after choosing an appropriate ansatz of the generalised bein (6.15).

70 CHAPTER 6. GEOMETRY OF E₃₍₃₎×R⁺EFT AND F-THEORY