Iosif Bena
email: iosif.bena@cea.fr

Gregory Giecold
email: gregory.giecold@cea.fr

Mariana Graña
email: mariana.grana@cea.fr

Nick Halmagyi
email: nicholas.halmagyi@cea.fr

Francesco Orsi
email: francesco.orsi@cea.fr

Supersymmetric Consistent Truncations of IIB on T 1,1

)u 1 +(2-2p)u 3 ( * ω p ) ∧ Ω ′ ∧ (g 5 + A 1 ) * 10 ω p ∧ Ω ′ ∧ (g 5 + A 1 ) = e (-4+2p)u 1 +(8-2p)u 3 ( * ω p ) ∧ Ω ′ * 10 ω p ∧ J ′ 1 ∧ J ′

Introduction

The study of type II supergravity on the conifold has provided much insight into the low-energy limit of string theory. The explicit Ricci flat metric on the singular conifold, its small resolution and its deformation were all computed in [START_REF] Candelas | Comments on conifolds[END_REF] and immediately after the advent of the AdS-CFT correspondence [START_REF] Maldacena | The large N limit of superconformal field theories and supergravity[END_REF][START_REF] Gubser | Gauge theory correlators from non-critical string theory[END_REF][START_REF] Witten | Anti-de sitter space and holography[END_REF] it was realized that the singular conifold, suitably embedded in IIB string theory admits a near horizon limit that contains an AdS 5 factor, and describes the supergravity dual of a certain superconformal field theory [START_REF] Klebanov | Superconformal field theory on threebranes at a calabi-yau singularity[END_REF]. This system and its generalizations have proved to be a fertile ground for the study of the gauge/gravity duality. In particular the "warped deformed conifold" non-conformal deformation found in [START_REF] Klebanov | Supergravity and a confining gauge theory: Duality cascades and chisb-resolution of naked singularities[END_REF] (which we will denote KS) is an explicit supergravity realization of confinement and chiral symmetry breaking. The solution found in [START_REF] Chamseddine | Non-Abelian BPS monopoles in N = 4 gauged supergravity[END_REF][START_REF] Maldacena | Towards the large N limit of pure N = 1 super Yang Mills[END_REF] (which we will denote CV/MN) is the holographic dual of a field theory whose infra-red limit is four-dimensional N = 1 Yang-Mills theory. The rich physics of the above supergravity solutions led to the proposal of an anzatz (which we will call the PT ansatz) [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] that describes the solutions interpolating between the KS and the CV/MN flows. Following the perturbative analysis of [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF], a fully backreacted supersymmetric solution interpolating between the KS and the CV/MN solutions was constructed in [START_REF] Butti | The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds[END_REF] (since this solution is dual to the baryonic branch of the warped deformed conifold we will denote it BB). Recently an interesting conceptual development regarding this system was provided [START_REF] Maldacena | The Unwarped, resolved, deformed conifold: Fivebranes and the baryonic branch of the Klebanov-Strassler theory[END_REF].

The current work constitutes a step towards improving our understanding of the spectrum of both supersymmetric and nonsupersymmetric deformations around the warped deformed conifold and its baryonic branch (KS and BB). A thorough understanding of these deformations is important for many problems in string cosmology and phenomenology and there has been much work on computing the spectrum of fluctuations around KS (for example [START_REF] Berg | Bulk Dynamics in Confining Gauge Theories[END_REF][START_REF] Berg | Glueballs vs. Gluinoballs: Fluctuation Spectra in non-AdS/non-CFT[END_REF][START_REF] Dymarsky | Gravity Multiplet on KS and BB Backgrounds[END_REF][START_REF] Benna | On Normal Modes of a Warped Throat[END_REF][START_REF] Dymarsky | I-odd Sector of the Klebanov-Strassler Theory[END_REF][START_REF] Gordeli | On I-even Singlet Glueballs in the Klebanov-Strassler Theory[END_REF]) but to our knowledge certain SU(2) × SU(2) invariant modes which lie within our ansatz have not been considered. These modes are crucial for constructing a supersymmetric five-dimensional theory and thus are needed to arrange the spectrum into supermultiplets.

The PT ansatz was constructed so as to contain a variety of conifold solutions in IIB supergravity and indeed families of new solutions were found within this ansatz [START_REF] Butti | The baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds[END_REF]. However this ansatz is deficient is several ways, firstly it does not contain one-form or two-form fields in five dimensions and secondly there are seven SU(2) × SU(2) invariant scalar modes which are absent. By systematically including all singlets under this symmetry we are able to construct an explicitly supersymmetric action in five dimensions. This has been an open problem for some time (it is ten years since the PT ansatz was written down and six years since explicit solutions were found) but recent progress in Kaluza-Klein reductions (see [START_REF] Gauntlett | Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions[END_REF][START_REF] Gauntlett | Consistent supersymmetric Kaluza-Klein truncations with massive modes[END_REF] for early work and [START_REF] Cassani | Type IIB supergravity on squashed Sasaki-Einstein manifolds[END_REF][START_REF] Gauntlett | Universal Kaluza-Klein reductions of type IIB to N=4 supergravity in five dimensions[END_REF][START_REF] Liu | Consistent massive truncations of IIB supergravity on Sasaki-Einstein manifolds[END_REF][START_REF] Skenderis | A Consistent truncation of IIB supergravity on manifolds admitting a Sasaki-Einstein structure[END_REF] for more recent additions) has demonstrated clearly that a promising strategy is at hand. These works were focused on finding infinite classes of reductions on manifolds using minimal information about the internal manifolds (namely the Sasaki-Einstein structure) whereas we will focus solely on T 1,1 . See also [START_REF] Herzog | Emergent Quantum Near-Criticality from Baryonic Black Branes[END_REF] for recent consistent (non-supersymmetric) truncations on T 1,1 .

One central motivation for the current work is to compute a superpotential in five dimensions for a reduction which includes PT. Such a superpotential would provide an organizing principle for the spectrum of supersymmetric and non-supersymmetric modes [START_REF] Borokhov | Non-Supersymmetric Deformations of the Dual of a Confining Gauge Theory[END_REF]. Recently, several of the current authors [START_REF] Bena | On the Existence of Meta-stable Vacua in Klebanov-Strassler[END_REF] have computed a certain sector of the spectrum around the warped deformed conifold in search of the modes sourced by a stack of anti-D3 branes placed at the tip of the deformed conifold. Having a superpotential at hand for this sub-sector was a crucial step in the calculation and also considerable help in the physical interpretation. With this superpotential it is manifestly clear which modes are BPS and which modes are not.

There exists a very interesting conjecture that a stack of anti-D3 branes at the tip of the warped deformed conifold will create a meta-stable vacuum and ultimately decay by brane-flux annihilation to the BPS vacuum [START_REF] Kachru | Brane/flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory[END_REF]. In the work [START_REF] Bena | On the Existence of Meta-stable Vacua in Klebanov-Strassler[END_REF] we have carefully imposed IR boundary conditions compatible with anti-D3 branes and have found a seemingly unphysical singularity. We have interpreted this singularity as evidence that the IR boundary conditions of anti-D3 branes are incompatible with the UV boundary conditions of KS. Since there must exist good IR boundary conditions we conclude that the UV boundary conditions must be changed. Establishing this conjecture to be true is probably tantamount to constructing the supergravity solution dual to this state explicitly. Then to demonstrate that such a candidate solution is in fact a metastable vacuum, it behooves one to check that indeed it does not have runaway directions. Rigorously, this would entail checking all modes but this not a reasonable task in general. Having this consistent truncation may be helpful in this regard in checking at least a closed subsector should such a solution ever be constructed. However recently there has been an example found where a nonsupersymmetric solution [START_REF] Pope | An SU(4) Invariant Compactification of d = 11 Supergravity on a Stretched Seven Sphere[END_REF] is stable within a consistent truncation but has runaway directions which lie outside the truncation [START_REF] Bobev | Supergravity Instabilities of Non-Supersymmetric Quantum Critical Points[END_REF]. This is a general limitation of consistent truncations. This paper is organized as follows: In section two we present our ansatz and proceed to derive the five-dimensional action. Then in section three we demonstrate that this is consistent with N = 4 supersymmetry and compute the embedding tensor. In section four we consider various additional consistent truncations including the PT ansatz. The reduction techniques used in this paper are very similar to those used in [START_REF] Gauntlett | Consistent supersymmetric Kaluza-Klein truncations with massive modes[END_REF] and the manifest supersymmetry is demonstrated following closely the strategy of [START_REF] Cassani | Type IIB supergravity on squashed Sasaki-Einstein manifolds[END_REF][START_REF] Gauntlett | Universal Kaluza-Klein reductions of type IIB to N=4 supergravity in five dimensions[END_REF]. In an attempt to minimize the amount of labor required for interpreting our work, we have tried to use notation quite similar to [START_REF] Gauntlett | Universal Kaluza-Klein reductions of type IIB to N=4 supergravity in five dimensions[END_REF] for the bulk of the computation. The same day we submitted this paper to the arxiv a paper appeared by D. Cassani and A. Faedo [START_REF] Cassani | A Supersymmetric consistent truncation for conifold solutions[END_REF] which has some overlap with this work.

2 The Kaluza-Klein Reduction on T 1,1

Our Kaluza-Klein reduction on T 1,1 is best thought of as a consistent truncation of the standard reduction of IIB on T 5 (where of course only massless modes are retained), followed by a very specific gauging which preserves N = 4 supersymmetry in five-dimensions. This gauging is due to the curvature of T 1,1 as well as the topological flux which we include.

To perform this Kaluza-Klein reduction, we can simply dimensionally reduce many parts of the ten-dimensional action, namely the kinetic terms for the dilaton-axion, three-forms and metric but the five-form kinetic terms and the Chern-Simons terms require much more care. We will proceed by deriving five-dimensional equations of motion for components of the five-form and then reconstructing a five-dimensional action from these. Since our reduction is based on a symmetry principle, it is guaranteed to be consistent.

The Ansatz

We will motivate our full ansatz by first understanding the structures we wish to preserve on the metric of T 1,1 . Then we will make an ansatz for the three-forms and five-form and solve the Bianchi identities. The dilaton and axion are additional fields which have trivial Bianchi identities.

The Metric

Our ansatz for the ten-dimensional metric is

ds 2 10 = e 2u 3 -2u 1 ds 2 5 + e 2u 1 +2u 2 E ′ 1 E ′ 1 + e 2u 1 -2u 2 E ′ 2 E ′ 2 + e -6u 3 -2u 1 E 5 E 5 , (1)
where

E 1 = 1 √ 6 σ 1 + iσ 2 , E 2 = 1 √ 6 Σ 1 + iΣ 2 , (2)
E ′ 1 = E 1 E ′ 2 = E 2 + vE 1 , (3)
E 5 = g 5 + A 1 , g 5 = 1 3 σ 3 + Σ 3 , (4)
(σ i , Σ j ) are internal left-invariant SU(2) one-forms (see appendix B), the scalar fields u j are real, v is complex and A 1 is a real one-form, all in the reduced five-dimensional theory. The particular parameterization we have chosen for the scalar fields in (1) might seem convoluted but is motivated by getting canonical kinetic terms in the five-dimensional action. This is the most general SU(2) × SU(2) invariant metric we can put on T1,1 and consequently, by introducing a radial co-ordinate in five dimensions and allowing for the fields to have radially dependent profiles this ansatz contains the most general SU(2) × SU(2) invariant metric on the conifold (singular, resolved or deformed).

In terms of G-structures, this is a U(1) structure on T 1,1 , namely we have restricted to all U(1) neutral fields where the U(1) in question is embedded in SO(5) as follows:

U(1) ⊂ SU(2) D ⊂ SO(4) ⊂ SO(5) (5)
and SU(2) D is diagonally embedded in SO(4). As can be seen from tables 1 and 2 in appendix A, this leaves 16 scalars, 9 one-forms and 1 graviton. Under this U(1) the vielbeins transform as

E 1 → e iα E 1 , E 2 → e -iα E 2 , E 5 → E 5
and from this it is simple to observe that the invariant fundamental forms are

J 1 = i 2 E 1 ∧ E 1 , J 2 = i 2 E 2 ∧ E 2 , Ω = E 1 ∧ E 2 , Ω = E 1 ∧ E 2 , (6)
as well as g 5 . From (3) we see that these forms behave nicely under the exterior derivative

dg 5 = 2(J 1 + J 2 ), dJ 1,2 = 0, (7)
dΩ = 3iΩ ∧ g 5 but it turns out to be prudent to introduce a twisted set of fundamental forms which behave nicely under the Hodge star operation (our conventions for Hodge dualizing are given in appendix C).

J ′ 1 = i 2 E ′ 1 ∧ E ′ 1 , J ′ 2 = i 2 E ′ 2 ∧ E ′ 2 , Ω ′ = E ′ 1 ∧ E ′ 2 , Ω ′ = E ′ 1 ∧ E ′ 2 . (8)
In fact this truncation can also be thought of as the K-invariant truncation where in standard co-ordinates on the conifold, K is a particular Z 2 symmetry acting on the conifold as

K : (ψ, θ 2 ) → (ψ + π, -θ 2 ). (9)
Consequently, since our ansatz contains all modes invariant under this symmetry it must be consistent 1 .

The Three-Forms

The three-forms in our ansatz are

H (3) = H 3 + H 2 ∧ (g 5 + A 1 ) + H 11 ∧ J 1 + H 12 ∧ J 2 + + M 1 ∧ Ω + M 0 Ω ∧ (g 5 + A 1 ) + c.c , (10)
F (3) = P (J 1 -J 2 ) ∧ (g 5 + A 1 ) + G 3 + G 2 ∧ (g 5 + A 1 ) + G 11 ∧ J 1 + G 12 ∧ J 2 + + N 1 ∧ Ω + N 0 Ω ∧ (g 5 + A 1 ) + c.c (11)
where we have included a topological term in F (3)

P (J 1 -J 2 ) ∧ (g 5 + A 1 ) (12)
which is proportional to the volume form on the topologically nontrivial S 3 ⊂ T 1,1 . One can also include an independent topological term for the NS flux2 but by using the IIB SL(2, Z) symmetry, this can always be rotated to a frame where the charge is just in F (3) .

To establish the spectrum from our ansatz we must first solve the Bianchi identities. From

dH (3) = 0 ( 13
)
we find

H 3 = dB 2 + 1 2 (db -2B 1 ) ∧ F 2 , H 2 = dB 1 , H 11 = d(b + b) -2B 1 , (14)
H 12 = d(b -b) -2B 1 , 3iM 1 = DM 0 = dM 0 -3iA 1 M 0 where F 2 = dA 1 . Then from dF (3) = -F (1) ∧ H (3) (15)
we find

G 3 = dC 2 -a dB 2 + 1 2 (dc -adb -2C 1 + 2aB 1 ) ∧ F 2 , G 2 = dC 1 -a dB 1 , G 11 = d(c + c) -2C 1 -a d(b + b) -2B 1 -P A 1 , (16)
G 12 = d(c -c) -2C 1 -a d(b -b) -2B 1 + P A 1 , 3iN 1 = DN 0 + M 0 da = dN 0 -3iA 1 N 0 + M 0 da
and a is the RR axion C (0) . From these relations we discover that the fluxes contribute 8 scalars

(b, b, c, c, M 0 , M 0 , N 0 , N 0 ) (17)
a pair of two-form potentials (B 2 , C 2 ) [START_REF] Gordeli | On I-even Singlet Glueballs in the Klebanov-Strassler Theory[END_REF] and two one-form potentials

(B 1 , C 1 ). ( 19
)
Using these fields one can of course write the three-form field strengths in terms of two-form potentials:

H (3) = dB (2) , ⇒ B (2) = B 2 + 1 2 bF 2 + B 1 ∧ (g 5 + A 1 ) + (b + b)J 1 + (b -b)J 2 + ( 1 3i M 0 Ω + c.c.), (20)
F (3) = P (J 1 -J 2 ) ∧ (g 5 + A 1 ) + dC (2) -a dB (2) , ⇒ C (2) = C 2 + 1 2 cF 2 + C 1 ∧ (g 5 + A 1 ) + (c + c)J 1 + (c -c)J 2 + ( 1 3i N 0 Ω + c.c). (21)

The Five-Form

We take the five-form to be manifestly self-dual

F (5) = e Z e 8(u 3 -u 1 ) vol 5 + e Z J ′ 1 ∧ J ′ 2 ∧ (g 5 + A 1 ) +K ′ 1 ∧ J ′ 1 ∧ J ′ 2 -e -8u 1 ( * 5 K ′ 1 ) ∧ (g 5 + A 1 ) +K ′ 21 ∧ J ′ 1 ∧ (g 5 + A 1 ) + e -4u 2 +4u 3 ( * 5 K ′ 21 ) ∧ J ′ 2 +K ′ 22 ∧ J ′ 2 ∧ (g 5 + A 1 ) + e 4u 2 +4u 3 ( * 5 K ′ 22 ) ∧ J ′ 1 + L ′ 2 ∧ Ω ′ + c.c ∧ (g 5 + A 1 ) + e 4u 3 ( * 5 L ′ 2 ) ∧ Ω ′ + c.c , (22)
where we have defined the primed forms such as K ′ 1 in Appendix E. Due to this self-duality of the five-form, the Bianchi identity for the five-form must be disentangled from the equation of motion

dF (5) = H (3) ∧ F (3)
and we find (see appendix E)

e Z = Q -2P b + 4i 3 M 0 N 0 -M 0 N 0 (23)
K 1 = Dk + 2(bDc -bDc) + 2i 3 M 0 N 1 -N 0 M 1 -M 0 N 1 + N 0 M 1 (24)
K 21 = Dk 11 + 1 2 Db ∧ Dc + Db ∧ Dc + D b ∧ Dc (25)
K 22 = Dk 12 + 1 2 Db ∧ Dc -Db ∧ Dc -D b ∧ Dc (26)
where

Dk = dk + 4cB 1 -QA 1 -2(k 11 + k 12 ), Dk 11 = dk 11 + P B 2 , Dk 12 = dk 12 -P B 2 , (27) Db = db -2B 1 , D b = d b, Dc = dc -C 1 , Dc = dc -P A 1
and Q is a constant corresponding to the D3 Page charge. So from the five-form we get:

scalar -k 1 -form -(k 11 , k 12 ) (28) 2 -form -(L 2 , L 2 )
Deriving the equations of motion for the five-form is one of the more involved steps in the analysis, the results are presented in appendix E. The equations of motion are of course necessary to construct the five-dimensional Lagrangian, to which we now turn.

The Five-Dimensional Lagrangian

There are several subtleties in producing a five-dimensional Lagrangian whose equations of motion match those of the ten dimensional theory, largely due to the Chern-Simons terms in ten dimensions. We have checked that the Lagrangian we present below indeed reproduces the ten dimensional equations of motion with the most non-trivial task being to check the flux equations of motion.

The five-dimensional action is the sum of several terms

L = L gr + L s,kin + L g,kin + L pot + L CS (29)
which correspond to the five-dimensional Einstein-Hilbert term, the scalar kinetic terms, the kinetic terms for the gauge fields and two-forms, the scalar potential and the Chern-Simons terms. We find the scalar kinetic terms to be

L s,kin = - 1 2 e -4(u 1 +u 2 )-φ H ′ 11 ∧ * H ′ 11 - 1 2 e -4(u 1 -u 2 )-φ H 12 ∧ * H 12 -4e -4u 1 -φ M ′ 1 ∧ * M ′ 1 - 1 2 e -4(u 1 +u 2 )+φ G ′ 11 ∧ * G ′ 11 - 1 2 e -4(u 1 -u 2 )+φ G 12 ∧ * G 12 -4e -4u 1 +φ N ′ 1 ∧ * N ′ 1 -8du 1 ∧ * du 1 -4du 2 ∧ * du 2 -12du 3 ∧ * du 3 -e -4u 2 Dv ∧ * Dv - 1 2 e -8u 1 K 1 ∧ * K 1 - 1 2 dφ ∧ * dφ - 1 2 e 2φ da ∧ * da, (30)
where we have twisted some of the one-forms

H ′ 11 = H 11 -|v| 2 H 12 -4 Im (vM 1 ), (31)
M ′ 1 = M 1 + i 2 vH 12 , (32)
G ′ 11 = G 11 -|v| 2 G 12 -4 Im (vN 1 ), (33)
N ′ 1 = N 1 + i 2 vG 12 ( 34
)
and φ is the dilaton. The kinetic terms for the gauge fields are

L g,kin = - 1 2 e -8u 3 F 2 ∧ * F 2 - 1 2 e 4u 1 -4u 3 -φ H 3 ∧ * H 3 - 1 2 e 4u 1 +4u 3 -φ H 2 ∧ * H 2 - 1 2 e 4u 1 -4u 3 +φ G 3 ∧ * G 3 - 1 2 e 4u 1 +4u 3 +φ G 2 ∧ * G 2 -4e 4u 3 1 + |v| 2 e -4u 2 L 2 ∧ * L 2 + 4e -4u 2 +4u 3 v 2 L 2 ∧ * L 2 + c.c - 1 2 e 4u 2 +4u 3 1 + |v| 2 e -4u 2 2 K 22 ∧ * K 22 - 1 2 e -4u 2 +4u 3 K 21 ∧ * K 21 +|v| 2 e -4u 2 +4u 3 K 22 ∧ * K 21 + 2e 4u 3 1 + |v| 2 e -4u 2 ivK 22 ∧ * L 2 + c.c -2e -4u 2 +4u 3 ivK 21 ∧ * L 2 + c.c (35)
where the somewhat off-diagonal last four lines come from the five-form.

The scalar potential has several contributions which we distinguish for clarity:

L pot = -V gr + V H (3) + V F (3) + V F (5) , (36)
V gr = -12e -4u 1 -2u 2 +2u 3 1 + |v| 2 + e 4u 2 + 9|v| 2 e -4u 2 +8u 3 +2e -8u 1 -4u 3 e 4u 2 + e -4u 2 (1 -|v| 2 ) 2 + 2|v| 2 , (37)
V H (3) = 4e -4u 1 +8u 3 -φ |M 0 | 2 + 2e -4u 2 [Im (M 0 v)] 2 , (38)
V F (3) = 1 2 e -4u 1 +8u 3 +φ 8|N ′ 0 | 2 + e 4u 2 P 2 + e -4u 2 P (|v| 2 -1) + 4 Im (N ′ 0 v) 2 , (39)
V F (5) = 1 2 e 2Z e -8u 1 +8u 3 (40)
where

N ′ 0 = N 0 - i 2 P v. (41)
As expected this scalar potential is almost but not quite a sum of squares. The only term which spoils this property is V gr . Finally the gravitational term is of course

L gr = R vol 5 ( 42
)
where R is the Ricci scalar in Einstein frame. The Chern-Simons terms are particularly long and unspectacular so we will not write them explicitly. In the ungauged case which we deal with below, they are somewhat simpler and also extremely crucial so we will present them explicitly there.

Manifest N = 4 Supersymmetry

A particularly insightful aspect of the works [START_REF] Cassani | Type IIB supergravity on squashed Sasaki-Einstein manifolds[END_REF][START_REF] Gauntlett | Universal Kaluza-Klein reductions of type IIB to N=4 supergravity in five dimensions[END_REF] was the construction of manifest N = 4 supersymmetry (by which we mean 16 supercharges). In that case, the reason this was unexpected was that this particular gauging of N = 4 supergravity does not have a vacuum which preserves all the supercharges, the maximally supersymmetric vacuum is an AdS 5 which preserves only N = 2. Still more surprisingly, despite the fact that we have generalized the reduction considered there by including 5 new scalars and one new vector, the N = 4 supersymmetry is still present in our work. We find that these extra fields constitute just one additional N = 4 vector multiplet compared to those works.

The scalar coset

In five dimensional, N = 4 supergravity there are the following multiplets

• graviton multiplet: (g µν , 6 × A µ , φ)

• vector multiplet: (A µ , 5 × φ) so clearly our reduction has the bosonic field content of N = 4 gauged supergravity coupled to three vector multiplets 3 . The scalar coset of this five-dimensional N = 4 gauged supergravity is [START_REF] Awada | N=4 Maxwell-Einstein Supergravity in five-dimensions and its SU(2) gauging[END_REF] SO(5, 3)

SO(5) × SO(3) × SO(1, 1) (43)
and a particular basis for this coset was given in [START_REF] Lu | M-theory/heterotic duality: A Kaluza-Klein perspective[END_REF] eq. (3.31). In the conventions of that paper we take our coset element to be

V = e 2 3/2 φ 1 H 1 e 2 -3/2 φ 2 H 2 e 2 -3/2 φ 3 H 3 e x 1 E 3 2 e x 2 E 3 1 e x 3 E 2 1 .e x 7 U 1 1 e x 8 U 2 1 e x 9 U 3 1 e x 10 U 1 2 e x 11 U 2 2 e x 12 U 3 2 .e x 4 V 23 e x 5 V 13 e x 6 V 12 (44)
and the scalar Kinetic terms are

L s,kin = -3Σ -2 dΣ ∧ * dΣ + 1 8 Tr(dM ∧ * dM -1 ) (45)
where

M = TrVV T . ( 46
)
Explicitly we find

-Tr(dM ∧ * dM -1 ) = 2 dφ 2 1 + dφ 2 2 + dφ 2 3 +4e -φ 2 +φ 3 dx 2 1 + 4e -φ 1 +φ 3 (dx 2 -x 1 dx 3 ) 2 + 4e -φ 1 +φ 2 dx 2 3 +4e φ 1 +φ 2 (dx 4 + x 7 dx 8 + x 10 dx 11 ) 2 +4e φ 1 +φ 3 dx 5 + x 7 dx 9 + x 10 dx 12 -x 1 (dx 4 + x 7 dx 8 + x 10 dx 11 ) 2 +4e φ 1 dx 2 7 + e φ 2 dx 8 -x 3 dx 7 2 +4e φ 3 dx 9 -(x 2 -x 1 x 3 )dx 7 -x 1 dx 8 2 +4e φ 1 dx 2 10 + e φ 2 dx 11 -x 3 dx 10 2 +4e φ 3 dx 12 -(x 2 -x 1 x 3 )dx 10 -x 1 dx 11 2 +4e φ 2 +φ 3 dx 6 + x 2 dx 4 + x 2 x 7 dx 8 + x 2 x 10 dx 11 -x 3 dx 5 +(x 8 -x 3 x 7 )dx 9 + (x 11 -x 3 x 10 )dx 12 2 . ( 47
)
It may be helpful to describe how this basis (which we will refer to as the "heterotic basis") is related to a more common basis in the gauged supergravity literature [START_REF] Schon | Gauged N = 4 supergravities[END_REF][START_REF] Samtleben | Lectures on Gauged Supergravity and Flux Compactifications[END_REF] (which we will refer to as the "gsg basis") where generators of SO(5, 3) are given by

(t M N ) Q P = δ Q [M η N ]P (48)
M, N . . . = 1, . . . , 8 and η = diag{+ + + + + ---}. Of course only a subset of the t M N generate the coset SO(5, 3)/(SO(5) × SO(3)). The two basis (where the Heterotic basis is completed to a full set of generators of SO(5, 3)) are related by conjugation with C:

C = D 1 + D 2 + D 3 + E 44 + E 55 , (49)
where

D i = E i,i -E i,i+5 + E i+5,i + E i+5,i+5 / √ 2 i = 1, . . . , 3 (50)
and where E ij is a matrix with 1 in the i-th row and j-th column and zero's elsewhere. This is most easily seen by relating η and η where

η16 = η61 = -1 η27 = η72 = -1 η38 = η83 = -1 η44 = η55 = 1 (51)
and V T ηV = η. Matching (47) to our scalar kinetic terms [START_REF] Bobev | Supergravity Instabilities of Non-Supersymmetric Quantum Critical Points[END_REF] obtained from dimensional reduction we find the rather nonlinear (but invertible) mapping

e 2u 3 = Σ -4u 2 = φ 1 -4u 1 -φ = φ 2 -4u 1 + φ = φ 3 √ 2 v = x 7 + ix 10 a = x 1 b -b = x 3 b + b = -x 4 - 1 2 x 3 (x 2 7 + x 2 10 ) c -c = x 2 (52) c + c = -x 5 - 1 2 x 2 (x 2 7 + x 2 10 ) 2 √ 2 3 M 0 = -x 8 -x 3 x 7 + i(x 11 -x 3 x 10 ) = -(x 8 -ix 11 ) + x 3 (x 7 -ix 10 ) 2 √ 2 3 N 0 = -x 9 -(x 2 -x 1 x 3 )x 7 -x 1 x 8 + i x 12 -(x 2 -x 1 x 3 )x 10 -x 1 x 11 = -(x 9 -ix 12 ) + (x 2 -x 1 x 3 )(x 7 -ix 10 ) + x 1 (x 8 -ix 11 ) k = x 6 + x 2 x 4 + 1 2 x 2 x 3 (x 2 7 + x 2 10 ) + 1 2
x 2 x 7 x 8 + x 10 x 11 + x 9 (x 8x 3 x 7 ) + x 12 (x 11x 3 x 10 ) .

3.2

The gauge kinetic and Chern-Simons terms in the ungauged reduction

In ungauged N = 4 supergravity, the kinetic and Chern-Simons terms for the gauge fields take a particularly simple form:

L g,kin + L CS = - 1 2 Σ -4 H 0 2 ∧ * H 0 2 - 1 2 Σ 2 M M N H M 2 ∧ * H N 2 + 1 √ 2 η M N A 0 1 ∧ H M 2 ∧ H N 2 ( 53
)
where H M 2 = DA M are field strengths for the gauge fields. Indeed this is what we find by setting the topological fluxes (P, Q) to zero and altering the differential relations [START_REF] Chamseddine | Non-Abelian BPS monopoles in N = 4 gauged supergravity[END_REF] to

dg 5 = 0, dJ 1,2 = 0, dΩ = 0. ( 54
)
With these alterations, massaging the Lagrangian [START_REF] Pope | An SU(4) Invariant Compactification of d = 11 Supergravity on a Stretched Seven Sphere[END_REF] into this form allows one to identify the correct basis to take for the nine gauge fields, however first we must integrate out the pair of twoforms (B 2 , C 2 ). The central difference between the gauged reduction and the ungauged reduction is the Bianchi identities and their solution: instead of ( 14), ( 16) and ( 23)-( 26) we have for H (3) :

H 3 = dB 2 -B 1 ∧ F 2 , H 2 = dB 1 , H 11 = d(b + b), H 12 = d(b -b), 3iM 1 = dM 0 , (55)
for

F (3) G 3 = dC 2 -adB 2 -(C 1 -aB 1 ) ∧ F 2 , G 2 = dC 1 -a dB 1 , G 11 = d(c + c) -a d(b + b), G 12 = d(c -c) -a d(b -b), 3iN 1 = dN 0 -adM 0 , (56)
and for F (5)

K 1 = dk + 2(bdc -bdc) + 2i 3 M 0 N 1 -N 0 M 1 -M 0 N 1 + N 0 M 1 , K 21 = dk 11 + (b + b)dC 1 -(c + c)dB 1 , K 22 = dk 12 + (b -b)dC 1 -(c -c)dB 1 , L 2 = dD 1 + 1 3i M 0 dC 1 -N 0 dB 1 . (57)
We find that before integrating out (B 2 , C 2 ), the Chern-Simons terms are

L top = -A 1 ∧ K 22 ∧ K 21 + K 1 ∧ (-C 1 ∧ H 2 + B 1 ∧ G 2 ) + 4L 2 ∧ L 2 +K 21 ∧ d(b -b) ∧ C 1 -d(c -c) ∧ B 1 +K 22 ∧ d(b + b) ∧ C 1 -d(c + c) ∧ B 1 + (4i/3)L 2 ∧ (M 0 C 1 -N 0 B 1 ) + c.c -dC 2 ∧ S 2 + dB 2 ∧ T 2 (58)
where

S 2 = k + 4 9 Re(M 0 N 0 ) dB 1 -(b 2 -b2 + 1 9 |M 0 | 2 )dC 1 -(b -b)dk 11 -(b + b)dk 12 - 8 3 Im M 0 dD 1 (59) T 2 = k - 4 9 Re(M 0 N 0 ) dC 1 + (c 2 -c2 + 1 9 |N 0 | 2 )dB 1 -(c -c)dk 11 -(c + c)dk 12 - 8 3 Im N 0 dD 1 . (60)
First we introduce Lagrange multiplers ( B 1 , C 1 )

∆L = C 1 ∧ d H 3 + B 1 ∧ d G 3 (61)
where

H 3 = dB 2 , G 3 = dC 2 ,
and then we integrate out ( H 3 , G 3 ) and after some algebra we find

L g,kin = - 1 2 e -8u 3 F 2 ∧ * F 2 - 1 2 e -4(u 1 -u 3 )-φ H 2 ∧ * H 2 - 1 2 e -4(u 1 -u 3 )+φ G 2 ∧ * G 2 - 1 2 e 4(u 1 +u 3 )-φ H 2 ∧ * H 2 - 1 2 e 4(u 1 +u 3 )+φ G 2 ∧ * G 2 -4e 4u 3 1 + |v| 2 e -4u 2 L 2 ∧ * L 2 + 4e -4u 2 +4u 3 v 2 L 2 ∧ * L 2 + c.c - 1 2 e 4u 2 +4u 3 1 + |v| 2 e -4u 2 2 K 22 ∧ * K 22 - 1 2 e -4u 2 +4u 3 K 21 ∧ * K 21 +|v| 2 e -4u 2 +4u 3 K 22 ∧ * K 21 + 2e 4u 3 1 + |v| 2 e -4u 2 ivK 22 ∧ * L 2 + c.c -2e -4u 2 +4u 3 ivK 21 ∧ * L 2 + c.c , (62)
L top = -A 1 ∧ dk 12 ∧ dk 11 + 4dD 1 ∧ dD 1 -dB 1 ∧ d C 1 -dC 1 ∧ d B 1 , (63)
Identifying ( 62) with (53) we construct the SO(5, 3) vector of one-form potentials:

A 0 = -A 1 / √ 2, A 1 = -k 11 / √ 2, A 2 = B 1 / √ 2, A 3 = C 1 / √ 2, A 4 = 2Im (D 1 ), A 5 = 2Re (D 1 ), A 6 = k 12 / √ 2, A 7 = C 1 / √ 2, A 8 = B 1 / √ 2. ( 64
)
So we have now shown that the ungauged theory, corresponding to a particular consistent truncation on the five-torus T 5 has N = 4 supersymmetry. Of course this is not the theory of most interest to us but was a necessary step in developing the gauged theory, which corresponds to a consistent truncation on T 1,1 .

The embedding tensor for the gauged theory

Having successfully demonstrated the manifest supersymmetry of the ungauged theory, the content of the gauged theory can be neatly summarized in the embedding tensor. For our purposes, the embedding tensor has components (f M N P , ξ M N ) which can be computed from the scalar kinetic terms:

DM M N = dM M N + 2A P f Q P (M M N )Q + 2A 0 ξ P (M M N )P . (65)
We have found the heterotic basis (44) to be computationally efficient but the embedding tensor is most naturally expressed in the "gsg basis" (48) where it is completely antisymmetric (with all indices lowered):

f M N P = f [M N P ] , ξ M N = ξ [M N ] . (66)
Note that our expressions (64) are in the heterotic basis. After some work explicitly computing (65), we find that the non-vanishing components in the basis (48) are

f 123 = -f 128 = f 137 = f 178 = 2, ( 67
) ξ 23 = -ξ 28 = ξ 37 = ξ 78 = -Q/ √ 2, ( 68
) ξ 45 = -3 √ 2 (69) ξ 36 = ξ 68 = √ 2P (70)
and permutations. From this one can read off the covariant field strength of N = 4 gauged supergravity:

H M = dA M + 1 2 f M N P A N ∧ A P + 1 2 ξ M P A 0 ∧ A P (71)
and for example we have

H 1 + H 6 = d(A 1 + A 6 ) -2 √ 2A 7 ∧ A 8 , H 7 = dA 7 , (72)
H 8 = dA 8
here we have used the basis of gauged fields in the "heterotic basis" (64). From this we see the same Heisenberg algebra which was observed in [START_REF] Cassani | Type IIB supergravity on squashed Sasaki-Einstein manifolds[END_REF][START_REF] Gauntlett | Universal Kaluza-Klein reductions of type IIB to N=4 supergravity in five dimensions[END_REF]. The only additional gaugings in our ansatz arise from the topological flux we have turned on (70). So the additional vector multiplet we have included has enhanced the complexity of the embedding tensor somewhat indirectly through the additional degrees of freedom required to allow for non-trivial topology and thus the flux P .

The Scalar Potential

A useful check of our computations is to compute the scalar potential from the gauged supergravity formula

V = 1 2 f M N P f QRS Σ -2 1 12 M M Q M N R M P S - 1 4 M M Q η N R η P S + 1 6 η M Q η N R η P S + 1 8 ξ M N ξ P Q Σ 4 M M P M N Q -η M P η N Q + 1 6 √ 2f M N P ξ QR ΣM M N P QR . ( 73
)
where

M M N P QR = ǫ abcde V a M V b N V c P V d Q V e R . (74)
a = 1, . . . , 5 are SO(5) indices and V is the coset element (44). To work with the SO(5) indices it is best to transform to the "gsg" basis for the coset. Note that the three separate terms in this expression are distinguished by the power of Σ = e 2u 3 and each such term is easily identified in [START_REF] Lu | M-theory/heterotic duality: A Kaluza-Klein perspective[END_REF]. After some calculation, we find agreement between the two expressions.

4 Further Consistent Truncations

The PT Ansatz

As mentioned in the introduction, one central motivation for the current work is to understand in detail the Kaluza-Klein reduction employed in [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF]. Since we have the most general SU(2) × SU( 2) invariant reduction, it is guaranteed that the reduction of [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] lies within ours but the former has no vector fields and so clearly cannot be supersymmetric. In the scalar sector it is obtained from our truncation by setting the following fields to zero

(a, Im M 0 , Re N 0 , c, c, k, Im v) → 0 (75)
leaving nine scalars. Although it is conceivable that this scalar sector alone could be supersymmetrized in some other way, we in fact find that the supersymmetrization of this ansatz requires further scalar fields to be included. We now demonstrate that indeed the PT truncation is consistent. Partial results in this direction were presented in the nice work [START_REF] Berg | Bulk Dynamics in Confining Gauge Theories[END_REF]. The most striking feature of the PT ansatz is that there is a distinct asymmetry between the RR and NS three-forms. One can understand this as a direct consequence of setting the axion to zero and satisfying the equation of motion for the axion (90). Following this logic explicitly in our new truncation, requires as a first step, certain choices in setting the source for the axion to zero:

-d( * e 2φ da) = e 4(u 1 -u 3 ) H 3 ∧ * G 3 + e -4(u 1 -u 3 ) H 2 ∧ * G 2 +e -4(u 1 +u 2 ) H ′ 11 ∧ * G ′ 11 + e -4(u 1 -u 2 ) H 12 ∧ * G 12 +4e -4u 1 M ′ 1 ∧ * N ′ 1 + c.c. + 4e -4u 1 +8u 3 M 0 N ′ 0 + c.c. -4e -4u 1 -4u 2 +8u 3 Im(M 0 v) P (1 -|v| 2 ) -4Im(N ′ 0 v) vol 5 . ( 76
)
We achieve this by setting

(Im M 0 , Re N 0 , Im v, c, c) → 0 (77) along with (C 2 , C 1 ) → 0. ( 78
)
Further inspection of the equations of motion which arise from the ten-dimensional three form equations of motion ( 91) and (92) reveal that in addition we must have

(k, B 2 , B 1 , k 11 , k 12 , L 2 , L 2 ) → 0. ( 79
)
From our formalism these steps are quite straightforward but one equation requires additional work. Since we have set B 1 → 0 we must set the various source terms in (120) to zero and this would appear to give a differential constraint amongst several of the remaining scalars (b, b, Re v, Re M 0 ):

0 = H 11 (1 -|v| 2 )e -4u 2 + H 12 e 4u 2 -|v| 2 (1 -|v| 2 )e -4u 2 + 2|v| 2 +4 Im (vM 1 ) 1 -(1 -|v| 2 )e -4u 2 . ( 80
)
However if we take the exterior derivative of the Hodge star of (80) we in fact recover a linear combination of ( 121),( 122),(123). This is the only non-trivial step in showing that the ansatz employed in [START_REF] Papadopoulos | Complex geometry of conifolds and 5-brane wrapped on 2-sphere[END_REF] is indeed a consistent truncation.

Since it has appeared quite straightforward within our N = 4 truncation to show the consistency of the PT ansatz, it is natural to wonder if there is an internal symmetry at work and we now follow a brief digression with regards to this idea. Such a symmetry must differentiate between the NS and RR three-forms and as such the best candidate is worldsheet parity reversal under which:

Ω p : (g, φ, B (2) , C (0) , C (2) , C (4) ) → (g, φ, -B (2) , -C (0) , C (2) , -C (4) ).
A geometric symmetry is also needed and the best candidate appears to be reversal of all internal co-ordinates:

σ : (σ 1 , Σ 1 , g 5 ) → -(σ 1 , Σ 1 , g 5 ), (σ 2 , Σ 2 ) → (σ 2 , Σ 2 ), which translates to σ : (g 5 , J 1 , J 2 , Im Ω) → -(g 5 , J 1 , J 2 , Im Ω), Re Ω → Re Ω.
At this point we see that vol 5 and vol T 1,1 transform with opposite sign:

Ω p • σ : (vol 5 , vol T 1,1 ) → (-vol 5 , vol T 1,1 ) (81)
so we are forced to append five-dimensional parity P 5 and finally

J = Ω p • σ • P 5 (82)
appears to be our best candidate for a symmetry principle which restricts one to the PT ansatz. However J does not commute or anti-commute with the exterior derivative, to be more precise it anti-commutes with the external exterior derivative but commutes with the internal one. This means that even if terms in the potential have equal J -charge, the field strength will not. As a result we conclude that there is no symmetry principle which restricts one to the PT ansatz, this is of course not a disaster at all because in the current work we have provided an explicitly supersymmetric embedding of the PT ansatz into a consistent truncation which is based on a symmetry principle.

The I Truncation

Another interesting truncation is the restriction to modes which are invariant under the so-called I symmetry:

I = Ω p • (-1) F L • σ, σ : (θ 1 , φ 1 , θ 2 , φ 2 ) → (θ 2 , φ 2 , θ 1 , φ 1 ). ( 83
)
This is the Z 2 symmetry which is broken away from the origin of the baryonic branch in the Klebanov-Strassler gauge theory [START_REF] Gubser | Symmetry breaking and axionic strings in the warped deformed conifold[END_REF]. This truncation of our N = 4 reduction eliminates (b, c) from the three form fluxes and the metric mode which is the coefficient of

E 1 E 1 -E 2 E 2 .
In addition only two vector fields remain, namely A 1 and the combination k 11 + k 12 . It seems plausible that this is the bosonic content of an N = 2 gauged supergravity in five dimensions with one vector multiplet and three hypermultiplets, it would be interesting to develop this further.

Conclusions

In this work we have constructed a five-dimensional gauged supergravity theory by explicit dimensional reduction on T 1,1 , including the entire set of modes which are singlets under the global SU(2) × SU(2) . The motivation for the current work is to better understand the physics of the linearized spectrum, both supersymmetric and non-supersymmetric, around the warped deformed conifold and the baryonic branch. Since the PT ansatz can be embedded within our theory, it is clear that the resolved, deformed and singular conifolds can all be found as solutions and there is also some possibility that new solutions may exist within our extended ansatz. While the five-dimensional, N = 4 theory we have constructed is uniquely specified by the number of vector multiplets and the embedding tensor, we also have explicit uplift formulas to ten-dimensional IIB supergravity. This is something which is often quite difficult to obtain, for example a closely related system is the SU(2) × U(1) gauged supergravity constructed in [START_REF] Corrado | Orbifolds and flows from gauged supergravity[END_REF] which contains in its solution space the Klebanov-Witten flow [START_REF] Klebanov | Superconformal field theory on threebranes at a calabi-yau singularity[END_REF], but explicit uplift formula are not available and one is forced to work directly in ten dimensions [START_REF] Halmagyi | The Complex geometry of holographic flows of quiver gauge theories[END_REF]. In fact it would be interesting to see if the Heisenberg algebra found here arises as a contraction of the SU(2) gauging in [START_REF] Corrado | Orbifolds and flows from gauged supergravity[END_REF].

There are several direct generalizations of the current work which could provide new insight into the physics of lower dimensional gauged supergravities and string theory. There exists a family of Einstein manifolds related to T 1,1 called T p,q , all of which can be viewed as U(1) fibrations over S 2 × S 2 however these do not admit a covariantly constant Killing spinor and thus appear to not preserve any supersymmetry. It would be interesting to determine whether reduction on these manifolds results in a non-supersymmetric theory or a supersymmetric theory with no (canonical) supersymmetric vacuum. Another interesting direction is to consider reductions of IIA on T 1,1 and compare the resulting embedding tensor of the gauged supergravity to the one found here. By T -duality, the spectrum of the ungauged theory is identical to that here, the only difference must lie in the embedding tensor. This will presumably shed some light on the work [START_REF] Halmagyi | Missing Mirrors: Type IIA Supergravity on the Resolved Conifold[END_REF] where evidence was presented that KS cannot have a mirror which is even locally geometric.

Finding a superpotential for this N = 4 supersymmetric theory we have presented is a small step further than we have completed in the current work but would be extremely useful and should have direct application to the physics of flux backgrounds. We will return to these issues in a forthcoming publication.

A G-Structures

Here we collate the branching rules required to compute the spectrum of singlets under the U(1) structure group. This provides an alternative way to understand the spectrum in our IIB supergravity ansatz:

field SO(5) → SO(4) → SU (2) D U (1) → U (1) neutral field g mn 15 → 1 0 + 4 + 1 0 + 9 → 1 0 + 2 × 2 + 1 0 + 3 × 3 v, v, u 1 , u 2 , u 3 → 1 0 + 2(1 1 + 1 -1 ) + 1 0 + 3(1 0 + 1 1 + 1 -1 ) B mn 10 → 4 + 6 → (2 + 2) + (1 0 + 1 0 + 1 0 + 3) b, b, M 0 , M 0 → 2(1 1 + 1 -1 ) + (1 0 + 1 0 + 1 0 + (1 0 + 1 1 + 1 -1 )) C mn 10 → 4 + 6 → (2 + 2) + (1 0 + 1 0 + 1 0 + 3) c, c, N 0 , N0 → 2(1 1 + 1 -1 ) + (1 0 + 1 0 + 1 0 + (1 0 + 1 1 + 1 -1 )) C mnpq 5 → 1 0 + 4 → 1 0 + (2 + 2) k → 1 0 + 2(1 1 + 1 -1 ) φ 1 0 → 1 0 → 1 0 → 1 0 φ a 1 0 → 1 0 → 1 0 → 1 0 a
Table 1: Decomposition of scalar fields under the structure group.

field

SO(5) → SO(4) → SU (2) D U (1) → U (1) neutral field g µm 5 → 1 0 + 4 → 1 0 + (2 + 2) → 1 0 + 2(1 1 + 1 -1 ) A 1 B µm 5 → 1 0 + 4 → 1 0 + (2 + 2) → 1 0 + 2(1 1 + 1 -1 ) B 1 C µm 5 → 1 0 + 4 → 1 0 + (2 + 2) → 1 0 + 2(1 1 + 1 -1 ) C 1 C µmnp 10 → 4 + 6 → (2 + 2) + (1 0 + 1 0 + 1 0 + 3) k 11 , k 12 , D 1 , D 1 → 2(1 1 + 1 -1 ) + (1 0 + 1 0 + 1 0 + (1 0 + 1 1 + 1 -1 )) B µν 1 0 → 1 0 → 1 0 → 1 0 B 2 C µν 1 0 → 1 0 → 1 0 → 1 0 C 2

B The One-Forms

In our ansatz in section 2.1.1 we have used the following standard invariant one-forms on T 1,1 (see [START_REF] Minasian | On the geometry of non-trivially embedded branes[END_REF][START_REF] Herzog | Remarks on the warped deformed conifold[END_REF]):

σ 1 = c ψ/2 dθ 1 + s ψ/2 s θ 1 dφ 1 σ 2 = s ψ/2 dθ 1 -c ψ/2 s θ 1 dφ 1 σ 3 = 1 2 dψ + c θ 1 dφ 1 (84) Σ 1 = c ψ/2 dθ 2 + s ψ/2 s θ 2 dφ 2 Σ 2 = s ψ/2 dθ 2 -c ψ/2 s θ 2 dφ 2 Σ 3 = 1 2 dψ + c θ 2 dφ 2 ,
where c ψ/2 = cos ψ 2 etc.

C Hodge Dualizing

In signature (d -1, 1) the Hodge star squares to

* d * d ω r = (-1) 1+r(d-r) ω r (85)
where ω r is an r-form. So in d = 5

* 5 * 5 ω r = -ω r (86)
and in d = 10 * 10 * 10 ω r = (-1) 1+r ω r .

Here we set up some conventions for embedding the five-dimensional Hodge star in ten dimensions:

D IIB Conventions

The d = 10, IIB action in Einstein frame is 4 [43]

S E IIB = 1 2 d 10 x √ GR - 1 2 dφ ∧ * dφ - 1 2 e 2φ dC (0) ∧ * dC (0) - e -φ 2 H (3) ∧ * H (3) - e φ 2 F (3) ∧ * F (3) - 1 4 F (5) ∧ * F (5) - 1 2 C (4) ∧ H (3) ∧ F (3) . (89)
The full set of equations of motion are

d( * e 2φ F (1) ) = -e φ H (3) ∧ * F (3) (90) d( * e φ F (3) ) = F (5) ∧ H (3) (91) d( * e -φ H (3) ) = e φ F (1) ∧ * F (3) + F (3) ∧ F (5) (92)
d * dφ = e 2φ F (1) ∧ * F (1) - 1 2 e -φ H (3) ∧ * H (3) + 1 2 F (3) ∧ * F (3) (93)
F (5) = * F (5) (94) dF (5) = H (3) ∧ F (3) (95) R M N = 1 2 ∂ M φ∂ N φ + 1 2 e 2φ ∂ M C (0) ∂ N C (0) + 1 96 F M P QRS F P QRS N + e -φ 4 H P Q M H N P Q - 1 12 g M N H P QR H P QR + e φ 4 F P Q M F N P Q - 1 12 g M N F P QR F P QR (96)

E The Five-Form

Due to the self-duality in ten dimensions the most involved part of reconstructing the fivedimensional action is the equations of motion for components of the five-form. Here we summarize various steps we have taken. To facilitate computing the exterior derivative, we first write the ansatz [START_REF] Gauntlett | Universal Kaluza-Klein reductions of type IIB to N=4 supergravity in five dimensions[END_REF] in terms of untwisted fundamental forms

F 5 = e Z e 8(u 3 -u 1 ) vol 5 + e Z J 1 ∧ J 2 ∧ (g 5 + A 1 ) +K 1 ∧ J 1 ∧ J 2 -e -8u 1 ( * K 1 ) ∧ (g 5 + A 1 ) +K 21 ∧ J 1 ∧ (g 5 + A 1 ) + e -4u 2 +4u 3 ( * K 21 ) ∧ J 2 +K 22 ∧ J 2 ∧ (g 5 + A 1 ) + e 4u 2 +4u 3 ( * K 22 ) ∧ J 1 + L 2 ∧ Ω + c.c ∧ (g 5 + A 1 ) + e 4u 3 ( * L 2 ) ∧ Ω + c.c (97)
where the unprimed forms are given in terms of the primed ones by

K 1 = K ′ 1 (98) K 21 = K ′ 21 -|v| 2 K ′ 22 + 4 Im vL ′ 2 (99) K 22 = K ′ 22 (100) L 2 = L ′ 2 - iv 2 K ′ 22 . (101)
The Hodge star creats a bit of a mess as well so we have defined some new fields

K 1 = K ′ 1 = K 1 (102) K 21 = K ′ 21 = K 21 -|v| 2 K 22 -4Im (vL 2 ) (103)
K 22 = K ′ 22 -|v| 2 e -8u 2 K ′ 21 + 4e -4u 2 Im (vL ′ 2 ) = 1 + |v| 2 e -4u 2 2 K 22 -|v| 2 e -8u 2 K 21 + 4e -4u 2 1 + |v| 2 e -4u 2 Im (vL 2 ) (104) L 2 = L ′ 2 - iv 2 e -4u 2 K ′ 21 = L 2 1 + |v| 2 e -4u 2 + iv 2 1 + |v| 2 e -4u 2 K 22 - iv 2 e -4u 2 K 21 -v 2 e -4u 2 L 2 (105)
and then the five-dimensional equations of motion are

DL 2 -3ie 4u 3 * L 2 = -M 0 G 3 + N 0 H 3 -H 2 ∧ N 1 + M 1 ∧ G 2 K 21 ∧ F 2 + d e 4(u 2 +u 3 ) * K 22 -2e -8u 1 * K 1 = H 11 ∧ G 3 + H 3 ∧ G 11 (106)
K 22 ∧ F 2 + d e 4(-u 2 +u 3 ) * K 21 -2e -8u 1 * K 1 = H 12 ∧ G 3 + H 3 ∧ G 12 (107)
L 2 ∧ F 2 + D(e 4u 3 * L 2 ) = M 1 ∧ G 3 + H 3 ∧ N 1 (108) -d e -8u 1 * K 1 = H 3 ∧ G 2 -H 2 ∧ G 3 (109)
where the two-form L 2 is charged

DL 2 = dL 2 -3iA 1 ∧ L 2 . (110)
The five-dimensional kinetic terms which we get from these equations are rather off-diagonal

L F (5) ,kin = -4e 4u 3 1 + |v| 2 e -4u 2 L 2 ∧ * L 2 + 4e -4u 2 +4u 3 v 2 L 2 ∧ * L 2 + c.c - 1 2 e 4u 2 +4u 3 1 + |v| 2 e -4u 2 2 K 22 ∧ * K 22 - 1 2 e -4u 2 +4u 3 K 21 ∧ * K 21 +|v| 2 e -4u 2 +4u 3 K 22 ∧ * K 21 + 2e 4u 3 1 + |v| 2 e -4u 2 ivK 22 ∧ * L 2 + c.c -2e -4u 2 +4u 3 ivK 21 ∧ * L 2 + c.c . (111)
The Bianchi identities are

de Z = P (H 12 -H 11 ) + 4 M 1 N 0 -M 0 N 1 + c.c (112) e Z F 2 + dK 1 + 2K 21 + 2K 22 = H 11 ∧ G 12 + H 12 ∧ G 11 + 4 M 1 N 1 + c.c. ( 113
) dK 21 = P H 3 + H 11 ∧ G 2 -H 2 ∧ G 11 (114) dK 22 = -P H 3 + H 12 ∧ G 2 -H 2 ∧ G 12 (115)
which we have solved in ( 23)-( 26).

F The Three-Forms

The three-forms are given in terms of twisted fundamental forms by

H (3) = H 3 + H 2 ∧ (g 5 + A 1 ) + H ′ 11 ∧ J ′ 1 + H 12 ∧ J ′ 2 + M ′ 1 ∧ Ω ′ + M 0 Ω ′ ∧ (g 5 + A 1 ) + c.c -4 Im (M 0 v)J ′ 1 ∧ (g 5 + A 1 ) (116) F (3) = P (J ′ 1 -J ′ 2 ) ∧ (g 5 + A 1 ) + G 3 + G 2 ∧ (g 5 + A 1 ) + G ′ 11 ∧ J ′ 1 + G 12 ∧ J ′ 2 + N ′ 1 ∧ Ω ′ + N ′ 0 Ω ′ ∧ (g 5 + A 1 ) + c.c -P |v| 2 + 4 Im (N ′ 0 v) J ′ 1 ∧ (g 5 + A 1 ), (117)
where

H ′ 11 = H 11 -|v| 2 H 12 -4 Im (vM 1 ) M ′ 1 = M 1 + i 2 vH 12 G ′ 11 = G 11 -|v| 2 G 12 -4 Im (vN 1 ) (118)
N ′ 1 = N 1 + i 2 vG 12 N ′ 0 = N 0 - i 2 P v.
The equations of motion are

d e 4(u 1 -u 3 )-φ * H 3 = -e Z G 3 + G 2 ∧ K 1 -G 12 ∧ K ′ 21 -G ′ 11 ∧ K ′ 22 -4 N ′ 1 ∧ L ′ 2 + c.c + 4e 4u 3 N ′ 0 * L ′ 2 + c.c. +P 1 -|v| 2 -4Im (N ′ 0
v) e -4(u 2 -u 3 ) * K ′

21

-P e 4u 2 +4u 3 ) * K ′ 22 + e 4(u 1 -u 3 )+φ da ∧ ( * 5 G 3 ) (119) d e 4(u 1 +u 3 )-φ * H 2 = 2(1 -|v| 2 )e -4(u 1 +u 2 )-φ * H ′ 11 + 2e -4(u 1 -u 2 )-φ * H 12 +e 4(u 1 -u 3 )-φ * H 3 ∧ F 2 + 8e -4u 1 -φ Im (v * M ′ 1 ) +G 3 ∧ K 1 + e -4(u 2 -u 3 ) G ′ 11 ∧ * K ′ 21 + e 4(u 2 +u 3 ) G 12 ∧ * K ′

G Dilaton-Axion

For the dilaton axion equations of motion is necessary to construct the follows quantities:

F (1) ∧ * 10 F (1) = J ′ 1 ∧ J ′ 2 ∧ (g 5 + A 1 ) da ∧ * da (129)
H (3) ∧ * 10 H (3) = J ′ 1 ∧ J ′ 2 ∧ g 5 ∧ e 4(u 1 -u 3 ) H 3 ∧ * H 3 + e 4(u 1 +u 3 ) H 2 ∧ * H 2 +e -4(u 1 +u 2 ) H ′ 11 ∧ * H ′ 11 + e -4(u 1 -u 2 ) H 12 ∧ * H 12 +8e -4u 1 M ′ 1 ∧ * M ′ 1 + 8e -4u 1 +8u 3 |M 0 | 2 vol 5 +16e -4u 1 -4u 2 +8u 3 [Im (M 0 v)] 2 vol 5 (130)

F (3) ∧ * 10 F (3) = J ′ 1 ∧ J ′ 2 ∧ (g 5 + A 1 ) ∧ e 4(u 1 -u 3 ) G 3 ∧ * G 3 + e 4(u 1 +u 3 ) G 2 ∧ * G 2 +e -4u 1 -4u 2 G ′ 11 ∧ * G ′ 11 + e -4u

22 +4e 4u 3 N ′ 1 ∧ * L ′ 2 + 11 = 21 -G 3 1 =′ 2 +H ′ 11 +e 4u 2 11 = K ′ 22 ∧ 1 = 1 -

 22121121312112112211 c.c. + e 4(u 1 +u 3 )+φ da ∧ * G 2 (120) d e -4(u 1 -u 2 )-φ * H 12 = -4e -4u 1 -φ Im ( * M ′ 1 ∧ Dv) + 12e -4u 1 +8u 3 -φ Re (M 0 v)vol 5+ P (1 -|v| 2 ) -4Im (N ′ 0 v) e Z e 8(u 3 -u 1 ) vol 5 + e -8u 1 G ′ 11 ∧ * K 1 +e 4(u 2 +u 3 ) G 2 ∧ * K ′ 22 -G 3 ∧ K ′ 21 + e -4(u 1 -u 2 )+φ da ∧ * G 12 (121) d e -4(u 1 +u 2 )-φ * H ′ -P e Z e 8(u 3 -u 1 ) vol 5 + e -8u 1 G 12 ∧ * K 1 + e -4(u 2 -u 3 ) G 2 ∧ * K ′ ∧ K ′ 22 + e -4(u 1 +u 2 )+φ da ∧ * G ′ 11 (122) D e -4u 1 -φ * M ′ -3ie -4u 1 +8u 3 -φ M 0 vol 5 +6e -4u 1 -4u 2 +8u 3 -φ vIm (vM 0 )vol 5 -1 2i e -4(u 1 +u 2 )-φ * H ′ 11 ∧ Dv e Z e 8(u 3 -u 1 ) N ′ 0 vol 5 + e -8u 1 N ′ 1 ∧ * K 1 -G 3 ∧ L ′ 2 + e 4u 3 G 2 ∧ * L ′ 2 + e -4u 1 +φ da ∧ * N ′ 1 (123)d e 4(u 1 -u 3 )+φ * G 3 = e Z H 3 -K 1 ∧ H 2 + K ′ 22 ∧ H ′ 11 + K ′ 21 ∧ H 12 +4e -4u 2 +4u 3 Im (vM 0 ) * K ′ 21 + 4 L ′ 2 ∧ M ′ 1 + c.c. -4e 4u 3 * L ′ 2 M 0 + c.c.(124)d e 4(u 1 +u 3 )+φ * G 2 = e 4(u 1 -u 3 )+φ * G 3 ∧ F 2 + 2(1 -|v| 2 ) e -4(u 1 +u 2 )+φ * G ′ 11 +2e -4(u 1 -u 2 )+φ * G 12 -8e -4u 1 Im (v * N ′ 1 ) + K 1 ∧ H 3 -4e 4u 3 M ′ 1 ∧ * L c.c. + e -4u 2 +4u 3 * K ′ 21 ∧ +4u 3 * K ′ 22 ∧ H 12 (125) d e -4(u 1 -u 2 )+φ * G 12 = -4e -4u 1 +φ Im ( * N ′ 1 ∧ Dv) + 12e -4u 1 +8u 3 +φ Re (N ′ 0 v)vol 5 K ′ 21 ∧ H 3e 4(u 2 +u 3 ) * K ′ 22 ∧ H ′ 2e -8u 1 * K 1 ∧ H ′ 11 +4Im (M 0 v)e Z e 8(u 3 -u 1 ) (126) d e -4(u 1 +u 2 )+φ * G ′ H 3e -4u 2 +4u 3 * K ′ 21 ∧ H 2e -8u 1 * K 1 ∧ H 12 (127) D e -4u 1 +φ * N ′ -3ie -4u 1 +8u 3 +φ N ′ 0 vol 5 -1 2i e -4u 1 -4u 2 ( * G ′ 11 ) ∧ Dv + 3v2e -4u 1 -4u 2 +8u 3 +φ P |v| 2 + 4Im (N ′ 0 v) vol 5 +L ′ 2 ∧ H 3e 4u 3 * L ′ 2 ∧ H 2e -8u 1 * K 1 ∧ M ′ M 0 e Z e -8(u 1 -u 3 ) vol 5 (128)

′ 1 +

 1 c.c. + 4e -4u 1 +8u 3 M 0 N ′ 0 + c.c. -4e -4u 1 -4u 2 +8u 3 Im(M 0 v) P (1 -|v| 2 ) -4Im(N ′ 0 v) vol 5(131)then (90) and (93) give the relevant equations of motion.

Table 2 :

 2 Decomposition of form fields under the structure group.

  1 +4u 2 G 12 ∧ * G 12 + 8e -4u 1 N ′ 1 ∧ * N ′ 1 +e -4u 1 +8u 3 8|N ′ 0 | 2 + e 4u 2 P 2 + e -4u 2 P (|v| 2 -1) + 4 Im (N ′ 0 v) 2 vol 5 H (3) ∧ * 10 F (3) = J ′ 1 ∧ J ′ 2 ∧ (g 5 + A 1 ) ∧ e 4(u 1 -u 3 ) H 3 ∧ * G 3 + e 4(u 1 +u 3 ) H 2 ∧ * G 2 +e -4(u 1 +u 2 ) H ′ 11 ∧ * G ′ 11 + e -4(u 1 -u 2 ) H 12 ∧ * G 12 +4e -4u 1 M ′ 1 ∧ * N

See the introduction of[START_REF] Bobev | Supergravity Instabilities of Non-Supersymmetric Quantum Critical Points[END_REF] for a review of the argument why a truncation to a sector invariant under a symmetry is consistent and a discussion of its limitations

In a forthcoming publication we will analyze more formal geometric and non-geometric properties of the general embedding tensor

The half maximal gauged supergravity theories in four and five dimensions are quite rigid and are summarized nicely in[START_REF] Dall'agata | General matter coupled N = 4 gauged supergravity in five dimensions[END_REF][START_REF] Schon | Gauged N = 4 supergravities[END_REF]. Another very useful source of information is the nice lecture notes[START_REF] Samtleben | Lectures on Gauged Supergravity and Flux Compactifications[END_REF]

= (-1) p e (-10+2p)u 1 +(2-2p)u

( * ω p ) ∧ (g 5 + A 1 )where * 10 is the Hodge star in d = 10 and * is the Hodge star in d = 5.

our conventions are taken from[START_REF] Polchinski | The string dual of a confining four-dimensional gauge theory[END_REF] page 12.

Acknowledgements

We would like to thank Davide Cassani, Anatoly Dymarsky, Jerome Gauntlett, Jim Liu, Boris Pioline, Dan Waldram and Nicholas Warner for useful conversations. This work was supported in part by the DSM CEA-Saclay, by the ANR grant 08-JCJC-0001-0, and by the ERC Starting Independent Researcher Grant 240210 -String-QCD-BH.


domain: chim sdu

text:
C Petuya

A Patt

J M Simon
email: jmsimon@u-bourgogne.fr

Sylvain Picaud

J M Salazar

A Desmedt
email: arnaud.desmedt@u-bordeaux.fr

Molecular Selectivity of CO-N 2 Mixed Hydrates: Raman Spectroscopy and GCMC Studies

This paper reports a novel quantitative investigation concerning the COselectivity properties for mixed CO-N2 hydrates. The study was developed by combining Raman Scattering experiments and Grand Canonical Monte Carlo simulations. Both approaches advocate in favor to say that mixed CO-N 2 hydrates are CO-selective and quantitatively compare with regard to the measured and calculated molecular selectivities.

Intriguingly, this preferential CO encapsulation is enhanced by decreasing the CO molar fraction of the gas mixture. Here, an original explanation to this peculiar behavior is provided in terms of a structural change of the clathrates from SI to SII and in terms of the thermodynamics of mixtures. This last issue was clearly evidenced by using the Ideal Adsorption Solution Theory, which gives results that compare favorably with those obtained from Raman Scattering and Grand Canonical Monte Carlo simulations.

I. Introduction

Gas clathrate hydrates -commonly known as gas hydrates -are crystalline compounds formed by H-bonded water molecules organized on a network of small and large cages, inside of which gaseous molecules are encapsulated. [START_REF] Sloan | Clathrate Hydrates of Natural Gases[END_REF] Nowadays, these materials have been a research subject of various scientific communities: physical chemistry, astrophysics, geosciences and environmental sciences. [START_REF]Gas Hydrates 1: Fundamentals, Characterization and Modeling[END_REF][START_REF]Gas Hydrates 2: Geoscience Issues and Potential Industrial Applications[END_REF] Clathrate hydrates are particularly relevant in many chemical engineering applications, such as flow assurance and gas separation technology. [START_REF] Sloan | Clathrate Hydrates of Natural Gases[END_REF][START_REF] Koh | Natural Gas Hydrates: Recent Advances and Challenges in Energy and Environmental Applications[END_REF] The methane hydrate naturally occurs on Earth, mainly on the ocean floor and in the frozen lands near the poles. [START_REF] Sloan | Clathrate Hydrates of Natural Gases[END_REF][START_REF]Gas Hydrates 2: Geoscience Issues and Potential Industrial Applications[END_REF] Natural gas hydrates are also supposed to be present in astrophysical environments like comets, planets and/or in their satellites. [START_REF] Mousis | Volatile Inventories in Clathrate Hydrates Formed in the Primordial Nebula[END_REF][START_REF] Mousis | Methane Clathrates in the Solar System[END_REF] For instance, the lack of noble gases in the atmosphere of Titan [START_REF] Mousis | Removal of Titan's Atmospheric Noble Gases by their Sequestration in Surface Clathrates[END_REF][START_REF] Choukroun | Is Titan's Shape Caused by its Meteorology and Carbon Cycle?[END_REF][START_REF] Tobie | Episodic Outgassing as the Origin of Atmospheric methane on Titan[END_REF][START_REF] Thomas | Clathrate Hydrates as a Sink of Noble Gases in Titan's Atmosphere[END_REF] , the composition of the interior lakes of Enceladus [START_REF] Bouquet | Possible Evidence for a Methane Source in Enceladus' Ocean[END_REF] or the N 2 depletion measured in the comet 67 P/Churyumov-Gerasimenko 12 may be interpreted with the help of gas hydrate formation. In view of these applications, one of the major properties of interest concerning gas hydrates is their ability to selectively encapsulate molecular species. Such selective trapping has been studied for mixed gas hydrates such as: CO 2 -CH 4 13,14 , CO 2 -N 2 [START_REF] Petuya | Selective Trapping of CO 2 Gas and Cage Occupancy in CO 2 -N 2 and CO 2 -CO Mixed Gas Hydrates[END_REF][START_REF] Chazallon | Selectivity and CO 2 Capture Efficiency in CO 2 -N 2 Clathrate Hydrates Investigated by in-situ Raman Spectroscopy[END_REF] , CO-CO 2 [START_REF] Petuya | Selective Trapping of CO 2 Gas and Cage Occupancy in CO 2 -N 2 and CO 2 -CO Mixed Gas Hydrates[END_REF] , or CO-N 2 [START_REF] Lectez | A ∼32-70 K Formation Temperature Range for the Ice Grains Agglomerated by Comet 67P/Churyumov-Gerasimenko[END_REF][START_REF] Petuya | Revealing CO-Preferential Encapsulation in the Mixed CO-N 2 Clathrate Hydrate[END_REF][START_REF] Patt | A Grand Canonical Monte Carlo Study of the N 2 , CO, and Mixed N 2 -CO Clathrate Hydrates[END_REF] . Of particular interest are the carbon monoxide and di-nitrogen species since they are abundant in astrophysical contexts. [START_REF] Lodders | Solar System Abundances and Condensation Temperatures of the Elements[END_REF][START_REF] Lunine | Thermodynamics of Clathrate Hydrates at Low and High Pressures with Application to the Outer Solar System[END_REF][START_REF] Mousis | Clathration of Volatiles in the Solar Nebula and Implications for the Origin of Titan's Atmosphere[END_REF] Moreover, in the industrial context, these molecules are important for gas separation applications. [START_REF]Gas Hydrates 2: Geoscience Issues and Potential Industrial Applications[END_REF] Even though CO and N 2 share very similar chemical properties (steric hindrance, isoelectronic, mass number), the mixed CO-N 2 gas hydrates are clearly COselective. This selective encapsulation has been reported by Raman scattering experiments [START_REF] Petuya | Revealing CO-Preferential Encapsulation in the Mixed CO-N 2 Clathrate Hydrate[END_REF] and by molecular simulations [START_REF] Lectez | A ∼32-70 K Formation Temperature Range for the Ice Grains Agglomerated by Comet 67P/Churyumov-Gerasimenko[END_REF][START_REF] Patt | A Grand Canonical Monte Carlo Study of the N 2 , CO, and Mixed N 2 -CO Clathrate Hydrates[END_REF] in different temperature-pressure domains. Worthwhile to mention is, that pure clathrates of CO and N 2 are first formed in the sI-structure and they transform, in time, to the thermodynamically stable phase sII. [START_REF] Zhu | Encapsulation Kinetics and Dynamics of Carbon Monoxide in Clathrate Hydrate[END_REF][START_REF] Petuya | Guest Partitioning in Carbon Monoxide Hydrates by Raman Spectroscopy[END_REF][START_REF] Petuya | Guest Partitioning and Metastability of the Nitrogen Gas Hydrate[END_REF][START_REF] Petuya | Ageing and Langmuir Behavior of the Cage Occupancy in the Nitrogen Gas Hydrate[END_REF] In the case of the mixed CO-N 2 hydrates [START_REF]The two clathrate structures existing for the CO-N 2 hydrate differ by the type of cage forming the sI and sII structures. Three types of cages are met: the small 5 12 cages are composed of 12 pentagons of H-bonded water molecules, and the large 5 12 6 2 and 5 12 6 4 cages are made of 12 pentagons and 2 and 4 hexagons, respectively. The structure sI is a cubic unit cell constituted of two 5 12 and six 5 12 6 2 cages and the structure sII of sixteen 5 12 and height 5 12 6 4 cages[END_REF] , this structural change promotes the CO molecular selectivity [START_REF] Petuya | Revealing CO-Preferential Encapsulation in the Mixed CO-N 2 Clathrate Hydrate[END_REF] . In this paper, this property is documented for the first time by combining Grand Canonical Monte Carlo (GCMC) simulations and Raman Scattering experiments. Especially, the influence of the chemical composition of the gas mixture on the molecular selectivity is investigated.

Additionally, an original thermodynamic interpretation based on the Ideal Adsorption Solution Theory (IAST) is provided to help explaining the tendency of CO to be favorably encapsulated.

II. Methods

Sample preparation. The CO-N 2 clathrate hydrate was formed by using ultra-pure water (Milli-Q quality) and CO-N 2 gas mixture (gas purity of CO and N 2 gas >99.997%) done with a homemade set up composed of two syringe pumps including the PM High Pressure pump (described in the Raman spectroscopy section) with which the two gases are mixed. Gas mixtures are consisting of 10 %, 25 %, 50 % 67 % and 75 % of CO which correspond to a

[CO]/[N 2 ] gas concentration ratio of 0.11, 0.33, 1, 2.03 and 3 respectively. First, the water was injected inside the sample cell and then cooled until ice formation. The temperature cell was maintained at 270 K (± 0.1 K) using a modified temperature controlled stage (Linkam Scientific Instruments Ltd., UK). Next, 200 bar of CO-N 2 pressure was applied on ice during (high resolution). The calibration of the spectrometer was done using the 520.7 cm -1 mode of a silicon sample and the 2348.4 cm -1 mode of a neon light. The sample temperatures were maintained at the desired value (± 0.1 K) during the acquisition by using a modified temperature controlled stage (Linkam Scientific Instruments Ltd., UK), including a homemade high-pressure optical cell equipped with a 2 mm thick sapphire optical window. The sample pressures were controlled by a PM High Pressure pump (Top Industrie, Vaux-le-Penil, France) which can contain 100 cm 3 of gas up to 500 bars.

Calculations strategy and details. Monte Carlo simulations were carried out in the Grand Canonical ensemble (𝜇, 𝑉, 𝑇), using the software GIBBS [START_REF] Ungerer | Applications of Molecular Simulation in the Oil and Gas Industry: Monte Carlo Methods[END_REF] and following the strategy already presented elsewhere. [START_REF] Patt | A Grand Canonical Monte Carlo Study of the N 2 , CO, and Mixed N 2 -CO Clathrate Hydrates[END_REF] Imposing the chemical potentials, or fugacities as is the case here, of CO and N 2 allows their population in the system to fluctuate through insertion and deletion Monte Carlo moves. In addition, the number of water molecules was kept constant. All species were subjected to translational and rotational Monte Carlo moves. This implies that the hydrate structures were considered as flexible. Moreover, the identity swap move between CO and N 2 molecules was also used to accelerate the convergence of the simulations. This Monte Carlo move is based on the substitution of a molecule of a certain type by a molecule of another type, at the same position. The frequencies of the different Monte Carlo moves have been set to 10% for translation, 10% for rotation, 40% for insertion/deletion, and 40%

for identity swap. The initial configurations of structures I and II (sI and sII, respectively)

were obtained from the work of Takeuchi et al., [START_REF] Takeuchi | Water Proton Configurations in Structure I, II, and H Clathrate Hydrates Unit Cells[END_REF] and were made of 2x2x2 elementary cells.

Thus, the sI system included 368 water molecules, forming 64 cages, and the sII system was composed of 192 cages for 1088 water molecules. In other words, the cubic simulation boxes [START_REF] Piper | The Adsorption of Carbon Monoxide on Graphite[END_REF] , and the TraPPE force field for N 2 [START_REF] Potoff | Vapor Liquid Equilibria of Mixtures Containing Alkanes, Carbon Dioxide, and Nitrogen[END_REF] (their parameters are given in Table S1). The cutoff for intermolecular interactions was set to half the length of the simulation box and long-range corrections were considered. And, the Lorentz-Berthelot mixing rules were used to derive the interaction between different Lennard-Jones sites.

III. Results and Discussion

In the present investigation, the selective property of the mixed gas hydrates is defined with the help of the molecular selectivity, 𝛽. This quantity is defined as the molar concentration ratio (or equivalently the molar fraction ratio) of the CO and N 2 in the gaseous and hydrate phase:

𝛽 = [ ] [ ] [ ] [ ] = (1)
where [A] z is the concentration of CO or N 2 in the z-phase, 𝑥 and 𝑦 are the molar fraction of A in the gaseous and hydrate phase, respectively.

𝑅 = = [ ] [ ] (2)
where I A is the Raman intensity and 𝛼 is the Raman scattering cross section of specie A.

Mixed hydrates with various CO-N 2 compositions are formed after two and a half days of pressurization at 200 bar and 270 K (these values are in the hydrate stability P-T region as shown in Figure S1). The Raman signature of the trapped molecules (𝜈 = 2135.5 cm -1 , 𝜈 = 2323.9 cm -1 ) is observed at lower frequency than that of the gas phase (𝜈 = 2141 cm -1 , 𝜈 = 2328.2 cm -1 ), as shown in Figure S2. As reported in previous investigations for the CO [START_REF] Petuya | Guest Partitioning in Carbon Monoxide Hydrates by Raman Spectroscopy[END_REF] , N 2 [START_REF] Petuya | Guest Partitioning and Metastability of the Nitrogen Gas Hydrate[END_REF] and CO/N 2 [START_REF] Petuya | Revealing CO-Preferential Encapsulation in the Mixed CO-N 2 Clathrate Hydrate[END_REF] hydrates, this frequency shift is associated to the confinement within the water cages. The frequency lowering upon confinement is related with the small sizes of the CO and N molecules with respect to the cage diameter. [START_REF] Petuya | Revealing CO-Preferential Encapsulation in the Mixed CO-N 2 Clathrate Hydrate[END_REF][START_REF] Petuya | Guest Partitioning in Carbon Monoxide Hydrates by Raman Spectroscopy[END_REF][START_REF] Petuya | Guest Partitioning and Metastability of the Nitrogen Gas Hydrate[END_REF] The molecular selectivity, 𝛽, is obtained by calculating the ratios of the Raman areas, equation ( 2), for the hydrate (R hyd ) and the gas phase (R gas ):

𝛽 = (3)
In this equation, it is assumed that Raman cross sections ratio of equation ( 2) are similar in the hydrate and gas phases, as reported for CO 2 -based mixed hydrates. [START_REF] Petuya | Selective Trapping of CO 2 Gas and Cage Occupancy in CO 2 -N 2 and CO 2 -CO Mixed Gas Hydrates[END_REF] Such an assumption could be done thanks to the small size of CO and N2 molecules compared to the cage diameter. Henceforth, the 𝛽 value allows determining the species preferentially trapped within the hydrate cages. Specifically, three cases are possible:

 𝛽 > 1, the hydrate is CO selective (i.e. the CO to N 2 concentration ratios within the hydrate phase is greater than the one within the gas phase),  𝛽 = 1, hydrate selectivity is not observed,  𝛽 < 1, the hydrate is N 2 selective.

Figure 1 shows the variation of 𝛽 measured at equilibrium gas compositions along the isotherm 150 K and along the isobar at 200 bar. Notice that 𝛽 is always greater than 1 irrespective of the pressure, the temperature and gas composition. A slight dependence of the selectivity is observed with the pressure (increase) and with the temperature (decrease), as reported in the insets of Figure 1 in the most selective case y CO = 0.1. More surprisingly, the CO-selectivity abruptly increases for low CO gas molar fraction. Previous measurements at 200 bar and 270 K [START_REF] Petuya | Revealing CO-Preferential Encapsulation in the Mixed CO-N 2 Clathrate Hydrate[END_REF] suggest that this can be associated to a structural change (from sII to sI when 𝑦 increases) for a gas composition of 𝑦 ~ 0.2.

For elucidating the unusual increase of 𝛽 at low CO gas molar fraction, we studied at the molecular level, the thermodynamic properties for mixed CO-N 2 clathrates. For this, we used the Grand Canonical Monte Carlo technique, allowing obtaining the hydrate composition (CO and N 2 molecules) by imposing the fugacity of each species. Once equilibrium is reached, the values of 𝛽 are calculated from the molar fraction expression given in equation (1). The reported values of 𝛽 are within a statistical uncertainty below 3%. Moreover, these values did not present a size dependence of the simulation box.

In agreement with experiments, previous works [START_REF] Lectez | A ∼32-70 K Formation Temperature Range for the Ice Grains Agglomerated by Comet 67P/Churyumov-Gerasimenko[END_REF][START_REF] Patt | A Grand Canonical Monte Carlo Study of the N 2 , CO, and Mixed N 2 -CO Clathrate Hydrates[END_REF] have shown that CO molecules are preferentially encapsulated. These studies were focused on low temperatures (50 -150) K and fugacities below 10 bar. Under these conditions and for gaseous molar fractions of 0.25, 0.5 and 0.75, those simulations did not exhibit any significant trend associated to the gas composition. In the present work, GCMC calculations are developed for a series of gas compositions characterized by 𝜉 = 𝑓 /(𝑓 + 𝑓 ) ranging from 0.01 to 0.95. . These calculations were performed at 𝑇 = 150 𝐾 and with a constant sum of fugacities, 𝑓 + 𝑓 , equal to 60 bar. These values (150 K and 60 bar) have been chosen according to the thermodynamic conditions used in the experiments. One should note that, for these conditions, the mixed CO-N 2 fluid is not ideal and this non-ideality depends on the composition of the fluid. This implies that the imposed fugacity of each species cannot be directly translated into partial pressure. However, for the composition corresponding to a non- It is worth noting that, in this range, the dependence of the molecular selectivity on the pressure is very small compared to its dependence on the fluid composition, as shown by the inset on the left panel of Figure 1.

For providing a thermodynamic interpretation of the observed (experimental and numerical) gas composition dependence on the molecular selectivity, we used the Ideal Adsorption Solution Theory (IAST). [START_REF] Myers | Thermodynamics of Mixed-Gas Adsorption[END_REF] This is justified by the analogy between enclathration and adsorption formulated by van der Waals and Platteeuw. [START_REF] Van Der Waals | Advances in Chemical Physics[END_REF] Indeed, IAST is a thermodynamic method for predicting mixture compositions of adsorbed phases from exclusively pure (single component) isotherms. Therefore, we also used GCMC simulations to compute the, single-guest isotherms for the pure CO and N 2 hydrates at 150 K. The results of these simulations were then used as inputs for the IAST predictions of the corresponding mixed CO-N 2 hydrate composition. Molar fraction of CO in the hydrate phase for the sI and sII structures, is depicted on the left panel of Figure 2 with respect to 𝜉 . If the gaseous phase behaves as an ideal solution, the ratio, 𝜉 , is equal to the molar fraction of CO in this phase. First, we observe that the predicted mixed hydrate compositions obtained from IAST are in excellent agreement with GCMC simulations for both structures. Thermodynamically, this implies that the CO-N 2 mixtures trapped inside hydrates can be considered as ideal mixtures. The comparison for sI is spot-highlighted, whereas a slight deviation can be noted for sII. Moreover, for any given partial fugacity, the CO molar fraction of sII is higher than for sI. This suggests that the molecular selectivity behaves similarly. Notice that 𝛽 is calculated from the gaseous molar fraction (Eq. 1) and not from fugacity ratio (excepting ideal gases). Given the thermodynamic conditions, which are beyond the critical points of CO and N 2 , the gas phase cannot be assumed to be ideal. The real composition is then determined for each imposed partial fugacity. This was done, by simulating (with GCMC) the fluid mixture CO-N 2 at conditions imposed to the hydrate. Once the mixture was equilibrated, the molar fraction of CO (𝑦 )

was calculated and compared to 𝜉 . On Figure 2 (right), we can observe that the measured molar fraction for CO is greater than its ideal counterpart, for all partial fugacities, indicating that the fluid phase is not ideal.

In Figure 3 are given the molecular selectivities by using the real composition of the gas phase. The figure shows, that the mixture is CO selective in agreement with experimental results. Moreover, the results show that 𝛽 decreases as the gas composition is enriched in CO.

Notice that, 𝛽 is always higher for sII than for sI. The predictions from IAST and GCMC simulations for 𝛽 exhibit a remarkable agreement (Figure 3). The comparison between experimental and computed molecular selectivities shows that they spread across similar ranges of values: 1.0 ≤ 𝛽 ≤ 3.5. The relevant difference lies in the way CO selectivity change with composition; Experimentally is observed a drastic decrease (jump) from 3.5 to 1.5 at low 𝑦 (while simulated data show a monotonic decrease for structures sI and sII). By considering the simulation curves in Figure 3 the experimental point at 𝛽 =3.5 is located above the sII GCMC curve and the point at 𝛽 =1.5 below the sI GCMC curve. This suggests that the experimental jump can be originated by a structural change from sI to sII when the CO content in the gas phase is diminished. This ansatz is supported by the experimental results obtained at 200 bar and 270 K. 17

IV. Concluding Remarks

The molecular selectivity of the mixed CO-N 2 hydrate has been investigated by combining Raman scattering and modeling (Grand Canonical Monte Carlo simulation and IAST). Both experimental and simulation approaches yield to determine that the carbon monoxide composition of the hydrate phase was higher than that of the gaseous counterpart by a factor up to 3.5, irrespective of the CO-N 2 gas mixture composition. Intriguingly, the CO-selectivity smoothly increases when the N 2 content increases in the CO-N 2 gas mixture. An explanation

2. 5

 5 Figure S1 line (3)(2)) and the day after the isobaric data are collected from 150 K to 270 K

were 24 .

 24 06 and 34.62 Å long in every direction, for sI and sII respectively, and periodic boundary conditions were applied. The simulations were performed during 10 7 Monte Carlo steps, with an equilibration stage of 10 6 steps. These last number of steps was proven to be sufficiently large to ensure only very small fluctuations of the number of encapsulated molecules, due to the quite small size of the simulated systems and to the use of the identity swap moves. The potential energy was calculated as the sum of the classical pairwise Coulomb and Lennard-Jones contributions of the interaction sites located on the H 2 O, CO and N 2 molecules. The Ewald summation technique was used to compute the Coulomb interactions. The repulsive-dispersive part of the interaction energy was calculated using the following molecular models: the TIP4P-Ewald model for H 2 O 29 , the potential from the work of Piper et al. for CO

Figure 1 .

 1 Figure 1. Variation of the molecular selectivity parameter β with the molar fraction of carbon

  ideal fluid (y CO =0.75), we performed a series of NPT Monte Carlo simulations for various pressures between 60 and 200 bar, and we compared the corresponding calculated density of the gas phase with the density obtained in the present GCMC calculations performed at f CO +f N2 = 60 bar. Very similar densities where calculated when the total pressure has been fixed at 190 bar in the NPT simulations. This indicates that the real total pressure in the simulations is below 190 bar irrespective of the gas phase compositions considered in our GCMC simulations. The simulation conditions thus always fall in the range of pressures explored by the experiments [60 -200] bar.

Figure 2 .

 2 Figure 2. Molar fractions of carbon monoxide in the clathrate phase, for both structures I and

Figure 3 .

 3 Figure 3. Variation of the molecular selectivity (𝛽) with the molar fraction of carbon

  ). Thus, during all the experiments, the gas mixture is at equilibrium.

	Raman spectroscopy. Raman spectra were recorded with a Labram microspectrometer
	(Horiba Jobin Yvon, Villeneuve d'Ascq, France) using a 514 nm wavelength laser as
	excitation source. A 50X objective permitted us to focus the incident laser beam and to collect
	the Raman scattering. The Raman scattering was dispersed by a holographic grating of 2400
	lines/mm (high resolution) and analyzed by a Peltier-cooled CCD detector (Andor,
	Belfast,UK). Thus, the Raman bands can be measured with a spectral resolution of 0.8 cm -1

to this increase is given by the ideal mixing rules of the IAST theory. Worth noting, is the abrupt jump in the selectivity at low CO concentration (typically, below 20%). This resultsupported by experimental and theoretical results -is associated to a structural change of the mixed hydrate from sI to sII. A novel feature of this study is the use of the IAST for predicting the molecular selectivity in the mixed CO-N 2 hydrate. This approach successfully compares with both Raman spectroscopy and GCMC results. Thanks to the combined approaches, these results provide, not exclusively, new information regarding the selectivity properties of mixed CO-N 2 hydrates, but also a coherent thermodynamic description. A predictive modeling strategy is thus available for studying gas hydrates in natural environments (atmosphere of planets and comets) and for molecular sieving.

ASSOCIATED CONTENT Supporting Information.

The following files are available free of charge.

Experimental details for sample preparation, Raman scattering and additional Raman spectra.

Calculation details and strategy. (PDF)

AUTHOR INFORMATION

The authors declare no competing financial interests.

ACKNOWLEDGMENT

This paper falls in the frame of MI2C project ANR-15CE29-0016 funded by the French ANR "Agence Nationale de la Recherche". The authors thank IFPEN for providing freely the Monte-Carlo code GIBBS.