Wrapped NS5-Branes, Consistent Truncations and In\"on\"u-Wigner Contractions

We construct consistent Kaluza-Klein truncations of type IIA supergravity on (i) $\Sigma_2\times S^3$ and (ii) $\Sigma_3\times S^3$, where $\Sigma_2 = S^2/\Gamma$, $\mathbb{R}^2/\Gamma$, or $\mathbb{H}^2/\Gamma$, and $\Sigma_3 = S^3/\Gamma$, $\mathbb{R}^3/\Gamma$, or $\mathbb{H}^3/\Gamma$, with $\Gamma$ a discrete group of symmetries, corresponding to NS5-branes wrapped on $\Sigma_2$ and $\Sigma_3$. The resulting theories are a $D=5$, $\mathcal{N}=4$ gauged supergravity coupled to three vector multiplets with scalar manifold $SO(1,1)\times SO(5,3)/(SO(5)\times SO(3))$ and gauge group $SO(2)\times\left(SO(2)\ltimes_{\Sigma_2}\mathbb{R}^4\right)$ which depends on the curvature of $\Sigma_2$, and a $D=4$, $\mathcal{N}=2$ gauged supergravity coupled to one vector multiplet and two hypermultiplets with scalar manifold $SU(1,1)/U(1)\times G_{2(2)}/SO(4)$ and gauge group $\mathbb{R}^+\times\mathbb{R}^+$ for truncations (i) and (ii) respectively. Instead of carrying out the truncations at the 10-dimensional level, we show that they can be obtained directly by performing In\"on\"u-Wigner contractions on the 5 and 4-dimensional gauged supergravity theories that come from consistent truncations of 11-dimensional supergravity associated with M5-branes wrapping $\Sigma_2$ and $\Sigma_3$. This suggests the existence of a broader class of lower-dimensional gauged supergravity theories related by group contractions that have a 10 or 11-dimensional origin.


Introduction
Supergravity theories in 10/11 dimensions are low-energy approximations of string/M-theory, and the studies of these theories provide invaluable insight into the rich structure of their high-energy counterparts. The construction of solutions of higher-dimensional supergravity theories, however, is a difficult task. A particularly powerful framework that has been developed over the years to tackle this problem is consistent Kaluza-Klein (KK) reductions.
These truncations reduce the higher-dimensional equations of motion to a set of lowerdimensional equations obtainable from a lower-dimensional supergravity theory, which are much easier to solve.
In this paper, we present two new consistent KK truncations of D = 10 type IIA supergravity on (i) Σ 2 × S 3 , where Σ 2 = S 2 , R 2 , H 2 or a quotient thereof, to a gauged N = 4 supergravity theory in D = 5, and (ii) Σ 3 × S 3 , where Σ 3 = S 3 , R 3 , H 3 or a quotient thereof, to a gauged N = 2 supergravity theory in D = 4, at the level of the bosonic fields. The S 3 factor common to both truncations corresponds to the standard S 3 truncation of type IIA supergravity to the maximal ISO(4) gauged supergravity in D = 7. Within this D = 7 theory is a "vacuum" solution that uplifts to the NS5-brane near horizon, linear dilaton solution.
The further truncations on Σ 2 and Σ 3 then correspond to the worldvolume of the NS5-brane wrapping these geometries. In particular, Σ 2 and Σ 3 are interpreted as a Slag/Kähler 2-cycle and Slag 3-cycle of a Calabi-Yau 2-and 3-fold (CY 2 , and CY 3 ) respectively. A motivation for truncation (i) stems from the supergravity solution in [1] and [2], which describes the near horizon limit of NS5-branes wrapping an S 2 , embedded inside a CY 2 . The dual D = 4 theory can be viewed as the IR limit of the little string theory compactified on S 2 with a topological twist. For truncation (ii), we similarly note the supergravity solution presented in [3], which describes the near-horizon limit of NS5-branes wrapping an S 3 , embedded inside a CY 3 . The dual D = 3 theory can be viewed as the IR limit of the little string theory compactified on S 3 with a topological twist.
The topological twist is an important feature of wrapped brane solutions. Schematically, the Killing spinor equation on the worldvolume of a brane wrapped on a cycle Σ is d + ω (1) − A (1) ǫ = 0, with ω (1) the spin connection on Σ, and A (1) the gauge field that couples to the R-symmetry current. This equation, in general, does not admit covariantly constant spinors, in which case supersymmetry is broken. An elegant solution to this, as pioneered in [4,5] and applied to constructing supergravity solutions by [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], is to set the gauge field to be equal to the spin connection on the cycle -the "twist", so that the Killing spinor equation admits covariantly constant spinors. This topological twist will be fully incorporated in both our truncations in order to preserve supersymmetry.
To carry out truncations (i) and (ii), the straightforward method would be to first reduce the type IIA theory on S 3 to obtain the maximal ISO(4) gauged supergravity in D = 7, and then further reducing on a Riemann surface Σ 2 to obtain the D = 5 theory, or on a Slag 3-cycle Σ 3 to obtain the D = 4 theory. Instead, we will show that the truncations can be carried out consistently starting from the D = 5 and D = 4 theories obtained from an M5-brane wrapping Σ 2 and Σ 3 , which we will discuss below, by performing Inönü-Wigner (IW) contractions. In terms of the 11-dimensional supergravity theory where the M5-brane lives, the IW contraction corresponds to the group contraction which takes S 4 → S 3 × R, where S 4 is the internal 4-sphere of the M5-brane. The opening of an isometry direction along R allows for the truncation of the 11-dimensional theory to the type IIA theory, as well as the interpretation of the M5-brane becoming the NS5-brane. This was shown in [23] as corresponding to a consistent transition from the D = 7 maximal SO(5) gauged supergravity theory to the D = 7 maximal ISO(4) gauged supergravity theory. Our full truncation procedure is summarised in figure 1. The key message from figure 1 is that by virtue of the consistency of the IW contraction, once the theory describing an M5-brane wrapped on a calibrated cycle is known, the theory describing an NS5-brane wrapping on the same cycle can be obtained accordingly. To be concrete, we will first describe our procedure for truncation (i). We start from the KK truncation in D = 11, first by reducing on S 4 to the maximal SO(5) gauged supergravity in D = 7 and then further reducing on the Riemann surface Σ 2 . The result is a gauged D = 5, N = 4 (16 supercharges) supergravity theory coupled to three vector multiplets with gauge group SO(2) × ISO(3), corresponding to the aforementioned wrapped M5-brane truncation on a Riemann surface described in [24] and [25]. Here, at the 5-dimensional level, we perform the IW contraction given in [23] to obtain a new D = 5, N = 4 gauged supergravity theory coupled to three vector multiplets with scalar manifold SO(1, 1)×SO(5, 3)/(SO(5)×SO (3)).
The scalar manifold of this new D = 5 theory is the same as the D = 5 theory in [24,25].
However, this should not come as a surprise since the IW contraction procedure retains the same number of degrees of freedom. Along with the stringent condition set by N = 4 supersymmetry, this ensures that the scalar manifold must remain the same. As we will show, the gauge group of the truncated D = 5 theory is SO(2) × G A 100 5,17 when Σ 2 = R 2 /Γ, and SO(2) × G A 0 5,18 when Σ 2 = S 2 /Γ or H 2 /Γ, where G A 100 5,17 and G A 0 5,18 are two, five-dimensional matrix groups whose Lie algebras are listed in [30]. The groups G A 100 5,17 and G A 0 5,18 are isomorphic to SO(2) ⋉ Σ 2 R 4 , where the action of the semi-direct product depends on the curvature of Σ 2 . As a consequence of the appearance of these unconventional gauge groups, the precise details of the gauging, such as the embedding tensors, as well as the vacuum structure of the theory, are entirely different from that of [24,25]. The method for truncation (ii) proceeds analogously. We first truncate the D = 7 maximal theory on a Slag 3-cycle Σ 3 as described in [26] to obtain a D = 4, N = 2 gauged supergravity theory coupled to a single vector multiplet and two hypermultiplets with gauge group U(1) × R + . Then, at the 4-dimensional level, we perform the same IW contraction and obtain a new D = 4, N = 2 gauged supergravity theory coupled to one vector multiplet and two hypermultiplets with scalar manifold SU(1, 1)/U(1) × G 2(2) /SO(4) and gauge group R + × R + 1 . Similar to the D = 5 case, the scalar manifold of our new D = 4 theory is the same as the D = 4 theory in [26], but the precise details of the gauging and the vacuum structure of the theories are entirely different.
The plan of the rest of the paper is as follows. In Section 2 we review the S 4 reduction of 11-dimensional supergravity to the maximal SO(5) gauged supergravity in D = 7, and how it relates to the S 3 reduction of type IIA to the maximal ISO(4) theory through the IW contraction. Following this, in Sections 3 and 4, we review the theories describing wrapped M5-branes on Riemann surfaces and Slag 3-cycles, and construct the analogous theories describing wrapped NS5-branes using the IW contractions introduced in Section 2. There, we will also consider consistent sub-truncations of the wrapped NS5-brane theories, and present new and reproduce old solutions. We conclude with a few final remarks in Section 5, and collect some useful results in the appendices. 1 For Σ 3 = S 3 , truncation (ii) corresponds to a KK truncation of type IIA on S 3 × S 3 . We note that the resulting D = 4, N = 2 theory is not related to the D = 4, N = 4 Freedman-Schwarz model [27] which can also be obtained from reducing type IIA on S 3 × S 3 [28] (for more details see [28,29]), as the precise details of the two truncation procedures are different.
2 Review of S 4 and S 3 reductions and the D = 7 maximal ISO(4) gauged theory In this section, we will review the D = 7 maximal SO(5) gauged supergravity and the D = 7 maximal ISO(4) gauged supergravity, which arise from consistent KK truncations of D = 11 supergravity and type IIA supergravity respectively, and remind readers how the two theories are related through the IW contraction.
The D = 7 maximal SO(5) gauged supergravity can be obtained by performing a Pauli reduction of D = 11 supergravity on S 4 . The details of this truncation, including the explicit demonstration of its consistency, are given in [31,32]. The bosonic field content of the theory is comprised of a metric, SO(5) Yang-Mills one-forms A ij (1) transforming in the 10 of SO(5), three-forms S i (3) transforming in the 5 of SO (5), and fourteen scalar fields given by a symmetric unimodular matrix T ij that parametrises the coset SL(5, R)/SO (5). The Lagrangian of the bosonic sector of the theory is given by 2 where g is the coupling constant. The scalar potential is given by and Ω (7) denotes the Chern-Simons terms for the Yang-Mills fields, which will not be important for our discussions here. 2 Our convention for the Hodge dual is * (dx m1 ∧ · · · ∧ dx mp ) = 1 where ǫ n1···nD with lowered indices is numerical, and q = D − p.
Any solution to the D = 7 maximal theory lifts to a solution of D = 11 supergravity, and the uplift formulae are provided in [31,32]. Most notably, the AdS 7 vacuum solution with A ij (1) = S i (3) = 0 and T ij = δ ij uplifts to the maximally supersymmetric AdS 7 × S 4 solution, which describes the near horizon limit of a stack of M5-branes. In [6], a half-maximal supersymmetric AdS 5 × H 2 /Γ solution was found, with Γ a Fuchsian subgroup of H 2 . The uplift of this solution has the form AdS 5 × H 2 /Γ × S 4 , with the S 4 non-trivially fibred over H 2 /Γ. The solution is dual to an N = 2 superconformal field theory in four dimensions that arises from the non-compact part of M5-branes wrapping a Riemann surface that is embedded in a CY 2 . In [10], a supersymmetric AdS 4 × H 3 /Γ solution preserving 8 supercharges was found. This uplifts to The solution is dual to N = 2 superconformal field theory in three dimensions that arises from the non-compact part of M5-branes wrapping a Slag 3-cycle that is embedded in a CY 3 .
These solutions in [6] and [10] motivate the constructions of consistent truncations of the D = 7 maximal theory in [24][25][26], which we will briefly review in Sections 3.1 and 4.1.
We are interested in the analogous story involving NS5-branes. The natural setting for this is the D = 7 maximal ISO(4) gauged supergravity theory, which can either be obtained by performing a Pauli reduction of type IIA supergravity on S 3 3 , interpreted as the internal 3-sphere of a stack of NS5-branes [34,35], or by taking an IW contraction of the D = 7 maximal theory which brings the SO(5) gauge group to ISO(4) [23]. The IW contraction procedure, as outlined in [23], involves decomposing the SO(5) vector indices in a 4 + 1 split, then rescaling all the fields by a contraction parameter k which is taken to zero at the end so that the gauge group becomes ISO (4). To be precise, the decomposition and rescaling of the fields in the maximal theory is given by: where A, B ∈ {1, . . . , 4}. Compared to [23], our 5 th index is their 0 th index. We note that there is an error in the (0, 0) component (our (5,5) component) of the decomposition of T ij given in [23], which rendered det T = 1. We have fixed this issue in (2.4).
After substituting (2.4) into the equations of motion of the SO(5) gauged theory and taking the limit k → 0, one obtains the D = 7 maximal ISO(4) gauged supergravity, whose equations of motion are recorded in appendix A. In terms of the D = 11 and type IIA theories, the IW contraction corresponds to taking the S 4 on which the D = 11 theory is reduced on, and turning it into S 3 × R, with R an isometry direction. To see this, let µ i , i ∈ {1, . . . , 5}, be the embedding coordinates of S 4 in R 5 given by The IW contraction in (2.4), now interpreted as a set of singular rescalings of the metric and 4-form flux in the D = 11 theory, is accompanied by an additional rescaling of the µ i coordinates [23]. We split µ i into µ A and µ 5 with A ∈ {1, . . . , 4}, and rescale The original constraint (2.5) in the limit k → 0 now reads withμ 5 unconstrained. The resulting topology is S 3 × R, withμ A parameterising the S 3 , andμ 5 parameterising R.

3-formsS A
(3) transforming in the 4 of SO(4), a 3-formS (3) , four scalarsτ A in the 4 of SO(4), and ten scalar fields given byΦ and a symmetric unimodular matrixT AB parametrising the coset SL(4, R)/SO (4). By defining the Yang-Mills field strength the following combinations of fundamental fields and making use of (A.5) to integrateS (3) as , (2.11) the Lagrangian of the bosonic sector is given by (2.12) The scalar potential isṼ andΩ (7) denotes the Chern-Simons terms depending only onÃ AB (1) andÃ 5A (1) , which again, will not be important for our following discussions. There is a consistent truncation of this maximal theory to a half-maximal SO(4) gauged theory obtained by setting where the removal of theÃ 5A (1) fields breaks the ISO(4) gauge group to SO(4). We will call this the half-maximal truncation, and it will be used in our next sections. In the context of the type IIA theory in 10 dimensions, the half-maximal truncation corresponds to the removal of the RR sector.
Any solution to the D = 7 maximal theory lifts to a solution of D = 10 type IIA supergravity, and the uplift formulae are provided in [34,35]. Most notably, the linear dilaton solution withÃ AB (1) =S (3) = 0 andT AB = δ AB preserves 16 supercharges and uplifts to the supersymmetric D = 10 solution, which describes the near horizon limit of a stack of NS5-branes. Similar to the M5 case, supersymmetric solutions corresponding to NS5branes wrapping calibrated cycles, like an S 2 in CY 2 and an S 3 in CY 3 , were constructed in [1,2] and [3] respectively. The uplift of these solutions to the type IIA theory are dual to compactifying little string theory on the calibrated cycles with a topological twist, and the geometry of the solutions has the internal S 3 non-trivially fibred over the cycles. These solutions motivate our construction of the consistent truncations of the maximal D = 7 theory on the calibrated cycles. By virtue of the consistency of the IW contraction, we can obtain such truncations by directly applying (2.4) to the corresponding truncations describing M5-branes wrapping on the appropriate cycles. This will be done in the next two sections. We will also make use of the half-maximal truncation in (2.14) to obtain the corresponding truncations of the NSNS sector of the type IIA/B theory, or equivalently, the type I theory.
3 Consistent KK truncations on Riemann surfaces in CY 2

Review of wrapped M5-branes on Riemann surfaces in CY 2
We first summarise the consistent KK ansatz used in [24] for M5-branes wrapped on Riemann surfaces. The ansatz for the D = 7 metric is given by where φ is a real scalar field defined on the five-dimensional spacetime. We denote {ē m ; m ∈ {0, . . . , 4}} and {ē a ; a ∈ {1, 2}} as the orthonormal frames for both ds 2 5 and ds 2 (Σ 2 ) respectively, and letω m n andω a b be the corresponding spin connections. The metric of the Riemann surface is normalised to R ab = lg 2 δ ab , with l = 0, ±1. The SO(5) fields are decomposed via SO(5) → SO(2)×SO(3), where the vector index of SO(5) accordingly decomposes as i = (a, α), with a ∈ {1, 2} and α ∈ {3, 4, 5}. The decomposition of the fields are given by This incorporates the spin connectionω ab in the expression for A ab (1) , which corresponds to the topological twist condition that ensures the preservation of supersymmetry on the wrapped M5-brane with worldvolume R 1,3 × Σ 2 . The ansatz is also comprised of three scalar fields ψ 1α , another three scalar fields ψ 2α , a 1-form A (1) and three 1-forms A αβ (1) , all defined on the D = 5 spacetime. The 3-form fields S i
We will call these D = 5 fields the NS5 fields, which are distinguished notationally from the M5 fields by a tilde.
We can now substitute the ansatz directly into the D = 7 equations of motion to obtain a D = 5 theory. However, as explained in Section 1, it is quicker, and perhaps more instructive to utilise the IW contraction that connects the SO(5) and ISO(4) theories. To do so, we must identify our NS5 fields in terms of the M5 fields presented in the previous section using the IW contraction procedure.

Field redefinitions
In order to make contact with the canonical language of D = 5, N = 4 supergravity in the next section, we find it convenient to make the following field redefinitions. We first replace (T −1σ (1) ) α by introducing two one-formsÃ α (1) and two Stueckelberg scalar fieldsξ α , and we note that the SO(2) gauge symmetry is non-linearly realised byΞ. Substituting this into (B.1.4), we deduce that

3.4)
and we note again that the SO(2) gauge symmetry is non-linearly realised byÃ α (1) . We also need to dualiseH (3) . There are two ways of doing this, the first being to integrate (B.1.10) directly, and the second, easier way is to add to the original Lagrangian the term withB (1) a Lagrange multiplier that enforces the Bianchi identity dH (3) =F (2) ∧F (2) .
TreatingH (3) as a fundamental field, the variation of the total Lagrangian L (5) + L dual (5) with respect toH (3) yields the algebraic relatioñ which we will substitute back into the total Lagrangian. Finally, we also find it convenient to redefineK Making use of the above redefinitions, the kinetic terms for the vectors can be rewritten as We note that the positive signs in theH (3) andG α (3) terms do not indicate the presence of ghosts, as when we consider the dualised fields (3.3.3) and (3.3.6) which encodes the fundamental degrees of freedom, we obtain a sign flip from applying the Hodge star twice.
The topological terms greatly simplify to Up to total derivatives, these can be rewritten as , which, as we will see, is the form in which the N = 4 supersymmetry is manifest.

D = 5, N = 4 supersymmetry
In this section we first summarise the general structure of N = 4 gauged supergravity in D = 5, coupled to n = 3 vector multiplets, mostly following the conventions of [36] (which generalised the results in [37]). For a cleaner presentation, we will remove all tildes from the NS5 fields for the rest of this section.
The scalar manifold is given by SO(1, 1)×SO(5, 3)/(SO(5)×SO(3)), with the SO(1, 1) part described by a real scalar field Σ, and the coset SO(5, 3)/(SO(5) × SO(3)) parametrised by where η is the invariant metric tensor of SO(5, 3). Global SO(5, 3) transformations are chosen to act on the right, while local SO(5) × SO(3) transformations act on the left via The coset can also be parametrised by a symmetric positive definite matrix with M M N an element of SO(5, 3). We can raise indices using η, and the inverse of M M N , which we denote by M M N , is given by We will work in a basis in which η is not diagonal, but instead given by (3.4.5) In order to work in a basis in which η is diagonal with the first five entries −1 and the last three entries +1, as in [36], we can apply a similarity transformation using the matrix which satisfies U = U T = U −1 and det U = 1. To evaluate the expression for the N = 4 scalar potential (3.4.13), we will also need the following antisymmetric tensor with the indices m 1 , . . . , m 5 running from 1 to 5.
The general D = 5, N = 4 gauged theory [36] is specified by a set of embedding tensors and ξ M , which specify both the gauge group in SO(1, 1) × SO(5, 3) as well assigning specific vector fields to the generators of the gauge group. The covariant derivative is given by 5 are the generators for SO(5, 3), t 0 is the generator for SO(1, 1), we have raised indices using η, and ∇ µ is the Levi-Civita connection. To ensure the closure of the gauge algebra, the embedding tensors must satisfy the following algebraic constraints Associated with the vector fields A 0 (1) and A M (1) , we also need to introduce two-form gauge fields B (2)0 and B (2)M . In the ungauged theory these appear on-shell as the Hodge duals of the fields strengths of the vectors. In the gauged theory the two-forms are introduced as off-shell degrees of freedom, but the equations of motion ensure that the suitably defined covariant field strengths are still Hodge dual. In particular, the two-forms appear in the covariant field strengths for the vector fields, H 0 (2) and H M (2) , via Using the N = 4 language of [36] 6 , the Lagrangian of the bosonic sector of the theory can be written as Here the terms involving the generators differ by a factor of two with the analogous expression in [36].
However, the generators that we use in (3.4.19) below, also differ by a factor of two, so our covariant derivative stays the same as [36]. 6 Note that we have multiplied the Lagrangian in [36] by a factor of two.
The scalar kinetic energy terms are given by (3.4.12) and the scalar potential is given by The kinetic terms for the vectors, which also involve two-form contributions via (3.4.10), are given by In order to present the topological part of the Lagrangian in (3.4.11), we introduce the calligraphic index M = (0, M) which allows us to package the 9 vector fields and 9 two-forms into the A M (1) and B (2)M , each transforming in the fundamental representation of SO(1, 1) × SO(5, 3). In the conventions of this paper 7 , we have Here the symmetric tensor d MN P = d (MN P) has non-zero components In an orthonormal frame, we take ǫ 01234 = +1 so that ǫ = vol 5 . We assume that [36] have taken ǫ 01234 = −1 and then the expression for the topological term given here agrees with that in [36] up to an overall factor of 2. and the only non-zero components of X MN P are given by

Scalar manifold
We take the generators of SO(5, 3) to be given by the 8 × 8 matrices with a non-diagonal η given in (3.4.5). In order to parametrise the coset SO(5, 3)/(SO(5) × SO (3)), we exponentiate a suitable solvable subalgebra of the Lie algebra. Following, for example [39], the three non-compact Cartan generators H i and the twelve positive root generators, with positive weights under H i , are given by 8 We note that Tr  (3)). This is achieved by first defining In addition, we defineĤ = −H 1 − H 2 which commutes with the above two generators. We introduce three scalar fields {ϕ 1 , ϕ 2 , ρ} to form the following coset representative

4.22)
8 To compare with (3.31) of [39] we should make the identifications ( where the 2 × 2 matrix V parametrises the coset SL(2, R)/SO (2) in the standard upper We can identify the scalar fields in the 2 × 2 matrix T αβ in the truncated theory as Collecting our results, the appropriate parametrisation of the coset SO(5, 3)/(SO (5)×SO (3)) is given by where we identify ϕ 2 and ϕ 3 as The remaining SO(1, 1) part of the scalar manifold is described by a real scalar field Σ,

Gauge group
In this section, we will demonstrate that the gauge group of the reduced D = 5 theory where the action of the semi-direct product depends on the 5,17 and G A 0 5,18 are two five-dimensional matrix Lie groups with Lie algebras A 100 5,17 and A 0 5,18 respectively. We will elaborate more on this in the discussion below. The compact SO(2) subgroup of the gauge group is generated by

4.28)
which is associated with the gauge field A (1) , and the non-compact part of the gauge group, which are associated with the gauge fields A α (1) , V α (1) and A (1) respectively (see (D.6)). We note that the one-form B (1) does not participate in the gauging. The generators in (3.4.29) satisfy the following commutation relations Rather remarkably, the algebra associated to l = 0 is not isomorphic to that associated to l = ±1. These two distinct algebras belong to two different families of five-dimensional real Lie algebras, namely A spq 5,17 and A p 5,18 , which are listed in [30]. The subscripts m and n in A p m,n denote the dimension of the Lie algebra and the n-th algebra on the list of [30] respectively, and the superscript p in A p m,n denotes the continuous parameter(s) on which the algebra can depend on. Specifically, when l = 0, the algebra is described by A 100 5,17 , and its minimal matrix group representation is given by [40] while the algebras for the l = ±1 cases are described by A 0 5,18 and their minimal matrix group representations are given by [40] where θ, x 1 , x 2 , x 3 , x 4 are the real parameters. We also provide an explicit representation of the generators (3.4.29) in appendix C, which after exponentiation recovers (3.4.31) and (3.4.32).
To understand the structure of the gauge group, let's focus on l = 0, and consider two elements where i ∈ {1, 2, 3, 4}. From this, we observe that this group is isomorphic to SO(2) ⋉ R 4 .
When l = ±1, the above map becomes a bit more complicated, but the overall SO(2) ⋉ R 4 structure remains the same. Putting it all together, we conclude that the gauge group of the

Embedding tensors
The components of the embedding tensor are specified by with those of our reduced theory via With the above identifications, we find that the Lagrangian of our D = 5 theory is equivalent to the canonical N = 4 Lagrangian. We have presented a few details of this calculation in appendix D.
To make the N = 2 supersymmetry explicit, we will rewrite the 5-dimensional theory in the canonical form. A review of the canonical Lagrangian of generic D = 5, N = 2 supergravity coupled to matter is given in [41][42][43], and we will follow mostly the conventions of [43].
Using the N = 2 language of [43], the Lagrangian of the bosonic sector of the theory can be written as Here, A I (1) , with I ∈ {0, 1, 2}, label the graviphoton as well as the two vector fields of the two vector multiplets, and φ x , with x ∈ {1, 2}, denotes the two real scalar fields of the two vector multiplets that parametrise a very special real manifold SO(1, 1) × SO(1, 1). The q X , with X ∈ {1, . . . , 8}, are the eight real scalar fields of the two hypermultiplets that parametrise the quaternionic Kähler manifold SO(4, 2)/(SO(4) × SO (2)). Within the covariant derivatives, K x I and k X I are sets of three Killing vectors on the very special real manifold and on the quaternionic Kähler manifold respectively. The structure constants of the gauge group are given byf I JK . We explain below how our truncated Lagrangian can be recast in the form of (3.5.2) with gauging only in the hypermultiplet sector.
We start with the vector multiplets. The very special real geometry is determined by a real, symmetric, constant tensor C IJK which specifies the embedding of SO(1, 1) × SO (1, 1) in a three-dimensional space with coordinates h I through (3.

5.4)
Defining h I = C IJK h J h K , the kinetic terms for the vectors are defined by Indices can be lowered and raised using a IJ and its inverse a IJ . Moreover, the pull-back of a IJ gives the metric for the scalar fields φ x With these definitions, we can immediately identify the field strengths and vector fields as where G (2) = dB (1) , and the non-zero components of both symmetric tensors a IJ and C IJK are The scalar coset of the vector multiplet sector is identified to be SO(1, 1) × SO(1, 1), and g xy = δ xy . In (E.11), we provide the canonical expression of the ungauged kinetic term for the scalar fields, which allows us to identify the two scalar fields within the very special real manifold to be, with the embedding coordinates given by (3.5.10) We now turn to the hypermultiplets. The explicit parametrisation of the quaternionic Kähler manifold SO(4, 2)/(SO(4)×SO (2)) can be found in appendix E. We identify the coordinates on this quaternionic Kähler manifold to be q X = ϕ 1 , 3φ + 1 4 log Φ, ρ, ξ, ψ 11 , ψ 21 , ψ 12 , ψ 22 . The two scalar fields ϕ 1 and ρ are introduced to parametrise the coset T αβ as we explain in appendix E. The associated metric on this quaternionic Kähler manifold can be found in (E.12). The Killing vectors k X I , which are responsible for the gauging, are given by The gauging only involves two vectors, A (1) and A (1) , and the gauge group is SO(2) × SO (2) when l = 0, and SO(2) × R + when l = ±1. To completely demonstrate the N = 2 supersymmetry, it remains to check that the potential matches with L pot N =2 , which is given in (E.21). We will show that this is indeed the case in appendix E.
We have shown that our truncation is an N = 2 supergravity theory in D = 5 comprised of one gravity multiplet, two vector multiplets and two hypermultiplets with SO (2)  where we ignored the common factor of SO (1, 1). Under the half-maximal truncation, the gauge group becomes SO(2) × R + when l = ±1, with R + ⊂ G A 0 5,18 , and SO(2) × SO(2) when l = 0, with the second factor SO(2) ⊂ G A 100 5,17 .

Further consistent sub-truncations
For the D = 5, N = 2 theory, there are two further sub-truncations of interest that arise from keeping sectors invariant under the subgroups of SO(2) 1 × SO(2) 2 . As we will see, these sub-truncations do not form supergravity theories.

SO(2) 1 invariant sector
We consider keeping only the fields that are singlets under SO(2) 1 . This amounts to setting B (1) = 0, ψ aα = 0 and A (1) = 0 in the equations of motion. For l = 0, we find that Ξ acts as a Stueckelberg field for A (1) , as it appears in the gauge invariant combination (1) . Its equation of motion is implied by the equation of motion for A (1) .
The remaining fields, taking into account of Ξ being pure gauge, consists of the metric, the scalars T αβ , φ, λ, Φ and the 1-form A (1) .

Some solutions
To demonstrate the consistency of our truncation, we will reproduce the one-parameter is given by where z is a radial variable, and the function x(z) is defined as Here, c is a real integration constant that, for l = 1, parameterises the different flows from the UV to the IR. The values of the scalar fields are The above solutions can easily be uplifted back to D = 10 using our truncation procedure.
For l = 0 in the SO(2) 2 invariant sector, we report the following domain wall solution

Review of wrapped M5-branes on Slag 3-cycles in CY 3
As in Section 3, we will first summarise the consistent KK ansatz used in [26] to describe M5-branes wrapped on Slag 3-cycles. The D = 7 metric is given by  = (a, α), with a ∈ {1, 2, 3} and α ∈ {4, 5}, and the fields under this decomposition are given by This ansatz incorporates the spin connectionω ab in the expression for A ab (1) , which corresponds to the topological twist condition that ensures the preservation of supersymmetry on the wrapped M5-brane with worldvolume R 1,2 × Σ 3 . The ansatz is also comprised of a scalar field β, two scalar fields θ α and a vector field A (1) , all defined in D = 4. The 3-form fields

NS5-branes wrapped on Slag 3-cycles in CY 3
For NS5-branes wrapped on Slag 3-cycles, we will, as we did for Riemann surfaces, write down the ansatz in terms of the maximal ISO (4)  and ds 2 (Σ 3 ) respectively, and letω m n andω a b be the corresponding spin connections. We normalise the metric of the Slag 3-cycle to satisfyR ab = lg 2 δ ab with l = 0, ±1. The fields are decomposed via SO(4) → SO(3) 1 × SO(3) 2 → SO(3) 1 , and are given bỹ

(4.2.2)
This again incorporates the spin connectionω ab in the expression forÃ ab (1) , which corresponds to the topological twist condition that ensures the preservation of supersymmetry on the wrapped NS5-brane. The 3-form fields are taken to bẽ   Using (2.4), we make the following identification of the M5 and NS5 fields With this, the 4-form Lagrangian given by

2.8)
whereR is the Ricci scalar of the D = 4 metric, the kinetic terms are the potential terms are 2.10) and the topological terms are L top (4) = 6(θG (3) Any solution of the equations of motion in B.2 can be uplifted to type IIA supergravity. This can be done by first using (4.2.1)-(4.2.4) to uplift to the ISO(4) gauged theory in D = 7, then using the uplift formulae in [23] which connect the ISO(4) gauged theory and the type IIA theory.
As we shall show, this theory is a D = 4, N = 2 supergravity theory coupled to a vector multiplet and two hypermultiplets with R + × R + gauging in the hypermultiplet sector, and a scalar manifold given by the symmetric space, SU(1, 1)/U(1) × G 2(2) /SO(4).

D = 4, N = 2 supersymmetry
In this section, we will show that our theory corresponds to the bosonic sector of D = 4, N = 2 supergravity coupled to a vector multiplet and two hypermultiplets with Abelian gauging in the hypermultiplet sector and scalar manifold SU(1, 1)/U(1) ×G 2(2) /SO(4). This will be done by recasting our Lagrangian (4.3.15) into the canonical form (see [44,45]) corresponds to the gauged hypermultiplet, and L pot (4) = −g 2 V , with V the scalar potential. For clarity, we have removed tildes from our NS5 fields for the rest of this section.
In (4.4.2), z is a complex coordinate, and G zz a Kähler metric with Kähler potential K V on SU(1, 1)/U(1), F I (2) = dA I (1) , I ∈ {0, 1} are the Abelian field strengths of the vectors in the graviton and vector multiplet, and N IJ is a z-dependent matrix. Taking X I to be the homogeneous coordinates on SU(1, 1)/U(1), and F to be a prepotential, the Kähler potential K V and matrix N IJ are given by 4.4) and where n V = 1 is the number of vector multiplets. Finally, for our case, where the gauging is restricted to the hypermultiplet sector, the scalar potential is given by Here, the six scalars P x I , x ∈ {1, 2, 3}, are moment maps defined by the relation where ω x (1) is the Sp(1) ∼ = SU(2) ⊂ SU(2) × Sp(2) = Hol(G 2(2) /SO(4)) part of the spin connection associated with the quaternionic Kähler metric h uv , and K x (2) is its curvature.
The 2-forms K x (2) are related to a triplet of complex structures J x by (J x ) u v = δ uw K x w v that satisfy the quaternionic algebra (4.4.9) For the vector multiplets, we introduce homogeneous coordinates X I = (1, z 2 ) on SU(1, 1)/U(1), with z the axion-dilaton defined in (4.3.13). The required pre-potential is given by Accordingly, the Kähler potential is 4.11) and the corresponding metric is (4.4.12) Next, we compute that is identical to that of the D = 4 theory obtained in [26].
We now consider the hypermultiplets. The explicit parameterisation of the coset G 2(2) /SO (4) is given in appendix F. We identify the coordinates on G 2(2) /SO(4) to be q u = (ϕ 1 , ϕ 2 , τ, θ, Θ, C, G, Γ) , (4.4.14) and the quaternionic Kähler metric h uv is given in (F.9). In this form, it is apparent that the two requisite Killing vectors of G 2(2) /SO(4), along which we have an Abelian R + × R + gauging. All that remains is to match the scalar potential, which is done at the end of appendix F.

Half-maximal truncation and D = 4, N = 1 supersymmetry
From (2.14), the half-maximal truncation of our D = 4, N = 2 theory is obtained by setting  [46]. The main step is to perform the following field redefinitions, In terms of these complex variables, the Lagrangian in (4.3.15) is now recast into the canonical form where the three complex scalar fields z α form the bosonic part of the three chiral multiplets in the reduced theory, and we removed the tildes from the 4-dimensional metric for notational convenience. The Kähler metric on the homogeneous space [SU(1, 1)/U(1)] 3 is given by where K is the Kähler potential The scalar potential P is given by where the holomorphic superpotential W is The fact that W is independent of z 3 is crucial for the construction of a subclass of AdS 3 × R solutions, as we will show later. An equivalent writing of the scalar potential P is where the Kähler covariant derivative is defined as ∇ α (·) = ∂ α (·) + (·)∂ α K.
We have shown that our truncated theory is an N = 1 supergravity theory in D = 4 coupled to three chiral multiplets with scalar manifold [SU(1, 1)/U(1)] 3 . We recognise that one of the SU(1, 1)/U(1) factors, the one parameterised by z 1 = β +ie ϕ 0 , is directly inherited from the N = 2 theory. In terms of the rest of the scalar manifold, we find that the truncation to the half-maximal sector decomposes Equivalently, the non-linearly realised global symmetry of the scalar manifold under the half-maximal truncation, neglecting the common factor of SU(1, 1), is 1) . (4.5.10)

Further consistent sub-truncations
For the D = 4, N = 1 theory, we notice that it is consistent to further set β = θ = 0.
There are two sub-truncations of particular interest (a) setting λ = 0 with l = 0, and (b) setting Γ = 0. For truncation (a), the remaining fields consist of the metric, φ, Φ and Γ.
The scalar potential is independent of Γ, and the shift symmetry, Γ → Γ + c allows the construction of a class of supersymmetric AdS 3 × R solutions, which we will discuss in detail in the next section. For truncation (b), the remaining fields consist of the metric, φ, λ and Φ. The l = ±1 cases cover the solutions in [3], whereas the l = 0 case covers the NS5-brane near-horizon, linear dilaton solution in D = 10.

Some solutions
We consider a particular type of ansatz for the half-maximal truncation as in [47] describing warped AdS 3 × R, which in canonical variables is given by where ds 2 (AdS 3 ) is the metric on AdS 3 of radius L. Applying the Killing spinor equations given in [48] to our ansatz, we obtain the following set of BPS equations where κ = ±1 denotes the chirality condition, C r = i 6 (∂ α K∂ r z α − ∂ᾱK∂ rzᾱ ) is the Kähler connection, and ω = ω(r) is a phase that appears in the expression for the Killing spinors.
As discussed in Section 4.6, we can further truncate the D = 4 theory by setting Re(z 1 ) = Re(z 2 ) = 0 and Im(z 1 ) = Im(z 2 ) (or equivalently setting β = θ = λ = 0) with l = 0. Within this sub-truncation, we can construct analytic, supersymmetric solutions to (4.7.2), given by z 1 = z 2 = ic 0 , z 3 = g cos ω(2α 0 r − gr 2 sin ω) + i(α 0 − gr sin ω) 2 , e A = |α 0 − gr sin ω| w 0 gL , cos ω = w 0 κ , where w 0 ∈ (0, 1], and c 0 and β 0 are real constants. When w 0 = 1, this simplifies to the following supersymmetric AdS 3 × R, linear axion solution Using the uplift formulae in [34], and taking Σ 3 = R 3 for convenience, we find that this linear axion solution in type IIA supergravity is given by where and S 3 is the unit 3-sphere. We can interpret this as the dyonic, Freund-Rubin AdS 3 × S 3 solution of a D = 6 supergravity theory obtained by reducing the IIA theory on R 4 . In the 6-dimensional picture, this solution corresponds to the near horizon of the (anti)-self-dual string [49]. From this, it is clear that the solution preserves half of the supersymmetries.
For completeness, we also report the domain wall solution for l = 0, e 5φ = gr , ds 2 4 = r 2 ds 2 (R 1,2 ) + dr 2 , theories obtained from wrapped M5-brane truncations on the appropriate cycles using a singular limit procedure, known as the Inönü-Wigner contraction. The two new theories obtained in sections 3 and 4 can be viewed as direct "cousins" of the 5 and 4-dimensional theories corresponding to the truncations associated with M5-branes wrapping Σ 2 [24,25] and Σ 3 [26], in the sense that they have the same amount of supersymmetry and field content, but as we have shown, the precise details of the gauging and the vacuum structures of the theories are entirely different.
There are more examples of wrapped NS5-brane truncations that can be obtained using our method. From the catalogue of wrapped M5-brane solutions given in [50], we observe that it is possible to obtain wrapped NS5-brane truncations on: (1) Σ 2 × Σ ′ 2 , a product of two Riemann surfaces inside two CY 2 spaces 10 ; (2) Σ 2 × Σ 3 with Σ 2 and Σ 3 a Riemann surface and a Slag 3-cycle inside a CY 2 and CY 3 respectively; (3) a Kähler 4-cycle in a CY 3 . IW contractions are of course not limited to just the SO(5) and ISO(4) gauged supergravities in D = 7. For example, [52][53][54] obtained the variants of the SO(8) gauged N = 8 supergravity theories in D = 4 with gauge groups ISO (7), interpreted as the IW contraction of the original SO(8) gauge group, as well as SO(p, q) (p + q = 8) and its IW contraction about SO(p). It is well-known that the SO(8) gauged theory can be obtained by a consistent truncation of M-theory on S 7 , as demonstrated in [55]. By interpreting the S 7 as the internal 7-sphere of a stack of M2-branes, the SO(8) gauged theory can be seen as the natural arena for wrapped M2-brane truncations. As such, the existence of the contracted ISO(7) gauged theory suggests that the IW contractions can be used to relate wrapped D2-brane truncations from the corresponding wrapped M2-brane truncations in a similar fashion to the relation between wrapped NS5-brane and M5-brane truncations.
These correspondences between the M2 and D2 truncations are purely within M-theory and its type IIA descendent, but can be seen to be related to the S 6 truncation of massive IIA, which yields the dyonic ISO(7) gauged supergravity in D = 4 [56]. By setting the Romans mass to zero, the dyonic theory becomes the electric ISO(7) theory in [52].
The consistency of the IW contraction procedure also opens up the question of which lower-dimensional gauged supergravity theories are related via the IW contraction (or potentially other such procedures), and whether such relations are actually contingent on there being a higher-dimensional picture, as there is in our case. This can perhaps be answered more systematically using the language of generalised geometry along the lines discussed in [25,[57][58][59], which we would like to address in the future.
A Equations of motion for D = 7 maximal ISO(4) gauged supergravity In this section, we apply the IW contractions to obtain the equations of motion of maximal ISO(4) gauged supergravity in D = 7 from those of the maximal SO(5) gauged theory.
For clarity, we will write down the SO(5) equations of motion. It is also more convenient to work with the scalar matrix M AB ≡Φ 1/4T AB instead ofT AB . We note that M AB is not independent ofΦ, as det M =Φ. We begin with the S j (3) equations, Using the notation defined in (2.8)-(2.10), we find that (A.1) yields the following two equa- Next, we consider the non-Abelian Bianchi identities These yield Following this, we consider the Yang-Mills equations The (k, l) = (5, 5) component gives a 0 = 0 identity, the (k, l) = (A, 5) components givẽ ∧ * G C We now consider the scalar equations, which are given by which is the limit of Q as k → 0, we find that the (5, 5), (A, 5) and (A, B) components of the scalar equations respectively yield Finally, we consider the Einstein equations which is the limit of X as k → 0, we find, after some algebra, that , (B. 1.4) and .

D Matching with N = 4 supergravity
In this section, we present some of the details on how to match the truncated D = 5 theory of section 3 with the canonical language of N = 4 theory in [36].
The parametrisation of the coset SO(5, 3)/(SO(5) × SO (3)) is given in (3.4.25), and we find that the Lie algebra valued Maurer-Cartan one form is given by where ϕ 2 , ϕ 3 are defined in (3.4.26) and are the ungauged versions of X (1) , P a (1) , Q α  (3)) is equal to The scalar manifold of the reduced theory is SO(1, 1) × SO(5, 3)/(SO(5) × SO(3)), and the ungauged kinetic term of all of the scalar fields can be recast into where the SO(1, 1) part of the scalar manifold is described by the real scalar field Σ, via To incorporate the gauging, we need to use the covariant derivative given in (3.4.8) which we denote as D = d + gA with Now we can decompose the gauged version of the Maurer-Cartan one form DV · V −1 = P + Q. In particular we have P = P (0) + g 2 V · A · V −1 + (V · A · V −1 ) T , which lies in the complement of the Lie algebra of SO(5) × SO(3). Finally, we find that the gauged scalar kinetic terms are recovered precisely after evaluating − 1 2 Tr( * P ∧ P). We provide here the explicit expression of the matrix M M N which is defined in (3.4.3), where and we also define To calculate the N = 4 scalar potential (3.4.13), we have to make use of the embedding tensor specified in (3.4.35) and we find the following non-vanishing contributions Combining these contributions, we are able to recover the scalar potential in our truncated theory (3.2.12).
We now turn to the vector sector. .
Combining these contributions, we recover the topological part of the Lagrangian in our truncated theory (3.3.10).
We are now ready to show that the scalar potential of our truncated theory is consistent with N = 2 supersymmetry. The scalar potential for a general N = 2, D = 5 gauged supergravity with no tensor multiplets and no Fayet-Iliopoulos terms is given by [43] L pot N =2 = 4g 2 4 P · P − 2 P x · P x − 2W x W x − 2N A N A . (E.21) The first two terms involve the moment maps for the Killing vectors k X I defined via where J is the triplet of complex structures, with P and P x are defined to be P ≡ 1 2 h I P I , P x ≡ − √ 3 2 ∂ x h I P I , (E. 23) and indices are raised and lowered using the metrics g xy and a IJ as discussed in section 3.5.
Here, J is represented in terms of the Pauli matrices σ, while J is represented by K. This means that We now turn to the term W x in the scalar potential, which is defined to be wheref P JI are the structure constants of the gauge algebra. For our truncated theory, the algebra is Abelian, sof P JI = 0, and hence W x = 0. The final term in the scalar potential is given by After explicitly evaluating the terms in (E.21) using the ingredients in this appendix as well as those in section 3.5, we recover the scalar potential in our truncated theory.
To match the potential terms, we will require the moment maps defined in (4.4.8), which in turn require the spin connection and curvature associated to the metric h uv . First, we define the vielbeins Let M u v = E u v − E v u , with E u v the 8 × 8 matrix with 1 in the u th row and v th column be the generators of so (8). The spin connection is then given by (F.14) These generators satisfy the canonical commutation relations In terms of these generators, the spin connection is where we defined The curvatures associated to ω x (1) and ∆ x (1) are defined by