Guillaume Legendre
email: guillaume.legendre@dauphine.fr

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces

. We propose in this Note a different approach which allows to construct two second-order in time numerical schemes. In a metric space framework, we show that the schemes are well defined and prove the convergence for one of them under some regularity assumptions. For the particular case of a Fokker-Planck gradient flow in the Wasserstein space, we obtain (theoretically and numerically) the second-order convergence.

Résumé

. Nous proposons dans cette Note une approche différente permettant de construire deux schémas numériques d'ordre deux en temps. Dans le cadre d'un espace métrique, nous montrons que les schémas sont bien définis et prouvons la convergence de l'un d'entre eux sous des hypothèses de régularité. Pour le cas particulier d'un flot de gradient Fokker-Planck dans l'espace de Wasserstein, nous obtenons (théoriquement et numériquement) la convergence à l'ordre deux.

Background on gradient flows and relationship with the implicit Euler scheme

Let f : R d → R be a smooth convex function and x be a point in R d ; a gradient flow starting from x is a curve (x t ) t≥0 solution to the Cauchy problem x t = -∇f (x t ) for t > 0, x 0 = x. A topic that has received considerable attention lately is the extension of this definition to a Polish metric space (X , d) and a functional F : (X , d) → R ∪ {+∞} (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Optimal transport. Old and new[END_REF][START_REF] Ambrosio | A user's guide to optimal transport[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF] for instance). An important advance was in particular the work of Jordan, Kinderlehrer and Otto [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], who introduced the following numerical scheme: given a time step τ > 0 and x in X , define recursively the sequence (x τ n ) n∈N such that x τ 0 = x and, for any natural integer n, x τ n+1 is selected as a minimizer of the functional

x → P JKO F (x; x τ n , τ ) := 1 2τ d 2 (x τ n , x) + F (x). (1)
When the metric space is a Hilbert space and the functional F is smooth enough, the above scheme amounts to the implicit Euler scheme, i.e.,

x τ n+1 -x τ n τ = -∇F (x τ n+1
). As a consequence, the Jordan-Kinderlehrer-Otto method (abbreviated as JKO from now on) can be seen as a variational generalization of the implicit Euler scheme in a metric space. Coupled with a suitable interpolation, the sequence constructed by this scheme defines, as the time step τ tends to zero, a (metric) gradient flow, the convergence being of order one in τ (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Theorem 4.0.4] for a sharp estimate in a particular setting). As a natural numerical counterpart of the theoretical procedure, the JKO scheme was subsequently used to compute the gradient flow in applications (see [START_REF] Kinderlehrer | Approximation of parabolic equations using the Wasserstein metric[END_REF][START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF][START_REF] Benamou | Discretization of functionals involving the Monge-Ampère operator[END_REF] for instance).

However, while the JKO scheme has theoretical advantages, its first-order convergence may be judged insufficient in practice. When the gradient flow is regular enough, one may seek to replace it with a secondorder alternative, or even combine a second-order scheme with the JKO scheme in order to construct a time step selection mechanism (see [12, Chapter 17, Section 17.2] for instance). The purpose of this Note is to propose two second-order schemes, which are both easy to implement (one of them directly uses the JKO scheme), and provide several preliminary theoretical and numerical results pertaining to them.

Second-order numerical schemes on metric spaces

Let (X , d) be a Polish metric space. We recall (see [2, page 37]) that a curve γ : [0, 1] → X is called a (constant speed) geodesic provided that d(γ(t), γ(s)) = |t -s| • d(γ(0), γ(1)), and that the space (X , d) is called geodesic if, for any couple of points (x, y) ∈ X 2 , there exists a geodesic γ connecting them, that is, such that γ(0) = x and γ(1) = y. We will denote by Geod x,y the set of all geodesics connecting x and y. Moreover, the set {γ( 12 ) | γ ∈ Geod x,y } will be called the set of midpoints of the couple (x, y) and will be denoted x+y 2 (see [START_REF] Burago | A course in metric geometry[END_REF]Chapter 2] for a definition in general metric spaces and [START_REF] Agueh | Barycenters in the Wasserstein space[END_REF] in the particular case of Wasserstein spaces).

Definition of the schemes

Given a time step τ , we define the Variational Implicit Midpoint (VIM) scheme, starting from x in X , for the (gradient flow of the) functional F by setting x τ 0 = x and, for any natural integer n, recursively choosing

x τ n+1 ∈ argmin y∈X P V IM F (y; x τ n , τ ), (2)
where

∀(x, y) ∈ X 2 , P V IM F (y; x, τ ) = inf d 2 (x, y) 2τ + 2F (u) u ∈ x + y 2 . ( 3
)
In the particular case of a Hilbert space and a smooth functional F , the point x τ n+1 is effectively a solution of the equation

x-x τ n τ +∇F ( x τ n +x
2 ) = 0, which characterizes the implicit midpoint rule method, a one-stage implicit Runge-Kutta method of second order (see [START_REF] Hairer | Solving ordinary differential equations I. Nonstiff problems[END_REF]Chapter II,Section 7]).

We now prove that the VIM scheme is well-defined by establishing the existence of a solution to problem (2). Theorem 2.1 Let (X , d) be a Polish metric space such that, for any x in X , the set

y∈X x + y 2 is closed. (4)
Assume that the functional F is lower semicontinuous, bounded from below, and such that, for all r > 0, and c ∈ R, the set {y ∈ X | F (y) ≤ c, d(x, y) ≤ r} is compact for some x in X . Then, there exists x in X such that P V IM F (x; x, τ ) = inf y∈X P V IM F (y; x, τ ). Proof Let (x n , u n ) n∈N be a minimizing sequence for inf y∈X P V IM F (y; x, τ ). In particular, the sequence 1 2 d 2 (x, x n ) n∈N is bounded and so is (d(x, u n )) n∈N . On the other hand, the sequence (F (u n )) n∈N is also bounded, so that, by hypothesis, the sequence (u n ) n∈N lives in a compact set and thus converges, up to a subsequence, to some u. In addition, since ∪ y∈X

x+y 2 is closed, there exists x in X such that u ∈ x+x 2 . Recalling that u n ∈ x+xn 2 , we obtain that d(x, x) = 2 d(x, u) = 2 lim n→∞ d(x, u n ) = lim n→∞ d(x, x n ).
The lower semicontinuity property of F then allows to conclude.

Remark 1 The assumptions on the functional F in the above theorem are classical (see [START_REF] Ambrosio | A user's guide to optimal transport[END_REF]Section 4.2.2] for instance), but can be weakened. In particular, we may not suppose F to be bounded from below.

One may observe that, for any natural integer n, any point x in X and any midpoint u in

x τ n +x 2 , it holds 1 2τ d 2 (x τ n , x) + 2 F (u) = 2 τ d 2 (x τ n , u) + 2 F (u) = 2 P JKO F u; x τ n , τ 2 . ( 5
)
As a consequence, another interpretation of the VIM scheme is to view the midpoint u as computed by a JKO scheme with a halved time step τ 2 and to extrapolate it as follows. Recall that a geodesic space (X , d) is called non-branching (see [START_REF] Ambrosio | A user's guide to optimal transport[END_REF]Definition 3.15] or [3, Theorem 11.2.10]) if, for any t ∈]0, 1[, a constant speed geodesic γ is uniquely determined by its initial point γ(0) and by the point γ(t). In particular, if u ∈ x+y1 2 ∩ x+y2 2 then y 1 = y 2 . By definition, the 2-extrapolate of the couple (x, u) (denoted 2u -x hereafter) is the unique point y such that u ∈ x+y 2 . On the contrary, when there exists no such point, the 2-extrapolate is defined as the point y with largest distance to x such that u is on a geodesic from x to y. When the space is non-branching and complete, using the properties of the distance between two points on a geodesic and the fact that any Cauchy sequence converges, we obtain that, for any fixed points x and u, the set {d(x, y) | ∃γ ∈ Geod x,y , u ∈ γ} is closed. The 2-extrapolate is thus well-defined and unique. It moreover satisfies d(x, u) ≤ d(x, 2 u -x) ≤ 2 d(x, u).

These considerations lead to the definition of the Extrapolated Variational Implicit Euler (EVIE) scheme, in which x τ n+1 is now the 2-extrapolate 2u -x τ n of the point u computed by the JKO scheme with time step τ 2 . Implementation-wise, the last scheme has the obvious advantage of using the JKO scheme. From a theoretical point of view, the existence and uniqueness of its solution in a non-branching space is assured under the same hypotheses that those required for the JKO scheme. While adaptations would certainly allow to define it in branching spaces, uniqueness may be lost, even when the JKO scheme possesses a unique solution at each step.

Convergence of the VIM scheme

In this section, we prove the convergence of the VIM scheme in a metric space setting by showing that a natural interpolation of its discrete solution converges to a gradient flow written in the integral form of an Energy Dissipation Inequality (EDI) (see [START_REF] Ambrosio | A user's guide to optimal transport[END_REF]Section 4.2]). More precisely, we recall that, given a functional F : (X , d) → R ∪ {+∞} with domain D(F ) = {x ∈ X , F (x) < +∞}, and a point x ∈ D(F ), the curve (x t ) t≥0 in X is a gradient flow in the EDI sense starting at x provided that it is a locally absolutely continuous curve, x 0 = x and

∀s ≥ 0, F (x s ) + 1 2 s 0 |x r | dr + 1 2 s 0 |∇F | 2 (x r ) dr ≤ F (x), (6)
a.e. t > 0, ∀s ≥ t, F (x s ) + 1 2

s t |x r | dr + 1 2 s t |∇F | 2 (x r ) dr ≤ F (x t ), (7)
where, for any point x in D(F ), the slope of F at x is

|∇F | (x) = lim sup z→x (F (x) -F (z)) + d(x, z) = max lim sup z→x F (x) -F (z) d(x, z) , 0 ,
and the metric derivative of x at r is

|x r | = lim h→0 d(x r+h , x r ) |h| . ( 8
)
The time step τ being fixed with 0 < τ < τ , we introduce a variational interpolation à la De Giorgi of the discrete solution of the VIM scheme by defining the curve (x τ t ) t≥0 as follows:

-x τ 0 = x, -for n ∈ N, x τ (n+1)τ ∈ argmin y∈X P V IM F (y; x τ nτ , τ ), -for n ∈ N and t ∈]nτ, (n + 1)τ [, x τ t ∈ argmin y∈X P V IM F (y; x τ nτ , t -nτ ).
For such a map, we define the discrete speed Dsp τ : [0, +∞) → [0, +∞) by

Dsp τ t = d(x τ nτ , x τ (n+1)τ ) τ for t in (nτ, (n + 1)τ ),
and the discrete slope Dsl τ : [0, +∞) → [0, +∞) by

Dsl τ t = d(x τ nτ , x τ t ) t -nτ
for t in (nτ, (n + 1)τ ).

In order to avoid unnecessary technicalities in the forthcoming analysis, we will suppose from now on that midpoints are unique, that is

∀(x, y) ∈ X 2 , the set x + y 2 is a singleton. ( 9
)
We finally introduce a notion of slope of F at the midpoint of (x, y) by setting

∇ M F (x, y) = lim sup z→y F x+y 2 -F x+z 2 + d x+y 2 , x+z 2 .
Theorem 2.2 Let T > 0 be fixed and (X , d) be a Polish metric space satisfying hypotheses (4) and [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF]. We moreover assume that (i) F is lower semicontinuous, bounded from below, and such that

∀r > 0, ∀c ∈ R, ∀x ∈ X the set {y ∈ X | F (y) ≤ c, d(x, y) ≤ r} is compact, ( 10
) (ii) F has the following continuity property if x n → x, and sup{|∇F |(x n ), E(x n )} < ∞ then F (x n ) → F (x); ( 11
) (iii) ∇ M F : D(F ) × D(F ) → [0, ∞)
∇ M F (x, x) ≤ lim inf n→∞ ∇ M F (x n , y n ),
(iv) if any two of the elements x, y, x+y 2 belong to D(F ), then the third also does and

F (x) + F (y) -2 F ( x+y 2 ) d 2 (x, y) ≤ H, ( 12
)
where H is a constant independent of x and y.

Then, for some τ > 0, the set of curves {(x τ t ) t∈[0,T ] ; 0 ≤ τ ≤ τ } is relatively compact (with respect to the local uniform convergence) and any limit curve is a gradient flow in the EDI formulation (6)-( 7).

Remark 2 Assumptions (i) and (ii) are classical and used in [START_REF] Ambrosio | A user's guide to optimal transport[END_REF]Assumption 4.13], whereas (iii) is a generalization of the second assumption in [START_REF] Ambrosio | A user's guide to optimal transport[END_REF]Assumption 4.13]. Item (iv) is however a regularity property specific to our setting; in particular it is satisfied when both F and -F are λ-convex in the sense of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Section 2.4 page 49]. On the other hand, it can be weakened to

F (x) + F (y) -2 F ( x+y 2 ) = o(d(x, y))
. Proof We will follow closely and adapt where necessary the method of proof given in [2, Subsection 4.2.2]. The proof is divided into several steps, starting with the derivation of some key properties of the variational interpolation (x τ t ) t≥0 introduced above. (I) First, we shall show that, the positive real number τ and the natural integer n being fixed, the

function (0, 1] θ → 1 2θτ d 2 (x τ nτ , x τ (n+θ)τ ) + 2 F x τ nτ +x τ (n+θ)τ 2
is locally Lipschitz and that its derivative is given by

d dθ 1 2θτ d 2 (x τ nτ , x τ (n+θ)τ ) + 2 F x τ nτ + x τ (n+θ)τ 2 = - 1 2θ 2 τ d 2 (x τ nτ , x τ (n+θ)τ ). (13)
Indeed, let 0 < θ 0 < θ < θ 1 ≤ 1. By the definition of the variational interpolation, one has

1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ0)τ ) + 2 F x τ nτ + x τ (n+θ0)τ 2 ≤ 1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ1)τ ) + 2 F x τ nτ + x τ (n+θ1)τ 2 , so that 1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ0)τ ) + 2 F x τ nτ + x τ (n+θ0)τ 2 - 1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ1)τ ) -2 F x τ nτ + x τ (n+θ1)τ 2 ≤ (θ 1 -θ 0 )τ 2θ 0 θ 1 τ 2 d 2 (x τ nτ , x τ (n+θ1)τ ).
Arguing symmetrically, one also has

1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ0)τ ) + 2 F x τ nτ + x τ (n+θ0)τ 2 - 1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ1)τ ) -2 F x τ nτ + x τ (n+θ1)τ 2 ≥ (θ 1 -θ 0 )τ 2θ 0 θ 1 τ 2 d 2 (x τ nτ , x τ (n+θ0)τ ),
and the function is thus locally Lipschitz. Moreover, dividing by θ 1 -θ 0 the last two inequalities and passing to the limits θ 0 ↑ θ and θ 1 ↓ θ, we obtain the proposed expression for the derivative.

(II) Next, we establish that, with the same notations as above,

θ → d(x τ nτ , x τ (n+θ)τ ) is non decreasing, θ → F x τ nτ +x τ (n+θ)τ 2
is non increasing and that it holds

∀θ ∈ (0, 1], ∇ M F (x τ nτ , x τ (n+θ)τ ) ≤ d(x τ nτ , x τ (n+θ)τ ) θτ . ( 14
)
Let 0 < θ 0 < θ 1 ≤ 1. By the respective minimality properties of x τ (n+θ0)τ and x τ (n+θ1)τ , we have

1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ0)τ ) + 2 F x τ nτ + x τ (n+θ0)τ 2 ≤ 1 2θ 0 τ d 2 (x τ nτ , x τ (n+θ1)τ ) + 2 F x τ nτ + x τ (n+θ1)τ 2 , 1 2θ 1 τ d 2 (x τ nτ , x τ (n+θ1)τ ) + 2 F x τ nτ + x τ (n+θ1)τ 2 ≤ 1 2θ 1 τ d 2 (x τ nτ , x τ (n+θ0)τ ) + 2 F x τ nτ + x τ (n+θ0)τ 2 .
Adding the last two inequalities, we get

1 θ 0 - 1 θ 1 d 2 (x τ nτ , x τ (n+θ0)τ ) -d 2 (x τ nτ , x τ (n+θ1)τ ) ≤ 0, so that d(x τ nτ , x τ (n+θ0)τ ) ≤ d(x τ nτ , x τ (n+θ1)τ ).
From this, we now have

1 2θ 1 τ d 2 (x τ nτ , x τ (n+θ0)τ ) + 2 F x τ nτ + x τ (n+θ1)τ 2 ≤ 1 2θ 1 τ d 2 (x τ nτ , x τ (n+θ1)τ ) + 2 F x τ nτ + x τ (n+θ1)τ 2 ,
which implies, using a previous inequality, that F

x τ nτ +x τ (n+θ 1 )τ 2 ≤ F x τ nτ +x τ (n+θ 0 )τ 2 .
Finally, by the definition of x τ (n+θ)τ , one has

∀y ∈ X , 1 2θτ d 2 (x τ nτ , x τ (n+θ)τ ) + 2 F x τ nτ + x τ (n+θ)τ 2 ≤ 1 2θτ d 2 (x τ nτ , y) + 2 F x τ nτ + y 2 ,
so that, by the very definition of the set of midpoints between two points,

2 θτ d 2 x τ nτ , x τ nτ + x τ (n+θ)τ 2 + 2 F x τ nτ + x τ (n+θ)τ 2 ≤ 2 θτ d 2 x τ nτ , x τ nτ + y 2 + 2 F x τ nτ + y 2 .
Hence, we get Using the reverse triangle inequality and taking the upper limit as y tends to x τ (n+θ)τ , we obtain that

F x τ nτ +x τ (n+θ)τ 2 -F x τ
∇ M F (x τ nτ , x τ (n+θ)τ ) ≤ 1 θτ lim sup y→x τ (n+θ)τ d x τ nτ , x τ nτ + x τ (n+θ)τ 2 + d x τ nτ , x τ nτ + y 2 = 2 θτ d x τ nτ , x τ nτ + x τ (n+θ)τ 2 = 1 θτ d(x τ nτ , x τ (n+θ)τ ).
(III) We now prove, that for any integers n and m such that 0 ≤ n < m one has,

1 2 mτ nτ (Dsp τ r ) 2 dr + 2 m-1 k=n F x τ kτ + x τ (k+1)τ 2 -2 m-1 k=n F (x τ kτ ) = - 1 2 mτ nτ (Dsl τ r ) 2 dr. ( 15
)
From ( 13), we infer that

∀k ∈ N, 1 2τ d(x τ kτ , x τ (k+1)τ ) + 2 F x τ kτ + x τ (k+1)τ 2 -2 F (x τ kτ ) = - 1 0 d 2 (x τ kτ , x τ (k+θ)τ ) 2θ 2 τ dθ,
which yields, by a change of variable and the respective definitions of the discrete speed and slopes, Therefore, for t ≤ T and T = nτ , it holds

∀k ∈ N, ∀t ∈ (kτ, (k + 1)τ ), τ 2 (Dsp τ t ) 2 + 2 F x τ kτ + x τ (k+1)τ 2 -2 F (x τ kτ ) = - 1
d 2 (x τ t , x τ 0 ) ≤ T 0 Dsp τ r dr 2 ≤ T T 0 (Dsp τ r ) 2 dr ≤ 2T 1 -2Hτ (F (x τ 0 ) -inf F ), ( 18
)
which shows that, for any T , the set {x τ t } 0≤t≤T is bounded (uniformly with respect to τ ). Using [START_REF] Kinderlehrer | Approximation of parabolic equations using the Wasserstein metric[END_REF], we conclude that it is relatively compact. Moreover, further exploitation of inequality (17) shows that for t = nτ < mτ = s, one has

d 2 (x τ t , x τ s ) ≤ s t Dsp τ r dr 2 ≤ 2(s -t) 1 -2Hτ (F (x τ 0 ) -inf F ), (19)
which gives equicontinuity and, by the Arzelà-Ascoli theorem, the relative compactness of the set of curves {(x τ t ) 0≤t≤T ; 0 ≤ τ ≤ τ } with respect to the local uniform convergence. (V) We finally pass to the limit. Let (τ n ) n∈N be a decreasing sequence tending to zero such that (x τn t ) converges to a limit curve x t locally uniformly as n tends to infinity. Using inequality (19), this curve is absolutely continuous and satisfies

∀0 ≤ t < s, s t |x r | dr ≤ lim inf n→+∞ s t (Dsp τn r ) 2 dr. ( 20
)
In addition, set t and, ∀n ∈ N, let N (n) ∈ N be such that N (n)τ n < t ≤ (N (n) + 1)τ n , so that N (n)τ n tends to t as n tends to infinity. Using the assumption (iii) on the lower semicontinuity of ∇ M F and ( 14), we have on the one hand

∇ M F (x t , x t ) ≤ lim inf n→+∞ ∇ M F (x τn N (n)τn , x τn t ) ≤ lim inf n→+∞ d(x τn N (n)τn , x τn t ) t -N (n)τ n = lim inf n→+∞ Dsl τn t .
On the other hand, it follows from ( 12) that

∀t > 0, ∇ M F (x t , x t ) = lim sup z→xt F (x t ) -F xt+z 2 + d x t , xt+z 2 = lim sup z→xt 2 F (x t ) -F xt+z 2 + d(x t , z) = lim sup z→xt (2F (x t ) -(F (x t ) + F (z))) + d(x t , z) = lim sup z→xt (F (x t ) -F (z)) + d(x t , z) = |∇F | (x t ).
By Fatou's lemma and (17), we thus obtain that, for

τ n ≤ 1/(2H), ∀t < s, s t |∇F | 2 (x r ) dr ≤ lim inf n→+∞ s t (Dsl τn r ) 2 dr ≤ 2 (F (x τ 0 ) -inf F ). ( 21
)
Passing to the limit in (17) using ( 20) and (21) for t = 0 and s arbitrary yields [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF]. Note that the same technique does not work for t > 0, since F is not necessarily continuous. However, from (21), it follows that, for almost every t > 0 (taken as fixed from now on), there exists a subsequence τ n k with sup k |∇F | (x

τn k t ) < ∞.
By assumption [START_REF] Mirebeau | Discretization of the 3D Monge-Ampere operator, between wide stencils and power diagrams[END_REF], we then obtain (7).

Second-order schemes for the Fokker-Planck equation as a gradient flow in the Wasserstein space

As a complement to the above results, we consider in this final section a particular gradient flow in the one-dimensional case, for which the results of Section 2.2 do not apply but which provides evidence that the VIM and EVIE schemes have good numerical properties. We refer to [START_REF] Mirebeau | Discretization of the 3D Monge-Ampere operator, between wide stencils and power diagrams[END_REF][START_REF] Benamou | Discretization of functionals involving the Monge-Ampère operator[END_REF] for possible approaches in higher dimensions compatible with these schemes.

The space X is now set to be the Wasserstein space of order 2, that is the set of probability measures on (R, B(R)) with finite second-order moment, P 2 (R), endowed with the 2-Wasserstein distance W 2 (see [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]Chapter 5] or [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF][START_REF] Villani | Optimal transport. Old and new[END_REF]). Note that, from now on, the variable x will denote the position over the real line.

Consider a probability measure ν in P 2 (R). When ν is absolutely continuous with respect to the Lebesgue measure, with probability density ρ, the functional F is defined by

F (ν) = E(ρ) + S(ρ) with E(ρ) = R V (x)ρ(x) dx and S(ρ) = σ 2 2 R ρ(x) log(ρ(x)) dx. ( 22
)
Here, the term E(ρ) corresponds to a potential energy, the function V being the potential in question, and the term S(ρ) corresponds to an internal energy, the scalar σ being a positive real number. Otherwise, we set F (ν) = +∞. Correspondingly, if the measures ν 1 and ν 2 are absolutely continuous with respect to the Lebesgue measure, with respective densities ρ 1 and ρ 2 , we denote by W 2 (ρ 1 , ρ 2 ) the Wasserstein distance between ν 1 and ν 2 and by Geod ρ1,ρ2 the set of geodesics connecting ν 1 to ν 2 .

For any smooth potential V , it is well-known (see [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF]) that the gradient flow in P 2 (R) of the functional F is a curve t → ν(t) in P 2 (R), such that, at almost any time t, the measure ν(t) is absolutely continuous with respect to the Lebesgue measure, with probability density ρ(t, •), and that ρ is solution of the Fokker-Planck partial differential equation of the form

∂ρ ∂t (t, x) = ∂ ∂x [V (x)ρ(t, x)] + σ 2 2 ∂ 2 ρ ∂x 2 (t, x), ( 23
)
which is itself related to the following stochastic differential equation

dX(t) = -V (X(t)) dt + σ dW (t), (24)
the stochastic process W being a standard one-dimensional Wiener process.

In what follows, we consider quadratic potentials of the form V (x) = θ (x-µ) 2

2

, where θ and µ are given constants. We denote by N (a, b 2 ) the normal (Gaussian) distribution with mean a and variance b 2 and also, by abusing the notation, the associated density (with respect to the Lebesgue measure)

ρ(x) = 1 √ 2πb 2 e -(x-a) 2 2b 2 . Proposition 3.1 If ρ(0, •) = N (µ 0 , σ 2 0 )
, any intermediary state of the semi-discrete VIM and EVIE schemes has a Gaussian distribution and the error of both schemes has order O(τ 2 ). Proof We will prove the assertion for the VIM scheme, the proof for the EVIE one being similar. The potential V being quadratic, the solution of ( 24) is an Ornstein-Uhlenbeck process, given by X(t) = X(0) e -θt + µ(1 -e -θt ) + σ √ 2θ e -θt W (e 2θt -1). In particular, the exact evolution from ρ n = N µ n , σ 2 n after τ time units is

ρ exact n+1 = N µ n e -θτ + µ(1 -e -θτ ), e -2θτ σ 2 n + σ 2 2θ 1 -e -2θτ .
On the other hand, ρ n (x)dx being absolutely continuous with respect to the Lebesgue measure, the set Geod ρn,ρ is a singleton for any ρ in P 2 (R) (and so is the set ρn+ρ 2 ). Hence, the unique geodesic γ ρ in Geod ρn,ρ is determined by a map T : R → R (which is the gradient of a convex function, see [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF][START_REF] Villani | Optimal transport. Old and new[END_REF]), γ ρ (t) being the push-forward of ρ n by x → (1 -t) x + t T (x). The minimization in (2) can thus be expressed in terms of T , for instance 1 2

d 2 (ρ n , ρ) = R (T (x)-x) 2 2
ρ n (x) dx. After some tedious (but rather straightforward) computations, one obtains that the unique element ρ

V IM n+1 minimizing 1 2τ d 2 (ρ n , ρ) + 2F ρn+ρ 2 also has a normal distribution, that is ρ V IM n+1 = N µ n+1 , σ 2 n+1 , with µ n+1 = 1 + θτ 2 -1 µ n 1 -θτ 2 + µθτ 2 and σ n+1 = 1 + θτ 2 -1 σ 2 n + σ 2 τ 1 + θτ 2 -σnθτ 2 .
Consequently, ρ V IM n+1 is an approximation of ρ exact n+1 exact to the second order in τ and, by the usual Gronwall inequalities, the error of the VIM scheme is globally of order O(τ 2 ).

We performed numerical simulations to support these results. Roughly speaking, we employed a continuous piecewise affine discretization for the cumulative distribution functions of elements in P 2 (R), which amounts to replace the space P 2 (R) by a discrete one, denoted P 2 (R) m and defined as follows. Let M ∈ N * and m = (m 1 , . . . , m M ) be an ordered mass distribution such that m k > 0 for any integer k in {1, . . . , M } and M k=1 m k = 1, fixed once and for all. An element in P 2 (R) m with density ρ X is then represented by a vector X = (x 0 , . .

. , x M ) ∈ R M +1 , such that x 0 < x 1 < • • • < x M -1 < x M , by setting ρ X constant and equal to m k x k+1 -x k on the segment [x k , x k+1 ).
Approximating the integral defining the discrete potential energy E(ρ X ) by using the composite Simpson quadrature rule (see [START_REF] William | Numerical recipes: the art of scientific computing[END_REF]Chapter 4

, Subsection 4.1.3]), one obtains E m (X) = M k=0 m k 6 V (x k ) + 4V x k + x k+1 2 + V (x k+1 ) . ( 25
)
On the other hand, the discrete internal energy S(ρ X ) can be computed exactly and has the simple form

S m (X) = M k=0 m k x k+1 -x k (x k+1 -x k ) log m k x k+1 -x k = M k=0 m k log(m k ) - M k=0 m k log(x k+1 -x k ). (26)
To compute the Wasserstein distance W 2 between the discrete densities ρ X1 and ρ X2 , we note that the map T is the unique piecewise linear function on any interval [x 1 k , x 1 k+1 ], which associates x 1 k with x 2 k and x 1 k+1 with x 2 k+1 , respectively. Straightforward computations then show that the squared distance

W 2 2 (ρ X1 , ρ X2 ) = R (T (x) -x)ρ X1 (x) dx reduces to W 2 2 (ρ X1 , ρ X2 ) = 1 3 M k=0 m k x 2 k -x 1 k 2 + x 2 k+1 -x 1 k+1 x 2 k -x 1 k + x 2 k+1 -x 1 k+1 2 . ( 27
)
For the numerical results presented in Figure 1, we have set σ = 1, θ = 1 2 , µ = 5, M = 32, a uniform mass step m k = 1 M , the final time equal to 1, and the projection on the discrete space of a standard Gaussian distribution as the initial datum. Second-order in time convergence is confirmed for both schemes.

The internal energy S(ρ X ) constitutes a barrier enforcing the constraint x k < x k+1 . As such the (continuous in time) dynamics of the probability density can also be viewed as a dynamics of the vector X which follows a, cumbersome but regular, system of ordinary differential equations. The proposed schemes provide a consistent and second-order in time discretization of this system, and we have obtained the following result. and of the VIM and EVIE schemes (solid lines) as functions of the length of the time step τ . The error is the W 2 distance between the numerical solution computed by the schemes on grids containing respectively 4, 7, 12, 20, 33, 54, 90 and 148 time steps and a numerical reference solution obtained using the VIM scheme on a finer grid (244 time steps). The solutions of the VIM and EVIE schemes are indistinguishable to machine precision. We observe that any of the second-order schemes on a grid with four steps is as accurate as the standard JKO scheme on a grid with 90 steps. Similarly, the VIM or EVIE scheme on a grid with seven steps is as good as the JKO scheme on a grid with 148 steps. The numerical estimation of the order of convergence of the JKO scheme and of the VIM and EVIE schemes are given in the legend. Right: CPU times (in seconds) with respect to the number of time steps (the implementation uses MATLAB Version: 8.6.0.267246 (R2015b) on a 8 core Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz HP Thinkstation.).

2 (

 2 The result follows from summing from k = n to m -1. (IV) Combining relation (15) with hypothesis[START_REF] William | Numerical recipes: the art of scientific computing[END_REF]

Corollary 3 . 1

 31 The scheme (2) discretized using (25), (26), and (27) converges at second order in time to the exact solution of the semi-discrete gradient flow of the functional F in the Wasserstein space of order 2.10 -3 10 -2 10 -1

Figure 1 .

 1 Figure1. Results of numerical simulation for the case considered in Section 3. Left: errors of the JKO scheme (dotted line) and of the VIM and EVIE schemes (solid lines) as functions of the length of the time step τ . The error is the W 2 distance between the numerical solution computed by the schemes on grids containing respectively 4, 7, 12, 20, 33, 54, 90 and 148 time steps and a numerical reference solution obtained using the VIM scheme on a finer grid (244 time steps). The solutions of the VIM and EVIE schemes are indistinguishable to machine precision. We observe that any of the second-order schemes on a grid with four steps is as accurate as the standard JKO scheme on a grid with 90 steps. Similarly, the VIM or EVIE scheme on a grid with seven steps is as good as the JKO scheme on a grid with 148 steps. The numerical estimation of the order of convergence of the JKO scheme and of the VIM and EVIE schemes are given in the legend. Right: CPU times (in seconds) with respect to the number of time steps (the implementation uses MATLAB Version: 8.6.0.267246 (R2015b) on a 8 core Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz HP Thinkstation.).

Acknowledgements

G. Turinici acknowledges support from the Agence Nationale de la Recherche (ANR), projects EMAQS ANR-2011-BS01-017-01 and CINE-PARA 2015-19.


Andrea Brini
email: a.brini@imperial.ac.uk

Vafa Famously

Quantum topology and curve counting invariants

published or not. The documents may come

1. Introduction 1 2. A 0-dimensional aperçu: matrix models, the topological recursion and enumerative geometry 2 3. The GOV correspondence for S 3 6 4. The GOV correspondence for Clifford-Klein 3-manifolds 9 5. Conclusions 14 References 15 Contents

Introduction

In a series of works [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF][START_REF] Ooguri | Knot invariants and topological strings[END_REF][START_REF]World sheet derivation of a large N duality[END_REF], Gopakumar, Ooguri and Vafa postulated and gave strong evidence for an identity between two rather different physical theories: on the one hand, U(N ) Chern-Simons theory on S 3 [START_REF] Witten | Quantum field theory and the Jones polynomial[END_REF], and on the other, topologically A-twisted string theory on the resolved conifold, Tot(O ⊕2 P 1 (-1)). From the physics side, such identification is a concrete example of 't Hooft's idea [START_REF] Hooft | A Planar Diagram Theory for Strong Interactions[END_REF] that perturbative gauge theories with classical gauge groups in the 1/N expansion should be equivalent to some first-quantised string theory on a given background. The topological nature of both Chern-Simons theory and the Amodel topological string allow for detailed checks of the correspondence, which might be regarded as a simplified setting for gauge/string dualities for the type II superstring, such as the AdS/CFT correspondence [START_REF] Martin | The large N limit of superconformal field theories and supergravity[END_REF].

Mathematically, the implications of the Gopakumar-Ooguri-Vafa (GOV) correspondence are perhaps even more striking: the correspondence ties together, in a highly non-trivial way, two different theories of geometric invariants having a priori little resemblance to each other. Witten's original intuition [START_REF] Witten | Quantum field theory and the Jones polynomial[END_REF] asserts that Chern-Simons observables give rise to topological invariants of 3-manifolds and links therein, in view of the Schwarz-type topological invariance of the quantum theory and its relation to rational conformal field theory. This leads in particular to the skein relations typical of knot invariants such as the HOMFLY-PT and Kaufmann invariants, and more generally, to the Reshetikhin-Turaev invariants arising in the representation theory of quantum groups and modular tensor categories [START_REF] Reshetikhin | Invariants of three manifolds via link polynomials and quantum groups[END_REF][START_REF] Yu | Ribbon graphs and their invariants derived from quantum groups[END_REF]. The A-type topological string side, in turn, also enjoys a mathematical definition -of a quite different flavour. The logarithm of the partition function for a given Calabi-Yau target X is a formal generating function of various virtual counts of curves in X, either via stable maps [START_REF] Kontsevich | Enumeration of rational curves via Torus actions[END_REF] or ideal sheaves [START_REF] Donaldson | Gauge theory in higher dimensions[END_REF]. In particular, the GOV correspondence asserts that the asymptotic expansion of the Reshetikhin-Turaev invariant associated to the quantum group U q (sl N ) at large N and fixed q N equates the formal Gromov-Witten potential of X in the genus expansion. This has a B-model counterpart due to recent developments in higher genus toric mirror symmetry a [START_REF] Bouchard | Remodeling the B-model[END_REF][START_REF]Computation of Open GromovWitten Invariants for Toric CalabiYau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture[END_REF], where the same genus expansion can be phrased in terms of the Eynard-Orantin topological recursion [START_REF] Eynard | Invariants of algebraic curves and topological expansion[END_REF] on the Hori-Iqbal-Vafa mirror curve of X [START_REF] Hori | D-branes and mirror symmetry[END_REF][START_REF] Hori | Mirror symmetry[END_REF].

The GOV correspondence has had a profound impact in both communities involved. In Gromov-Witten / Donaldson-Thomas theory, it has laid the foundations of the use of large N dualities to solve the topological string on toric backgrounds [START_REF] Aganagic | The topological vertex[END_REF][START_REF] Aganagic | All loop topological string amplitudes from Chern-Simons theory[END_REF][START_REF] Maulik | Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds[END_REF] and to obtain all-genus results that went well beyond the existing localisation computations at the time, as well as some remarkable results for the intersection theory on moduli spaces of curves [START_REF] Mariño | Framed knots at large N[END_REF] and, via the relation of Chern-Simons theory to random matrices, an embryo of the remodeling proposal [START_REF] Bouchard | Remodeling the B-model[END_REF][START_REF] Marino | Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants[END_REF][START_REF] Mariño | Open string amplitudes and large order behavior in topological string theory[END_REF]. In the other direction, the integral structure of BPS theory invariants leads to highly non-trivial constraints for the structure of quantum knot invariants [START_REF] Labastida | Knots, links and branes at large N[END_REF][START_REF] Liu | Proof of the Labastida-Mario-Ooguri-Vafa conjecture[END_REF][START_REF] Marino | String theory and the Kauffman polynomial[END_REF].

Since the original correspondence of [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF][START_REF] Ooguri | Knot invariants and topological strings[END_REF] was confined to the case where the gauge group is the unitary group U (N ), the base manifold is S 3 , and the knot is the trivial knot, a natural question to ask is whether a similar connection could be generalized to more general classical gauge groups [START_REF] Bouchard | Counting higher genus curves with crosscaps in Calabi-Yau orientifolds[END_REF][START_REF]Topological open string amplitudes on orientifolds[END_REF]89], knots other than the unknot [START_REF] Brini | Torus knots and mirror symmetry[END_REF][START_REF] Labastida | Polynomial invariants for torus knots and topological strings[END_REF][START_REF] Labastida | Knots, links and branes at large N[END_REF] (see also [START_REF] Aganagic | Topological Strings, D-Model, and Knot Contact Homology[END_REF] for a significant generalisation, in a rather different setting), as well as categorified/refined invariants of various types [START_REF] Aganagic | Knot Homology and Refined Chern-Simons Index[END_REF][START_REF] Gukov | Khovanov-Rozansky homology and topological strings[END_REF]. A further natural direction would be to seek the broadest generalisation of the correspondence in its original form beyond the basic case of the threesphere; this would require to provide a description of the string dual of Chern-Simons both in terms of Gromov-Witten theory and the Eynard-Orantin theory on a specific spectral curve setup. This program was initiated in [START_REF] Aganagic | Matrix model as a mirror of Chern-Simons theory[END_REF] (see also [START_REF] Brini | Chern-Simons theory on L(p, q) lens spaces and Gopakumar-Vafa duality[END_REF][START_REF] Halmagyi | Large N duality, lens spaces and the Chern-Simons matrix model[END_REF][START_REF] Halmagyi | The spectral curve of the lens space matrix model[END_REF]) for the case of lens spaces, and its most general frame of validity has been recently described in [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF]. This short survey is meant as an overview of the results of [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF], putting them into context and highlighting their implications for quantum invariants and Calabi-Yau enumerative geometry alike. It is structured as follows: we first give a lightning review of the three-way relation between matrix models, intersection theory on M g,n , and the Eynard-Orantin recursion. We then move on to discuss the review the uplift of these notions to the GOV correspondence for S 3 . Finally, we discuss its generalisation to the setting of Clifford-Klein 3-manifolds and outline the proof of the B-model side of the correspondence for the case of the Le-Murakami-Ohtsuki invariant [START_REF] Thang | On a universal perturbative invariant of 3-manifolds[END_REF].

Acknowledgements. I would like to thank my collaborators Gaëtan Borot, Bertrand Eynard, Luca Griguolo, Marcos Mariño, Domenico Seminara and Alessandro Tanzini for sharing their insights and educating me on various aspects of the Chern-Simons/topological strings correspondence as well as for most enjoyable collaborations on this topic. This paper is a write-up of material presented at a talk at the 2016 AMS Von Neumann Symposium on Topological Recursion and its Inuence in Analysis, Geometry, and Topology, July 48, 2016, at Charlotte (NC); I wish to express my thanks to Motohico Mulase and Chiu-Chu Melissa Liu for the wonderful scientific environment at the workshop and for giving me the opportunity to present my work. This research was partially supported by a fellowship under the ERC-2015-CoG "GWT".

A 0-dimensional aperçu: matrix models, the topological recursion and enumerative geometry

As a warm-up for the description of the triangle of relations linking 3D TQFT, curve-counting invariants and the topological recursion, let us consider the much more basic setting of 0D QFTs -namely, matrix models. For any N, r ∈ Z + , let U (N ) denote the rank-N unitary group, H N be its Lie algebra of hermitian matrices, and for t ∈ C , x in a formal neighbourhood of the origin of C r we let dµ N (x) be a family of formally finite Ad-invariant measures, absolutely continuous with respect to the standard gaussian measure dµ

(0) N = dµ N (0) on H N R N 2 .
The typical setup here is

(1) dµ N (x) = N N N i=1 dM ii N i<j dReM ij dImM ij e -N 2t TrM 2 + j xj TrM j = N N dµ (0) N e j xj TrM j
where N N ∈ C , and the exponential deformation in the last equality should be treated as formal. For fixed t, we will denote by Z N (x) the formal total mass of dµ N (x),

(2)

Z N (x) = H N dµ N (x) ∈ C[[x]],
and for any collection of positive integers k 1 , . . . , k h , h ≥ 1, we write

(3) W k1,...,k h ;N (x) ∂ i ki x k 1 ,...,x k h log Z N (x) ∈ C[[x]
] for the normalised cumulants. Eqs. ( 1) and ( 2) define respectively a formal hermitian 1-matrix model and its partition function, and we are particularly interested in their formal behaviour as N → ∞. It is a matter of book-keeping in the application of Wick's theorem to Eqs. ( 2) and (3) to show that both ln Z N and W k1,...,km have a formal connected ribbon graph (fat-graph) expansion [START_REF] Brezin | Planar Diagrams[END_REF][START_REF] Hooft | A Planar Diagram Theory for Strong Interactions[END_REF] (4) ln

Z N (x) = g≥0 N 2-2g F g (x), (5) W k1,...,k h ;N (x) = g≥0 N 2-2g-h W g,h;k1,...,k h (x).
The 1/N expansion of Eqs. ( 4) and ( 5) lends itself to two interpretations.

• The first one stems from the observation that, as the dual of a ribbon graph is a polygonulation of a connected, oriented topological 2-manifold, this 1/N expansion can be regarded as a genus expansion in a sum over random polygonulations of a Riemann surface. The 1/N expansion gives then an answer to an enumeration problem: it is easy to show in particular, when

N N = (t/N ) N (N +1)/2 (π/N ) -N 2 /2 2 -N
, that the coefficients of the formal expansion in x have an enumerative meaning as a count of topologically inequivalent dual graphs. One consequence is that, as (metric) ribbon graphs can be used as a means to describe simplicial decompositions of the moduli space M g,n of pointed Riemann surfaces via the Strebel correspondence, the enumerative content of Eq. ( 4) provides a combinatorial answer to topological questions on this moduli space [START_REF] Distler | A Critical matrix model at c = 1[END_REF][START_REF] Harer | The cohomology of the moduli space of curves[END_REF][START_REF] Penner | Perturbative series and the moduli space of Riemann surfaces[END_REF].

• The second (physical) interpretation comes from a special case of the observation, due to 't Hooft, that the perturbative 1/N expansion of a U (N ) gauge theory with only adjoint fields takes the shape of the expansion in the string coupling of a first quantised closed string theory, possibly with probe branes insertions [START_REF] Hooft | A Planar Diagram Theory for Strong Interactions[END_REF]; in a sense the observation above that the enumeration problem of fatgraphs is related to intersection-theoretic problems on moduli spaces of curves is the most basic example of this phenomenon. Indeed, 1-matrix models are possibly the most basic avatars of such QFTs, and it is only natural to ask what is the string dual of the formal 1-matrix model? This is a sharp, and already non-trivial question in the context of gauged matrix models, as can be seen in the following example.

Example 2.1. Take the simplest example of a gaussian measure dµ N (x) = dµ

(0)
N , with a normalisation factor given by ( 6)

N N [Vol g (U (N ))] -1
where the metric g is the Ad-invariant metric induced by the Killing form on H N . The gaussian integration is trivial, and yields (7)

H N dµ (0) N = 2πt N N 2 /2
The normalisation factor in Eq. ( 6) is less trivially computed as a product of volumes of spheres [START_REF] Macdonald | The volume of a compact Lie group[END_REF] and it takes the form of a Barnes double gamma function:

(8) Vol g (U (N )) = (2π) N (N +1)/2 G 2 (N + 1)
where

G 2 (x + 1) = Γ(x)G 2 (x), G 2 (1) = 1.
Computing the large N asymptotics of ln Z N is an exercise in the use of the all-order Stirling formula for the Gamma function; the result is

(9) ln Z N (t) = N 2 2 log(t) - 3 2 - 1 12 log N + ζ (-1) + g≥2 B 2g 2g(2g -2) N 2-2g .
We point out three features of this example.

i. In this case we can give a full answer to the question of finding a string dual for the matrix model. Indeed, Eq. ( 9) coincides with the perturbative expansion of the c = 1 string theory at the self-dual radius [START_REF] Ghoshal | c = 1 string as the topological theory of the conifold[END_REF][START_REF] Gross | One-dimensional string theory on a circle[END_REF][START_REF] Periwal | Topological closed string interpretation of Chern-Simons theory[END_REF], upon identifying the string coupling constant g s and the cosmological constant µ as g s = t/N , t = iµ.

ii. The coefficients of N 2-2g in the genus expansion of Eq. ( 9) have a particular geometrical significance of their own: by the Harer-Zagier formula [START_REF] Harer | The Euler characteristic of the moduli space of curves[END_REF], they coincide with the Euler characteristic of the moduli space of genus g-curves with no marked points and g > 1, ( 10)

χ(M g,0 ) = B 2g 2g(2g -2) = (-1) g-1 2 2g-1 (2g -1)(2g -3)! 2 2g-1 -1 Mg,1 ψ 2g-2 1 λ g
This is perhaps the simplest setting where an answer to a topological problem on the moduli space of Riemann surfaces (respectively, in the second equality, an intersection theoretic problem on M g,n ) is encoded into a formal 1-matrix model 1 . This is not an accident, and the rest of this paper will be devoted to non-trivial extensions of this phenomenon.

iii. A specular (in a precise sense) point of view to the previous statement is given by the study of the large N expansion in Eq. ( 9) using loop equations [START_REF] Brezin | Planar Diagrams[END_REF][START_REF] Migdal | Loop Equations and 1/N Expansion[END_REF]; an infinite set of differential constraints on the cumulants of Eq. (3) (see [START_REF] Eynard | Random matrices[END_REF][START_REF] Marino | Chern-Simons theory, matrix models, and topological strings[END_REF] for reviews). Its upshot in our case is that the planar free energy, F 0 , in Eq. ( 4) can be recovered as an integral transform of Wigner's semi-circle function,

F 0 (t) = -t 2 √ t 0 z 2 ρ(z)dz, ρ(z) 1 2πi z 2 -4t. ( 11
)
Equivalently, for t ∈ C consider the spectral curve (C t , dλ t ) given by the smooth, genus zero affine plane curve C t = {(z, y) ∈ C 2 |y 2 = z 2 -4t} endowed with the meromorphic differential dλ t = ydz. Then F 0 (t) is recovered from the rigid special Kähler geometry relations [START_REF] Strominger | Special geometry[END_REF] (12)

t = 1 2πi A dλ, ∂F 0 ∂t = 1 2 B dλ,
where the integrals over

A ∈ H 1 (C t , Z), B ∈ H BM 1 (C t , Z) denote respectively a contour integral encircling the cut [-2 √ t, 2 √ t]
of y(z) in the z plane, and the sum of principal value integrals from -∞ -to -2 √ t and from -2 √ t to -∞ + ; here -∞ ± refers to the two pre-images of z = -∞ under the branched covering map z :

C t → C.
By the looks of it, Point ii) above relates the formal 1/N expansion of the gauged gaussian matrix model to the computation of a particular class of intersection numbers on the moduli space of curves; in physics language, these are a subset of observables in a simple example of an A-type topologically twisted string theory 2 . Point iii) does the same thing (at the leading order) for something superficially rather different: special geometry governs the dependence on complex (vector) moduli of the prepotential in type IIB compactification, and is precisely what is captured by the planar limit of a B-type topologically twisted string theory. This can actually be made more precise: it was shown in [START_REF] Ghoshal | c = 1 string as the topological theory of the conifold[END_REF] that the c = 1 string at the self-dual 1 The distinction between gauged and ungauged matrix models is somewhat in the eye of the beholder: the same result would be obtained via the large N study of the (ungauged) Penner matrix model in a double-scaling limit; see [START_REF] Distler | A Critical matrix model at c = 1[END_REF]. 2 In particular, these numbers are closely related to the equivariant Gromov-Witten theory of the affine line -a deformation of topological gravity by linear insertions of Chern classes of the Hodge bundle.

radius reproduces the genus expansion of the topological A-model on the singular conifold, which is its own self-mirror on the B-side.

A-model: intersection theory on M g,n .

The triangle of relations we found in this example between formal matrix models, intersection theory on M g,n and special geometry on a family of curves is part of a more general story. For the general formally deformed measure dµ N (x), we write [START_REF] Bershadsky | Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes[END_REF] W g,h (z 1 , . . . , z h ; x)

h i=1 z -1 i k1,...,kn W g,h;k1,...,k k (x) h i=1 z -ki i
for the generating function of the cumulants of Eq. ( 4) at the g th order in the 1/N expansion of the formal 1-matrix model of Eq. ( 1). At the leading (planar) order in 1/N , the first loop equation reduces to

(14) W 0,1 (z) = 1 2t P (z) - 1 2t M (z) σ(z),
where

σ(z) = (z -b 1 (x, t))(z -b 2 (x, t)
) and the moment function M (z; x) as well as the branchpoints b i (x, t) are entirely determined by P (z) and t. The second planar loop equation fixes the two-point function to take the form [START_REF] Akemann | Higher genus correlators for the Hermitian matrix model with multiple cuts[END_REF] (15

) W 0,2 (z 1 , z 2 ) = - 1 2(z 1 -z 2 ) 2 + σ(z 1 ) 2(z 1 -z 2 ) 2 σ(z 1 ) σ(z 2 ) - σ (z 1 ) 4(z 1 -z 2 ) σ(z 1 )σ(z 2 ) .
In two remarkable papers [START_REF]Intersection numbers of spectral curves[END_REF][START_REF]Invariants of spectral curves and intersection theory of moduli spaces of complex curves[END_REF] (see also [START_REF] Dunin-Barkowski | Identification of the Givental formula with the spectral curve topological recursion procedure[END_REF]), Eynard established the following Theorem 2.1. There exist explicit generating functions of tautological classes

Λ g,n (x) ∈ C[[x]] ⊗ R • (M g,n ), B g,n (z 1 , . . . , z n ; x) ∈ C[[x; z 1 , . . . z n ]] ⊗ R • (M g,n ) such that W g,n (z 1 , . . . , z n ; x) = Mg,n Λ g,n (x)B g,n (x, z 1 , . . . , z h ), F g (x) = Mg,0 Λ g,0 (x). ( 16
)
In Theorem 2.1, the classes Λ g,n (x) and B g,h (x, z 1 , . . . , z h ) are entirely determined by knowledge of W 0,1 and W 0,2 in Eqs. ( 14) and (15). Theorem 2.1 encodes the solution of an intersection theoretic problem on M g,n into the knowledge of the correlators W g,n (z 1 , . . . , z n ; x) of an associated formal 1-matrix model. In fact, the applicability of Theorem 2.1 goes far beyond the realm of (multi-)matrix models: it applies to any spectral curve setup and higher order correlators defined through the Eynard-Orantin topological recursion as in the following section.

2.2. B-model: the Eynard-Orantin topological recursion. Theorem 2.1 gives a means to compute the 1/N expansion of the correlators of the 1-matrix model in terms of intersection numbers on M g,n , which in general is hardly a simplification from the computational point of view. An independent way to compute them was found in 2004 by Eynard, and further developed in subsequent works with Chekhov and Orantin, in terms of a recursive solution of the loop equations, as in the following Theorem 2.2 ([29, 37, 41]). Let C t be the projectivisation of the plane curve given by the zero locus of y 2 -σ(z) in C 2 with coordinates (z, y), and write dE w (z) ∈ Ω 1 (C t \ {w = z}) for the logarithmic derivative of the prime form on C t , normalised as A dE w = 0; here A is an oriented loop winding once counterclockwise around the segment [b 1 (x, t), b 2 (x, t)]. Then, for 2g -2 + h > 0 the following recursions hold:

W g,h+1 (z 0 , z 1 . . . , z h ) = qi Res z=qi dE z0 (z) 2M (z)σ(z) W g-1,h+2 (z, z, z 1 , . . . , z h ) + g l=0 J⊂H W (g-l) |J|+1 (z, z J )W (l) |H|-|J|+1 (z, z H\J ) , (17) (18)
F g = qi Res z=qi dE z0 (z) 2M (z)σ(z) z dz W g,1 (z ) Here W 0,1 (q i ) = 0, I ∪ J = {z 1 , . . . , z h }, I ∩ J = ∅,
denotes omission of the terms (h, I) = (0, ∅) and (g, J), and the primitive in Eq. ( 18) is independent of its base point.

Eq. ( 18) and Theorem 2.2 together make up the topological recursion: it determines recursively all the higher orders of the correlators W g,h starting from (g, h) = (0, 1), (0, 2) in Eqs. ( 14) and [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF], and consequently the all-order expansion of the partition function. Its frame of applicability goes far beyond the formal analysis of single-cut 1-matrix models, encompassing also multi-cut solutions, multi-matrix models, and non-polynomial potentials. The overall picture that emerges can be codified by the diagram of Figure 1. How much of this story is specific to matrix models? It should be noticed that 't Hooft's original argument in no way depended on the dimensionality of the gauge theory we are doing perturbation theory on, as long as the interacting theory has a weakly coupled limit given by a free Gaussian theory, a sensible UVregularisation scheme exists, and renormalisation does not lead to a factorially divergent perturbation series at each order in the topological expansion. A natural, if perhaps bold, question that could be asked is then: "Can the top left corner in Figure 1 be replaced by a higher dimensional (topological) gauge theory?".

Evidence that this is not a mere speculation comes from a non-trivial example, found by Gopakumar, Ooguri and Vafa in [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF][START_REF] Ooguri | Knot invariants and topological strings[END_REF], where the entire setup of Figure 1 carries forth verbatim to higher dimension. Let M be a smooth, closed, oriented real 3-manifold and K be a link in it. For a smooth

U (N ) gauge connection A on M let (19) CS[A] M Tr A ∧ dA + 2 3 A ∧ A ∧ A
be the Chern-Simons functional of A. It was Witten's realisation that the formal functional integrals [94]

Z CS N (M, k) D[A] exp k 2πi CS[A] , W CS N,ρ (M, K, k) Z -1 N D[A] Tr ρ (Hol K (A)) exp k 2πi CS[A] (20)
lead to (framed) topological invariants of M and K in the form of the Reshethikin and Turaev's sl N -quantum invariants; here k ∈ Z * is the Chern-Simons level, and the coloring ρ is an irreducible representation of U (N ).

Their large N expansion can be worked out entirely explicitly in different ways, for example by resorting to the surgery techiniques of [START_REF] Witten | Quantum field theory and the Jones polynomial[END_REF]: for the partition function we find [START_REF] Brini | E 8 spectral curves[END_REF] ln

Z CS N (M, k) = g≥0 g 2g-2 s F CS g (t)
where

g s = 2πi(k + N ) -1 , t = g s N , and (22)
F CS g (t) = |B 2g | 2g(2g -2)! Li 3-2g (e -t ) + (-1) g B 2g B 2g-2 2g(2g -2)(2g -2)!
for g ≥ 2, with similar formulas for g = 0, 1 [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF]. If we trust that a higher dimensional analogue of the discussion of the previous section exists, the two questions we are tasked to answer are:

(1) characterise the large N A-model dual of Chern-Simons theory in terms of a precise intersection theoretic problem on M g,n ;

(2) characterise its mirror, large N B-model dual in terms of the topological recursion on a specific spectral curve setup.

A-model.

The A-model dual found in [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF] is a sort of q-deformation of the setup of Example 2.1, and it has an explicit presentation in terms of an intersection theory problem on a moduli space of stable maps. First off, it was found in [START_REF]Chern-Simons gauge theory as a string theory[END_REF] that the partition function of U (N ) Chern-Simons theory on any closed three manifold M is equal to the partition function of the open topological A-model on the total space of the cotangent bundle T * M , with N Lagrangian A-branes wrapping its zero section. The proposal of [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF] is to relate the latter at large N to the ordinary, closed A-model/Gromov-Witten theory on a target space obtained from T * S 3 via a complex deformation to a normal singular variety and a bimeromorphic resolution of its singularities (the conifold transition), as follows. When S 3 has radius one in the canonical metric, we have an obvious isometry T * S 3 R 3 × S 3 SL(2, C) given by the decomposition of a special linear matrix A into radial (positive definite) and polar (unitary) part:

(23) A = U e H , U ∈ SU(2), H ∈ H 0 (2, C).
In particular this endows T * S 3 with a complex structure given by its presentation as a quadric hypersurface det A = 1 in Mat(2, C) C 4 .

We now perform the following two operation on this A-model target space:

(1) First we vary the radius of the base unit sphere to give a flat family ψ : X = GL(2, C) → C * via the determinant map, whose fiber X [µ] at a point µ with Im µ = 0 and Re µ > 0 is isomorphic to the cotangent bundle T * S 3 [µ] . Notice that we are doing nothing here either from the point of view of Chern-Simons theory or the open topological A-model on the cotangent bundle, as a homeomorphism of the base (resp. a complex deformation of the total space) leave invariant the CS partition function (resp. the A-model partition function).

(2) The second is to add the locus of non-invertible matrices to form: [START_REF] Brini | Crepant resolutions and open strings[END_REF] ψ :

Mat(2, C) -→ C .
The singular fiber X [0] above µ = 0 is the singular quadric det A = 0: this is the singular conifold of Example 2.1. The latter admits a canonical toric minimal resolution

(25) π : X -→ X [0] , X (ρ(A), v) ∈ X [0] × P 1 , ρ(A)v = 0 ,
where π is the projection to the first factor. The point A = 0 is singular in X [0] , and its fiber is a complex projective line with [v 1 : v 2 ] as homogeneous coordinates. Using coordinate charts on P 1 exhibits X as the total space of O(-1) ⊕ O(-1) → P 1 , i.e. the resolved conifold.

The proposal of [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF] is that

(26) F CS g (t) = GW X g (t)
where we have identified the 't Hooft parameter on Chern-Simons theory with the A-model Kähler parameter dual to the curve class of the base O(-1) ⊕2 P 1 . In Eq. ( 26), GW X g denotes the genus-g primary Gromov-Witten potential of X,

(27) GW X g (t) = d≥0 e td [Mg,0( X,d)] vir 1
where M g ( X, d) is the stack of degree d stable maps from genus g curves to X and [M g ( X, d)] vir denotes its virtual fundamental cycle.

There are a number of reasons, at various degrees of rigour, to take this proposal seriously:

(1) Firstly, it holds true asymptotically in the t → 0 limit: indeed on the CS side, the partition function is known to reduce in this limit to the gauged gaussian matrix model of Example 2.1 [START_REF] Periwal | Topological closed string interpretation of Chern-Simons theory[END_REF]; and on the GW side, the t → 0 regime corresponds to the small volume limit (the singular conifold). The claim then follows from the conifold/c = 1 duality we discussed in Example 2.1;

(2) There are also heuristic physics arguments in favour of the duality. One hinges on viewing Eq. ( 26) as an open/closed duality in the A-model, the equality can be interpreted as a brane/flux duality in the physical type IIA superstring: in principle, both sides compute superpotential observables in effective N = 1 theories, obtained either as the world-volume theory of wrapped D6-branes in T * S 3 of via turning on internal RR field strengths on the resolved side [START_REF] Bershadsky | Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes[END_REF][START_REF] Vafa | Superstrings and topological strings at large N[END_REF]. These string configurations can then be related by lifting them to M-theory [START_REF] Atiyah | An M theory flop as a large N duality[END_REF].

(3) A further microscopic derivation of the duality can also be given [START_REF]World sheet derivation of a large N duality[END_REF] upon relating the open partition function on the deformed conifold and the closed partition function on the resolved conifold by their UV completion in terms of a gauged linear σ-model: here the t → 0 limit leads to a coexistence of Coulomb/Higgs phases on the closed string world-sheet, with the holes of the open theory arising from integrating out the contribution of Coulomb regions.

(4) What is more, the equality of Eq. ( 26) is a sharp mathematical statement about RTW invariants and GW invariants that can be rigorously proven, or disproven, by an explicit calculation. The l.h.s. can be rigorously shown to lead to the expansion of Eq. ( 22) from an explicit MacLaurin expansion of the finite sums appearing in the surgery formula for the Reshetikhin-Turaev invariant [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF]; and on the other hand the r.h.s. is amenable to a direct localisation analysis in Gromov-Witten theory, first performed in [START_REF] Faber | Hodge integrals and Gromov-Witten theory[END_REF][START_REF] Graber | Localization of virtual classes[END_REF]. The end result is an exact agreement of the two generating functions.

(5) Finally, a natural question is how the story generalises when we incorporate links in Chern-Simons theory. It was first proposed in [START_REF] Ooguri | Knot invariants and topological strings[END_REF] that when K = is the trivial knot, Eq. ( 26) should generalise to an equality between the colored HOMFLY invariants of Section 3 and open GW potentials for a suitable choice of Lagrangian A-branes L:

(28) W CS g,h, d S 3 , t, = GW X,L g,h, d (t)
where in the l.h.s. we consider a knot invariant obtained as a power sum (instead of Schur functions, as in Section 3) holonomy invariant specified by a vector of integers d 1 , . . . , d h , we take its connected part, and we expand it at large N as in Eq. ( 5), and in the r.h.s. we take the generating function of open GW invariants with boundary on the fixed locus of a real involution (a real bundle on the equator of the base P 1 ), with fixed genus g and number of holes h for the source of curves and winding numbers d 1 , . . . , d h around S 1 ⊂ P 1 . Indeed, the line of reasoning of [START_REF] Vafa | Superstrings and topological strings at large N[END_REF] carries through to this setting, and also in this case a localisation definition/computation of open GW invariants can be performed [START_REF] Sheldon | Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc[END_REF], confirming the prediction of [START_REF] Ooguri | Knot invariants and topological strings[END_REF].

B-model.

Since the A-model target space is a (non-compact) Calabi-Yau threefold, we would expect a geometric mirror picture in terms of rigid special Kähler geometry of a family of local CY3s, akin to what happened in Example 2.1 for the case of matrix models. Not only is this case, but the analogy with the spectral curve setup of random matrices is even more poignant -the special geometry relations on

0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 Figure 2.
The fan (left) of the resolved conifold: its skeleton is given by four rays, labelled v i in the picture, whose tips lie on an affine hyperplane at unit distance from the origin of Z3 . The intersection of the fan with the hyperplane is shown on the right (the toric diagram); superimposed is the pq-web diagram.

the local CY3 mirror (which, in this particular case, is just the deformed conifold geometry of the previous section) can be shown to reduce to Seiberg-Witten-type relations on a family of genus zero spectral curves, given by the Hori-Iqbal-Vafa mirror of X:

(29) t = 1 2πi A ln yd ln x, ∂GW X 0 ∂t = 1 2 B ln yd ln x
where A, B are homology 1-cycles relative to the principal divisors x = 0, y = 0 of the plane curve defined by P X (x, y) = 1 + x + y + e t = 0; here P X is the Newton polynomial the diagram in Figure 2. What is more, motivated by the study of the chiral boson theory of [START_REF] Aganagic | Topological strings and integrable hierarchies[END_REF], it was first suggested in [START_REF] Mariño | Open string amplitudes and large order behavior in topological string theory[END_REF], and then postulated in full detail in [START_REF] Bouchard | Remodeling the B-model[END_REF], that the higher genus B-model potentials (and, by mirror symmetry, the higher genus open GW invariants of X) should be computed by the topological recursion of Theorem 2.2 and Eq. ( 18) upon identiying ydx = W 0,1 (x). This prediction has been subsequently proved in full rigour in [START_REF]Computation of Open GromovWitten Invariants for Toric CalabiYau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture[END_REF] for all toric Calabi-Yau manifolds that are symplectic quotients 3 .

The GOV correspondence for Clifford-Klein 3-manifolds

We have seen how the picture of Figure 1 generalises verbatim to the higher dimensional setting of Chern-Simons theory on S 3 : the top-right corner has a concrete A-model picture in terms of the Gromov-Witten theory of the resolved conifold, and the bottom-left corner is encoded by the topological recursion on its Hori-Iqbal-Vafa mirror curve. The emerging picture is not only beautiful and unexpected; from a practical point of view, it has deeply affected the relation between quantum invariants, curve-counting invariants, and the topological recursion. Appealing and impactful as it has been, this is however just one example -placed in the classical setting of U (N ) theories, encompassing the simplest closed 3-manifold, and with the simplest knot therein. It is then natural to seek a strict generalisation of the GOV correspondence to other4

(1) knots: in its strictest sense, the GOV correspondence has been shown to carry through to the case of torus knots both in terms of an enumerative theory of open GW invariants on the A-side [START_REF] Diaconescu | Large N duality, lagrangian cycles, and algebraic knots[END_REF] and the topological recursion on a specific spectral curve setup on the B-side [START_REF] Brini | Torus knots and mirror symmetry[END_REF]. For general knots, it would seem that substantially new ideas are needed both on the A-and the B-side of topological string theory [START_REF] Aganagic | Topological Strings, D-Model, and Knot Contact Homology[END_REF]; the role of the topological recursion in particular is however less clear in this setting [START_REF] Gu | Knot Invariants from Topological Recursion on Augmentation Varieties[END_REF].

(2) (classical) gauge groups: SO(N ) and Sp(N ) Chern-Simons theory at large N , which compute in particular the Kauffmann invariant of links, can also fit in the picture of the previous section by considering suitable orientifolds of the resolved conifold [89], for which an operative definition of unoriented invariants can be given either by localisation or via the topological vertex [START_REF] Bouchard | Counting higher genus curves with crosscaps in Calabi-Yau orientifolds[END_REF][START_REF]Topological open string amplitudes on orientifolds[END_REF].

(3) 3-manifolds: this is possibly the boldest generalisation -replace altogether S 3 by an arbitrary closed 3-manifold M . At face value this boils down to solving the problem for arbitrary knots on S 3 : the partition function of a link L in M would then be recovered by the surgery formula from the partition function of a 2-component link L L M in S 3 , where L M is a framed link M can be obtained from (this always exists by Lickorish's theorem). However, both the surgery formula in Reshetikhin-Turaev theory [START_REF] Søren | Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula[END_REF][START_REF] Witten | Quantum field theory and the Jones polynomial[END_REF], and functional localisation in Chern-Simons theory [START_REF] Beasley | Non-Abelian localization for Chern-Simons theory[END_REF][START_REF] Blau | Chern-Simons Theory on Seifert 3-Manifolds[END_REF][START_REF] Kallen | Cohomological localization of Chern-Simons theory[END_REF] lead to an expression of the partition function as a sum over contribution labelled by flat connections (classical vacua) on M :

(30) Z CS N (M, k) = v∈Hom(π1(M ),U (N ))/U (N ) Z CS N,v (M, k).
It was first proposed in [START_REF] Aganagic | Matrix model as a mirror of Chern-Simons theory[END_REF] that the finer invariants given by the individual summands Z CS N,v may also be interpreted as the A/B-model partition function on a background specified by v; notice that this is a more refined object to deal with than the partition function of the link L M , where all these contributions are summed over. That a dual curve counting theory exists is encouraged by the successful test of this proposal for the case of L(p, 1) lens spaces in [START_REF] Aganagic | Matrix model as a mirror of Chern-Simons theory[END_REF][START_REF] Halmagyi | The spectral curve of the lens space matrix model[END_REF]. The case of more general 3-manifolds was considered in [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF][START_REF] Borot | Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces[END_REF][START_REF] Brini | Chern-Simons theory on L(p, q) lens spaces and Gopakumar-Vafa duality[END_REF], and we review it below. Equivalently, by Hamilton's theorem, it is a spherical space form, M = S 3 /Γ for Γ a freely acting finite isometry group of S 3 w.r.t. its canonical metric; and by Perelman's elliptisation theorem, if follows that these coincide with the orientable 3-manifolds having finite fundamental group. The classification of the possible Γ goes back to Hopf [START_REF] Hopf | Zum Clifford-Kleinschen Raumproblem[END_REF]: these are central extensions of the left-acting finite subgroups of SL 2 (C), which admit an ADE classification; see [15, Appendix A] for more details. We restrict henceforth for simplicity of exposition to the case where the central extension is trivial and Γ is one such SL 2 (C)-subgroup; most of our arguments to follow will be unaffected by this. Now since |Γ| < ∞, the sum in Eq. ( 30) truncates at finite N . Denote by V Γ,N the finite set of gaugeequivalent flat connections and by V Γ = lim N →∞ V Γ,N be its direct limit with respect to the composition of morphisms given by the embedding U(N ) → U(N + 1). Then [START_REF] Coates | Wall-crossings in toric Gromov-Witten theory. I. Crepant examples[END_REF] V

Γ = N R+1 , V Γ,N = (N 0 , . . . , N R ) ∈ N R+1 , N 0 + R i=1 D i N i = N .
where R is the number of nodes in the simply-laced Dynkin diagram associated to Γ, and D i are the respective Dynkin indices. When N → ∞, we thus consider a CS vacuum [A] t parametrized by t N i g s for i ∈ {0, . . . , R}, and in particular the rank is encoded in the 't Hooft parameter t = N g s . The resulting partition functions at N = ∞ have a standard 1/N expansion [START_REF] Diaconescu | Large N duality, lagrangian cycles, and algebraic knots[END_REF] ln

Z CS N,v(t) (g s ) = g≥0 g 2g-2 s F CS g (t 0 , . . . , t R )
with free energies depending now on R + 1 't Hooft parameters.

These manifolds also carry a natural class of knots with them. For S 3 , the standard GOV correspondence focused on the unknot -which could be regarded as the fibre knot of the Hopf fibration on S 2 . Now Clifford-Klein manifolds are also Seifert-fibred, and in the ADE case they can be regarded as Seifert fibrations over an ADE P 1 -orbifold with one (resp. three) exceptional fibres for Dynkin type A (resp. D and E). As for S 3 , we will similarly be interested in the 1/N expansion of the RTW invariant for the knots K f running around these exceptional fibres:

(33) W CS N,v(t) (K f , g s , d) = g≥0 g 2g-2 s F g,h (t 0 , . . . , t R ; d)
The quest is to find now a Calabi-Yau threefold geometry X Γ with special Lagrangians L Γ , as well as a spectral curve setup Σ Γ for each such Γ, such that open/closed GW theory on ( X Γ , L Γ ) and the topological recursion on Σ Γ lead to Eqs. ( 32) and ( 33). This program was completed in [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF], generalising ideas on the lens space case in [START_REF] Aganagic | Matrix model as a mirror of Chern-Simons theory[END_REF].

A-model from geometric transition.

On the A-model the idea is fairly simple: we take seriously the geometric transition argument of Section 3.2 and apply it to the setting at hand. To this aim, notice that the Γ-action on the resolved conifold X only acts on the first factor of Eq. ( 25), giving a fibrewise action on p : X → P 1 (the second factor in Eq. ( 25)); the fibre over a point z ∈ P 1 is isomorphic to a Gorenstein surface singularity of the same ADE type of Γ. The topological A-string on this target geometry can be studied in several phases, two of which are distinguished:

• in the orbifold chamber, we are looking at a theory of twisted stable maps on the Calabi-Yau stack

X Γ orb [ X/Γ]
. This is the maximally singular phase containing the Γ-orbifold of the conifold point, which is the natural point of expansion for the dual Chern-Simons theory;

• in the large radius chamber, we take a crepant resolution X Γ res of the singularities of (the coarse moduli space of) X Γ orb obtained by canonically resolving the surface singularity C 2 /Γ fibrewise, and we are looking at the ordinary GW theory of Y Γ res . We have four remarks about the resulting target space geometry.

R1: in either case the target supports at least a (C ) 2 torus action; this is the product of the 1-torus action that rotates the base P 1 and the fibrewise 1-torus action inherited from the scalar action on the C 2 -fibre of X. In particular, as for the usual toric case, we will specialise to a resonant 1-subtorus by imposing that its action is trivial on the canonical bundle of the target; this acts with compact fixed 0-and 1-dimensional fixed loci. This in turn allows to define equivariant GW invariants by localisation on either X Γ res or X Γ orb . R2: As we already mentioned, the Γ-action is fibrewise and it covers the trivial action on the base P 1 ; this circumstance will be important in a moment.

R3: For all Γ, there are natural anti-holomorphic involutions whose fixed loci define Lagrangians for both X Γ res and X Γ orb . These generalise the toric Lagrangian branes of [START_REF] Aganagic | Mirror symmetry, D-branes and counting holomorphic discs[END_REF][START_REF] Brini | Open orbifold Gromov-Witten invariants of [C 3 /Zn]: localization and mirror symmetry[END_REF][START_REF] Ooguri | Knot invariants and topological strings[END_REF], and the residual Calabi-Yau C action of R1 above allows to define/compute open GW invariants by localisation. In particular, the methods of [START_REF] Brini | Open orbifold Gromov-Witten invariants of [C 3 /Zn]: localization and mirror symmetry[END_REF] carry through verbatim to the setting at hand.

R4:

The holomorphic involution of [89] turns out to commute with the Γ-action, for all ADE types, and an orientifold theory can be defined in the same way.

4.

3. B-model from geometric engineering. R1-R2 above define completely what would be the top-right (A-model) corner of a diagram like Figure 1 for the case at hand. Now notice that in R1 above, for type D and E the torus action we highlighted does not extend to a full three-torus action on X Γ since Γ is non-abelian. In particular, we cannot resort to the toric mirror symmetry methods of [START_REF] Hori | D-branes and mirror symmetry[END_REF][START_REF] Hori | Mirror symmetry[END_REF] to find a spectral curve setup for these cases, unlike for type A. A way out on physical grounds is however pointed at by R2: since the action is fibrewise, the topological A-string on these backgrounds is known to geometrically engineer in a suitable limit a 4-dimensional N = 2 gauge theory with simply-laced compact gauge group corresponding to the Dynkin type of Γ (and no adjoint hypermultiplets, since P 1 has genus zero) [START_REF] Sheldon | M-theory, topological strings and spinning black holes[END_REF]; so it is expected in this degenerate limit, which corresponds to a divisor at infinity in the stringy Kähler moduli space of X Γ , to have a mirror picture in terms of spectral curves of Seiberg-Witten type [START_REF] Sheldon | M-theory, topological strings and spinning black holes[END_REF][START_REF] Seiberg | Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory[END_REF]. As a matter of fact, even away from this limit the A-model is still expected to give rise to a gauge theory with eight supercharges, albeit in one dimension higher -namely N = 1 pure super Yang-Mills theory on R 4 × S 1 , with the "field-theory limit" of [START_REF] Sheldon | M-theory, topological strings and spinning black holes[END_REF] corresponding to the fifth-dimensional circle shrinking to zero-size. Now for type A N , the five-dimensional theory also enjoys a description in terms of spectral curves of Seiberg-Witten/Hori-Iqbal-Vafa type: these are the spectral curves of the periodic Ruijsenaars system (relativistic Toda chain) with N sites [START_REF] Nekrasov | Five dimensional gauge theories and relativistic integrable systems[END_REF]; the 4d limit corresponds to the non-relativistic limit. Furthermore, it has long been known that for all ADE types the relevant SW curves should coincide with the spectral curves of the Lie-theoretic generalisation of the non-relativistic Toda chain [START_REF] Martinec | Integrable systems and supersymmetric gauge theory[END_REF] specialised to ADE Lie algebras. Putting all this together, it is then natural to look for a relativistic deformation of [START_REF] Martinec | Integrable systems and supersymmetric gauge theory[END_REF] to supply the spectral curves defining a candidate B-model mirror for the non-toric geometries of the previous section, and these can be computed from the setup of [START_REF] Fock | Loop groups, Clusters, Dimers and Integrable systems[END_REF][START_REF] Williams | Double Bruhat Cells in Kac-Moody Groups and Integrable Systems[END_REF]. We refer the reader to [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF]Section 3.6] for a more detailed review; the upshot is that the spectral curves can be computed in the following two steps:

(1) fix ρ to be a minimal irreducible G Γ -module, where G Γ is the simply-connected simple Lie group over C of ADE type specified by Γ, and consider the characteristic polynomial of a group element g in the representation ρ

(34) p Γ ρ (g) det ρ (id ρ x -g) : T /Weyl(G Γ ) → C[x]
Here T Γ is the Cartan torus of G Γ . We can decompose this on the Weyl character ring upon expanding the determinant in a polynomial with coefficients given by anti-symmetric characters of the representation ρ, and then write the latter as polynomials of the fundamental characters u i :

(35) p Γ ρ (g) = dim ρ n=0 (-1) dim ρ-n χ ∧ dim ρ-n ρ (g)x n , χ ∧ n ∈ Z[u 1 , . . . , u R ]
(2) Now let u k be the character of the maximally dimensional fundamental representation. Then the spectral curve is defined by the family of non-compact Riemann surfaces given by the polynomial equation (36) p Γ ρ x; u i + δ i,k (y + u 0 y -1 = 0 along with a choice of recursion differential given by the Poincaré one form W 0,1 = ln yd ln x, and higher order generating functions obtained by the topological recursion on Eq. [START_REF] Dunin-Barkowski | Identification of the Givental formula with the spectral curve topological recursion procedure[END_REF].

The resulting web of relations is pictured in Figure 3.

4.4.

The LMO invariant and the topological recursion. Our proposal fills all the vertices in the diagram of Figure 3: the sl N -Reshetikhin-Turaev knot invariants of fibre knots in S 3 /Γ in the 1/N expansion around a fixed flat connection would equate the open Gromov-Witten potentials of ( X Γ , L Γ ) on one hand, and the Eynard-Orantin invariants of Eq. (36) on the other; a similar statement applies to the RT invariant of S 3 /Γ itself, the closed GW potnetial of X Γ , and the topological recursion free energies.

However this is all conjectural for the moment -so how do we prove this? One way to proceed is to resort to the relation of Chern-Simons theory on Seifert fibred spaces (such as the Clifford-Klein manifolds) and matrix models. It was shown by Mariño in [START_REF] Marino | Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants[END_REF] that Witten's surgery formula for these spaces allows to rewrite the CS partition function as well as the Wilson loops around certain classes of knots as a matrix integral; in particular, the quantum invariants Eqs. ( 32) and (33) of S 3 /Γ and fibre knots therein around the exceptional fibres fall squarely in this category [START_REF] Beasley | Non-Abelian localization for Chern-Simons theory[END_REF][START_REF] Blau | Chern-Simons Theory on Seifert 3-Manifolds[END_REF][START_REF] Marino | Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants[END_REF]. When v = 0 is the trivial flat connection, the resulting matrix model turns out to be a trigonometric deformation of the gauged gaussian matrix model of Example 2.1: it is a canonical ensemble with gaussian 1-body interaction and a sum of q-deformed Vandermonde 2-body interactions, with coefficients determined by the orders of the exceptional fibres of the Seifert fibration [START_REF] Bar-Natan | A rational surgery formula for the LMO invariant[END_REF][START_REF] Borot | Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces[END_REF][START_REF] Brini | Torus knots and mirror symmetry[END_REF][START_REF] Marino | Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants[END_REF]: this restriction gives rise to the so-called LMO invariant. This presentation is amenable to a large N analysis via loop equations, akin to that of Point iii) in Example 2.1, leading for all Γ to a singular integral equation to be solved by the input datum of the recursion -the planar disk function W 0,1 . Such an analysis was performed in [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF][START_REF] Borot | Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces[END_REF][START_REF] Brini | Torus knots and mirror symmetry[END_REF]; the strategy, and ensuing results are as follows.

(1) It can readily be shown that these are single-cut matrix models due to the gaussian nature of the 1-body potential, and that the (exponentiated) planar resolvent W 0,1 (x) is never a log-algebraic function of its argument except when Γ is the trivial group. However, a strategy introduced in [START_REF] Brini | Torus knots and mirror symmetry[END_REF] for torus knots and then employed on a full scale on [START_REF] Borot | Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces[END_REF] is to considered a symmetrised version of the resolvent, which leads to the same large N eigenvalue density as the original resolvent on the physical cut and may be such that the sheet transitions given by crossing the cuts close to a finite group.

(2) The previous step can be performed for any Seifert 3-manifold as that is the level of generality to which the ideas of [START_REF] Marino | Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants[END_REF] apply. It can be shown however that the only Seifert spaces for which the group of sheet transitions is finite, and whose planar resolvent thus gives rise to algebraic spectral curves, are precisely the Clifford-Klein 3-manifolds: there is no hope to extract an algebraic setup for the large N /B-model curve, even for the restriction to the trivial flat connection, for parabolic and hyperbolic Seifert 3-manifolds. In the elliptic case, instead, e W0,1 is a root of an algebraic equation in C × C :

(37) P Γ (x, e W0,1 ; t) = 0 where t is the 't Hooft parameter g s N . It should be noticed that these symmetrisation is not unique; however the freedom of choice here is in bijection with the freedom of choice of an irreducible G Γmodule on the Toda side: in either case (see [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF] and [START_REF] Mcdaniel | A Lie-theoretic Galois theory for the spectral curves of an integrable system[END_REF] respectively) it can be shown that these choices do not affect the calculation of the partition function, so we will henceforth drop the subscript ρ from Eq. ( 36).

(3) For these cases, a computational tour-de-force leads to determine in full detail the spectral curve for type A [START_REF] Brini | Torus knots and mirror symmetry[END_REF], D [START_REF] Borot | Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces[END_REF], and E 6 [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF]; substantial information can be extracted for E 7,8 , with a full solution available in all cases in the limit t → 0.

(4) Now, proving the B-model side of the GOV correspondence for the LMO invariant amounts to finding a restriction u i (t) of the Toda hamiltonians of the ADE relativistic Toda chain such that (38) p Γ x; u i + δ i,k (y + u 0 y -1 | ui=ui(t) = P Γ (x, y; t)

A detailed analysis of both spectral curves setup shows5 that this is indeed the case for type A, D and E 6,7 (see also [START_REF] Brini | E 8 spectral curves[END_REF] for progress on the E 8 case), and for all ADE types in the small 't Hooft coupling limit t → 0 [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF].

(5) Finally, having a matrix integral representation for the partition function of Chern-Simons theory on these spaces allows to rigorously derive the topological recursion of Theorem 2.2 and Eq. ( 18) for the cumulants of the resulting distribution. On the other hand, on the B-model side, the topological recursion can be either regarded as the definition of the higher genus open/closed topological Bstring on a family of curves, or, from a physics standpoint, it could be derived from the chiral boson theory on the spectral curve obtained from dimensional reduction of the BCOV Kodaira-Spencer theory of gravity [START_REF] Bershadsky | Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes[END_REF][START_REF] Dijkgraaf | Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity[END_REF]. Since the two theories boil down to the same recursion with the same input datum, we reach the remarkable conclusion that they give rise to the same invariants to all orders in 1/N and arbitrary colourings of the invariants. This yields an all-genus proof of the B-model side of the GOV correspondence for this type of manifolds, restricted to the LMO invariant.

Conclusions

We have reviewed how the strict generalisation of the GOV correspondence that bears all the ingredients of the simplest original setting of [START_REF] Gopakumar | On the gauge theory/geometry correspondence[END_REF][START_REF] Ooguri | Knot invariants and topological strings[END_REF], including a geometric A-model theory of open/closed Gromov-Witten invariants and an all-genus B-model theory governed by the topological recursion on a specific spectral curve setup, can be given for the case of Clifford-Klein 3-manifolds -and these alone, according to our remarks in the previous section. This opens severals directions for future research, including an extension to the setting of refined/categorified invariants (particularly on the B-model side), quantum integrability, and the relation to gauge theories. We single out here three more topics in particular on which we hope to report in the near future.

B-model general flat backgrounds. Since most of the analysis of the previous section was restricted to the study of the LMO invariant, it is natural to ask how to extend the GOV strings correspondence to a general Chern-Simons vacuum, thus completing the proof of the B-model version of the GOV correspondence. On the B-model side, the family of relativistic Toda spectral curves was constructed in [START_REF] Borot | Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems[END_REF] for type ADE 6,7 , with the missing E 8 case to appear in [START_REF] Brini | E 8 spectral curves[END_REF]. The more difficult bit here is to provide a complete large N analysis of the matrix model, although this might be possible by suitably rewriting the finite sums expressions of [START_REF] Blau | Chern-Simons Theory on Seifert 3-Manifolds[END_REF][START_REF] Marino | Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants[END_REF] in terms of ordinary eigenvalue models on the real line. Establishing an explicit solution of the loop equations for this matrix model in terms of the topological recursion applied on the corresponding Toda spectral curve would give a full proof of the B-side of the GOV correspondence. This would also shed light on some of the difficulties encountereed in the analysis of arbitrary flat backgrounds to the case of general lens spaces and non-SU(2) abelian quotients of the resolved conifold [START_REF] Brini | Chern-Simons theory on L(p, q) lens spaces and Gopakumar-Vafa duality[END_REF].

Non-toric "remodeling-the-B-model". Another line of developement consists of extending the remodeling proposal of [START_REF] Bouchard | Remodeling the B-model[END_REF] to the non-toric setting at hand for type D and E. A first step here would be to fully spell out the computation of the disk functions as in [START_REF]Open topological strings and integrable hierarchies: Remodeling the A-model[END_REF][START_REF] Brini | Open orbifold Gromov-Witten invariants of [C 3 /Zn]: localization and mirror symmetry[END_REF] for the case at hand, and then derive the topological recursion from the analysis of the descendent theory. A promising route would be to derive the J-and Rcalibrations for the quantum cohomology of X Γ from the steepest descent analysis of the Toda differential, as in [START_REF] Brini | Crepant resolutions and open strings[END_REF], and then retrieve the topological recursion from Givental's R-action on the associated cohomological field theory [START_REF] Dunin-Barkowski | Identification of the Givental formula with the spectral curve topological recursion procedure[END_REF][START_REF] Fang | On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds[END_REF]. This would lead to a proof of the remodeling conjecture on an important class of examples, beyond the toric case.

The quantised McKay correspondence. One of the more intriguing consequences of our work is that several limits of the B-model geometries of Section 4.4 would provide a unified construction of spectral curves relevant for both non-toric mirror symmetry and the theory of Frobenius manifolds. In particular, the u 0 → 0 limit, corresponding to the limit where the Kähler volume of the base P 1 in X Γ is sent to infinity, gives rise to a family of 1-dimensional Landau-Ginzburg models for the stack [C 2 /Γ] and its crepant resolution: this would finally grant access to a host of explicit computations on the descendent theory, which are likely to be instrumental in the proof of the quantum McKay correspondence in full generality [START_REF] Bryan | The crepant resolution conjecture[END_REF][START_REF] Coates | Wall-crossings in toric Gromov-Witten theory. I. Crepant examples[END_REF][START_REF] Ruan | The cohomology ring of crepant resolutions of orbifolds[END_REF]. This would notably include the higher genus theory, by following the arguments employed for the type A case in [START_REF] Brini | Crepant resolutions and open strings[END_REF]. Moreover, Dubrovin's almost duality would relate these mirrors to the LG formulation of the Frobenius manifold structure on the orbit spaces of extended affine Weyl groups and ordinary Weyl groups associated to simply-laced root systems upon considering, respectively, the relativistic and non-relativistic limit of the Toda curves of Section 4.4. We plan to report on this in future work.

  : moduli of curves B-model: topological recursion

Figure 1 .

 1 Figure 1. Large N duality and mirror symmetry in d = 0

4. 1 .

 1 CS theory on Clifford-Klein 3-manifolds. We start by recalling the following Definition 4.1. A Clifford-Klein 3-manifold (M, g) is a closed oriented smooth 3-manifold M admitting a smooth metric g of everywhere strictly positive Ricci curvature.

  r o r s y m m e t r y 5d ADE SW theory A-model on Y Γ Chern-Simons/WRT invariant of S 3 /Γ B-model: spectral curve of ADE relativistic Toda

Figure 3 .

 3 Figure 3. The GOV correspondence for S 3 /Γ.

See[START_REF] Fang | On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds[END_REF] for a recent generalisation of this result to the case of orbifolds.

The following list leaves out one important generalisation, tying together categorification on the knot theory side[START_REF] Gukov | Khovanov-Rozansky homology and topological strings[END_REF], refined BPS counting on the A-model side[START_REF] Choi | The refined BPS index from stable pair invariants[END_REF][START_REF] Iqbal | The Refined topological vertex[END_REF][START_REF] Sheldon | M-theory, topological strings and spinning black holes[END_REF], and some version of the β-deformation on the B-model side[START_REF] Aganagic | Quantum Geometry of Refined Topological Strings[END_REF][START_REF] Aganagic | Knot Homology and Refined Chern-Simons Index[END_REF][START_REF] Brini | The uses of the refined matrix model recursion[END_REF]; we will comment briefly on this at the end of this paper.

Technically, the polynomials almost never match on the nose, but it can always be shown that upon restriction the Chern-Simons spectral curve arises as a non-trivial reducible component of a degenerate limit of the Toda curve.