Jonathan Gubspun

Marc Georgelin

Julien Deschamps

Marc Leonetti

Clément De Loubens

Roberto Trozzo

F Edwards-Levy

Perturbations of the flow induced by a microcapsule in a capillary tube

I. INTRODUCTION

Controlling the delivery of molecules is of prime importance in various domains as medicine, biology, nutrition or cosmetics. Microencapsulation of active materials within a sub-millimetric shell is a promising technique to take up this challenge [1,[START_REF] Edwards-Lévy | Serum albumin: structure, functions and health impact[END_REF]. For example, encapsulation has allowed workers to develop new treatments of chronic heart disease by controlling the spatio-temporal release of growth factors in the blood stream [START_REF] Banquet | [END_REF] or to create artificial red blood cells [4]. The delivery is achieved by hydrodynamic stresses or external stimuli inducing dynamic of deformations of the microcapsule and transfer through its membrane or even its breakdown [5]. The mechanical properties of the membrane have so to be tuned according to the hydrodynamic conditions encountered in the final application.

Microcapsule membranes can be made of cross-linked polymer [6] or complex of polyelectrolytes [7,8]. These membranes show a variety of elastic behaviors that can be varied with the kind of constituents and their * leonetti@irphe.univ-mrs.fr concentration [9]. Polysiloxane [10] and nylon microcapsules [11,12] break for minute deformation while Human Serum Albumin (HSA) microcapsules can be deformed elastically up to 180 % of stretching [13], for example. Up to now, experimental efforts have been made to characterize accurately the elastic properties of the membrane. Compression between two plates [14] and simple flows as elongation [9,11], shear [12,15,16], Poiseuille [17,18], flow through constriction [19] or centrifugal flow [20] have been used to deform microcapsules and measure their shear elastic modulus in the small deformation regime and moderate in the case of Poiseuille flow. More recently, the non-linear elastic model and the viscoelastic properties of the membrane were deduced in the regime of large deformations [13] and from the dynamic of tank-treading in simple shear flow [21] for HSA microcapsules. Numerical simulations play a major role to identify the relevant mechanical properties of the membrane [22] as its constitutive law [13,23], its viscosity [24] or its bending modulus [25]. Experimental and numerical studies on microcapsules in flow have focused a strong attention to the shape and its dynamics but not on the flow around it and notably on the perturbation of the applied flow. However, it plays a major role to understand the interactions between several soft particles in a three-dimensional suspension [26] but also in a confined channel [27]. These perturbations can have dramatic effects on the formation of clusters as observed for drops [28] and biological suspensions of soft particles like blood [29].

In confined flows, microcapsules are sufficiently soft to deform and move through channel of comparable dimensions such as blood vessels. When the microcapsule is small compared to the size of the channel, its shape is slightly deformed and the flow approximately the same than a Poiseuille flow. On the contrary, when the confinement is such that the size of the object is comparable to the size of the channel or even bigger, both the flow and the microcapsule shape may be largely modified. The latter situation has been studied for bubbles, drops or solid particles. Taylor was one of the first to report the streamlines of the outer flow perturbed by the bubble [30]. Then theoretical developments have been conducted for solid particles when immersed in the center or off-center of a Poiseuille flow [31,32] and also quasi-spherical drops [33][34][35]. Ho & Leal observed experimentally recirculation inside a drop and the existence of stagnation rings on its front and rear [36]. The axisymmetric recirculation of the fluid has also been studied between two adjacent drops spaced by a typical distance of the same order than the drop size inside the capillary. This analysis has been extended numerically for different capillary numbers, viscosities and confinements with similar qualitative results [37].

Whereas the flow has been studied around simple objects such as droplets or bubbles, no detailed investigations have been reported regarding the flow generated by soft particles as microcapsules with a comparison between experimental, analytical and numerical results. While the shear is transmitted across the fluid interface of bubbles and droplets, the solid membrane of microcapsules should induce major differences in term of fluid-structure interactions. Moreover, the deformation of microcapsules in narrow tube presents an inversion of the membrane curvature at its rear, called 'parachutelike' shape in the studies on vesicles [38][39][40][41][42] and capsules [17,43,44], whereas previous studies on drops and bubbles [30,[33][34][35][36][37]45] have not considered this situation [46,47].

In this paper, we study the effects of deformable microcapsules on the surrounding flow in a capillary channel. After mechanical characterization using an elongation flow, a successive approach is developed: i) determination of several families of characteristic shapes of capsules in capillaries varying the capillary number and the confinement and ii) determination of the flow field around these microcapsules measured by microparticles tracking velocimetry (PTV). We study microcapsules made of cross-linked proteins (HSA), the only ones which have been fully characterized: constitutive law in all the regimes of deformation, Poisson coefficient, shear elastic modulus and shear membrane viscosity. To discuss the results on the measured flow, numerical simulations are performed using the same deformations and microcapsule velocity as in experiments to calculate the flow field using the boundary element method for Stokes flow conditions. Note that we don't consider the full fluid-structure interaction but underformable axisymmetric solid body with no slip velocity at the membrane. We analyze the evolution of the velocity of the microcapsule with its deformation. The topology of the perturbation of flow field is emphasized according to the curvature of the rear of the particle, as the amplitude and spatial extent of this perturbation.

II. MATERIALS AND METHODS

A. Microcapsule preparation

Microcapsules were prepared by interfacial crosslinking of human serum albumin (HSA) with terephthaloyl chloride (see Andry et al. [6] for details). HSA, organic solvents (chloroform and cyclohexane), crosslinking agent (terephtaloyl chloride) and surfactants (sorbitan trioleate and polysorbate) were purchased from LFB Biomedicaments, SDF, Acros Organics and Sigma, respectively. HSA was dissolved in a pH 5.9 phosphate buffer at the concentration 7.5 g / 100 mL . This aqueous solution was emulsified in cyclohexane containing 2% (w/v) sorbitan trioleate at a stirring speed of 625 rpm. A 2.5% (w/v) solution of terephthaloyl chloride in chloroform:cyclohexane (1:4 v/v) was then added to the emulsion and the cross-linking reaction was allowed to develop for 30 min. The reaction was stopped by dilution of the reaction medium with cyclohexane. The microcapsules were separated from the organic phase by centrifugation and washed successively with cyclohexane, with water containing 2% (w/v) polysorbate and finally thrice with pure water. Microcapsules were then diluted of about thousand times and mixed very gently for at least 24 hours in 98 % glycerol (VWR) to allow the internal phase to equilibrate with the external one by diffusion through the pores of the membrane. The membrane is permeable on long time scale but may be considered as impermeable on short time scale, namely the volume is conserved during the experiments as shown in [48].

Finally, we obtain a diluted suspension of spherical microcapsules whose radii range from 120 to 230 µm, embedded in glycerol.

B. Mechanical characterization of microcapsule with elongation flow

Beyond their softness, the main advantage of viscoelastic membranes made of human albumin compared to others (ovalbumin, alginate, alginate-albumin mixing for example) is their full characterization supporting by previous studies from the small to large deformations providing the constitutive law [13]. The membrane mechanical response is in excellent agreement with the generalized Hooke law with a 2D Poisson coefficient ν s equal to 0.4. Viscous dissipation inside the human albumin membrane determined in [21] is not relevant in the present paper as we study axisymmetric stationary shapes without inter-facial flow.

We measure separately the evolution of G s with the radius a by studying the deformations of several microcapsules under a planar elongation flow in the linear regime of deformations with the same technique described in [9]. As shown in [9,13,48], the shear elastic modulus G s of HSA microcapsules increases with the radius. The size dependance of G s has been explained in considering that all HSA molecules initially available in the primary droplet, finally are cross-linked in the capsule membrane [48]. For the parameters of microcapsule preparation given above, G s increases from 0.05 N/m to 0.5 N/m when a increases from 120 to 230 µm. The evolution of G s with the radius can be fitted by an exponential law:

G s = 1.9 × 10 -3 exp (a × 2.6 × 10 -2 ) (1)
with the radius a in µm and the shear elastic modulus G s in N/m. The dilatation elastic coefficient K can be deduced with the equation K = G s (1 + ν s )/(1 -ν s ) with ν s equal to 0.4 [21] . Equation 1 allows to determine the capillary number which evaluates the hydrodynamic shear stress at membrane versus the elastic response. With it and the confinement, readers have all the necessary parameters to perform new numerical studies with different constitutive laws and compare to our experimental shapes and capsule velocity. The glycerol with microcapsules is pushed in the capillary tube with an home made syringe pump designed with an actuator PI 235.52S. The glass tube is immersed in a Petri dish full of glycerol to match the optical index of its inner and outer media and prevent any optical distortion (Fig. 2). A thin layer of oil covers the glycerol to prevent water absorption. The entire system is rigid to reach in less than 100 ms a stationary flow in the capillary. The Petri dish is mounted on top of an inverted microscope Olympus IX70. Microcapsules are observed in the midplane of the channel with a magnification ×10 (1.70 µm/pix) and a high speed camera Photron SA3 (5000 i/s, 1024×512 pixels).

In the reference frame of the microcapsule and far from it, V is axisymmetric and satisfies the Poiseuille equation:

V(r) = 2 V 1 - r 2 R 2 -V caps e z ( 2
)
where V is the mean flow velocity, V caps the capsule velocity. r and z are the radial and axial coordinates, e r and e z the respective unit vectors (Fig. 3).

D. Flow measurements & numerical simulations

For each microcapsule, initial size and capillary radius are measured at rest. As the microcapsule is flowing in the tube we extract the profile of its stationary deformed shape in the plane (e r , e z ) and measure the radial extension of the microcapsule L r , the position of its center of mass and its velocity V caps (Fig. 3). V caps is measured by tracking the front of the microcapsule in the reference frame of the laboratory and ranges from 0.5 to 20 mm/s. The flow in presence of isolated microcapsules is characterized by Particle Tracking Velocimetry (PTV). Polystyrene microspheres (Bangs Laboratories Inc.) with a radius ranging from 8 to 16 µm are seeded in glycerol at a concentration of 2.4 mg/mL. The size, very small compared to capsule radius has also been chosen to be detectable by PTV without measurable effect on the flow. The x10 objective has a large depth of field and integrates particles on a layer which is approximately 25 % of the diameter of the channel. All visible particles are taken into account to characterize the flow. Moreover the microcapsule move too quickly (V caps up to 20 mm/s) to track the capsule along the tube. In these conditions, the entire velocity field cannot be determined in the reference frame of the laboratory by one experiment. Thus to obtain the full velocity field around the microcapsule it is necessary to stack partial informations field extracted from several observations of the same microcapsule in the same conditions. Thus the microcapsule is pushed and sucked back up to 10 times in the capillary for a given flow rate to reconstruct the flow field in its reference frame. Particles are tracked and trajectories are extracted with Matlab (MathWorks). Finally, all the trajectories are stacked together to calculate the velocities in the axial v z and radial v r directions over 20 boxes along e r and 40 along e z (Fig. 4).

The microcapsule velocity in a capillary with a flow rate πR 2 V has been measured for 21 microcapsules. Each microcapsule is observed for different flow rates. The mean velocity V of the fluid in the capillary is deduced from the flow rate applied by the syringe pump or measured by tracking the fastest particles. Both methods were compared and match perfectly.

Experiments were carried out at a room temperature of 22±0.5 • C controlled inside the Petri dish with a precision of 0.1 • C. The viscosity of the glycerol η was measured with a rheometer ThermoScientifc Haake Mars III in a cone-plate geometry (2 • cone, 60 mm diameter) as a function of the temperature and ranged from 0.99 to 1.18 Pa.s.

The flow field is also calculated by numerical simulations by considering different shapes of microcapsules extracted from the experiments. Instead of solving the full complex fluid structure interaction problem of a deformable object in a capillary tube, the microcapsule is considered as a solid body translated in a Poiseuille flow with a velocity V caps . Briefly, the flow is computed by the boundary element method for Stokes flow conditions [49,[START_REF] Pozrikidis | Modeling and simulation of capsules and biological cells[END_REF]. The code has been validated in several configurations (axi / non-axisymmetric) in a capillary for rigid bodies and vesicles [START_REF] Trozzo | [END_REF] and in free space for vesicles [52][53][54].

III. MICROCAPSULE SHAPES IN A CAPILLARY

The shape of microcapsules moving along capillaries varies according two dimensionless parameters [44]:

• the capillary number Ca = η V G s (3)
• the confinement ratio

β = a R (4)
While the confinement β is a geometrical parameter, the capillary number Ca is the ratio of viscous stress η V /a on the membrane elastic response G s /a. For β << 1, the shape is quasi-spherical. As the capillary number (the microcapsule velocity) increases, the curvature at the rear changes its sign with the so-called 'parachute-like' shape already observed with alginatealbumine capsules [17] but with an unknown membrane material (constitutive law), and only with macroscopic capsules (diameter = 4 mm). Note also in figure 6 of [17] that all capsules are not axisymmetrical while it is the case here.

The confinement can be larger than unity contrary to a rigid particle, a scarcely studied range. The seminal work [17] focuses mainly on β ≤ 0.87. For β > 0.9, the lubrication film between the membrane and the wall is expected to play an important role. The array of shapes (Fig. 5) provides a good representation of all shapes which can be observed. These microcapsules are sufficiently soft to deform and enter in pores smaller than their own sizes (case β > 1). The results show an increasing of the lubrication film thickness with Ca as expected from studies on droplets and red blood cells moving in a capillary [43,55]. However, the range of data is too restricted to infer the power exponent. The mean flow velocity V (equation 2) is given to provide all necessary data for any numerical studies. The shear elastic modulus can be deduced from equation 1.

IV. MICROCAPSULE VELOCITY

We investigate the effects of flow parameters on the velocity of the capsule. Whereas the confinement ratio β is usually defined with the shape of the capsule at rest (equation 4), we calculate the local confinement L r /R by considering the radial extension of the capsule under flow (Fig. 3). The velocity of the capsule relative to the mean flow velocity V caps / V as a function of L r /R follows the same master curve for all the microcapsules (β ranges from 0.35 to 1.1) and large variations of L r /R (from 0.3 to 0.9, Fig. 6). It is interesting to compare this evolution with analytical or asymptotical theoretical results obtained for rigid particles in capillaries. For small values of confinement, the microcapsule is nearly spherical. In this case, the velocity of a solid sphere of radius a moving at the center of a cylindrical capillary [31] is given by:

V caps V = 2 - 4 3 β 2 for β 1 (5)
When β tends to 0, the velocity of the sphere is maximal and equal the velocity of the flow in the center of the capillary, i.e. 2 V . For deformable spheres, as microcapsules, if we replace β by the local confinement under flow L r /R in the previous equation, we obtain an excellent agreement with experiments up to β ≤ 0.6, as the shape of the microcapsules are quasi-spherical for small confinements.

As β tends to 1, the shape is strongly deformed even at small capillary number (or velocity) at the rear of the microcapsule but also on its lateral part which is facing the capillary wall. Then, the transverse size L r is smaller than the initial microcapsule size a and the capillary radius R. As the capillary number increases, the axial length L z of the microcapsule extends and becomes larger than the capillary diameter highlighting a long lubrication film. This is amplified for β > 1 (Fig. 5-d). The local confinement under flow L r /R is so the relevant parameter in this limit, because the thin film of thickness e ≈ R -L r and length L z exerts a viscous force of about 2πRηL z V caps /e. This force is counterbalanced by the pressure difference acting on the rear and the front of the microcapsule πR 2 L z (dp/dz). Finally, the only geometrical parameter that plays a role on the equilibrium, given by 2ηV caps ≈ dp/dz(1 -L r /R), is the local confinement L r /R. This analysis allows us to understand why the experimental evolution of V caps / V with L r /R corresponds perfectly with the analytical solutions obtained for rigid sphere on a large range of confinement [56], β ranging from 0.1 to 0.9. These results match also asymptotic analytical results [32] in the limit β → 1:

V caps V = 1 1 - 4 3 1 -β β + 46 15
1 -β β 2 for β → 1 (6)
Finally, the local confinement L r /R allows us to find the same evolution of the velocity of the microcapsule V caps as the one obtained for rigid spheres [56], albeit large axial extension L z . This latter has counterbalanced effects on the pressure force pushing the capsule and the viscous stress in the thin film between the capsule and the capillary. Experimental results are compared with asymptotic analysis [31,32] and analytical results [56] for hard spheres with radii equal to Lr.

0

V. FLOW AROUND MICROCAPSULES

In this section, we focus on the perturbed flow due to the presence of a microcapsule moving along the capillary in its stationary state. We report the experimental results obtained by PTV and extract the essential facts. We compare the results with numerical simulations at constant shape provided by experiments with no-slip condition (rigid body simulations) [49][START_REF] Pozrikidis | Modeling and simulation of capsules and biological cells[END_REF][START_REF] Trozzo | [END_REF][52][53][54]. This is relevant as the capsules have an axisymmetrical shape in the stationary state. In all generality, the flow profile is noted V(r, z) = (v r , v z ) in the reference frame of the microcapsule.

A. recirculation region

In the reference frame of the microcapsule, the fluid moves forward in the central region and backward close to the channel wall (Fig. 4). The velocity of the fluid in the axial direction of the capillary v z (r, z) shows a sign inversion between these two regions. At the front and the rear of the soft object, the flow presents two recirculation regions characterized by a non-nul radial component of the flow velocity v r (r, z) and the presence of stagnation rings on the rear and the front of the microcapsule (Fig. 4,7). The size of these inversion zones is defined by λ(z) the radius for which v z (r, z) equals zero, i.e. v z (λ, z) = 0. Far from the microcapsule, λ is deduced analytically by subtracting the velocity of the microcapsule to the velocity profile of a Poiseuille flow:

λ(z) = R 1 - V caps 2 V 1/2 for z → ±∞ (7)
The numerical and experimental evolution of λ with the axial direction z is compared in Fig. 7. Note that the experimental flow field has been interpolated to obtain a better precision on the position of λ. As expected λ is constant far from the microcapsule with only 9 % of difference between experimental and theoretical value (equation 7). That is in agreement with the integration experimental results over the depth of field of the objective (see section II D). Close to the microcapsule, the radius of the inversion zones λ(z) changes notably (Fig. 7). The regions of positive (red) and negative (blue) values of the flow velocity in the axial direction v z are also highlighted in Fig. 8 with numerical simulations done for the same microcapsule (β = 0.95) showing 'bullet-like' and 'parachute-like' shapes as its velocity V caps increases. At the front of the microcapsule, the curvature of the membrane being always the same whatever the confinement or the capillary number are, the radius of the inversion zone λ exhibits the same behavior with only one recirculation region and one stagnation ring. This flow characteristic is captured both numerically and experimentally by PTV and was also observed for drops or bubbles [30,33,36,37,46,47]. In contrast, at the rear of the microcapsule, it is noteworthy to see that the position as well as the number of stagnation rings of the capsule vary with the shape (Fig. 7 &8). The shape of these recirculation domains is closely related to the variations of curvature of the membrane in this region. Three stagnation rings on the microcapsule can be observed when the rear curvature starts to invert and a small additional recirculation region is generated in the vicinity of the curvature inversion (Fig. 7-top & 8-b) and λ can have multiple values for the same abscissa z. The lack of particles close to the membrane does not provide enough information to capture experimentally these small variations of λ. Increasing again the curvature of the rear, we observe experimentally and numerically that the two recalculations regions finally collapse in one (Fig. 7-bottom &8-c). For high velocity, the backward region penetrates inside the inverted rear of the microcapsule to almost reach the axis of the channel. Thus only one stagnation ring remains close to the centerline (Fig. 8-d).

B. Perturbation region

The presence of a single microcapsule in a Poiseuille flow perturbs the flow by creating recirculation regions at the rear and the front (Fig. 7). We determine the effects of the local confinement on the amplitude and the spatial extent of this perturbation. In this way, the averaged radial velocity <| v r |> along the capillary radius is calculated as a function of the axial The microcapsule is the same as presented in Fig. 3 position z. Experimental and numerical results show that <| v r |> vanishes far from the microcapsule and two distinct perturbation zones appear close to it (Fig. 9). To quantify the perturbation regions, the maximum amplitude of < |v r | >, called < |v r | > max is determined at the rear and also to the front of the microcapsule. The experimental data are previously smoothed by a local parabolic interpolation. Then we measure the distance l z from one end of the capsule to <| v r |>= 0.15 < |v r | > max (Fig. 9) at the front and the rear. This cut-off value was chosen to slightly exceed the level of noise of experimental data.

The dimensionless amplitude of the perturbation <| v r |> max /2 V increases linearly with the local confinement L r /R (Fig. 10). Again, experimental data are underestimated of about 9 % compared to numerical ones due to the depth of field of the objective. For moderate confinements (L r /R ≈ 0.7), the amplitude is the same for both sides of the microcapsule, i.e. about 12.4 % of the mean velocity of the Poiseuille flow. However, the amplitude of the rear perturbation region increases more steeply than the front one to reach up to 0.09 compared to 0.075 for a local confinement L r /R = 0.9.

While the amplitude varies with L r /R, the length of the perturbation l z shows no variations neither with the capillary number nor with the local confinement L r /R. l z is smaller upstream than downstream. For numerical simulations l z /R is equal to 0.87 ± 0.02 for the rear region and 0.70 ± 0.02 for front regions whatever Ca and a/R. These values are supported by experiments: l z /R equal to 0.87±0.03 for rear regions and 0.77±0.04 for the front. In figure 9 we notice that the amplitude of the perturbation zone abruptly increases from the capsule rear while it slowly increases and partially covers its front. As l z is measured from both ends of the capsule, this implies that l z is smaller at the front than at the rear of the capsule. Our results are in good agreement with theoretical developments for rigid spheres [57]. A distance of about R up and downstream is expected for the velocity of a cylindrical Poiseuille flow to recover unperturbed when locally disturbed.

VI. CONCLUSION

The velocity field around microcapsules in capillary tubes of comparable size has been explored. PTV measurements of experimental observations were compared successfully with numerical simulations developed from experimental microcapsule profile and velocity. Three aspects of the velocity fields have been investigated: the microcapsule velocity, the recirculation and disturbance regions. Some aspects of the flow are similar to the case of rigid spheres, drops or bubbles, whereas when the curvature of the rear of the microcapsule is inverted, the topology of the recirculation region has never been observed for such elongated shapes with no-slip condition at the interface due to the specificity of microcapsules.

The microcapsule velocity V c is found to depend only on its radial extension L r /R and unexpectedly not on its axial length L z whereas it varies in our experiment up to four times its radial extension (Fig. 5-d and Ca = 0.018). This is explained by the simultaneous increasing pressure drop and shear force in the thin film governed by lubrication. For this reason, the experimental capsule velocities are in good agreement with the analytical expression for confinement 0 < L r /R < 0.6 and numerical simulation for L r /R > 0.65, both considering the motion of a rigid sphere [31,32,56].

Since the microcapsule velocity varies between the mean flow rate and the maximal velocity of Poiseuille flow, the flow exhibits a front and back recirculation regions. Similarly to drops and bubbles [30,33,36,37,46,47], two stagnation rings were observed experimentally at the rear and the front of the microcapsule surface for low capillary numbers. With the rising of the capillary number, the radius of the ring at the front does not change while experimental results show a strong decreasing of the radius at the rear. This is due to the large change of shape at the rear. These are in agreement with numerical studies even if for moderate capillary numbers, the picture is more complex with the emergence of three stagnation rings at the rear (Fig. 8-b).

In addition to the recirculation regions, the velocity field is disturbed over a given distance which is a key parameter when multiple objects interact hydrodynamically. This is a preliminary before understanding microcapsules interaction in a Poiseuille flow. Indeed, drops, vesicles and red blood cells can assemble in a cluster in a Poiseuille flow as shown numerically [28,58,59] for example. We expect a similar effect with microcapsules. If the range of perturbed region does not vary with the capillary number and confinement, the amplitude of perturbed flow varies linearly with the confinement and more steeply for the rear part. Such a variation should affect the interaction between two microcapsules moving along a channel.

FIG. 1 .FIG. 2 .

 12 FIG. 1. Up -Schematic diagram of the experimental setup with an elongation flow: a microfluidic cross chamber. Bottom -Example of the deformation of a capsule with an initial radius of 37 µm. The variation of the deformation with the elongation shear stress provides the shear elastic modulus. See text and [9] for details.

C.

  Experimental setup for Poiseuile flow An axisymmetric Poiseuille flow is generated in glass capillary tubes (Hampton research) of 80 mm length and 150 to 200 µm nominal inner radii R. Glass tubes are glued on steel needles (Harvard apparatus) and plugged to a 10 mL glass syringe (Poulten and Graph GmbH).

FIG. 3 .

 3 FIG. 3. Parameters used to characterize the shape of microcapsules at rest and under flow. The geometry of the capillary tube is axisymmetric (er, ez). The radii of the microcapsule a and the capillary tube R are measured at rest (left). The velocity Vcaps, radial extension Lr, axial extension Lz and center of mass of the deformed microcapsule are measured under flow (right).

FIG. 4 .

 4 FIG. 4. Velocity field measured by PTV in the frame of reference of the microcapsule. The velocity field is measured from 7 movies of the same microcapsule pushed in the same conditions ; a = 159µm, 2R = 403µm, β = 0.8 (confinement, see Eq. 4), V = 3.65 mm/s, Gs = 0.12 N/m, Ca = 0.032.

FIG. 5 .

 5 FIG. 5. Experimental shapes of microcapsules for different confinement ratio β = a/R and capillary numbers Ca. When Ca increases, the curvature of the microcapsule rear inverts. The width of each box is equal to the capillary diameter 2R of 384 µm for column a and 403 µm for columns b, c and d. Mean flow velocity V from top to bottom for each column: a. 0.67 mm/s, 2.59 mm/s, 3.73 mm/s, 5.91 mm/s. b. 1.25 mm/s, 3.65 mm/s, 6.54 mm/s, 12.7 mm/s. c. 1.95 mm/s, 4.04 mm/s, 6.68 mm/s, 13.92 mm/s. d. 3.98 mm/s, 6.74 mm/s, 10.02 mm/s, 13.72 mm/s.

FIG. 7 .

 7 FIG.7. Velocity field and recirculation regions determined experimentally and numerically for two different microcapsules. Experimental (blue) and numerical (red) results are shown on the lower and upper half part of the capillary tube respectively. The size of the inversion zone λ(z) is defined as the distance from the centerline to vz(λ, z) = 0. Blue circles shows λ obtained from experiments while red squares from numerical simulations. The size of the image is 1162×403 µm 2 . Top: a = 195µm, β = 0.95, V = 4.04 mm/s, Vcaps = 4.98 mm/s, Ca = 0.01 ; Bottom: a = 231µm, β = 1.10, V = 13.72mm/s, Vcaps = 16.44 mm/s, Ca = 0.018.

FIG. 8 .

 8 FIG.8. Numerical simulations of the velocity field around the same microcapsule (a = 195µm, β = 0.95) for different mean flow velocities V and position of the inversion zone λ (red circles). Red and blue arrows show positive and negative axial flow velocity vz respectively. From top to bottom V equal to 1.95mm/s; 4.04mm/s; 6.68mm/s and 13.92mm/s. Note : same capsule than row c in figure4.

FIG. 9 .

 9 FIG.9. Front and rear perturbation regions are calculated numerically and experimentally from the mean of the radial velocity <| vr |> for each cross section z. These perturbations are characterized by the local maxima of the mean radial velocity <| vr |>max and the length lz between one microcapsule end to the threshold 0.15 <| vr |>max. The microcapsule is the same as presented in Fig.3

AKNOWLEDGMENTS

This work has benefited of financial support from the ANR CAPSHYDR (11-BS09-013-02), from Labex MEC (ANR-11-LABX-0092), from A*MIDEX (ANR-11-IDEX-0001-02) and from CNES. This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01).


R Pandharipande

D Zvonkine

Cohomological field theories with non-tautological classes

A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.

Introduction 0.1 Moduli of curves

Let M g,n be the moduli space of Deligne-Mumford stable curves of genus g with n markings [START_REF] Deligne | The irreducibility of the space of curves of given genus[END_REF]. There are natural forgetful morphisms dropping the last marking, p : M g,n+1 → M g,n ,

and boundary morphisms q : M g-1,n+2 → M g,n , r : M g1,n1+1 × M g2,n2+1 → M g,n ,

where n = n 1 + n 2 and g = g 1 + g 2 . The images of both q and r lie in the boundary ∂M g,n ⊂ M g,n .

Stability requires 2g -2 + n > 0.

The cohomology and Chow groups of the moduli space of curves are H * (M g,n , C) and A * (M g,n , C) .

See [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of Moduli[END_REF][START_REF] Pandharipande | A calculus for the moduli space of curves[END_REF] for a survey of results and open questions.

Cohomological field theories

The starting point for defining a Cohomological Field Theory [START_REF] Kontsevich | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF] is a triple of data (V, η, 1 ¯) where

• V = V 0 ⊕ V 1 is
a finite dimensional C-vector space1 with a Z 2 -grading,

• η is an even nondegenerate quadratic form -a non-degenerate 2-form on V which is symmetric on V 0 , skew-symmetric on V 1 , and satisfies

η(V 0 , V 1 ) = η(V 1 , V 0 ) = 0 , • 1 ¯∈ V 0 is a distinguished element.
Given a C-basis {e i } of V , the symmetric form η can be written as a matrix η jk = η(e j , e k ) .

The inverse matrix is denoted, as usual, by η jk . We will only consider bases which respect the grading of V . A Cohomological Field Theory consists of a system Ω = (Ω g,n ) 2g-2+n>0 of even tensors Ω g,n ∈ H * (M g,n , C) ⊗ (V * ) ⊗n .

The tensor Ω g,n associates a cohomology class in H * (M g,n , C) to vectors v 1 , . . . , v n ∈ V assigned to the n markings. Let Ω g,n (v 1 , . . . , v n ) denote the associated cohomology class in H * (M g,n , C). For vectors v i ∈ V v i of pure grading v i ∈ {0, 1}, the even condition is

deg Ω g,n (v 1 , . . . , v n ) = v 1 + • • • + v n mod 2 ,
where deg denotes the cohomological degree in H * (M g,n , C).

In order to define a Cohomological Field Theory, the system Ω = (Ω g,n ) 2g-2+n>0 must satisfy the CohFT axioms:

(i) Each tensor Ω g,n is S n -invariant (in the Z 2 -graded sense) for the natural action of the symmetric group S n on H * (M g,n , C) ⊗ (V * ) ⊗n obtained by simultaneously permuting the n marked points of M g,n and the n factors of V * .

(ii) The tensor q * (Ω g,n ) ∈ H * (M g-1,n+2 , C) ⊗ (V * ) ⊗n , obtained via pull-back by the boundary morphism

q : M g-1,n+2 → M g,n ,
is required to equal the contraction of Ω g-1,n+2 by the bi-vector j,k

η jk e j ⊗ e k inserted at the two identified points:

q * (Ω g,n (v 1 , . . . , v n )) = j,k
η jk Ω g-1,n+2 (v 1 , . . . , v n , e j , e k ) in H * (M g-1,n+2 , C) for all v i ∈ V .

The tensor r * (Ω g,n ), obtained via pull-back by the boundary morphism2 r : M g1,n1+1 × M g2,n2+1 → M g,n ,

is similarly required to equal the contraction of Ω g1,n1+1 ⊗ Ω g2,n2+1 by the same bi-vector:

r * (Ω g,n (v 1 , . . . , v n )) = j,k
η jk Ω g1,n1+1 (v 1 , . . . , v n1 , e j ) ⊗ Ω g2,n2+1 (e k , v n1+1 , . . . , v n ) in H * (M g1,n1+1 , C) ⊗ H * (M g2,n2+1 , C) for all v i ∈ V .

(iii) The tensor p * (Ω g,n ), obtained via pull-back by the forgetful map

p : M g,n+1 → M g,n , is required to satisfy Ω g,n+1 (v 1 , . . . , v n , 1 ¯) = p * Ω g,n (v 1 , . . . , v n )
for all v i ∈ V . In addition, the equality

Ω 0,3 (v 1 , v 2 , 1 ¯) = η(v 1 , v 2 ) is required for all v i ∈ V . Definition 1 A system Ω = (Ω g,n ) 2g-2+n>0 of tensors Ω g,n ∈ H * (M g,n , C) ⊗ (V * ) ⊗n
satisfying (i), (ii), and (iii) is called a Cohomological Field Theory (CohFT) with unit.

The simplest example of a Cohomological Field Theory with unit is given by the trivial CohFT,

V = V 0 = Q , η(1, 1) = 1 , 1 ¯= 1 , Ω g,n (1, . . . , 1) = 1 ∈ H 0 (M g,n , C) .
A more interesting example is given by the total Chern class

c(E) = 1 + λ 1 + . . . + λ g ∈ H * (M g,n , C) of the rank g Hodge bundle E → M g,n , V = V 0 = Q , η(1, 1) = 1 , 1 ¯= 1 , Ω g,n (1, . . . , 1) = c(E) ∈ H * (M g,n , C) . Definition 2 For a CohFT Ω = (Ω g,n ) 2g-2+n>0 , the topological part ω of Ω is defined by ω g,n = [Ω g,n ] 0 ∈ H 0 (M g,n , C) ⊗ (V * ) ⊗n .
The topological (or degree 0) part [ ] 0 of Ω is simply obtained from the canonical summand projection

[ ] 0 : H * (M g,n , C) → H 0 (M g,n , C) .
If Ω is a CohFT with unit, then ω is also a CohFT with unit. The topological part of the CohFT obtained from the total Chern class of the Hodge bundle is the trivial CohFT.

A Topological Field Theory (TopFT) with unit is a CohFT θ with unit of the form

θ g,n ∈ H 0 (M g,n , C) ⊗ (V * ) n 2g-2+n>0
.

The topological part of a CohFT with unit is a TopFT with unit.

Tautological cohomology

The subrings of tautological classes on the moduli spaces of curves,

RH * (M g,n , C) ⊂ H * (M g,n , C) ,
have been extensively studied -see [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of Moduli[END_REF][START_REF] Graber | Constructions of nontautological classes on moduli spaces of curves[END_REF][START_REF] Janda | Relations on M g,n via equivariant Gromov-Witten theory of P 1[END_REF][START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF] for definitions, generators, and Pixton's conjectured set of relations.

Definition 3 A CohFT Ω takes the value γ ∈ H * (M g,n , C) if γ lies in the image of Ω g,n : V ⊗n → H * (M g,n , C) .
As a consequence of the Givental-Teleman classification [START_REF]Gromov-Witten invariants and quantization of quadratic Hamiltonians[END_REF][START_REF]Semisimple Frobenius structures at higher genus[END_REF][START_REF] Teleman | The structure of 2D semi-simple field theories[END_REF], all semisimple CohFTs with unit take values in the tautological cohomology of the moduli spaces of curves.

Constructions

There are several constructions of CohFTs with unit -Gromov-Witten theory, Witten's r-spin class, and the Chern characters of the Verlinde bundle all define CohFTs with unit, see [START_REF] Marian | The Chern character of the Verlinde bundle over M g,n[END_REF][START_REF] Pandharipande | Cohomological field theory calculations[END_REF][START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF][START_REF] Pandharipande | Tautological relations via r-spin structures[END_REF]. Moreover, once a CohFT is found, others can be constructed via the action of the Givental group, see [START_REF] Pandharipande | Relations on M g,n via 3-spin structures[END_REF][START_REF] Shadrin | BCOV theory via Givental group action on cohomological fields theories[END_REF][START_REF] Teleman | The structure of 2D semi-simple field theories[END_REF].

Our perspective here is different. We would like to construct CohFTs by hand. The task is difficult since an infinite amount of compatibility (involving all higher genera) is required. Our first result concerns minimal classes.

Definition 4 A class γ ∈ H * (M g,n , C) is minimal if q * (γ) = 0 and r * (γ) = 0
for all boundary maps to M g,n of type q and r.

In other words, γ ∈ H * (M g,n , C) is minimal if the restriction of γ to every boundary component of M g,n vanishes. For example, the Poincaré dual of a point in H 2(3g-3+n) (M g,n , C) is always a minimal class.

Definition 5 A minimal class γ ∈ H * (M g,n , C) satisfies the parity condition if either the degree of γ is even or if both the degree of γ and the number n of marked points are odd.

Theorem 6 Let γ ∈ H * (M g,n , C) be a minimal class that satisfies the parity condition. Then there exists a CohFT with unit Ω γ which takes the value γ.

More precisely, for every minimal class γ satisfing the parity condition, we construct a canonical CohFT Ω γ taking the value γ.

There exist non-tautological cohomology classes on the moduli space of curves. The simplest is perhaps

0 = φ ∈ H 11,0 (M 1,11 , C) ∼ = C
defined via the discriminant modular form, see [START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of Moduli[END_REF]Section 2] for an exposition. Since φ is a class of odd cohomological degree,

φ / ∈ RH * (M 1,11 , Q) .
Since no boundary component of M 1,11 has nonvanishing odd cohomology, φ is a minimal class (which also satisfies the parity condition).

Corollary 7

The CohFT with unit Ω φ takes values outside of the tautological cohomology of the moduli spaces of curves.

The CohFT Ω φ is the first known example of a CohFT with unit taking non-tautological values. Whether the Gromov-Witten theory of a nonsingular projective variety X can ever take non-tautological values is an interesting question. For semisimple X, the Gromov-Witten CohFT must take values in the tautological ring. Perhaps the simplest non-semisimple variety X is a curve of higher genus. However, the CohFTs obtained from the Gromov-Witten theories of target curves have been proven to take values in tautological cohomology by Janda [START_REF] Janda | Gromov-Witten theory of target curves and the tautological ring[END_REF].

On the other hand, the Gromov-Witten theory of a nonsingular projective variety X may produce classes outside of the tautological ring

R * (M g,n , C) ⊂ A * (M g,n , C)
in Chow. Simple examples can be found in the case of higher genus target curves X. The class of the moduli point3

[X, p 1 , . . . , p n ] ∈ A 0 (M g,n , C) occurs as push-forward to M g,n of n i=1 ev * i (p i ) ∩ M g,n (X, [X]) vir ∈ A 0 (M g,n (X, [X]), C) .
However, the virtual class has the possibility of being better behaved in cohomology.

Minimal classes of even degree

Minimal classes have played an important role in the study of the tautological ring. For example, the tautological class,

λ g λ g-1 ∈ H 4g-2 (M g , C) ,
which appears in the socle evaluation of RH * (M g ) for g ≥ 2, is well-known to be minimal. While we expect the existence of non-tautological minimal classes of even cohomological degree, we do not know any examples at the moment. Since all of the even degree cohomology in genus 1 is tautological [START_REF] Petersen | The structure of the tautological ring in genus one[END_REF], non-tautological minimal classes of even degree do not exist on M 1,n . D. Petersen has provided a proof of the non-existence of non-tautological minimal classes of even degree on M 2,n which appears in the Appendix. So the search for non-tautological minimal classes of even degree should start in genus 3.

Acknowledgments

The construction presented here was found in Paris (at Café Nuance) in preparation for the colloquium of R.P. at Jussieu in December 2017. The results were also discussed at the conference Hurwitz cycles on the moduli of curves at Humboldt University in Berlin in February 2018 (organized with G. Farkas). We thank J. Schmitt for discussions about minimal classes and D. Petersen for contributing the Appendix on minimal classes in low genus. R.P. was partially supported by SNF grant 200021-143274, ERC grant AdG-320368-MCSK, SwissMAP, and the Einstein Stiftung.

Construction of Ω γ

Let γ ∈ H * (M h,m , C) be a minimal class satisfying the parity condition. If γ ∈ H * (M 0,3 , C), then the trivial CohFT takes the value γ. For the proof of Theorem 6, we may therefore assume

γ ∈ H * (M h,m , C) , (h, m) = (0, 3) .
Then, since γ is minimal, the cohomological degree of γ must be positive,

γ ∈ H >0 (M h,m , C) , (h, m) = (0, 3) .
To construct a canonical CohFT with unit Ω γ which takes the value γ, we start with the topological field theory ω γ associated with the Frobenius algebra structure of H * (X, C), where X is a genus m curve4 . We then modify ω γ by hand by adding higher degree classes to obtain Ω γ . We begin with a careful description of ω γ .

State space

The state space (V, η, 1 ¯) of both Ω γ and ω γ is described as follows.

• Let V be the Z 2 -graded C-vector space of dimension 2m + 2 with basis given by the vectors

a, b 1 , b 2 , . . . , b m , c 1 , c 2 , . . . , c m , d . (1)
and grading

a = d = 0, b i = c i = 1.
• Let η be the non-degenerate graded-symmetric 2-form on V defined by

η(a, d) = η(d, a) = 1 , η(b i , c i ) = -η(c i , b i ) = 1 ,
and η vanishes on all other pairs of basis vectors.

• Let 1 ¯= a.

Let B be the set of 2m + 2 basis vectors [START_REF] Bergström | Siegel modular forms of degree three and the cohomology of local systems[END_REF]. The span of

a, b 1 , . . . , b m
in V is a maximal isotropic subspace with respect to η. We will use the notation

η(v 1 , v 2 ) = v 1 , v 2 .
The bi-vector dual to η can be written explicitly as

a ⊗ d + d ⊗ a - m i=1 (b i ⊗ c i -c i ⊗ b i ) . ( 2
)
1.2 The algebra structure

The vector space V carries an algebra structure given by the following multiplication rules.

• a is the (left and right) unit of the algebra,

• b i ⋆ c i = -c i ⋆ b i = d,
• all pairwise products of basis elements vanish, except in the two cases above.

Because the above algebra is isomorphic to the cohomology algebra of a genus m surface, there is a natural Z-grading which lifts the Z 2 -grading:

a = 0 , b i = c i = 1 , d = 2 .
Remark 8 The algebra structure on V is not semisimple. Indeed, all elements of the basis B except a are nilpotents.

The values of ω γ

Since ω γ g,n takes values in H 0 (M g,n , C) which is canonically C, we view the values of ω γ g,n as complex numbers.

Proposition 9

The TopFT ω γ has the following evaluation on basis elements of B.

•

In genus g = 0, ω γ 0,n (d, a, . . . , a n-1
) = 1 and

ω γ 0,n (b i , c i a, . . . , a n-2 ) = 1 for 1 ≤ i ≤ m .
The evaluations obtained from these by permuting the entries are equal to 1 or -1 as determined by the grading. All other evaluations vanish.

• In genus g = 1, ω γ 1,n (a, . . . , a) = 2 -2m and all other evaluations vanish. • In genus g ≥ 2, all evaluations vanish, ω γ g,n = 0.

Proof. By the axioms of a TopFT, the genus 0 evaluation

ω γ 0,n (v 1 , . . . , v n ) is equal to η(v 1 ⋆ • • • ⋆ v n , 1 ¯), which is the coefficient of d in the product v 1 ⋆ • • • ⋆ v n .
Using the Z-grading of the algebra, the coefficient vanishes unless the sum of the gradings of v 1 , . . . , v n equals 2, whence the result.

For the higher genus cases, we evaluate the 0-cohomology class ω γ g,n (v 1 , . . . , v n ) on a point of the moduli space that corresponds to a genus g curve with g nonseparating nodes, in other words, a rational curve with g pairs of identified points. Using the genus 0 case, only one node is possible without vanishing, therefore ω γ vanishes for g ≥ 2. The factor 2 -2m (the Euler characteristic of a genus m curve) in the g = 1 case occurs via CohFT axiom (ii) and the definition of the symmetric form η and its inverse bi-vector. ♦

1.4 Full CohFT Ω γ Recall γ ∈ H * (M h,m , C) is a minimal class satisfying γ ∈ H >0 (M h,m , C) , (h, m) = (0, 3) .
For now let's assume that the degree of γ has the same parity as the number of marked points m.

Definition 10

The values of the full CohFT (Ω γ g,n ) 2g-2+n>0 on basis vectors from B are defined by the following rules.

• Let g = h and n ≥ m. Let p : M h,n → M h,m be the forgetful map. Then we have

Ω γ g,n (b 1 , b 2 , . . . b m , a, . . . , a n-m ) = p * γ . (3)
• The evaluations obtained from the above by permutations of the entries are equal to p * γ or -p * γ according to the grading.

• In all other cases, namely, either g = h, or g = h, but n < m, or g = h and n ≥ m, but the basis vectors on which Ω γ is evaluated are not obtained by a permutation of (3), we have

Ω γ g,n (v 1 , . . . , v n ) = ω γ g,n (v 1 , . . . , v n ) .
The tensor Ω γ g,n is even, because we have assumed that deg γ and m have the same parity. Certainly Ω γ takes the value γ since, by definition,

Ω γ h,m (b 1 , . . . , b m ) = γ ∈ H * (M h,m , C) .
To complete the proof of Theorem 6, we must check that Ω γ satisfies all of the required axioms for a CohFT with unit. CohFT axiom (i), invariance under the symmetric group, and CohFT axiom (iii), compatibility with the forgetful map p, both follow immediately from the construction.

CohFT axiom (ii) for Ω γ

Our first remark is that (3) of Definition 10 can be rewritten equivalently as

Ω γ g,n (b 1 , . . . , b m , a, . . . , a n-m ) = ω γ g,n (b 1 , . . . , b m , a, . . . , a n-m ) + p * γ. Indeed, ω γ g,n (b 1 , . . . , b m , a, . . . , a n-m
) = 0 for all g and n as shown in Proposition 9.

(If g ≥ 2 then all correlators vanish. If g = 0 we need either a d-insertion or a b-and a c-insertion to get a nonvanishing correlator, which we don't have here.

Finally, if g = 1 a nonvanishing correlator has a-insertions only. This could be the case here if m = 0. But since γ is a cohomology class in M h,m , the case h = 1, m = 0 is ruled out.)

In order to check CohFT axiom (ii) for Ω γ , we must prove the compatibility of Ω γ under pull-back for the boundary morphisms of type q and r. Since CohFT axiom (ii) holds for ω γ , we must only study the effect of the correction term p * (γ).

Consider first the boundary morphism

q : M g-1,n+2 → M g,n .
There are two cases where the correction term has the possibility of appearing: ) .

• If g -1 = h,
Then q * (Ω γ h,n (v 1 , . . . , v n )) vanishes since γ is minimal. Thus the minimality of γ ensures that the compatibility still holds.

Consider next the boundary morphism

r : M g1,n1+1 × M g2,n2+1 → M g,n ,
where n = n 1 + n 2 and g = g 1 + g 2 .

On the right-hand side, the correction term appears in

Ω γ h,n (b 1 , . . . , b m , a, . . . , a n-m
) and terms obtained from this by permutations of entries. We enumerate the cases where the correction term appears on the right-hand side as follows. Using Proposition 9, there are exactly four cases, up to permuting the n 1 entry vectors of the first factor and the n 2 entry vectors in the second factor. In the formulas below, a vector with a hat denotes a skipped entry, while an underlined vector originates from the sum η jk e j ⊗ e k :

•

Ω γ h,n1+1 (b 1 , . . . , b m , a, . . . , a n1-m , a) × Ω γ 0,n2+1 (d, a, . . . , a n2
),

•

Ω γ h,n1+1 (b 1 , . . . , b i , . . . , b m , a, . . . , a n1-m+1 , b i ) × Ω γ 0,n2+1 (c i , b i , a, . . . , a n2-1
),

• Ω γ 0,n1+1 (a, . . . , a

n1 , d) × Ω γ h,n2+1 (a, b 1 , . . . , b m , a, . . . , a n2-m
),

• Ω γ 0,n1+1 (b i , a, . . . , a

n1-1 , c i , ) × Ω γ h,n2+1 (b i , b 1 , . . . , b i , . . . , b m , a, . . . , a n2-m+1
).

To show that these are the only cases, we use two simple remarks:

(i) Since each term in the bi-vector η -1 always contains a basis vector of type c or d as a factor, the correction term can appear only in one of the two factors.

(ii) The factor without correction contains an entry of c or d type and thus can only be of genus 0 so as not to vanish.

We can now prove the equality

r * (Ω g,n (v 1 , . . . , v n )) = j,k η jk Ω g1,n1+1 (v 1 , . . . , v n1 , e j ) ⊗ Ω g2,n2+1 (e k , v n1+1 , . . . , v n ) .
First, from the analysis above, we see that both sides vanish unless g = h and Let p : M h,n → M h,m be the map which forgets the marked points carrying the basis vector a. Consider the image of the map p • r. We have

r * Ω g,n (v 1 , . . . , v n ) = ±(p • r) * γ.
By the minimality of γ, this class vanishes whenever the image of p • r is a boundary stratum. In order for p • r to be onto, p has to contract one of the two irreducible components of the curve. A component is contracted if it has genus 0 and all of its marked points, except perhaps one, carry the basis vector a. This leaves us with exactly the same four cases as in the enumeration above. In the first two cases, both r * Ω g,n (v 1 , . . . , v n ) and j,k

η jk Ω g1,n1+1 (v 1 , . . . , v n1 , e j ) ⊗ Ω g2,n2+1 (e k , v n1+1 , . . . v n ) are equal to ±(p • r) * γ ⊗ 1 ,
the sign determined by the order of the vectors, which is the same on both sides. Similarly, in the last two cases, both

r * Ω g,n (v 1 , . . . , v n ) and j,k η jk Ω g1,n1+1 (v 1 , . . . , v n1 , e j ) ⊗ Ω g2,n2+1 (e k , v n1+1 , . . . v n ) are equal to ±1 ⊗ (p • r) * γ .
Again, the sign is determined by the order of the vectors, which is the same on both sides. The proof of Theorem 6 is complete. ♦

Remark 11

The strategy can be summarized as follows. We start with a Topological Field theory ω γ which depends only on the number of markings m, γ ∈ H >0 (M h,m , C) .

We then define Ω γ by adding a correction term to ω γ in the particular cases

Ω γ h,n (b 1 , . . . , b m , a, . . . , a n-m ) = ω γ h,n (b 1 , . . . , b m , a, . . . , a n-m ) + p * n,m (γ) , (4)
up to permutations of the entries. To verify the CohFT axioms for Ω γ , we use the minimality of γ for lower moduli spaces and the exact solution of the TopFT ω γ of Proposition 9 for higher moduli spaces. It is important that the correction (4) occurs only for insertions in the maximal isotropic subspace spanned by a, b 1 , . . . , b m .

Remark 12 If γ is a minimal class of even degree, it is possible to remove the grading from the construction. The basis of the vector space V remains the same, but the quadratic form η becomes symmetric and, in general, all the signs related to the grading disappear. In this case, it is not necessary to require the number of marked points m to have the same parity as γ.

2 Deformations

General theory

Let Ω be a CohFT with unit and state space (V, η, 1 ¯).

Definition 13 A system of tensors

Λ = Λ g,n : V ⊗n → H * (M g,n , C) 2g-2+n>0
defines a first order deformation of Ω if

Ω g,n + ǫΛ g,n 2g-2+n>0
satisfies the axioms of a CohFT with unit modulo ǫ 2 = 0.

If the system of tensors Λ = (Λ g,n ) 2g-2+n>0 satisfies the further condition Λ 0,3 = 0 , then the first order deformation Λ of Ω preserves the TopFT structure. The proof of Theorem 6 shows that every minimal class

γ ∈ H >0 (M h,m , C)
whose degree has the same parity as m yields a first order deformation of ω γ which preserves the TopFT structure. We can write the CohFT axiom conditions for the deformation

Ω g,n + ǫΛ g,n 2g-2+n>0
more explicitly:

(i) Each tensor Λ g,n is S n -invariant(in the Z 2 -graded sense) for the natural action of the symmetric group S n .

(iiq) The tensor q * (Λ g,n ) ∈ H * (M g-1,n+2 , C) ⊗ (V * ) ⊗n , obtained via pull-back by the boundary morphism

q : M g-1,n+2 → M g,n ,
is required to equal the contraction of Λ g-1,n+2 by the bi-vector j,k

η jk e j ⊗ e k .

(iir) The tensor r * (Λ g,n ), obtained via pull-back by the boundary morphism

r : M g1,n1+1 × M g2,n2+1 → M g,n ,
is required to equal j,k

η jk Ω g1,n1+1 (v 1 , . . . , v n1 , e j ) ⊗ Λ g2,n2+1 (e k , v n1+1 , . . . , v n ) + j,k η jk Λ g1,n1+1 (v 1 , . . . , v n1 , e j ) ⊗ Ω g2,n2+1 (e k , v n1+1 , . . . , v n ) .
(iii) The tensor p * (Λ g,n ), obtained via pull-back by the forgetful map

p : M g,n+1 → M g,n , is required to satisfy Λ g,n+1 (v 1 , . . . , v n , 1 ¯) = p * Λ g,n (v 1 , . . . , v n ) for all v i ∈ V .
The simplest method of constructing deformations of Ω is via Givental's R-matrix action. Other deformations are, in general, hard to find.

Isotropic deformations of the CohFT ω m

Consider the topological field theory ω m defined in Sections 1.1 and 1. and carries the non-degenerate graded-symmetric 2-form on η defined by

η(a, d) = η(d, a) = 1, η(b i , c i ) = -η(c i , b i ) = 1
with values on all other pairs of basis vectors defined to vanish. Definition 14 A first order deformation of ω m defined by Λ is isotropic if

Λ g,n (v 1 , . . . , v n-1 , c i ) = Λ g,n (v 1 , . . . , v n-1 , d) = 0
for all g, n, v j ∈ B, and

1 ≤ i ≤ n.
Theorem 15 Let Λ define an isotropic first order deformation of ω m which preserves the TopFT structure. Then,

Λ g,n (v 1 , . . . , v n ) ∈ H * (M g,n , C) for v j ∈ {b 1 , . . . , b m } is always a minimal class.
Proof. Let (v 1 , . . . , v n ) be an n-tuple of vectors satisfying v j ∈ {b 1 , . . . , b m } for all j. In order to prove that the class

Λ g,n (v 1 , . . . , v n ) ∈ H * (M g,n , C)
is minimal, we must study the pull-backs under the boundary morphisms of type q and r.

• The vanishing of q * (Λ g,n (v 1 , . . . , v n )) under

q : M g-1,n+2 → M g,n
is immediate from axiom (iiq) of Section 2.1: each term of the bi-vector (2) always includes a basis element of type c or d.

• The vanishing of r * (Λ g,n (v 1 , . . . , v n )) under r : M g1,n1+1 × M g2,n2+1 → M g,n ,
where n = n 1 + n 2 and g = g 1 + g 2 , follows directly from axiom (iir), Proposition 9 for ω m , and the isotropic property of Λ.

Since Λ g,n (v 1 , . . . , v n ) vanishes under all boundary restrictions, the class is minimal. ♦ Theorems 6 and 15 show minimal classes γ satisfying deg γ = m mod 2 and the isotropic deformations of ω m are essentially equivalent notions. Deformations of ω m via Givental's R-matrix action will in general not be isotropic. Also, the R-matrix action will produce only deformations taking values in the tautological cohomology -so the deformations obtained from minimal classes

γ ∈ H >0 (M h,m , C)
will not always be obtained via the R-action.

Appendix: Minimal cohomology classes on M g,n in low genus by D. Petersen

A1. Minimal cohomology

In this appendix, we make some remarks regarding minimal cohomology classes on M g,n . Cohomology will always be taken with Q-coefficients. We begin by recalling a useful result from Deligne's mixed Hodge theory.

Proposition 1 Let X be a smooth projective variety, Z ⊂ X a closed subvariety 5 of pure codimension c with complement U , Z → Z any resolution of singularities. There are exact sequences

H k-2c ( Z)(-c) → H k (X) → Gr W k H k (U ) → 0 and 0 → Gr W k H k c (U ) → H k (X) → H k ( Z).
Proof. The first exact sequence follows by combining [4, Corollaire

. The second one is just the Poincaré dual of the first one. ♦

We apply Proposition 1 to the case where X = M g,n , U = M g,n , and Z is the normalization of the boundary, so that each component 6 of Z is either of the form

M g ′ ,n ′ +1 × M g-g ′ ,n-n ′ +1 or M g-1,n+2 . Definition 2 Let H k min (M g,n ) ⊆ H k (M g,n
) denote the subspace of minimal cohomology classes.

Proposition 3 For all k, g, and n, there is an isomorphism

Gr W k H k c (M g,n ) ∼ = H k min (M g,n ) .
Proof. Immediate from the second exact sequence of Proposition 1. ♦ Proposition 4 There is a perfect pairing between

Gr W k H k (M g,n ) and H 2(3g-3+n)-k min (M g,n ) .
5 Z need not be irreducible. 6 For some components, Z is 2-fold cover of the normalization, but Proposition 1 still holds.

Proof. There is a perfect pairing between H k (M g,n ) and H

2(3g-3+n)-k c

(M g,n ) by Poincaré duality, which induces a perfect pairing between their respective associated graded for the weight filtration. With the previous proposition, the result follows. ♦

In fact the perfect pairing of Proposition 4 can be described more explicitly. Take any class α ∈ Gr W k H k (M g,n ), and lift it (non-canonically) to a class

α ∈ H k (M g,n ). If β ∈ H 2(3g-3+n)-k min (M g,n ) is a class of complementary degree, then the cup product α ∪ β ∈ H 2(3g-3+n) (M 2,n ) ∼ = Q is in fact well defined: if α ′ is a different choice of lift, then the difference α -α ′
is pushed forward from the boundary, so its integral against β is zero since β vanishes on the boundary. The reader familiar with the λ g λ g-1 -pairing on the tautological ring of M g will find this construction familiar; indeed, what makes the λ g λ g-1 -pairing work is precisely that λ g λ g-1 is a minimal class on M g .

A2. Minimal classes in genus zero

Proposition 5 The point class in H 2(n-3) (M 0,n ) is the only minimal class in H * (M 0,n ).
Proof. There are very many ways to see this; here is one. By Proposition 4, the claim is equivalent to Gr W k H k (M 0,n ) = 0 for k > 0. Consider the compactification M 0,n ⊂ C n-3 ⊂ P n-3 . By the first exact sequence of Proposition 1, there is a surjection H k (P n-3 ) → Gr W k H k (M 0,n ). This map factors through H k (C n-3 ), which vanishes for k > 0. ♦

A3. Minimal classes in genus one

Proposition 6 The point class in H 2n (M 1,n ) is the only minimal class of even degree in H * (M 1,n ).

Proof. By Proposition 4, this is equivalent to the claim that Gr W 2k H 2k (M 1,n ) is nontrivial only for k = 0, which is exactly [START_REF] Petersen | The structure of the tautological ring in genus one[END_REF]Theorem 1.1]. ♦

As pointed out in the body of this paper, the existence of odd cohomology in H * (M 1,n ) for n ≥ 11 implies that there exist plenty of odd minimal classes in genus one. For any positive integer k there is a rational Hodge structure S[k] which is "attached" to the space S k of cusp forms for SL(2, Z) of weight k. Over C, the Hodge structure becomes canonically the direct sum of the spaces of holomorphic and antiholomorphic cusp forms:

S[k] ⊗ Q C ∼ = S k ⊕ S k .
The Hodge structure S[k] is pure of weight k -1, and the Hodge numbers of the two summands above are (k -1, 0) and (0, k -1). Arguments similar to those of [19, Section 2 and 3] allow one to prove the following more refined statement.

Proposition 7 There is an isomorphism

Gr W k H k (M 1,n ) ∼ = S[k + 1] ⊗ Ind Sn S k ×S n-k sgn k ⊗ 1 n-k for all n ≥ k. When n < k, Gr W k H k (M 1,n ) = 0.
We omit the proof.

A4. Minimal classes from tautological classes

Before moving on to genus two, let us point out an obvious source of classes in Gr W k H k (M g,n ): every tautological class on M g,n is of pure weight, being the class of an algebraic cycle. Thus every tautological class in RH k (M g,n ) must "give rise to" a minimal class in degree 2(3g -3 + n) -k, where we put quotation marks to emphasize that the vector spaces Gr W k H k (M g,n ) and H 2(3g-3+n)-k min (M g,n ) are dual to each other and a vector in one of them does not correspond canonically to any vector in the other one.

By a result of Buryak-Shadrin-Zvonkine [START_REF] Buryak | Top tautological group of M g,n[END_REF], the tautological ring of M g,n vanishes above cohomological degree 2(g -1) for n > 0, and RH 2(g-1) (M g,n ) is spanned by the classes ψ g-1 1 , ψ g-1 2 , . . . , ψ g-1 n . These classes should correspond to certain minimal classes in the cohomology of M g,n , and this is in fact explained in [START_REF] Buryak | Top tautological group of M g,n[END_REF]: let

α s = λ g λ g-1 ψ 1 ψ 2 . . . ψ s • • • ψ n
where the hat means an omitted factor. In [2, Section 2] the authors show that the classes α s are minimal, and that the pairing between the n classes α 1 , . . . , α n and ψ g-1 1 , . . . , ψ g-1 n is perfect. The authors of [START_REF] Buryak | Top tautological group of M g,n[END_REF] also proposed a generalization of the Faber conjecture: that the ring RH * (M g,n ) is level of type n, i.e. that a class in the tautological ring vanishes if and only if its product with any class of complementary degree vanishes. We expect this to be false. However, if we assume this statement for the moment, then every nonzero α ∈ RH k (M g,n ) pairs nontrivially with a class in RH 2(3g-3+n)-k min (M g,n ), and the collection of minimal classes on M g,n that "'corresponds to" RH * (M g,n ) is just the ideal in RH * (M g,n ) generated by the n minimal classes α 1 , . . . , α s .

Remark 8

In genus zero and one, every cohomology class (resp. every even degree cohomology class) on M g,n is tautological. Moreover, the tautological ring of M g,n is trivial in these cases. This is another way of seeing Propositions 5 and 6.

A5. More interesting examples in genus two

When g = 2, the tautological ring RH * (M 2,n ) has rank n in cohomological degree 2 and vanishes above this, and, as explained in the previous subsection, one finds a subspace of n minimal classes in H 4+2n min (M 2,n ). However, in genus two it is not true that every class in Gr W 2k H 2k (M 2,n ) is tautological. The first time one sees a pure even degree class on M 2,n is when n = 20, in which case one finds the class constructed by Graber-Pandharipande [START_REF] Graber | Constructions of nontautological classes on moduli spaces of curves[END_REF] ) is also spanned by a copy of the representation [2, 2, . . . , 2]. But it is also known that the Graber-Pandharipande class (and its S 20 -conjugates) are the only nontautological even degree classes on M 2,20 , in the sense that the span of these classes and the tautological ring is the whole even cohomology. In particular, the classes in H 24 min (M 2,20 ) must be tautological. The assertions of this paragraph are proven in [START_REF] Petersen | Tautological rings of spaces of pointed genus two curves of compact type[END_REF][START_REF] Petersen | The Gorenstein conjecture fails for the tautological ring of M 2,n[END_REF].

By what we have said so far, there must exist a tautological minimal class on M 2,20 , very different from the "obvious" examples given by the classes α s from the previous subsection.

Problem 9 Find a geometric construction7 of a minimal class in R 12 (M 2,20 ).

As we will see shortly, the class must in fact be the pushforward of a class in R 11 (M 1,22 ).

Hain and Looijenga [10, Conjecture 5.4] at one point conjectured that the ideal of minimal classes in the tautological ring of M g is principal, generated by λ g λ g-1 ; this was part of a proposed generalization of Faber's conjecture on M g . The obvious generalization to incorporate marked points would be that the ideal of minimal classes in R * (M g,n ) is generated by α 1 , . . . , α n . From what we have said here, such a conjecture is false (and fails "for the first time" on M 2,20 ).

For n > 20, one finds a larger and larger number of nontautological classes, and one could hope to find a nontautological even minimal class in genus two. Unfortunately, this is not possible: Proposition 10 Every class in H 2k min (M 2,n ) is pushed forward along the boundary map from H 2k-2 (M 1,n+2 ). In particular, every even minimal cohomology class in genus two is tautological.

Proof. From Proposition 1 we get the short exact sequence

H k-2 (M 1,n+2 ) → H k (M 2,n ) → H k (M ct 2,n ) .
This shows that classes not pushed forward from M 1,n+2 map injectively into H k (M ct 2,n ). Since the map from Gr W k H k c (M 2,n ) = H k min (M 2,n ) factors through H k c (M ct 2,n ), it is enough to argue that a minimal class of even degree which is in the image of H k c (M ct 2,n ) → H k (M ct 2,n ) is actually zero. Now we apply the results of [START_REF] Petersen | Tautological rings of spaces of pointed genus two curves of compact type[END_REF]. From the study of the decomposition theorem in that paper we see that if f : M ct 2,n → M ct 2 is the forgetful map then there are isomorphisms

H k c (M ct 2,n ) ∼ = p+q=k H p c (M ct 2 , R q f * Q), H k (M ct 2,n ) ∼ = p+q=k H p (M ct 2 , R q f * Q),
compatible with the map H k c (M ct 2,n ) → H k (M ct 2,n ), and that the sheaves R q f * Q decompose as direct sums of local systems on M ct 2 and on Sym 2 M 1,1 associated to representations of the symplectic group.

The cohomology of local systems on M ct 2 = A 2 is known from [START_REF] Petersen | Cohomology of local systems on the moduli of principally polarized abelian surfaces[END_REF], where it is in particular shown that the map H k c (A 2 , V λ ) → H k (A 2 , V λ ) can only be nontrivial in the middle degree k = 3. In particular a minimal class on M ct 2,n which lies in a summand corresponding to such a local system must be of odd degree, as the sheaves R q f * Q have vanishing cohomology for odd q because of the hyperelliptic involution. On the other hand the cohomologies of local systems on Sym 2 M 1,1 will never give rise to nontrivial minimal classes; those summands of H * (M ct 2,n ) restrict isomorphically to corresponding summands in the cohomology of the preimage of Sym 2 M 1,1 , i.e. to the cohomology of M ct 2,n \ M rt 2,n . ♦

Remark 11 An important ingredient in the previous proof is that we understand completely the cohomology of local systems in genus two. In genus three there is only partial conjectural information based on point counts [START_REF] Bergström | Siegel modular forms of degree three and the cohomology of local systems[END_REF]. Assuming conjectural formulas for the cohomology of local systems in genus three, it seems plausible that one can obtain a similar result also when g = 3: any minimal even class on M 3,n is pushed forward from M 2,n+2 . Since there are even non-tautological classes in genus two, this does not rule out the existence of minimal even non-tautological classes in genus three. The same conjectural formulas suggest that the first case where one finds nontautological classes is M 3,18 .

Remark 12 Just as in genus one, there are lots of odd minimal classes on M 2,n , which can be described in terms of automorphic forms. Although there is no statement as simple as Proposition 7, it turns out that every vector-valued cusp eigenform for Sp(4, Z) of weight ≥ 3 gives rise to cohomology classes8 on M 2,n , and these are always going to minimal. There are also classes coming from "endoscopy".

  the correction term could appear in the left-hand side. However CohFT axiom (ii) with the bi-vector (2) includes basis elements of type c or d at one of the last two markings, while the correction term only appears when all basis elements are of types a and b. So the correction never appears and the compatibility holds.• If g = h, n ≥ m,the correction term appears in the right-hand side whenever the markings (v 1 , . . . , v n ) are a permutation of (b 1 , b 2 , . . . , b m , a, . . . , a n-m

v 1 ,

 1 . . . , v n is a permutation of b 1 , . . . , b m , a, . . . , a n-m . Now assume g = h and v 1 , . . . , v n is a permutation of b 1 , . . . , b m , a, . . . , a n-m .

  2. The state space V has basis B = (a, b 1 , b 2 , . . . , b m , c 1 , c 2 , . . . , c m , d) ,

  : the class in H 22 (M 2,20 ) of bielliptic curves whose markings are switched pairwise by the bielliptic involution is nontautological, and its restriction to H 22 (M 2,20 ) is nontrivial. In fact Gr W 22 H 22 (M 2,20 ) is spanned by the bielliptic class and its conjugates under the S 20 -action, and equals a copy of the representation [2, 2, . . . , 2] of S 20 . By Proposition 4, this shows that H 24 min (M 2,20

Often CohFTs are defined over the field Q. However, for our examples here, we will require C.

Here, we assume the n 1 markings of first factors are {1, . . . , n 1 } and the n 2 markings of the second factor are {n 1 + 1, . . . , n 1 + n 2 = n}. Such a marking distribution can always be achieved by Sn-action (with a possible sign change due to the grading).

Moduli points are known not to be always tautological in Mg,n. For example, since M 1,11 has a holomorphic differential form, moduli points of M 1,11 are not always tautological by[START_REF] Srinivas | Rational equivalence of 0-cycles on normal varieties over C, Proceedings of Algebraic geometry -Bowdoin[END_REF].

The TopFT ω γ depends only on m -the number of markings of the moduli space associated to γ. In Section 2.2, we will use the notation ω m for ω γ .

The problem is formulated in Chow.

For an introduction to these constructions, see[START_REF] Faber | Tautological and non-tautological cohomology of the moduli space of curves in Handbook of Moduli[END_REF] Section 3].