I Vladoiu
email: vladoiu@physics.pub.ro

M Stafe
email: stafe@physics.pub.ro

C Negutu

I M Popescu

Influence of the pulse number and fluence of a nanosecond laser on the ablation rate of metals, semiconductors and dielectrics

Keywords: ablation rate, threshold fluence PACS: 42.62.-b, 61.82.Bg, 61.82.Fk, 61.82.Ms, 81.20.-n

We investigated the pulsed laser ablation of metallic (Al), semiconductor (Si), and wide bandgap dielectric (LiNbO 3 ) targets in air at normal atmospheric conditions by using 4.5 ns pulses at 532 nm wavelength. We determined the dependence of the ablation rate on the pulse number and laser fluence. The number of consecutive laser pulses hitting the target on the same area was between 5 and 40, and the laser fluence was varied in the range of 10-250 J/cm 2 by changing the irradiated area at the target surface. We find that the ablation rate of the three targets is approximately constant when the pulse number is smaller than 15. Further increase of the pulse number leads to a decrease of the ablation rate, the fastest decrease of the ablation rate with pulse number being observed for the dielectric target. The dependence of the ablation rate on the laser fluence indicates two different regimes. In the first regime, which is for values of the fluence smaller than the threshold value (~70 J/cm 2 for Al, ~90 J/cm 2 for Si, and ~180 J/cm 2 for LiNbO 3 ), the ablation rate increases approximately logarithmically with the fluence. In the second regime, characterized by values of the fluence greater than the threshold value, there is a steep increase of the ablation rate. This sudden jump of the ablation rate at the threshold fluence is due to the transition from normal vaporisation to phase explosion, and to the changes in the dimensionality of the plasma-plume hydrodynamics from one-dimensional to three-dimensional.

Introduction

The efficiency of the pulsed laser ablation (PLA) is described by the ablation rate which gives the maximum thickness of the layer ablated during a laser pulse. The ablation rate is a key parameter for the continuously growing fields of applications of PLA, such as optoelectronics, micromechanics, pulsed laser deposition, and laser induced plasma spectrometry [START_REF] Bauerle | Laser processing and chemistry[END_REF][START_REF] Popescu | Aplicatii ale laserelor[END_REF][START_REF] Von Allemen | Laser-Beam Interaction with Materials[END_REF][START_REF] Bauerle | Proceedings of an International Conference Laser Processing and Diagnostics[END_REF][START_REF] Ready | Effects of High-power Laser Radiation[END_REF][START_REF] Ursu | Interactiunea radiatiei laser cu metalele[END_REF][START_REF] Simon | [END_REF][START_REF]Laser Ablation and Desorption[END_REF][9][START_REF]Laser ionization mass analysis[END_REF].

The characteristics of the laser beam, ablated material, and ambient atmosphere strongly influences the ablation rate [START_REF] Bauerle | Laser processing and chemistry[END_REF][START_REF] Popescu | Aplicatii ale laserelor[END_REF][START_REF] Von Allemen | Laser-Beam Interaction with Materials[END_REF][START_REF] Bauerle | Proceedings of an International Conference Laser Processing and Diagnostics[END_REF][START_REF] Ready | Effects of High-power Laser Radiation[END_REF][START_REF] Ursu | Interactiunea radiatiei laser cu metalele[END_REF][START_REF]Laser Ablation and Desorption[END_REF][START_REF] Stafe | [END_REF][12][13][14][15][16][17][18][19][20][21][22][23]. Previous experiments on PLA in multipulse regime indicated that the ablation rate of metals decreases exponentially with pulse number [START_REF] Bauerle | Laser processing and chemistry[END_REF]17]. This was attributed mainly to the superposition of two effects: less efficient transport of the ablated material with increasing crater depth, and enhanced attenuation of the incident pulses in the laser produced plasma plume [START_REF] Bauerle | Laser processing and chemistry[END_REF]18]. In the case of semiconductors and dielectrics it was demonstrated that the ablation rate increases logarithmically with the laser fluence [START_REF] Bauerle | Laser processing and chemistry[END_REF]12]. Our previous experiments on PLA indicated this logarithmic behaviour in the case of metals and, additionally, allowed us to identify a threshold fluence above which there is a sudden jump of the ablation rate [START_REF] Stafe | [END_REF]. The jump of the ablation rate at the threshold fluence was related to the superposition of two effects: the phase explosion, which refers to the transition from a superheated liquid to a mix of vapours and droplets that occurs when material surface pass the thermodynamic critical temperature, and the change of the dimensionality of the plasma plume hydrodynamics from one-dimensional to threedimensional [START_REF] Stafe | [END_REF]12,14,24,[START_REF] Lu | [END_REF].

Here, we provide further insight into PLA by investigating the efficiency of the ablation of three materials with very different thermal and optical properties (metal-Al, semiconductor-Si, and wide bandgap dielectric-LiNbO 3 ) in open air. We determine the dependence of the ablation rate of the three materials on the pulse number and the laser fluence at 532 nm wavelength in two steps. First, by fixing the targets in the focal plane of the focusing lens, we analyse the dependence of the ablation rate on the number of consecutive pulses that hit the target in the same area. The analysis indicates the maximum number of pulses for which the ablation rate is approximately constant. Second, by setting a small number of pulses and varying the laser fluence by changing the diameter of the irradiated area on the target surface, we demonstrate two different domains in the dependence of the ablation rate on the fluence. The sudden jump of the ablation rate at the limit of the two domains appears for a given laser fluence (referred to as the threshold fluence), which is demonstrated to depend on the thermal and optical properties of the material.

Experiment

The laser system employed here consists of a Q-switched Nd:YAG laser that works in TEM00 mode, having 1.4 cm -1 line-width (~170 longitudinal modes) at 532 nm wavelength of the second harmonic. The laser pulses with 4.5 ns duration and 180 mJ/pulse energy are emitted at 10 Hz repetition rate.

The laser pulses are focused at normal incidence by a quartz lens (f/10, f=10 cm) on the surface of the thick targets (thickness larger than 1 mm) which are placed in the focal plane at normal atmospheric conditions, the estimated Rayleigh length being ~200 µm. A number of laser pulses, which varies between 5 and 40, hit the target producing craters that are characterised by their depth and diameter. The diameter of the craters is varied in the range 0.03 to 0.16 cm by moving the targets away from the focal plane with a translation step of 2 mm by using a micro-mechanical stage. We move the targets toward lens system in order to avoid the air breakdown that would lead to loss of the pulse energy into the ignited plasma plume.

The diameter of the craters drilled at different positions of the target approximate the diameter of the irradiated area [22,27]. Then, the laser fluence corresponding to a given target position along its path away from the focal plane is derived by dividing the energy pulse by the irradiated area.

The depth of the craters that are drilled at different target positions is used to calculate the ablation rate by dividing the depth of the crater by the number of laser pulses that were used to drill the craters.

The depth and diameter of the craters were measured by using a Leica Microsystems Heidelberg GmbH confocal microscope. The relative errors of the craters depth and craters diameter that we measured are within 5%.

Results and discussion

The dependence of the crater depth on the pulse number (N) for of Al, Si and LiNbO 3 targets is presented in Figure 1. The targets are fixed in the focal plane, while the number of consecutive laser pulses hitting the target in the same area increase from 5 to 40. Fig. 1 indicates that the depth of the crater (h) increases linearly with pulse number when this number goes from zero to ~15; the slope of the dash-dotted lines that fit the experimental data gives the ablation rate. Consequently, during first stages of PLA in multi-pulse regime, the ablation rate is approximately constant, being ~8.3 µm/pulse for Al and ~2.7 µm/pulse for Si (at a fluence of ~150 J/cm 2 ), and ~1.7 µm/pulse for LiNbO 3 (at a fluence of ~250 J/cm 2 ).

Further increase of the pulse number leads to a non-linear increment of the crater depth. The fitting curves (solid lines in Fig. 1) indicate that the ablation rate ceases to be constant: it decays with increasing pulse number. The fitting curves are described by the equation

d N h h 1 = , (1)
where 1 h denotes the crater depth corresponding to one single pulse. d indicates the deviation of the fitting curve from the dash-dotted straight line that fits the first 3 (for LiNbO 3 ) or 4 (for Al and Si) experimental points indicated in the graphs of Fig. 1: the closer to unit is d the closer to a constant is ablation rate when increasing the pulse number. The value of the d coefficient in eq. ( 1) is ~0.94 for Al and Si, and ~0.69 for LiNbO 3 , indicating that the ablation rate decays much stronger with pulse number in the case of the dielectric target as compared to the metallic and semiconductor targets. The data indicate that in the case of dielectric material the ablation rate corresponding to the 40 th pulse is ~50% smaller than its value for the first pulses, whereas in the case of metallic and semiconductor targets is only ~15% smaller.

The decrease of the ablation rate with pulse number can be related to the superposition of two effects which are determined by the increment of the crater depth with pulse number. First, the increase of the crater depth leads to the confinement of the plasma plume within the crater. This is asserted by comparing the plasma hydrodynamic length and the crater depth: for a nanosecond pulse, the speed of the ablated species is m/s 10 10 ~4 3 ÷ [START_REF] Bauerle | Laser processing and chemistry[END_REF]. Thus, during a laser pulse, the ablation products travel away from the target ~20÷200 µm; this means that the hydrodynamic length is comparable with the depth of the crater after irradiation with 40 laser pulses. The plasma confinement leads to strong attenuation of the laser beam by scattering and inverse Bremsstrahlung absorption into the plasma plume and to less efficient transport of the ablated species, and enhanced recondensation of the material.

Second, the increase of the crater depth with pulse number leads to the increase of the inclination of the irradiated area relative to the original target surface. Thereby, the incidence angle of the laser beam at the irradiated area increases with pulse number, leading to the decrease of the surface transmittance and of the effective laser fluence at the walls of the crater, and consequently to the increase of energy loss by heat conduction [18]. The much smaller thermal diffusivity of the dielectric target as compared to the metallic and semiconductor targets (see table 1) leads to the much faster decrease of the ablation rate of LiNbO 3 with pulse number.

By setting a small number of laser pulses for hitting the target, which imply a linear increase of the crater depth with pulse number, we study the dependence of the ablation rate on the laser fluence by changing the target position relative to the focal plane. Fig. 2 depicts the dependence of the ablation rate on laser fluence, the graphs exhibiting two different regimes. In the first regime, increasing the laser fluence to a certain value ( (5) in the case of LiNbO 3 target. Here, the fluence is given in J/cm 2 .

Eqs. [START_REF] Von Allemen | Laser-Beam Interaction with Materials[END_REF][START_REF] Bauerle | Proceedings of an International Conference Laser Processing and Diagnostics[END_REF][START_REF] Ready | Effects of High-power Laser Radiation[END_REF] indicate that the ablation threshold fluence of Al (~0.2 J/cm 2 ) is much smaller than in the case of silicon (4 J/cm 2 ) and LiNbO 3 (6.5 J/cm 2 ). This may be correlated with the optical absorption coefficient being larger for the metallic target than for the semiconductor and dielectric targets (table 1).

The comparison of the values of e h ∆ given in equations (3-5) and the corresponding laser fluences indicate a much higher efficiency of the ablation of Al as compared to silicon and lithium niobate. Further information on the efficiency of the three materials can be derived by comparing the ablation rate of the three materials at a given laser fluence. At 40 J/cm 2 fluence, the ablation rate of Al (~5.5 µm/pulse) is ~5 times larger than Si (~1.2 µm/pulse), and ~11 times larger than 3 (~0.5 µm/pulse). The larger ablation rate obtained in the case of metallic target as compared to silicon and dielectric targets may be related to the large values of the optical absorption coefficient and thermal diffusivity of Al. Figure 3 gives a comparison of the ablation rates of the three materials as a function of optical and thermal parameters of the materials. The figure indicates that the ablation rate increases with the optical absorption coefficient (Fig. 3a), and thermal diffusivity (Fig. 3b).

In the second regime, increasing the laser fluence above the threshold value th F , the ablation rate has a sudden jump and increases further steeply (dash-dotted lines in Fig. 2). The threshold fluence is ~70 J/cm 2 in the case of Al, ~90 J/cm 2 for Si, and ~180 J/cm 2 in the case of LiNbO 3 . Figure 4 presents the dependence of the threshold fluence on the optical absorption coefficient, indicating that the threshold fluence decreases with increasing the optical absorption coefficient of the material.

The jump in the ablation rate at the threshold fluence originates in the superposition of the phase explosion with the change in the plasma plume hydrodynamics. At the threshold fluence there is a transition from normal vaporisation to phase explosion at the critical temperature of the material. For all of the targets, the transition is indicated by the enhanced ejection and re-deposition of droplets around the craters [START_REF] Lu | [END_REF]. The smaller value of the threshold fluence in the case of the metallic target as compared to the semiconductor and dielectric targets could be related to the large absorption coefficient of Al which facilitates reaching of the critical temperature during PLA, resulting in the large amount of melt that is observed at threshold around the craters as re-solidified droplets.

The dimensionality of the plasma hydrodynamics changes at the threshold fluence from one-dimensional to three-dimensional regime. The one-dimensional regime characterizes a high density and strong absorbent plasma plume which expands in axial direction, whereas the three-dimensional regime characterizes a less absorbent and low density plasma plume which expands both in axial and radial directions [START_REF] Von Allemen | Laser-Beam Interaction with Materials[END_REF]. The jump in the ablation rate appears when the laser beam diameter (defined as the central Airy disk) becomes the same order of magnitude with the hydrodynamic length of the plasma plume. This relationship between the beam diameter and the hydrodynamic length of the plasma at threshold is demonstrated by a simple calculus as follows. The hydrodynamic length of the plasma is given by [START_REF] Bauerle | Laser processing and chemistry[END_REF]15]

τ v = h l ( 6
)
where τ is the pulse duration and

M T k / v γ =
is the expansion velocity of the plasma plume considered as an ideal gas with adiabatic coefficient 3 / 5 = γ , temperature T , and atomic mass M, while k is the Boltzmann constant. Considering a plasma temperature of ~10 5 K for Al [14,15], ~⋅ 5 10 4 K for Si [30], and ~10 4 K for LiNbO 3 [31] obtained when irradiating the targets with nanosecond pulses, eq. ( 6) gives a plasma hydrodynamic length of ~50 µm. This length is similar with the diameter of the laser beam corresponding to the targets positions where the threshold fluence is reached.

Conclusion

We investigated the effects of pulse number and laser fluence on the ablation rate of metals (Al), semiconductors (Si), and dielectrics (LiNbO 3 ) by using nanosecond laser pulses at 532 nm wavelength in air at normal atmospheric conditions. The number of consecutive laser pulses hitting the target on the same area was varied from 5 to 40, whereas the laser fluence was varied by changing the diameter of the irradiated area at the target surface. The dependence of the ablation rate on the pulse number indicate that the ablation rate is approximately constant during the first ~15 pulses, being ~8.3 µm/pulse for Al, ~2.7 µm/pulse for Si, and ~1.7 µm/pulse for LiNbO 3 when placing the targets in the focal plane. Further increase of the pulse number lead to a decay of the ablation rate, the most significant (~50%) being in the case of dielectric target. The decrease of the ablation rate with pulse number was attributed to the superposition of two phenomena: the enhanced attenuation of the laser beam in the plasma plume which is confined within the crater, and the decay of the surface transmittance and of the effective laser fluence at the walls of the crater due to the gradual increase of the incidence angle of the laser beam at the irradiated area with pulse number. The much faster decrease of the ablation rate with pulse number obtained in the case of LiNbO 3 as compared to the metallic and semiconductor targets was related to the small thermal diffusivity of LiNbO 3 .

The dependence of the ablation rate on the laser fluence indicates that the ablation rate of the three materials increases logarithmically with the fluence when the fluence is smaller than a threshold value. The threshold value depends on the target, being ~70 J/cm 2 for Al, ~90 J/cm 2 for Si and ~180 J/cm 2 for LiNbO 3 . The relative efficiency of the ablation of the three materials was analysed here by comparing the ablation rate at a given laser fluence. The data indicate that, for a fixed value of laser fluence of ~40 J/cm 2 , the ablation rate is ~5.5 µm/pulse for Al, ~1.2 µm/pulse for Si, and ~0.5 µm/pulse for LiNbO 3 . The larger ablation rate obtained in the case of metallic target as compared to semiconductor and dielectric targets may be related to the large values of the optical absorption coefficient and thermal diffusivity of Al.

Increasing the fluence above the threshold value, lead to a sudden jump of the ablation rate followed by its steep increase. The jump of the ablation rate at the threshold fluence was attributed to the superposition of two effects: the transition from normal vaporisation to phase explosion regime at the critical temperature of the material, and the changes of dimensionality of the plasma hydrodynamics from one-dimensional to three-dimensional.

Figure 1 .

 1 Figure 1. Crater depth as a function of pulse number for: (a) Al, (b) Si, and (c) LiNbO 3 . The solid lines represent the fitting curves described by eq. (1), and the dash-dotted lines represent the fitting curves for the linear regime which holds when pulse number is smaller than ~15 for Al and Si, and ~10 for LiNbO 3 .

Figure 2 .Figure 3 .

 23 Figure 2. Ablation rate vs. laser fluence for: (a) Al, (b) Si, and (c) LiNbO 3 . The solid lines represent the fitting curves for the logarithmic regime, and the dash-dotted lines represent the fitting curves for the steep regime (above the threshold fluence)

Fig. 4

 4 Fig. 4 Threshold fluence as a function of optical absorption coefficient of the materials. The dash-dotted lines are depicted for guiding the eyes.

Table I :

 I Optical absorption coefficient, α (at 532 nm wavelength), and thermal diffusivity, D, of the targets

	Material	α (cm -1 )	D (cm 2 /s)
	Al	1.5×10 6	0.98
	Si	0.9×10 4	0.84
	LiNbO 3	< 1	0.02


Jesper Lykke Jacobsen
email: jesper.jacobsen@ens.fr

Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

We present an efficient algorithm for computing the partition function of the qcolouring problem (chromatic polynomial) on regular two-dimensional lattice strips. Our construction involves writing the transfer matrix as a product of sparse matrices, each of dimension ∼ 3 m , where m is the number of lattice spacings across the strip. As a specific application, we obtain the large-q series of the bulk, surface and corner free energies of the chromatic polynomial. This extends the existing series for the square lattice by 32 terms, to order q -79 . On the triangular lattice, we verify Baxter's analytical expression for the bulk free energy (to order q -40 ), and we are able to conjecture exact product formulae for the surface and corner free energies.

Introduction

The q-colouring problem consists in assigning to each of the vertices of a graph G any one of q different colours, in such a way that adjacent vertices carry different colours. The number of possible colourings (possibly zero) is known as the chromatic polynomial P G (q) of the graph. Although this definition supposes that q is a positive integer, it is not hard to show [1,2] that P G (q) is indeed a polynomial in q, and so the number of colours can be regarded as a formal variable.

The history of the q-colouring problem is long and interesting, and we refer the reader to [3] for an extensive list of references. The case where G is a planar graph has attracted particular interest. A well-known result is then the four-colouring theorem [4] which states that P G (4) > 0 for any planar G. Other results exploit the extension of P G (q) to non-integer values of q. One interesting question is whether there exists, in the planar case, some q c so that P G (q) > 0 for all q ∈ [q c , ∞). This statement has been established as a theorem [5] for q c = 5, and the so-called Birkhoff-Lewis conjecture [5] that this extends to q c = 4 is widely believed to be true.

Apart from these general results, the case where G is a regular planar latticetypically square or triangular-has also been intensively studied. In particular, a long series of papers [6,3,7,8,9,10,11,12] is devoted to the study of the location and properties of chromatic zeroes, P G (q) = 0, in the complex q-plane for a variety of boundary conditions. Some of the mechanisms responsible for the generation of real chromatic zeroes in the region q ∈ [0, 4] have even been understood analytically [13,14]. One should also mention that there exists a family of planar graphs with real chromatic roots converging to q = 4 from below [15].

An important technical ingredient in the studies of chromatic zeroes is the possibility to build up P G (q), for regular lattice strips of a given width m, by means of a transfer matrix T . In order to elucidate the amazingly intricate behaviour of the chromatic polynomial in the thermodynamical limit, it is clearly desirable to be able to access as large m as possible in these computations.

In the present paper we show how to improve the constructions of [16,3,7,8,11] by writing T as a product of sparse matrices, each of dimension ∼ 3 m . This is implemented in an efficient way, that enables us to compute exactly P G (q) for rectangles as large as 20 × 21. Using the finite-lattice method [17] this results in series expansions-in powers of 1/q-of the bulk, surface and corner free energies of the chromatic polynomial. To give an idea of the improvement on existing results, the bulk free energy series on the square lattice was taken to order q -36 in [18], and very recently extended to order q -47 in [11]. We here add a further 32 terms to this latter series, taking it to order q -79 . The surface and corner free energies are similarly obtained to orders q -79 and q -78 .

The case of the triangular lattice-which does not appear to have been studied previously in terms of series expansions of the free energy-turns out to be particularly interesting. Baxter [19,6] has shown that for this lattice the chromatic polynomial possesses an integrable structure. This permitted him to derive an analytical expression for the bulk free energy, in the form of an infinite product formula, valid in the region q ∈ (-∞, 0] ∪ [4, ∞). We obtain in this case the bulk, surface and corner free energies to order q -40 . Our results validate Baxter's result for the bulk free energy, and permit us to conjecture also the exact infinite product formulae for the surface and corner free energies.

Chromatic polynomial

We first recall how the chromatic polynomial appears as a special case of the partition function of the q-state Potts model. Given a finite undirected graph G = (V, E) with vertex set V and edge set E, the Hamiltonian of the Potts model is given by

H = -J (ij)∈E δ σ i ,σ j , ( 1
)
where J is the coupling constant, and a spin σ i ∈ {1, 2, . . . , q} is defined at each vertex i ∈ V . The Kronecker delta function δ σ i ,σ j equals 1 if σ i = σ j , and 0 otherwise. Setting v = e J -1, the partition function can be rewritten as

Z G (q, v) = {σ} (ij)∈E 1 + vδ σ i ,σ j (2) = A⊆E q k(A) v |A| , (3)
where k(A) is the number of connected components in the subgraph G A = (V, A), and |A| is the number of elements in the edge subset A ⊆ E. In the rewriting (3)-known as the Fortuin-Kasteleyn (FK) representation [20]-q does no longer need to be an integer, and in particular can be used as an expansion parameter. For q → ∞, only the term with A = ∅-whence k(A) = |V |-contributes, and thus one can develop Z G (q, v) in powers of 1/q. The chromatic polynomial appears as the special case v = -1,

P G (q) = Z G (q, -1) , (4)
as is evident from (2).

Transfer matrix formalism

Thanks to the FK representation (3), the partition function for a lattice strip of width m can be built using a transfer matrix T of dimension independent of q [16,3]. In this section we briefly review the necessary formalism, with the minor modification that we rewrite things in terms of q -1 rather than q, since this is more convenient for our series expansion purposes. We then introduce several improvements:

(i) A factorisation of T into sparse matrices that requires only the use of non-nearestneighbour and almost non-nearest-neighbour partitions.

(ii) An efficient labelling of non-nearest-neighbour partitions by natural numbers.

(iii) The use of modular arithmetic instead of arbitrary-precision integer arithmetic.

The combination of these ingredients will yield an efficient algorithm that allows, in particular, to add further terms to the series expansions of [18,11].

Let us start by noting that, in view of (3), T can be taken to act on states which are partitions of the vertex set {1, 2, . . . , m} corresponding to a single horizontal layer of spins. The role of T is to propagate the system one lattice spacing upwards in the vertical direction. The states describe how the connected components of G A = (V, A) partition the m vertices of a layer, where A is the intersection of A with the set of edges which are below the layer under consideration. For instance, when m = 3, the five possible states are the partitions

(1)(2)(3) , ( 12
)(3) , ( 13
)(2) , ( 1
)(23) , ( 123
) . (5)
We shall see shortly how the information contained in these states permits us to account for the non-locality of the Boltzmann weight q k(A) . Our objective is to write the chromatic polynomial on a rectangular piece, of width m and height n, of the square or triangular lattice in the form

P G (q) = e 0 |T last T n-1 |e 0 ( 6
)
where the basis state |e 0 corresponds to the all-singleton partition (1)(2) • • • (m). Each transfer matrix T adds one layer to the lattice, consisting of m -1 horizontal edges and m vertical edges-and for the triangular lattice also m -1 diagonal edges. A special matrix T last will be required to deal with the last layer of m -1 horizontal edges. The notation e 0 | means projection on the all-singleton state, and is not meant to imply the definition of a scalar product on the space of partitions (although such a scalar product can indeed be defined).

Let us observe that it would be quite wasteful to construct the matrix T explicitly. Not only would this require memory resources of the order of (dim T ) 2 for its storage, but computing the product T |v , where |v represents an arbitrary linear combination of all possible basis states, would take a time proportional to (dim T ) 2 . If, on the other hand, we write T as a product of matrices each corresponding to the addition of a single edge, these matrices will turn out to be sparse (with only one or two non-zero entries per column), and so both time and memory requirements are linear in the dimension of these matrices. In order to proceed, we first study these single-edge matrices in some detail.

Single-edge transfer matrices

Since we are going to construct a series expansion in powers of 1/q, it will be useful to pull out a trivial factor in (3) and study instead

ZG (q, v) = q -|V | Z G (q, v) = A⊆E q k(A)-|V | v |A| . (7)
The lowest-order term, corresponding to A = ∅, is then 1. It is useful to think of the factor q k(A)-|V | in the following way: whenever the addition of an edge to the set A joins up two connected component the result is a factor of q -1 . The single-edge transfer matrix that adds a horizontal edge between vertices i and i + 1 can now be written

H i,i+1 = 1 + vJ i,i+1 , (8)
where the first (resp. second) term represents the possibility of that edge being absent from (resp. present in) the subset A. The join operator J i,i+1 acts on a partition of {1, 2, . . . , m} as the identity if i and i + 1 belong to the same block. If they belong to two different blocks, the action of J i,i+1 is to join up the blocks and attribute a factor q -1 . We now make a remark that will turn out to be very important in what follows. For the case of the chromatic polynomial (i.e., v = -1) H i,i+1 is idempotent. It projects out states in which i and i + 1 belong to the same block. It turns out useful to introduce the normalised projection operator P i,i+1 which acts on a partition as the identity if i and i + 1 belong to different blocks, and as zero if i and i + 1 are in the same block. We have then

H i,i+1 P i,i+1 = H i,i+1 = (H i,i+1 ) 2 . ( 9
)
The transfer matrix corresponding to a vertical edge between vertices i and i ′ can be thought of as the following sequence of operations:

(i) Insert i ′ into the partition as a singleton.

(ii) Apply H i,i ′ .

(iii) Eliminate i from the partition.

(iv) Relabel i ′ as i.

Alternatively, the vertical edge as can be represented as the operator

V i = vq -1 + D i , (10)
where the first (resp. second) term represents the situation where that edge is present in (resp. absent from) the subset A. The detach operator D i acts by transforming i into a singleton. The advantage of the rewriting is that the relabelling i ′ → i is done automatically, but one should keep in mind that i does not represent the same vertex on the lattice before and after the action of V i . Note that although the vertical-edge transfer matrix

V i is related to H i,i ′ , it is not a projector in the chromatic case v = -1.
We next show how to combine the observation that H i,i+1 is a projector with the sparse matrix decomposition of T in order to achieve a very efficient transfer matrix formalism. Although this can be done for any regular lattice, we shall discuss only the cases of the square and triangular lattices.

Square lattice

One can use the operators H i,i+1 and V i to build up the lattice, by writing T as a suitable product of such operators. An obvious construction is to first lay down all the horizontal edges of a layer, and the add all vertical edges that connect to the next layer:

T = m i=1 V i m-1 i=1 H i,i+1 (any v) . (11)
Since the next transfer matrix in the product (6) will first lay down a row of horizontal edges, we may use (9) and write instead, for v = -1,

T = m-1 i=1 P i,i+1 m i=1 V i m-1 i=1 H i,i+1 (v = -1 only) . ( 12
)
Although the construction ( 12) takes advantage of the sparse matrix decomposition, it is not maximally efficient for reasons that we now point out. Consider first the case of generic v. Successive applications of T to |e 0 , as in (6), will generate a number of basis states corresponding to all possible partitions of {1, 2, . . . , m} that are consistent with planarity. These are known in the literature as non-crossing partitions, and their number is given by the Catalan number

C m = (2m)! m!(m + 1)! = 1, 2, 5, 14, 42, 132, . . . , (13)
which grows asymptotically as C m ∼ m -3/2 4 m for m ≫ 1.

For v = -1 we can use (12), and so once a layer is completed the number of basis states equals the number of non-crossing non-nearest-neighbour partitions of {1, 2, . . . , m}. Their number is given [3] by the Motzkin number M m-1 , where

M k = ⌊k/2⌋ j=0 k 2j C j = 1, 1, 2, 4, 9, 21, . . . . (14)
For instance, when m = 4, the four possible such partitions read

(1)(2)(3)(4) , ( 1
)(24)(3) , ( 13
)(2)(4) , ( 14
)(2)(3) .
The asymptotic behaviour is now

M k ∼ k -3/2 3 k for k ≫ 1.
This looks like a considerable improvement. However, the intermediate stages that appear when building up the product ( 12) still require all nearest-neighbour partitions, and so the time necessary for a complete multiplication with T still grows like 4 m . Fortunately, we can write the sparse matrix decomposition of T in a way which (essentially) only makes use of non-crossing non-nearest-neighbour partitions. This is accomplished by utilising the property ( 9) maximally. Indeed we can write, for v = -1,

T = T m T m-1 . . . T 3 T 2 T 1 (15)
with

T i =    m-1 j=1 j =i P j,j+1    V i H i,i+1 for i = 1, 2, . . . , m -1 T m = m-1 j=1 P j,j+1 V m . ( 16
)
After the multiplication by T i , the non-nearest-neighbour constraint is now in action everywhere, except possibly between sites i and i + 1. The number of partitions where i and i + 1 do belong to the same block is M m-2 , since for such partitions the contraction of i and i + 1 into a single point clearly leads to a non-crossing non-nearest-neighbour partition on m -1 points. The sparse-matrix factorisation ( 15)-( 16) is a central result of this paper. It implies that, for the square lattice, the number of basis states needed to compute the chromatic polynomial using ( 6) and ( 15) is M m-1 when a layer has just been completed, and

M m-1 + M m-2 in the intermediate stages. Both numbers grow like 3 m .
It remains to describe how the matrix T last lays down the last layer of horizontal edges. As before this is done one edge at a time

T last = T last m T last m-1 . . . T last 3 T last 2 T last 1 (17)
In this product, each factor adds a horizontal bond between i and i + 1. Having done that, vertex i and those to the left of it are of no use any longer, so we might as well detach them as singletons. This leads to

T last i = m-1 j=i+1 P j,j+1 D i H i,i+1 for i = 1, 2, . . . , m . (18)
Note that after the action of T last , only the all-singleton state |e 0 remains. The chromatic polynomial of G appears as the coefficient of this state, leading finally to (6).

Triangular lattice

The triangular lattice can be drawn as a square lattice with diagonals. These diagonals can be constructed, within the transfer matrix approach, by applying the operators H i,i+1 just after site i has been propagated from one layer to the next by means of V i , but before V i+1 has been employed. We have again the factorisation ( 15), but now with

T 1 = m-1 j=1 P j,j+1 V 1 H 1,2 T i = m-1 j=1 P j,j+1 V i H i-1,i H i,i+1 for i = 2, 3, . . . , m -1 T m = m-1 j=1 P j,j+1 V m H m-1,m . (19)
Note that, due to the existence of the diagonal edges, the leftmost factor in each expression is the product over all nearest-neighbour projectors. Therefore, the number of basis states required in any intermediate stage of the construction is M m-1 .

The expressions ( 17)-( 18) for T last are identical to those for the square lattice.

Ordering the states

At this point, one would be ready to write a transfer matrix for the chromatic polynomial using only non-crossing non-nearest-neighbour (NCNNN) partitions -up to a small subtlety for the square lattice-as intermediate states, by storing the partitions encountered using standard hashing techniques. One can however reduce further-albeit only by a constant factor-time and memory requirements by establishing a bijection between the natural numbers {1, 2, . . . , M m-1 } and the set of M m-1 NCNNN partitions. Using this, the vectors appearing in the computation of ( 6) can be realised simply as standard arrays, indexed by the natural numbers, and each basis state can be mapped back and forth between its representation as a natural number and as a partition on which the operators H i,i+1 , V i and P i,i+1 can be made to act. This idea was previously implemented in the context of the planar Potts model (i.e., for non-crossing partitions) by Blöte and Nightingale [16], and we extend it here by adding the non-nearest-neighbour constraint.

An important step in establishing the bijection is to produce an ordering of the NCNNN partitions for a given m. To this end, consider the first point in the partition. Two situations may arise:

(i) If point 1 is a singleton, the restriction of the partition to the points {2, 3, . . . , m} is again NCNNN.

(ii) If point 1 is not a singleton, let k ∈ {0, 1, . . . , m -3} be the smallest integer so that point 3 + k belongs to the same block as 1. Then the restriction of the partition to points {2, 3, . . . , 2 + k} and the restriction to points {3 + k, 4 + k, . . . , m} are two independent NCNNN partitions of respectively k + 1 and mk -2 points.

This argument proves in particular the recursion formula (with p = m -1)

M p = M p-1 + p-2 k=0 M k M p-k-2 (20)
with initial values M 0 = M 1 = 1. Suppose now, as an induction hypotheses, that we have defined an ordering of NCNNN partitions of fewer than m points. The ordering of NCNNN partitions of 1 or 2 points is trivial, since in these cases there is only one such partition. We can then order NCNNN partitions of m points as follows:

(i) Place first the partitions in which point 1 is a singleton. The ordering of these partitions follows inductively from the ordering of the partitions of points {2, 3, . . . , m}.

(ii) Place next the partitions in which point 1 is not a singleton, in the order of increasing values of the integer k defined above. The ordering of the partitions with a fixed value of k follows inductively from the ordering of the partitions of points {2, 3, . . . , 2 + k} and {3 + k, 4 + k, . . . , m}, using the former (resp. latter) as the most (resp. least) significant 'bit'.

For instance, the ordering for m = 5 points reads:

(1)(2)(3)(4)( 5) , (1)(2)(35)(4) , (1)(24)(3)( 5) , (1)(25

)(3)(4) , ( 13
)(2)(4)(5) , ( 135
)(2)(4) , ( 14
)(2)(3)(5) , ( 15
)(2)(3)(4) , ( 15
)(24) (3)
. The mapping from integers to partitions can now be constructed by inferring the value of k from (20) and proceeding recursively. The inverse mapping is constructed similarly by obtaining k explicitly from the partition.

Finally, for the square lattice, we need a bijection between the integers {1, 2, . . . , M m-1 + M m-2 } and partitions in which points i and i + 1 are allowed to be in the same block. Again this follows from our ability to order such partitions. To this end, we place first the M m-1 partitions in which i and i + 1 are not in the same block, and next the M m-2 partitions in which i and i + 1 are in the same block. The remaining ingredients of the bijection can be taken over directly from the NCNNN case.

Implementational details

We wish to compute the chromatic polynomial, normalised as in (7),

PG (q) = q -|V | Z G (q, -1) , ( 21
)
where G is an m × n rectangular section of the square or triangular lattice (the latter being considered as a square lattice with added diagonals). The result is a polynomial of degree |V | -1 in the variable q -1 with constant term 1. To obtain this by the transfer matrix method, each NCNNN state must have a coefficient (weight) which is a polynomial of degree d = |V | -1 with integer coefficients. These coefficients can be stored quite simply as an integer array of length d + 1. Since the goal is to produce a series expansion in powers of q -1 of the various free energies, there is actually no need to compute the whole polynomial PG (q). We shall see below that one knows in advance to which degree d max the expansion can be taken. It suffices therefore to truncate all polynomials intervening in the coefficients to that order. Such a truncation leads to non-negligible savings of memory resources.

A practical problem arises because the integer coefficients rapidly become very large. This is effectively handled by using modular arithmetic, i.e., by computing the coefficients PG (q) modulo various primes p i . Since the transfer process only uses additions and subtractions, it is convenient to use 32-bit unsigned integers in the computations and to choose p i < 2 31 in order to avoid overflow before each modulo operation can be carried out. Using the Chinese remainder theorem, the partial results PG (q) mod p i can be combined to give PG (q) mod i p i . Once the latter stabilises upon using further primes we know that we have arrived at the complete result PG (q), with no modulo. The results reported below made necessary the use of up to 16 different primes.

Results

We first review briefly how the series expansion for the free energies can be obtained from the chromatic polynomials on finite m × n rectangles G m,n by use of the finitelattice method [17]. The goal is to obtain a development for the free energy per vertex in the form

f m,n (q) ≡ log P G (q) mn (22) = log q + ∞ k=0 a k q k + ∞ k=0 b k q k 1 m + 1 n + ∞ k=0 c k q k 1 mn ,
exactly valid, order by order in q -1 , for all m and n sufficiently large. The quantities within square brackets are referred to as the bulk, surface and corner free energies, f b (q), f s (q) and f c (q).

Finite-lattice method

We discuss first the case of the square lattice. The free energies introduced above are given by the finite-lattice method [17] as

f b (q) = log q + (r,s)∈B(k) α k (r, s) log PG(r,s) (z) + O(z 2k-3 ) , ( 23
)
f s (q) = (r,s)∈B(k) β k (r, s) log PG(r,s) (z) + O(z 2k-3 ) , (24)
f c (q) = (r,s)∈B(k) γ k (r, s) log PG(r,s) (z) + O(z 2k-3 ) , (25)
where the set B(k) are r × s rectangles G(r, s) such that r ≤ s and r + s ≤ k.

For convenience-and to facilitate the comparison with previous work [18,11]-the expansion variable q -1 has here been changed to

z = 1 q -1 . ( 26
)
The error term O(z 2k-3 ) in ( 23)-( 25) is due to certain classes of convex polygons [17] which do not fit into any of the rectangles in B(k).

The finite-lattice weights for f b are given by [17]

α k (r, s) = (2 -δ r,s ) (δ r+s,k -3δ r+s,k-1 + 3δ r+s,k-2 -δ r+s,k-3 ) ( 27
)
while those for f s and f c read [21]

β k (r, s) = (2 -δ r,s ) 2 -r -s 2 δ r+s,k + 3(r + s) -2 2 δ r+s,k-1 - 3(r + s) + 2 2 δ r+s,k-2 + r + s -2 2 δ r+s,k-3 , (28)
γ k (r, s) = (2 -δ r,s )((r -1)(s -1)δ r+s,k + (1 + r + s -3rs)δ r+s,k-1 + (3rs + r + s -1)δ r+s,k-2 -(r + 1)(s + 1)δ r+s,k-3 ) . ( 29
)

Square lattice series

Using the transfer matrix formalism we can compute PG(r,s) for lattice strips of width r ≤ m max . Without resorting to super computer facilities or massive parallel computations, we have found that m max = 20 was feasible on a standard workstation. This corresponds to a maximum dimension of the transfer matrices of M 19 + M 18 = 24 735 666 on the square lattice. Setting k = 41 above, this gives then the free energies f b , f s and f c correctly to order z 78 . For the case of f b and f s an extra term can be obtained by adding manually the term µ 2mmax z 4mmax-1 to f b [18], and the term -m max µ 2mmax z 4mmax-1 to f s [11], where [18] ∞ p=4

µ p x 2p = x 8 2 -2x 2 -x 2 √ 1 -4x 2 (1 -4x 2 )(2 + x 2 ) + x 12 3 -4x 2 -4 √ 1 -4x 2 (1 -4x 2 ) 2 . ( 30
)
We thus obtain the series to order z 79 in these cases.

The results for all three series are given in Table 1. They extend those of [11] by 32 new terms.

It is possible to analyse ‡ the extended series using the method of differential approximants. The goal is to obtain the locations z ℓ and exponents λ ℓ of the singularities 1. Large-q series for the bulk, surface and corner free energies of the chromatic polynomial on the square lattice, in terms of the variable z = 1/(q -1). We give the coefficients of exp(f b ) = (q-1) of the free-energy series, which is supposed to behave like ∼ (zz ℓ ) λ ℓ as z → z ℓ .

k α k β k γ k 0 1 1 1 1 0 1 0 2 0 0 0 3 1 -1 1 4 0 -1 0 5 0 0 0 6 0 1 0 7 1 - 1
2 q ∞ k=0 α k z k , exp(f s ) = ∞ k=0 β k z k , and exp(f c ) = ∞ k=0 γ k z k . k α k β k γ k 35 -
Proceeding exactly as described in section 5.5.3 of [11], one obtains for the bulk free energy the first-order (K = 1) approximants z 1 = 0.24(4) ± 0.40(5)i λ 1 = -1.8( 7) z 2 = 0.64( 7)

λ 2 = -1.6(7) z 3 = -1.00(1) λ 3 = 0.99(6) (31)
while the second-order (K = 2) approximants read z 1 = 0.24(4) ± 0.40(3)i λ 1 = -2.4(7) z 2 = 0.62( 10)

λ 2 = -1.3(16) (32)
Somewhat disappointingly, these results hardly improve on those reported in [11], despite of the substantial extensions of the series. The same is true for the reanalysis of the surface and corner free energies.

Triangular lattice series

For the triangular lattice, we can use exactly the same finite-lattice weights as for the square lattice. Taking again m max = 20 and k = 41 corresponds to a maximum dimension of the transfer matrices of M 19 = 18 199 284. The error term in ( 23)-( 25) is now O(z k ), and we know of no special tricks for obtaining an extra term. Therefore, we obtain in this case the three series to order z 40 included. In a couple of important papers, Baxter [19,6] has shown that the triangularlattice chromatic polynomial is an integrable system. In particular, he has set up the Bethe Ansatz equations and derived three different analytical expressions for the bulk free energy f b . The question which one of the three expressions dominates for a given real-or even complex-value of q turns out to be very intricate [6], and has recently been the subject of a careful reanalysis [8]. The expression which is dominant for real q ∈ (-∞, 0] ∪ [4, ∞) turns out to be

e f b = - 1 x ∞ j=1 (1 -x 6j-3 )(1 -x 6j-2 ) 2 (1 -x 6j-1 ) (1 -x 6j-5 )(1 -x 6j-4 )(1 -x 6j )(1 -x 6j+1 ) (33)
where the previous expansion variable ( 26) has now been traded for x which is defined by

q = 2 -x - 1 x . (34)
As expected, our series expansion validates (33) to order x 40 included. The analytic computation of f s and f c was not attempted in [19,6]. Presumably f s could be derived by a substantial adaptation of Baxter's work in order to impose non-periodic boundary conditions in his transfer matrix formalism. On the other hand, the derivation of f c appears to be a much harder problem, since this would require precise knowledge of the correspondence between boundary states in the Bethe setup and the initial lattice model, as well as the ability to compute scalar products in the Bethe formalism.

The series expansions that we obtain for f s and f c read as follows e fs = 1 -2x + x 2 + 2x

It is seen that the coefficients are numerically much smaller than those for the square lattice (see Table 1). This in itself is not surprising, since the radius of convergence is 1 in the x variable (triangular-lattice case), while that of the z variable (square-lattice case) is ≃ 1 2 (and presumably exactly 1 2 in view of [26]). What is truly surprising is that the coefficients are integers! It thus looks reasonable to try to conjecture exact expressions.

In analogy with other problems where both bulk and surface properties have been worked out, one would expect the periodicity of 6 in the product formula (33) to be doubled to 12 in the corresponding surface quantities. Taking this as a clue suggests trying for e fs and e fc product formulae of the form

∞ j=1 11 k=0 (1 -x 12j-k ) α k (37)
for some exponents α k to be determined. Knowledge of the coefficients of x 1 , . . . , x 12 from (35)-(36) allows us to determine α 0 , . . . , α 11 ; we can then test our conjectural formula against the coefficients of x 13 , . . . , x 40 .

In this way we have obtained the conjecture

e fs = ∞ j=1 (1 -x 12j-11 )(1 -x 12j-6 )(1 -x 12j-5 )(1 -x 12j-4 ) (1 -x 12j-9 )(1 -x 12j-8 )(1 -x 12j-7 )(1 -x 12j-2 ) 2 (38)
for the surface free energy, and

e fc = ∞ j=1 (1 -x 12j-11 ) -1 (1 -x 12j-10 )(1 -x 12j-9 ) 3 (1 -x 12j-8 ) 2 × (1 -x 12j-7 ) 4 (1 -x 12j-6 ) -3 (1 -x 12j-5 )(1 -x 12j-4 ) -3 × (1 -x 12j-3 ) 5 (1 -x 12j-2 ) 2 (1 -x 12j-1 ) 2 (1 -x 12j ) (39)
for the corner free energy. The validation of ( 38)-(39) by 40 -12 = 28 independent checks means that both conjectures are established beyond any reasonable doubt.

4.4. Limits q → 0 and q → 4 It is obvious that the product formulae (33), ( 38) and (39) converge for |x| < 1, since this is true for each individual factor in the product (by taking logarithms). It remains to discuss the limiting cases x → 1 (i.e., q → 0) and x → -1 (i.e., q → 4). The latter case-four-colouring the vertices of the triangular lattice-can be shown to be equivalent to other physically interesting problems, such as three-colouring the edges of the hexagonal lattice [22,23], or fully-packed loops on the hexagonal lattice with fugacity n loop = 2 [24].

In the x → 1 case, we write x = 1ǫ for ǫ ≪ 1. Then, to leading order, we have for each factor in the products 1x α ≃ αǫ. This leads to lim

x→1 e f b = - ∞ j=1 (6j -3)(6j -2) 2 (6j -1) (6j -5)(6j -4)(6j)(6j + 1) = 3 3/2 Γ 1 3 9 (2π) 5 ≃ 3.770919 • • • , (40)
lim x→1 e fs = ∞ j=1 (12j -11)(12j -6)(12j -5)(12j -4) (12j -9)(12j -8)(12j -7)(12j -2) 2 = 1 2π Γ 5 12 Γ 1 4 Γ 1 6 2 ≃ 0.305639 • • • , (41)
In the x → -1 case, we write x = -1 + ǫ for ǫ ≪ 1. When α is even, we have then 1x α ≃ αǫ as before. And when α is odd, 1x α ≃ 2 to leading order. This leads to lim It is remarkable that f c diverges in both limits x → ±1.

x→-1 e f b = ∞ j=1 (3j -1) 2 (3j -2)(3j) = 3Γ 1 3 3 4π 2 ≃ 1.460998 • • • , (43)

Discussion

In this paper we have used the finite-lattice method to obtain large-q series for the bulk, surface and corner free energies-denoted f b , f s and f c -for chromatic polynomials on regular two-dimensional lattices. Long series could be obtained due to a technical improvement of the transfer matrix method, which is specific to the antiferromagnetic zero-temperature limit of the Potts model, and amounts to obtaining a factorisation of T in terms of sparse matrices, each of dimension ∼ 3 m , where m is the strip width. Although this applies in principle to any regular lattice, we have focussed here on the most commonly studied lattices (square and triangular).

For the square lattice the improvement allowed us to add 32 new terms to the existing series [11]. Unfortunately, the reanalysis of the singularities of this series, using the method of differential approximants, did not lead to any significant improvement over [11]. The focus in this paper was therefore rather on the triangular lattice, where the series permitted us to conjecture the exact analytical formulae (38)-(39) for f s and f c , and to confirm the correctness of Baxter's [19,6] result (33) for f b .

It should be noticed that going to m max = 20 leads to 79 terms in the square-lattice series, but only to 40 terms for the triangular lattice. Longer series could be obtained in the latter case by using a version of the finite-lattice method which is specifically adapted to the triangular lattice [25]. However, since our series have enabled us to conjecture the exact (infinite-order) results (38)-(39) the motivation for extracting further terms of the series is admittedly rather scarce. Indeed, the series reported here already provide 40 -12 = 28 independent checks of the proposed expressions (38)-(39).

It would be interesting to elucidate the physical mechanism by which f c diverges in the limits q → 0 and q → 4.

It is regrettable that the series obtained here do not provide any information about the regions where the chromatic polynomial is a critical system-that is, q ∈ [0, 3] for the square lattice [26,13,14,3,7,10] and q ∈ [0, 4] for the triangular lattice [19,6,8,10]. The reason is that q = ∞ seems to be the only viable expansion point for the finitelattice method.

It is possible that the fact that non-nearest neighbour constraint can be made local conceals some deeper representation theoretical meaning. It is well-known that the generic representation theory of the Temperley-Lieb algebra, which is central to the algebraic formulation of the planar Potts model, undergoes profound modifications when q = 4 cos(π/(p + 1)) 2 with p integer (or rational) [27]. It might be that similar modifications occur upon taking the particular value v = -1 for the temperature variable. It should be noticed that some steps toward an algebraic formulation of the chromatic polynomial have been taken in [28].

Table

				4
	8	3	-8	12
	9	4	-16	20
	10	3	-16	28
	11	3	-12	67
	12	11	-41	208
	13	24	-138	484
	14	8	-210	753
	15	-91	47	750
	16	-261	849	679
	17	-290	1471	2320
	18	254	-493	10020
	19	1671	-8052	30548
	20	3127	-19901	68832
	21	786	-19966	108744
	22	-13939	37556	65229
	23	-49052	223807	-236055
	24	-80276	508523	-739289
	25	21450	321314	101404
	26	515846	-2052462	7201383
	27	1411017	-8417723	26255714
	28	1160761	-13374892	43505098
	29	-4793764	10841423	-17552274
	30 -20340586	112595914	-291420026
	31 -29699360	260687001	-674637832
	32 33165914	70989018	27442
	33 256169433 -1341964856 4426763291
	34 495347942 -4108283969 12910062402

Table 1 .

 1 (continued.)

		127736296	-3304416038	9737827437
	36	-3068121066	14960606999	-49131891078
	37	-7092358808	58237169596	-184470253912
	38	-1024264966	65268280922	-183956055539
	39	35697720501	-162368154719	621518352427
	40	91243390558	-767619757924	2660084937207
	41	25789733672	-975329692910	3075466954690
	42	-420665229170	1872486336165	-7500763944932
	43	-1089052872105	9701425034093	-34822638005931
	44	-238516756366	12262136381593	-38841312202313
	45	5101697398582	-24192583755347	104412348649015
	46	12146149238921	-118764516172484	448320847685638
	47	-598931311074	-130312353695974	421063156900936
	48	-63338329084478	346985943489639	-1507607631713074
	49	-125863684143541	1412474002216034	-5446631048235906
	50	71523371777335	1098760305055003	-3234717862390686
	51	790320197578279	-5183269098276812	22683201852542681
	52	1200257719380100	-16304598291388925	64013086293590143
	53	-1721302814683702	-5448655170720498	8336294391279875
	54	-9694394826555237	75813721738614619	-329171853716322290
	55	-10463394611604378	182868102163636571	-718400221457685146
	56	29575960135787439	-33129258404821172	310674267788777570
	57	115472492262862427	-1051108217408277081	4563895897602278816
	58	85395764100885186	-2017022924530473193	7953356311623456864
	59	-421383549369627730	1373959176501915516	-7763050320815504003
	60	-1343046574902992458	13749175266084267367	-60017851952733958335
	61	-736061882807769534	22595026341744522020	-90527220474610273313
	62	5291983916641725051	-23294356475365888405	121713573010967365453
	63	15613586538385554242	-172293246256852139137	768987768617382802911
	64	8244136015720127003	-268117182162694379708	1123758858144026736112
	65	-61540273845219665190	298116598787375322521	-1553871017012870302628
	66 -186576584391637858059	2127014580990734112356	-9884276597620937431531
	67 -119156912656052312977	3423100771562123299025 -15168542715543012752367
	68 700439920487427255618 -3325599173839179924758 18496951095018457216592
	69 2311567965527717105415 -26543042275747617969190 130030721001519590513331
	70 1784944419294044372980 -45519479688835787926364 210058368547617077899237

Table 1 .

 1 (continued.)

  3 -2x 4x 6 + 2x 7 + 2x 8 -4x 9 -2x 11 + 7x12 -8x 14 + 4x 15 -5x 16 + 12x 17 -12x 19 + 4x 20 -10x 21 + 24x 22 -2x 23 -19x 24 + 2x 25 -15x 26 + 46x 27 -10x 28 -24x 29x 30 -24x 31 + 77x 32 -22x 33 -25x 34 -16x 35 -35x 36 + 118x 37 -41x 38 -14x 39 -45x 40 + O(x 41 ) ,(35)e fc = 1 + x -3x 3 -5x 4 -6x 5 + 2x 6 + 11x 7 + 21x 8 + 9x 9 -12x 10 -41x 11 -33x 12 + 72x 14 + 82x 15 + 40x 16 -102x 17 -169x 18 -139x 19 + 103x 20 + 273x 21 + 319x 22 -50x 23 -398x 24 -612x 25 -111x 26 + 517x 27 + 1066x 28 + 454x 29 -537x 30 -1686x 31 -1067x 32 + 398x 33 + 2500x 34 + 2079x 35 + 76x 36 -3521x 37 -3641x 38 -1138x 39 + 4643x 40 + O(x 41 ) .

‡ We are grateful to J. Salas for providing the analysis mentioned in this paragraph.

Acknowledgements

This work was supported by the Agence Nationale de la Recherche (grant ANR-06-BLAN-0124-03). The author expresses his gratitude to J. Salas and A.D. Sokal for a careful reading of the manuscript; and in particular to J. Salas for reanalysing the square-lattice series along the lines of [11].


Jacques Monod

Sandra Legout
email: legout@pasteur.fr

Keywords:

Introduction

2010 is the centenary anniversary of the birth of Jacques Monod, who together with François Jacob and André Lwoff received the Nobel Prize in 1965 in recognition of their outstanding contribution to molecular biology. There exists abundant literature concerning the various stages of his life and his scientific achievements [START_REF] Fantini | Pour une éthique de la connaissance / Jacques Monod[END_REF][START_REF] Lwoff | Origins of molecular biology : a tribute to Jacques Monod[END_REF][START_REF] Monod | Diauxie et respiration au cours de la croissance des cultures de B. coli[END_REF][START_REF] Monod | From enzymatic adaptation to allosteric transitions[END_REF][START_REF] Morange | Histoire de la biologie moléculaire[END_REF][START_REF] Ullmann | Origins of molecular biology: a tribute to Jacques Monod[END_REF], but the early stages of his scientific career are generally not emphasized. It so happens that, between 1942 and 1956, Jacques Monod published 16 articles in the "Annales de l'Institut Pasteur" [START_REF] Legout | The "Annales de l'Institut Pasteur", 1887-2007: a glimpse into history[END_REF]. His first publications in the "Annales" were not symbolic of his work, but they represent a decisive turning point in his intellectual pursuit which was to lead to the most prestigious recognition in the international scientific world. Thus, when he decided to begin his thesis, Monod had already acquired wide experience working in many laboratories. This propensity to move from one laboratory to another without settling in reflects the state of mind of Monod at the time, for he was also tempted by a career in music, for which he was extremely gifted. Jacques Monod took his time, but finally chose to pursue his career in biological research.

During the Second World War, Monod joined the French Resistance. His laboratory harbored a secretarial office for printing up anti-Nazi pamphlets. Although his commitment considerably slowed down his scientific activities, he managed to finalize his thesis dissertation, which he defended in 1941 under the title " Recherches sur la croissance des cultures bactériennes". In his memoir published in 1942 [START_REF] Monod | Recherche sur la croissance des cultures bactériennes[END_REF] he reported on an unknown phenomenon which he termed "diauxic growth" to account for the fact that when bacteria are placed in a medium containing both glucose and lactose, growth is continuous until depletion of glucose; then, after a pause, growth resumes with the use of lactose. André Lwoff then pointed out to him that this phenomenon was certainly a particular example "of enzymatic adaptation". This work also led to a publication in the "Annales de l'Institut Pasteur" in 1942 [START_REF] Monod | Diauxie et respiration au cours de la croissance des cultures de B. coli[END_REF]. Since he was being spied upon by the Gestapo due to his active role in the French Resistance, he left the Sorbonne and took refuge at the Institut Pasteur in André Lwoff's laboratory to study enzymatic adaptation. After the liberation, he engaged in the 1 st French Army of General de Lattre de Tassigny. He officially joined the Institut Pasteur as Head of a Laboratory in 1945, where he worked until his death in 1976.

Publication in the "Annales"

Under what circumstances did Monod choose to publish his work in the "Annales de l'Institut Pasteur"? The "Association des Microbiologistes de Langue Française" (AMLF), which later became the French Society for Microbiology, was created in 1938 [START_REF] Grimont | A brief history of the French Society for Microbiology[END_REF]. Scientific reports presented in the format of research papers (i.e. including methods, tables and figures) were usually published in the "Annales". During the War, the AMLF had difficulty in organizing congresses, and the Institut Pasteur held monthly meetings on its premises. André Lwoff, who was an AMLF member, encouraged Monod to present his data. Until then, Jacques Monod's works had always been presented during formal sessions of the French Academy of Sciences. He thus discovered a more casual atmosphere and an audience which greatly appreciated the originality of his discoveries. This is how 10 of his contributions were published in the "Annales de l'Institut Pasteur", including a paper with Elie Wollman (Table 1). The initial objective of the "Annales" was to publish research work performed in the Institut Pasteur laboratories (both on the mainland and overseas) (3). It was thus within this framework that Jacques Monod published 6 other articles (Table 1).

While Monod regularly contributed papers to the "Annales" between 1942 and 1947, only three papers were published in the journal thereafter (Table 1). The explanation for this is simple. French continued to be the main language of the "Annales" and Monod's work, highly specialized in genetics and molecular biology, addressed an international readership.

He thus published in Anglo-Saxon journals. Two of those three papers published during the 1950s included results from projects funded by foreign institutions (Table 1: papers 14 and 16). Since the second one was also sponsored by a French institution, it was important to publish the results in a French journal. In the third article, Monod returned to the source of his works: bacterial growth. In that paper, Monod set out the theoretical and experimental bases of a method of continuous culture (Table 1: paper 15). It is understandable that this particular work was published in the Annales, since Monod had been teaching microbiology courses since 1945. The experimental possibilities offered by this new technique were to be widely used thereafter in French and foreign laboratories.

After his death, several of his papers that had been published in the "Annales" were reproduced in a volume of selected papers edited by André Lwoff and Agnes Ullmann (4, Table 1), bearing witness to the importance of his contributions. He himself cited three of the papers published in the "Annales" (Table 1) in his Nobel Lecture (8).

Conclusion

Jacques Monod's papers published in the " Annales de l'Institut Pasteur " came just after his work on bacterial growth and were a continuation of his thesis. Since they were at the origin of the scientific work which led him to the Nobel Prize, they constitute pioneering research and are important in understanding the future orientation of his research, which was determinant for the continuation of his career: in particular, the discovery of the transcriptional regulation system, in which he analyzed control of genetic expression in the lac operon, in Escherichia coli and the theory of allosteric interactions [START_REF] Monod | From enzymatic adaptation to allosteric transitions[END_REF].

2 .

 2 From the Sorbonne to the Institut Pasteur Monod, who obtained a science degree (licence) in Paris in 1931, was granted a scholarship so as to learn microbiological techniques at Strasbourg University (France), in the laboratory of Edouard Chatton, a specialist in protozoa. The following year he returned to Paris on a Commercy fellowship to join the "Laboratoire d'évolution des êtres organisés" (Laboratory of evolution of living organisms), directed by his former professor at the Sorbonne, Maurice Caullery, and continued working with ciliates. Then, in 1934, he became an assistant in the laboratory of Zoology (Faculty of Sciences, Paris) directed by Charles Perez. It was to be his first paid permanent position. But the job did not suit him well--he worked in a small dilapidated room, with poor facilities and little exchange with his colleagues. To get away from his dreary laboratory he joined Paul-Emile Victor in a scientific expedition to Greenland on the "Pourquoi pas?" He then had the opportunity of learning genetics. He obtained a Rockefeller grant, in 1936 to work on eye pigmentation genes of Drosophila in the laboratory of Thomas H. Morgan at the California Institute of Technology in Pasadena (Fig. 1). He discovered another way to perform research founded on collaborative work and critical and fruitful discussions. Back in France, he carried on research for a while at the Institute of Physicochemical Biology, where he continued to work on the model developed in USA. Having kept his position of assistant at the Sorbonne, Jacques Monod came back to work in the laboratory of Charles Perez in 1937, where he began his thesis on bacterial growth using Escherichia coli as a model system. This choice resulted form the influence of two men. As a student in zoology, it was during a training course in the Laboratory of Marine Biology in Roscoff (France) in 1929 that Jacques Monod met Georges Teissier for the first time. The latter was a specialist in biometrics, and oriented Jacques Monod's research towards the study of the kinetics of the growth of bacteria. It was also in Roscoff that Monod met André Lwoff. Lwoff, who was head of the Service of Microbial Physiology at the Institut Pasteur, was well known for his work on growth factors. He convinced Monod to give up the ciliate model for bacteria and provided him with a culture of E. coli.

Table 1

 1 Work of Jacques Monod published in the "Annales de l'Institut Pasteur"

	167
	168
		Type	Authors	Title	Year, volume,
					pages
	1.	AMLF report	Monod, J.	Influence de l'amide de l'acide	(1942) 68, 435-
		7 May 1942		nicotinique, de l'aneurine et de	438
				l'acide ascorbique sur la
				croissance d'Escherichia coli
	2.	AMLF report	Monod, J.	Sur le phénomène de lyse lié à	(1942) 68, 445-
		5 March 1942		l'inanition carbonée	451
	3.	ALMF report	Monod, J.	Diauxie et respiration au cours de	(1942) 68, 548-
		2 July 1942 a		la croissance des cultures de B.	450
				coli
	4.	AMLF report	Monod, J.	Inflence de la concentration des	(1943) 69, 179-
		4 February		substrats sur la rapidité	181
		1943		d'adaptation chez le B. coli
	5.	AMLF report	Monod, J.	Sur la non-additivité d'action de	(1944) 70, 57-59
		1 July 1943		certains enzymes bactériens
	6.	AMLF report	Monod, J.	Remarque sur le problème de la	(1944) 70, 60-61
		1 July 1943		spécificité des enzymes
				bactériens
	7.	AMLF report	Monod, J.	Inhibition de l'adaptation	(1944) 70, 381-
		1 December		enzymatique chez B. coli en	384
		1943 a,b		présence de 2-4 dinitrophénol
	8.	AMLF report	Monod, J.	Sur la nature du phénomène de	(1945) 71, 37-38
		2 December		diauxie
		1943 a
	9.	AMLF report	Morel. M.,	Sur l'utilisation du saccharose par	(1946) 72, 647-
		8 November	Monod, J.	Proteus vulgaris	656
		1945
	10. Research	Monod, J.,	Mutation et adaptation	(1946) 72, 868-
		paper a, b	Audureau,	enzymatique chez Escherichia	878
			A.	coli-mutabile
	11. Research	Monod, J.	Sur une mutation spontanée	(1946) 72, 879-
		paper a		affectant le pouvoir de synthèse	890
				de la méthoinine chez une
				bactérie coliforme
	12. Research	Lwoff, A.,	Essai d'analys edu rôle de	(1947) 73, 323-
		paper	Monod J.	l'anhydride carbonique dans la	347
				croissance microbienne
	13. AMLF report	Monod, J.,	L'inhibition de la croissance et de	(1947) 73, 937-
		3 April 1947	Wollman, E. a	l'adaptation enzymatique chez les	956
				bactéries infectées par le
				bacteriophage
	14. Research	Monod, J.,	De l'amylomaltase d'Escherichia	(1950) 78, 65-77
		paper a, c	Torriani, A.	coli
	15. Research	Monod, J.	La technique de culture continue:	(1950) 79, 390-
		paper a		théorie et applications	410
	16. Research	Rickenberg,	La galactoside-perméase	(1956) 91, 829-
		paper a, b,d	H., Cohen	d'Escherichia coli	857

Acknowledgements

The author wishes to thank Prof. Agnes Ullmann for critical reading of the manuscript.


Henning Samtleben
email: henning.samtleben@ens-lyon.fr

MAXIMAL SUPERSYMMETRY AND NON-ABELIAN GAUGE GROUPS 1

I review recent progress in the construction and classification of maximally supersymmetric theories with non-abelian gauge groups. The algebraic framework is based on the underlying exceptional symmetry groups. This has applications for supergravity theories describing flux compactifications as well as for the recently constructed three-dimensional superconformal theories describing the dynamics of multiple M2-branes.

Introduction

Supersymmetric theories in four dimensions are invariant under symmetries that close into the superextension of the Poincaré algebra

{Q i α , Qj β } = δ i j γ µ αβ P µ , (1)
where the indices µ and α, β refer to space-time vectors and spinors, respectively. The indices i, j = 1, . . . , N label the number of supercharges. As a result, the field content of these theories falls into representations of the superalgebra (1). Algebraic consistency restricts the maximal number of supercharges to N = 4 and N = 8 for global and local supersymmetry, respectively. The resulting supermultiplets are collected in table 1, the states being organized by the helicity group SO (2).

helicity -2 - The N = 4 supermultiplet combines 6 scalar fields with their fermionic superpartners and the two helicity states of a massless vector field. The most general theory with global N = 4 supersymmetry combines an arbitrary number n of these multiplets, whose gauge fields realize a gauge group G of dimension n. These are the super Yang-Mills theories [1]. Their classification thus reduces to the classification of Lie algebras g of a given dimension, i.e. to the solution of the Jacobi identities

f [ab c f d]c e = 0 . (2)
The situation is quite different for maximal local supersymmetry, i.e. for theories realizing the N = 8 supermultiplet of table 1. As this multiplet includes the spin-2 states of the space-time metric, only a single copy can be described by an interacting theory. It then becomes a non-trivial question which gauge groups can be realized by the 28 gauge fields while preserving maximal local supersymmetry. Indeed, after the original construction of N = 8 supergravity [2] with abelian gauge group U (1) 28 and the first nonabelian version [3] with gauge group SO(8), a variety of other viable noncompact and non-semisimple gauge groups were discovered [4,5,6]. A convenient framework to achieve a general classification uses the so-called embedding tensor [7,8,9]. In the following I will review this formalism which -generalizing (2) -reduces the classification of maximal supergravities to a set of algebraic equations for the embedding tensor. I sketch the emerging structure of non-abelian tensor fields in these theories and finish with an example of a non-trivial application of the formalism also in theories with global supersymmetry.

Gauged supergravity

A distinguished property of the abelian N = 8 supergravity is the underlying global symmetry group G 0 = E 7(7) [2]. Its 70 scalar fields parametrize the coset space manifold E 7(7) /SU (8) while its 28 vector fields combine with their magnetic duals into the fundamental 56-dimensional representation of E 7 (7) , which we denote by A M µ , (M = 1, . . . , 56) . On the physical fields of table 1, this symmetry thus acts in a non-linear and non-local way; infinitesimally, we parametrize this action as

δ Λ Φ = Λ α t α • Φ , (3)
for an arbitrary physical field Φ, parameter Λ and with t α denoting the generators of e 7 (7) = Lie E 7(7) (i.e. α = 1, . . . , 133 parametrizing the adjoint representation), closing into

[t α , t β ] = f αβ γ t γ . (4)
In this version of the theory, none of the matter fields is charged under the U (1) 28 gauge group. Non-abelian deformations of the theory are obtained by selecting a subset of generators t α and promoting their action (3) into a local symmetry. The choice of generators is conveniently parametrized as

X M ≡ Θ M α t α , (5)
by a constant embedding tensor Θ M α , which encodes the embedding of the gauge group into the global E 7 (7) of the ungauged theory. The action of the generators X M can be promoted to a local symmetry by introducing standard covariant derivatives2

D µ ≡ ∂ µ + A M µ X M . (6)
Not every choice of generators (5) leads to a consistent theory. In particular, the generators X M must close into a subalgebra. Moreover, it is only for very particular subalgebras that the deformation is compatible with maximal supersymmetry. Fortunately, all consistency conditions translate into a simple set of algebraic equations for the embedding tensor Θ M α . Closure of the gauge algebra requires

Θ P β Θ N α (t β ) M N + Θ P β Θ M γ f βγ α = 0 , (7)
which can be interpreted as generalization of the Jacobi identities (2). Supersymmetry imposes a linear constraint on the embedding tensor

Θ M α = (P 912 Θ) M α , (8)
where the projector refers to the tensor product decomposition

56 ⊗ 133 = 56 ⊕ 912 ⊕ 6480 , (9)
of the fundamental and the adjoint representation of E 7 (7) , corresponding to the two indices of the embedding tensor. This leaves 912 free parameters in Θ M α , subject to the 8778 quadratic equations (7). It can be shown [9] that every solution to the system of algebraic equations ( 7), ( 8) defines a consistent theory with maximal local supersymmetry and gauge group spanned by the generators (5). In particular, the scalar potential that appears in the gauged theory, takes the form

V = Θ M α Θ N β V M N αβ (φ) , V M N αβ (φ) ≡ (t α ) P R (t β ) Q S M P Q M RS + 7 (t α ) P Q (t β ) Q P M M N , (10)
bilinear in the embedding tensor Θ M α . The scalar dependence arises through the symmetric positive definite matrices M = VV T , with V denoting the E 7(7) /SU (8) coset representative, see [9] for details.

The universal form of the Lagrangian of the gauged theory and in particular of its scalar potential (10) is of particular importance for the study of the low energy effective actions associated to particular flux compactifications of higher dimensional theories. E.g. in flux compactifications of IIA/IIB supergravity on (twisted) tori, the flux parameters show up as deformation parameters among the components of the embedding tensor and can be uniquely identified by purely group-theoretical means, see [11] and references therein. The expression (10) then allows to read off the full scalar potential of the effective theory upon specifying the particular choice of Θ M α . The framework sketched here can equally well be applied to the construction of gauged supergravity theories in other space-time dimensions and with a lower number of supersymmetries, in which case the constraints ( 7), ( 8) are replaced by equations covariant under the relevant global symmetry group.

Non-abelian tensor fields

Apart from the minimal couplings induced by the covariant derivatives ( 6), the field strengths of the vector fields acquire the standard non-abelian modification

F M µν = ∂ µ A M ν -∂ ν A M µ + X [N P ] M A N µ A P ν , (11)
with X M N P ≡ Θ M α t α N P . While in standard Yang-Mills theory this defines covariant objects, here the object (11) in general does not transform covariantly under local gauge transformations. Again, the reason is, that vector fields a priori transform in a given fundamental representation of the global symmetry group; as a consequence, the "structure constants" X M N P do not satisfy the standard Jacobi identities. The natural covariant object turns out to be the following combination [9]

H M µν = F M µν + Z M α B µν α , (12)
with

Z M α ≡ -1 2 Ω M N Θ N α
, upon the introduction of two-form tensor fields B µν α , transforming in the adjoint representation of E 7 (7) . The new field strength H M µν transforms covariantly under the combined set of gauge transformations

δA M µ = D µ Λ M -Z M α Ξ µ α , δB µν α = 2 D [µ Ξ ν] α + 2(t α ) M K Ω KN Λ M H N µν -A M [µ δA N ν] , (13)
which exhibit Stückelberg-type couplings between vector fields and antisymmetric two-forms, as is familiar from massive deformations of supergravities, e.g. [12]. Similarly, the construction of a covariant field strength for the two-forms requires the introduction of three-form tensor fields, etc. The final result is a full hierarchy of non-abelian tensor fields which (schematically) extends (13) to

δB µν = 2D [µ Ξ ν] + . . . -Y Φ µν , δC µνρ = 3D [µ Φ νρ] + . . . -W Σ µνρ , δD µνρσ = 4D [µ Σ νρσ] + . . . , (14)
where we have omitted all E 7(7) -indices, see [13,14] for details. The intertwining tensors Z, Y and W are functions of the embedding tensor and determine the precise E 7 (7) representation content in which the higher rank p-forms must appear. For the N = 8 theory, the required forms B, C and D transform in the 133, the 912, and the 133 + 8645, respectively. This gives agreement with the predictions obtained from analyzing the branchings of the infinite-dimensional representations of the underlying very extended Kac-Moody algebra [15,16]. Moreover, this representation assignment indicates that the higher rank forms do not represent new propagating degrees of freedom but magnetic duals to the physical field content: the two-forms are dual to the 133 E 7(7) Noether currents, while the 912 three-forms are related to the deformation parameters combined in the embedding tensor as

D [µ C νρσ] M α + . . . = V M N αβ (φ) Θ N β , (15)
with the scalar dependent matrix V M N αβ (φ) from (10). The 8778 fourforms D in turn can be interpreted as Lagrange multipliers for the quadratic constraints (7) [13].

Global supersymmetry in three dimensions

In certain cases, the embedding tensor formalism also provides a valuable framework for theories with global supersymmetry [17,18]. In three dimensions, maximally supersymmetric Yang-Mills theory is obtained by dimensional reduction from ten dimensions [1]. However, another N = 8 theory can be constructed by gauging the free theory of n scalar multiplets {Φ Ik , Ψ Ȧk }, labeled by k, with I and Ȧ labeling SO(8) vectors and spinors, respectively. The SO(n) Noether currents

J kl µ = Φ I[k ∂ µ Φ l]I + Ψ Ȧ[k γ µ Ψ l] Ȧ , (16)
allow for the introduction of dual vector fields which can be used to gauge (a subgroup of) the global SO(n) group. In three dimensions, the embedding tensor thus is of the form Θ M α → Θ kl,mn . The constraints (7), (8) in this case take the form Θ kl,mn = Θ [kl,mn] , 0 = Θ kl,pr Θ mn,rq -Θ mn,pr Θ kl,rq -Θ kl,mr Θ nr,pq + Θ kl,nr Θ mr,pq , (17) which reproduces the equations underlying the three-algebra structure of Bagger and Lambert [19], to which SO(4) is the only solution with Θ kl,mn = ǫ klmn . For N < 8, in contrast the corresponding equations admit a variety of possible solutions.

Table 1 :

 1 Maximal supermultiplets.

			3 2	-1 -1 2	0 + 1 2	+1 + 3 2	+2
	N = 4:			1	4	6	4	1
	N = 8:	1	8 28 56 70 56 28	8	1

To be as general as possible, we allow for connections combining electric and magnetic gauge fields. It turns out that for consistent choice of ΘM α (see below) this gives rise to sensible theories which allow for an off-shell formulation even in the presence of magnetic charges[10].

Acknowledgments I am most grateful to E. Bergshoeff, B. de Wit, O. Hohm, A. Le Diffon, H. Nicolai, D. Roest, E. Sezgin and M. Trigiante for many discussions and collaboration on the topics presented. This work is supported by the Agence Nationale de la Recherche.