Supersymmetric V-systems

We construct ${\mathcal N}=4 \,$ $\, D(2,1;\alpha)$ superconformal quantum mechanical system for any configuration of vectors forming a V-system. In the case of a Coxeter root system the bosonic potential of the supersymmetric Hamiltonian is the corresponding generalised Calogero-Moser potential. We also construct supersymmetric generalised trigonometric Calogero-Moser-Sutherland Hamiltonians for some root systems including $BC_N$.


Introduction
Calogero-Moser Hamiltonian is a famous example of an integrable system [9,38,43] which is related to a number of mathematical areas (see e.g. [11]). Generalised Calogero-Moser systems associated with an arbitrary root system were introduced by Olshanetsky and Perelomov [40], [41]. N = 2 supersymmetric quantum Calogero-Moser systems were constructed in [21] and considered further in [7]. They were generalised to classical root systems in [8] and to an arbitrary root system in [5].
A motivation for construction of N = 4 Calogero-Moser system goes back to the work [25] on a conjectural description of near-horizon limit of Reissner-Nordström black hole where appearance of su(1, 1|2) superconformal Calogero-Moser model was suggested. Though we also note more recent different considerations of near extremal black holes in [35]. Another motivation to study supersymmetric (trigonometric) Calogero-Moser-Sutherland systems comes from the relation of these systems with conformal blocks and possible generalisation of these relations to the supersymmetric case [29].
Wyllard gave an ansatz for N = 4 supercharges in [45]. In general Wyllard's ansatz depends on two potentials F and W . He constructed su(1, 1|2) N particle Calogero-Moser Hamiltonian for a single value of the coupling parameter c = 1/N as bosonic part of his supersymmetric Hamiltonian with W = 0. Wyllard argued that his ansatz does not produce superconformal Calogero-Moser Hamiltonians for general values of c. Necessary differential equations for F and W were derived in [45]. Thus potential F satisfies generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations (in the form of [36]) as it was pointed out in [1]. Wyllard's potential F has the form F = γ∈A (γ, x) 2 log(γ, x), (1.1) where A is the root system A N −1 . Examples based on root systems A = G 2 , B 3 were also considered in [45]. Solutions F to WDVV equations of this type appear also in Seiberg-Witten theory [36] and in theory of Frobenius manifolds [10]. More generally, Veselov introduced the notion of a ∨-system in [44]. ∨-systems form special collections of vectors in a linear space, which satisfy certain linear algebraic conditions. A logarithmic prepotential (1.1) corresponding to a collection of vectors A satisfies WDVV equations if A is a ∨-system. The class of ∨-systems contains Coxeter root systems, deformations of generalized root systems of Lie superalgebras, special subsystems in and restrictions of such systems [18,42]. A complete description of the class remains open (see [19] and references therein).
Several attempts have been made to construct supersymmetric mechanics such that the corresponding Hamiltonian has bosonic potential of Calogero-Moser type with a reasonably general coupling parameter(s). Wyllard's ansatz for N = 4 supercharges was extended to other root systems in [23], [24] where solutions for a small number of particles were studied both for W = 0 and W = 0. In particular, su(1, 1|2) superconformal Calogero-Moser systems related to A = A 1 ⊕ G 2 , F 4 and subsystems of F 4 were derived. Superconformal su(1, 1|2) Calogero-Moser systems for the rank two root systems were derived in [3] via suitable action in the superspace. For the WDVV equations arising in the superfield approach we refer to [31].
A many-body model with D(2, 1; α) supersymmetry algebra with α = − 1 2 was considered in [12]. This model was obtained by a reduction from matrix model and it incorporates an extra set of bosonic variables ("U(2) spin variables") which enter the bosonic potential of the corresponding Hamiltonian. One-dimensional version of such a model was considered in [14] and, for any α, in [2], [13]. A generalisation of the many-body classical spin superconformal model for any value of the parameter α was proposed in [30]. Within D(2, 1; α) supersymmetry ansatz of [30] a class of bosonic potentials was obtained in [17]. The potential F has the form (1.1) for a root system A. Then W is a twisted period of the Frobenius manifold on the space of orbits corresponding to the root system A. Such polynomial twisted periods were described in [17], they exist for special values of parameter α. Although the corresponding bosonic potentials are algebraic this class does not seem to contain generalised Calogero-Moser potentials associated with A.
Recently a construction of type A N −1 supersymmetric (classical) Calogero-Moser model with extra spin bosonic generators and N N 2 fermionic variables (for any even N ) was presented in [32]. The ansatz for supercharges is more involved and extra fermionic variables appear due to reduction from a matrix model. A related quantum N = 4 supersymmetric spin A N −1 Calogero-Moser system was studied recently in [16]. Furthermore, a simpler ansatz for supercharges for the spin classical A N −1 Calogero-Moser system was presented in [33]. This model has 1 2 N N(N + 1) fermionic variables and the supersymmetry algebra is osp(N |2). Most recently classical supersymmetric osp(N |2) Calogero-Moser systems were presented in [34]; these models have nonlinear Hermitian conjugation property of matrix fermions and supercharges are cubic in fermions.
In the current work we present two constructions of supersymmetric N = 4 quantum mechanical system starting with an arbitrary ∨-system. In the case of a Coxeter root system A the bosonic part of the Hamiltonian is the Calogero-Moser Hamiltonian associated with A introduced by Olshanetsky and Perelomov in [41], which we get in two different gauges: the potential and potential free ones. In the latter case the Hamiltonian is not formally self-adjoint; this gauge comes from the radial part of the Laplace-Beltrami operator on symmetric spaces [4,26,41]. The superconformal algebra is D(2, 1; α) where α depends on the ∨-system and is ultimately related with the coupling parameter in the resulting Calogero-Moser type Hamiltonian. We use original ansatz for the supercharges [45], [23] based on the potentials F , W and we take W = 0. In the special case when α = −1 the superalgebra D(2, 1; −1) contains the superalgebra su(1, 1|2) as its subalebra, and our first ansatz on the su(1, 1|2) generators reduces to the one considered in [23,24]. It was emphasised in [24] that such quantum models with W = 0 are non-trivial with bosonic potentials proportional to squared Planck constant, though they were not considered in more detail in [24]. Thus we extend considerations in [24] for W = 0 to the case of superconformal algebra D(2, 1; α), and we get in this framework quantum Calogero-Moser type systems associated with an arbitrary ∨-system, which includes Olshanetsky-Perelomov generalisations of the Calogero-Moser system with arbitrary invariant coupling parameters.
We also consider generalised trigonometric Calogero-Moser-Sutherland systems related to a collection of vectors A with multiplicities. We include these Hamiltonians in the supersymmetry algebra provided that extra assumptions on A are satisfied which are similar to WDVV equations for the trigonometric version of the potential F . We show that these assumptions can be satisfied when A is an irreducible root system with more than one orbit of the Weyl group, that is BC N , F 4 and G 2 cases. A related solution of WDVV equations for the root system B N was obtained in [27].
The structure of the paper is as follows. We recall the definition of the Lie superalgebra D(2, 1; α) in Section 2. We give two types of representations of this superalgebra in Sections 3, 4. Starting with any ∨-system we get two corresponding supersymmetric Hamiltonians. In Section 5 we present them explicitly. We consider supersymmetric trigonometric Calogero-Moser-Sutherland systems in Section 6.
Let us recall the definition of the family of Lie superalgebras D(2, 1; α), which depends on a parameter α ∈ C (see e.g. [20,Section 20]). The algebra has 8 odd generators Q abc and 9 even generators T ab = T ba , I ab = I ba , J ab = J ba (a, b, c = 1, 2). Elements T ab , I ab and J ab generate three pairwise commuting sl(2) algebras.
For example, ǫ f (a Q |cd|b) = 1 2 ǫ f a Q cdb + ǫ f b Q cda . We also have relations for all a, b, c, d, e, f = 1, 2. Let us rename generators as follows: We will use ǫ ab and ǫ ab to lower and raise indices, e.g. Q a = ǫ ab Q b ,Q a = ǫ abQ b . We consider N (quantum) particles on a line with coordinates and momenta (x j , p j ), j = 1, . . . , N to each of which we associate four fermionic variables {ψ aj ,ψ j a |a = 1, 2}. We will also write x = (x 1 , . . . , x N ), p = (p 1 , . . . , p N ).
We assume the following (anti)-commutation relations (a, b = 1, 2; j, k = 1, . . . , N): Thus one can think of p k as p k = −i ∂ ∂x k . We introduce further fermionic variables by They satisfy the following useful relations: We will be assuming throughout that summation over repeated indices takes place (even when both indices are either low or upper indices) unless it is indicated that no summation is applied.
Let F = F (x 1 , . . . , x N ) be a function such that where F rjk = ∂ 3 F ∂xr∂x j ∂x k for any r, j, k = 1, . . . , N. We assume that all the derivatives F rjk are homogeneous in x of degree -1. Furthermore, we assume that F satisfies the following Witten-Dijkgraaf-Verlinde-Verlinde equations (WDVV) equations for any r, j, k, m, n = 1, . . . , N.
The following relations for arbitrary operators A, B, C will be useful: We are going to present two representations of D(2, 1; α) algebra using F .
Let us firstly check relations (2.3), (2.4) involving generators J ab and I ab .
Proof. We consider the commutator which implies the statement.
We will use the following relations: Thus by using the first relation in (3.11) [I 11 , I 12 ] = ψ k b ψ bk = iI 11 . Similarly, . Hence, by using the latter relation in (3.11) [I 22 , I 12 ] =ψ bkψk b = −iI 22 , and hence the statement follows.
In what follows, we will use the following relation: Lemma 3.4 (cf. [23]). Let Q abc , J ab be as above. Then the relations (2.4b) hold.
Proof. Firstly let us note that the sum of the last two terms in (3.14) is anti-symmetric in a and b and J ab = J ba . Therefore we have by applying (3.14) Therefore we get from (3.13) and (3.15) that as required in (2.4b). Further, we consider which coincides with the corresponding relation in (2.4b). The remaining relations can be proven similarly. (3.20) By reordering terms in (3.20) we obtain Note that F lmnψ l cψ an ψ cm = 0 if c is fixed such that c = a. Hence (3.21) can be rearranged as −2F lmnψ l aψ am ψ an which is also equal to −F lmnψ l bψ bm ψ an . Therefore Therefore, with the help of (3.13) we get which matches with (2.4c).
Let us now consider the generator Q 11a . Firstly, it is immediate that [I 11 , Q 11a ] = 0, as required. In addition, we have by (3.19) that as required for (2.4c). The remaining relations in (2.4c) can be checked similarly.
In the following theorem we will use the identity We will use the following relations. We have by (2.11) and (2.13) and similarly, where the Hamiltonian H is given by Similarly, using (3.26) we obtain Note thatψ al ψ n c F lnr p r + ψ r cψ ak F rkj p j = 1 2 δ ln δ ac F lnr p r . Then, after canceling out terms and simplifying we have In particular, we note that using the symmetry of F ljk we have that Using the symmetry ∂ r F ljk = ∂ l F rjk and F ljk = F kjl it follows from (3.31), (3.32) and (3.33) that the sum of expressions in (3.31) and (3.32) vanishes if a = c. Therefore we get from (3.31), (3.32), (3.33) that Note that here a = a. Therefore the right-hand side of (3.34) equals Therefore in total expression (3.30) becomes Finally, let us consider the term {B, B ′ }. We first show that By using (3.24) we obtain Then using the symmetry of F lmn under the swap of l and m we obtain Note that by (2.6), (2.8) we have Further on by (2.10) we have F rjk F rmn = F rnk F rmj and therefore some terms in the righthand side of (3.38), (3.39) enter the relation Then by using (3.38)-(3.40) and the symmetry of F rjk under the swap of r and j we obtain Note that for c = a we have C = 0, since F rjk F lmn δ jl ψ br ψ n cψ m bψ ak = 0 by using (2.10). Further on, if c = a then by using (2.10) we have which is equal to 1 4 F rjk F klm ψ br ψ j bψ l dψ dm because of relations (3.35). This proves that C = 0. Then the term {B, B ′ } takes the following form: Therefore, the statement follows. Proof. Firstly, we have that which is the corresponding relation (2.2). Lemma 3.8. Let Q abc , I ab , T ab , J ab be as above. Then relations (2.1) hold.
Proof. Firstly let us consider where a is complimentary to a. Note that we can assume now that a = f . Therefore Therefore by formula (2.9) as required for the corresponding relation (2.1).
Let us now note that Hence the right-hand side of (2.1) for {Q 21a , Q 12b } is By considering various values of a, b ∈ {1, 2}, expression (3.45) takes the form Note that by (2.12), (2.13) we have Note also that ψ ar ψ al ∂ r F lmn = 0 using the symmetry of ∂ r F lmn under the swap of r and l.
Then ψ ar ψ dl ψ m dψ cn ∂ r F lmn = 0 and hence Similarly, Therefore using the symmetry of F rjk under the swap of j and r, and that of F lmn under the swap of l and m we obtain Further, note that for any b ∈ {1, 2} we have by using (2.10) that F lmr F rjk ψ dl ψ m d ψ bj = 0. Hence the right-hand side of (3.50) vanishes. Therefore it follows that Therefore by considering various values of a, b ∈ {1, 2} and by using Lemma 3.8 and Theorem 3.6 we obtain the following: which are the corresponding relations (2.5).
Similarly we have By Lemma 3.4 we have Therefore by considering various values of a, b ∈ {1, 2} we obtain:

The second representation
Let now the supercharges be of the form  (3.37)). Therefore in total, we get that and hence the statement follows. Proof. Firstly, we have that as required. Moreover we have [F rmn p r , x j p j ] = −iF rmn p r + ix j ∂ j F rmn p r = −2iF rmn p r . Then it is easy to see that [H, D] = iH, as required. Further on, [K, D] = − 1 2 [x 2 , x j p j ] = iK, which is the corresponding relation (2.2).
We note that since I and J keep the same form as in the first representation, the statement of the Lemmas 3.2, 3.3 hold.
and the right-hand side of (2.1) becomes (cf. (3.46)) {Q 21a , Q 12b } = x r p r ǫ ab + 4iαψ (arψbr) − 2i(1 + α)ǫ ab (ψ 2rψ1r − ψ 1rψ2r ) = −2iψ arψbr + x r p r ǫ ab + 2i(1 + 2α)ψ brψar , which is equal to (4.8) as required. The remaining relations can be checked similarly. Let us recall that from the proof of Lemma 3.9 (formula (3.53)) we have Therefore an analogue of (3.54) takes the form  The proof of the lemma is the same as the proof of Lemma 3.10 for the first representation since I ab and J ab keep the same form, and the proof of commutation relations with H in Lemma 3.10 relies only on relations (2.1) which express H as the anticommutator of the supercharges Q a andQ a .

Hamiltonians
We now proceed to explicit calculations of Hamiltonians appearing in Theorem 3.6 and Theorem 4.1. We start with a Coxeter root system case.

Coxeter systems.
In this case we take R to be a Coxeter root system in V ∼ = R N [28]. More exactly, let R be a collection of vectors which spans V and is invariant under orthogonal reflections about all the hyperplanes (γ, x) = 0, γ ∈ R, where (·, ·) is the standard scalar product in V . We also assume that R can be decomposed as a disjoint union of its subsets R + and −R + such that each subsystem R + and R − contains no collinear vectors. Furthermore, let us assume that squared length (γ, γ) = 2 for any γ ∈ R, and that R is irreducible. Non-equal choices of length of roots in the cases when the Coxeter group has two orbits on R are covered by considerations in Subsection 5.2 below.
The corresponding function F has the form where λ ∈ C. It is established in [37], [44] that F satisfies generalized WDVV equations (2.10). Recall the following property.
where h is the Coxeter number of R.
Lemma 5.1 has the following corollary.
Lemma 5.2. Let F be given by (5.1). Then Proof. Let γ ∈ R have coordinates γ = (γ 1 , . . . , γ N ). By Lemma 5.1 we have The following identity will be useful below: It follows from the observation that the left-hand side is non-singular at all the hyperplanes (β, x) = 0, β ∈ R + . Let us choose now Then hλ = −(2α+1), so by Lemma 5.2 function F satisfies the required condition (2.9). Thus it leads to D(2, 1; α) superconformal mechanics with the Hamiltonians given by Theorems 3.6, 4.1. We now simplify these Hamiltonians for the root system case.
Theorem 5.3. Let function F be given by (5.1). Then the Hamiltonian H given by (3.27) is supersymmetric with the superconformal algebra D(2, 1; α), where α is given by (5.3). The rescaled Hamiltonian H 1 = 4H has the form where ∆ = −p 2 is the Laplacian in V and the fermionic term Proof. By formula (3.27) we have that where potential Let us firstly simplify U. We have Then because of identity (5.2). The statement follows from formulas (5.5), (5.6).
The following theorem can be easily checked directly.
Proposition 5.5. Hamiltonians H 1 , H 2 from Theorems 5.3, 5.4 satisfy gauge relation The proof follows immediately by making use of the identity (5.2).
Remark 5.6. We note that the Hamiltonian H 2 is not self-adjoint under hermitian involution defined by Note that since F rjk ψ k aψ r bψ bj = F rjk (ψ r bψ bj ψ k a −ψ r a δ kj ) we may express (Q a ) † in terms ofQ a (see (4.2)) as follows (Q a ) † =Q a − iF lmnψ l a δ nm . We then have (4.7). Then supersymmetry algebra constraint {Q a , (Q c ) † } = −2δ a c H leads to restrictions α = − 1 2 , or α = − h+2 4 . In both cases the bosonic part of the Hamiltonian H can be seen to be zero.

5.2.
General ∨-systems. Let us consider a finite collection of vectors A in V ∼ = C N such that the corresponding bilinear form Let us recall what it means that A is a ∨-system [44]. We can assume by applying a suitable linear transformation to A that for any u, v ∈ V . In this case A is a ∨-system if for any γ ∈ A and for any two-dimensional plane π ⊂ V such that γ ∈ π one has β∈A∩π (β, γ)β = µγ, for some µ = µ(γ, π) ∈ C.
Let F = F A (x 1 , . . . , x N ) be the corresponding function where λ ∈ C. Then F satisfies generalised WDVV equations (2.10) (see [44]). Furthermore, the condition Therefore this leads to D(2, 1; α) superconformal mechanics with the Hamiltonians given by Theorems 3.6, 4.1, which we present explicitly in the following theorem.
Theorem 5.7. Let function F be given by (5.7). Then the Hamiltonian H given by (3.27) is supersymmetric with the superconformal algebra D(2, 1; α), where α = − 1 2 (λ + 1). The rescaled Hamiltonian H 1 = 4H has the form where ∆ = −p 2 is the Laplacian in V and the fermionic term Furthermore, the Hamiltonian H given by (4.4) is also supersymmetric with the superconformal algebra D(2, 1; α), where α = − 1 2 (λ + 1) and the rescaled Hamiltonian H 2 = 4H has the form The proof is similar to the one in the Coxeter case. The following proposition can also be checked directly.

Trigonometric version
In this section we consider prepotential functions F = F (x 1 , . . . , x N ) of the form where A is a finite set of vectors in V ∼ = C N , c α ∈ C are some multiplicities of these vectors, and function f is given by so that f ′′′ (z) = coth z.
We are interested in the supercharges of the form Q a = p r ψ ar + iF rjk ψ br ψ j bψ ak , Q c = p lψ l c + iF lmn ψ l dψ dm ψ n c , a, c = 1, 2, which is analogous to the first representation considered in Section 3.
Function F should satisfy conditions for all r, j, m, n = 1, . . . , N but we no longer assume conditions (2.9). Then we have the following statement on supersymmetry algebra.
Theorem 6.1. Let us assume that F satisfies conditions (6.2). Then for all a, b = 1, 2 we have where the Hamiltonian H is given by Furthermore, the rescaled Hamiltonian H 1 = 4H has the form where ∆ = −p 2 is the Laplacian in V and the fermionic term The proof of the first part of the theorem is the same as the proof of Theorem 3.6 together with the proof of the relevant part of Lemma 3.8. The proof of formula (6.3) is similar to the proof of Theorem 5.3.
Let us now consider supercharges of the form Q a = p r ψ ar + iF rjk ψ br ψ j bψ ak , Q c = p lψ l c + iF lmnψ l dψ dm ψ n c , a, c = 1, 2, which is analogous to the second representation considered in Section 4. Then we have the following statement on supersymmetry algebra. Theorem 6.2. Let us assume that F satisfies conditions (6.2). Then for all a, b = 1, 2 we have where the Hamiltonian H is given by Furthermore, the rescaled Hamiltonian H 2 = 4H, has the form
The proof of the first part of the theorem is the same as the proof of Theorem 4.1 together with the proof of the relevant part of Lemma 4.4. Then formula (6.6) can be easily derived from the form (6.5) of H.
Let us now assume that A = R is a crystallographic root system, and that the multiplicity function c(α) = c α , α ∈ R is invariant under the corresponding Weyl group W . For a general root system R the corresponding function F does not satisfy equations (6.2). For example, if R = A N −1 then relations (6.2) do not hold. But for some root systems and collections of multiplicities relations (6.2) are satisfied.
In the rest of this section we consider such cases when prepotential F satisfying (6.2) does exist. The corresponding root systems R have more than one orbit under the action of the Weyl group W . We start by simplifying the corresponding Hamiltonians H 1 given by (6.3). Proposition 6.3. Let us assume that prepotential F given by (6.1) for a root system R with invariant multiplicity function c satisfies (6.2). Then Hamiltonian (6.3) can be rearranged as Φ = Φ + const, with Φ given by (6.4) and R + is a positive subsystem in R.
Indeed, it is easy to see that for the crystallographic root system R the term β,α∈R β ∼α is non-singular at tanh(α, x) = 0 for all α ∈ R, hence it is constant. One can show that the Hamiltonian H 1 given by (6.3) simplifies to the required form. We now show that solutions to equations (6.2) exist for the root systems R = BC N , R = F 4 and R = G 2 , with special collections of invariant multiplicities.
Let R + be a positive subsystem in the root system R. For a pair of vectors a, b ∈ V we define a 2-form B (a,b) R + has good properties with regard to the action of the corresponding Weyl group W . Namely, the following statement takes place.
for any w ∈ W .
Proof. Let us choose a simple root α ∈ R + . It is sufficient to prove the statement for w = s α . Let us rewrite B (a,b) It is easy to see that for any β, γ ∈ R B β,γ (s α a, s α b) = B sαβ,sαγ (a, b) (6.10) since (u, s α v) = (s α u, v) for any u, v ∈ V . Let us now apply s α to equality (6.8). Since by the relation (6.10). This proves the first equality in (6.9). In order to prove the second equality (6.9) let us notice that in fact Hence s α B (a,b) Let us derive some conditions for a function F to satisfy equations of the form (6.2). Let F i be the N × N matrices of third derivatives of F , (F i ) lm = ∂ 3 F ∂x i ∂x l ∂xm , and for any vector a = (a 1 , . . . , a N ) ∈ V let us denote F a = N i=1 a i F i . Theorem 6.5. Let a, b ∈ V . Then the equations and therefore Hence the equations [F a , F b ] = 0 are equivalent to α,β∈R c α c β B α,β (a, b)(α, β) coth(α, x) coth(β, x)α ⊗ β = 0, which can be easily checked to be equivalent to It is easy to see that the sum in the left-hand side of the equality (6.12) is non-singular at tanh(α, x) = 0 for all α ∈ R + , hence this sum is always constant. In an appropriate limit in a cone coth(α, x) → 1 for all α ∈ R + , and therefore the equality (6.12) is equivalent to the equality Let e i , i = 1, . . . , N be the standard orthonormal basis in V . We may express B (a,b) for some scalars g ij = g ij (a, b). Then linear independence of the basis vectors and condition (6.11) give rise to N 2 equations g ij (a, b) = 0. If A N −1 ⊂ R then by Proposition 6.4 we should have that g ij (a, b) = ±g σ(i)σ(j) (σ(a), σ(b)) for any transposition σ ∈ S N which acts on vectors a, b by the corresponding permutation of coordinates. This shows that the condition (6.11) reduces to a single equation g ij = 0 for any fixed i, j and general a, b ∈ V . For convenience we will write below B e i ,e j (a, b) as B ij (a, b). Theorem 6.6. Let R = BC N . Let the positive half of the root system BC N be where η ∈ C × is a parameter. Let r be the multiplicity of vectors ηe i , and let s be the multiplicity of vectors 2ηe i . Let q be the multiplicity of vectors η(e i ± e j ). Then the function satisfies conditions (6.2) if and only if r = −8s − 2(N − 2)q. The corresponding supersymmetric Hamiltonians given by (6.6), (6.7) take the form and with Φ given by where d mtk = d mtk (ǫ) = δ mk + ǫδ tk , and Φ = Φ + const.
By Proposition 6.4, g ij = 0 for all 1 ≤ i < j ≤ N if and only if r = −8s − 2(N − 2)q. The form of the Hamiltonians H 2 , H 1 follows from Theorem 6.2 and Proposition 6.3 respectively. Then the statement follows.
Remark 6.7. We note that for the multiplicity s = 0 Theorem 6.6 is contained in [27]. Indeed, Theorem 2.3 in [27] states that the function F given by formula (6.14) with root system R = B N satisfies WDVV equations. It also follows from the proof of Theorem 2.3 in [27] that the corresponding metric is proportional to the standard metric δ ij . Therefore WDVV equations are equivalent to equations (6.2).
By Proposition 6.4, g ij = 0 for all 1 ≤ i < j ≤ 4 if and only if r = −2q or r = −4q. The form of the Hamiltonians H 2 , H 1 follows from Theorem 6.2 and Proposition 6.3. Then the statement follows. where η ∈ C × is a parameter. Let s be the multiplicity of the short roots α i , i = 1, 2, 3 and let r be the multiplicity of the long roots α j , j = 4, 5, 6. Then the function satisfies conditions ( Proof. The coefficient at e 1 ∧ e 2 in the form B R + given by (6.8), (6.13) is where (α i ∧ α j , e 1 ∧ e 2 ) = det(c 1 , c 2 ) where c k are the column vectors c k = ((α i , e k ), (α j , e k )) ⊺ , k = 1, 2, and 2c α i c α j (α i , α j )B α i ,α j (a, b)(α i ∧ α j , e 1 ∧ e 2 ).
By Proposition 6.4, g ij = 0 for all 1 ≤ i < j ≤ 3 if and only if s = −3r or s = −9r. The form of the Hamiltonians H 1 , H 2 follows from Theorem 6.2 and Proposition 6.3 respectively. Then the statement follows.
Remark 6.10. The bosonic part of the supersymmetric Hamiltonians (6.6), (6.7) becomes Calogero-Moser Hamiltonian in the rational limit. For example let us consider the case of the root system BC N and let us introduce rescaled multiplicities s = η 2 s, q = η 2 q and r = η 2 r in Theorem 6.6. Then in the limit η → 0 bosonic parts of Hamiltonians H 1 and H 2 given by (6.15), (6.16) become the rational B N Hamiltonians H b,r 1 , H b,r 2 with two independent coupling parameters, namely, where l = 2((N − 2) q + 2 s). Thus supersymmetric Hamiltonians (6.15), (6.16) can be viewed as η-deformation of the rational superconformal Hamiltonians considered in Theorems 5.3, 5.4 for the root system R = B N .

Concluding remarks
Since work [45] there were extensive attempts to define superconformal N = 4 Calogero-Moser type systems for sufficiently general coupling parameters and suitable superconformal algebras. Some low rank cases were treated in [23], [24]. A number of works were devoted to the superconformal extensions of Calogero-Moser systems where extra spin type variables had to be present (see [15] for a discussion and the review). In the current work we presented superconformal extensions of the ordinary Calogero-Moser system with scalar potential as well as its generalisations for an arbitrary ∨-system, which includes Olshanetsky-Perelomov generalisations of Calogero-Moser systems with arbitrary invariant coupling parameters. The superconformal algebra is D(2, 1; α) where parameter α is related to the coupling parameter(s). It is crucial for our considerations that we deal with quantum rather than classical Calogero-Moser type systems.
We also presented supersymmetric non-conformal deformations of the Calogero-Moser type systems related with the root system B N (which may be thought of as the Calogero-Moser system with boundary terms) as well as with some other exceptional root systems. It would be very interesting to see if there are any relations of considered systems with black holes (cf. [25] for the conjectural relation with supersymmetric Calogero-Moser systems and e.g. [35], [39] and references therein for non-conformal deformations of AdS 2 black hole geometry).
All our considerations are also extended to non-self-adjoint gauge of the Calogero-Moser type Hamiltonians. There has been considerable interest in such non-self-adjoint but PT symmetric bosonic Hamiltonians (see e.g. [22] and references therein). It would be interesting to see whether these Hamiltonians play a role in the context of supersymmetry.
It may also be interesting to clarify integrability of considered supersymmetric Hamiltonians.