Superconformal Mechanics: Critical Scaling and Dualities of the 1D sigma-models
Francesco Toppan
TEO, CBPF (MCTI) Rio de Janeiro, Brazil
Workshop on Spinnining Particles in QFT San Cristobal de las Casas, Mexico November 05-08, 2013
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 1 / 57 Based on:
Z. Kuznetsova and F. T., JMP 53, 043513 (2012), arXiv:1112.0995.
S. Khodaee and F.T., JMP 53, 103518 (2012), arXiv:1208.3612.
L. Baulieu and F. T., NPB 855, 742 (2012), arXiv:1108.4370.
+ D-module reps for global SQM (last 10 years) F. T., talk at XXIXth ICGTMP, Tianjin, China, August 2012. In the proceedings (also arXiv:1302.3459). M. Gonzales, S. Khodaee, O. Lechtenfeld and F.T., JMP 54 (2013) 072902. N. Aizawa, Z. Kuznetsova and F.T., JMP 54 (2013) 093506. N. A. L. Holanda and F. T., work in preparation.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 2 / 57 Main results:
Existence of a critical scaling dimension for N = 4, 7, 8 finite SCA’s.
Constraints on superconformal mechanics in the Lagrangian setting.
Possible applications:
Test particles near BH horizon (Britto-Pacumio et al. hep-th/9911066),
CFT1/AdS2 correspondence,
Computation of BH entropy via l.w.r.’s and ladder operators,
Higher-spin theories in the world-line framework,
Cosmology: supersymmetric mini-superspace. Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 3 / 57 The bosonic Conformal Mechanics is based on the sl(2) algebra with D-module rep given by
H = ∂t ,
D = −(t∂t + λ), 2 K = −(t ∂t + 2λt).
λ is the scaling dimension.
For any λ we have
[D, H] = H, [D, K ] = −K , [H, K ] = 2D.
(D is the dilatation operator).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 4 / 57 Finite SCA’s from Kac’s list of simple Lie superalgebras:
N = 0 : sl(2), N = 1 : osp(1, 2), N = 2 : sl(2|1), N = 3 : osp(3|2), N = 4 : A(1, 1), D(2, 1; α), N = 5 : B(2, 1), N = 6 : A(2, 1), B(1, 2), D(3, 1), N = 7 : B(3, 1), G(3), N = 8 : D(4, 1), D(2, 2), A(3, 1), F(4), ... : ...,
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 5 / 57 Properties of finite SCA’s:
Even sector Geven : sl(2) ⊕ R (R is the R-symmetry).
Odd sector Godd : 2N generators.
The dilatation operator D induces the grading
G = G−1 ⊕ G 1 ⊕ G0 ⊕ G 1 ⊕ G1. − 2 2
The sector G1 (G−1) containes a unique generator given by H (K ). The G0 sector is given by the union of D and the R-symmetry S subalgebra (G0 = {D} {R}). The odd sectors G 1 and G 1 are spanned by the supercharges Qi ’s 2 − 2 and their superconformal partners Qei ’s, respectively. The invariance under the global supercharges Qi ’s and the generator K implies the invariance under the full superconformal algebra G.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 6 / 57 Comment:
The N -extended global supersymmetry algebra
{Qi , Qj } = 2δij H, i, j = 1,..., N ,
[H, Qi ] = 0,
is a subalgebra of the finite SCA’s. It is of course the superalgebra of the (N -Extended) SUSY QM.
Question: can we extend the sl(2) D-module reps to SCA’s D-module reps?
... we already know the SQM’s D-module reps ... Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 7 / 57 D-module reps of N -Extended SQM “root” supermultiplets obtained via 1D SUSY versus Clifford algebras connection: Qi , Qj = 2δij H, versus Γi , Γj = 2ηij .
The root operators Qi can be expressed as 0 γi d Qi = , Hγ˜i 0 d
where γi and γ˜i are antidiagonal blocks entering the Clifford algebra generators 0 γi 2 Γi = , (Γi ) = +1. γ˜i 0
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 7 / 57 Root supermultiplets:
A “minimal” root supermultiplet (d, d) contains d bosonic fields of scaling dimension λ and d fermionic fields of scaling dimension λ + 1/2.
N = 1 1 N = 9 16 N = 17 256 N = 25 4096 N = 2 2 N = 10 32 N = 18 512 N = 26 8192 N = 3 4 N = 11 64 N = 19 1024 N = 27 16384 N = 4 4 N = 12 64 N = 20 1024 N = 28 16384 N = 5 8 N = 13 128 N = 21 2048 N = 29 32768 N = 6 8 N = 14 128 N = 22 2048 N = 30 32768 N = 7 8 N = 15 128 N = 23 2048 N = 31 32768 N = 8 8 N = 16 128 N = 24 2048 N = 32 32768
The mod 8 property of these reps of the N -extended supersymmetry is in consequence of the Bott’s periodicity of Clifford algebras.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 8 / 57 Comparison: D-modules versus superspace
For large N the superspace is highly reducible. There is no known general rule on how to impose constraints and get all minimal representations.
Number of fields (d). (a): number of (bosonic) component fields in N (a) (b) unconstrained N = 1 1 1 superspace; N = 2 2 2 (b): number of (bosonic) N = 4 8 4 component fields in a N = 8 128 8 minimal representation. N = 16 215 128 N = 32 231 32768
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 9 / 57 Beyond root supermultiplets: the dressing transformations How to obtain, e.g., the real supermultiplet: (φ; ψ1, ψ2; F) ↔ (1, 2, 1)? Notations
(d1, d2,..., dl ) - the field content of the multiplet l is the length of the multiplet
At a given N the admissible (d1, d2,..., dl ) supermultiplets are recovered from the root supermultiplet (d, d) via a dressing transformation: −1 Qi → Qbi = SQi S For length-3 supermultiplets the (d, d) → (d − k, d, k) dressing matrix S has the form S(k) = diag(H, H,..., H, 1,..., 1; 1 ..., 1) (in this case H enters the diagonal k times). For length-4 supermultiplets (d − k, d − m, k, m) a dressing matrix has the form S(k,m) = diag(H,..., H, 1 ..., 1; H ..., H, 1,..., 1) with H entering the upper diagonal block k times and the lower block m times.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 10 / 57 Admissible dressings For any N , all length-3 multiplets (d − k, d, k) are representations of the N -extended supersymmetry. length-4 irreps are present only for N = 3, 5, 6, 7 and for N ≥ 9, length-5 irreps appear starting from N = 10. The results, up to N = 8 are reported in the following table
N length-2 length-3 length-4 1 (1,1) 2 (2,2) (1,2,1) 3 (4,4) (3,4,1), (2,4,2), (1,4,3) (1,3,3,1) 4 (4,4) (3,4,1), (2,4,2), (1,4,3) 5 (8,8) (7,8,1), (6,8,2), (5,8,3), (4,8,4) (1,5,7,3), (3,7,5,1) (3,8,5), (2,8,6), (1,8,7) (1,6,7,2), (2,7,6,1), (2,6,6,2), (1,7,7,1) 6 (8,8) (7,8,1), (6,8,2), ..., (1,8,7) (1,6,7,2), (2,7,6,1), (2,6,6,2), (1,7,7,1) 7 (8,8) (7,8,1), (6,8,2), ..., (1,8,7) (1,7,7,1) 8 (8,8) (7,8,1), (6,8,2), ..., (1,8,7)
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 11 / 57 From Global susy D-modules to SCA D-modules Input:
- A global N -Extended supermultiplet with component fields (d1, d2, d3,...). Comment: the scaling dimensions of the component fields are 1 (λ, λ + 2 , λ + 1,...); the overall scaling dimension of the supermultiplet is λ. In application to SCM, λ ∈ R. - A diagonal action of the D, H, K ∈ sl(2) generators on the component fields, with the given scaling dimensions.
Requirement:
Closure of the (anti)commutation relations on a given finite SCA (to be determined). Comment: the generators are obtained by repeatedly applying anti-commutators involving K and Qi ’s. E.g.: Qei = [K , Qi ], Sij = {Qei , Qj }.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 12 / 57 For i 6= j, the Sij generators belong to the R-symmetry algebra.
The closure on a finite SCA would require, in particular, that the commutators
l [Sij , Qk ] = αijk Ql , close on the susy generators Qi ’s (no extra odd generators).
We get a set of non-trivial relations for the consistency of SCA’s D-module reps, involving the scaling dimension λ. Comment: the Sij generators, for i > j, are not necessarily linear independent. The number of linearly independent generators allow to identify the R-sector of the algebra.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 13 / 57 Most relevant cases: N = 4, 7, 8 finite SCA’s
N = 4: simple SCA’s are A(1, 1) and the exceptional superalgebras D(2, 1; α), for α ∈ C\{0, −1}.
Superalgebra isomorphism for α’s connected via an S3 group transformation:
(1) (3) (5) 1+α α = α, α = −(1 + α), α = − α , (2) 1 (4) 1 (6) α α = α , α = − (1+α) , α = − (1+α) .
A(1, 1) can be regarded as a degenerate superalgebra recovered from D(2, 1; α) at the special values α = 0, −1. For α real (α ∈ R) a fundamental domain under the action of the S3 group can be chosen to be the closed interval
α ∈ [0, 1].
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 14 / 57 Some points in the interval are of special significance:
i) - the extremal point α = 0 corresponds to the A(1, 1) superalgebra,
ii) - the extremal point α = 1 correspond to the D(2, 1) superalgebra, belonging to the D(m|n) = osp(2m|2n) classical series,
1 iii) - the midpoint α = 2 corresponds to the F(4) subalgebra 1 D(2, 1; 2 ) ⊂ F(4),
1 iv) - the point α = 3 corresponds to the G(3) subalgebra 1 D(2, 1; 3 ) ⊂ G(3).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 15 / 57 Over C, there are four finite N = 8 SCA’s and one finite N = 7 SCA.
The finite N = 8 superconformal algebras are: i) the A(3, 1) = sl(4|2) superalgebra, possessing 19 even generators and bosonic sector given by sl(2) ⊕ sl(4) ⊕ u(1), ii) the D(4, 1) = osp(8, 2) superalgebra, possessing 31 even generators and bosonic sector given by sl(2) ⊕ so(8), iii) the D(2, 2) = osp(4|4) superalgebra, possessing 16 even generators and bosonic sector given by sl(2) ⊕ so(3) ⊕ sp(4), iv) the F(4) exceptional superalgebra, possessing 24 even generators and bosonic sector given by sl(2) ⊕ so(7).
The finite N = 7 superconformal algebra is the exceptional superalgebra G(3), possessing 17 even generators and bosonic sector given by sl(2) ⊕ g2.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 16 / 57 Existence of critical scaling dimension λ’s:
Results: D-module reps are induced, with the identifications:
N = 4: D(2, 1; α) reps are recovered from the (k, 4, 4 − k) supermultiplets, with a relation between α and the scaling dimension given by α = (2 − k)λ.
N = 8: for k 6= 4, all four N = 8 finite superconformal algebras are 1 recovered, at the critical values λk = k−4 , with the identifications: D(4, 1) for k = 0, 8, F(4) for k = 1, 7, A(3, 1) for k = 2, 6 and D(2, 2) for k = 3, 5.
1 N = 7: the global supermultiplet (1, 7, 7, 1) induces, at λ = − 4 , a D-module representation of the exceptional superalgebra G(3).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 17 / 57 Explicit non-critical N = 3 case; B(1, 1) = osp(3, 2) from (1, 3, 3, 1):
H = 18 · ∂t ,
D = −18 · t∂t − Λ, 2 K = −18 · t ∂t − 2tΛ,
Q1 = (−E38 + E47 − E52 + E61)∂t + E16 − E25 + E74 − E83,
Q2 = (E28 − E46 − E53 + E71)∂t + E17 − E35 − E64 + E82,
Q3 = (−E27 + E36 − E54 + E81)∂t + E18 − E45 + E63 − E72,
Qe1 = (−E38 + E47 − E52 + E61)t∂t + (E16 − E25 + E74 − E83)t +
−E52(2λ + 2) + (−E38 + E47)(2λ + 1) + E612λ,
Qe2 = (E28 − E46 − E53 + E71)t∂t + (E17 − E35 − E64 + E82)t +
−E53(2λ + 2) + (E28 − E46)(2λ + 1) + E712λ,
Qe3 = (−E27 + E36 − E54 + E81)t∂t + (E18 − E45 + E63 − E72)t +
−E54(2λ + 2) + (−E27 + E36)(2λ + 1) + E812λ,
S1 = E34 − E43 + E78 − E87,
S2 = −E24 + E42 − E68 + E86,
S3 = E23 − E32 + E67 − E76, 3 1 1 1 for Λ = diag(λ, λ + 1, λ + 1, λ + 1, λ + 2 , λ + 2 , λ + 2 , λ + 2 ), Si = ijk Sjk . Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 18 / 57 The (anti)commutation relations of the B(1, 1) superalgebra reads as
[H, K ] = 2D, [D, H] = H, [D, K ] = −K , 1 1 [D, Qi ] = 2 Qi , [D, Qei ] = − 2 Qei , [H, Qei ] = Qi , [K , Qi ] = Qei , {Qi , Qj } = 2δij H, {Qei , Qej } = −2δij K , {Qi , Qej } = −2δij D + ijk Sk , [Si , Qj ] = −ijk Qk , [Si , Qej ] = −ijk Qek , [Si , Sj ] = −ijk Sk .
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 19 / 57 D-modules and superconformal mechanics in Lagrangian framework. The generators of the D-module representations act as graded Leibniz derivatives on functions of the component fields. Example:
d D(ϕ ϕ ) = (Dϕ )ϕ + ϕ (Dϕ ) = − (ϕ ϕ ) − λ (ϕ ϕ ), A B A B A B dt A B A·B A B
(the scaling dimension of ϕA · ϕB is λA·B = λA + λB).
Manifest, global N = 4, supersymmetric action for (k, N , N − k):
L = Q4Q3Q2Q1[F(~x)].
F(~x) is an arbitrary function (named the prepotential) of the propagating bosons ~x. k > 0 is the dimensionality of the target manifold.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 20 / 57 Comment: correct dimensionality of a standard kinetic term, with quadratic time-derivatives of the propagating bosons.
Global N = 8: four supercharges are manifestly realized. The invariance under the remaining four supercharges (Qj , for j = 5, 6, 7, 8) is imposed via a constraint
d Q L = P j dt j for some functions Pj of the component fields and their time-derivatives. As a result the prepotential F(~x) is no longer unconstrained.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 21 / 57 Explicit computations prove that the invariance under the global N = 8 supersymmetry is achieved if F satisfies the harmonic condition
k F = 0, where “k ” stands for the k-dimensional Laplacian operator.
The superconformal mechanics is derived by constructing first a global supersymmetric Lagrangian and then requiring the action of K to produce a time-derivative:
d K L = M, dt for some function M of the component fields and their time-derivatives.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 22 / 57 An N = 4 SCM system: n± inhomogeneous (1, 4, 3) chiral supermultiplets interacting
Let us take n inhomogeneous N = 4 (1, 4, 3) supermultiplets I I I I (x ; ψ , ψi ; gi ), where I = 1,..., n) and i = 1, 2, 3. n+ supermultiplets have positive chirality (I = 1,..., n+), n− = n − n+ have negative chirality (I = n+ + 1,..., n).
The supertransformations are:
I I I I Q4x = ψ , Qi x = ψi , I I I I Q4ψ = x˙ , Qi ψ = −gi , I I I I I I Q4ψj = gj , Qi ψj = δij x˙j + sIijk (gk + ck ), I ˙I I ˙I ˙I Q4gj = ψj , Qi gj = −δij ψ − sIijk ψk .
The signs sI = ±1 define the chirality. I The inhomogeneous parameters ck are arbitrary.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 23 / 57 Up to a total derivative, the Lagrangian L = Q4Q3Q2Q1(F) is
X I J J ˙I J ˙I J I I L = sIFIJ −x˙ x˙ + ψ ψ + ψi ψi − gi (gi + ci ) + IJ X 1 F −s ψK ψJ (gI + cI) + gK ψJ ψI) + IJK I i i i 2 ijk k j i IJK X 1 F ψLψK ψJ ψI IJKL 6 ijk k j i IJKL
(here F ≡ ∂ F and similarly for the higher order derivatives). I ∂xI The summation over i, j, k is understood.
This action is globally N = 4-invariant.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 24 / 57 By inserting the prepotential X F = AMN xM ln xN , MN
(AMN is an arbitrary constant matrix), the system of n± interacting supermultiplets is superconformally invariant under the A(1, 1) superalgebra (each supermultiplet has conformal weight λ = −1). The constraint induced by K implies, for r = 2, 3, 4,
M (r) (r) x FMI1...Ir = γ FI1...Ir , γ = 1 − r.
These equations are satisfied for the above choice of F.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 25 / 57 The real action (for β real) Z Z S = dtL = dt(xβx˙ 2) is scale-invariant and dimensionless provided that the scaling dimension λ for the field x satisfies the condition 1 λ = − β + 2
(the scaling dimension of t is assumed to be [t] = −1).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 26 / 57 Global N = 8 (1, 8, 7) model:
Unique propagating boson x, fermions ψ, ψj and auxiliary fields gj (j = 1, 2,..., 7). The action is Z 1 S = dt{(ax + b)[x˙ 2 − ψψ˙ − ψ˙ ψ + g 2] + a[ψψ g − C g ψ ψ ]}, j j j j j 2 ijk j j k for some real coefficients a, b. The totally antisymmetric coupling constants Cijk are the octonionic structure constants: C123 = C147 = C165 = C257 = C354 = C367 = 1.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 27 / 57 The scale-invariance and the dimensionless of the action requires the homogeneity of the Lagrangian. Therefore, either a = 0 or b = 0.
For a = 0 (for b 6= 0) the scaling dimension λ of x (corresponding to 1 β = 0) is λ = − 2 . This value, however, does not coincide with the critical scaling dimension for the N = 8 supermultiplet with k = 1.
For b = 0 and a 6= 0 we obtain a nontrivial Lagrangian, due to the presence of the cubic term. The scaling dimension λ is nowrecovered for β = 1. 1 We obtain for this value the critical scaling dimension λ = − 3 of the N = 8 k = 1 supermultiplet. At this critical value the (1, 8, 7) supermultiplet induces a D-module representation of the F(4) superconformal algebra and the model is F(4)-invariant.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 28 / 57 The S3 α-orbit of D(2, 1; α) and the constraints on multiparticle superconformal mechanics
The combined properties of having different α’s producing isomorphic N = 4 superconformal algebras, together with the set of critical relations between α and the scaling dimension λ of the (k, 4, 4 − k) N = 4 supermultiplets (with k = 0, 1, 2, 3, 4), given by
α = (2 − k)λ,
produce highly non-trivial constraints on the admissible N = 4 superconformal mechanics models and their scaling dimension.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 29 / 57 The N = 4 superconformal invariance for several (at least two, let’s say (k, 4, 4 − k) and (k 0, 4, 4 − k 0)) interacting supermultiplets requires that they should carry a D-module representation for the same D(2, 1; α) superalgebra.
Given two supermultiplets with k 0 6= k, this requirement produces a) a constraint on the admissible values for α and b) a constraint on the common scaling dimensions.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 30 / 57 (4, 4, 0) ⊕ (4, 4, 0): λ ∈ R, 3 1 √ (4, 4, 0) ⊕ (3, 4, 1): λ = , , ± 2, 2 3 1 (4, 4, 0) ⊕ (2, 4, 2): λ = 0, , 2 (3, 4, 1) ⊕ (3, 4, 1): λ ∈ R, √ 1 1 3 (4, 4, 0) ⊕ (1, 4, 3): λ = 1, − , − ± , 2 2 2 (3, 4, 1) ⊕ (2, 4, 3): λ = 0, 1, √ √ (4, 4, 0) ⊕ (0, 4, 4): λ = −1 ± 5, 1 ± 5, √ √ 1 5 1 5 (3, 4, 1) ⊕ (1, 4, 3): λ = − ± , ± 2 2 2 2 (2, 4, 2) ⊕ (2, 4, 2): λ ∈ R, √ 1 1 3 (3, 4, 1) ⊕ (0, 4, 4): λ = −1, , ± , 2 2 2 (2, 4, 2) ⊕ (1, 4, 3): λ = 0, −1, 1 (2, 4, 2) ⊕ (0, 4, 4): λ = 0, − , 2 (1, 4, 3) ⊕ (1, 4, 3): λ ∈ R, 3 1 √ (1, 4, 3) ⊕ (0, 4, 4): λ = − , − , ± 2, 2 3 (0, 4, 4) ⊕ (0, 4, 4): λ ∈ R.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 31 / 57 The decomposition of N = 8 D-module representations into N = 4 supermultiplets can (partly) explain the N = 8 critical scalings.
(7, 8, 1): the (7, 8, 1) supermultiplet gets decomposed into (4, 4, 0) ⊕ (3, 4, 1). Its critical scaling dimension is therefore 3 1 constrained to be either λ = 2 or λ = 3 (its actual value). (6, 8, 2): the (4, 4, 0) ⊕ (2, 4, 2) decomposition of the (6, 8, 2) supermultiplet implies that its critical scaling dimension can only be 1 found at λ = 0 or λ = 2 (its actual value). (5, 8, 3): the (5, 8, 3) supermultiplet admits the decompositions (4, 4, 0) ⊕ (1, 4, 3) and (3, 4, 1) ⊕ (2, 4, 2). Their combination uniquely implies a possible critical scaling dimension at λ = 1 (its actual value). (4, 8, 4): there is no N = 8 superconformal D-module based on it. This supermultiplet admits the three N = 4 decompositions (4, 4, 0) ⊕ (0, 4, 4), (3, 4, 1) ⊕ (1, 4, 3) and (2, 4, 2) ⊕ (2, 4, 2). There is, however, no common scaling dimension λ belonging to both (4, 4, 0) ⊕ (0, 4, 4) and (3, 4, 1) ⊕ (1, 4, 3).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 32 / 57 A particularly interesting case involves the irrational solution of (3, 4, 1) ⊕ (1, 4, 3).
The value αFD√ coincides with the golden mean√ conjugate 1 1 Φ = − 2 (1 − 5) (the golden mean ϕ = 2 (1 + 5) belongs to its S3-orbit). This case admits four compatible solutions for the scaling dimension λ (given by ±φ and ±Φ).
Comment: there exists an N = 4 superconformal mechanics of interacting supermultiplets whose common scaling dimension is given by the golden ratio.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 33 / 57 Link between the N = 8 critical scalings and the harmonic functions.
A global N = 8-invariant action depends on a D-dimensional harmonic function Φ (DΦ = 0).
R i j Its bosonic sector is the 1D σ-model S = dt(gij x˙ x˙ ). Its D-dimensional target metric is conformally flat: gij = δij Φ.
A conformal factor Φ = r b, dependent on the radial coordinate r alone, is harmonic if the equation b + D = 2 is satisfied.
On the other hand, S is scale invariant and dimensionless if the scaling dimension λ = [xi ] satisfies (b + 2)λ − 1 + 2 = 0, [t] = −1. 1 We recover the critical scaling dimensions λ = D−4 .
The scalar curvature R of the target, computed from Φ, is 1 R = r D−4[(D − 1)(D − 2)2(D − 6)]. 4 Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 34 / 57 Table:
D Φ R λcr S G 1 0 − − − 4 − D(4, 1) 1 1 r 0 − 3 + F(4) 1 2 1 0 − 2 + A(3, 1) −1 3 −1 3 r − 2 r −1 + D(2, 2) 4 r −2 −6 − − − 5 r −3 −18r 1 + D(2, 2) −4 1 6 r 0 2 + A(3, 1) −5 75 3 1 7 r 2 r 3 + F(4) −6 4 1 8 r 126r 4 + D(4, 1)
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 35 / 57 Dual supermultiplets:
N = 8 supermultiplets dually related under the exchange D ↔ 8 − D. Critical scalings related via λD ↔ −λ8−D. The same superconformal algebra is recovered.
Let Ψ ≡ ΨD be a superconformal multiplet, Let Ψe ≡ Ψ8−D be a dual counterpart. R A pairing < ·|· > exists s.t. the integral dt < Ψe|Ψ > is superconformally invariant.
Generalized superconformal action: Z Sα,β,γ(Ψ, Ψ)e = αSD(Ψ) + βS8−D(Ψ)e + γ dt < Ψe|Ψ > .
Comment: this construction naturally leads to superconformal models with double target manifolds.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 36 / 57 The classical superconformal action induces, in the path integral formulation, a superconformally invariant theory with partition function
ZZ R R Z(α, β, γ; J, Je) = DΨDΨee−Sα,β,γ (Ψ,Ψ)+e dt
At least formally the measure DΨDΨe is superconformally invariant.
Both the action and the measure are dimensionless. Indeed
1 [DΨ] = DλD − 8(λD + 2 ) + (D − 8)(λD + 1) = 4 − D is compensated by [DΨ]e = D − 4.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 37 / 57 Conformal symmetry in Minkowski D-dimensional spacetime (D − 1, 1):
ds2 = 0 is preserved.
For D 6= 2 the conformal algebra is so(D, 2).
Special cases:
0 + 1 conformal algebra: so(1, 2) ≈ sl(2).
1 + 1 conformal algebra: witt ⊕ witt (holo+antiholo)
witt a.k.a. centerless Virasoro algebra.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 38 / 57 witt algebra:
[Ln, Lm] = (n − m)Ln+m, n, m ∈ Z sl(2) algebra:
[D, H] = H, [D, K ] = −K , [H, K ] = 2D.
sl(2) ⊂ witt for n, m = 0, ±1: sl(2) ≡ {L±1, L0}.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 39 / 57 parabolic versus hyperbolic/trigonometric D-module reps and homogeneous versus inhomogeneous D-module reps.
n+1 n par : Ln = −t ∂t − λnt , 1 hyp : L = − enµt (∂ + λ ) . n µ t n µ dimensional parameter. witt closure for λn = nλ + γ.
hom : δn(ϕ) = Lnϕ = anϕ˙ + bnϕ,
inhom : δn(ϕ) = Lnϕ = anϕ˙ + bn.
Let xn = an∂t , yn = bn. In matrix form hom Ln = (xn + yn) , x y Linhom = n n . n 0 0 witt closure for λn = nλ + γ. Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 40 / 57 Combining par, hyp and hom, inhom we have four distinct actions of the witt generators on the bosonic field ϕ(t).
Special choices for γ
par : γ = λ, hyp : γ = 0.
In both cases bn = a˙ n. One of the Ln operators, proportional to ∂t , is the “Hamiltonian”: L−1 for par and L0 for hyp.
Dimensional analysis: hom, par: no dimensional parameter, inhom, par: one dimensional parameter, λ, hom, hyp: one dimensional parameter, µ, inhom, hyp: two dimensional parameters, µ and λ.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 41 / 57 D = 1 conformal actions:
2 Let L = g · ϕt + h.
Up to total derivatives, for hom:
2 δL ≡ [gAt + 2gB + gϕBϕ]ϕt + [2gBt ϕ + hϕA + Nϕ]ϕt + [hϕBϕ + Nt ].
For par, B = λAt . System to be solved
At [(1 + 2λ)g + λgϕφ] = 0,
2λgAtt ϕ + hϕA + Nϕ = 0,
λhϕϕAt + Nt = 0.
2 2 Similar system for hyp. On the other hand, the relation Att = n µ A, only exists for hyp case.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 42 / 57 The Inhom case variation is
2 δL ≡ [gAt + Bgϕ]ϕt + [2gBt + hϕA + Nϕ]ϕt + [hϕB + Nt ].
The system to be solved is
At [g + λgϕ] = 0,
2λgAtt + hϕA + Nϕ = 0,
λhϕAt + Nt = 0.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 43 / 57 The D = 1 sl(2)-invariant actions (n = 0, ±1):
Homogeneous − parabolic case: power-law Lagrangian
− 1+2λ 2 1 L = C1ϕ λ ϕt + C2ϕ λ
for λ 6= 0. No dimensional parameter. Scalings: [λ] = 0, [ϕ] = λ, [g] = −1 − 2λ, [h] = 1, [C1,2] = 0 and [S] = 0, where S = R dtL.
Homogeneous − hyperbolic case:
− 1+2z 2 2 2 − 1 L = C1 ϕ z ϕt + µ z ϕ z .
([ϕ] = z, [z] = 0, [g] = −1 − 2z, [h] = 1, [µ] = 1, [C1] = 0).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 44 / 57 Inhomogeneous − parabolic case:
− 1 ϕ 2 ϕ L = C1e λ ϕt + C2e λ .
([ϕ] = [λ] = λ, [C1] = −1 − 2λ, [C2] = 1).
Inhomogeneous − hyperbolic case:
− 1 ϕ 2 2 2 L = C1e z ϕt + µ z .
([ϕ] = [z] = z, [C1] = −1 − 2z, [µ] = 1).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 45 / 57 The D = 2 witt ⊕ witt-invariant actions:
The homogeneous case (no dimensional parameter, power-law Lagrangian):
C1 1 L = ϕ ϕ + C ϕ λ . ϕ2 + − 2 RR (S = dz+dz−L, [C1,2] = 0, [ϕ] = 2λ, [λ] = 0).
The inhomogeneous case (the Liouville equation):
1 ϕ L = ϕ+ϕ− + C2e λ .
([ϕ] = [λ] = λ, [C1] = −2λ, [C2] = 2).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 46 / 57 (super)conformal Galilei algebra in D + 1 dimensions:
The 0 + 1 representation
2 H = ∂t , D = −t∂t − λ, K = −t ∂t − 2λt.
is replaced by
d X λ 7→ λb = λ + ` xi ∂xi . i=1
(` specifies the (2` + 1)d dimensional abelian ideal which carries a spin ` representation of the sl(2, R) subalgebra).
The algebra contains Pi and Mij .
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 47 / 57 Two inequivalent N = 2 supersymmetric extensions:
G(2,2) and G(1,2,1). Besides the 4 bosonic and 4 fermionic generators entering the sl(2|1) d(d−1) subalgebra and the 2 so(d) generators Mij we have, for both the truncated superalgebras G(2,2) and G(1,2,1), (n) the Pi generators for n = 0, 1,..., 2` (their number is d × (2` + 1)) and (n) the Xa,i generators for n = 0, 1,..., 2` − 1 and a = 0, 1 (their number is 2d × 2`). (n) For G(2,2) the Ji generators are obtained for n = 0, 1,..., 2` (their number is d × (2` + 1)). For G(1,2,1) they are obtained for n = 0, 1,..., 2` − 2 (their number is 1 d × (2` − 1) for ` ≥ 2 ). 1 Therefore the G(2,2) superalgebra contains, for ` ≥ 2 , 2d extra (n) generators in the Ji set with respect to the G(1,2,1) superalgebra.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 48 / 57 The G(2,2) D-module rep is given by
H = (e11 + e22 + e33 + e44)∂t , 1 D = −(e + e + e + e )(t∂ + λ + `x ∂ ) − (e + e ), 11 22 33 44 t i xi 2 33 44 2 K = −(e11 + e22 + e33 + e44)(t ∂t + 2λt + 2`txi ∂xi ) − t(e33 + e44),
R = −2(`xi ∂xi + λ)(e12 − e21 + e34 − e43) + (e34 − e43),
Q0 = e13 + e24 + (e31 + e42)∂t ,
Q1 = e14 − e23 − (e32 − e41)∂t ,
S0 = (e13 + e24)t + (e31 + e42)(t∂t + 2λ + 2`xi ∂xi ),
S1 = (e14 − e23)t − (e32 − e41)(t∂t + 2λ + 2`xi ∂xi ), (n) n Pi = (e11 + e22 + e33 + e44)t ∂xi , (n) n Ji = (e12 − e21 + e34 − e43)t ∂xi , (n) n X0,i = (e31 + e42)t ∂xi , (n) n X1,i = −(e32 − e41)t ∂xi ,
Mij = (e11 + e22 + e33 + e44)(xi ∂xj − xj ∂xi ).
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 49 / 57 The G(1,2,1) D-module rep is given by
H = (e11 + e22 + e33 + e44)∂t , 1 D = −(e11 + e22 + e33 + e44)(t∂t + λ + `xi ∂x ) − (2e22 + e33 + e44), i 2 K = −(e + e + e + e )(t2∂ + 2λt + 2`tx ∂ ) − t(2e + e + e ), 11 22 33 44 t i xi 22 33 44 R = e34 − e43,
Q0 = e13 + e42 + (e24 + e31)∂t ,
Q1 = e14 − e32 − (e23 − e41)∂t , S = (e + e )t + (e + e )(t∂ + 2λ + 2`x ∂ ) + e , 0 13 42 24 31 t i xi 24 S = (e − e )t − (e − e )(t∂ + 2λ + 2`x ∂ ) − e , 1 14 23 23 41 t i xi 23 (n) P = (e + e + e + e )tn∂ , i 11 22 33 44 xi (n) J = e · tn∂ , i 21 xi (n) X = (e + e )tn∂ , 0,i 24 31 xi (n) X = (e − e )tn∂ , 1,i 41 23 xi M = (e + e + e + e )(x ∂ − x ∂ ). ij 11 22 33 44 i xj j xi
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 50 / 57 Besides the sl(2|1) subalgebra, the non-vanishing (anti)commutators of G(2,2) are
( ) ( − ) [ , n ] = n 1 , H Pi nPi ( ) ( ) [ , n ] = −( − `) n , D Pi n Pi ( ) ( + ) [ , n ] = −( − `) n 1 , K Pi n 2 Pi ( ) ( − ) [ , n ] = n 1 , Qa Pi nXa,i ( ) ( ) [ , n ] = ( − `) n , Sa Pi n 2 Xa,i ( ) ( ) [ , n ] = ` n , R Pi 2 Ji ( ) ( − ) [ , n ] = n 1 , H Xa,i nXa,i
(n) 1 (n) [D, X ] = −(n − ` + )X , a,i 2 a,i ( ) ( + ) [ , n ] = −( − ` + ) n 1 , K Xa,i n 2 1 Xa,i ( ) ( ) ( ) { , n } = δ n − n , Qa Xb,i abPi abJi ( ) ( + ) ( + ) { , n } = δ n 1 − n 1 , Sa Xb,i abPi abJi ( ) ( ) [ , n ] = −( ` + ) n , R Xa,i 2 1 abXb,i
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 51 / 57 and
( ) ( − ) [ , n ] = n 1 , H Ji nJi ( ) ( ) [ , n ] = −( − `) n , D Ji n Ji ( ) ( + ) [ , n ] = −( − `) n 1 , K Ji n 2 Ji ( ) ( − ) [ , n ] = − n 1 , Qa Ji abnXb,i ( ) ( ) [ , n ] = − ( − `) n , Sa Ji ab n 2 Xb,i ( ) ( ) [ , n ] = − ` n , R Ji 2 Pi
[Mij , Mkl ] = δjk Mil + δil Mjk + δjl Mki + δik Mlj , ( ) ( ) ( ) [ n , ] = δ n − δ n , Pk Mij ik Pj jk Pi ( ) ( ) ( ) [ n , ] = δ n − δ n , Xa,k Mij ik Xa,j jk Xa,i ( ) ( ) ( ) [ n , ] = δ n − δ n . Jk Mij ik Jj jk Ji
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 52 / 57 Central extensions (G(2,2))
Two types, consistent with the graded Jacobi identities: 1) for any d and half-integer ` (mass central extension)
(m) (n) (m) (n) [Pi , Pj ] = δij δm+n,2`ImM, [Ji , Jj ] = δij δm+n,2`ImM, (m) (n) {Xa,i , Xb,j } = δab δij δm+n,2`−1αmM. 2) for d = 2 and integer ` (exotic central extension) (m) (n) (m) (n) [Pi , Pj ] = ij δm+n,2`ImΘ, [Ji , Jj ] = ij δm+n,2`ImΘ, (m) (n) {Xa,i , Xb,j } = δab ij δm+n,2`−1αmΘ, (12 = −21 = 1).
The structure constants Im, αm are common for the two extensions; they are given by m Im = c0(−1) m!(2` − m)!, αm = Im/(2` − m),
where c0 is an arbitrary number depending on ` but independent of m. Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 53 / 57 Conclusions and open problems
The critical D-module representations induce invariant actions in the Lagrangian framework for classical superconformal mechanics.
The quantization based on the functional approach via path integral is also naturally encoded.
The superconformal invariance of the measure requires considering dually related supermultiplets leading to sigma-model actions with double target manifolds.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 54 / 57 Open questions inspired by the operatorial quantization:
A D-module rep induces a specific lowest weight representation (there exists a l. w. vector |w > s.t. H|w >= 0 and Qi |w >= 0 for i = 1,..., N ) at the critical scaling dimension λcr .
Are the l.w.r.’s induced by a D-module rep unitary for some given real form of the d = 1 SCA?
Is there an intrinsic, representation-theoretical characterization, to single out a l.w.r. induced by a D-module, from a generic l.w.r. of the corresponding superalgebra?
Are the unitary l.w.r.’s which are not induced by a D-module rep defining a quantum mechanics which does not admit a classical counterpart and a Lagrangian description?
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 55 / 57 Further developments:
Test particles near BH horizon,
CFT1/AdS2 correspondence,
Computation of BH entropy via l.w.r.’s and ladder operators,
Higher-spin theories in the world-line framework,
Cosmology: supersymmetric mini-superspace.
Conformal topological theories (via twisted supersymmetry).
Extension to affine superalgebras.
Extension to Galileian superconformal theories.
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 56 / 57 Thanks for the attention!
Francesco Toppan (CBPF) Superconformal Mechanics Spinning Particles in QFT 57 / 57