Francesco Toppan (CBPF, Rio de Janeiro, Brazil) A 3D Superconformal QM with sl (2 | 1) dynamical symmetry Talk at 10 th Mathematical Physics Meeting: School and Conference on Modern Math. Phys. Belgrade, Sept. 09 - 14, 2019 Based on: I.E. Cunha & F.T., preprint CBPF-NF-002/19 arXiv:1906.11705[hep-th]

Previous works (three methods): Quantization of world-line superconformal actions ( 1 D sigma-models): I. E. Cunha, N. L. Holanda & F.T., PRD (2017), arXiv:1610.07205 Symmetries of Matrix PDEs: F.T. & M. Valenzuela, Adv. Math. Phys. (2018), arXiv:1705.04004 Direct approach: • N. Aizawa, Z. Kuznetsova & F.T., JMP (2018), arXiv:1711.02923 • N. Aizawa, I. E. Cunha, Z. Kuznetsova & F.T., JMP (2019), arXiv:1812.00873

1 F. Calogero (1969) - sl 2 -invariance, x 2 potential. de Alfaro-Fubini-Furlan (1976) - oscillator term addition (discrete, grounded from below spectrum, ground state) . Conformal Mechanics in the new Millennium (motivations): Holography: AdS 2 − CFT 1 test particle close to RN BH horizon (Britto-Pacumio et al. 1999). AdS 2 holography and SYK models (Maldacena and Stanford 2016).

Contents: • Construction of the 3 D SCQM model • Construction of the 3 D β -deformed oscillator • Determination of the sl (2 | 1) lwr’s. • Alternative selections of Hilbert spaces (following Miyazaki-Tsutsui ’02 and F´ eh´ er-Tsutsui-F¨ ul¨ op ’05) • Spectra and zigzag patterns of vacuum energies. • Interpolating linear/quadratic regimes for energy degeneracies • Orthonormal eigenstates from associated Laguerre polynomials and spin-spherical harmonics. • Dimensional reductions. • Comment on larger algebraic structures.

The 3 D SCQM model: Natural Ansatz for N = 2 susy ( a = 1 , 2): � � 1 ∂ − β √ / Q a = γ a r 2 N F / r . 2 � x 2 1 + x 2 2 + x 2 β is a real parameter, r = 3 the radial coordinate, while / ∂ = ∂ i h i and / r = x i h i are written in terms of quaternions ( h i ); γ a are Clifford matrices s.t. [ γ a , h i ] = 0; N F is the Fermion Parity Operator. N = 2 supersymmetric quantum mechanics: { Q a , Q b } = 2 δ ab H , [ H , Q a ] = 0 . The 4 × 4 matrix supersymmetric Hamiltonian H is given by − → S · − → � � 2 ∇ 2 + 2 β L + β ( β +1) ( − 1 ) I 2 0 r 2 2 r 2 = H − → S · − → 2 ∇ 2 − 2 β L + β ( β − 1) ( − 1 0 ) I 2 r 2 2 r 2 x 3 is the three-dimensional Laplacian, − → where ∇ 2 = ∂ 2 x 1 + ∂ 2 x 2 + ∂ 2 S 2 and − → is the spin- 1 L is a orbital angular momentum.

The Hamiltonian H is Hermitian. Since the spin is 1 2 , the total angular momentum − → J = − → L + − → S of the quantum-mechanical system is half-integer. The Hamiltonian is non-diagonal; on the other hand, due to 2 − − 2 − − − → L · − → 2( − 1 → → → 2 ) = 1 2( j ( j + 1) − l ( l + 1) − 3 S = J L S 4) , it gets diagonalized in each sector of given total j and orbital l angular momentum. In each such sector it corresponds to a constant kinetic term plus a 1 diagonal potential term proportional to r 2 .

sl (2 | 1) superconformal algebra: DFF construction: Introduce the conformal partner of H as the rotationally invariant operator K of scaling dimension [ K ] = − 1: 1 2 r 2 I 4 K = Verify whether the repeated (anti)commutators of the operators Q a and K close the superconformal algebra sl (2 | 1). Itis so!. Four extra operators ( Q a , D , R ) have to be added. D is the (bosonic) dilatation operator which, together with H , K , close the sl (2) subalgebra, two fermionic operators Q a and R is the u (1) R -symmetry bosonic operator of sl (2 | 1): [ D , H ] = − 2 iH , [ D , K ] = 2 iK , [ H , K ] = iD , [ D , Q a ] = − iQ a , [ D , Q a ] = iQ a , [ H , Q a ] = iQ a [ K , Q a ] = − iQ a , { Q a , Q b } = 2 δ ab H , { Q a , Q b } = 2 δ ab K , { Q a , Q b } = δ ab D + ǫ ab R , [ R , Q a ] = − 3 i ǫ ab Q b , [ R , Q a ] = − 3 i ǫ ab Q b , with the antisymmetric tensor ǫ ab normalized so that ǫ 12 = 1.

Deformed oscillator: By setting H osc = H + K , we obtain the 4 × 4 matrix deformed oscillator Hamiltonian H osc whose spectrum is discrete and bounded from below. By construction, the sl (2 | 1) dynamical symmetry of the H Hamiltonian acts as a spectrum-generating superalgebra for the H osc Hamiltonian. The explicit expression is − 1 2 ∇ 2 · I 4 + 1 L )) + 1 2 r 2 ( β 2 · I 4 + β N F (1 + 4 · I 2 ⊗ � 2 r 2 · I 4 . S · � H osc =

Appearance of two-component spherical harmonics: j = l + δ 1 2 , δ = ± 1 . for In the given j , l sector we get S = 1 α = δ ( j + 1 � L · � 2 α, with 2) − 1 . The energy eigenstates of the system are described with the help of the two-component Y j , l , m ( θ, φ ) spin spherical harmonics given by m − 1 � j + 1 δ 2 (1 − δ ) + δ mY 2 δ ( θ, φ ) 2 1 j − 1 , Y j , j − 1 2 δ, m ( θ, φ ) = √ 2 j − δ + 1 � m + 1 j + 1 2 (1 − δ ) − δ mY 2 δ ( θ, φ ) 2 j − 1 where Y n l ( θ, φ ) (for n = − l , − l + 1 , . . . , l ) are the ordinary spherical harmonics. The spin spherical harmonics Y j , j − 1 2 δ, m ( θ, φ ) are the eigenstates for the compatible observable operators � J · � J , � L · � L , J z , with eigenvalues j ( j + 1), ( j − 1 2 δ )( j − 1 2 δ + 1), m , respectively.

Creation (annihilation) operators: a † b = Q b − iQ b . a b = Q b + iQ b , Indeed, we obtain 1 1 } = 1 2 { a 1 , a † 2 { a 2 , a † = 2 } , H osc together with [ H osc , a b † ] = a † [ H osc , a b ] = − a b , b . For completeness we also present the commutators [ a 1 , a † 1 ] = [ a 2 , a † 3 · I 4 + 4 · I 2 ⊗ � S · � 2 ] = L − 2 β N F . / γ b ( I 4 · ( ∂ r ∓ r ) − 2 L − β r r I 2 ⊗ � S · � a ± √ = r N F ) . b 2 r They can be factorized as a ± = ( I 4 · ( ∂ r ∓ r ) − 2 L − β / r r I 2 ⊗ � S · � a ± γ b a ± , √ b = with r N F ) . 2 r

Lowest weight vectors: A lowest weight state Ψ lws is defined to satisfy a − b Ψ lws = 0 . Due to the factorization, in both b = 1 , 2 cases, this is tantamount to satisfy a − Ψ lws = 0. The vectors a + 1 v and a + 2 v , with v belonging to the lowest weight representation, differ by a phase. Therefore, the action of a + 1 , a + 2 produces the same ray vector characterizing a physical state of the Hilbert space. We search for solutions Ψ ǫ j ,δ, m ( r , θ, φ ) of the form Ψ ǫ f ǫ j ,δ ( r ) · e ǫ ⊗ Y j , j − 1 ǫ = ± 1 . j ,δ, m ( r , θ, φ ) = 2 δ, m ( θ, φ ) , with The sign of ǫ (no summation over this repeated index) refers to the bosonic (fermionic) states with respective eigenvalues ǫ = +1 � 1 � ( ǫ = − 1) of the Fermion Parity Operator N F ; we have e +1 = 0 � 0 � and e − 1 = . 1

Solutions: Solutions are obtained for r γ ( j ,δ,ǫ ) e − 1 2 r 2 , f ǫ j ,δ ( r ) = where α + βǫ = δ ( j + 1 γ ( j ,δ,ǫ ) ( β ) = 2) + βǫ − 1 . The corresponding lowest weight state energy eigenvalue E j ,δ,ǫ ( β ) from H osc ( β )Ψ ǫ E j ,δ,ǫ ( β )Ψ ǫ j ,δ, m ( r , θ, φ ) = j ,δ, m ( r , θ, φ ) is δ ( j + 1 2) + βǫ + 1 E j ,δ,ǫ ( β ) = 2 . Since E j ,δ,ǫ ( β ) does not depend on the quantum number m , this energy eigenvalue is (2 j + 1) times degenerate.

Alternative Hilbert spaces Without loss of generality we can restrict the real parameter β to belong to the half-line β ≥ 0 since the mapping β ↔ − β is recovered by a similarity transformation which exchanges bosons into fermions: SH osc ( β ) S − 1 = H osc ( − β ) S = σ 1 ⊗ I 2 . with To the following j , δ, ǫ, m quantum numbers, j ∈ 1 2 + N 0 , δ = ± 1 , ǫ = ± 1 , m = − j , − j + 1 , . . . , j , is associated an sl (2 | 1) lowest weight vector and its induced rep. Two choices to select the Hilbert space naturally appear: i : the wave functions can be singular at the origin, case but they need to be normalized, case ii : the wave functions are assumed to be regular at the origin.

Case i corresponds in restricting the admissible lowest weight representations to those satisfying the necessary and sufficient condition 2 γ ( j ,δ,ǫ ) ( β ) + 3 > 0 . The normalizability condition is equivalent to the requirement E j ,δ,ǫ ( β ) > 0 for the lowest weight energy E j ,δ,ǫ ( β ). Case ii corresponds in restricting the admissible lowest weight representations to those satisfying the condition ≥ β ≥ 0 . γ ( j ,δ,ǫ ) ( β ) 0 for The single-valuedness of the wave functions at the origin implies that γ ( j ,δ,ǫ ) ( β ) = 0 can only be realized with vanishing ( l = 0) orbital angular momentum. At β = 0 one recovers the vacuum state of the undeformed oscillator. For the deformed β > 0 oscillator the strict inequality follows γ ( j ,δ,ǫ ) ( β ) > 0 for β > 0

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