New deformations of N = 4 and N = 8 supersymmetric mechanics

Abstract. This is a review of two different types of the deformed N = 4 and N = 8 supersymmetric mechanics. The first type is associated with the worldline realizations of the supergroups S U(2|1) (four supercharges), as well as of S U(2|2) and S U(4|1) (eight supercharges). The second type is the quaternionKähler (QK) deformation of the hyper-Kähler (HK) N = 4 mechanics models. The basic distinguishing feature of the QK models is a local N = 4 supersymmetry realized in d = 1 harmonic superspace.


Introduction
Supersymmetric Quantum Mechanics (SQM) [1] is the d = 1 supersymmetric theory.It: -Catches the basic features of higher-dimensional supersymmetric theories via the dimensional reduction; -Provides superextensions of integrable models like Calogero-Moser systems, Landautype models, etc.
In this Talk, two different types of deformations of N = 4 SQM models will be outlined.
• The appearance of the Wess-Zumino terms for the bosonic fields, of the type ∼ im(żz − z ż).
• At the lowest energy levels, wave functions form atypical su(2|1) multiplets, with unequal numbers of the bosonic and fermionic states.
As the next step, analogous deformations of N = 8 SQM were studied.The flat N = 8 superalgebra, admits two deformations with the minimal number of extra bosonic generators.
Not all of the admissible multiplets of the flat N = 8 SQM have S U(2|2) analogs.It is most important that the so called "root" N = 8 multiplet (8, 8, 0) does not.Meanwhile, all other flat N = 8 multiplets and their invariant actions can be obtained from the root one and its general action through the appropriate covariant truncations [16].How to construct a deformed version of the (8, 8, 0) multiplet?It becomes possible in the models based on world-line realizations of the supergroup S U(4|1).This multiplet is described by a chiral S U(4|1) superfield with the additional S U(4|1) covariant constraints on the component fields The d = 1 field content is just 8 = 2+6 real bosonic fields (φ, y IJ ) in the S U( 4) representation (1 ⊕ 6) and 4 complex fermionic fields χ L in the fundamental of S U( 4).
The invariant action has the very simple form where, in the bosonic limit, Besides this action, there were constructed two more invariant actions which are not equivalent to each other.One of them exhibits the relevant superconformal symmetry OS p(8|2).
It was also found that there exists another (twisted) multiplet (8, 8, 0), with the bosonic fields in 4 of S U( 4).

QK N = 4 SQM as a deformation of HK SQM models
Another type of deformations of N = 4 SQM models proceeds from the general Hyper-Kähler (HK) subclass of the latter.The deforrmed models are N = 4 supesymmetrization of the Quaternion-Kähler (QK) d = 1 sigma models [18].Both HK and QK N = 4 SQM models can be derived from N = 4, d = 1 harmonic superspace approach [19].
HK manifolds are bosonic targets of sigma models with rigid N = 2, d = 4 supersymmetry [20].After coupling these models to local N = 2, d = 4 supersymmetry in the supergravity framework the target spaces are deformed into the so called Quaternion-Kähler (QK) manifolds [21].QK manifolds are also 4n dimensional, but their holonomy group is a subgroup of S p(1) × S p(n).The deformation parameter is just Einstein constant κ, and in the "flat" limit κ → 0, the appropriate HK manifolds are recovered.
What about N = 4 Quaternion-Kähler SQM?The main question was as to how to ensure, in one or another way, a local supersymmetry and local S U(2) automorphism symmetry.
In our recent paper with Luca Mezincescu [18], it was shown how to construct N = 4 SQM with an arbitrary QK bosonic target.Like in constructing N = 4 HK SQM, the basic tool is d = 1 harmonic superspace.Let us give a brief account of this approach.
The starting point is the ordinary Its harmonic extension is introduced as : Its main feature is the existence of the analytic basis, in which one can single out an analytic subspace and define the analytic superfields: An important tool is the harmonic derivatives: The basic building block of N = 4 SQM models is N = 4, d = 1 multiplet (4, 4, 0).It is described off-shell by an analytic superfield q +a (ζ, u): (a) D + q +a = D+ q +a = 0 (Grassmann analyticity), (b) D ++ q +a = 0 (Harmonic analyticity), The free off-shell action reads: The general nonlinear d = 1 sigma model action is constructed as: In the bosonic sector it yields HKT ("Hyper-Kähler with torsion") sigma model.In components, the torsion appears in a term quartic in fermions.How to construct general HK N = 4, d = 1 sigma models?No torsion appears in this case, the geometry involves only Riemann curvature tensor.The answer was found in [22].
The basic superfields are still real analytic, q +A (ζ, u) = f iA u + i + ... , i = 1, 2, A = 1, . . .2n , they encompass just 4n fields f iA (t) parametrizing the target bosonic manifold, (q + A ) = Ω AB q + B , with Ω AB = −Ω BA a constant symplectic metric.The linear constraint D ++ q +A = 0 is promoted to a nonlinear one The superfield action is bilinear as in the free case, the whole interaction appearing only due to nonlinear deformation of the q +A -constraint.The object L +4 is an analytic hyper-Kähler potential [23]: every L +4 produces the component HK metric g iA kB ( f ) and, vice versa, each HK metric originates from some HK potential L +4 .The harmonic superspace approach supplies the natural arena for defining N = 4 QK SQM.The new features of these models as compared to their HK prototypes are as follows.
By analogy with the N = 2, d = 4 case we postulate that local N = 4, d = 1 supersymmetry preserves the Grassmann analyticity, The explicit structure of the minimal set of analytic parameters is as follows Here, b(t), τ (ik) (t) and λ i (t), λi (t) are arbitrary local parameters, bosonic and fermionic.How to generalize the (4, 4, 0) superfields q +A (ζ, w) to local supersymmetry?The simplest possibility is to keep the linear constraint It is covariant under the transformations δD ++ = −Λ ++ D 0 and δq +a = Λ 0 q +a , with To construct invariant actions, one also needs the transformations of the integration measures µ H := dtdwd 2 θ + d 2 θ − , µ (−2) := dtdwd 2 θ + and the covariant derivative The extra superfields Q +r (ζ, w) , r = 1, 2, . . .2n , encompassing n off-shell multiplets (4, 4, 0), obey the same linear harmonic constraint D ++ Q +r = 0 and transform under local N = 4 supersymmetry in the same way as q +a .The basic part of the total invariant action reads where γ = ±1 .The new object is a supervielbein E. It is harmonic-independent, D ++ E = D −− E = 0 , and possesses the following local N = 4 supersymmetry transformation law EPJ Web of Conferences 191, 06004 (2018) https://doi.org/10.1051/epjconf/201819106004QUARKS-2018 This is not the end!One more important term should be added to S (2) : Thus the simplest locally N = 4 supersymmetric action reads Why should the "cosmological constant" term S β be added?
To answer this question, we pass to the bosonic limit: The auxiliary fields M, M and µ fully decouple.Also, e(t) is an analog of d = 1 vierbein, so it is natural to choose the gauge e = 1 .
Then the bosonic Lagrangian becomes At β 0 L ik can be eliminated by its algebraic equation of motion, while D serves as the Lagrange multiplier giving rise to the constraint relating f ia and F ir : Assuming that f ia starts with a constant (compensator!), one uses local S U(2) freedom, δ f ia = τ i l f la , to gauge away the triplet from f ia , Then the constraint can be solved as (and analogously for γ = −1).The final form of the bosonic action for γ = 1 is The case of γ = −1 is recovered by the replacement |β| → −|β| .
These actions describe d = 1 nonlinear sigma models on non-compact and compact maximally "flat" 4n dimensional QK manifolds, respectively, Thus N = 4 mechanics constructed is just a superextension of these QK sigma models.

Generalizations
The basic step in generalizing to N = 4 mechanics with an arbitrary QK manifold is to pass to nonlinear harmonic constraints The invariant superfield action looks the same as in the HP n case The bosonic action precisely coincides with d = 1 reduction of the general QK sigma model action derived from N = 2, d = 4 supergravity-matter action in [24].This coincidence proves that we have indeed constructed the most general QK N = 4 mechanics.
One more possibility is to consider the following generalization of the HP n action S loc (q, Q) = µ H √ EF (X, Y, w − ), X := √ E (q +a q − a ) , Y := √ E (Q +r Q − r ), ( 14) When E=const, it is reduced to the particular form of the HKT action µ H F (q +A , q −B , w ± ), while for F (X, Y, w − ) = γ X − Y + β just to HP n action.So the target geometry associated with S loc (q, Q) is expected to be a kind of QKT, i.e. "Quaternion-Kähler with torsion" [25].To date, not too much known about such geometries...

Summary and Outlook
Two different deformations of N = 8 supersymmetric mechanics based on the supergroups S U(2|2) and S U(4|1) as a generalization of the S U(2|1) mechanics were sketched.N = 4, d = 1 harmonic superspace methods were used to construct a new class of N = 4 supersymmetric mechanics models, those with d = 1 Quaternion-Kähler sigma models as the bosonic core.
A few generalizations of QK mechanics were proposed, in particular "Quaternion-Kähler with torsion" (QKT) models.

Some further lines of study:
(a).To construct the Hamiltonian formalism for the new class of mechanical systems, including N = 4 supercharges.To perform quantization, at least for the simplest case of HP n mechanics, to find the energy spectrum.