Olaf Lechtenfeld

A bstract: Quantum correlation functions in supersymmetric field theories can be encoded in the Nicolai map. The infinitesimal inverse map provides their response to a change in a coupling constant through a “flow equation”. We investigate the flow in the gravitational constant κ for four-dimensional Poincaré supergravity via its superconformal formulation. By expressing the full superconformal off-shell action as a supervariation we derive a “flow operator” in the effective vierbein theory, whose exponentiation yields the gauge-invariant part of an all-order (inverse) Nicolai map for supergravity. Alas, the BRST gauge fixing adds to this functional differential flow operator a multiplicative term, which ruins its derivational property and therefore jeopardizes its connection with the Nicolai map. We argue and present an example, however, that such multiplicative flow contributions might be rewritable as derivational ones with the help of Wick’s theorem in the free effective theory where the Nicolai-transformed correlators are ultimately evaluated. At least in the Landau gauge, the supergravity flow equation is shown to be regular at κ =0, allowing for a perturbative expansion around Minkowski geometry. Therewith, we have overcome two of the three obstacles met in an earlier attempt towards a Nicolai map for Poincaré supergravity. Finally, we compare with the direct on-shell construction of a Nicolai map to leading order in κ .

We establish a relationship between the Davey-Stewartson II (DS II) integrable system in ( 2 + 1 ) dimensions and quasiexactly solvable planar interacting Dirac Hamiltonians that exhibit the super-Klein tunneling (SKT) effect. The Dirac interactions are constructed from the real and imaginary parts of breather solutions of the DS II system. In this framework, the SKT effect arises when the energy is tuned to match the constant background of the soliton, while the resulting Dirac Hamiltonians simultaneously support bound states embedded in the continuum. By imposing the SKT boundary conditions, we employ Darboux transformations to construct a general three-parameter family of DS II breather solutions that can be mapped to Dirac Hamiltonians. At the initial soliton time, the corresponding Dirac systems form a massless two-parameter family of Hermitian models with nontrivial electrostatic potentials. As the soliton time evolves, the systems become P T -symmetric and develop a nontrivial imaginary mass term. Finally, when the soliton time is taken to be imaginary, the construction yields Hermitian Dirac systems that lack time-reversal symmetry. In all cases, we identify the emergence of quasisymmetry transformations that preserve the SKT subspace of states while not commuting with the full Hamiltonian.

A bstract: We investigate the possibility of a Nicolai map for minimal supergravity in four dimensions. Such a map would allow for the computation of quantum supergravity correlation functions in terms of flat-space correlators in an effective nonlocal bosonic theory with the help of a nonlinear field transformation, the inverse Nicolai map. Such a map is guaranteed for off-shell global supersymmetry, but local supersymmetry presents at least three obstacles for the construction. Their effects are analyzed in detail, in an attempt to set up a Nicolai map to leading order in the gravitational coupling. We find indications that the conformal factor of the metric obstructs the off-shell construction, suggesting that the unimodular variant of supergravity may do better. The on-shell supersymmetry approach, successful for super-Yang-Mills theory in its critical dimensions, also fails, because the graviton self-interaction cannot be written as a supervariation. Nevertheless, by brute force we obtain a four-parameter first-order Nicolai map fulfilling the free-action condition. For the acid test of determinant matching, however, one needs to push the general ansatz and the perturbative expansion to the second order and to the quantum level.

A bstract: It is long known that quantum Calogero models feature intertwining operators, which increase or decrease the coupling constant by an integer amount, for any fixed number of particles. We name these as “horizontal” and construct “vertical” intertwiners, which change the number of interacting particles for a fixed but integer value of the coupling constant. The emerging structure of a grid of intertwiners exists only in the algebraically integrable situation (integer coupling) and allows one to obtain each Liouville charge from the free power sum in the particle momenta by iterated intertwining either horizontally or vertically. We present recursion formulæ for the intertwiners as a factorization problem for partial differential operators and prove their existence for small values of particle number and coupling. As a byproduct, a new basis of non-symmetric Liouville integrals appears, algebraically related to the standard symmetric one.

Supersymmetric field theories can be characterized by their Nicolai map, which is a nonlinear and nonlocal field transformation to their free-field limit. The systematic construction of such maps has recently been outlined for actions with power more than two in the fermions, which produces a perturbative expansion in loop-decorated fermionic tree diagrams. We thoroughly investigate the nonlinear ℂP ¹ sigma model in (3+1)-dimensional Minkowski space as a paradigmatical example. We construct and test a chiral form of the Nicolai map, to third order in the coupling, including all (regularized) quantum parts. In addition, all trees with one or two edges are summed up. The free-action condition determines only the one-edge part of the map. We resolve the fermion loop decoration of the Nicolai trees by injecting an auxiliary vector field and present the ensuing classical Nicolai map to second order in a dimensionful coupling.

In 1977 Lüscher found a class of SO(4)-symmetric SU(2) Yang–Mills solutions in Minkowski space, which have been rederived 40 years later by employing the isometry S3 ≅ SU(2) and conformally mapping SU(2)-equivariant solutions of the Yang–Mills equations on (two copies of) de Sitter space dS4≅R×S3. Here we present the noncompact analog of this construction via AdS3 ≅ SU(1, 1). On (two copies of) anti-de Sitter space AdS4≅R×AdS3 we write down SU(1,1)-equivariant Yang–Mills solutions and conformally map them to R1,3. This yields a two-parameter family of exact SU(1,1) Yang–Mills solutions on Minkowski space, whose field strengths are essentially rational functions of Cartesian coordinates. Gluing the two AdS copies happens on a dS3 hyperboloid in Minkowski space, and our Yang–Mills configurations are singular on a two-dimensional hyperboloid dS3∩R1,2. This renders their action and the energy infinite, although the field strengths fall off fast asymptotically except along the lightcone. We also construct Abelian solutions, which share these properties but are less symmetric and of zero action.

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE ₃ with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE ₃ is a ℤ₂ extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE ₃, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.