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 $\textit{change the number of interacting particles}$ 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.

Chile

Francisco Correa

Luis Inzunza

Germany

Olaf Lechtenfeld

Coupling and particle number intertwiners in the Calogero model

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 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.

Nicolai maps for supersymmetric sigma models

We construct Nicolai maps for supersymmetric Yang–Mills theory in four and ten spacetime dimensions in the light-cone gauge, where the elimination of non-propagating degrees of freedom causes nonlocal and four-fermi interactions in the Lagrangian. The presence of the latter used to be an obstruction to the Nicolai map, which has recently been overcome at the price of quantum corrections to the map. No gauge-fixing or ghost terms arise in this formulation, since only physical transverse degrees of freedom occur. We present an explicit form of the Nicolai map to second order in the gauge coupling. In four dimensions, a ‘chiral’ choice of the map leaves one of the two transverse gauge-field modes invariant, which forces the classical part of the map (on the other mode) to become a polynomial (quadratic in the gauge coupling, cubic in the gauge field)! In the power series expansion for the ten-dimensional map however, cancellations at each order in the coupling are systematic but incomplete, still leaving an infinite power series for the Nicolai map (on all eight transverse modes). Nevertheless, the existence of a polynomial variant is conceivable, also for the maximal <math><mi>N</mi><mo>=</mo><mn>4</mn></math> theory in four dimensions.

Nicolai maps for super Yang–Mills on the light cone

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.

Don Zagier

A hyperbolic Kac-Moody Calogero model

Nicolai maps offer an alternative description of supersymmetric theories via nonlinear and nonlocal transformations characterized by the so-called ‘free-action’ and ‘determinant-matching’ conditions. The latter expresses the equality of the Jacobian determinant of the transformation with the one obtained by integrating out the fermions, which so far have been considered only to quadratic terms. We argue that such a restriction is not substantial, as Nicolai maps can be constructed for arbitrary nonlinear sigma models, which feature four-fermion interactions. The fermionic effective one-loop action then gets generalized to higher loops and the perturbative tree expansion of such Nicolai maps receives quantum corrections in the form of fermion loop decorations. The ‘free-action condition’ continues to hold for the classical map, but the ‘determinant-matching condition’ is extended to an infinite hierarchy in fermion loop order. After general considerations for sigma models in four dimensions, we specialize to the case of ℂP N  symmetric spaces and construct the associated Nicolai map. These sigma models admit a formulation with only quadratic fermions via an auxiliary vector field, which does not simplify our analysis.

Lorenzo Casarin

Maximilian Rupprecht

Nicolai maps with four-fermion interactions

Adding a topological theta term to the action of <math><mi>N</mi><mo>=</mo><mn>1</mn></math> <math><mi>D</mi><mo>=</mo><mn>4</mn></math> super Yang–Mills theory modifies its Nicolai map. For the BPS value of the theta angle a chiral version of the map emerges, which allows for a considerable simplification compared to the non-chiral formulation. We exhibit these improvements to all orders in perturbation theory and compute the map to fourth order in the coupling on the Laudau-gauge hypersurface. The second-order contribution vanishes, and antisymmetrizations are more manifest. All checks are verified to third order.

An improved Nicolai map for super Yang–Mills theory

We find a two-parameter family of solutions of the Yang–Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts of it with non-compact coset spaces with SO(1,3) isometry. The interior of the lightcone is foliated with hyperbolic space <math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>≅</mo><mtext>SO</mtext><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>/</mo><mtext>SO</mtext><mo>(</mo><mn>3</mn><mo>)</mo></math>, while the exterior of the lightcone employs de Sitter space dS<math><mmultiscripts><mrow><mo>≅</mo></mrow><mprescripts></mprescripts><mrow><mn>3</mn></mrow><none></none></mmultiscripts><mtext>SO</mtext><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo><mo>/</mo><mtext>SO</mtext><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>. The lightcone itself is parametrized by SO(1,3)/ISO(2) in a nilpotent fashion. Equivariant reduction of the SO(1,3) Yang–Mills system on the first two coset spaces yields a mechanical system with inverted double-well potential and the foliation parameter serving as an evolution parameter. Its known analytic solutions are periodic or runaway except for the kink. On the lightcone, only the vacuum solution remains. The constructed Yang–Mills field strength is singular across the lightcone and of infinite action due to the noncompact cosets. Its energy-momentum tensor takes a very simple form, with energy density of opposite signs inside and outside the lightcone.

Kaushlendra Kumar

Gabriel Picanço Costa

Jona Röhrig

Yang–Mills solutions on Minkowski space via non-compact coset spaces

In rigidly supersymmetric quantum theories, the Nicolai map allows one to turn on a coupling constant (from zero to a finite value) by keeping the (free) functional integration measure but subjecting the fields to a particular nonlocal and nonlinear transformation. A recursive perturbative construction of the Nicolai-transformed field configuration expresses it as a power series in the coupling, with its coefficient function at order n being a sum of particular tree diagrams. For a quantum-mechanical example, the size of these tree diagrams (under a certain functional norm) is estimated by the <math><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math>st power of the field size, and their number grows like <math><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mn>4.967</mn></mrow><mrow><mspace width="0.25em"></mspace><mi>n</mi></mrow></msup></math>. Such an asymptotic behaviour translates to a finite convergence radius for the formal perturbative expansion of the Nicolai map, which establishes its non-perturbative existence. The known factorial growth of the number of Feynman diagrams for quantum correlators is reproduced by the combinatorics of free-field Wick contractions as usual. We expect our results to extend to higher dimensions, including super Yang–Mills theory.

Supersymmetric large-order perturbation with the Nicolai map

The Nicolai map is a field transformation that relates supersymmetric theories at finite couplings g with the free theory at g = 0. It is obtained via an ordered exponential of the coupling flow operator integrated from 0 to g. Allowing multiple couplings, we find that the map in general depends on the chosen integration contour in coupling space. This induces a large functional freedom in the construction of the Nicolai map, which cancels in all correlator computations. Under a certain condition on the coupling flow operator the ambiguity disappears, and the power-series expansion for the map collapses to a linear function in the coupling. A special role is played by topological (theta) couplings, which do not affect perturbative correlation functions but also alter the Nicolai map. We demonstate that for certain ‘magical’ theta values the uniqueness condition holds, providing an exact map polynomial in the fields and independent of the integration contour. This feature is related to critical points of the Nicolai map and the existence of ‘instantons’. As a toy model, we work with   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric quantum mechanics. For a cubic superpotential and a theta term, we explicitly compute the one-, two- and three-point correlation function to one-loop order employing a graphical representation of the (inverse) Nicolai map in terms of tree diagrams, confirming the cancellation of theta dependence. Comparison of Nicolai and conventional Feynman perturbation theory nontrivially yields complete agreement, but only after adding all (1PI and 1PR) contributions.

Is the Nicolai map unique?

We analyze the integrability of the <math><mi>N</mi></math>-extended supersymmetric Calogero–Moser model. We explicitly construct the Lax pair <math><mo>{</mo><mi>L</mi><mo>,</mo><mi>A</mi><mo>}</mo></math> for this system, which properly reproduces all equations of motion. After adding a supersymmetric oscillator potential we reduce the latter to solving <math><mover><mrow><mi>U</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mi>A</mi><mspace width="0.2em"></mspace><mi>U</mi></math> for the time evolution operator <math><mi>U</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>. The bosonic variables, however, evolve independently of U on closed trajectories, as is required for superintegrability. To visualize the structure of the conserved currents we derive the complete set of Liouville charges up to the fifth power in the momenta, for the <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supersymmetric model. The additional, non-involutive, conserved charges needed for a maximal superintegrability of this model are also found.