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.

Germany

Federico Arrighi

Saurish Khandelwal

Olaf Lechtenfeld

Towards a Nicolai map for supergravity

We propose a novel approach to study hyperbolic Kac-Moody algebras, and more specifically, the Feingold-Frenkel algebra 𝔉, which is based on considering the tensor algebra of level-one states before descending to the Lie algebra by converting tensor products into multiple commutators. This method enables us to exploit the presence of mutually commuting coset Virasoro algebras, whose number grows without bound with increasing affine level. We present the complete decomposition of the tensor algebra under the affine and coset Virasoro symmetries for all levels $\textit{ℓ}$ ≤ 5, as well as the maximal tensor ground states from which all elements of 𝔉 up to level five can be (redundantly) generated by the joint action of the affine and coset Virasoro generators, and subsequent conversion to multi-commutators, which are then expressed in terms of transversal and longitudinal DDF states. We comment on the deep relations between the algebra 𝔉 and Einstein gravity in four space-time dimensions, and outline novel directions for future work.

Axel Kleinschmidt

Switzerland

Hannes Malcha

Hermann Nicolai

From tensor algebras to hyperbolic Kac-Moody algebras

We clarify and extend our earlier work [1, 2] where it was shown how to amend a scheme originally proposed by M. Gell-Mann to identify the three families of quarks and leptons of the Standard Model with the 48 spin- $$ \frac{1}{2} $$ fermions of $\textit{N}$ = 8 supergravity that remain after absorption of eight Goldstinos, a scheme that in its original form is dynamically realized at the SU(3) × U(1) stationary point of gauged $\textit{N}$ = 8 supergravity. We explain how to deform and enlarge this symmetry at the kinematical level to the full Standard Model symmetry group SU(3) $\textit{×}$SU(2) $\textit{×}$U(1) , with the correct charge and chiral assignments for all fermions. The framework also leaves room for an extra U(1) symmetry. This symmetry enhancement is achieved by embedding the Standard Model symmetries into (a quotient group of) K(E ), the ‘maximal compact subgroup’ of the maximal rank hyperbolic Kac-Moody symmetry E , and an infinite prolongation of the SU(8) $\textit{R}$-symmetry of $\textit{N}$ = 8 supergravity. This scheme, which is also supposed to encompass quantum gravity, cannot be realized within the framework of space-time based (quantum) field theory, but requires space-time and related geometrical concepts to be ‘emergent’. We critically review the main hypotheses underlying this construction.

Poland

Krzysztof Meissner

Standard model symmetries and K(E

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 $\textit{ℂP}$ 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 compute the Nicolai map for the supersymmetric Yang-Mills theory, in the light-cone gauge, to the second order in the coupling constant for all critical dimensions (d = 3, 4, 6, 10). The process of integrating out unphysical degrees of freedom in this gauge, produces a four fermion interaction term. We show that, to the order investigated here, this term is harmless. We demonstrate the existence of a particularly ‘simple’ map in d = 4 in the light-cone gauge and address the issue of uniqueness in the context of the map. We also investigate the map in the light-cone superspace in d = 4.

India

Nipun Bhave

Saurabh Pant

Nicolai maps and uniqueness in the light-cone gauge

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

We examine possible experimental signatures that may be exploited to search for stable supermassive particles with electric charges of  $${{\mathcal {O}}}(1)$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>   in future underground experiments, and the upcoming JUNO experiment in particular. The telltale signal providing a unique signature of such particles, would be a correlated sequence of three or more nuclear recoils along a straight line, corresponding to the motion of a non-relativistic ( $$\beta \lesssim 10^{-2}$$  <math> <mrow> <mi>β</mi> <mo>≲</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>  ) particle that could enter the detector from any direction. We provide some preliminary estimates for the expected event rates.

Searching for supermassive charged gravitinos in underground experiments

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

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.