SO(8) supergravity and the magic of machine learning

Abstract

Using de Wit-Nicolai D = 4 N = 8 SO(8) supergravity as an example, we show how modern Machine Learning software libraries such as Google's TensorFlow can be employed to greatly simplify the analysis of high-dimensional scalar sectors of some M-Theory compactifications. We provide detailed information on the location, symmetries, and particle spectra and charges of 192 critical points on the scalar manifold of SO(8) supergravity, including one newly discovered N = 1 vacuum with SO(3) residual symmetry, one new potentially stabilizable non-supersymmetric solution, and examples for "Galois conjugate pairs" of solutions, i.e. solution-pairs that share the same gauge group embedding into SO(8) and minimal polynomials for the cosmological constant. Where feasible, we give analytic expressions for solution coordinates and cosmological constants.As the authors' aspiration is to present the discussion in a form that is accessible to both the Machine Learning and String Theory communities and allows adopting our methods towards the study of other models, we provide an introductory overview over the relevant Physics as well as Machine Learning concepts. This includes short pedagogical code examples. In particular, we show how to formulate a requirement for residual Supersymmetry as a Machine Learning loss function and effectively guide the numerical search towards supersymmetric critical points. Numerical investigations suggest that there are no further supersymmetric vacua beyond this newly discovered fifth solution.At the moment, the N = 8 Supergravity Theory is the only candidate in sight. There are likely to be a number of crucial calculations within the next few years that have the possibility of showing that the theory is no good. If the theory survives these tests, it will probably be some years more before we develop computational methods that will enable us to make predictions and before we can account for the initial conditions of the universe as well as the local physical laws. These will be the outstanding problems for theoretical physics in the next twenty years or so.But to end on a slightly alarmist note, they may not have much more time than that.At present, computers are a useful aid in research, but they have to be directed by human minds. If one extrapolates their recent rapid rate of development, however, it would seem quite possible that they will take over altogether in theoretical physics. So, maybe the end is in sight for theoretical physicists, if not for theoretical physics. S. Hawking, Conclusion of his 1981 Inaugural lecture [1]"Is the end in sight for theoretical physics?"

$$ \mathcal{N} $$ = 2 supersymmetric S-folds

“…By implementing a generalised S-S ansatz in E 7(7) -EFT, we uplifted the AdS 4 vacuum to a new family of AdS 4 × S 1 × S 5 S-folds of type IIB supergravity with hyperbolic monodromies M(k) = −S T k (with k ≥ 3 ) along S 1 . The residual SU(2) × U(1) symmetry and N = 2 supersymmetry of the AdS 4 vacuum are realised on the S-folds: the internal S 5 is deformed into a product of S 2 and (squashed) S 3 with SU(2) × U(1) σ × U(1) φ isometries and a warping factor, whereas the 4 The two terms in B2 and C2 which are proportional to σ1 ∧ σ3 can be eliminated by means of a gauge transformation of the form B2 → B2 − d 2 √ 2 ∆ 4 e −η sin(2θ) cos θ cos ψ σ2 , C2 → C2 + d 2 √ 2 ∆ 4 e η sin(2θ) cos θ sin ψ σ2 , (3.45) where we have shifted the coordinate φ → ψ + π 4 . However, since these terms are generated by the generalised S-S ansatz discussed in Section 3.1, we will retain them here.…”

Properties of the new $$ \mathcal{N} $$ = 1 AdS4 vacuum of maximal supergravity

“…The numerical data for the N = 1 critical point, S1384096, in [1] point towards additional symmetry, which we identify as a discrete Z 2 subgroup of the SO(8) gauge group. The resulting SO(3) × Z 2 -invariant truncation of the N = 8 supergravity can be constructed analytically.…”

“…Recently, a new numerical approach based on Machine Learning (ML) software libraries, such as Google's TensorFlow [30], was employed in [1] to simplify the analysis of the potential resulting in the total of 192 critical points together with a precise information about those points that includes the mass spectra of small fluctuations and unbroken (super)symmetries. It is expected that this list of critical points should be nearly complete.…”

“…Perhaps the most interesting result of the search in [1] is a discovery of yet another N = 1 supersymmetric critical point, S1384096, which is invariant under a triality symmetric SO(3) subgroup of the SO(8) gauge group. Moreover, this point gives rise to the only new AdS 4 solution that is perturbatively stable.…”

“…

The recent comprehensive numerical study of critical points of the scalar potential of fourdimensional N = 8, SO(8) gauged supergravity using Machine Learning software in [1] has led to a discovery of a new N = 1 vacuum with a triality-invariant SO(3) symmetry. Guided by the numerical data for that point, we obtain a consistent SO(3) × Z 2 -invariant truncation of the N = 8 theory to an N = 1 supergravity with three chiral multiplets.

…”