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CQG+ ‘Insight’ Article – A Kind Of Magic by Leron Borsten & Alessio Marrani

A Kind Of Magic

The road from Dunsink to the exceptional symmetries of M-theory


By Leron Borsten and Alessio Marrani 


Our journey starts in the fall of 1843 at the Dunsink Observatory[1], presiding from its hill-top vantage over the westerly reaches of Dublin City, seat to the then Astronomer Royal Sir William Rowan Hamilton. In the preceding months Hamilton had become preoccupied by the observation that multiplication by a complex phase induces a rotation in the Argand plane, revealing an intimate link between two-dimensional Euclidean geometry and the complex numbers ℂ. Fascinated by this unification of geometry and algebra, Hamilton set about the task of constructing a new number system that would do for three dimensions what the complexes did for two. After a series of trying failures, on October 16th 1843, while walking from the Dunsink Observatory to a meeting of the Royal Irish Academy on Dawson Street, Hamilton surmounted his apparent impasse in a moment of inspired clarity: rotations in three dimensions require a four-dimensional algebra with one real and three imaginary units satisfying the fundamental relations i= j= k= ijk = -1. The quaternions ℍ were thus born. Taken in that instant of epiphany, Hamilton etched his now famous equations onto the underside of Broome bridge, a cave painting illuminated not by campfire, but mathematical insight and imagination.  Like all great mathematical expressions, once seen they hang elegant and timeless, eternal patterns in the fixed stars merely chanced upon by our ancestral explorers.

Borsten_and_Marrani

Leron Borsten (left) and Alessio Marrani (right) stood before Hamilton’s fundamental relations, Broome bridge Dublin. Leron is currently a Schrödinger Fellow in the School of Theoretical Physics, Dublin Institute for Advanced Studies. Alessio is currently a Senior Grantee at the Enrico Fermi Research Centre, Roma.

This discovery set in motion a subtle dance intertwining algebra and symmetry. It invites diverse interpretations, but that which suits our purpose best is the realisation of the three families of classical simple Lie algebras, 𝔰𝔳(n+1), 𝔰𝔲(n+1), 𝔰𝔭(n+1), as the infinitesimal isometries of the real, complex and quaternionic projective spaces, ℝℙn, ℂℙn, ℍℙn. The classical Lie algebras are unified as rotations in real, complex and quaternionic universes. Yet, the five remaining exceptional simple Lie algebras, 𝔤2, 𝔣4, 𝔢6, 𝔢7, 𝔢8, are left unaccounted for and, from this perspective at least, geometrically enigmatic.

To remedy this shortcoming we must first return to Dublin, 1843. Hamilton’s college friend, John T. Graves, on receiving word of the quaternions was struck by their seemingly conjured existence, writing to Hamilton on October 26th: “If with your alchemy you can make three pounds of gold, why should you stop there?”. But two months later to the day, Graves reciprocated, sharing with Hamilton a further generalisation of the real ℝ, complex ℂ, and quaternionic ℍ numbers: the octonions 𝕆. Endowed with seven imaginary units, ei,  the octonions constitute the largest example of what are now known as the normed division algebras: ℝ, ℂ, ℍ, 𝕆. The multiplication rules of ei are governed by the Fano plane, as described in figure 1. Using the Cayley-Dickson doubling procedure we can build each algebra from two copies of its predecessor. However, with each doubling a property is lost. In particular, the octonions, unlike their well-mannered older siblings, are non-associative. This makes 𝕆 capricious and uncooperative, but also exceptional.

Borsten_Fig_2

Figure 1: The Fano plane. Following the lines gives the multiplication rules of the imaginary octonions (going against the arrows, one picks up a minus sign).

Returning to the simple Lie algebras, the sequence ℝℙn, ℂℙn, ℍℙn cries out for the inclusion of 𝕆ℙn. The octonionic projective line 𝕆ℙ1 reproduces the classical Lie algebra 𝔰𝔳(8) already accommodated by ℝℙ7. However, the isometries of the next rung on the ladder, the Cayley plane 𝕆ℙ2, do indeed yield an exceptional algebra, namely 𝔣4. However, the non-associativity of 𝕆 renders 𝕆ℙn a bona fide projective space for ≤ 2 only and consequently the sequence ends here, hence the singular status of 𝔣4. It would seem, naively, that the remaining exceptional algebras do not fit into this story. However, when viewed in the right way, the Cayley plane realisation of 𝔣4 can be generalised by considering not one, but two algebras, ℝ ⊗ 𝕆, ℂ ⊗ 𝕆, ℍ ⊗ 𝕆, and 𝕆 ⊗ 𝕆 yielding precisely the exceptional Lie algebras 𝔣4, 𝔢6, 𝔢7, and 𝔢8. Allowing in this construction the two algebras to vary over all ℝ, ℂ, ℍ, 𝕆, we obtain what has come to be known as the Freudenthal-Rosenfeld-Tits magicsquare of Lie algebras, as depicted in figure 2. Since these early discoveries, the octonions have been found time and time again lurking in the corners where geometry meets algebra.

Borsten_Fig_1

Figure 2: The Freudenthal-Rosenfeld-Tits magic square of Lie algebras. The exceptional algebras appear in the octonionic row/column. Each entry can be realised as a symmetry of supergravity.

Although it would be fair to say that the octonions have yet to cement themselves in the annals of physics they have over the years appeared in a variety of suggestive guises. One such occurrence takes place in M-theory, an ambitious, albeit tentative, approach to the challenges of quantum gravity and unification. Being fundamentally non-perturbative, M-theory remains largely mysterious. A vital piece of the puzzle in our present understanding is the notion of “U-duality”. The five consistent ten-dimensional superstring theories and eleven-dimensional supergravity are interconnected through a web of U-duality relations, leading to the conjecture that they merely represent disparate glimpses of a single overarching framework living in D = 11 spacetime dimensions: M-theory. To make contact with our daily four-dimensional experience one can compactify. Although phenomenologically irrelevant, the simplest example of a compactification is given by taking one dimension to form a circle. If the radius of the circle is small enough, this dimension becomes essentially undetectable. By compactifying on an n-torus, that is taking n dimensions as circles, we can descend to D = 11 – n dimensions. The low-energy effective field theory limit of M-theory compactified on an n-torus is the unique maximally supersymmetric D = 11 – n supergravity theory. In this limit, the U-dualities of M-theory are reflected in the global symmetries of the corresponding supergravity theory. In particular, for D = 5, 4, 3 or n = 6, 7, 8 the global symmetry algebras are given by the exceptional  Lie algebras sitting in the octonionic row/column of the magic square[2]. We have overlooked a subtlety here. On compactifying eleven-dimensional supergravity to D = 5, 4, 3 the 𝔢6, 𝔢7, 𝔢8 symmetries are initially hidden, revealing themselves only once a judicious choice of (generalised) electromagnetic duality transformations[3] has been applied.

What happens when some other choice of dualisations is made? Well, the manifest symmetries are typically different in each case. What we demonstrate in our paper is that there exists a choice of dualisations for which a fascinating generalisation of the magic square makes an unexpected appearance. On complexifying the normed division algebras, which we will continue to denote ℝ, ℂ, ℍ, 𝕆, two new algebras in the sequence emerge: the three-dimensional ternions 𝕋, nestled tightly between ℂ and ℍ, and the six-dimensional sextonions 𝕊 sitting half-way from ℍ to 𝕆. Including 𝕋 and 𝕊 in the magic square construction reveals two further half-levels, obscured from view between the oft-visited floors of the ℝ, ℂ, ℍ, 𝕆 edifice. In particular, the 𝕋 ⊗ 𝕆 and 𝕊 ⊗ 𝕆 entries yield the non-reductive exceptional Lie algebras 𝔢 and 𝔢, living half-lives somewhere in-between 𝔢6, 𝔢7, and 𝔢8. Remarkably, the entire extended magic square, and so implicitly our enlarged ℝ, ℂ, 𝕋, ℍ, 𝕊, 𝕆 family, is realised through the symmetry algebras of supergravity.

A kind of magic, if you will.


[1] Now a constituent of the Astronomy and Astrophysics section of the School of Cosmic Physics, Dublin Institute for Advanced Studies.

[2] For all the group theory and supergravity aficionados, please note we are not paying attention to the particular real forms that appear here and throughout.

[3] Not to be confused with the U-dualities of M-theory.


Further reading

Normed division algebras and the magic square:

J.C. Baez, The Octonions, Bull. Am. Math. Soc. 39 (2002), 145–205.

Supergravity, global symmetries and the magic square:

E. Cremmer, B. Julia, and J. Scherk, “Supergravity theory in 11 dimensions,” Phys. Lett. B76 (1978) 409–412.

E. Cremmer and B. Julia, “The SO(8) supergravity,” Nucl. Phys. B159 (1979) 141.

E. Cremmer, B. Julia, H. Lu, and C. Pope, “Dualization of dualities. 1.,” Nucl.Phys. B523 (1998) 73–144, arXiv:hep-th/9710119

B. Julia, “Group disintegrations,” in Superspace and Supergravity, S. Hawking and M. Rocek, eds., Nuffield Gravity Workshop, pp. 331–350. Cambridge University Press (1980).

M. Günaydin, G. Sierra, and P. K. Townsend, “Exceptional supergravity theories and the magic square,” Phys. Lett. B133 (1983) 72.

L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “A magic square from Yang-Mills squared,” Phys.Rev.Lett. 112 (2014) 131601, arXiv:1301.4176

Sextonions and the extended magic square:

B.W. Westbury, “Sextonions and the magic square,” Journal of the London Mathematical Society 73 (2006) no. 2, 455–474, arxiv:math/0411428 

J.M. Landsberg and L. Manivel, “The sextonions and 𝔢,” Advances in Mathematics 201 (2006) 2 no. 1, 143-179, arxiv:math/0402157

A. Marrani and P. Truini, “Sextonions, Zorn matrices, and 𝔢,” Letters in Mathematical Physics 107 (2017) no.10, 1859-1875, arXiv:1506.04604

Thursday 7th December: STP Seminar – “Quantum Aspects of Black Hole and Fuzzy Sphere in String Theory”

Title: Quantum Aspects of Black Hole and Fuzzy Sphere in String Theory

Speaker: Yoshifumi Hyakutake (Ibaraki University, Japan)

Abstract: One of important directions in superstring theory is to reveal quantum nature of black hole. In this talk we embed Schwarzschild black hole into superstring theory or M-theory, which we call a smeared black hole, and resolve quantum corrections to it. Then we boost the smeared black hole along the 11th direction and construct a smeared quantum black 0-brane in 10 dimensions. Quantum aspects of the thermodynamic for these black objects are investigated in detail. We also discuss fuzzy configurations which will correspond to the microscopic description of the black hole.

Time: Thursday 7th December 2017, 2.00pm.

Place: Lecture Room, School of Theoretical Physics, DIAS, 10 Burlington Road, Dublin 4.

2017-10 – An Algebraic Approach to Minimal Models in CFTs

DIAS-STP-17-10

An Algebraic Approach to Minimal Models in CFTs

D. O’Connor and V. Filev

This preprint is available on Arxiv.org

This preprint is available for download :

Details provided:

  • Date of publication : 06/10/17

To obtain copies of any of the preprints in the archives, please contact us and specify the preprint number(s), author(s), title(s) and number of copies wanted.

Thursday 2nd November: STP Seminar – “Top Mass from Asymptotic Safety”

Title: Top Mass from Asymptotic Safety

Speaker: Astrid Eichhorn (University of Heidelberg)

Abstract: I will introduce the key ideas underlying the asymptotic safety scenario, which is mainly explored as a model of quantum gravity and will review its current status. I will then highlight that it might at the same time provide an ultraviolet completion for the Standard Model of particle physics. First hints indicate that such a setting might even reduce the number of free parameters of the Standard Model, and turn the top mass as well as the low-energy value of the Abelian gauge coupling into predictable quantities. Within simple approximations of the Renormalization Group flow for gravity and matter, the theoretical values for these quantities obtained from asymptotic safety lie in the vicinity of the observed values.

Time: Thursday 2nd November 2017, 2.30pm.

Place: Lecture Room, School of Theoretical Physics, DIAS, 10 Burlington Road, Dublin 4.

Thursday 26th October: STP Seminar – “Towards understanding the endpoint of the superradiant instability: Kerr black holes with synchronised hair”

Title: Towards understanding the endpoint of the superradiant instability: Kerr black holes with synchronised hair

Speaker: Eugen Radu (Aveiro University)

Abstract: A 50 year-old lingering question in black hole (BH) physics is the endpoint of the Kerr BH superradiant instability, triggered by massive, bosonic fields.In a recent breakthrough, East and Pretorius reported long term numerical evolutions of this instability, using a Proca field to trigger it.Evolutions terminate in stationary states of the vector field condensate synchronised with a rotating BH horizon. We show these end points are fundamental states of Kerr BHs with synchronised Proca hair. We also propose a universal (i.e. field-spin independent), analytic model for the subset of BHs that possess a quasi-Kerr horizon, and show the model is accurate for hairy BHs that may emerge dynamically from superradiance.

Time: Thursday 26th October 2017, 2.30pm.

Place: Lecture Room, School of Theoretical Physics, DIAS, 10 Burlington Road, Dublin 4.

Hallowe’en Night 2017 – Tuesday 31st October, 18:30-20:00 : Don’t Be Afraid of the Dark (Matter)

Don’t Be Afraid of the Dark (Matter)

Dublin Institute for Advanced Studies (DIAS) will hold a special event this Hallow’en Night exploring the spooky dark matter that surrounds us.

Tuesday 31st October has been designated as International Dark Matter Day – to coincide with Hallowe’en. DIAS’ event will take place from 6.30pm to 8pm at 10 Burlington Road, Dublin 4.

Dark Matter

The evidence for Dark Matter (and the even more mysterious dark energy) has been steadily accumulating since the 1930s and we now believe that less than 5% of the universe is made of the “normal” matter that physicists study in the laboratory. The nature of the dark matter is one of the greatest mysteries in modern physics and will be discussed by the Directors of the DIAS schools of Cosmic Physics, Professor Luke Drury, and Theoretical Physics, Professor Werner Nahm as a dialogue between astrophysics and particle physics.

The event is free to attend, and you can register now at Eventbrite

15th December – School of Theoretical Physics Statutory Public Lecture 2017

**Advance booking is required here**

Dublin Institute for Advanced Studies – School of Theoretical Physics Statutory Public Lecture 2017

Friday 15th December 2017 at 6.00 p.m.

The Physics and Astrophysics of Merging Neutron-Star Binaries

By: Prof. Dr. Luciano Rezzolla (Goethe University of Frankfurt)

Location: Edmund Burke Theatre (Room 1008), Arts Building, Trinity College Dublin

Abstract: I will argue that if black holes represent one the most fascinating implications of Einstein’s theory of gravity, neutron stars in binary system are arguably its richest laboratory, where gravity blends with astrophysics and particle physics. I will discuss the rapid recent progress made in modelling these systems and show how the inspiral and merger of a binary system of neutron stars is more than a strong source of gravitational waves. Indeed, while the gravitational signal can provide tight constraints on the equation of state for matter at nuclear densities, the formation of a black-hole–torus system can explain much of the phenomenology of short gamma-ray bursts, while the ejection of matter during the merger can shed light on the chemical enrichment of the universe.

About Professor Rezzolla : Prof. Dr. Luciano Rezzolla is presently the Chair of Theoretical (Relativistic) Astrophysics and Director at the Institute for Theoretical Physics (ITP) of the Goethe University of Frankfurt, Germany. He is also Senior Fellow at the Frankfurt Institute of Advanced Studies (FIAS).

Dunsink Observatory Public Open Night – Maths Week 2017

Speaker: Dr Samuel Kováčik, Government of Ireland Post-Doctoral Fellow funded by the Irish Research Council and based in the School of Theoretical Physics, DIAS.

Title of Talk: Sir Hamilton and the story of making things up   (Sir William Rowan Hamilton (1805-1865))

Date : Wednesday 18th October 2017 at Dunsink Observatory

Advance booking for this talk is required here

Samuel Kovacik bio: Born in Bratislava (Slovakia), where he Graduated in Theoretical Physics, from Comenius University Bratislava, 2012.
He was then awarded a Ph.D. in Theoretical physics, from Comenius University Bratislava, 2016. This was followed by a Postdoctoral scholarship in DIAS from 2016.
He now holds a Government of Ireland Post-doctoral Fellowship position awarded by the Irish Research Council. Samuel is interested in the Research of Quantum Space(time) and he was honoured by the opportunity to give a TEDx talk in 2015. He spends some part of his free time on sports and the other on popularising science.

STP Vacancy – Junior Post-doctoral Scholarship(s)

The School of Theoretical Physics of the Dublin Institute for Advanced Studies invites applications for one or possibly two junior post-doctoral scholarships in theoretical physics. Applicants are welcome from all fields of theoretical and mathematical physics; however, preference for one of the positions will be given to candidates with a proven research record in gauge/gravity, lattice gauge theory or non-commutative geometry. Experience in Monte Carlo simulations would be an advantage.

Closing date for applications is December 3rd 2017.

Please apply with a CV, research statement, list of publications and
three letters of recommendation via the DIAS website:

https://jobs.stp.dias.ie/

2017-08 – Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

DIAS-STP-17-08

Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

P. Feehan

This preprint is available on Arxiv.org

This preprint is available for download :

Details provided:

  • Date of publication : 31/08/17

To obtain copies of any of the preprints in the archives, please contact us and specify the preprint number(s), author(s), title(s) and number of copies wanted.