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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

Science Week Event: Evening, 16th Nov. 2017, 5 Merrion Sq. – limitied places remaining

Earthquakes and other Geohazards

Despite its lack of large earthquakes, Ireland holds a special place in the history of Earthquake studies thanks to the pioneering work of Robert Mallet in the mid-19th century. Come see the past and present of live earthquake monitoring worldwide, from the heart of Dublin.

Location: Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2

Date: 16th November 2017

Time: 19:00 – 20:00 and 20:30 – 21:30

There are a limited number of tickets for this FREE event. Please go to:

https://www.eventbrite.ie/e/earthquakes-and-other-geohazards-tickets-39513975364

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.

M7.3 Iran earthquake, 12th November 2017

A magnitude 7.3 earthquake occurred on the 12th November 2017 at 18:18 UTC in the northwest of Iran. For more details please see

this post on the INSN homepage.

Science Week 2017 Press Release

Press Release

Thursday 9th November 2017

DIAS School of Cosmic Physics announce programme of events for Science Week 2017

 

Space, telescopes, meteors earthquakes and other geohazards will form part of the exciting line-up of the Dublin Institute for Advanced Studies (DIAS) School of Cosmic Physics 2017 Science Week programme.

Science Week takes place nationwide from 12th to 19th November and the DIAS School of Cosmic Physics is offering a schedule of free interactive events and outreach activities led by their researchers.

Events and activities taking place at DIAS School of Cosmic Physics during Science Week include:

·         ‘Space and Telescopes of the Future’ a public talk by space enthusiast and radio broadcaster, Seanie Morris at Dunsink Observatory, Castleknock, Dublin 15, Sunday 12th November 2017 at 7.30pm: https://www.dias.ie/2017/11/06/science-week-at-dunsink-observatory/

  • Daily talks on meteors and space travel, by Dr Jonathan Mackey and other DIAS Researchers for primary and secondary school students at Dunsink Observatory.
  • Talks on meteors for groups from the Irish Girl Guides and secondary school students from DIAS, Astronomy & Astrophysics PhD Candidate, Sam Green at Dunsink Observatory.

All of the public events will be offered on a ‘first come, first served’ basis and members of the public are encouraged to book early.

Announcing details of their Science Week programme today, Dr Eucharia Meehan, Director of DIAS, said: “We’re delighted to announce our programme of events for this year’s Science Week.

“Science Week is an ideal opportunity for the public to engage with the work of our researchers  and this year we have some exciting talks from researchers in the School of Cosmic Physics.

“We would encourage anyone with a curiosity about science, space and the natural world to explore with the Dublin Institute for Advanced Studies this Science Week.”

 

ENDS

 

For further information, contact:

Eva Dowling, Alice PR & Events, Tel: 01-5582151 / 083-1496045, Email: media@alicepr.com

Notes to Editors:

DIAS was founded by Éamon de Valera in 1940 as a research centre focused on three disciplines: Celtic Studies, Theoretical Physics and Cosmic Physics.  In addition to conducting and publishing research, the organisation runs the Dunsink Observatory and coordinates a number of national initiatives on behalf of government.  Further information is available at www.dias.ie.

Follow DIAS on social media:

  • Twitter: @DIAS_Dublin

Facebook: www.facebook.com/DIASDublin

Science Week at Dunsink Observatory

Kick Off Science Week at Dunsink Observatory

Seanie Morris talks about Space and Telescopes of the Future
Sunday November 12th at 7:20 pm

Free Event, ample free parking

Advance booking required: Book here



Family Open Night, November 22nd

Big Bear Planetarium at Dunsink Observatory: “BACK TO MARS FOR GOOD”

Using state-of-the-art Fulldome 360-degree 3D digital projection technology with stunning graphics and advanced computer simulations. This is a  fully interactive learning environment which is educational and fun. Discover the many fascinating facts about our Universe and learn from qualified Astro Officers.


Starts at 7:00 pm,  Free Event and ample Free Parking

Advance booking required:  Fully Booked

26th October 2017 – DIAS Annual Report Published

PRESS RELEASE 

Thursday, 26th October 2017

DIAS annual report

The Dublin Institute for Advanced Studies (DIAS) has today (26.10.17) published its annual report for 2016.

The report documents the key activities undertaken by DIAS last year, including global research, involvement in major national scientific initiatives, and public engagement campaigns.

DIAS was founded by Éamon de Valera in 1940 as a fundamental research centre focused on three disciplines: Celtic Studies, Theoretical Physics and Cosmic Physics.  In addition to conducting and publishing research, the organisation runs the Dunsink Observatory and coordinates a number of national initiatives on behalf of government.

Some of the key highlights in the 2016 DIAS annual report include:

  • Attracting international talent to Ireland through competitive processes – DIAS funded 85 researchers last year. Of these, 30 per cent were Irish, with 70 per cent from other countries of origin.
  • 115 international research visitors to DIAS from 20 countries.
  • Success in securing competitive research funding including a €2.8m award to initiate the development of the first permanent sea-floor seismic, tsunami and marine acoustic observatory in the Irish NE Atlantic. The project is entitled “Insitu Marine Laboratory for Geosystems Research” (iMARL).
  • Further development of the Irish Script on Screen initiative to the point that 365 manuscripts are now available online for researchers nationally and internationally. This online resource had over four million hits in 2016.
  • High-level publications across high-impact journals (including Nature).
  • Ongoing development of the iCRAG Centre for geosciences, in which DIAS is a leading partner.
  • DIAS’s key role in the establishment of the iLOFAR (the European low-frequency radio telescope) site.
  • High-profile international collaborations, including with the European Space Observatory, the European Space Agency, the James Webb Space Telescope endeavour, the High Energy Stereoscopic System (HESS) and the Cherenkov Telescope Array (CTA).
  • 5,500 visitors to the Dunsink Observatory; and the installation of Meteor cameras at Dunsink.
  • DIAS’s ongoing coordination of the Irish National Seismic Network, and the Seismology in Schools initiative.
  • DIAS’s partnership with the Department of Foreign Affairs and Trade to run the national office for the Preparatory Commission for the Comprehensive Nuclear-Test-Ban Treaty Organisation. This office marked its twentieth anniversary in 2016.

Commenting today (26.10.17) on the publication of the annual report, Dr. Vincent Cunnane, Chairman of DIAS, said: “For DIAS, 2016 began with the deployment of Ocean Bottom Seismometers off the coast of Donegal and ended with a lecture on why experts disagree about the physics of climate change.

“In addition to our growing research output, we had a significant increase in public engagement over the year – through our own lectures and events, and our involvement in initiatives like Science Week and Culture Night.  We also continued to enhance our international reputation, as can be seen from the high-level global partnerships in which we were involved and the high rates of interest from international researchers in conducting research at DIAS.

“In 2016, DIAS continued to contribute to the national and global knowledge pool, and to influence future developments in the study of our cosmos, our planet and our identity.  We have been building on this work this year, and will be launching a new strategic plan in the coming months.”

DIAS’s 2016 annual report is available to download at: https://dias.ie/reports

ENDS

 

For further information, contact: Martina Quinn, Alice PR & Events, Tel: 01-5582151 / 087-6522033, Email: media@alicepr.com

17th November – School of Celtic Studies Statutory Public Lecture 2017

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.