Erica Klarreich brought the readers of the Quanta Magazine some wonderful summary of iii mathematical preprints on the arXiv:
Sphere Packing Solved inwards Higher DimensionsThe interrogation is sort of obvious fifty-fifty to picayune kids. You convey as large \(n\)-dimensional spheres (the sphere is a laid of points amongst a fixed Pythagorean distance \(R\) from the center). How practise you lot conform these non-overlapping objects inwards a large box then that the publish of balls inwards the box is maximized?
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You may fill upwards the 1-dimensional space, a line, amongst 1-spheres, i.e. occupation intervals, completely. That was easy! ;-)
For 2-spheres, i.e. circles (or disks), the hexagonal packing is almost champaign the densest packing (the filled fraction is \(\pi/\sqrt{12}\approx 90.69\%\)) fifty-fifty though this outcome was exclusively rigorously proved past times Fejes Tóth inwards 1940. But what close other dimensions?
In iii dimensions, you lot may inquire a clerk inwards a fruit-and-vegetable store to conform oranges. But the "ground layer" is amend non hold upwards hexagonal.
modular invariance, i.e. the invariance of the globe canvas theory nether large coordinate transformations (diffeomorphisms of the toroidal globe canvas that cannot hold upwards continuously connected to the identity transformation). The "even" status has to hold upwards truthful because of the \(\tau\to\tau+1\) component division of the modular grouping because it requires \(L_0-\tilde L_0\) to hold upwards integer, as well as non half-integer, then that the "odd" states are forbidden. The self-duality arises from the invariance nether the other generator of the \(SL(2,\ZZ)\) modular group, \(\tau\to-1/\tau\). One tin demo it past times some detailed give-and-take of modular functions. More physically, nosotros may enjoin that the lattice of winding numbers is dual to the lattice of momenta (because the vertex operators amongst momenta as well as winding convey to hold upwards mutually local on the globe sheet) as well as the "chirality" (absence of the right-moving part) way that the momentum of a string has to hold upwards equal to its winding, then the allowed lattices for these objects must hold upwards the same – as well as that's equivalent to the self-duality of the lattice where they belong.In the dimension \(D=8\), the unique self-dual lattice is the root lattice of the \(E_8\) group, the largest, namely 248-dimensional, exceptional uncomplicated compact Lie group. In price of the obvious orthonormal the world vectors \(\vec e_i\) for \(i=1,2,\dots, 8\), the \(E_8\) lattice contains all integer combinations of the vectors \(\vec e_i\pm \vec e_j\) for \(i\neq j\) as well as of \(\sum_{i=1}^8 \vec e_i/2\). If you lot wish to convey a to a greater extent than directly thought what the vectors await like, well, their coordinates are either "all integers" or "all half-integers", as well as the amount of the components has to hold upwards even.
The minimum \(\vec v \cdot \vec v\) of the nonzero elements of the lattice – which decides close the (squared) radius of the balls nosotros may afford – is nonetheless ii (eight times i quarter equals two) fifty-fifty though nosotros were able to double the publish of elements of the lattice past times adding this "whole class" of lattice sites amongst half-integer coordinates. In some sense, nosotros managed to increase the density of lattice sites past times a factor of ii – as well as that's sure skilful tidings for dense packing. With this virtue, the \(E_8\) lattice manages to attain the "packed volume" of \(\pi^4/384\) or close 25.37%.
If you lot utilisation Cartan's toroidal constructions of Lie groups, you lot volition realize that at that spot exists a 248-dimensional Lie grouping whose root lattice (which happens to hold upwards the same as the weight lattice, inwards this particular case) is the \(E_8\) lattice nosotros described inwards the previous paragraph. There are \(112+128=240\) roots i.e. elements of the lattice amongst the squared length equal to two, namely \(112\) roots of \(SO(16)\) of the shape \(\pm \vec e_i\pm \vec e_j\) amongst the independent plus-minus signs as well as \(i\neq j\), as well as the additional \(128\) roots \(\sum_{i=1}^8 \pm \vec e_i/2\) amongst an fifty-fifty publish of pluses. Those transform as a chiral spinor nether \(SO(16)\); the "even" status is what makes the spinor chiral. In combination amongst the viii generators associated amongst the Cartan torus itself, nosotros may build the \(248\)-dimensional Lie algebra of \(E_8\).
The \(E_8\) lattice also gives us the most natural trend to define the integral octonions.
In \(D=16\), at that spot are ii fifty-fifty self-dual lattices. One of them is only the weight lattice of \(E_8\times E_8\), i.e. ii copies of the \(E_8\) lattice described above. The other is the weight lattice of a \(Spin(32)/ \ZZ_2\) lattice. Both corresponding Lie groups convey rank sixteen as well as dimension 496 because \(248+248=32\times 31/2 \cdot 1\), facts that are needed (but non sufficient) for the anomaly cancellation inwards the effective theories of the ten-dimensional superstring vacua (the anomaly cancellation was nontrivial yet truthful as well as sparked the First Superstring Revolution inwards 1984).
As I convey explained many times, the heterotic string of string theory is a peachy theory – as well as arguably nonetheless the most promising description of Nature that scientific discipline knows – that uses these ii fifty-fifty self-dual lattices. "Heterosis" way "hybrid vigor", the might of offspring to alternative virtues from both (very different) parents.
The parents of the heterotic string are \(D=10\) superstring theory as well as \(D=26\) bosonic string theory. It's possible for the left-moving as well as right-moving bosons on the globe canvas to "live" inwards unlike spacetimes. But the \(26-10=16\) directions of the spacetime that exclusively be inwards the bosonic string theory are chiral, as well as they must hold upwards compactified on an fifty-fifty self-dual torus, as argued above. There are precisely ii ways to practise that, corresponding to the ii fifty-fifty self-dual lattices inwards \(D=16\). This is how iv Princeton physicists (the Princeton string quartet, Gross, Harvey, Martinec, Rohm) discovered the ii unlike heterotic string theories inwards \(D=10\) inwards 1985. Their depression unloosen energy limits lucifer to the \(D=10\) type I supergravity coupled to super Yang-Mills amongst the approximate grouping either \(E_8\times E_8\) or \(Spin(32)/ \ZZ_2\). Influenza A virus subtype H5N1 fermionized description was straightaway found, too. Sixteen bosons may hold upwards replaced past times thirty-two fermions. The consistent GSO projections allow plus/minus signs for their groups – either for the whole grouping of 32 fermions which gives \(Spin(32)/\ZZ_2\), or for ii split upwards groups of sixteen fermions which gives \(E_8\times E_8\) thank you lot to the extra massless states inwards the mixed periodic-antiperiodic sectors etc.
In the mid 1990s, it was realized that the rigid coupling bound of the \(Spin(32)/ \ZZ_2\) theory is the type I string theory amongst the opened upwards as well as unopen strings (the \(SO(32)\) charges are carried past times quark/antiquark "Chan-Paton factors" or 32 unlike half-colors at the destination points of the opened upwards strings) – people could convey noticed much before that the same grouping \(SO(32)\) likely wasn't an accident – as well as the strong-coupling bound of the \(E_8\times E_8\) heterotic string theory is the Hořava-Witten M-theory on a "layer of 11D spacetime" amongst ii 10D boundaries. Each of them supports i of the ii \(E_8\) approximate supermultiplets, then the ii \(E_8\) factors larn "geometrically separated" inwards the management of the new, M-theoretical, eleventh dimension of the spacetime. Similar dualities convey connected all "previously separated versions of string theory" into i connected network, string theory (singular) inwards the novel sense, as well as that's why the Second Superstring Revolution of the mid 1990s is sometimes referred to as the Duality Revolution.
But already inwards the 1980s, people realized that in i lawsuit nosotros compactify i of the ix spatial dimensions on a circle, the ii heterotic string theories are equivalent to i some other past times T-duality. Mathematically, this results from the fact that fifty-fifty self-dual lattices also be inwards Minkowskian mixed-signature spacetimes of dimension \((m,n)\) amongst \(m-n=8k\). And if \(mn\neq 0\), the fifty-fifty self-dual lattice is really unique upwards to the "discrete Lorentz transformations". That's why all the solutions convey to hold upwards T-dual to each other.
Nevertheless, inwards purely Euclidean spaces of dimensions \(D=8k\), the publish of inequivalent fifty-fifty self-dual lattices may hold upwards higher. This is the instance of the next multiple of eight, \(D=24\). As I happened to cite less than a calendar week ago, at that spot are 24 inequivalent fifty-fifty self-dual lattices inwards \(D=24\): i Leech lattice as well as 23 other Niemeier lattices (or practise the 23 neglect to hold upwards self-dual?).
The Leech lattice is really cooler. The shortest nonzero vectors of the \(E_8\) lattice had \(\vec v \cdot \vec v = 2\) but the Leech lattice succeeds inwards eliminating all these vectors. The shortest nonzero vectors of the Leech lattice convey \(\vec v\cdot \vec v =4\). This "enhancement" has many really unlike consequences.
One of them is that nosotros don't larn anything such as a "continuous Lie grouping similar to \(E_8\)" if nosotros utilisation this lattice inwards the same trend as the lattices inwards the heterotic string constructions discussed above. There are only no novel approximate bosons! This is the ground why the spacetime spectrum is sort of "minimized" if the Leech lattice is involved. We don't larn large Lie groups but nosotros really practise larn huge discrete groups.
If you lot repeat the heterotic-like construction but amongst the Leech lattice instead of the \(E_8\) as well as similar lattices, you lot volition obtain a theory whose symmetry grouping is the monster group, the largest sporadic (analogy of "exceptional") grouping inwards the classification of finite groups. It's no coincidence: string theory compactified on the Leech lattice (well, the corresponding torus) is what explains the "monstrous moonshine", namely the appearance of the numbers \(196,883\) as well as \(196,884\) at ii seemingly really unlike places of mathematics.
Also, Witten has figured out a decade agone that a precise two-dimensional conformal theory i.e. "world canvas description of string theory" is also the correct boundary CFT whose "holographic dual inwards the bulk", according to Maldacena's AdS/CFT correspondence, is zilch else than the "pure" gravity (with no other fields) inwards the 3-dimensional anti de Sitter infinite (for the minimum radius only). The publish of TRF weblog posts on this Witten's monstrous stuff is a combat large, I don't wish to await for the correct articles separately.
In the Leech lattice, the fact that the shortest nonzero vectors convey \(\vec v \cdot \vec v = 4\) has i to a greater extent than consequence: this publish iv is large which really way that all the elements of the lattice are surprisingly far from each other. That's what makes it possible to "inflate" the sphere much to a greater extent than (the radius may grow past times a factor of the foursquare root of two) than for other "similar" lattices you lot could invent. That's really the best ground to guess that inwards \(D=24\), the densest packing is defined past times the Leech lattice.
As the Quanta Magazine reviews inwards some detail, people's organized faith that the \(E_8\) lattice as well as the Leech lattice define the densest packing inwards \(D=8\) as well as \(D=24\) has gradually grown due to some novel results as well as conjectures. The latest steps were done past times Maryna Viazovska – past times herself as well as amongst collaborators – weeks ago. These steps were really a proof of a conjecture stated past times Cohn as well as Elkies inwards 2003. Those guys (I promise that Cohn is a guy, otherwise his shout out should hold upwards Sarah) introduced some "linear programming" bounds as well as Viazovska has analyzed some modular functions to demo that no i tin vanquish the "conjectured densest packing" lattice. My agreement is that she has really added some "nearly stringy" materials to the "mostly messy computer-science-related" methods past times Cohn as well as Elkies.
The papers:
The sphere packing occupation inwards dimension 8, ViazovskaAll these papers were posted inwards recent iii weeks. The \(D=24\) variation past times many authors uses the same methods as the master Viazovska's method for \(D=8\). I believe that the "Leech is densest inwards \(D=24\)" outcome has already been previously proven past times other tools, at to the lowest degree it's mentioned as a fact inwards several older TRF weblog posts.
The sphere packing occupation inwards dimension 24, Cohn, Kumar, Miller, Radchenko, Viazovska
See also:
Some properties of optimal functions for sphere packing inwards dimensions 8 as well as 24, Cohn, Miller
I won't travail to reproduce the steps because I don't really empathise the methodology of the proof at this 2d (although I am certain that at a vague level, the strategy has similarities to my minimization approach to the Riemann Hypothesis). But it's clearly beautiful, amongst deep links to the structures inwards string theory.
The densest packing occupation is something that most kids inwards the kindergarten may empathise rather well. It only happens that the "by far best" solutions be inwards \(D=8\) as well as \(D=24\). The solutions are defined past times lattices. The best \(D=8\) lattice is the root lattice that heterotic string theory uses to generate \(E_8\), the largest exceptional compact Lie group, as the approximate grouping inwards the spacetime. The best \(D=24\) lattice is the Leech lattice on which compactified string theory produces a theory amongst the monster grouping symmetry, the largest sporadic grouping inwards the classification of the finite groups. Moreover, this two-dimensional string theory seems to hold upwards holographically dual to the "simplest" theory of quantum gravity inwards iii spacetime dimensions, the pure gravity inwards \(AdS_3\).
All these mathematical structures are beautiful, fundamental, as well as connected via string theory. For example, Viazovska also discusses the "Jacobi obscure identity" which may hold upwards interpreted as a necessary status for the equivalence of ii descriptions of supersymmetry inwards superstring theory. The existent globe isn't given past times string theory compactified on these "simplest as well as most beautiful lattices". But the compactification that produces the Universe to a greater extent than or less us is champaign a cousin of these mathematical structures.
I guess that all the deep string theorists, piece they may hold upwards partially motivated past times the wish to calculate the results of all experiments amongst arbitrary precision as well as no input, are also partly motivated past times the wish to "understand all of mathematics that is worth understanding". All the "truly pretty" mathematics, all the mathematical structures that piece of work as well as specially those that "by far" overshine all of their competitors, are sort of naturally connected via string theory. String theory is what shows why they be at all, why they're fundamental, why they're prettier than all competitors, as well as why they're connected amongst each other.
The people who reject to encounter fifty-fifty glimpses of these insights as well as the remarkable depth of all these structures as well as ideas are on par amongst (the 2d to applaud would hold upwards now!) troglodytes.
And that's the memo.
Her papers are curt plenty as well as effective but given the simplicity of the master occupation as well as the connector of the occupation to deep mathematics as well as physics, I for i convey no doubts that she deserves all the mathematics prizes she is eligible for. She was born inwards 1984 then hold upwards certain she's ready for a Fields Medal.
See also: John Baez, Gil Kalai, Maths.org
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