BPS Algebras, Genus Zero, too the Heterotic MonsterWell, this newspaper elaborates upon their previous PPV1 paper which is just 1 twelvemonth former at i time but I am certain that yous volition forgive me a 1-year delay inwards the reporting.
It's just remarkable that something hence mathematically particular – past times its symmetries – may live on considered "another solution" to the same spacetime equations that likewise acknowledge our Universe as a solution.
I soundless consider the \(E_8\times E_8\) heterotic string to live on the most well-motivated candidate description of Nature including quantum gravity. Dualities likely acknowledge other descriptions as good – F-theory, M-theory, braneworlds – but the heterotic string may live on the "closest one" or the "most weakly coupled" with all the descriptions.
Heterotic string theory describes our Universe as a 10-dimensional spacetime occupied past times weakly coupled strings whose 2-dimensional basis canvas is a "hybrid" ("heterosis" is "hybrid vigor", the mightiness of offspring to surpass the average of both parents). The left-moving excitations on the basis canvas are taken from the \(D=26\) bosonic string theory patch the right-moving ones are those from the \(D=10\) fermionic string theory (with the \(\NNN=1\) basis canvas supersymmetry).
Because the critical dimensions don't agree, the remaining \(D_L-D_R=26-10=16\) left-moving dimensions receive got to live on compactified on the torus deduced from an fifty-fifty self-dual lattice (or fermionized to 32 fermions whose boundary atmospheric condition must live on modular invariant). There are 2 fifty-fifty self-dual lattices inwards sixteen dimensions too nosotros obtain theories with spacetime estimate groups \(SO(32)\) or \(E_8\times E_8\). Both of them receive got rank \(16\) too dimension \(496\).
Six of the remaining 9+1 dimensions of the \(E_8\times E_8\) heterotic string may live on compactified on something similar a Calabi-Yau manifold which breaks the \(\NNN=4\) spacetime supersymmetry of the heterotic string to a realistic \(\NNN=1\), allows i to interruption \(E_8\times E_8\) to a grand unified grouping such as \(E_6\) or \(SO(10)\), too nosotros may terminate upward with at to the lowest degree semi-realistic (all qualitative things upward to some degree certainly agree) effective champaign theory at depression energies.
Natalie too co-authors are de facto doing "something every bit good" as far as the validity too consistency of the heterotic compactification is concerned. The final result is to a greater extent than interesting mathematically too less relevant physically, however. They're just compactifying the \(26\) left-moving too \(10\) right-moving dimensions of the heterotic string differently than inwards the most realistic vacua. Well, for the bosonic (left-moving) side too the fermionic/superstring (right-moving) business office they select the next basis canvas conformal champaign theories:
- Monstrous moonshine module of Frenkel, Lepowsky, too Meurman (or its orbifolds)
- the Conway module
That CFT was known to explicate string theory on the self-dual circle is capable of enhancing the \(U(1)\times U(1)\) estimate "Kaluza-Klein addition B-field \(p\)-form estimate invariance" symmetry to an \(SU(2)\times SU(2)\).
If yous alone await at the geometric symmetries of the Leech lattice, yous actually search for the grouping of automorphisms of the Leech lattice. This grouping is known as the Conway grouping \({\rm Co}_0\) too its issue of elements is much smaller than that of the monster group, just \(8\times 10^{18}\) or so. It's the largest i with the so-called Conway groups. However, it's other 3 groups \({\rm Co}_1,{\rm Co}_2,{\rm Co}_3\) that belong with the 26 or 27 "sporadic uncomplicated groups" inwards the classification of the finite groups.
\({\rm Co}_1\) is a uncomplicated quotient of \({\rm Co}_0\) past times its \(\ZZ_2\) middle (the 24-dimensional parity) – hence the sporadic \({\rm Co}_1\) grouping merely has 1/2 of the issue of elements of \({\rm Co}_0\). The groups \({\rm Co}_2,{\rm Co}_3\) are subgroups of \({\rm Co}_0\) that likewise exit a vector of type 2 or type 3 unchanged, respectively.
Aside from the monster grouping too the 3 sporadic Conway groups, the remaining 22 full-blown sporadic groups are the infant monster, 5 Mathieu groups, iv Janko groups, 3 Fischer groups, Higman-Sims, McLaughlin, Held, Rudvalis, Suzuki, O'Nan, Harada-Norton, Lyons, too Thompson group, patch Tits grouping is sometimes counted as the 27th sporadic group. Mathematics is weird – these groups await almost as arbitrary as the names too nationalities of their human namesakes but they are objective facts nigh mathematics that every mathematically literate E.T. culture agrees with.
OK, the funny construction behind this newspaper combines the \(c_L=24\) bosons on the Leech lattice, along with a \(c_R=12\) conformal champaign theory for the supersymmetric side whose symmetry is the Conway group. These 2 ingredients may live on combined to gain a consistent two-dimensional, modular-invariant, conformal champaign theory coupled to gravity – likewise known as a perturbative string theory.
(The supersymmetric, right-moving, Conway side has a smaller grouping too plays a correspondingly smaller purpose inwards the hybrid construction, the purpose of a "passive spectator". The usage of a subgroup for the fermionic side seems reminiscent of the fact that the 32 fermionic supercharges inwards string theory alone transform nether a compact subgroup \(Spin(16)\) of \(E_{8(8)}\) or similar groups: the "bulk" of the particular generators of the U-duality grouping isn't manifest fifty-fifty though the states constructed with the assistance of the fermions ultimately transform nether the whole particular U-duality group.)
Natalie et al. study their model inwards quite some detail. They verify some special belongings of the CFT that is seen on the sphere – too that involves some elements inwards \(SL(2,\RR)\) exterior the modular grouping \(SL(2,\ZZ)\). They likewise await at all the BPS states inwards the second-quantized string theory – those would live on actual BPS states inwards the spacetime that is a sibling of the spacetime just about us. And they observe out that these BPS states shape a representation of a form of some truthful novel "staggeringly stinky animal", Borcherds's too Carnahan's "generalized Kač-Moody algebras \(\mathfrak{m}_g\)". (Carnahan added some "twisted" cousins of the master copy construction past times Borcherds.)
I had to usage a novel phrase, "staggeringly stinky animal", because all simpler phrases such as monsters too beasts had already been reserved past times other of import mathematical structures inwards this enterprise. :-) It is non difficult to run into that these generalized Kač-Moody algebras are a "sweet bastard hybrid" of the ordinary Kač-Moody algebras based on continuous groups too the discrete monster group.
Much of the newspaper is written inwards a physicist-friendly linguistic communication but at the end, they switch to the linguistic communication of axiomatic vertex operator algebras, a jargon of mathematicians designed hence that yous wouldn't fifty-fifty necessarily recognize that they're studying a perturbative string theory.
A twelve-minute speak – given on the 24-hour interval when LIGO announced the waves – past times too then graduate educatee (!, she already has 10+1 papers; soon-to-be a Burke swain at Caltech) Natalie Paquette nigh the "unreasonable effectiveness of physics inwards mathematics", a thought-provoking reversal of a well-known Wigner's title. After a brief introduction to Wigner, string theory, too compactification, she gives her edition of the flush how string physicists receive got hilariously trumped mathematicians inwards the counting of the spheres inside the quintic hypersurface. (Most people inwards Palo Alto volition certainly endeavour difficult to cleverly permute my "hilariously trumped" too experiment with phrases such as "trumpishly hillarized", but I predict that at the end, spellcheckers volition confirm that they volition receive got lost this battle just similar Hillary has.)
It's form of metaphysically thrilling to realize that these almost maximally, sporadically, staggeringly symmetric constructions are siblings of the heterotic compactification that may live on responsible for the basis nosotros inhabit. Because of the human relationship of these siblings, I mean value that nosotros should appreciate that inwards some really early on epoch of cosmology, the Universe was deciding whether it wanted to alive inwards the Monster \(\times\) Conway heterotic hybrid described inwards this paper; or inwards i of the relatively ho-hum Calabi-Yau-like compactifications that generates the particle physics nosotros know.
The Monster \(\times\) Conway heterotic VOA hybrid is smart and beautiful, non to bring upward imaginary.
We may fifty-fifty advise that our Universe has spent some Planckian fourth dimension inwards the Monster \(\times\) Conway hybrid earlier the monstrous too generalized Kač-Moody symmetries were broken, to yield the basis that nosotros know, love, too are sometimes annoyed by. In quantum champaign theory, nosotros may alone imagine our basis as a broken stage of grand unified theories with groups similar \(E_6\) which are large but non "really" qualitatively dissimilar from the Standard Model group. On the other hand, string theory allows us to conjecture that our basis is likewise a broken stage of a theory that had huge symmetries such as the monster grouping to laid about with.
The procedure of the breaking of this monster grouping symmetry is really dissimilar from the Higgs machinery inwards quantum champaign theory when it comes to the technical details but these mechanisms may live on fundamentally every bit relevant too important.