Jacques Distler together with iii co-authors (CDTZ, Austin together with Maryland) convey published an impressively technical newspaper "classifying" sure exotic beasts,
Tinkertoys for the \(E_8\) Theory.Their newspaper is accompanied past times a cool interactive website which you lot should investigate. In particular, if you're bespeak why they didn't include "all the stuff" from the website inwards their paper, endeavour this subpage on four punctured spheres. Pick a combination of the 4 parameters together with press "go". What you lot acquire are approximately diagrams of S-duality frames (1,025,438 of them) where you lot come across 2 circles amongst approximately objects together with left-right arrows connecting them. You tin mail away click at the objects together with acquire additional information on the "fixtures".
If you lot don't sympathise what the information agency together with how to role them, don't worry. vii billion people don't sympathise it together with my gauge is that thirty people inwards the earth may bargain amongst the information – to the extent that they would convey a jeopardy to discovery a typo if 1 were artificially introduced. ;-)
What did they do? They discussed approximately 4-dimensional quantum champaign theories amongst enhanced supersymmetry – the \(\NNN=2\) gauge theories inwards \(D=4\) – that may endure derived from a master copy theory, the six-dimensional \((2,0)\) theory – past times operations that require lots of grouping theory involving \(E_8\), the largest exceptional Lie grouping (that the laymen sure know every bit the Garrett Lisi grouping after a famous crackpot surfer), every bit good every bit geometric operations introducing together with curing singularities inside a two-sphere (injured past times 3 or 4 punctures).
Now, all this scientific discipline may endure formally considered "non-gravitational quantum champaign theory amongst lots of geometry together with grouping theory". So there's no full-blown string/M-theory inwards it. On the other hand, everything virtually their structure is completely stringy – inwards the feel that basically everyone who understands these quantum champaign theories together with operations amongst them is a string theorist.
They're elaborating upon the "class-S" structure past times Davide Gaiotto from Apr 2009 every bit good every bit Gaiotto, Moore, together with Neitzke from July 2009. Now, I know Jacques Distler inwards somebody – every bit good every bit from tons of interactions on the web. I've spent years side past times side to Davide Gaiotto together with Andy Neitzke (who co-authored papers amongst AN) at Harvard, together with years side past times side to Greg Moore at Rutgers. So all this materials has been pushed largely past times vivid minds whom I know real well. I believe that their expertise is rare together with I yell back it's nontrivial to detect grad students who would endure able together with willing to acquire the necessary materials fifty-fifty at the score of a "useful brute forcefulness assistant".
The six-dimensional \((2,0)\) theory is a non-gravitational local quantum champaign theory inwards 5+1 dimensions (well, a household unit of measurement of such theories). So formally, it's every bit non-gravitational or non-stringy every bit you lot tin mail away get. However, almost all its definitions together with relationships that are known are tightly incorporated into the wisdom of string/M-theory. Note that \((2,0)\) refers to the amount of supersymmetry. In 5+1 dimensions, the supercharges transform every bit 5+1-dimensional spinors. Those spinors of \(Spin(5,1)\) include 2 inequivalent chiral representations, just similar inwards \(Spin(4)\approx SU(2)\times SU(2)\) – which has the same types of spinors together with where nosotros "cancelled" the temporal together with spatial dimensions – then different the representative of \(Spin(3,1)\) or \(D=4\), it's non plenty to state "how much SUSY". You must state "how much left-handed SUSY" together with "how much right-handed SUSY". The \((2,0)\) issue agency that it's enhanced supersymmetry but both supersymmetries convey the same chirality inwards half dozen dimensions.
There are numerous relations of this champaign theory to string/M-theory. It may endure obtained every bit a low-energy restrict of M5-brane dynamics inside M-theory; or every bit approximately approximation of type IIB string theory on a singularity inside a 4-dimensional Euclidean space. It follows that the large \(N\) restrict of the \(SU(N)\) \((2,0)\) theories is Maldacena-dual to a \(AdS_7\times S^4\) compactification of M-theory. That's the principal link to the higher-dimensional physics.
But the \((2,0)\) has lots of cool relationships to lower-dimensional (compared to six) physics. It has lots of descendants. The \((2,0)\) theory may endure formally associated amongst an ADE gauge grouping – either \(SU(N)\) or \(SO(2N)\) or \(E_6,E_7,E_8\). The outset 2 are infinite families, the latter iii are isolated exceptions. Distler et al. studied the largest isolated exception.
The real being of these theories – which is "somewhat" hypothetical together with if you lot deny the being of string/M-theory, you lot could in all probability consistently deny the real being of the \((2,0)\) theories every bit good – is plenty to attempt out many marvelous facts virtually quantum champaign theories together with string theory vacua. In particular, the \(\NNN=4\) gauge theory inwards \(D=4\) has the \(SL(2,\ZZ)\) S-duality group. This duality is obvious if you lot realize that the \(D=4\) theory may endure obtained past times compactifying the \((2,0)\) theory on a 2-torus: the duality is zero else than the grouping of "large" coordinate transformations on the two-torus.
Distler et al., together with the predecessors, take in a to a greater extent than complicated compactification than the unproblematic two-torus. One replaces the two-torus past times a to a greater extent than full general Riemann surface, peculiarly a unproblematic two-sphere, 1 that may convey "punctures", together with at that topographic point may endure a "partial topological twist" associated amongst every "puncture". At whatever rate, at that topographic point are many ways how to acquire rid of the 2 extra dimensions together with dimensionally cut the six-dimensional theory to a four-dimensional one.
One ends upward amongst many \(\NNN=2\) theories inwards \(D=4\) – the "Seiberg-Witten" amount of supersymmetry inwards a four-dimensional spacetime. Depending on the types of topological twists etc., 1 gets many possible "descendants" of the six-dimensional theory. Some of them work, approximately of them don't work.
I can't explicate all the details – 1 argue is that I manifestly neglect to sympathise all the details – but I desire to brand a full general big-picture claim that is besides philosophical that it doesn't appear inwards such papers, together with possibly the authors aren't sufficiently playful to invent it.
As I announced inwards the subtitle, this structure of the four-dimensional theories follows the logic of Darwin's development – amongst mutual ancestors together with mutations – piece the park bottom-up structure of quantum champaign theories (construction where nosotros listing possible pieces, fields, together with combine them together with their interaction) is the creationist mental attitude to build quantum champaign theories.
If the species were identified amongst four-dimensional quantum champaign theories, they could arise inwards 2 basic ways: creation together with evolution. Creation agency that God asks His assistant for the edifice blocks together with decides, every bit if he were an engineer, how to combine the edifice blocks into a four-dimensional theory amongst approximately fields together with interactions. So God decides it would endure fun to convey something similar an elephant, then He asks his assistant for a torso together with just constructs an elephant.
Gaiotto, Neitzke, Distler, Moore, together with others are approaching the nascence of the four-dimensional theories differently, past times the evolutionary path. They pick their ancestor, inwards this representative the \(E_8\) \((2,0)\) theory inwards half dozen dimensions, together with allow it live. They pick several clones of it together with mutate them – which happens inwards the ordinary life – together with they acquire diverse ancestors amongst a modified DNA.
Influenza A virus subtype H5N1 large departure is that inwards the creationist attitudes to the spectrum of the quantum champaign theories, you lot basically impose your weather condition "how the theory should comport at long distances" every bit the initial constraints that dictate everything. On the other hand, the stringy evolutionary mental attitude is different. You allow the choices that may occur at the fundamental, deoxyribonucleic acid score – similar the partial topological twists assigned to the punctures – together with what the theory looks similar at depression energies is a surprise.
Note that this is precisely how it industrial plant inwards the real-world biological evolution. Some deoxyribonucleic acid may endure mutated – deoxyribonucleic acid is to a greater extent than key than the anatomy of an brute – together with what the anatomy together with physiology of the mutated brute arises is a surprise. Whether such a novel brute species is "viable" is entirely determined later, past times natural selection.
The stringy mental attitude based on the \((2,0)\) theory only acknowledges that analogously to the DNA, approximately local choices inwards the Riemann surface of the 2 compactified dimensions are to a greater extent than fundamental, together with the long-distance behaviour of the resulting four-dimensional quantum champaign theory is an emerged, derived effect of the to a greater extent than key choices.
In biology, fifty-fifty if nosotros didn't know virtually DNA, it would endure to a greater extent than correct to assume that "something similar that exists" together with is to a greater extent than key (the to a greater extent than primary cause) than the anatomy of the brute species. This DNA-centered mental attitude is analogous to the stringy 1 together with those who desire to avoid the stringy root of the theories are analogous to those who insist on creationism. Both of them imagine that it's scientifically OK to imagine that the Creator starts amongst the long-distance or anatomic final result He wants to acquire – piece the stringy or evolutionary folks know that the development of species or four-dimensional QFTs is a scientific procedure that has to obey approximately laws of Nature, too.