Originally shared by Cliff Harvey on Google+ If you have a somewhat advanced physics reading level and arent already familiar with these subjects, I highly recommend this essay on the importance of N=4 super #YangMills in D=4, the theory that has been appropriately called the Hydrogen atom of the 21st centruy by Zvi Bern. Its one of those basic objects that keeps reappearing in different contexts, and not to mention is unique for purely mathematical and conceptual reasons too. It is one of those gems all theorists must know, like the harmonic oscillator. If the text is beyond you, Im just going to summarize a couple of the most important connections to appreciate here. motls.blogspot/2012/03/35-years-of-nnn4-yang-mills-theory.html The blog entry starts by outlining some of the numerological considerations showing that #Supersymmetric gauge theories can only be defined in particular numbers of dimensions. This is interesting and important, but no need to get caught up in details here. (Related: here is a helpful pdf summarizing the noteworthy #SUSY theories in different numbers of dimensions. Some others, especially D=10 will be referenced in the text) bolvan.ph.utexas.edu/~vadim/Classes/01f/396T/table.pdf One of the most amazing and insightful aspects of the theory is explained starting in the section Finiteness, and it ties together an otherwise mysterious aspect of ordinary #Electrodynamics with a fuller string-theoretical picture where the mystery can be better understood. The mystery is electromagentic duality: Why is it that Maxwells equations exhibit a duality (a symmetry) under exchange of electric and magnetic fields, and the exchange of charges for magnetic monopoles? From the naive Maxwellian perspective the meaning and significance of this obvious symmetry is unclear, but its really a special case of S-duality, the strength duality which relates strongly and weakly-coupled string theories. Its one of the most indispensable tools in the stringy arsenal, because the strong coupling regime is where calculations become impossible. While S-duality is not fully transparent in string theory in general, it is fully manifest in the (N=1 D=10) super Yang-Mills theory, which is the low energy limit of type I and heterotic string theories, and from which our N=4 D=4 theory is obtained by dimensional reduction. In this setting #SDuality manifests as part of a larger SL(2,Z) symmetry group, that can adjust not only the strength of the interactions, but their chirality (handedness). (To get deeper into the rich web of insights regarding electromagnetic/S-duality Ive been appreciating this document Electromagnetic duality for children) maths.ed.ac.uk/~jmf/Teaching/EDC.html The other major topic is the holographic duality: The ability to describe #StringTheory living on AdS spaces completely in terms of conformal field theories living on the boundary of the space. The N=4 D=4 SYM theory is the classic example, fully equivalent to type IIB string theory on the product space AdS_5 x S^5. In this case, the boundary theory is only 4 dimensional even though the full space is 10-D, because the 5-sphere part of the space shrinks away at the boundary. Even before this most specific manifestation of the duality was understood, tHooft realized that in the large-N limit of a #GaugeTheory, the topology of open strings could be seen emerging from the diagrammatics. He understood that the limit must be some string theory. This connection is, I think, one of the most breathtaking that modern physics has to offer us. The gauge/string duality is probably the most important principle weve ever discovered for understanding quantum gravity. This came into sharp focus only 15 years ago. (To see this particular point elaborated in more detail, here is a favorite reference of mine for learning about the holographic correspondence) arxiv.org/abs/hep-th/9905111 These points could have just been chosen to convey the importance of the N=4 D=4 SYM theory which has just celebrated its 35th brithday. But I think theyre also worth sharing to convey a much bigger idea. That so many of the principles and insights of string theory and quantum field theory possess a unity to their logic. They are a continuous and smoothly consistent set of implications, such that to try and cut one from the other and deny that intimate connection looks impossible to me without ruining the whole structure. Though I still have to fill in many gaps in my knowledge and, like Feynman, I wont be satisfied until I know how to demonstrate all the key steps on my own. But everything I have managed to learn about this network of knowledge leads me to this conclusion, despite some serious attempts to understand those who disagree.
Posted on: Fri, 03 Oct 2014 23:46:44 +0000
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