Witten's talk at Strings 2011 analyses a four dimensional knot QFT using S duality and supersymmetric localization, and then T duality, which extends the theory to a theory with five-branes on the addition of an $S^1$ direction. This $S^1$ results in a partition function
$\textrm{Tr} (-1)^{F} q^{k}$
in terms of instanton number $k$, using supersymmetry in the form of a so called quantum group, starting with the basic Hopf algebra generated by $(-1)^{F}$. This arithmetic index for Khovanov homology introduces the familiar braiding obtained from the interchange of two fermions. Witten concludes with a consideration of the reverse process, which reduces the full six dimensional UV completion right down to the two dimensional SCFT associated to the original Chern-Simons theory.
Now we should recall that S, T and U dualities are unified by qudit entanglement, so that the doubling of the three dimensional theory is seen as a fermionization of the three bosonic spatial directions. This unification must occur through a combination of the braid crossing forced by T duality with a matrix description of the electromagnetic S duality, which is a motivic process.
14 years ago
Paying attention, Mitchell?
ReplyDeleteI don't understand how your last paragraph works, but let me talk about the path upwards from 3 to 6 dimensions in terms of branes, and maybe we can find a common language?
ReplyDeleteYou start with a knotted Wilson loop in d=3 Chern-Simons. Then you think of this as actually a Wilson loop in d=4 N=4 Yang-Mills, on the d=3 boundary of some d=4 space (like "half of R^4"). Then you think of all this as the worldvolume theory of some D3-branes terminating on some 5-branes. So we still have knots existing in a d=3 space, but the variables are really from a d=4 theory (the 3+1 worldvolume of the D3-branes), and they are somehow immersed in a d=6 space (worldvolume of the 5-branes on which the D3-branes terminate).
The S-duality turns the 5-brane from NS5 to D5. The T-duality turns the d=4 variables into d=5 variables, and now the knots become (knot x circle) products. Finally, the half-line direction leading into the d=4 / d=5 space (i.e. away from the knot / knot x circle) is expanded into a cigar geometry (with the knot x circle living at the tip of the cigar), topologically a disk, and now we have the d=6 variables.
That's what I get if I stay focused on the initial "three-dimensionality" through the transformations leading up to 6 dimensions.
Section 5.1.1 of "Fivebranes and knots" describes the d=6 theory in purely field-theoretic terms (without the 11-dimensional embedding). Equation 5.3 looks important for the step up from 3 to 4 dimensions.
ReplyDeleteYes, that is pretty much the content of the talk, remembering to change Wilson lines into 't Hooft operators. Now in our motivic quantum geometry, all these classical spaces are emerging from quantum information, encoded directly into categorical braids/ribbons. Like in any circuit diagram, one oftens uses one strand per qudit, so that the three dualities work first with $3$ strand ribbons from (an extended) $B_3$. This $B_3$ is canonically represented in the noncommutative and nonassociative version of the $SL(2,C)$ matrix groups, which is roughly what I am eluding to in the final paragraph. Note that Witten's $R^2$ cigar (kind of) closes off the half space with a point, so that we could think of all three extra dimensions as $S^1$s, which is the usual situation in M theory entanglement.
ReplyDeleteAh, yes, equation 5.3 is two copies of $S^3$, so my extra 'point' could turn the $R^2 \times S^1$ into the complex Hopf fibration alternative to $S^2 \times S^1$. Those are the 'magnetic' degrees of freedom, and the left hand piece includes the original 'electric' spatial part. Then we have 2 qubit spaces hanging around, like for twistors.
ReplyDeleteOf course, Witten must know that everything is morally twistor geometry. I guess he just wants to study curvature without getting bogged down in more abstract homology.
ReplyDeleteLet me back up a little. Every level of Witten's construction consists of path integrals in field theories, basically. What we're looking for is an equivalent categorical, algebraic construction in which the enormous gauge redundancies of the field theories aren't present (i.e. Arkani-Hamed's "third theory").
ReplyDeleteIn going from 3 to 6 dimensions I see two things happening. First, the support for the (Wilson / 't Hooft) operator goes from 1-dimensional (knot) to 2-dimensional (knot x circle). This makes me think of the thickening of strands into ribbons, but with the opposite edges of the ribbon identified (i.e. that's the circle factor, transverse to the original knot). Second, there's a change in what gets integrated along / across this 1-or-2-dimensional object: it's originally a Chern-Simons connection, then it's a super-Yang-Mills connection-plus-scalars, then something from the d=5 field theory, then something from the d=6 field theory. So all of that has to be reduced to its algebraic essentials.
Yes, ribbons for the duality, correct. But when we move away from local gauge theories, we can't just use traditional path integrals, can we? That's why everything MUST be done in terms of categorical cohomology, where integrals are pairings between homology and cohomology. But because Witten knows how locality emerges, through the twistor geometry, he must realise that there are SOME path integrals that are closer to the motivic picture than others, which is what he is looking for. It is a step that will help convince ordinary stringers to do more category theory, when they finally get it.
ReplyDeleteContinuing with more elementary matters... Reading through "Fivebranes and knots", I finally found the outlines of how Khovanov homology is computed in six dimensions. You consider one of the d=6 (0,2) theories (there are infinite families of them) on the manifold M x D, where M is a d=4 manifold and D is the d=2 "cigar" manifold. The "knot x circle" surface operator then lives in the copy of M located at the tip of the cigar. Shrink the cigar to a half-line, and you are "taking an index" of the d=6 theory and moving to d=5 super-Yang-Mills on a half-space (M x half-line), with a (knot x circle) surface operator now living in the copy of M at the end of the half-line. M has itself all along been of the form (3-manifold x circle), so now you compactify d=5 SYM on that circle. There's a detail here that I still didn't get (to do with a change in the Yang-Mills gauge group, to the Langlands dual group), but there are two ways down from here to a knotted line operator, one leads to a knotted Wilson loop, one to a knotted 't Hooft operator, and they are electric-magnetic duals.
ReplyDeleteSo a step towards a categorical version of all this, would be to express the evaluation of the d=6 surface operator as an integration pairing of a differential form and a closed submanifold, as you say. The closed submanifold is simply the d=2 submanifold which provides the support for the surface operator, but what is the differential form? Section 5.1.3 of "Fivebranes and knots" says that some surface operators correspond, in the 11-dimensional realization, to M2-branes; section 5.1.4, that others correspond to M5-brane configurations (reviewed in references 88-91). Digging into these references, and their references, it's possible that we are talking about a self-dual 2-form, internal to the M5-brane, which is the restriction to its worldvolume of the "C-field" 3-form of 11-dimensional supergravity, the N=1 supersymmetry of which pairs up the graviton and this 3-form as bosonic partners of the fermionic "gravitino". But further research is required.
ReplyDeleteI have to point out one more thing, that AdS/CFT shows up here, in its d=7 form (AdS7/CFT6). The d=6 SCFT is equivalent to M-theory on a space of the form AdS7 x X4, X4 some 4-manifold. This 11-dimensional volume is where all the M-branes show up, as dual to various combinations of field operators from the SCFT. But as Brian Swingle points out, such AdS geometries can be interpreted in terms of scale-dependent RG flow of entanglement in the SCFT. So the possibility of reexpressing all of this in terms of quantum information really is quite good.
ReplyDeleteYes, amusingly I once wrote a PhD proposal (in Australia) about a 7d CSFT; back in 1996. Of course, nobody took me seriously. Also check out Moore's Strings11 talk on $(0,2)$ 6d theories. He was going to introduce higher categories, so they are in his slides. You are welcome to carry on worrying about the AdS jargon. Personally, I couldn't be bothered, because we don't need to think in these terms. We are talking about redefining the notion of manifold itself, using constructive number theory and categories.
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