So, a while ago, I found a couple of papers about a statement of the form 'the Moonshine VOA is made of 48 copies of the Ising model'
So, a while ago, I found a couple of papers about a statement of the form 'the Moonshine VOA is made of 48 copies of the Ising model'
Superficially, we can say that it seems to make sense at the level of the central charges, since the Moonshine VOA has c=24
Superficially, we can say that it seems to make sense at the level of the central charges, since the Moonshine VOA has c=24
And if we look in detail at the constructions, there are really products of fields of the copies, and an important use of the spin-flip involution, as well as of the Kramers-Wannier duality, and it really seems to check out
And if we look in detail at the constructions, there are really products of fields of the copies, and an important use of the spin-flip involution, as well as of the Kramers-Wannier duality, and it really seems to check out
Nevertheless, there is something that is extremely important in this story, which is a Vertex Operator Algebra (VOA) one: the fields need to be holomorphic (for one needs to take their modes to act upon other fields), and there is a priori no known holomorphic field of scaling dimensions (1/16,0) for the Ising model
Nevertheless, there is something that is extremely important in this story, which is a Vertex Operator Algebra (VOA) one: the fields need to be holomorphic (for one needs to take their modes to act upon other fields), and there is a priori no known holomorphic field of scaling dimensions (1/16,0) for the Ising model
This whole story is a bit mysterious, but my conclusion became that it is a bit premature to assume that such a field doesn't exist
This whole story is a bit mysterious, but my conclusion became that it is a bit premature to assume that such a field doesn't exist
If we look at what comes out of the parafermionic observables for O(n) models, and if we look at so-called (Kramers-Wannier) duality defect lines, we definitely find hints of the existence of such a field
If we look at what comes out of the parafermionic observables for O(n) models, and if we look at so-called (Kramers-Wannier) duality defect lines, we definitely find hints of the existence of such a field
From Fermions to Chiral Spin Fields
From Fermions to Chiral Spin Fields
The way the Ising free fermion is usually thought of as product of spin and disorders... but the way the connection with complex analysis is made is more evident via the parafermionic observable construction, which generalizes the constructions to other lattice models corresponding (at least conjecturally) to a continuum of central charges and SLE kappa values
The way the Ising free fermion is usually thought of as product of spin and disorders... but the way the connection with complex analysis is made is more evident via the parafermionic observable construction, which generalizes the constructions to other lattice models corresponding (at least conjecturally) to a continuum of central charges and SLE kappa values
And in some sense there are three classes of parafermionic observables introduced by Stas
And in some sense there are three classes of parafermionic observables introduced by Stas
The critical FK(q) observables for $q\in(0,4)$
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The critical FK(q) observables for q∈(0,4)
The dilute O(n) observables for $n\in(0,2)$
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The dilute O(n) observables for n∈(0,2)
The dense O(n) observables for $n\in(0,2)$
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The dense O(n) observables for n∈(0,2)
In terms of universality classes, the dense O(n) and FK(q) with $q=n^2$ should yield the same CLE...
In terms of universality classes, the dense O(n) and FK(q) with q=n2 should yield the same CLE...
And (at least on the honeycomb lattice)
And (at least on the honeycomb lattice)
On the square lattice and isoradial graphs
On the square lattice and isoradial graphs
This is not very surprising if we think that we will just somehow fill the space with loops and that configurations will have probability proportional to $n^{\#loops}$)
This is not very surprising if we think that we will just somehow fill the space with loops and that configurations will have probability proportional to n#loops)
What do we learn if we look at dense $O(\sqrt 2)$?
What do we learn if we look at dense O(2)?
There is a parafermionic observable of spin $\frac 1 {16}$, which is a (holomorphic)complexification of the boundary change operator that changes monochromatic (all + or all -) boundary conditions into free boundary conditions (and vice versa)
There is a parafermionic observable of spin 161, which is a (holomorphic)complexification of the boundary change operator that changes monochromatic (all + or all -) boundary conditions into free boundary conditions (and vice versa)
What is the intuitive claim behind all of that?
What is the intuitive claim behind all of that?
There should be some kind of 'duality defect' lines similar to the (usual) 'spin-flip defect' lines makes spins across them "pretend" that their neighbors are the opposite of what they are, but instead across them the spins should connect to their dual (in the Kramers-Wannier sense)
There should be some kind of 'duality defect' lines similar to the (usual) 'spin-flip defect' lines makes spins across them "pretend" that their neighbors are the opposite of what they are, but instead across them the spins should connect to their dual (in the Kramers-Wannier sense)
There is a nice paper of Aasen-Mong-Fendley about this, and the lattice construction of duality defect lines is very interesting, but ultimately I think there are flaws about it that cannot be fixed easily (the endpoints of the duality defect lines cannot be handled easily)
There is a nice paper of Aasen-Mong-Fendley about this, and the lattice construction of duality defect lines is very interesting, but ultimately I think there are flaws about it that cannot be fixed easily (the endpoints of the duality defect lines cannot be handled easily)
And if we think that 'duality disorder operators' would be the ends of the duality defect lines (the relevant statistics about the lattice models should basically only depend on the end points of defect lines, plus some minor topological data, like their windings around correlator insertions)
And if we think that 'duality disorder operators' would be the ends of the duality defect lines (the relevant statistics about the lattice models should basically only depend on the end points of defect lines, plus some minor topological data, like their windings around correlator insertions)
And now, if we take something like a product of spin and duality disorder (at close locations), the intuition would be that we get something fairly close to a chiral spin operator
And now, if we take something like a product of spin and duality disorder (at close locations), the intuition would be that we get something fairly close to a chiral spin operator
Boundary Value Problems to the Rescue
Boundary Value Problems to the Rescue
In many of these situations, things can look very unclear, perhaps because we start from a lattice model observable which counts loops and we have an infinite number of loops in the continuum (and that's the only place where we have true holomorphicity and clean CFT structures a priori)
In many of these situations, things can look very unclear, perhaps because we start from a lattice model observable which counts loops and we have an infinite number of loops in the continuum (and that's the only place where we have true holomorphicity and clean CFT structures a priori)
December 30th, 2024
December 30th, 2024
However, if we think of 'macroscopic holes' in our domain instead of 'microscopic loops', we get to boundary value problems with unique solutions, and we can probably prove things about these, and we can reason about the number of loops surrounding those without a problem and get things about them
However, if we think of 'macroscopic holes' in our domain instead of 'microscopic loops', we get to boundary value problems with unique solutions, and we can probably prove things about these, and we can reason about the number of loops surrounding those without a problem and get things about them
Loop Counting in the Macroscopic Hole Model
Loop Counting in the Macroscopic Hole Model
The nice piece of reasoning that works well on the discrete level is to say 'ok, we are discrete holomorphic, because if we xor a configuration ending at $z$ with two half edges leading to $w$ (think of the honeycomb lattice to simplify), then in the nontrivial case where we change the number of loops in the process, the modified winding will take care of this
The nice piece of reasoning that works well on the discrete level is to say 'ok, we are discrete holomorphic, because if we xor a configuration ending at z with two half edges leading to w (think of the honeycomb lattice to simplify), then in the nontrivial case where we change the number of loops in the process, the modified winding will take care of this
Now, if we start adding a complex phase for the loops surrounding a macroscopic hole, then to account for the change of phase cause by the destruction of a loop, we should in principle go to a multiple cover for the arriving path
Now, if we start adding a complex phase for the loops surrounding a macroscopic hole, then to account for the change of phase cause by the destruction of a loop, we should in principle go to a multiple cover for the arriving path