1. One-parameter Fractals in 3D non-unitary CFTs
Hirohiko SHIMADA (OIST)
cf. arXiv: ,
H. Shimada and S. Hikami

2. Random walks generated from in base-6
Triangular
lattice 3D
Cubic
lattice
■6■　The key ideas behind the bootstrap are the SUM RULE and PARTIAL WAVE EXPANSION.
Perhaps the most familiar SUM RULE is the Euler’s SUM RULE about the topology, which says that the number of Vertices minus Edges plus Faces should equal to 2.
For the regular polygons, if we let the number of edges [occurring] at one vertex and one face to be p and q, THEN we have this relation.
The only FIVE solutions of this Sum rule listed here correspond to the celebrated Plato’s classification.
In d≧2, always
The Brownian RW belongs to a “trivial”
Gaussian universality class (no self-interactions).

3. Self-avoiding walks on the hyper-cubic lattice in D=4
Simulations by N. Clisby
Phys. Rev. Lett. 2010
Self-avoiding walks on the hyper-cubic lattice in D=4
sufficiently long walks
■6■　The key ideas behind the bootstrap are the SUM RULE and PARTIAL WAVE EXPANSION.
Perhaps the most familiar SUM RULE is the Euler’s SUM RULE about the topology, which says that the number of Vertices minus Edges plus Faces should equal to 2.
For the regular polygons, if we let the number of edges [occurring] at one vertex and one face to be p and q, THEN we have this relation.
The only FIVE solutions of this Sum rule listed here correspond to the celebrated Plato’s classification.

4. 3D Self-avoiding walks (polymers) in D=2, 3 2D
The universality class of
polymers in a solution
The conformal symmetry fixes the fractal dimension:
double points in any projections !
arXiv:
Simulations by N. Clisby 2D
■6■　The key ideas behind the bootstrap are the SUM RULE and PARTIAL WAVE EXPANSION.
Perhaps the most familiar SUM RULE is the Euler’s SUM RULE about the topology, which says that the number of Vertices minus Edges plus Faces should equal to 2.
For the regular polygons, if we let the number of edges [occurring] at one vertex and one face to be p and q, THEN we have this relation.
The only FIVE solutions of this Sum rule listed here correspond to the celebrated Plato’s classification.
A special case of SLE with
(Theorem: Beffara 2008)

5. High-T expansion for the O(N) model N=1: Ising N=0: SAW
■6■　The key ideas behind the bootstrap are the SUM RULE and PARTIAL WAVE EXPANSION.
Perhaps the most familiar SUM RULE is the Euler’s SUM RULE about the topology, which says that the number of Vertices minus Edges plus Faces should equal to 2.
For the regular polygons, if we let the number of edges [occurring] at one vertex and one face to be p and q, THEN we have this relation.
The only FIVE solutions of this Sum rule listed here correspond to the celebrated Plato’s classification.
High-T expansion (Lattice) Continuum limit (QFT)
Paths / Loops open/closed particle-trajectories
“Length” along the loop particle-”proper-time”
Loop-weight = N of the O(N) model
(small-β)

6. 3D Ising high-T expansion =Grand canonical ensemble of
loop gases (loop weight N=1)
b=1
b’=3
Winter-Janke-Schakel (2010)
Metropolis plaquette update:
Acceptance prob.= min(1, [tanh b]^(b’-b))
■6■　The key ideas behind the bootstrap are the SUM RULE and PARTIAL WAVE EXPANSION.
Perhaps the most familiar SUM RULE is the Euler’s SUM RULE about the topology, which says that the number of Vertices minus Edges plus Faces should equal to 2.
For the regular polygons, if we let the number of edges [occurring] at one vertex and one face to be p and q, THEN we have this relation.
The only FIVE solutions of this Sum rule listed here correspond to the celebrated Plato’s classification.
Conformal bootstrap for the O(N) rank-2 sym. tensor (2-leg water-melon operator)
N=1
N=0
RG N=-2
Ising
SAW
LERW

7. Crossing symmetry sum rule: S, T, A sectors in the O(N) model
Traceless-symmetric tensor
Singlet
Anti-symmetric tensor
Conformal Bootstrap study for the O(N) model
3D, N=2, 3, … Kos, Poland, Simmons-Duffin, Vichi 2013, 2015
5D, (3D) Nakayama-Ohtsuki 2014

8. 3D O(N) Watermelon operator
Bootstrap using <ffff> and positivity works for 0<N<2.
The convergence of the bound slows down as N-->0.
The “kink “ apparently disappears in N<0.2.
NB. there is a pole at “N=0“ of the stress-tensor OPE coefficient^2.
A new singular minimum emerges in the current central charge:
A. Petkou, Annals Phys. 249 (1996) 180.
J. Kiskis et al., J. Stat. Phys. 73 (1993) 765.
In practice, it enable us to determine the fractal dim. of SAW: RG MC
arXiv:
In conclusion, there are variety of important 3D critical statistical models (solutions to the crossing), which are non-unitary and form one-parameter families (O(N), q-Potts, disordered models, anyons, etc.).
Are there natural parameterization by the fractal dimension as in 2D CFT ??
■5■ Here comes the recent important progress in the conformal bootstrap in ---3D---!, namely,
Slava Rychkov and others require some hundreds of inequalities coming from the Unitarity condition
and found a phenomenon that the low-lying dimensions in the Ising model lie at the kink of the boundary.
By the method of the linear optimization, they achieved 6-digits-precision on the exponents!
However, this key observation still lacks fundamental understanding………
For instance, we do not understand, why this part is a Straight line, and what this curve may characterize and so on!
Though, along this line, the dimensions related to multicritical Z2-invariant models seem to appear.
*****I will discuss this blue curve later using the Modular Invariance.
(Theorem: Beffara 2008)