The first part of the thesis deals with the introduction of an on-shell formalism for massless and massive particles. One of the objects resulting from this construction, the rotating polynomial, is then introduced into the distribution relation to derive a convex hull that bounds the EFT coefficients. We further investigate the intersection of the convex hull due to the positive expansion of residue and the half-moment curve.
In the second part, we turn our attention to topological defect lines in (1 + 1)d topological field theory with Haagerup melting ring. We first solve the F symbols of fusion categories in the Haagerup-Izumi family under the assumption of transparency. For is greater, the constraint does not appear for the range (x, y) displayed in the plot.
The sum of the Minkowski vectors comes from the s-channel (purple dots) and the u-channel (red dots). This means that the UV and IR EFTs populate different regions in the polytope, as shown on the right.
LIST OF TABLES
INTRODUCTION
In the fixed-momentum subspace, the action of the Pauli-Lubanski vector forms a Lie algebra whose corresponding Lie group is called the small group. The determinant of this matrix is the square of the four vector pµ, and so for massless pµ the matrix is singular. 1.4) These objects are Weyl spinors that transform individually under the U(1) part of the massless group.
One important property to get the most out of the shell formalism is uniformity. Combining the entries from unity and locality, we know the locations of the simple poles of the scattering amplitudes at the 2 → 2 tree level and the shape of the residue at that pole. In short, the correlation functions of the operators in TQFT do not depend on the geometry of the manifold on which it is defined.
More complex cobordisms can be obtained by composing these basic coborisms, and the resulting image under TQFT is just a composition of the respective linear mappings. For each element in the set HomC(X, Y), X is called the morphism domain and Y is called the codomain.
SCATTERING AMPLITUDES FOR ALL MASSES AND SPINS
Much of the nontrivial physics of scattering amplitudes stems from a simple question. 5These results can also be obtained by converting the usual polarization vector representation of the three-particle amplitude to the massive helical spinor basis. However, we will only be concerned with the parts of the four-particle amplitudes that are forced to exist by consistent factorization with respect to the three-particle amplitudes.
This means that the coupling constant appearing in the spin-s exchange channel must be identified with the graviton exchange constant. We will be interested in the most general form of the removed M3{α1···α2S},h2,h3, which is now a tensor in SL(2, C) Lorentz indices. In the next section, when we glue the three-point amplitudes to get a four-point one, it will be convenient to choose ζ as the spinor of the outer arms on the other side.
For simplicity, we will consider the case where all four particles have the same spin. This research will lead to the discovery of the Higgs and Super-Higgs mechanisms. We have thus verified that the 4pt massive amplitudes are an "infrared distortion" of the massive amplitudes, reproducing and unifying the different helicities in the HE limit.
First, in the massless limit, in addition to the universal couplings with gravity that we have.
THE EFT-HEDRON
It is again interesting to see the same objects appearing in the completely different, very general setting of the EFT hedron. Note that the pole residuals for the UV amplitude eq. (3.12), for example in the s-channel, are constants. That is, the coupling ak,0 is given by the residual of the Higgs pool in the sandu channel.
This arises naturally as the dispersive representation of the four-point amplitude at the forward boundary. Matching both sides of the equation above, we immediately see that the axes are positive. Thus we see that the convex hull of the Taylor vectors from the rotation polynomial gives a cyclic polytope.
The geometry of interest will thus be the intersection of the convex hull in eqn (3.145), with the cyclic plane XCyc. Unit vectors trivially meet this criterion due to the positivity of the Gegenbauer Taylor coefficients. Consider the ratio of the first term on the RHS of equation (3.197), with respect to the next three.
Note that the convex hull of such a deformed moment curve can be directly cut by the entire positivity of the transformed Hankel matrix:. Then we consider the positivity of Det[eq.(3.232)], where WI is the limit of the Minkowski sum. Note that most of the edges for the s- and u-channel cyclic polytope remain the surface for the Minkowski sum.
It is bounded by the Minkowski sum of the s-channel (purple dots) andu-channel (red dots) origin vectors. For gravity, the analysis is a straightforward extension of photon EFT: just put h= 2 in the polynomial basis. Note that we again find that the limit of the Minkowski sum is given by that of maximal k.
We see that the string theory EFTs are clustered near the low spin boundaries of the polygon. In other words, the low energy run drives the couplings outside the EFT hedron.
The notion of a transparent fusion category is introduced in Definition 4.3.1, from which several successive graph equivalences and F-symbol relations are derived to reduce the number of independent F-symbols from O(n6) to O(n2) and represent the pentagon identity. practically solvable. These relations are summarized in a system of constraints in Definition 4.4.1, and the solutions for the pentagon identity under said constraints provide a classification of F-symbols for transparent Haagerup-Izumi merger categories. As mentioned above, the given equivalent to the F symbols for several unitary Haagerup-Izumi fusion categories was obtained by Izumi [105], Evans and Gannon [75], and Grossman and Snyder [93] using Cuntz algebra techniques; Such constructions were further generalized by Evans and Gannon [74] to fusion categories that need not be unitary.
Recently, the F symbols for all fusion categories realizing the Haagerup fusion ring (G=Z3) with six simple objects have been computed using the Pentagon approach by Titsworth [163], and for the special case of Haagerup H3- the fusion category (in the nomenclature of Grossman and Snyder [93]) independently by Osborne, Stiegemann and Wolf [139]. First, it offers the direct pentagonal construction for Haagerup-Izumi fusion categories beyond the Haagerup case (G=Z3); in particular, the Haagerup-Izumi fusion categories classified in Theorem 4.5.2 have not appeared in the literature beyond G = Z5. Second, the special transparent gauge used in this paper—where all F-symbols involving at least one external invertible object take the value one—not only makes the independent F-symbols directly comparable to the Cuntz algebra datum for Izumi [105], Evans and Gannon [74, 75], and Grossman and Snyder [93], but also make the F symbols automatically tetrahedrally symmetric (A4 or S4tetrahedral invariant in the language of this paper) and unitary for pseudo-unitary fusion categories. 2.
2In the F symbols for the Haagerup, fusion categories were presented with six simple objects in non-transparent meters that do not enjoy tetrahedral symmetry. The present authors used the Mathematica package provided by Titsworth [163] to check that the F symbols in the present paper are indeed gauge equivalent to his. In physical applications, such a gauge satisfies the assumptions of various theoretical constructions - the Levin-Wen string-net models [119], large classes of statistical models (see [2] and references within) and the associated single chains [77] - and allows the more effective exploitation of the G=Z2n+1 symmetry.
Of course, for a given fusion ring, there may exist non-transparent fusion categories that evade the current approach. Note: The authors first obtained the F symbols for the Haagerup fusion categories with six simple objects from Titsworth [163]. This observation led the present authors to postulate that transparent fusion categories also exist for the subsequent Haagerup–Izumi fusion rings with G=Z2n+1.
The types of fusion category discussed in this chapter are the key fusion categories above ground field=C.3 The notation for array diagrams is as follows. This consistency condition is the pentagon identity (FLL6,L3,L4 .. 4.2). Solving the identity of the pentagon means constructing a central fusion category. If there are isomorphism classes of simple objects, then the pentagon identity is a set of O(n9) cubic polynomial equations for O(n6) variables, modulo O(n3) gauge freedom.
