This thesis discusses consequences of color-kinematics duality and applications towards computing quantum scattering amplitudes and classical radiation fields. Stemming from this duality, tree-level Bern-Carrasco-Johansson amplitude relations can be extended to one-loop integral coefficient relations for scattering in Yang-Mills theory. The double copy, which also follows from color-kinematics duality, allows for graviton scattering amplitudes to be found from scattering amplitudes in Yang-Mills theory. Additionally, a classical radiative double copy for obtaining gravitational waves in various theories is discussed. As a warm-up, a classical double copy of the Lienard-Wiechert potential in electrodynamics is found within a specific context, which allows for gravitational waves in linearized gravity to be found. Next, radiation in Yang-Mills and Yang-Mills-biadjoint-scalar theories is found, and the radiative double copy of these results allows for radiation in general relativity and Einstein-Yang-Mills theory, repsectively. In light of the recent detection of gravitational waves by the LIGO collaboration, this motivates the search of efficient analytic techniques for computing gravitational radiation. The double copy offers a way to apply methods from particle physics to gravitational-wave astronomy.
This dissertation reviews work in computing N = 4 super-Yang–Mills (sYM) and N = 8
maximally supersymmetric gravity (mSUGRA) scattering amplitudes in D = 4 spacetime
dimensions in novel ways.
After a brief introduction and overview in Ch. 1, the various techniques used to construct amplitudes in the remainder of the dissertation are discussed in Ch. 2. This includes several new concepts such as d log and pure integrand bases, as well as how to construct the amplitude using exactly one kinematic point where it vanishes. Also included in this chapter is an outline of the Mathematica package on shell diagrams and numerics.m (osdn) that was developed for the computations herein. The rest of the dissertation is devoted to explicit examples.
In Ch. 3, the starting point is tree-level sYM amplitudes that have integral representations with residues that obey amplitude relations. These residues are shown to have corresponding residue numerators that allow a double copy prescription that results in mSUGRA residues.
In Ch. 4, the two-loop four-point sYM amplitude is constructed in several ways, showcasing many of the techniques of Ch. 2; this includes an example of how to use osdn. The two-loop five-point amplitude is also presented in a pure integrand representation with comments on how it was constructed from one homogeneous
cut of the amplitude. On-going work on the two-loop n-point amplitude is presented at the end of Ch. 4.
In Ch. 5, the three-loop four-point amplitude is presented in the d log representation and
in the pure integrand representation.
In Ch. 6, there are several examples of four- through seven-loop planar diagrams that illustrate how considerations of the singularity structure of the amplitude underpin dual-conformal invariance. Taken with the previous examples, this is additional evidence that the structure known to exist in the planar sector extends to the full theory. At the end of this chapter is a proof that all mSUGRA amplitudes have a pole at infinity for (L ≥ 4)-loops.
Finally in Ch. 7, the current status of ultraviolet divergences in the five-loop four-point mSUGRA amplitude is addressed. This includes a discussion of ongoing work aimed at resolving the mSUGRA finiteness question.
The following Mathematica scripts are submitted with this dissertation:
• on shell diagrams and numerics.m with dependencies:
– all_trees *.m
– external_kinematics_*_point.m
– rational_external_*_point.m
where “*” is a wild-card string of any set of characters of any length – either an integer
or a number spelled out.
- 1 supplemental ZIP
Classically, the ground states of $\mathcal{N}=4$ Super Yang-Mills Theory (SYM) on $\mathbb{R}\times S^3/\Gamma$ where $\Gamma$ is one of the ADE subgroup of $SU(2)$ are flat Wilson lines winding around the ADE singularity. S-duality acts on this finite-dimensional ground state Hilbert space and its action is the same as the $S$ operator in a certain dual Chern-Simons theory on $T^2$. The dual Chern-Simons theory arises out of the only long-range interaction in a string/M theory construction by considering a stack of D3(M5) branes on ADE singularity. This SYM/Chern-Simons duality is verified by matching the ground state Hilbert spaces of both theories and by comparing the S-duality operators of both theories.
To one-loop order, the SYM ground state degeneracy is exact. A detailed computation using the superconformal index shows that each classical SYM ground state acquires the same supersymmetric Casimir energy. S-duality maps the SYM ground state Wilson lines to ground state t' Hooft lines taking values in the Langlands dual group. The number of t' Hooft lines are shown to agree with that of the Wilson lines. In addition, the t' Hooft lines have the same supersymmetric Casimir energy as the corresponding Wilson lines. These two facts provide a ground state test of S-duality.
The SYM/Chern-Simons duality has an important extension to the class S theory obtained from compactifying M5 branes on a Riemann surface $\mathcal{R}$. The ground states of class S theory on $\mathbb{R}\times S^3/\Gamma$ are dual to the states of the dual Chern-Simons theory on $\mathcal{R}$. In particular, we uncover a surprising result that there is only one unique ground state for the conformal $\mathcal{N}=2$ $SU(2)$ four-flavor theory on $\mathbb{R}\times S^3/\Gamma$. Finally, we apply the SYM/Chern-Simons duality to a non-Lagrangian class S theory and find that its ground states obey the fusion rule of the current algebra of the dual Chern-Simons theory.
If Person A delivers drugs to Person B at the latter’s request, Person A is liable for drug trafficking—a serious offense in many jurisdictions. However, the liability of Person B for drug trafficking is unclear as much may depend on Person B’s intention with the drugs. The Singaporean Courts recently had to grapple with this issue in Liew Zheng Yang v. Public Prosecutor and Ali bin Mohamad Bahashwan v. Public Prosecutor and other appeals. Prior to these two cases, the position in Singapore was clear—Person B should be liable for drug trafficking as an accessory to Person A, in line with Singapore’s strong stance against drug offenses. However, since these cases, the Singaporean Courts have taken a contrary position and held that Person B may not be liable if the drugs were for his/her own consumption.
This Article examines the law with respect to this drug conspiracy offense in Singapore, looking at its history, the primary legislation and similar cases. It also scrutinizes the judicial reasoning in the two cases above and considers whether this can be reconciled with the Courts’ prior position on the issue. In this analysis, the Article also investigates the position taken in other comparable common law jurisdictions—including the UK, Australia, Canada and the United States—and concludes that the Singaporean Courts’ reasoning in the aforementioned two cases may not be tenable and warrant a reexamination.
Embedded contact homology (ECH) capacities were defined by Hutchings and provide a family of obstructions to embeddings of four-dimensional symplectic manifolds. In Part I of this thesis, we prove that for a four-dimensional Liouville domain with all ECH capacities finite, the asymptotics of the ECH capacities recover the symplectic volume. This was joint work with Daniel Cristofaro-Gardiner and Michael Hutchings. In Part II of this thesis, we construct topological absolute gradings in Heegaard Floer homology and bordered Floer homology that satify all of the expected properties. This was joint work with Yang Huang. We also show that the isomorphism between Heegaard Floer homology and ECH preserves the absolute gradings.