On-Shell Gauge Invariant Three-Point Amplitudes

Assuming locality, Lorentz invariance and parity conservation we obtain a set of differential equations governing the 3-point interactions of massless bosons, which in turn determines the polynomial ring of these amplitudes. We derive all possible 3-point interactions for tensor fields with polarisations that have total symmetry and mixed symmetry under permutations of Lorentz indices. Constraints on the existence of gauge-invariant cubic vertices for totally symmetric fields are obtained in general spacetime dimensions and are compared with existing results obtained in the covariant and light-cone approaches. Expressing our results in spinor helicity formalism we reproduce the perhaps mysterious mismatch between the covariant approach and the light cone approach in 4 dimensions. Our analysis also shows that there exists a mismatch, in the 3-point gauge invariant amplitudes corresponding to cubic self-interactions, between a scalar field $\phi$ and an antisymmetric rank-2 tensor field $A_{\mu\nu}$. Despite the well-known fact that in 4 dimensions rank-2 anti-symmetric fields are dual to scalar fields in free theories, such duality does not extend to interacting theories.Comment: significantly revised, final version published in JHE

TT‾T\overline{T} deformation in SCFTs and integrable supersymmetric theories

We calculate the $\mathcal{S}$-multiplets for two-dimensional Euclidean $\mathcal{N}=(0,2)$ and $\mathcal{N} = (2,2)$ superconformal field theories under the $T\overline{T}$ deformation at leading order of perturbation theory in the deformation coupling. Then, from these $\mathcal{N} = (0, 2)$ deformed multiplets, we calculate two- and three-point correlators. We show the $\mathcal{N} = (0,2)$ chiral ring's elements do not flow under the $T\overline{T}$ deformation. For the case of $\mathcal{N} = (2,2)$, we show the twisted chiral ring and chiral ring cease to exist simultaneously. Specializing to integrable supersymmetric seed theories, such as $\mathcal{N} = (2,2)$ Landau-Ginzburg models, we use the thermodynamic Bethe ansatz to study the S-matrices and ground state energies. From both an S-matrix perspective and Melzer's folding prescription, we show that the deformed ground state energy obeys the inviscid Burgers' equation. Finally, we show that several indices independent of $D$-term perturbations including the Witten index, Cecotti-Fendley-Intriligator-Vafa index and elliptic genus do not flow under the $T\overline{T}$ deformation.Comment: 46 page

The High-dimensional Phase Diagram and the Large CALPHAD Model

When alloy systems comprise more than three elements, the visualization of the entire phase space becomes not only daunting but is also accompanied by a data surge. Addressing this complexity, we delve into the FeNiCrMn alloy system and introduce the Large CALPHAD Model (LCM). The LCM acts as a computational conduit, capturing the entire phase space. Subsequently, this enormous data is systematically structured using a high-dimensional phase diagram, aided by hash tables and Depth-first Search (DFS), rendering it both digestible and programmatically accessible. Remarkably, the LCM boasts a 97% classification accuracy and a mean square error of 4.80*10-5 in phase volume prediction. Our methodology successfully delineates 51 unique phase spaces in the FeNiCrMn system, exemplifying its efficacy with the design of all 439 eutectic alloys. This pioneering methodology signifies a monumental shift in alloy design techniques or even multi-variable problems

Self-duality under gauging a non-invertible symmetry

We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of $c=1$ CFTs, it is known that the theory is self-dual under gauging a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry, and has $\mathsf{Rep}(H_8)$ and $\mathsf{Rep}(D_8)$ fusion category symmetries as a result. We find that gauging the entire $\mathsf{Rep}(H_8)$ fusion category symmetry maps the orbifold theory at radius $R$ to that at radius $2/R$. At $R=\sqrt{2}$, which corresponds to two decoupled Ising CFTs (Ising$^2$ in short), the theory is self-dual under gauging the $\mathsf{Rep}(H_8)$ symmetry. This implies the existence of a topological defect line in the Ising$^2$ CFT obtained from half-space gauging of the $\mathsf{Rep}(H_8)$ symmetry, which commutes with the $c=1$ Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the $c=1$ Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the $\mathsf{Rep}(H_8)$ symmetry to form a bigger fusion category which is a $\mathbb{Z}_2$-extension of $\mathsf{Rep}(H_8)$. We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising$^2$ CFT. Finally, we show that the torus partition functions of the Monster$^2$ CFT and Ising$\times$Monster CFT are also invariant under gauging the $\mathsf{Rep}(H_8)$ symmetry.Comment: 58 pages, 4 figures, 5 tables, 2 Mathematica ancillary files; v2: minor edit

Root-TT‾T \overline{T} Deformed Boundary Conditions in Holography

We develop the holographic dictionary for pure $\mathrm{AdS}_3$ gravity where the Lagrangian of the dual $2d$ conformal field theory has been deformed by an arbitrary function of the energy-momentum tensor. In addition to the $T \overline{T}$ deformation, examples of such functions include a class of marginal stress tensor deformations which are special because they leave the generating functional of connected correlators unchanged up to a redefinition of the source and expectation value. Within this marginal class, we identify the unique deformation that commutes with the $T \overline{T}$ flow, which is the root-$T \overline{T}$ operator, and write down the modified boundary conditions corresponding to this root-$T \overline{T}$ deformation. We also identify the unique marginal stress tensor flow for the cylinder spectrum of the dual CFT which commutes with the inviscid Burgers' flow driven by $T \overline{T}$, and we propose this unique flow as a candidate root-$T \overline{T}$ deformation of the energy levels. We study BTZ black holes in $\mathrm{AdS}_3$ subject to root-$T \overline{T}$ deformed boundary conditions, and find that their masses flow in a way which is identical to that of our candidate root-$T \overline{T}$ energy flow equation, which offers evidence that this flow is the correct one. Finally, we also obtain the root-$T \overline{T}$ deformed boundary conditions for the gauge field in the Chern-Simons formulation of $\mathrm{AdS}_3$ gravity.Comment: 65 pages; LaTe

The Hitchhiker's Guide to 4d N=2\mathcal{N}=2 Superconformal Field Theories

Superconformal field theory with $\mathcal{N}=2$ supersymmetry in four dimensional spacetime provides a prime playground to study strongly coupled phenomena in quantum field theory. Its rigid structure ensures valuable analytic control over non-perturbative effects, yet the theory is still flexible enough to incorporate a large landscape of quantum systems. Here we aim to offer a guidebook to fundamental features of the 4d $\mathcal{N}=2$ superconformal field theories and basic tools to construct them in string/M-/F-theory. The content is based on a series of lectures at the Quantum Field Theories and Geometry School (https://sites.google.com/view/qftandgeometrysummerschool/home) in July 2020.Comment: v3: Improved discussion, fixed typos, added references v2: Typos fixed and added references. v1: 96 pages. Based on a series of lectures at the Quantum Field Theories and Geometry School in July 202

Aspects of Symmetries in Quantum Field Theories

We study the properties and applications of generalized symmetries in the quantum field theories. We explore how to use the spacetime symmetry and the internal symmetry to refine the Cardy formula in 2-dimensional conformal field theories. We use the technique of group theoretical fusion categories to study the physical implications of triality defects in 2d CFT. We also explore when the non-invertible symmetries are non-intrinsically non-invertible, that is, when they can be constructed from topological manipulations of invertible symmetries