We study BPS loop operators in a 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory with four hypermultiplets in the fundamental representation and one hypermultiplet in the anti-symmetric representation. The algebra of BPS loop operators in the $\Omega$-background provides a deformation quantization of the Coulomb branch, which is expected to coincide with the quantized K-theoretic Coulomb branch in the mathematical literature. For the rank-one case, i.e., $Sp(1) \simeq SU(2)$, we show that the quantization of the Coulomb branch, evaluated using the supersymmetric localization formula, agrees with the polynomial representation of the spherical part of the double affine Hecke algebra (spherical DAHA) of $(C_1^{\vee}, C_1)$-type. For higher-rank cases, where $N \geq 2$, we conjecture that the quantized Coulomb branch of the 4d $\mathcal{N}=2$ $Sp(N)$ gauge theory is isomorphic to the spherical DAHA of $(C_N^{\vee}, C_N)$-type . As evidence for this conjecture, we demonstrate that the quantization of an 't Hooft loop agrees with the Koornwinder operator in the polynomial representation of the spherical DAHA.

We show that transformation formulas of multiple $q$-hypergeometric series agree with wall-crossing formulas of $K$-theoretic vortex partition functions obtained by Hwang, Yi and the author \cite{Hwang:2017kmk}. For the vortex partition function in 3d $\mathcal{N}=2$ gauge theory, we show that the wall-crossing formula agrees with the Kajihara transformation \cite{kajihara2004euler}. For the vortex partition function in 3d $\mathcal{N}=4$ gauge theory, we show that the wall-crossing formula agrees with the transformation formula by Halln\"as, Langmann, Noumi and Rosengren \cite{Halln_s_2022}. Since the $K$-theoretic vortex partition functions are related with indices such as the $\chi_t$-genus of the handsaw quiver variety, we discuss geometric interpretation of Euler transformations in terms of wall-crossing formulas of handsaw quiver variety.

We investigate the cosmological phase transition dynamics in a supersymmetric left-right symmetric model based on the gauge group $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ that addresses the strong CP problem through extended parity symmetry and doublet-doublet splitting. We compute the finite temperature effective potential including one-loop Coleman-Weinberg corrections, thermal contributions, and daisy resummation to determine whether the $SU(2)_R \times U(1)_{B-L} \to U(1)_Y$ symmetry breaking transition can produce observable gravitational waves. For phenomenologically viable parameters satisfying current LHC constraints, we find that the phase transition is strongly first-order with nucleation temperature $T_n \sim 0.5 v_R$, transition strength parameter $\alpha \sim 0.01-0.3$, and inverse duration $\beta/H \sim 100$. The resulting stochastic gravitational wave background peaks at frequencies $f \sim 0.1-1$ Hz with amplitude $h^2\Omega_{GW} \sim 10^{-14}-10^{-12}$. We find that there is a parameter region where the gravitational wave spectrum overlaps with DECIGO/BBO sensitivity curves, providing a potentially observable signature connecting the theoretical solution to the strong CP problem with gravitational wave experiments.

We examine the fundamental Kazakov-Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on the graph similar to that in the small coupling region. We estimate the phase structure of the FKM model both in the small and large coupling regions by comparing it with the Gross-Witten-Wadia (GWW) model. We further validate the theoretical analysis through detailed numerical simulations.

We propose a method to compute the entanglement entropy (EE) using the tensor renormalization group (TRG) method. The reduced density matrix of a $d$-dimensional quantum system is represented as a $(d+1)$-dimensional tensor network. We develop an explicit algorithm for $d=1$ that enables the calculation of EE for single-interval subsystems of arbitrary size. We test our method in two-dimensional tensor network of the Ising model. The central charge is obtained as $c=0.49997(8)$ for $D=96$, which agrees with the theoretical prediction within an error, demonstrating the accuracy and reliability of our proposed method.

Graph neural networks (GNNs) have emerged as a state-of-the-art data-driven tool for modeling connectivity data of graph-structured complex networks and integrating information of their nodes and edges in space and time. However, as of yet, the analysis of social networks using the time series of people's mobile connectivity data has not been extensively investigated. In the present study, we investigate four snapshot - based temporal GNNs in predicting the phone call and SMS activity between users of a mobile communication network. In addition, we develop a simple non - GNN baseline model using recently proposed EdgeBank method. Our analysis shows that the ROLAND temporal GNN outperforms the baseline model in most cases, whereas the other three GNNs perform on average worse than the baseline. The results show that GNN based approaches hold promise in the analysis of temporal social networks through mobile connectivity data. However, due to the relatively small performance margin between ROLAND and the baseline model, further research is required on specialized GNN architectures for temporal social network analysis.

We investigate the cosmological phase transition dynamics in a supersymmetric left-right symmetric model based on the gauge group $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ that addresses the strong CP problem through extended parity symmetry and doublet-doublet splitting. We compute the finite temperature effective potential including one-loop Coleman-Weinberg corrections, thermal contributions, and daisy resummation to determine whether the $SU(2)_R \times U(1)_{B-L} \to U(1)_Y$ symmetry breaking transition can produce observable gravitational waves. For phenomenologically viable parameters satisfying current LHC constraints, we find that the phase transition is strongly first-order with nucleation temperature $T_n \sim 0.5 v_R$, transition strength parameter $\alpha \sim 0.01-0.3$, and inverse duration $\beta/H \sim 100$. The resulting stochastic gravitational wave background peaks at frequencies $f \sim 0.1-1$ Hz with amplitude $h^2\Omega_{GW} \sim 10^{-14}-10^{-12}$. We find that there is a parameter region where the gravitational wave spectrum overlaps with DECIGO/BBO sensitivity curves, providing a potentially observable signature connecting the theoretical solution to the strong CP problem with gravitational wave experiments.

$D_4$ triality invariants are modular forms as well as polynomial invariants for a fiber product of the modular group and the Weyl group of type $F_4$. We show that the ring of $D_4$ triality invariants satisfying a certain cusp condition is isomorphic to the ring of joint covariants of a binary cubic and a binary quadratic form.

A brief derivation of the lagrangian of the minimal supersymmetric theory is given and the complete expression of the lagrangian in terms of mass eigenstates is presented.

We introduce a new method to treat Majorana fermions on the GRACE system, which has already been developed for the computation of the matrix elements for the processes of the standard model. In the standard model, we already have included such particles as Dirac fermions, gauge bosons and scalar bosons in the system. On the other hand, there are four Majorana fermions called neutralinos in the minimal SUSY standard model (MSSM). In consequence, we have constructed a system for the automatic computation of cross-sections for the processes of the MSSM. It is remarkable that our system is also applicable for another model including Majorana fermions once the definition of the model file is given.

grc4f is a Monte-Carlo package for generating e+e- to 4-fermion processes in the standard model. All of the 76 LEP-2 allowed fermionic final state processes evaluated at tree level are included in version 1.1. grc4f addresses event simulation requirements at e+e- colliders such as LEP and up-coming linear colliders. Most of the attractive aspects of grc4f come from its link to the GRACE system: a Feynman diagram automatic computation system. The GRACE system has been used to produce the computational code for all final states, giving a higher level of confidence in the calculation correctness. Based on the helicity amplitude calculation technique, all fermion masses can be kept finite and helicity information can be propagated down to the final state particles. The phase space integration of the matrix element gives the total and differential cross sections, then unweighted events are Generated. Initial state radiation (ISR) corrections are implemented in two ways, one is based on the electron structure function formalism and the second uses the parton shower algorithm called QEDPS. The latter can also be applied for final state radiation (FSR) though the interference with the ISR is not yet taken into account. Parton shower and hadronization of the final quarks are performed through an interface to JETSET. Coulomb correction between two intermediate W's, anomalous coupling as well as gluon contributions in the hadronic processes are also included.

We introduce a new method to treat Majorana fermions and interactions with fermion-number violation on the GRACE system which has been developed for the automatic computation of the matrix elements for the processes of the standard model. Thus we have constructed a system for the automatic computation of cross-sections for the processes of the minimal SUSY standard model (MSSM).

For the study of reactions in High Energy Physics (HEP) automatic computation systems have been developed and are widely used nowadays. GRACE is one of such systems and it has achieved much success in analyzing experimental data. Since we deal with the cross section whose value can be given by calculating hundreds of Feynman diagrams, we manage the large scale calculation, so that effective symbolic manipulation, the treat of singularity in the numerical integration are required. The talk will describe the software design of GRACE system and computational techniques in the GRACE.

We examine a class of Calabi-Yau varieties of the determinantal type in Grassmannians and clarify what kind of examples can be constructed explicitly. We also demonstrate how to compute their genus-0 Gromov-Witten invariants from the analysis of the Givental $I$-functions. By constructing $I$-functions from the supersymmetric localization formula for the two dimensional gauged linear sigma models, we describe an algorithm to evaluate the genus-0 A-model correlation functions appropriately. We also check that our results for the Gromov-Witten invariants are consistent with previous results for known examples included in our construction.

We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in asymptotically Euclidean $AdS_2$ background using the matrix model description recently found by Saad, Shenker and Stanford [arXiv:1903.11115]. We show that the partition function of JT gravity is written as the expectation value of a macroscopic loop operator in the old matrix model of 2d gravity in the background where infinitely many couplings are turned on in a specific way. Based on this expression we develop a very efficient method of computing the partition function in the genus expansion as well as in the low temperature expansion by making use of the Korteweg-de Vries constraints obeyed by the partition function. We have computed both these expansions up to very high orders using this method. It turns out that we can take a low temperature limit with the ratio of the temperature and the genus counting parameter held fixed. We find the first few orders of the expansion of the free energy in a closed form in this scaling limit. We also study numerically the behavior of the eigenvalue density and the Baker-Akhiezer function using the results in the scaling limit.

Cumulative cultural evolution is what made humanity to thrive in various ecological and demographic environments. Solutions to the tasks that humans needed to solve could be mapped onto a task space which could take the form of either closed or open-ended fitness landscape, with the former being modeled more extensively than the latter in studies of cultural evolution. In this article, we modified a simulation by Arthur and Polak (2006) that modeled open-ended fitness landscape by using a computer simulation that builds logical circuits with circuits that were built in earlier trials. We used this simulation to clarify the nature of open-ended fitness landscape and to investigate whether the speed of accumulation of culture is increased by an increase in group size. The results indicated that group size increased the speed of accumulation but is limited than expected. Also, when two types of accumulation, invention and improvement, were distinguished the nature of the two differed. In improvement, the trajectory followed a convex function with productivity of one agent decreasing as group size increased. In invention, the trajectory showed a continuous pattern of rapid increase followed by a plateau.

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak's differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak's equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.

This thesis is driven by a central question: "What can we learn from random geometries about the structure of quantum spacetime?" In Chapter 2, we provide a partial review of the mathematical foundation of this thesis, random geometry. In Chapter 3, we use a construction coming from random geometry called Mating of Trees to build scale-invariant random geometries that appear in Liouville Quantum Gravity and have the potential to implement the UV fixed point predicted by Asymptotic Safety in two and three dimensions. In Chapter 4 we explore the random geometry formulation of JT gravity and how our understanding of random critical maps yields the discovery of a new family of deformations of JT gravity. Furthermore, the connection between JT gravity and matrix models leads us to delve deeper into the link between discrete geometry and hyperbolic surfaces, building upon the geometry of metric maps and irreducible metric maps in Chapter 5.