Numerous results in string theory have proven beneficial to our modern understanding of mathematics and vice versa, and my current research is precisely focused on the intersection of these two fascinating subjects. Specifically, there have been many constructions of gravity theories using mathematical frameworks. Of particular interest to me is the study of supergravity theories in three to six dimensions from M-theory or F-theory compactified on elliptically-fibered Calabi-Yau threefolds or fourfolds, as well as the topological and geometric properties of such Calabi-Yau varieties. Moreover, infinite dimensional von Neumann algebras of various types are used to understand the local algebras in quantum field theories. This leads to my interest on how to understand holographic quantum field theories and their gravity duals by studying quantum error correction using von Neumann algebras in toy models.
Calabi-Yau compactification and spectra
M-theory compactified on Calabi-Yau threefolds gives a low energy limit of 5d supersymmetric theories, the crepant resolutions, which are parametrized by the Kahler moduli, provide a powerful dictionary between geometry and physics: The Coulomb phases of the 5d theory are equivalent to the extended Kahler cone of the Calabi--Yau threefolds. It has been my continuous desire to understand the structure of the Coulomb branch of the five-dimensional N=1 theory with Lie group G and representation R geometrically engineered by an elliptic fibration for the same various choices of G. Using intersection theory, I computed the triple intersection numbers, which can be interpreted as the 5d Chern-Simons levels. As the triple intersection polynomials are the geometric interpretation of the prepotentials, I was able to compute the charged matter contents of the 5d theories. Furthermore, I have shown that by comparing the triple intersection numbers and the prepotentials, the number of 6d charged multiplets is completely fixed in all studied models except for the semi-simple groups without a Z2 quotient; I gave a short talk at String-Math 2018 on this topic, focusing on the role of Mordell--Weil torsion. By studying their 6d uplifts, I showed that the anomalies are all canceled via Green-Schwartz mechanism.
Topological and geometric properties of Calabi-Yau varieties
The Euler characteristic of an elliptic fibration plays a central role in many physics problems, including the gravitational anomalies in 6d and tadpole cancellation in 4d. This was my primary motivation for computing the closed form formulas for the Euler characteristics of elliptic fibrations that are used to geometrically engineer many gauge theories; I gave a short talk at Strings 2017 on this topic explaining the results. From the Euler characteristic, I computed the Hodge numbers of the elliptically-fibered Calabi--Yau threefolds for various gauge groups. Furthermore, I polished our methods to compute the topological invariants of the elliptic fourfolds. I computed for a large collection of elliptic and genus-one fibered fourfolds the closed form expressions for the characteristic classes, and showed that the A-genus (and thus the index of the Dirac operator) are independent on the choice of a gauge group.
Quantum Error Correction and Infinite-dimensional von Neumann Algebras
In my current project, I give an explicit construction of a quantum error correcting code (QECC) where the code and physical Hilbert spaces are infinite dimensional. I built a von Neumann algebra of type II$_1$ acting on the code Hilbert space and showed how it is mapped to the physical Hilbert space. Infinite-dimensional QECCs, such as the one I constructed, should help understanding the connection between entanglement wedge reconstruction and the JLMS formula, which states that the relative entanglement entropy of the boundary gives the bulk relative entropy. I believe that the methods used in my construction can be extended to an infinite-dimensional analog of the HaPPY code, yielding a holographic interpretation.
Monica Jinwoo Kang and Elliott Gesteau, To appear
Monica Jinwoo Kang and Eugene Tang, To appear
Monica Jinwoo Kang and David Kolchmeyer, Entanglement Wedge Reconstruction of Infinite-dimensional von Neumann algebras using Tensor Networks (arXiv:1910.06328)
Mboyo Esole, Monica Jinwoo Kang. 48 Crepant Paths to SU(2)xSU(3) (arXiv:1905.05174)
Monica Jinwoo Kang and David Kolchmeyer, Holographic Relative Entropy in Infinite-dimensional Hilbert Spaces (arXiv:1811.05482)
Mboyo Esole, Monica Jinwoo Kang. Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups (arXiv:1808.07054)
Mboyo Esole, Monica Jinwoo Kang. Characteristic numbers of crepant resolutions of Weierstrass models (arXiv:1807.08755)
Mboyo Esole, Monica Jinwoo Kang. Flopping and Slicing: SO(4) and Spin(4)-models (arXiv:1802.04802)
Mboyo Esole, Monica Jinwoo Kang, Shing-Tung Yau. Mordell-Weil Torsion, Anomalies, and Phase Transitions (arXiv:1712.02337)
Mboyo Esole, Ravi Jagadeesan, Monica Jinwoo Kang. The Geometry of G2, SO(7), and SO(8)-models (arXiv:1709.04913)
Mboyo Esole, Patrick Jefferson, Monica Jinwoo Kang. The Geometry of F4-Models (arXiv:1704.08251)
Mboyo Esole, Monica Jinwoo Kang, Shing-Tung Yau. A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations (arXiv:1410.0003)
Kyung-Yuen Ham, Young Cheol Jeon, Jinwoo Kang, Nam Kyun Kim, Wonjae Lee, Yang Lee, Sung Ju Ryu, and Hae-Hun Yang. IFP Rings and Near-IFP Rings J Korean Math. Soc. 45 (2008), No.3, pp. 727740.