arXiv subjects
Publications and source records attributed to Hee-Joong Chung.
We realize a homological block of a knot complement in $S^3$ for $G_{\mathbb{C}}=SL(2,\mathbb{C})$ as a half-index of a 3d $\mathcal{N}=2$ theory via an expression of the homological block as an inverted Habiro series by working out some examples, which we expect to extend to general knots. Also, by choosing a certain set of poles in the integral expression of the half-index, we obtain the colored Jones polynomial.
For $G=SU(N)$ at finite and large $N$, with a totally symmetric representation, we realize the homological block $F_K$ for a knot complement $S^3 \backslash K$, given in the form of the inverted Habiro series, as a half-index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ by studying some examples, which we expect to extend to general knots. From the half-index expression, it is also possible to realize the colored HOMFLY-PT polynomial by taking a certain set of poles. Through the half-index realization, we describe a method for obtaining the $G=SU(N)$ homological block and its $a$-deformed version for $S^3 \backslash K$ from a Habiro series expression for the colored HOMFLY-PT polynomial. We also discuss some properties of partition functions for arbitrary $N$.
We consider quiver forms that appear in the motivic Donaldson-Thomas generating series or characters of conformal field theories and relate them to 3d $\mathcal{N}=2$ theories on $D^2 \times_q S^1$ with certain boundary conditions preserving 2d $\mathcal{N}=(0,2)$ supersymmetry. We apply this to the 3d-3d correspondence and provide a Lagrangian description of 3d $\mathcal{N}=2$ theories $T[M_3]$ with 2d $\mathcal{N}=(0,2)$ boundary conditions for 3-manifolds $M_3$ in several contexts.
The notion of {\em entanglement entropy} in quantum mechanical systems is an important quantity, which measures how much a physical state is entangled in a composite system. Mathematically, it measures how much the state vector is not decomposable as elements in the tensor product of two Hilbert spaces. In this paper, we seek its arithmetic avatar: the theory of arithmetic Chern-Simons theory with finite gauge group $G$ naturally associates a state vector inside the product of two quantum Hilbert spaces and we provide a formula for the {\em von Neumann entanglement entropy} of such state vector when $G$ is a cyclic group of prime order.
We calculate homological blocks for a knot in Seifert manifolds when the gauge group is $SU(N)$. We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.
We prove an arithmetic path integral formula for the inverse p-adic absolute values of the Kubota-Leopoldt p-adic L-functions at roots of unity.
We consider the $S^2 \times_q S^1$ supersymmetric index of a 3d $\mathcal{N}=2$ theory $T[M_3]$ when $M_3$ is a plumbed 3-manifold. We engineer an effective description of $T[M_3]$ from the expression of the homological block for plumbed 3-manifolds as a $D^2 \times_q S^1$ partition function of a 3d $\mathcal{N}=2$ theory $T[M_3]$ with a boundary condition. We check that the supersymmetric index for such a $T[M_3]$ is invariant under the 3d Kirby moves.
We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition function at or around roots of unity $q=e^{2πi \frac{1}{K}}$ with rational level $K=\frac{r}{s}$ where $r$ and $s$ are coprime integers. From the exact expression for the $G=SU(2)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds at other roots of unity obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at other roots of unity. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the standard limit in the volume conjecture.
We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial.
We calculate the homological blocks for Seifert manifolds from the exact expression for the $G=SU(N)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mariño. For the $G=SU(2)$ case, it is possible to express them in terms of the false theta functions and their derivatives. For $G=SU(N)$, we calculate them as a series expansion and also discuss some properties of the contributions from the abelian flat connections to the Witten-Reshetikhin-Turaev invariants for general $N$. We also provide an expected form of the $S$-matrix for general cases and the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks.
We calculate partition function and correlation functions in A-twisted 2d $\mathcal{N}=(2,2)$ theories and topologically twisted 3d $\mathcal{N}=2$ theories containing adjoint chiral multiplet with particular choices of $R$-charges and the magnetic fluxes for flavor symmetries. According to Gauge-Bethe correspondence, they correspond to Heisenberg XXX and XXZ spin chain models. We identify the partition function as the inverse of the norm of the Bethe eigenstates. Correlation functions are identified as the coefficients of the expectation value of Baxter $Q$-operators. In addition, we consider correlation functions of 2d $\mathcal{N}=(2,2)^*$ theory and their relation to equivariant quantum cohomology and equivariant integration of cotangent bundle of Grassmann manifolds. Also, we study the ring relations of supersymmetric Wilson loops in 3d $\mathcal{N}=2^*$ theory and Bethe subalgebra of XXZ spin chain model.
The half-BPS boundary conditions preserving $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ supersymmetry in 3d $\mathcal{N}=4$ supersymmetric gauge theories are examined. The BPS equations admit decomposition of the bulk supermultiplets into specific boundary supermultiplets of preserved supersymmetry. Nahm-like equations arise in the vector multiplet BPS boundary condition preserving $\mathcal{N}=(0,4)$ supersymmetry and Robin-type boundary conditions appear for the hypermultiplet coupled to vector multiplet when $\mathcal{N}=(2,2)$ supersymmetry is preserved. The half-BPS boundary conditions are realized in the brane configurations of Type IIB string theory.
We apply ideas of Dijkgraaf and Witten on three-dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications.
Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of n-th power residue symbols. This formalism leads to a precise arithmetic analogue of a 'path-integral formula' for linking numbers.
In fivebrane compactifications on 3-manifolds, we point out the importance of all flat connections in the proper definition of the effective 3d N=2 theory. The Lagrangians of some theories with the desired properties can be constructed with the help of homological knot invariants that categorify colored Jones polynomials. Higgsing the full 3d theories constructed this way recovers theories found previously by Dimofte-Gaiotto-Gukov. We also consider the cutting and gluing of 3-manifolds along smooth boundaries and the role played by all flat connections in this operation.
We study open string configurations in Rindler space suspended from D-branes at finite distance from the Rindler horizon. This is a toy model for strings in the near horizon region of a black hole and has applications for the study of strings configurations of heavy quarks in the AdS/CFT duals of hot field theories, as well as other applications to the study of open strings ending on giant gravitons. We find that this setup produces very similar results to those that have been found in the AdS black hole setup, but it is much more tractable analytically. We also comment on some quantum applications of our studies to the understanding of the spectrum of strings ending on giant gravitons.