The full list for arXiv preprints is available here:
(note: authors are listed alphabetically)


1. Supersymmetric Gauged Matrix Models from Dimensionsional Reduction on a Sphere

Cyril Closset, Dongwook Ghim, Rak-Kyeong Seong


It was recently proposed that N=1 supersymmetric gauged matrix models have a duality of order four – that is, a quadrality – reminiscent of infrared dualities of SQCD theories in higher dimensions. In this note, we show that the zero-dimensional quadrality proposal can be infered from the two-dimensional Gadde-Gukov-Putrov triality. We consider two-dimensional N=(0,2) SQCD compactified on a sphere with the half-topological twist. For a convenient choice of R-charge, the zero-mode sector on the sphere gives rise to a simple N=1 gauged matrix model. Triality on the sphere then implies a triality relation for the supersymmetric matrix model, which can be completed to the full quadrality.

2. Machine Learning of Calabi-Yau Volumes

Daniel Krefl, Rak-Kyeong Seong
Phys. Rev. D 96, 066014 (2017)


We employ machine learning techniques to investigate the volume minimum of Sasaki-Einstein base manifolds of non-compact toric Calabi-Yau 3-folds. We find that the minimum volume can be approximated via a second order multiple linear regression on standard topological quantities obtained from the corresponding toric diagram. The approximation improves further after invoking a convolutional neural network with the full toric diagram of the Calabi-Yau 3-folds as the input. We are thereby able to circumvent any minimization procedure that was previously necessary and find an explicit mapping between the minimum volume and the topological quantities of the toric diagram. Under the AdS/CFT correspondence, the minimum volumes of Sasaki-Einstein manifolds correspond to central charges of a class of 4d N=1 superconformal field theories. We therefore find empirical evidence for a function that gives values of central charges without the usual extremization procedure.

3. Calabi-Yau Volumes and Reflexive Polytopes

Yang-Hui He, Rak-Kyeong Seong, Shing-Tung Yau


We study various geometrical quantities for Calabi-Yau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the Sasaki-Einstein base of the corresponding Calabi-Yau cone are calculated. By doing so, we conjecture new bounds for the Sasaki-Einstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.

4. Elliptic Genera of 2d (0,2) Gauge Theories from Brane Brick Models

Sebastian Franco, Dongwook Ghim, Sangmin Lee, Rak-Kyeong Seong
JHEP 1706:068,2017


We compute the elliptic genus of abelian 2d (0,2) gauge theories corresponding to brane brick models. These theories are worldvolume theories on a single D1-brane probing a toric Calabi-Yau 4-fold singularity. We identify a match with the elliptic genus of the non-linear sigma model on the same Calabi-Yau background, which is computed using a new localization formula. The matching implies that the quantum effects do not drastically alter the correspondence between the geometry and the 2d (0,2) gauge theory. In theories whose matter sector suffers from abelian gauge anomaly, we propose an ansatz for an anomaly cancelling term in the integral formula for the elliptic genus. We provide an example in which two brane brick models related to each other by Gadde-Gukov-Putrov triality give the same elliptic genus.

5. Quadrality for Supersymmetric Matrix Models

Sebastian Franco, Sangmin Lee, Rak-Kyeong Seong, Cumrun Vafa
JHEP 1707:053,2017


We introduce a new duality for N=1 supersymmetric gauged matrix models. This 0d duality is an order 4 symmetry, namely an equivalence between four different theories, hence we call it Quadrality. Our proposal is motivated by mirror symmetry, but is not restricted to theories with a D-brane realization and holds for general N=1 matrix models. We present various checks of the proposal, including the matching of: global symmetries, anomalies, deformations and the chiral ring. We also consider quivers and the corresponding quadrality networks. Finally, we initiate the study of matrix models that arise on the worldvolume of D(-1)-branes probing toric Calabi-Yau 5-folds.

6. Orbifold Reduction and 2d (0,2) Gauge Theories

Sebastian Franco, Sangmin Lee, Rak-Kyeong Seong
JHEP 1703:016,2017


We introduce Orbifold Reduction, a new method for generating 2d (0,2) gauge theories associated to D1-branes probing singular toric Calabi-Yau 4-folds starting from 4d N=1 gauge theories on D3-branes probing toric Calabi-Yau 3-folds. The new procedure generalizes dimensional reduction and orbifolding. In terms of T-dual configurations, it generates brane brick models starting from brane tilings. Orbifold reduction provides an agile approach for generating 2d (0,2) theories with a brane realization. We present three practical applications of the new algorithm: the connection between 4d Seiberg duality and 2d triality, a combinatorial method for generating theories related by triality and a 2d (0,2) generalization of the Klebanov-Witten mass deformation.

7. Brane Brick Models in the Mirror

Sebastian Franco, Sangmin Lee, Rak-Kyeong Seong, Cumrun Vafa
JHEP 1702:106,2017


Brane brick models are Type IIA brane configurations that encode the 2d N=(0,2) gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how 2d (0,2) triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry leads to a geometric unification of dualities in different dimensions, where the order of duality is n−1 for a Calabi-Yau n-fold. This makes us conjecture the existence of a quadrality symmetry in 0d. Finally, we comment on how the M-theory lift of brane brick models connects to the classification of 2d (0,2) theories in terms of 4-manifolds.

8. Brane Brick Models and 2d (0,2) Triality

Sebastian Franco, Sangmin Lee, Rak-Kyeong Seong
JHEP 1605:020,2016


We provide a brane realization of 2d (0,2) Gadde-Gukov-Putrov triality in terms of brane brick models. These are Type IIA brane configurations that are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. Triality translates into a local transformation of brane brick models, whose simplest representative is a cube move. We present explicit examples and construct their triality networks. We also argue that the classical mesonic moduli space of brane brick model theories, which corresponds to the probed Calabi-Yau 4-fold, is invariant under triality. Finally, we discuss triality in terms of phase boundaries, which play a central role in connecting Calabi-Yau 4-folds to brane brick models.

9. Consistency and Derangements in Brane Tilings

Amihay Hanany, Vishnu Jejjala, Sanjaye Ramgoolam, Rak-Kyeong Seong
Journal of Physics A 49 (2016) 355401


Brane tilings describe Lagrangians (vector multiplets, chiral multiplets, and the superpotential) of four dimensional N=1 supersymmetric gauge theories. These theories, written in terms of a bipartite graph on a torus, correspond to worldvolume theories on N D3-branes probing a toric Calabi-Yau threefold singularity. A pair of permutations compactly encapsulates the data necessary to specify a brane tiling. We show that geometric consistency for brane tilings, which ensures that the corresponding quantum field theories are well behaved, imposes constraints on the pair of permutations, restricting certain products constructed from the pair to have no one-cycles. Permutations without one-cycles are known as derangements. We illustrate this formulation of consistency with known brane tilings. Counting formulas for consistent brane tilings with an arbitrary number of chiral bifundamental fields are written down in terms of delta functions over symmetric groups.

10. Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0,2) Quivers

Sebastian Franco, Sangmin Lee, Rak-Kyeong Seong
JHEP 1602:047,2016


We introduce brane brick models, a novel type of Type IIA brane configurations consisting of D4-branes ending on an NS5-brane. Brane brick models are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. They fully encode the infinite class of 2d (generically) N=(0,2) gauge theories on the worldvolume of the D1-branes and streamline their connection to the probed geometries. For this purpose, we also introduce new combinatorial procedures for deriving the Calabi-Yau associated to a given gauge theory and vice versa.

11. 2d (0,2) Quiver Gauge Theories and D-branes

Sebastian Franco, Dongwook Ghim, Sangmin Lee, Rak-Kyeong Seong, Daisuke Yokoyama

JHEP 1509:072,2015


We initiate a systematic study of 2d (0,2) quiver gauge theories on the worldvolume of D1-branes probing singular toric Calabi-Yau 4-folds. We present an algorithm for efficiently calculating the classical mesonic moduli spaces of these theories, which correspond to the probed geometries. We also introduce a systematic procedure for constructing the gauge theories for arbitrary toric singularities by means of partial resolution, which translates to higgsing in the field theory. Finally, we introduce Brane Brick Models, a novel class of brane configurations that consist of D4-branes suspended from an NS5-brane wrapping a holomorphic surface, tessellating a 3-torus. Brane Brick Models are the 2d analogues of Brane Tilings and allow a direct connection between geometry and gauge theory.

12. Hilbert Series for Theories with Aharony Duals

Amihay Hanany, Chiung Hwang, Hyungchul Kim, Jaemo Park, Rak-Kyeong Seong

JHEP 1511:132,2015


The algebraic structure of moduli spaces of 3d N=2 supersymmetric gauge theories is studied by computing the Hilbert series which is a generating function that counts gauge invariant operators in the chiral ring. These U(N_c) theories with N_f flavors have Aharony duals and their moduli spaces receive contributions from both mesonic and monopole operators. In order to compute the Hilbert series, recently developed techniques for Coulomb branch Hilbert series in 3d N=4 are extended to 3d N=2. The Hilbert series computation leads to a general expression of the algebraic variety which represents the moduli space of the U(N_c) theory with N_f flavors and its Aharony dual theory. A detailed analysis of the moduli space is given, including an analysis of the various components of the moduli space.

13. Mass-deformed Brane Tilings

Massimo Bianchi, Stefano Cremonesi, Amihay Hanany, Jose Francisco Morales, Daniel Ricci Pacifici, Rak-Kyeong Seong
JHEP 1410:027,2014


We study renormalization group flows among N=1 SCFTs realized on the worldvolume of D3-branes probing toric Calabi-Yau singularities, thus admitting a brane tiling description. The flows are triggered by masses for adjoint or vector-like pairs of bifundamentals and are generalizations of the Klebanov-Witten construction of the N=1 theory for the conifold starting from the N=2 theory for the C^2/Z_2 orbifold. In order to preserve the toric condition pairs of masses with opposite signs have to be switched on. We offer a geometric interpretation of the flows as complex deformations of the Calabi-Yau singularity preserving the toric condition. For orbifolds, we support this interpretation by an explicit string amplitude computation of the gauge invariant mass terms generated by imaginary self-dual 3-form fluxes in the twisted sector. In agreement with the holographic a-theorem, the volume of the Sasaki-Einstein 5-base of the Calabi-Yau cone always increases along the flow.

14. Hilbert Series and Moduli Spaces of k U(N) Vortices

Amihay Hanany, Rak-Kyeong Seong
JHEP 1502:012,201vortices.jpg

We study the moduli spaces of k U(N) vortices which are realized by the Higgs branch of a U(k) supersymmetric gauge theory. The theory has 4 supercharges and lives on k D1-branes in a N D3- and NS5-brane background. We realize the vortex moduli space as a C* projection of the vortex master space. The Hilbert series is calculated in order to characterize the algebraic structure of the vortex master space and to identify the precise C* projection. As a result, we are able to fully classify the moduli spaces up to 3 vortices.

15. Hilbert Series for Moduli Spaces of Instantons on C^2/Z_n

Anindya Dey, Amihay Hanany, Noppadol Mekareeya, Diego Rodriguez-Gomez, Rak-Kyeong Seong
JHEP 1401:182,2014


We study chiral gauge-invariant operators on moduli spaces of G instantons for any classical group G on A-type ALE spaces using Hilbert Series (HS). Moduli spaces of instantons on an ALE space can be realized as Higgs branches of certain quiver gauge theories which appear as world-volume theories on Dp branes in a Dp-D(p+4) system with the D(p+4) branes (with or without O(p+4) planes) wrapping the ALE space. We study in detail a list of quiver gauge theories which are related to G-instantons of arbitrary ranks and instanton numbers on a generic A_{n-1} ALE space and discuss the corresponding brane configurations. For a large class of theories, we explicitly compute the Higgs branch HS which reveals various algebraic/geometric aspects of the moduli space such as the dimension of the space, generators of the moduli space and relations connecting them. In a large number of examples involving lower rank instantons, we demonstrate that HS for equivalent instantons of isomorphic gauge groups but very different quiver descriptions do indeed agree, as expected.

16. Double Handled Brane Tilings

Stefano Cremonesi, Amihay Hanany, Rak-Kyeong Seong
JHEP 1310:001,2013


We classify the first few brane tilings on a genus 2 Riemann surface and identify their toric Calabi-Yau moduli spaces. These brane tilings are extensions of tilings on the 2-torus, which represent one of the largest known classes of 4d N=1 superconformal field theories for D3-branes. The classification consists of 16 distinct genus 2 brane tilings with up to 8 quiver fields and 4 superpotential terms. The Higgs mechanism is used to relate the different theories.

17. New Directions in Bipartite Field Theories

Sebastian Franco, Daniele Galloni, Rak-Kyeong Seong
JHEP 1306:032,2013


We perform a detailed investigation of Bipartite Field Theories (BFTs), a general class of 4d N=1 gauge theories which are defined by bipartite graphs. This class of theories is considerably expanded by identifying a new way of assigning gauge symmetries to graphs. A new procedure is introduced in order to determine the toric Calabi-Yau moduli spaces of BFTs. For graphs on a disk, we show that the matroid polytope for the corresponding cell in the Grassmannian coincides with the toric diagram of the BFT moduli space. A systematic BFT prescription for determining graph reductions is presented. We illustrate our ideas in infinite classes of BFTs and introduce various operations for generating new theories from existing ones. Particular emphasis is given to theories associated to non-planar graphs.

18. Brane Tilings and Specular Duality

Amihay Hanany, Rak-Kyeong Seong
JHEP 1208:107,2012

We study a new duality which pairs 4d N=1 supersymmetric quiver gauge theories. They are represented by brane tilings and are worldvolume theories of D3 branes at Calabi-Yau 3-fold singularities. The new duality identifies theories which have the same combined mesonic and baryonic moduli space, otherwise called the master space. We obtain the associated Hilbert series which encodes both the generators and defining relations of the moduli space. We illustrate our findings with a set of brane tilings that have reflexive toric diagrams.

19. Supersymmetric Gauge Theories on the Five-Sphere

Kazuo Hosomichi, Rak-Kyeong Seong, Seiji Terashima



We construct Euclidean 5d supersymmetric gauge theories on the five-sphere with vector and hypermultiplets. The SUSY transformation and the action are explicitly determined from the standard Noether procedure as well as from off-shell supergravity. Using localization techniques, the path-integral is shown to be restricted to the integration over a generalization of instantons on CP^2 and the Coulomb moduli.

20. Brane Tilings and Reflexive Polygons

Amihay Hanany, Rak-Kyeong Seong



Reflexive polygons have attracted great interest both in mathematics and in physics. This paper discusses a new aspect of the existing study in the context of quiver gauge theories. These theories are 4d supersymmetric worldvolume theories of D3 branes with toric Calabi-Yau moduli spaces that are conveniently described with brane tilings. We find all 30 theories corresponding to the 16 reflexive polygons, some of the theories being toric (Seiberg) dual to each other. The mesonic generators of the moduli spaces are identified through the Hilbert series. It is shown that the lattice of generators is the dual reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver gauge theories with the lattice of generators being the toric diagram of the dual and vice versa.

21. Calabi-Yau Orbifolds and Torus Coverings

Amihay Hanany, Vishnu Jejjala, Sanjaye Ramgoolam, Rak-Kyeong Seong

JHEP 1109:116,2011


The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to the counting of orbifolds of toric Calabi-Yau singularities, with particular attention to Abelian orbifolds of C^D. By doing so, the work introduces a novel analytical method for counting Abelian orbifolds, verifying previous algorithm results. One identifies a p-fold cover of the torus T^{D-1} with an Abelian orbifold of the form C^D/Z_p, for any dimension D and a prime number p. The counting problem leads to polynomial equations modulo p for a given Abelian subgroup of S_D, the group of discrete symmetries of the toric diagram for C^D. The roots of the polynomial equations correspond to orbifolds of the form C^D/Z_p, which are invariant under the corresponding subgroup of S_Ds. In turn, invariance under this subgroup implies a discrete symmetry for the corresponding quiver gauge theory, as is clearly seen by its brane tiling formulation.

22. An Introduction to Counting Orbifolds

John Davey, Amihay Hanany, Rak-Kyeong Seong

Fortsch.Phys.59:677-682, 2011


We review three methods of counting abelian orbifolds of the form C^3/Gamma which are toric Calabi-Yau (CY). The methods include the use of 3-tuples to define the action of Gamma on C^3, the counting of triangular toric diagrams and the construction of hexagonal brane tilings. A formula for the partition function that counts these orbifolds is given. Extensions to higher dimensional orbifolds are briefly discussed.

23. Symmetries of Abelian Orbifolds

Amihay Hanany, Rak-Kyeong Seong

JHEP 1101:027,2011


Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of multiplicative sequences, one for each cycle in the Cycle Index of the permutation group. A multiplicative sequence is controlled by its values on prime numbers and their pure powers. Therefore, we pay particular attention to orbifolds of the form C^D/Gamma where the order of Gamma is p^alpha. We propose a generalization of these sequences for any D and any p.

24. Counting Orbifolds

John Davey, Amihay Hanany, Rak-Kyeong Seong

JHEP 1006:010,2010


We present several methods of counting the orbifolds C^D/Gamma. A correspondence between counting orbifold actions on C^D, brane tilings, and toric diagrams in D-1 dimensions is drawn. Barycentric coordinates and scaling mechanisms are introduced to characterize lattice simplices as toric diagrams. We count orbifolds of C^3, C^4, C^5, C^6 and C^7. Some remarks are made on closed form formulas for the partition function that counts distinct orbifold actions.


Statistical Mechanics/Condensed Matter Physics

25. Statistical Geometry and Topology of the Human Placenta

Rak-Kyeong Seong, Pascal Getreuer, Yingying Li, Theresa Girardi, Carolyn M. Salafia, Dimitri D. Vvedensky
Advances in Applied Mathematics, Modeling, and Computational Science
Fields Institute Communications Volume 66, 2013, pp 187-208

26. Statistical topology of radial networks: a case study of tree leaves

Rak-Kyeong Seong, Carolyn M. Salafia, Dimitri D. Vvedensky
Philosophical Magazine, Volume 92, Issue 1-3, 2012

27. Statistical thermodynamics and weighted topology of radial networks

Rak-Kyeong Seong, Dimitri D. Vvedensky