Showing papers on "Mirror symmetry published in 2021"

Coulomb and Higgs branches from canonical singularities. Part 0

Abstract: Five- and four-dimensional superconformal field theories with eight supercharges arise from canonical threefold singularities in M-theory and Type IIB string theory, respectively. We study their Coulomb and Higgs branches using crepant resolutions and deformations of the singularities. We propose a relation between the resulting moduli spaces, by compactifying the theories to 3d, followed by 3d $$ \mathcal{N} $$ = 4 mirror symmetry and an S-type gauging of an abelian flavor symmetry. In particular, we use this correspondence to determine the Higgs branch of some 5d SCFTs and their magnetic quivers from the geometry. As an application of the general framework, we observe that singularities that engineer Argyres-Douglas theories in Type IIB also give rise to rank-0 5d SCFTs in M-theory. We also compute the higher-form symmetries of the 4d and 5d SCFTs, including the one-form symmetries of generalized Argyres-Douglas theories of type (G, G′).

Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning

Abstract: We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.

Multi-dimensional wave steering with higher-order topological phononic crystal

Abstract: The recent discovery and realizations of higher-order topological insulators enrich the fundamental studies on topological phases. Here, we report three-dimensional (3D) wave-steering capabilities enabled by topological boundary states at three different orders in a 3D phononic crystal with nontrivial bulk topology originated from the synergy of mirror symmetry of the unit cell and a non-symmorphic glide symmetry of the lattice. The multitude of topological states brings diverse possibilities of wave manipulations. Through judicious engineering of the boundary modes, we experimentally demonstrate two functionalities at different dimensions: 2D negative refraction of sound wave enabled by a first-order topological surface state with negative dispersion, and a 3D acoustic interferometer leveraging on second-order topological hinge states. Our work showcases that topological modes at different orders promise diverse wave steering applications across different dimensions.

D-instantons in Type IIA string theory on Calabi-Yau threefolds

Abstract: Type IIA string theory compactified on a Calabi-Yau threefold has a hypermultiplet moduli space whose metric is known to receive non-perturbative corrections from Euclidean D2-branes wrapped on 3-cycles. These corrections have been computed earlier by making use of mirror symmetry, S-duality and twistorial description of quaternionic geometries. In this paper we compute the leading corrections in each homology class using a direct world-sheet approach without relying on any duality symmetry or supersymmetry. Our results are in perfect agreement with the earlier predictions.

Special Lagrangian submanifolds of log Calabi-Yau manifolds

Abstract: We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kahler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|−KY| is a smooth divisor with D2=d, then X=Y∖D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y= P2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kahler rotation, X can be compactified to the complement of a Kodaira type Id fiber appearing as a singular fiber in a rational elliptic surface πˇ:Yˇ→P1.

Three Dimensional Mirror Symmetry beyond $ADE$ quivers and Argyres-Douglas theories

Abstract: Mirror symmetry, a three dimensional $$ \mathcal{N} $$ = 4 IR duality, has been studied in detail for quiver gauge theories of the ADE-type (as well as their affine versions) with unitary gauge groups. The A-type quivers (also known as linear quivers) and the associated mirror dualities have a particularly simple realization in terms of a Type IIB system of D3-D5-NS5-branes. In this paper, we present a systematic field theory prescription for constructing 3d mirror pairs beyond the ADE quiver gauge theories, starting from a dual pair of A-type quivers with unitary gauge groups. The construction involves a certain generalization of the S and the T operations, which arise in the context of the SL(2, ℤ) action on a 3d CFT with a U(1) 0-form global symmetry. We implement this construction in terms of two supersymmetric observables — the round sphere partition function and the superconformal index on S2 × S1. We discuss explicit examples of various (non-ADE) infinite families of mirror pairs that can be obtained in this fashion. In addition, we use the above construction to conjecture explicit 3d $$ \mathcal{N} $$ = 4 Lagrangians for 3d SCFTs, which arise in the deep IR limit of certain Argyres-Douglas theories compactified on a circle.

Theta bases and log Gromov-Witten invariants of cluster varieties

Abstract: Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse.

Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics

Abstract: In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas’ GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $${\mathcal {H}}$$ closely related to Solomon’s space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $$C^{1,\alpha }$$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen’s theorem on the existence of $$C^{1,\alpha }$$ geodesics in the space of Kahler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM [26] and special Lagrangians in Landau–Ginzburg models [27].

Connecting 5d Higgs Branches via Fayet-Iliopoulos Deformations

Abstract: We describe how the geometry of the Higgs branch of 5d superconformal field theories is transformed under movement along the extended Coulomb branch. Working directly with the (unitary) magnetic quiver, we demonstrate a correspondence between Fayet-Iliopoulos deformations in 3d and 5d mass deformations. When the Higgs branch has multiple cones, characterised by a collection of magnetic quivers, the mirror map is not globally well-defined, however we are able to utilize the correspondence to establish a local version of mirror symmetry. We give several detailed examples of deformations, including decouplings and weak-coupling limits, in $(D_n,D_n)$ conformal matter theories, $T_N$ theory and its parent $P_N$, for which we find new Lagrangian descriptions given by quiver gauge theories with fundamental and anti-symmetric matter.

Completing the eclectic flavor scheme of the $\boldsymbol{\mathbb Z_2}$ orbifold

Abstract: We present a detailed analysis of the eclectic flavor structure of the two-dimensional $\mathbb Z_2$ orbifold with its two unconstrained moduli $T$ and $U$ as well as $\mathrm{SL}(2,\mathbb Z)_T\times \mathrm{SL}(2,\mathbb Z)_U$ modular symmetry. This provides a thorough understanding of mirror symmetry as well as the $R$-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the $(T,U)$ modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, CP and $R$-symmetries on the superpotential and Kaehler potential of the theory.

Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold

Abstract: We use numerical methods to obtain moduli-dependent Calabi-Yau metrics, and from them, the moduli-dependent massive tower of Kaluza-Klein states for the one-parameter family of quintic Calabi-Yau manifolds. We then compute geodesic distances in their K\"ahler and complex structure moduli space using exact expressions from mirror symmetry, approximate expressions, and numerical methods, and we compare the results. Finally, we fit the moduli dependence of the massive spectrum to the geodesic distance to obtain the rate at which states become exponentially light. The result is indeed of order 1, as suggested by the swampland distance conjecture. We also observe level crossing in the eigenvalue spectrum and find that states in small irreducible representations of the symmetry group tend to become lighter than states in larger irreducible representations.

Completing the eclectic flavor scheme of the ℤ2 orbifold

Abstract: We present a detailed analysis of the eclectic flavor structure of the two-dimensional ℤ2 orbifold with its two unconstrained moduli T and U as well as SL(2, ℤ)T × SL(2, ℤ)U modular symmetry. This provides a thorough understanding of mirror symmetry as well as the R-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the (T, U) modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, $$ \mathcal{CP} $$ and R-symmetries on the superpotential and Kahler potential of the theory.

3d N=4 Bootstrap and Mirror Symmetry

Abstract: We investigate the non-BPS realm of 3d ${\cal N} = 4$ superconformal field theory by uniting the non-perturbative methods of the conformal bootstrap and supersymmetric localization, and utilizing special features of 3d ${\cal N} = 4$ theories such as mirror symmetry and a protected sector described by topological quantum mechanics (TQM). Supersymmetric localization allows for the exact determination of the conformal and flavor central charges, and the latter can be fed into the mini-bootstrap of the TQM to solve for a subset of the OPE data. We examine the implications of the $\mathbb{Z}_2$ mirror action for the SCFT single- and mixed-branch crossing equations for the moment map operators, and apply numerical bootstrap to obtain universal constraints on OPE data for given flavor symmetry groups. A key ingredient in applying the bootstrap analysis is the determination of the mixed-branch superconformal blocks. Among other results, we show that the simplest known self-mirror theory with $SU(2) \times SU(2)$ flavor symmetry saturates our bootstrap bounds, which allows us to extract the non-BPS data and examine the self-mirror $\mathbb{Z}_2$ symmetry thereof.

Euclidean D-branes in Type IIB string theory on Calabi-Yau threefolds.

Abstract: We compute the contribution of Euclidean D-branes in type IIB string theory on Calabi-Yau threefolds to the metric on the hypermultiplet moduli space in the large volume, weak coupling limit. Our results are in perfect agreement with the predictions based on S-duality, mirror symmetry and supersymmetry.

Magnetic impurities along the edge of a quantum spin Hall insulator: Realizing a one-dimensional AIII insulator

Abstract: In this paper we construct a one-dimensional insulator with an approximate chiral symmetry belonging to the AIII class and discuss its properties. The construction principle is the intentional pollution of the edge of a two-dimensional quantum spin Hall insulator with magnetic impurities. The resulting bound states hybridize and disperse along the edge. We discuss under which circumstances this chain possesses zero-dimensional boundary modes on the level of an effective low-energy theory. The main appeal of our construction is the independence on details of the impurity lattice: the zero modes are stable against disorder and random lattice configurations. We also show that in the presence of Rashba coupling, which changes the symmetry class to A, one can still expect localized half-integer boundary excess charges protected by mirror symmetry although there is no nontrivial topological index. All of the results are confirmed numerically in a microscopic model.

Three-Dimensional Mirror Symmetry and Elliptic Stable Envelopes

Abstract: We consider a pair of quiver varieties $(X;X^{\\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $X \\times X^{\\prime} $ (the mother function) whose restrictions to $X$ and $X^{\\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond.

Abstract: Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated to anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary condition with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of $2\pi$. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

Signatures of dephasing by mirror-symmetry breaking in weak-antilocalization magnetoresistance across the topological transition in Pb 1 − x Sn x Se

Abstract: Many conductors, including recently studied Dirac materials, show saturation of coherence length on decreasing temperature. This surprising phenomenon is assigned to external noise, residual magnetic impurities, or two-level systems specific to noncrystalline solids. Here, by considering the SnTe-class of compounds as an example, we show theoretically that breaking of mirror symmetry deteriorates Berry's phase quantization, leading to additional dephasing in weak-antilocalization magnetoresistance (WAL-MR). Our experimental studies of WAL-MR corroborate these theoretical expectations in (111) Pb1-xSnxSe thin film with Sn contents x corresponding to both topological crystalline insulator and topologically trivial phases. In particular, we find the shortening of the phase coherence length in samples with intentionally broken mirror symmetry. Our results indicate that the classification of quantum transport phenomena into universality classes should encompass, in addition to time-reversal and spin-rotation invariances, spatial symmetries in specific systems.

Mirror symmetry for K3 surfaces

Abstract: For certain K3 surfaces, there are two constructions of mirror symmetry that appear very different. The first, known as BHK mirror symmetry, comes from the Landau–Ginzburg model for the K3 surface; the other, known as LPK3 mirror symmetry, is based on a lattice polarization of the K3 surface in the sense of Dolgachev’s definition. There is a large class of K3 surfaces for which both versions of mirror symmetry apply. In this class we consider the K3 surfaces admitting a certain purely non-symplectic automorphism of order 4, 8, or 12, and we complete the proof that these two formulations of mirror symmetry agree for this class of K3 surfaces.

3d Mirror Symmetry for Instanton Moduli Spaces

Abstract: We prove that the Hilbert scheme of $k$ points on $\mathbb{C}^2$ (Hilb$^k[\mathbb{C}^2]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kahler and equivariant parameters as well as inverting the weight of the $\mathbb{C}^\times_\hbar$-action. First, we find a two-parameter family $X_{k,l}$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb$^k[\mathbb{C}^2]$ is obtained via direct limit $l\to\infty$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted $\hbar$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$N$ sheaves on $\mathbb{P}^2$ with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

An investigation of the symmetrical properties in the invariant imbedding T-matrix method for the nonspherical particles with symmetrical geometry

Abstract: The Invariant Imbedding (IIM) T-matrix method is recognized as one of the most promising scattering models since it can perform the scattering simulation of the nonspherical particles in a semi-analytical way. However, because the T-matrix should be updated in each iterative process, its computational efficiency is an important issue in the actual scattering simulation. To alleviate this problem, the symmetrical properties for the nonspherical particles with symmetrical geometries are systematically investigated in this paper. Firstly, the symmetry of the U-matrix (an important matrix in the IIM T-matrix model) is derived for the particles with mirror symmetry with respect to the coordinate planes. In this case, the U-matrix is firstly decomposed into the sine and cosine components, and then its symmetrical properties are obtained by combining the spatial symmetry of the permittivity and the symmetry of the angular functions. In the second part, the symmetry of the U-matrix is derived for the particles with N-folds symmetrical geometry, and based on the symmetrical properties, the method to simplify the T-matrix iteration is further proposed. In this case, it can be found that by using the symmetry of the U-matrix, both the U-matrix and T-matrix can be rearranged into the block diagonal ones, and the calculation of the T-matrix can be decomposed into the iteration of several block sub-matrices, which can cut down the computational amount and memory consumption notably. Also, it can be seen that the derivation process also provides another point of view to understand the symmetry of T-matrix for the particles with N-folds symmetrical geometry.

Maximally Mutable Laurent Polynomials

Abstract: We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), that we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del~Pezzo surfaces. Furthermore we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anticanonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.

Line Defects in Three Dimensional Mirror Symmetry beyond Linear Quivers

Abstract: The map of half-BPS line defects under mirror symmetry has previously been worked out for 3d $\mathcal{N}=4$ linear quivers with unitary gauge groups, where these defects have a clear realization in terms of a brane picture in Type IIB String Theory. In this work, we initiate the study of line defects and the associated mirror maps for more general 3d $\mathcal{N}=4$ quiver gauge theories from a QFT approach, using the $S$-type operations introduced in \cite{Dey:2020hfe}. In particular, our construction does not rely on any String Theory realization of the quiver gauge theories and the defects. After discussing the general framework for the construction of these line defects and their mirror maps, we focus on quiver gauge theories of the $D$-type and the affine $D$-type with unitary gauge groups, as a concrete set of examples. Some of the line defects we study admit a Hanany-Witten description and we show that the associated mirror maps predicted by the Type IIB construction in these cases agree with the QFT computation. In addition, we study an example involving defects in an affine $D$-type theory, for which the dual theory is not directly realized by the Type IIB description. In a companion paper, we will discuss defects in infinite families of non-ADE quivers using the general construction developed in this paper.

Abelian mirror symmetry of $$ \mathcal{N} $$ = (2, 2) boundary conditions

Abstract: We evaluate half-indices of $$ \mathcal{N} $$ = (2, 2) half-BPS boundary conditions in 3d $$ \mathcal{N} $$ = 4 supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions.