Showing papers on "Mirror symmetry published in 2019"

Infinite Distance Networks in Field Space and Charge Orbits

Abstract: The Swampland Distance Conjecture proposes that approaching infinite distances in field space an infinite tower of states becomes exponentially light. We study this conjecture for the complex structure moduli space of Calabi-Yau manifolds. In this context, we uncover significant structure within the proposal by showing that there is a rich spectrum of different infinite distance loci that can be classified by certain topological data derived from an associated discrete symmetry. We show how this data also determines the rules for how the different infinite distance loci can intersect and form an infinite distance network. We study the properties of the intersections in detail and, in particular, propose an identification of the infinite tower of states near such intersections in terms of what we term charge orbits. These orbits have the property that they are not completely local, but depend on data within a finite patch around the intersection, thereby forming an initial step towards understanding global aspects of the distance conjecture in field spaces. Our results follow from a deep mathematical structure captured by the so-called orbit theorems, which gives a handle on singularities in the moduli space through mixed Hodge structures, and is related to a local notion of mirror symmetry thereby allowing us to apply it also to the large volume setting. These theorems are general and apply far beyond Calabi-Yau moduli spaces, leading us to propose that similarly the infinite distance structures we uncover are also more general.

The Swampland Distance Conjecture and towers of tensionless branes

Abstract: The Swampland Distance Conjecture states that at infinite distance in the scalar moduli space an infinite tower of particles become exponentially massless. We study this issue in the context of 4d type IIA and type IIB Calabi-Yau compactifications. We find that for large moduli not only towers of particles but also domain walls and strings become tensionless. We study in detail the case of type IIA and IIB 𝒩 = 1 CY orientifolds and show how for infinite Kahler and/or complex structure moduli towers of domain walls and strings become tensionless, depending on the particular direction in moduli space. For the type IIA case we construct the monodromy orbits of domain walls in detail. We study the structure of mass scales in these limits and find that these towers may occur at the same scale as the fundamental string scale or the KK scale making sometimes difficult an effective field theory description. The structure of IIA and IIB towers are consistent with mirror symmetry, as long as towers of exotic domain walls associated to non-geometric fluxes also appear. We briefly discuss the issue of emergence within this context and the possible implications for 4d vacua.

Observation of Topological Photocurrents in the Chiral Weyl Semimetal RhSi

Abstract: Weyl semimetals are crystals in which electron bands cross at isolated points in momentum space. Associated with each crossing point (or Weyl node) is an integer topological invariant known as the Berry monopole charge. The discovery of new classes of Weyl materials is driving the search for novel properties that derive directly from the Berry charge. The circular photogalvanic effect (CPGE), whereby circular polarized light generates a current whose direction depends on the helicity of the absorbed photons, is a striking example of a macroscopic property that emerges from Weyl topology. Recently, it was predicted that the rate of current generation associated with optical transitions near a Weyl node is proportional to its monopole charge and independent of material-specific parameters. In Weyl semimetals that retain mirror symmetry this universal photogalvanic current is strongly suppressed by opposing contributions from energy equivalent nodes of opposite charge. However, when all mirror symmetries are broken, as in chiral Weyl systems, nodes with opposite topological charge are no longer degenerate, opening a window of photon energies where the topological CPGE can emerge. In this work we test this theory through measurement of the photon-energy dependence of the CPGE in the chiral Weyl semimetal RhSi. The spectrum is fully consistent with a topological CPGE, as it reveals a response in a low-energy window that closes at 0.65 eV, in quantitative agreement with the theoretically-derived bandstucture.

Modular fluxes, elliptic genera, and weak gravity conjectures in four dimensions

Abstract: We analyse the Weak Gravity Conjecture for chiral four-dimensional F-theory compactifications with N = 1 supersymmetry. Extending our previous work on nearly tensionless heterotic strings in six dimensions, we show that under certain assumptions a tower of asymptotically massless states arises in the limit of vanishing coupling of a U(1) gauge symmetry coupled to gravity. This tower contains super-extremal states whose charge-to-mass ratios are larger than those of certain extremal dilatonic Reissner-Nordstrom black holes, precisely as required by the Weak Gravity Conjecture. Unlike in six dimensions, the tower of super-extremal states does not always populate a charge sub-lattice. The main tool for our analysis is the elliptic genus of the emergent heterotic string in the chiral N = 1 supersymmetric effective theories. This also governs situations where the heterotic string is non-perturbative. We show how it can be computed in terms of BPS invariants on elliptic four-folds, by making use of various dualities and mirror symmetry. Compared to six dimensions, the geometry of the relevant elliptically fibered four-folds is substantially richer than that of the three-folds, and we classify the possibilities for obtaining critical, nearly tensionless heterotic strings. We find that the (quasi-)modular properties of the elliptic genus crucially depend on the choice of flux background. Our general results are illustrated in a detailed example.

T[SU( N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences

Abstract: We study various duality webs involving the 3d FT[SU(N)] theory, a close relative of the T[SU(N)] quiver tail. We first map the partition functions of FT[SU(N)] and its 3d spectral dual to a pair of spectral dual q-Toda conformal blocks. Then we show how to obtain the FT[SU(N)] partition function by Higgsing a 5d linear quiver gauge theory, or equivalently from the refined topological string partition function on a certain toric Calabi-Yau three-fold. 3d spectral duality in this context descends from 5d spectral duality. Finally we discuss the 2d reduction of the 3d spectral dual pair and study the corresponding limits on the q-Toda side. In particular we obtain a new direct map between the partition function of the 2d FT[SU(N)] GLSM and an (N + 2)-point Toda conformal block.

Flipping the head of T [SU( N)]: mirror symmetry, spectral duality and monopoles

Abstract: We consider T [SU(N)] and its mirror, and we argue that there are two more dual frames, which are obtained by adding flipping fields for the moment map on the Higgs and Coulomb branch. Turning on a monopole deformation in T [SU(N)], and following its effect on each dual frame, we obtain four new daughter theories dual to each other. We are then able to construct pairs of 3d spectral dual theories by performing simple operations on the four dual frames of T [SU(N)]. Engineering these 3d spectral pairs as codimension-two defect theories coupled to a trivial 5d theory, via Higgsing, we show that our 3d spectral dual theories descend from spectral duality in 5d, or fiber base duality in topological string. We provide further consistency checks about our web of dualities by matching partition functions on the squashed sphere, and in the case of spectral duality, matching exactly topological string computations with holomorphic blocks.

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Abstract: In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian X=Grn−k(Cn), as well as the mirror dual Landau–Ginzburg model (Xˇ∘,W:Xˇ∘→C), where Xˇ∘ is the complement of a particular anticanonical divisor in a Langlands dual Grassmannian Xˇ=Grk((Cn)∗) and the superpotential W has a simple expression in terms of Plucker coordinates. Grassmannians simultaneously have the structure of an A-cluster variety and an X-cluster variety; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network or X-cluster chart ΦG:(C∗)k(n−k)→X∘ and a Plucker cluster or A-cluster chart ΦG∨:(C∗)k(n−k)→Xˇ∘. Here X∘ and Xˇ∘ are the open positroid varieties in X and Xˇ, respectively. To each X-cluster chart ΦG and ample boundary divisor D in X∖X∘, we associate a Newton–Okounkov body ΔG(D) in Rk(n−k), which is defined as the convex hull of rational points; these points are obtained from the multidegrees of leading terms of the Laurent polynomials ΦG∗(f) for f on X with poles bounded by some multiple of D. On the other hand, using the A-cluster chart ΦG∨ on the mirror side, we obtain a set of rational polytopes—described in terms of inequalities—by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates and then tropicalizing. Our first main result is that the Newton–Okounkov bodies ΔG(D) and the polytopes obtained by tropicalization on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of ) these Newton–Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton–Okounkov bodies, in the case in which the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians.

On the remodeling conjecture for toric Calabi-Yau 3-orbifolds

Abstract: The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) [arXiv:0709.1453, arXiv:0807.0597] relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera.

The non-archimedean SYZ fibration

Abstract: We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.

Mirror skin effect and its electric circuit simulation

Abstract: We analyze impacts of crystalline symmetry on the non-Hermitian skin effects. Focusing on mirror symmetry, we propose a novel type of skin effects, a mirror skin effect, which results in significant dependence of energy spectrum on the boundary condition only for the mirror invariant line in the two-dimensional Brillouin zone. This effect arises from the topological properties characterized by a mirror winding number. We further reveal that the mirror skin effect can be observed for an electric circuit composed of negative impedance converters with current inversion where switching the boundary condition significantly changes the admittance eigenvalues only along the mirror invariant lines. Furthermore, we demonstrate that extensive localization of the eigenstates for each mirror sector result in an anomalous voltage response.

Feynman integrals, toric geometry and mirror symmetry

Abstract: This expository text is about using toric geometry and mirror symmetry for evaluating Feynman integrals We show that the maximal cut of a Feynman integral is a GKZ hypergeometric series We explain how this allows to determine the minimal differential operator acting on the Feynman integrals We illustrate the method on sunset integrals in two dimensions at various loop orders The graph polynomials of the multi-loop sunset Feynman graphs lead to reflexive polytopes containing the origin and the associated variety are ambient spaces for Calabi-Yau hypersurfaces Therefore the sunset family is a natural home for mirror symmetry techniques We review the evaluation of the two-loop sunset integral as an elliptic dilogarithm and as a trilogarithm The equivalence between these two expressions is a consequence of (1) the local mirror symmetry for the non-compact Calabi-Yau three-fold obtained as the anti-canonical hypersurface of the del Pezzo surface of degree 6 defined by the sunset graph polynomial and (2) that the sunset Feynman integral is expressed in terms of the local Gromov-Witten prepotential of this del Pezzo surface

Intrinsic Mirror Symmetry

Abstract: We associate a ring R to a log Calabi-Yau pair (X,D) or a degeneration of Calabi-Yau manifolds X->B. The vector space underlying R is determined by the tropicalization of (X,D) or X->B, while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D is maximally degenerate, then we propose that Spec R is the mirror to X\D, while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj R. The main result in this paper is that R as defined is an associative, commutative ring with unit, with associativity the most difficult part.

Spin-Polarization Control Driven by a Rashba-Type Effect Breaking the Mirror Symmetry in Two-Dimensional Dual Topological Insulators.

Abstract: Three-dimensional topological insulators protected by both the time reversal (TR) and mirror symmetries were recently predicted and observed. Two-dimensional materials featuring this property and their potential for device applications have been less explored. We find that, in these systems, the spin polarization of edge states can be controlled with an external electric field breaking the mirror symmetry. This symmetry requires that the spin polarization is perpendicular to the mirror plane; therefore, the electric field induces spin-polarization components parallel to the mirror plane. Since this field preserves the TR topological protection, we propose a transistor model using the spin direction of protected edge states as a switch. In order to illustrate the generality of the proposed phenomena, we consider compounds protected by mirror planes parallel and perpendicular to the structure, e.g., ${\mathrm{Na}}_{3}\mathrm{Bi}$ and half-functionalized (HF) hexagonal compounds, respectively. For this purpose, we first construct a tight-binding effective model for the ${\mathrm{Na}}_{3}\mathrm{Bi}$ compound and predict that HF-honeycomb lattice materials are also dual topological insulators.

Extended Bloch theorem for topological lattice models with open boundaries

Abstract: While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and ${\mathcal{Z}}_{2}$ insulators with open boundaries of codimension one. In the case of the two-dimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states.

Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations

Abstract: We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author is trivial. We also show that these families are the completion of an analytic family, without reparametrization, and that they are formally versal as deformations of logarithmic schemes. Other applications include canonical one-parameter type III degenerations of K3 surfaces with prescribed Picard groups. As a technical result of independent interest we develop a theory of period integrals with logarithmic poles on finite order deformations of normal crossing analytic spaces.

Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves

Abstract: We introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably these structures include a tropical quantum field theory and an $L_{\infty}$-structure. The latter is an instance of Getzler's gravity algebra, and the $l_2$-bracket is a restriction of the Schouten-Nijenhuis bracket. We explain the relationship to string topology in the appendix (thanks to Janko Latschev).

Valley Hall phases in kagome lattices

Abstract: We report the finding of the analogous valley Hall effect in phononic systems arising from mirror symmetry breaking, in addition to spatial inversion symmetry breaking. We study topological phases of plates and spring-mass models in kagome and modified kagome arrangements. By breaking the inversion symmetry it is well known that a defined valley Chern number arises. We also show that effectively, breaking the mirror symmetry leads to the same topological invariant. Based on the bulk-edge correspondence principle, protected edge states appear at interfaces between two lattices with different valley Chern numbers. By means of a plane wave expansion method and the multiple scattering theory for periodic and finite systems, respectively, we computed the Berry curvature, the band inversion, mode shapes, and edge modes in plate systems. We also find that appropriate multipoint excitations in finite system gives rise to propagating waves along a one-sided path only.

From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals

Abstract: We study a surprising phenomenon in which Feynman integrals in $D=4-2\varepsilon$ space-time dimensions as $\varepsilon \to 0$ can be fully characterized by their behavior in the opposite limit, $\varepsilon \to \infty$. More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on $\varepsilon$ and are known to be governed by intersection numbers of twisted forms. They give rise to differential equations that can be obtained exactly as a truncating expansion in either $\varepsilon$ or $1/\varepsilon$. We use the latter for explicit computations, which are performed by expanding intersection numbers in terms of Saito's higher residue pairings (previously used in the context of topological Landau-Ginzburg models and mirror symmetry). These pairings localize on critical points of a certain Morse function, which correspond to regions in the loop-momentum space that were previously thought to govern only the large-$D$ physics. The results of this work leverage recent understanding of an analogous situation for moduli spaces of curves, where the $\alpha' \to 0$ and $\alpha' \to \infty$ limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

Mirror symmetry of 3D N = 4 gauge theories and supersymmetric indices

Abstract: We compute supersymmetric indices to test mirror symmetry of three-dimensional (3D) $\mathcal{N}=4$ gauge theories and dualities of half-Bogomol'ny Prasad Sommerfield enriched boundary conditions and interfaces in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory. We find the matching of indices as strong evidences for various dualities of the 3D interfaces conjectured by Gaiotto and Witten under the action of S-duality in Type IIB string theory.

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Abstract: We show that the naive counts of rational curves in any affine log Calabi-Yau variety $U$, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a compatible multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of $U$. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when $U$ is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich.

Hourglass Fermion in Two-Dimensional Material.

Abstract: The hourglass fermion, as an exotic quasiparticle protected by nonsymmorphic symmetry, has excited great research interest recently. However, its bulk counterpart in two-dimensional (2D) solid-state materials has seldom been studied. In this Letter, we propose a 2D rectangular lattice made of p_{x} and p_{y} orbitals with glide mirror symmetry but without inversion symmetry to realize the hourglass fermion. The glide mirror symmetry guarantees a Dirac nodal line, while the Rashba spin-orbital coupling splits it into two Weyl nodal lines and generates two pairs of hourglass fermion located at the glide mirror plane. Furthermore, based on first principles calculations, we predict a surface-supported 2D material Bi/Cl-SiC(111) to realize our proposal, making a huge-bandwidth hourglass cone. Moreover, the hourglass fermion exhibits a spin-momentum locking spin texture and also sustains a giant spin Hall conductivity. Our results demonstrate a general routine for designing an hourglass fermion in 2D materials, which will be easily extended to other surfaces with different adatoms and lattice symmetries.

Variations on S -fold CFTs

Abstract: A local SL(2, ℤ) transformation on the Type IIB brane configuration gives rise to an interesting class of superconformal field theories, known as the S-fold CFTs. Previously it has been proposed that the corresponding quiver theory has a link involving the T(U(N)) theory. In this paper, we generalise the preceding result by studying quivers that contain a T(G) link, where G is self-dual under S-duality. In particular, the cases of G = SO(2N), USp′(2N) and G2 are examined in detail. We propose the theories that arise from an appropriate insertion of an S-fold into a brane system, in the presence of an orientifold threeplane or an orientifold fiveplane. By analysing the moduli spaces, we test such a proposal against its S-dual configuration using mirror symmetry. The case of G2 corresponds to a novel class of quivers, whose brane construction is not available. We present several mirror pairs, containing G2 gauge groups, that have not been discussed before in the literature.

Twisted indices of 3d N = 4 gauge theories and enumerative geometry of quasi-maps

Abstract: We explore the geometric interpretation of the twisted index of 3d $$ \mathcal{N} $$ = 4 gauge theories on S1 × Σ where Σ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of solutions to generalised vortex equations on Σ, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d $$ \mathcal{N} $$ = 4 mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters.

Mirror-symmetry induced topological valley transport along programmable boundaries in a hexagonal sonic crystal.

Abstract: Valley states, labeling the frequency extrema in momentum space, carry a new degree of freedom (valley pseudospin) for topological transport of sound in sonic crystals. Recently, the field of valley acoustics has become a hotspot due to its potentials in developing various topological-insulator-based devices. In most previous works, topological valley transport is implemented at the interfaces of two connected artificial crystals. With respect to the interface, the mirror symmetry of crystal structures supports either even-mode or odd-mode valley states. In this work, we propose a physical insight of transforming one hexagonal crystal into a virtual lattice by utilizing the mirror operation of rigid or soft boundaries, which greatly reduces the dimension of the acoustic structure and provides a possible way to implement the programmable routing of topological propagation. We investigate two cases that the rigid and soft boundaries are introduced either at the edge or inside a single hexagonal crystal. Our results clearly demonstrate the high-transmission valley transport along the folded boundaries, where reflection or scattering is prohibited at the sharp bending or corners due to topological protection. Three functional devices are exemplified, which are single-crystal-based topological delay-line filter, delay-line switcher and beam splitter. Our work reveals the inherent relation between the field symmetries of valley states and structural symmetries of sonic crystals. Programmable routing of topological sound transport through boundary engineering provides a platform for developing integrated and versatile topological-insulator-based devices.

Mirror symmetry and line operators

Abstract: We study half-BPS line operators in 3d N=4 gauge theories, focusing in particular on the algebras of local operators at their junctions. It is known that there are two basic types of such line operators, distinguished by the SUSY subalgebras that they preserve; the two types can roughly be called "Wilson lines" and "vortex lines", and are exchanged under 3d mirror symmetry. We describe a large class of vortex lines that can be characterized by basic algebraic data, and propose a mathematical scheme to compute the algebras of local operators at their junctions --- including monopole operators --- in terms of this data. The computation generalizes mathematical and physical definitions/analyses of the bulk Coulomb-branch chiral ring. We fully classify the junctions of half-BPS Wilson lines and of half-BPS vortex lines in abelian gauge theories with sufficient matter. We also test our computational scheme in a non-abelian quiver gauge theory, using a 3d-mirror-map of line operators from work of Assel and Gomis.

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.