Discrete Torsion Research Articles

Non-invertible symmetries and higher representation theory II

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra \mathfrak{so}(N)𝔰𝔬(N), and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed ’t Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.

On the rationality and the code structure of a Narain CFT, and the simple current orbifold

In this paper, we discuss the simple current orbifold of a rational Narain CFT (Narain RCFT). This is a method of constructing other rational CFTs from a given rational CFT, by ‘orbifolding’ the global symmetry formed by a particular primary fields (called the simple current). Our main result is that a Narain RCFT satisfying certain conditions can be described in the form of a simple current orbifold of another Narain RCFT, and we have shown how the discrete torsion in taking that orbifold is obtained. Additionally, the partition function can be considered a simple current orbifold with discrete torsion, which is determined by the lattice and the B-field. We establish that the partition function can be expressed as a polynomial, with the variables substituted by certain q-series. In a specific scenario, this polynomial corresponds to the weight enumerator polynomial of an error-correcting code. Using this correspondence to the code theory, we can relate the B-field, the discrete torsion, and the B-form to each other.

Matrix factorizations, Reality and Knörrer periodicity

AbstractMotivated by periodicity theorems for Real ‐theory and Grothendieck–Witt theory and, separately, work of Hori and Walcher on the physics of Landau–Ginzburg orientifolds, we introduce and study categories of Real matrix factorizations. Our main results are generalizations of Knörrer periodicity to categories of Real matrix factorizations. These generalizations are structurally similar to (1,1)‐periodicity for ‐theory and 4‐periodicity for Grothendieck–Witt theory. We use techniques from Real categorical representation theory, which allow us to incorporate into our main results equivariance for a finite group and discrete torsion twists.

Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions

We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between these non-invertible topological defects and the one-form symmetry surface defects are determined. Interestingly, the fusion coefficients are generally not numbers, but 2+1d TQFTs, such as invertible SPT phases, $${\mathbb {Z}}_N$$ gauge theories, and $$U(1)_N$$ Chern-Simons theories. The associativity of these algebras over TQFT coefficients relies on nontrivial facts about 2+1d TQFTs. We further prove that some of these non-invertible symmetries are intrinsically incompatible with a trivially gapped phase, leading to nontrivial constraints on renormalization group flows. Duality and triality defects are realized in many familiar gauge theories, including free Maxwell theory, non-abelian gauge theories with orthogonal gauge groups, $${{{\mathcal {N}}}}=1,$$ and $${{{\mathcal {N}}}}=4$$ super Yang-Mills theories.

The fate of discrete torsion on resolved heterotic [formula omitted] orbifolds using (0,2) GLSMs

This paper aims to shed light on what becomes of discrete torsion within heterotic orbifolds when they are resolved to smooth geometries. Gauged Linear Sigma Models (GLSMs) possessing (0,2) worldsheet supersymmetry are employed as interpolations between them. This question is addressed for resolutions of the non–compact C3/Z2×Z2 and the compact T6/Z2×Z2 orbifolds to keep track of local and global aspects. The GLSMs associated with the non–compact orbifold with or without torsion are to a large extent equivalent: only when expressed in the same superfield basis, a field redefinition anomaly arises among them, which in the orbifold limit reproduces the discrete torsion phases. Previously unknown, novel resolution GLSMs for T6/Z2×Z2 are constructed. The GLSM associated with the torsional compact orbifold suffers from mixed gauge anomalies, which need to be cancelled by appropriate logarithmic superfield dependent FI–terms on the worldsheet, signalling H–flux due to NS5–branes supported at the exceptional cycles.

Decomposition in Chern–Simons theories in three dimensions

In this paper, we discuss decomposition in the context of three-dimensional Chern–Simons theories. Specifically, we argue that a Chern–Simons theory with a gauged noneffectively-acting one-form symmetry is equivalent to a disjoint union of Chern–Simons theories, with discrete theta angles coupling to the image under a Bockstein homomorphism of a canonical degree-two characteristic class. On three-manifolds with boundary, we show that the bulk discrete theta angles (coupling to bundle characteristic classes) are mapped to choices of discrete torsion in boundary orbifolds. We use this to verify that the bulk three-dimensional Chern–Simons decomposition reduces on the boundary to known decompositions of two-dimensional (WZW) orbifolds, providing a strong consistency test of our proposal.

Vibro-Impact Motions of a Three-Degree-of-Freedom Geartrain Subjected to Torque Fluctuations: Model and Experiments

Abstract In this study, a multimesh gear system subjected to torque fluctuations is employed as an example to study vibro-impacts of multidegree-of-freedom systems having multiple clearances. Such rotational systems are common in various automotive geared drivetrains where external torque fluctuations lead to contact loss at gear mesh interfaces to result in sequences of impacts. The specific configuration considered here is a three-axis, two-gear mesh drivetrain that is commonly used in engine timing gear systems, known for its vibro-impacts resulting in rattling noise. On the theoretical side, a discrete torsion model is developed and solved using a piecewise-linear solution method. Its predictions are compared to measurements from a three-axis geartrain to demonstrate its accuracy. The validated model is employed to characterize the sensitivity of vibro-impact motions and associated nonlinear behavior to key excitation parameters for two kinematic configurations. For the idler configuration where the middle gear is not subject to any external disturbance, double-sided impacts of one gear mesh were shown to induce separation on the other gear mesh such that vibro-impacts are localized in a single mesh. For the torque-split configuration, the ratio of the torques carried by the outputs was identified as a major parameter defining regions and types of rattle motions.

Gauge enhanced quantum criticality beyond the standard model

Standard lore ritualizes our quantum vacuum in the four-dimensional spacetime (4D) governed by one of the candidate Standard Models (SMs), while lifting towards some grand unification-like structure (GUT) at higher energy scales. In contrast, in our work, we introduce an alternative view that the SM is a low energy quantum vacuum arising from various neighbor vacua competition in an immense quantum phase diagram. In general, we can regard the SM arising near the gapless quantum criticality (either critical points or critical regions) between the competing neighbor vacua. In particular detail, we demonstrate how the $su(3)\ifmmode\times\else\texttimes\fi{}su(2)\ifmmode\times\else\texttimes\fi{}u(1)$ SM with 16n Weyl fermions arises near the quantum criticality between the GUT competition of Georgi-Glashow $su(5)$ and Pati-Salam $su(4)\ifmmode\times\else\texttimes\fi{}su(2)\ifmmode\times\else\texttimes\fi{}su(2)$. We propose two enveloping toy models. Model I is the conventional $so(10)$ GUT with a Spin(10) gauge group plus GUT-Higgs potential inducing various Higgs transitions. Model II modifies model I by adding a new 4D discrete torsion class of Wess-Zumino-Witten-like term built from GUT-Higgs field [which matches a nonperturbative global mixed gauge-gravity anomaly captured by a 5D invertible topological field theory ${w}_{2}{w}_{3}(TM)={w}_{2}{w}_{3}({V}_{\mathrm{SO}(10)})$], which manifests a beyond-Landau-Ginzburg quantum criticality between Georgi-Glashow and Pati-Salam models, with extra beyond the Standard Model excitations emerging near a hypothetical quantum critical region. If the internal symmetries were treated as global symmetries (or weakly coupled to probe background fields), we suggest a particular low-energy realization of model II as an analogous gapless 4D deconfined quantum criticality with new beyond the SM fractionalized fragmentary excitations of color-flavor separation, and gauge enhancement including a dark gauge force sector, altogether requiring a double fermionic Spin structure named DSpin. If the internal symmetries are dynamically gauged (as they are in our quantum vacuum), we suggest the model II's 4D criticality as a boundary criticality such that only appropriately gauge enhanced dynamical GUT gauge fields can propagate into an extradimensional 5D bulk. The phenomena may be regarded as SM deformation or ``morphogenesis.'' Although our derivation is based on kinematics and global anomaly matching constraints between UV parent and various candidate IR field theories, the constraints are still highly subtle and suggestive. Future verifications on the IR dynamics will be desirable.

Quantum symmetries in orbifolds and decomposition

In this paper, we introduce a new set of modular-invariant phase factors for orbifolds with trivially-acting subgroups, analogous to discrete torsion and generalizing quantum symmetries. After describing their basic properties, we generalize decomposition to include orbifolds with these new phase factors, making a precise proposal for how such orbifolds are equivalent to disjoint unions of other orbifolds without trivially-acting subgroups or one-form symmetries, which we check in numerous examples.

English

We introduce and develop a categorification of the theory of Real representations of finite groups. In particular, we generalize the categorical character theory of Ganter--Kapranov and Bartlett to the Real setting. Given a Real representation of a group $\mathsf{G}$, or more generally a finite categorical group, on a linear category, we associate a number, the modified secondary trace, to each graded commuting pair $(g, \omega) \in \mathsf{G} \times \hat{\mathsf{G}}$, where $\hat{\mathsf{G}}$ is the background Real structure on $\mathsf{G}$. This collection of numbers defines the Real $2$-character of the Real representation. We also define various forms of induction for Real representations of finite categorical groups and compute the result at the level of Real $2$-characters. We interpret results in Real categorical character theory in terms of geometric structures, namely gerbes, vector bundles and functions on iterated unoriented loop groupoids. This perspective naturally leads to connections with the representation theory of unoriented versions of the twisted Drinfeld double of $\mathsf{G}$ and with discrete torsion in $M$-theory with orientifold. We speculate on an interpretation of our results as a generalized Hopkins--Kuhn--Ravenel-type character theory in Real equivariant homotopy theory.

A generalization of decomposition in orbifolds

This paper describes a generalization of decomposition in orbifolds. In general terms, decomposition states that two-dimensional orbifolds and gauge theories whose gauge groups have trivially-acting subgroups decompose into disjoint unions of theories. However, decomposition can be, at least naively, broken in orbifolds if the orbifold has discrete torsion in the trivially-acting subgroup. (Formally, this breaks finite global one-form symmetries.) Nevertheless, even in such cases, one still sees rudiments of decomposition. In this paper, we generalize decomposition in orbifolds to include such examples of discrete torsion, which we check in numerous examples. Our analysis includes as special cases (and in one sense generalizes) quantum symmetries of abelian orbifolds.

Uncovering a spinor–vector duality on a resolved orbifold

Spinor–vector dualities have been established in various exact string realisations like orbifold and free fermionic constructions. This paper aims to investigate possibility of having spinor–vector dualities on smooth geometries in the context of the heterotic string. As a concrete working example the resolution of the T4/Z2 orbifold is considered with an additional circle supporting a Wilson line, for which it is known that the underlying orbifold theory exhibits such a duality by switching on/off a generalised discrete torsion phase between the orbifold twist and the Wilson line. Depending on this phase complementary parts of the twisted sector orbifold states are projected out, so that different blowup modes are available to generate the resolutions. As a consequence, not only the spectra of the dual pairs are different, but also the gauge groups are not identical making this duality less apparent on the blowup and thus presumably on smooth geometries in general.

S -folds, string junctions, and 4D N=2 SCFTs

$S$-folds are a nonperturbative generalization of orientifold 3-planes which figure prominently in the construction of four-dimensional (4D) $\mathcal{N}=3$ superconformal field theories (SCFTs) and have also recently been used to realize examples of 4D $\mathcal{N}=2$ SCFTs. In this paper, we develop a general procedure for reading off the flavor symmetry experienced by D3-branes probing 7-branes in the presence of an $S$-fold. We develop an $S$-fold generalization of orientifold projection which applies to nonperturbative string junctions. This procedure leads to a different 4D flavor symmetry algebra depending on whether the $S$-fold supports discrete torsion. We also show that this same procedure allows us to read off admissible representations of the flavor symmetry in the associated 4D $\mathcal{N}=2$ SCFTs. Furthermore, this provides a prescription for how to define F-theory in the presence of $S$-folds with discrete torsion.

Discrete curvature and torsion from cross-ratios

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

More on mathcal{N} =2 S-folds

We carry out a systematic study of 4d mathcal{N} = 2 preserving S-folds of F-theory 7-branes and the worldvolume theories on D3-branes probing them. They consist of two infinite series of theories, which we denote following [1, 2] by {mathcal{S}}_{G,mathrm{ell}}^{(r)} for ℓ = 2, 3, 4 and {mathcal{T}}_{G,mathrm{ell}}^{(r)} for ℓ = 2, 3, 4, 5, 6. Their distinction lies in the discrete torsion carried by the S-fold and in the difference in the asymptotic holonomy of the gauge bundle on the 7-brane. We study various properties of these theories, using diverse field theoretical and string theoretical methods.

Heterotic orbifolds, reduced rank and SO(2n + 1) characters

The moduli space of the maximally supersymmetric heterotic string in [Formula: see text]-dimensional Minkowski space contains various components characterized by the rank of the gauge symmetries of the vacua they parametrize. We develop an approach for describing in a unified way continuous Wilson lines which parametrize a component of the moduli space, together with discrete deformations responsible for the switch from one component to the other. Applied to a component that contains vacua with [Formula: see text] gauge-symmetry factors, our approach yields a description of all backgrounds of the component in terms of free-orbifold models. The orbifold generators turn out to act symmetrically or asymmetrically on the internal space, with or without discrete torsion. Our derivations use extensively affine characters of [Formula: see text]. As a by-product, we find a peculiar orbifold description of the heterotic string in 10 dimensions, where all gauge degrees of freedom arise as twisted states, while the untwisted sector reduces to the gravitational degrees of freedom.

Holography of massive M2-brane theory with discrete torsion

We investigate the gauge/gravity duality between the mathcal{N} = 6 mass-deformed ABJ theory with hbox {U}_k(N+l)times hbox {U}_{-k}(N) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1) times SO(4)/{mathbb Z}_ktimes SO(4)/{{mathbb {Z}}}_k isometry and the discrete torsion l. For chiral primary operators with conformal dimensions Delta =1,2, we obtain the exact vacuum expectation values using the holographic method in 11-dimensional supergravity and show that the results depend on the shapes of droplet pictures in LLM geometries. The frac{l}{sqrt{N}} contributions from the discrete torsion l for several simple droplet pictures in the large N limit are determined in holographic vacuum expectation values. We also explore the effects of the orbifolding {{mathbb {Z}}}_k and the asymptotic discrete torsion l, on the gauge/gravity duality dictionary and on the nature of the asymptotic limits of the LLM geometries.

On mirror maps for manifolds of exceptional holonomy

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.

Non-Perturbative Superpotentials and Discrete Torsion

We discuss the non-perturbative superpotential in $${{E}_{8}} \times {{E}_{8}}$$ heterotic string theory on a non-simply connected Calabi–Yau manifold X, as well as on its simply connected covering space $$\tilde {X}.$$ The superpotential is induced by the string wrapping holomorphic, isolated, genus zero curves. We show, in a specific example, that the superpotential is non-zero both on $$\tilde {X}$$ and on X avoiding the no-go residue theorem of Beasley and Witten. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus zero curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and their contributions do not cancel each other.

Non-BPS fractional branes with Bose-Fermi cancellation in asymmetric orbifolds

We study non-BPS D-branes in the type II string vacua with chiral space-time SUSY constructed based on the asymmetric orbifolds of mathrm{K}3cong {T}^4/{mathrm{mathbb{Z}}}_2 as a succeeding work of [21]. We especially focus on the fractional D-branes that contains the contributions from the twisted sector of the {mathrm{mathbb{Z}}}_2 -orbifolding.We show that the cylinder partition functions for these fractional branes do not vanish, as in the cases of ordinary non-BPS D-branes. This aspect is in a sharp contrast with the bulk-type branes studied in [21]. We then discuss the extensions of models by including the discrete torsion depending on the spin structures, and investigate whether we obtain the vanishing self-overlaps associated to the fractional branes. The existence/absence of bose-fermi cancellations both in the closed and open string spectra as well as the massless spectra in the twisted sectors crucially depend on the discrete torsion. We find that some choices of the torsion indeed realize the vanishing self-overlaps for the fractional branes, with keeping the vanishing torus partition function intact.