Topological Field Theory Research Articles

The Physical Meaning of the Holographic Principle

We show in this pedagogical review that far from being an apparent law of physics that stands by itself, the holographic principle is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Utilizing the notion of holographic screens, which are information encoding boundaries between physical subsystems, we demonstrate that the physical interaction is an information exchange during which information is strictly conserved. Then we use generalized holographic principle in order to flesh out a fully-general quantum theory of measurement in which the measurement produces finite-resolution, classical outcomes. Further, we show that the measurements are given meaning by quantum reference frames and sequential measurements induce topological quantum field theories. Finally, we discuss principles equivalent to the holographic principle, including Markov blankets and the free-energy principle in biology, multiple realizability and virtual machines in computer science, and active inference and interface theories in cognitive science. This appearance in multiple disciplines suggests that the holographic principle is not just a fundamental principle of physics, but of all of science.Quanta 2022; 11: 72–96.

Toward the group completion of the Burau representation

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or writhe is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an E 2 E_2 Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of C \mathbb {C} , and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on R + 3 \mathbb {R}^3_+ .

Rotationally Symmetric Yang-Mills Gauge Theory and Generalized Uncertainty Principle

Recently, it has been of considerably high interest to introduce the notion of minimal length in quantum mechanics and quantum field theories to get a proper description of quantum gravity. In this paper, we have attempted to unify the notion of minimal length with R×S^3 topological field theories which were introduced by Carmeli and Malin in 1985 [14]. We have collectively called this (R×S^3 )_H topological field theories where the “H” in the sub-script stands for Heisenberg’s notion of minimal length. A proper description for equations like Schrodinger’s equation, Klein-Gordon equation and QED, QCD Lagrangian on (R×S^3 )_H topology is derived. HIGHLIGHTS It has been of considerably high interest to incorporate existing theories with the notion of minimal length because it is thought to unify theories at large scale with the quantum world In this paper, we have done a similar unification by incorporating Rotaionally symmetric field theories with the notion of minimal length The derived equations work consistently under local and global gauge transformations

The Branes Behind Generalized Symmetry Operators

AbstractThe modern approach tom‐form global symmetries in ad‐dimensional quantum field theory (QFT) entails specifying dimension topological generalized symmetry operators which non‐trivially link withm‐dimensional defect operators. In QFTs engineered via string constructions on a non‐compact geometryX, these defects descend from branes wrapped on non‐compact cycles which extend from a localized source / singularity to the boundary . The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in . This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. We illustrate these general features in the context of 6D superconformal field theories, where we use the F‐theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3‐form symmetry detect a non‐invertible fusion rule for these operators. We also sketch how similar considerations hold for related systems.

From Three Dimensional Manifolds to Modular Tensor Categories

Using M-theory in physics, Cho et al. (JHEP 2020:115 (2020) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $$\text {SL}(2,{\mathbb {C}})$$ -flat connections and $$\text {SL}(2,{\mathbb {C}})$$ -adjoint Reidemeister torsions of a three manifold can be packaged together to produce a $$(2+1)$$ -topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular T-matrix and the quantum dimensions of a candidate modular data. The modular S-matrix follows from essentially a trial-and-error procedure. We find premodular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our examples. Our main result is a mathematical construction of the modular data of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The modular data of premodular categories from Seifert fibered spaces can be realized using Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a $${\mathbb {Z}}_2$$ -homology sphere, and condensation of bosons in the resulting properly premodular categories leads to either modular or super-modular tensor categories.

Abelian Duality in Topological Field Theory

We prove a topological version of abelian duality where the gauge groups are finite abelian. The theories are finite homotopy TFTs, topological analogues of the $p$-form $U(1)$ gauge theories. Using Brown-Comenetz duality, we extend the duality to finite homotopy TFTs of $\pi$-finite spectra.

Higher form symmetries TFT in 6d

Symmetries and anomalies of a d-dimensional quantum field theory are often encoded in a (d + 1)-dimensional topological action, called symmetry topological field theory (TFT). We derive the symmetry TFT for the 2-form and 1-form symmetries of 6d (1, 0) field theories, focusing on theories with a single tensor multiplet (rank 1). We implement this by coupling the low-energy tensor branch action to the background fields for the higher-form symmetries and by looking at the symmetry transformation rules on dynamical and background fields. These transformation rules also imply a mixing of the higher-form symmetries in a 3-group structure. For some specific and related higher rank cases, we also derive the symmetry TFT from the holographic dual IIA supergravity solutions. The symmetry TFT action contains a coupling between the 2-form symmetry and the 1-form symmetry backgrounds, which leads to a mixed anomaly between the 1-form symmetries of the 5d KK-theory obtained by circle compactification. We confirm this by a pure 5d analysis provided by the 5d effective low-energy Coulomb branch Lagrangian coupled to background fields. We also derive the symmetry TFT for 5d SU(p) supersymmetric gauge theories with Chern-Simons level q and for 5d theories without non-abelian gauge theory description at low-energy. Finally, we discuss the fate of the 2-form and 1-form symmetry of rank 1 6d field theories when coupled to gravity.

Duality defects in E8

We classify all non-invertible Kramers-Wannier duality defects in the E8 lattice Vertex Operator Algebra (i.e. the chiral (E8)1 WZW model) coming from ℤm symmetries. We illustrate how these defects are systematically obtainable as ℤ2 twists of invariant sub-VOAs, compute defect partition functions for small m, and verify our results against other techniques. Throughout, we focus on taking a physical perspective and highlight the important moving pieces involved in the calculations. Kac’s theorem for finite automorphisms of Lie algebras and contemporary results on holomorphic VOAs play a role. We also provide a perspective from the point of view of (2+1)d Topological Field Theory and provide a rigorous proof that all corresponding Tambara-Yamagami actions on holomorphic VOAs can be obtained in this manner. We include a list of directions for future studies.

On the Moore-Tachikawa varieties

Moore-Tachikawa varieties are certain Hamiltonian holomorphic symplectic varieties conjectured in the context of 2-dimensional topological quantum field theories. We discuss several constructions related to these varieties.

Field theory of higher-order topological crystalline response, generalized global symmetries and elasticity tetrads

I discuss aspects of higher-order topological field theory of crystalline insulators with no other symmetries. I show how the topology and geometry of the crystalline lattice is organized in terms of so-called elasticity tetrads which are ground state degrees of freedom labelling translational lattice topological charges, higher-form conservation laws and responses on sub-dimensional manifolds of the bulk insulator. The quasitopological responses obtained in this way depend on the lattice and its embedding in space, as expected for weak topology. In a topological crystalline insulator, they classify higher-order responses and global symmetries in a transparent fashion in generic dimensions. This hierarchy coincides with the dimensional hierarchy of topological terms, the multipole expansion, and anomaly inflow, related to a mixed number of elasticity tetrads and electromagnetic gauge fields. In the continuum limit of the elasticity tetrads, the semi-classical expansion in momentum space can be used to derive the higher-order or subdimensional topological responses to local U(1) symmetries, such as electromagnetic gauge fields, with explicit formulas for the higher-order quasi-topological invariants in terms of the elasticity tetrads and Green’s functions. The topological responses in arbitrary dimensions are readily generalized to parameter space to allow for e.g. multipole pumping. The simple results further bridge the recently appreciated connections between topological field theory, higher-form symmetries and gauge fields and their relation to fractonic excitations and topological defects with restricted mobility in the elasticity of crystalline insulators.

Universal Non‐Invertible Symmetries

AbstractIt is well‐known that gauging a finite 0‐form symmetry in a quantum field theory leads to a dual symmetry generated by topological Wilson line defects. These are described by the representations of the 0‐form symmetry group which form a 1‐category. We argue that for a d‐dimensional quantum field theory the full set of dual symmetries one obtains is in fact much larger and is described by a ‐category, which is formed out of lower‐dimensional topological quantum field theories with the same 0‐form symmetry. We study in detail a 2‐categorical piece of this ‐category described by 2d topological quantum field theories with 0‐form symmetry. We further show that the objects of this 2‐category are the recently discussed 2d condensation defects constructed from higher‐gauging of Wilson lines. Similarly, dual symmetries obtained by gauging any higher‐form or higher‐group symmetry also form a ‐category formed out of lower‐dimensional topological quantum field theories with that higher‐form or higher‐group symmetry. A particularly interesting case is that of the 2‐category of dual symmetries associated to gauging of finite 2‐group symmetries, as it describes non‐invertible symmetries arising from gauging 0‐form symmetries that act on ‐form symmetries. Such non‐invertible symmetries were studied recently in the literature via other methods, and our results not only agree with previous results, but our approach also provides a much simpler way of computing various properties of these non‐invertible symmetries. We describe how our results can be applied to compute non‐invertible symmetries of various classes of gauge theories with continuous disconnected gauge groups in various spacetime dimensions. We also discuss the 2‐category formed by 2d condensation defects in any arbitrary quantum field theory.

The BV action of 3D twisted R-Poisson sigma models

We determine the solution to the classical master equation for a 3D topological field theory with Wess-Zumino term and an underlying geometrical structure of a twisted R-Poisson manifold on its target space. The graded geometry of the target space departs from the usual QP structure encountered in the AKSZ construction of topological sigma models, the obstruction being attributed to the presence of the Wess-Zumino 4-form. Due to the inapplicability of the AKSZ construction in this case, we set up the traditional BV/BRST formalism for twisted R-Poisson sigma models in any dimension, which feature an open gauge algebra and constitute multiple stages reducible constrained Hamiltonian systems. An unusual feature of the theories is that they exhibit non-linear openness of the gauge algebra, in other words products of the equations of motion appear in them. Nevertheless, we find the BV action in presence of the 4-form twist in 3D, namely for a specific 4-form twisted (pre-)Courant sigma model. Moreover, we provide a complete set of explicit formulas for the off-shell nilpotent BV operator for untwisted R-Poisson sigma models in any dimension.

Higher central charges and topological boundaries in 2+1-dimensional TQFTs

A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge c_-c− has to vanish. In this paper, we consider conditions associated with ``higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.

Towards topological fixed-point models beyond gappable boundaries

We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the mathematical community as state-sum constructions or lattice topological quantum field theories. All of the established ansatzes for fixed-point models imply the existence of a gapped boundary as well as a commuting-projector Hamiltonian. Thus, they fail to capture topological phases without a gapped boundary or commuting-projector Hamiltonian, most notably chiral topological phases in $2+1$ dimensions. In this work, we present a more general fixed-point ansatz not affected by the aforementioned restrictions. Thus, our formalism opens up a possible way forward towards a microscopic fixed-point description of chiral phases and we present several strategies that may lead to concrete examples. Furthermore, we argue that our more general ansatz constitutes a universal form of topological fixed-point models, whereas established ansatzes are universal only for fixed-points of phases which admit topological boundaries.

Comments on summing over bordisms in TQFT

Recent works in quantum gravity, motivated by the “factorization problem” and “baby universes,” have considered sums over bordisms with fixed boundaries in topological quantum field theory (TQFT). We discuss this construction and observe a curious splitting formula for the total amplitude.

Topological disorder parameter: A many-body invariant to characterize gapped quantum phases

We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in (2+1)d. TDP is defined as the constant correction that appears in the ground state expectation value of a partial symmetry transformation applied to a connected spatial region $M$, the absolute value of which scales generically as $\exp(-\alpha l+\gamma)$ where $l$ is the perimeter of $M$ and $\gamma$ is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that $e^\gamma$ can be related to the quantum dimension of the symmetry defect, and provide a general formula for $\gamma$ when the entanglement Hamiltonian of the topological phase can be described by a (1+1)d conformal field theory (CFT). A special case of TDP is equivalent to the topological R\'enyi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish a new tool to detect quantum topological order.

Big Bang and Topology

In this paper, we discuss the initial state of the universe at the Big Bang. By using the ideas of Freedman in the proof of the disk embedding theorem for 4-manifolds, we describe the corresponding spacetime as a gravitational instanton. The spatial space is a fractal space (wild embedded 3-sphere). Then, we construct the quantum state from this fractal space. This quantum state is part of the string algebra of Ocneanu. There is a link between the Jones polynomial and Witten’s topological field theory. Using this link, we are able to determine the physical theory (action) as the Chern–Simons functional. The gauge fixing of this action determines the foliation of the spacetime and the smoothness properties. Finally, we determine the quantum symmetry of the quantum state to be the enveloped Lie algebra Uq(sl2(C)), where q is the fourth root of unity.

Reply to "Comment on 'Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory' ".

We reply to the Comment by Bryk etal. [Phys Rev. E 106, 036601 (2022)10.1103/PhysRevE.106.036601] on our paper [Phys. Rev. E 105, 024602 (2022)2470-004510.1103/PhysRevE.105.024602].

Comment on "Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory".

We discuss a field-theoretical approach to liquids, solids, and glasses, published recently [Phys. Rev. E 105, 034108 (2022)10.1103/PhysRevE.105.034108], which aims to describe these materials in a common formalism. We argue that such a formalism contradicts the known hydrodynamic theory of classical liquids. In particular, the authors miss the important particle-number conservation law and the density fluctuations as a hydrodynamic slow variable. This results in an exotic mechanism of hydrodynamic sound instead of the standard hydrodynamic one due to combined particle-number and momentum conservation, a fact well documented in fluid-mechanics textbooks.

Domain Walls Between 3d Phases of Reshetikhin–Turaev TQFTs

We study surface defects in three-dimensional topological quantum field theories which separate different theories of Reshetikhin–Turaev type. Based on the new notion of a Frobenius algebra over two commutative Frobenius algebras, we present an explicit and computable construction of such defects. It specialises to the construction in Carqueville et al. (Geom Topol 23:781–864, 2019. https://doi.org/10.2140/gt.2019.23.781. arXiv:1705.06085) if all 3-strata are labelled by the same topological field theory. We compare the results to the model-independent analysis in Fuchs et al. (Commun Math Phys 321:543–575, 2013. https://doi.org/10.1007/s00220-013-1723-0. arXiv:1203.4568) and find agreement.