Chern-Simons Theory Research Articles

Double Copy From Tensor Products of Metric BV■‐Algebras

AbstractField theories with kinematic Lie algebras, such as field theories featuring color–kinematics duality, possess an underlying algebraic structure known as BV■‐algebra. If, additionally, matter fields are present, this structure is supplemented by a module for the BV■‐algebra. The authors explain this perspective, expanding on our previous work and providing many additional mathematical details. The authors also show how the tensor product of two metric BV■‐algebras yields the action of a new syngamy field theory, a construction which comprises the familiar double copy construction. As examples, the authors discuss various scalar field theories, Chern–Simons theory, self‐dual Yang–Mills theory, and the pure spinor formulations of both M2‐brane models and supersymmetric Yang–Mills theory. The latter leads to a new cubic pure spinor action for 10‐dimensional supergravity. A homotopy‐algebraic perspective on colour–flavour‐stripping is also given, obtain a new restricted tensor product over a wide class of bialgebras, and it is also show that any field theory (even one without colour–kinematics duality) comes with a kinematic ‐algebra.

Q-operators are ’t Hooft lines

We study ’t Hooft lines in four-dimensional holomorphic-topological Chern-Simons theory. We relate them to Q-operators in the theory of integrable systems. We give a physical interpretation of the fundamental TQ and QQ relations satisfied by Q-operators and conventional transfer matrices.

Qudit stabilizer codes, CFTs, and topological surfaces

We study general maps from the space of rational conformal field theories (CFTs) with a fixed chiral algebra and associated Chern-Simons (CS) theories to the space of qudit stabilizer codes with a fixed generalized Pauli group. We consider certain natural constraints on such a map and show that the map can be described as a graph homomorphism from an orbifold graph, which captures the orbifold structure of CFTs, to a code graph, which captures the structure of self-dual stabilizer codes. By studying explicit examples, we show that this graph homomorphism cannot always be a graph embedding. However, we construct a physically motivated map from universal orbifold subgraphs of CFTs to operators in a generalized Pauli group. We show that this map results in a self-dual stabilizer code if and only if the surface operators in the bulk CS theories corresponding to the CFTs in question are self-dual. For CFTs admitting a stabilizer code description, we show that the full Abelianized generalized Pauli group can be obtained from twisted sectors of certain 0-form symmetries of the CFT. Finally, we connect our construction with SymTFTs, and we argue that many equivalences between codes that arise in our setup correspond to equivalence classes of bulk topological surfaces under fusion with invertible surfaces. Published by the American Physical Society 2024

Hall-like behaviour of higher rank Chern-Simons theory of fractons

Fracton phases of matter constitute an interesting point of contact between condensed matter and high-energy physics. The limited mobility property of fracton quasi-particles finds applications in many different contexts, including quantum information, spin liquids, elasticity, hydrodynamics, gravity and holography. In this paper we adopt a field theoretical approach to investigate the three dimensional action of a rank-2 symmetric tensor field invariant under the covariant fracton symmetry. The theory appears as a non-topological higher rank generalization of the ordinary Chern-Simons model, depending only on the traceless part of the tensor gauge field. After defining a field strength, a rank-2 traceless “electric” field and a “magnetic” vector field are identified, in analogy with the standard Chern-Simons ones. Once matter is introduced, a Hall-like behaviour with fractonic features emerges. In particular, our model shows a Hall-like dipole current, together with a vectorial “flux-attachment” relation for dipoles. This gives a possible starting point for a fracton-vortex duality. A gauge-fixing term is then introduced, from which propagators are computed and the counting of the degrees of freedom is performed. Finally, the energy-momentum tensor is shown to be conserved and the integrated energy density is proved to be zero, which reminds the topological nature of the standard Chern-Simons model.

Some lower dimensional quantum field theories reduced from Chern-Simons gauge theories

We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on S3 and the reduced quantum field theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding a topological theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function. Published by the American Physical Society 2024

Ground states of Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} theory on ADE singularities and dual Chern-Simons theory

In radial quantization, the ground states of a gauge theory on ADE singularities ℝ4/Γ are characterized by flat connections that are maps from Γ to the gauge group. We study Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} theory of type a1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{a}}_1 $$\\end{document} = su2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{su}(2) $$\\end{document} on a Riemann surface of genus g > 1, without punctures. The fundamental building block of Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} theory is the trifundamental Trinion theory — a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for Γ = ℤk, agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group Γ = Dic2, corresponding to Chern-Simons gauge group SO(8).

Higher gauge theory and integrability

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of four-dimensional Chern-Simons theory with defects, which provided a systematic framework for constructing two-dimensional integrable systems. In this article, we propose a novel approach to studying higher-dimensional integrable models employing techniques from higher category theory. Starting with higher Chern-Simons theory on the 4-manifold R×Y, we complexify and compactify the real line to CP1 and introduce the disorder defect ω=z−1dz. This procedure defines a holomorphic five-dimensional variant of higher Chern-Simons theory, which, when endowed with suitable boundary conditions, allows for the localization to a three-dimensional theory on Y. The equations of motion of the resulting model are equivalent to the flatness of a 2-connection (L,H), that we then use to construct the corresponding higher holonomies. We prove that these are invariants of homotopies relative boundary, which enables the construction of conserved quantities. The latter are labeled by both the categorical characters of a Lie crossed module and the infinite number of homotopy classes of surfaces relative boundary in Y. Moreover, we also demonstrate that the three-dimensional theory has left and right acting symmetries whose current algebra is given by an infinite-dimensional centrally extended affine Lie 2-algebra. Both of these conditions are direct higher homotopy analogs of the properties satisfied by the two-dimensional Wess-Zumino-Witten conformal field theory, which we therefore interpret as facets of integrable structures. Published by the American Physical Society 2024

Witten–Reshetikhin–Turaev invariants and indefinite false theta functions for plumbing indefinite H-graphs

Gukov–Pei–Putrov–Vafa conjectured the existence of [Formula: see text]-series whose radial limits are Witten–Reshetikhin–Turaev invariants and called them homological blocks. For weakly negative definite plumbed 3-manifolds, Gukov–Pei–Putrov–Vafa and Gukov–Manolescu constructed homological blocks. In this paper, we construct indefinite false theta functions which are candidates of homological blocks for some plumbed 3-manifolds which are not weakly negative definite. Moreover we prove that, for the Poincaré homology sphere, our indefinite false theta function coincides with the original homological block.

Knot Floer homology and surgery on equivariant knots

AbstractGiven an equivariant knot of order 2, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on . This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an application, we calculate equivariant correction terms which are invariants of a generalized version of the spin rational homology cobordism group, and define two knot concordance invariants. We also compute the action of the symmetry on the knot Floer complex of for several equivariant knots.

Chern-Simons theory, decomposition, and the A model

In this paper, we discuss how gauging one-form symmetries in Chern-Simons theories is implemented in an A-twisted topological open string theory. For example, the contribution from a fixed H/Z bundle on a three-manifold M, arising in a BZ gauging of H Chern-Simons, for Z a finite subgroup of the center of H, is described by an open string worldsheet theory whose bulk is a sigma model with target a Z-gerbe (a bundle of one-form symmetries) over T∗M, of characteristic class determined by the H/Z bundle. We give a worldsheet picture of the decomposition of one-form-symmetry-gauged Chern-Simons in three dimensions, and we describe how a target-space constraint on bundles arising in the gauged Chern-Simons theory has a natural worldsheet realization. Our proposal provides examples of the expected correspondence between worldsheet global higher-form symmetries, and target-space gauged higher-form symmetries.

Integrability in gravity from Chern-Simons theory

This paper presents a new perspective on integrability in theories of gravity. We show how the stationary, axisymmetric sector of General Relativity can be described by the boundary dynamics of a four-dimensional Chern-Simons theory. This approach shows promise for simplifying solution generating methods in both General Relativity and higher-dimensional supergravity theories. The four-dimensional Chern-Simons theory presented generalises those for flat space integrable models by introducing a space-time dependent branch cut in the spectral plane. We also make contact with twistor space approaches to integrability, showing how the branch cut defects of four-dimensional Chern-Simons theory arise from a discrete reduction of six-dimensional Chern-Simons theory.

Geometric actions for Lower Spin Gravity

We perform the Hamiltonian reduction of Lower Spin Gravity, the simplest bulk dual for a Warped Conformal Field Theory (WCFT), consisting in an SL(2) × U(1) Chern-Simons model. We identify the boundary action as the geometric action on coadjoint orbits of the Warped Virasoro group. We use this reduced action to compute one-loop contributions to the torus partition function and compare them to the Warped Virasoro characters.

Self-gravitating solutions in Yang–Mills–Chern–Simons theory coupled to 3D massive gravity

We study self-gravitating solutions of three-dimensional massive gravity coupled to the Yang–Mills–Chern–Simons gauge theory. Among these, there is a family of asymptotically Warped-Anti de Sitter black holes that come to generalize previous solutions found in the literature and studied in the context of WAdS3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}/CFT2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}. We also present self-gravitating solutions to three-dimensional Einstein–Yang–Mills theory, as well other self-gravitating solutions in the presence of higher-curvature terms.

Interpolating gauge-fixing for Yang–Mills–Chern–Simons theory in D=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=3$$\\end{document}

The Yang–Mills–Chern–Simons theory in three-dimensional Minkowski space-time is studied in a gauge-fixing scheme which interpolates between the covariant gauge and light-cone gauge, the interpolating gauge-fixing. The ultraviolet finiteness of the theory is proved via the Becchi–Rouet–Stora (BRS) algebraic renormalization procedure, which allows us to demonstrate the vanishing of all β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-functions and all anomalous dimensions to all orders in perturbation theory.

Classical Dynamical r-matrices for the Chern–Simons Formulation of Generalized 3d Gravity

AbstractClassical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for $$\mathfrak {su}(2)$$ su ( 2 ) and $$\mathfrak {sl}(2,{\mathbb {R}})$$ sl ( 2 , R ) . We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

Lens Factor Choice in IOL Power Calculation after Laser Refractive Surgery: The Right Constant for Advanced Lens Measurement Approach (ALMA).

Background/Objectives: To evaluate the advanced lens measurement approach (ALMA) formula accuracy using different lens constants available on the user group for laser interference biometry (ULIB) and IOL Con platforms. Methods: In this retrospective, comparative, case-series study, 150 eyes of 160 patients with previous myopic Photorefractive Keratectomy (PRK) or laser-assisted in situ keratomileusis (LASIK), who underwent uneventful cataract surgery and IOL implantation, were examined. The ALMA formula was evaluated to calculate the refractive prediction error (PE), analysing four different categories of lens constants: both nominal and optimized A-Constant for SRKT, which are available on the ULIB and IOL Con platforms. An additional analysis was carried out in this study, evaluating if a decreased ULIB optimized constant (DUOC) with different fixed factors (-1.2 -1.3 -1.4 -1.5) could improve refractive outcomes. Median absolute error (MedAE) and percentage of eyes within ±0.50 and ±1.00 diopters (D) of prediction error were measured as the main outcomes. Results: Comparing the lens factors available on ULIB and IOL Con platforms, the ALMA formula reported a lower MedAE and higher percentages of eyes with a refractive PE within 1.0 D using ULIB nominal constants (all p < 0.05). Using DUOC (-1.3), and there was a statistically significant improvement of both MedAE and of the percentages of eyes with PE within ±0.50 D with the ALMA method compared to nominal ULIB constants (all p < 0.05). Conclusions: The impact of different lens factors in the IOL power calculation after myopic LRS should be carefully evaluated. The ALMA formula, in the absence of optimized constants by zeroing the mean error, should be used by subtracting 1.3 from the optimized ULIB constants available on the IOL Con website. This finding suggests further studies to test which of these constants could work better with the other post-refractive surgery formulas.

Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra K is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of K⊗K¯ and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in C3 this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of C3. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy. Published by the American Physical Society 2024

What can abelian gauge theories teach us about kinematic algebras?

The phenomenon of BCJ duality implies that gauge theories possess an abstract kinematic algebra, mirroring the non-abelian Lie algebra underlying the colour information. Although the nature of the kinematic algebra is known in certain cases, a full understanding is missing for arbitrary non-abelian gauge theories, such that one typically works outwards from well-known examples. In this paper, we pursue an orthogonal approach, and argue that simpler abelian gauge theories can be used as a testing ground for clarifying our understanding of kinematic algebras. We first describe how classes of abelian gauge fields are associated with well-defined subalgebras of the diffeomorphism algebra. By considering certain special subalgebras, we show that one may construct interacting theories, whose kinematic algebras are inherited from those already appearing in a related abelian theory. Known properties of (anti-)self-dual Yang-Mills theory arise in this way, but so do new generalisations, including self-dual electromagnetism coupled to scalar matter. Furthermore, a recently obtained non-abelian generalisation of the Navier-Stokes equation fits into a similar scheme, as does Chern-Simons theory. Our results provide useful input to further conceptual studies of kinematic algebras.

Flows in the space of interacting chiral boson theories

We study interacting theories of N left-moving and N¯ right-moving Floreanini-Jackiw bosons in two dimensions. A parametrized family of such theories is shown to enjoy (nonmanifest) Lorentz invariance if and only if its Lagrangian obeys a flow equation driven by a function of the energy-momentum tensor. We discuss the canonical quantization of such theories along classical stress tensor flows, focusing on the case of the root-TT¯ deformation, where we obtain perturbative results for the deformed spectrum in a certain large-momentum limit. In the special case N=N¯, we consider the quantum effective action for the root-TT¯-deformed theory by expanding around a general classical background, and we find that the one-loop contribution vanishes for backgrounds with constant scalar gradients. Our analysis can also be interpreted via dual U(1) Chern-Simons theories in three dimensions, which might be used to describe deformations of charged AdS3 black holes or quantum Hall systems. Published by the American Physical Society 2024

Comparison of intraocular lens power prediction by American Society of Cataract and Refractive Surgery formulas and Barrett True-K TK in eyes with prior laser refractive surgery.

To evaluate the prediction accuracy of various intraocular lens (IOL) power calculation formulas on American Society of Cataract and Refractive Surgery (ASCRS) calculator and Barrett True-K total keratometry (TK) in eyes with previous laser refractive surgery for myopia. This retrospective study included eyes with history of myopic laser refractive surgery, which have undergone clear or cataractous lens extraction by phacoemulsification followed by IOL implantation. Those who underwent uneventful crystalline lens extraction were included. Eyes with any complication of refractive surgery or those with eventful lens extraction procedure and those who were lost to follow-up were excluded. Formulas compared were Wang-Koch-Maloney, Shammas, Haigis-L, Barrett True-K no-history formula, ASCRS average power, ASCRS maximum power on the ASCRS post-refractive calculator and the IOLMaster 700 Barrett True-K TK. Prediction error was calculated as the difference between the implanted IOL power and the predicted power by various formulae available on ASCRS online calculator. Forty post-myopic laser-refractive surgery eyes of 26 patients were included. Friedman's test revealed that Shammas formula, Barrett True-K, and ASCRS maximum power were significantly different from all other formulas (P < 0.00001 for each). Median absolute error (MedAE) was the least for Shammas and Barrett True-K TK formulas (0.28 [0.14, 0.36] and 0.28 [0.21, 0.39], respectively) and the highest for Wang-Koch-Maloney (1.29 [0.97, 1.61]). Shammas formula had the least variance (0.14), while Wang-Koch-Maloney formula had the maximum variance (2.66). In post-myopic laser refractive surgery eyes, Shammas formula and Barrett True-K TK no-history formula on ASCRS calculator are more accurate in predicting IOL powers.