Chern-Simons Theory Research Articles

Notes on the L∞-approach to local gauge field theories

It is well known that a Q-manifold gives rise to an L∞-algebra structure on the tangent space at a fixed point of the homological vector field. From the field theory perspective this implies that the expansion of a classical Batalin-Vilkovisky (BV) formulation around a vacuum solution can be equivalently cast into the form of an L∞-algebra. In so doing, the BV symplectic structure determines a compatible cyclic structure on the L∞-algebra. Moreover, L∞ quasi-isomorphisms correspond to so-called equivalent reductions (also known as the elimination of generalized auxiliary fields) of the respective BV systems. Although at the formal level the relation is straightforward, its implementation in field theory requires some care because the underlying spaces become infinite-dimensional. In the case of local gauge theories, the relevant spaces can be approximated by nearly finite-dimensional ones through employing the jet-bundle technique. In this work we study the L∞-counterpart of the jet-bundle BV approach and generalise it to a more flexible setup of so-called gauge PDEs. It turns out that in the latter case the underlying L∞-structure is analogous to that of Chern-Simons theory. In particular, the underlying linear space is a module over the space-time exterior algebra and the higher L∞-maps are multilinear. Furthermore, a counterpart of the cyclic structure turns out to be degenerate and possibly nonlinear, and corresponds to a compatible presymplectic structure the gauge PDE, which is known to encode the BV symplectic structure and hence the full-scale Lagrangian BV formulation. Moreover, given a degenerate cyclic structure one can consistently relax the L∞-axioms in such a way that the formalism still describes non-topological models but involves only finitely-generated modules, as we illustrate in the example of Yang-Mills theory.

D-instanton, threshold corrections, and topological string

In this note, we prove that the one-loop pfaffian of the non-perturbative superpotential generated by Euclidean D-branes in type II compactifications on orientifolds of Calabi-Yau threefolds is determined by the moduli integral of the new supersymmetric index defined by Cecotti, Fendley, Intriligator, and Vafa. As this quantity can be computed via topological string theory, Chern-Simons theory, matrix models, or by solving the holomorphic anomaly equation, this result provides a method to directly compute the one-loop pfaffian of the non-perturbative superpotential. The relation between the one-loop pfaffian, threshold corrections to the gauge coupling, and the one-loop partition function of open topological string theory is discussed.

Resurgence in complex Chern-Simons theory at generic levels

In this note we study the resurgent structure of sl(2, ℂ) Chern-Simons state integral model on knot complements S3\\41, S3\\52 with generic discrete level k ≥ 1 and with small boundary holonomy deformation. The coefficients of the saddle point expansions are in the trace field of the knot extended by the holonomy parameter. Despite increasing complication of the asymptotic series as the level k increases, the resurgent structure of the asymptotic series is universal: both the distribution of Borel plane singularities and the associated Stokes constants are independent of the level k.

Gauging Lie group symmetry in (2+1)d topological phases

We present a general algebraic framework for gauging a 0-form compact, connected Lie group symmetry in (2+1)d topological phases. Starting from a symmetry fractionalization pattern of the Lie group GG, we first extend GG to a larger symmetry group \tilde{G}G̃, such that there is no fractionalization with respect to \tilde{G}G̃ in the topological phase, and the effect of gauging \tilde{G}G̃ is to tensor the original theory with a \tilde{G}G̃ Chern-Simons theory. To restore the desired gauge symmetry, one then has to gauge an appropriate one-form symmetry (or, condensing certain Abelian anyons) to obtain the final result. Studying the consistency of the gauging procedure leads to compatibility conditions between the symmetry fractionalization pattern and the Hall conductance. When the gauging can not be consistently done (i.e. the compatibility conditions can not be satisfied), the symmetry GG with the fractionalization pattern has an ’t Hooft anomaly and we present a general method to determine the (3+1)d topological term for the anomaly. We provide many examples, including projective simple Lie groups and unitary groups to illustrate our approach.

Integrable Degenerate varvec{mathcal {E}}-Models from 4d Chern–Simons Theory

We present a general construction of integrable degenerate mathcal {E}-models on a 2d manifold Sigma using the formalism of Costello and Yamazaki based on 4d Chern–Simons theory on Sigma times {mathbb {C}}{P}^1. We begin with a physically motivated review of the mathematical results of Benini et al. (Commun Math Phys 389(3):1417–1443, 2022. https://doi.org/10.1007/s00220-021-04304-7) where a unifying 2d action was obtained from 4d Chern–Simons theory which depends on a pair of 2d fields h and {mathcal {L}} on Sigma subject to a constraint and with {mathcal {L}} depending rationally on the complex coordinate on {mathbb {C}}{P}^1. When the meromorphic 1-form omega entering the action of 4d Chern–Simons theory is required to have a double pole at infinity, the constraint between h and {mathcal {L}} was solved in Lacroix and Vicedo (SIGMA 17:058, 2021. https://doi.org/10.3842/SIGMA.2021.058) to obtain integrable non-degenerate mathcal {E}-models. We extend the latter approach to the most general setting of an arbitrary 1-form omega and obtain integrable degenerate mathcal {E}-models. To illustrate the procedure, we reproduce two well-known examples of integrable degenerate mathcal {E}-models: the pseudo-dual of the principal chiral model and the bi-Yang-Baxter sigma -model.

3 definitions of BF theory on homology 3-spheres

3-dimensional BF theory with gauge group G (= Chern-Simons theory with non-compact gauge group TG) is a deceptively simple yet subtle topological gauge theory. Formally, its partition function is a sum/integral over the moduli space mathcal{M} of flat connections, weighted by the Ray-Singer torsion. In practice, however, this formal expression is almost invariably singular and ill-defined.In order to improve upon this, we perform a direct evaluation of the path integral for certain classes of 3-manifolds (namely integral and rational Seifert homology spheres). By a suitable choice of gauge, we sidestep the issue of having to integrate over mathcal{M} and reduce the partition function to a finite-dimensional Abelian matrix integral which, however, itself requires a definition. We offer 3 definitions of this integral, firstly via residues, and then via a large k limit of the corresponding G × G or Gℂ Chern-Simons matrix integrals (obtained previously). We then check and discuss to which extent the results capture the expected sum/integral over all flat connections.

Singular supertranslations and Chern-Simons theory on the black hole horizon

We construct the standard and dual supertranslation charges on the future horizon of the Schwarzschild black hole, using the first-order formulation of gravity with the Holst action. The Dirac bracket algebra of standard and dual supertranslation charges is shown to exhibit a central term in the presence of singularities in the two-sphere function associated with supertranslation. We show that one can cancel this anomalous term and restore the asymptotic symmetry algebra by introducing a gravitational Chern-Simons theory on the horizon. This demonstrates that consistency of the asymptotic symmetry algebra requires a new structure on the horizon.

Line operators in Chern-Simons-Matter theories and Bosonization in Three Dimensions II: Perturbative analysis and all-loop resummation

We study mesonic line operators in Chern-Simons theories with bosonic or fermionic matter in the fundamental representation. In this paper, we elaborate on the classification and properties of these operators using all loop resummation of large N perturbation theory. We show that these theories possess two conformal line operators in the fundamental representation. One is a stable renormalization group fixed point, while the other is unstable. They satisfy first-order chiral evolution equations, in which a smooth variation of the path is given by a factorized product of two mesonic line operators. The boundary operators on which the lines can end are classified by their conformal dimension and transverse spin, which we compute explicitly at finite ’t Hooft coupling. We match the operators in the bosonic and fermionic theories. Finally, we extend our findings to the mass deformed theories and discover that the duality still holds true.

Non-Abelian fusion, shrinking, and quantum dimensions of Abelian gauge fluxes

Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group.

S-Dual of Maxwell–Chern-Simons Theory

We discuss the dynamics of three-dimensional Maxwell theory coupled to a level-k Chern-Simons term. Motivated by S-duality in string theory, we argue that the theory admits an S-dual description. The S-dual theory contains a nongauge one-form field, previously proposed by Deser and Jackiw [Phys. Lett. 139B, 371 (1984).PYLBAJ0370-269310.1088/1126-6708/1999/10/036] and a level-k U(1) Chern-Simons term, Z_{MCS}=Z_{DJ}Z_{CS}. The couplings to external electric and magnetic currents and their string theory realizations are also discussed.

Arithmetic Chern–Simons theory with real places

The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern–Simons invariant with coefficient [Formula: see text] [Formula: see text] associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [17]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3] and the abelian cyclic gauge group with coefficient [Formula: see text] in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3]. Our non-trivial examples (with non-abelian gauge group and general coefficient [Formula: see text]) will be given by a simple twisting argument based on examples of [F. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas and M. Taylor, Cup products in the étale cohomology of number fields, New York J. Math. 24 (2018) 514–542].

Embedding integrable superspin chain in string theory

Using results on topological line defects of 4D Chern-Simons theory and the algebra/cycle homology correspondence in complex surfaces S with ADE singularities, we study the graded properties of the sl(m|n) chain and its embedding in string theory. Because of the Z2-grading of sl(m|n), we show that the (m+n)!/m!n! varieties of superspin chains with underlying super geometries have different cycle homologies. We investigate the algebraic and homological features of these integrable quantum chains and give a link between graded 2-cycles and genus-g Rieman surfaces Σg. Moreover, using homology language, we yield the brane realisation of the sl(m|n) chain in type IIA string and its uplift to M-theory. Other aspects like graded complex surfaces with sl(m|n) singularity as well as super magnons are also described.

A Chern-Simons model for baryon asymmetry

In search of a phenomenological model that would describe physics from Big Bang to the Standard Model (SM), we propose a model with the following properties (i) above an energy about Λcr>1016 GeV there are Wess-Zumino supersymmetric preons and Chern-Simons (CS) fields, (ii) at Λcr∼1016 GeV spontaneous gauge symmetry breaking takes place in the CS sector and the generated topological mass provides an attractive interaction to equal charge preons, (iii) well below 1016 GeV the model reduces to the standard model with essentially pointlike quarks and leptons, having a radius ∼1/Λcr≈10−31 m. The baryon asymmetry turns out to have a fortuitous ratio nB/nγ≪1.

Slow-rotating charged black hole solution in dynamical Chern-Simons modified gravity

Chern-Simons (CS) gravity is a modified theory of Einstein's general relativity. The CS theory arises from the low energy limit of string theory which involves anomaly correction to the Einstein-Hilbert action. The CS term is given by the product of the Pontryagin density with a scalar field. In this study, we derive a charged slowly rotating black hole (BH) solution. The main incentives of this BH solution are axisymmetric and stationary and form distortion of the Kerr-Newman BH solution with a dipole scalar field. Additionally, we investigate the asymptotic correction of the metric with the inverse seventh power of the radial distance to the BH solution, This indicates that it will escape any meaningful constraints from weak field experiments. To find this kind of BH by observations, we investigate the propagation of the photon near the BH and we show that the difference between the left-rotated polarization and the right-handed one could be observed as stronger than the case of the Kerr-Newman BH. Finally, we derived the stability condition using the geodesic deviations.

On mathcal{N} = 4 supersymmetry enhancements in three dimensions

We introduce a class of 3d theories consisting of strongly-coupled mathcal{N} = 4 systems coupled to mathcal{N} = 3 Chern-Simons gauge multiplets, which exhibit mathcal{N} = 4 enhancements when a peculiar condition on the Chern-Simons levels is met. An example is the SU(N)3 Chern-Simons theory coupled to the 3d TN theory, which enhances to mathcal{N} = 4 when 1/k1 + 1/k2 + 1/k3 = 0. We also show that some but not all of these mathcal{N} = 4 enhancements can be understood by considering M5-branes on a special class of Seifert manifolds. Our construction provides a large class of mathcal{N} = 4 theories which have not been studied previously.

Edge states and SUSY in (2+1)-dimensional Maxwell Chern–Simons theory

In a [Formula: see text]-dimensional Maxwell–Chern–Simons theory coupled with a fermion and a scalar, which has [Formula: see text] SUSY in the absence of the boundary, supersymmetry is broken on the insertion of a spatial boundary. We show that only a subset of the boundary conditions allowed by the self-adjointness of the Hamiltonian can preserve partial ([Formula: see text]) supersymmetry, while for the remaining boundary conditions SUSY is completely broken. In the latter case, we demonstrate two distinct SUSY-breaking mechanisms. In one scenario, the SUSY-breaking boundary conditions are not consistent with the supersymmetry transformations. In another scenario, despite the boundary conditions being consistent with the SUSY transformations, unpaired fermionic edge states in the domain of the Hamiltonian leads to the breaking of the supersymmetry.

Quantum codes, CFTs, and defects

We give a general construction relating Narain rational conformal field theories (RCFTs) and associated 3d Chern-Simons (CS) theories to quantum stabilizer codes. Starting from an abelian CS theory with a fusion group consisting of n even-order factors, we map a boundary RCFT to an n-qubit quantum code. When the relevant ’t Hooft anomalies vanish, we can orbifold our RCFTs and describe this gauging at the level of the code. Along the way, we give CFT interpretations of the code subspace and the Hilbert space of qubits while mapping error operations to CFT defect fields.

Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling

We consider a simple model for QCD dynamics in which DGLAP integro-differential equation may be solved analytically. This is a gauge model which possesses dominant evolution of gauge boson (gluon) distribution and in which the gauge coupling does not run. This may be N=4 supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with a lower level of supersymmetry, or topological Chern–Simons field theories. We maintain only one term in the splitting function of unintegrated gluon distribution and solve DGLAP analytically for this simplified splitting function. The solution is found using the Cauchy integral formula. The solution restricts the form of the unintegrated gluon distribution as a function of momentum transfer and of Bjorken x. Then, we consider an almost realistic splitting function of unintegrated gluon distribution as an input to DGLAP equation and solve it by the same method which we have developed to solve DGLAP equation for the toy-model. We study a result obtained for the realistic gluon distribution and find a singular Bessel-like behavior in the vicinity of the point x=0 and a smooth behavior in the vicinity of the point x=1.

Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory

We show that the approaches to integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations are closely related, at least classically. Following a suggestion of Kevin Costello, we start from holomorphic Chern-Simons theory on twistor space, defined with the help of a meromorphic (3,0)-form Ω. If Ω is nowhere vanishing, it descends to a theory on 4d space-time with classical equations of motion equivalent to the anti-self-dual Yang-Mills equations. Examples include a 4d analogue of the Wess-Zumino-Witten model and a theory of a Lie algebra valued scalar with a cubic two derivative interaction. Under symmetry reduction, these yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces holomorphic Chern-Simons theory to the 4d Chern-Simons theory with disorder defects studied by Costello & Yamazaki. Finally we show that a similar reduction by a single translation leads to a 5d partially holomorphic Chern-Simons theory describing the Bogomolny equations.

Learning knot invariants across dimensions

We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The three-dimensional invariant of interest is the Jones polynomial J(q)J(q), and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t)Kh(q,t), smooth slice genus gg, and Rasmussen’s ss-invariant. We find that a two-layer feed-forward neural network can predict ss from \text{Kh}(q,-q^{-4})Kh(q,−q−4) with greater than 99%99% accuracy. A theoretical explanation for this performance exists in knot theory via the now disproven knight move conjecture, which is obeyed by all knots in our dataset. More surprisingly, we find similar performance for the prediction of ss from \text{Kh}(q,-q^{-2})Kh(q,−q−2), which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts gg from \text{Kh}(q,t)Kh(q,t) with similarly high accuracy, and we discuss the extent to which the machine is learning ss as opposed to gg, since there is a general inequality |s| ≤2g|s|≤2g. The Jones polynomial, as a three-dimensional invariant, is not obviously related to ss or gg, but the network achieves greater than 95%95% accuracy in predicting either from J(q)J(q). Moreover, similar accuracy can be achieved by evaluating J(q)J(q) at roots of unity. This suggests a relationship with SU(2)SU(2) Chern—Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network’s performance.