Chern-Simons Theory Research Articles

Quantum chaos in 2D gravity

We present a quantitative and fully non-perturbative description of the ergodic phase of quantum chaos in the setting of two-dimensional gravity. To this end we describe the doubly non-perturbative completion of semiclassical 2D gravity in terms of its associated universe field theory. The guiding principle of our analysis is a flavor-matrix theory (fMT) description of the ergodic phase of holographic gravity, which exhibits \mathrm{U}(n|n)U(n|n) causal symmetry breaking and restoration. JT gravity and its 2D-gravity cousins alone do not realize an action principle with causal symmetry, however we demonstrate that their universe field theory, the Kodaira-Spencer (KS) theory of gravity, does. After directly deriving the fMT from brane-antibrane correlators in KS theory, we show that causal symmetry breaking and restoration can be understood geometrically in terms of different (topological) D-brane vacua. We interpret our results in terms of an open-closed string duality between holomorphic Chern-Simons theory and its closed-string equivalent, the KS theory of gravity. Emphasis will be put on relating these geometric principles to the characteristic spectral correlations of the quantum ergodic phase.

Homotopy Double Copy of Noncommutative Gauge Theories

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of L∞-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity.

Weak gravitational lensing and shadow cast by rotating black holes in axionic Chern–Simons theory

We investigate the impact of the axionic coupling parameter on the bending angle of light and the shadow cast by slowly rotating black holes in Chern–Simons modified gravity. We utilize the Ishihara et al. method to derive the deflection angle of light for an observer and source located at finite distances from a lens object in an asymptotically flat spacetime, using the Gauss–Bonnet theorem. The deflection angle exhibits an increasing trend up to a certain point, followed by a decrease as a function of the impact parameter, with the presence of the axion matter field causing the observed increase. Additionally, we calculate the Einstein ring radius as a direct application of the weak deflection angle. We also investigate the effect of the axion matter field on the time delay of light and analyze its impact on the shadow cast by slowly rotating black holes. Our findings reveal a significant effect of the axionic coupling parameter on the black hole’s shadow.

Higher Chern-Simons based on (2-)crossed modules

We present higher Chern-Simons theories based on (2-)crossed modules. We start from the generalized differential forms in Generalized Differential Calculus and define the corresponding generalized connections which consist of higher connections. Then we establish the generalized Chern-Simons forms to get the higher Chern-Simons actions. Finally, we develop the higher second Chern forms and Chern-Weil theorems.

Kinematic Lie Algebras from Twistor Spaces.

We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying BV^{▪}-algebra, extending the ideas of Reiterer [A homotopy BV algebra for Yang-Mills and color-kinematics, arXiv:1912.03110.]. Conversely, we show that any theory with a BV^{▪}-algebra features a kinematic Lie algebra that controls interaction vertices, both on shell and off shell. We explain that the archetypal example of a theory with a BV^{▪}-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The BV^{▪}-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann Chern-Simons theories come with BV^{▪}-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.

Boundary modes in the Chamon model

We study the fracton phase described by the Chamon model in a manifold with a boundary. The new processess and excitations emerging at the boundary can be understood by means of a diagrammatic framework. From a continuum perspective, the boundary theory is described by a set of scalar fields in similarity with standard KK-matrix Chern-Simons theory. The continuum theory recovers the gapped boundaries of the lattice model once we include sufficiently strong interactions that break charge conservation. The analysis of the perturbative relevance of the leading interactions reveals a regime in which the Chamon model can have a stable gapless fractonic phase at its boundary.

Keeping matter in the loop in dS3 quantum gravity

We propose a mechanism that couples matter fields to three-dimensional de Sitter quantum gravity. Our construction is based on the Chern-Simons formulation of three-dimensional Euclidean gravity, and it centers on a collection of Wilson loops winding around Euclidean de Sitter space. We coin this object a Wilson spool. To construct the spool, we build novel representations of \U0001d530\U0001d532(2). To evaluate the spool, we adapt and exploit several known exact results in Chern-Simons theory. Our proposal correctly reproduces the one-loop determinant of a free massive scalar field on S3 as GN → 0. Moreover, allowing for quantum metric fluctuations, it can be systematically evaluated to any order in perturbation theory.

Chiral Loop Quantum Supergravity and Black Hole Entropy

Recent work has shown that local supersymmetry on a spacetime boundary in N-extended AdS supergravity in chiral variables implies coupling to a boundary OSp(N|2)C super Chern–Simons theory. Consequently there has been a proposal to define and calculate the entropy S for the boundary, in the supersymmetric version of loop quantum gravity, for the minimal case N=1, via this super Chern–Simons theory. We give an overview of how supergravity can be treated in loop quantum gravity. We review the calculation of the dimensions of the quantum state spaces of UOSp(1|2) super Chern–Simons theory with punctures, and its analytical continuation, for the fixed quantum super area of the surface, to OSp(1|2)C. The result is S=aH/4 for large (super) areas. Lower order corrections can also be determined. We begin also a discussion of the statistical mechanics of the surface degrees of freedom by calculating the grand canonical partition function at zero chemical potential. This is a new result.

Peacock patterns and resurgence in complex Chern–Simons theory

The partition function of complex Chern–Simons theory on a 3-manifold with torus boundary reduces to a finite-dimensional state-integral which is a holomorphic function of a complexified Planck’s constant tau in the complex cut plane and an entire function of a complex parameter u. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear q-difference equation. We further conjecture that a distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte–Gaiotto–Gukov. We provide proofs of our statements regarding the q-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic textbf{4}_1 and textbf{5}_{2} knots.

Modified Villain formulation of Abelian Chern-Simons theory

We formulate $U(1)_k$ Chern-Simons theory on a Euclidean spacetime lattice using the modified Villain approach. Various familiar aspects of continuum Chern-Simons theory such as level quantization, framing, the discrete 1-form symmetry and its 't Hooft anomaly, as well as the electric charge of monopole operators are manifest in our construction. The key technical ingredient is the cup product and its higher generalizations on the (hyper-)cubic lattice, which recently appeared in the literature. All unframed Wilson loops are projected out by a peculiar subsystem symmetry, leaving topological, ribbon-like Wilson loops which have the correct correlation functions and topological spins expected from the continuum theory. Our action can be obtained from a new definition of the theta term in four dimensions which improves upon previous constructions within the modified Villain approach. This bulk action coupled to background fields for the 1-form symmetry is given by the Pontryagin square, which provides anomaly inflow directly on the lattice.

Two-fold refinement of non simply laced Chern-Simons theories

Inspired by the two-parameter Macdonald's deformation of the formulae for non simply laced simple Lie algebras, we propose a two-fold refinement of the partition function of the corresponding Chern-Simons theory on S3. It is based on a two-fold refinement of the Kac-Peterson formula for the volume of the fundamental domain of the coroot lattice of non simply laced Lie algebras. We further derive explicit integral representations of the two-fold refined Chern-Simons partition functions. We also present the corresponding generalized universal-like expressions for them. With these formulae in hand, one can try to investigate a possible duality of the corresponding Chern-Simons theories with hypothetical two-fold refined topological string theories.

Relationship of the Hennings and Witten–Reshetikhan–Turaev invariants for higher rank quantum groups

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen et al. [On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146(1) (2009) 151–163; Three-manifold invariants associated with restricted quantum groups, Math. Z. 272(3–4) (2012) 987–999] which relates the Hennings and WRT invariants for [Formula: see text] and [Formula: see text] for arbitrary rational homology three-spheres.

The Hamilton–Jacobi analysis for higher-order Chern–Simons gravity

The Hamilton–Jacobi [ H J ] analysis for higher-order Chern–Simons gravity is performed. The complete set of H J Hamiltonians is identified and a fundamental H J differential is constructed, from which the characteristic equations are obtained. In addition, the symmetries of the theory are identified and the results are compared with other approaches reported in the literature. • We report the Hamilton–Jacobi [HJ] analysis for higher order Chern-simons theory. • We report a fundamental HJ differential. • The complete Hamiltonians are identified. • The generalized HJ brackets are constructed. • The gauge transformations are identified.

Non-invertible symmetries of class S theories

We study the non-invertible symmetries of class mathcal{S} theories obtained by compactifying the type {mathfrak{a}}_{p-1} 6d (2,0) theory on a genus g Riemann surface with no punctures. After setting up the general framework, we describe how such symmetries can be classified up to genus 5. Of central interest to us is the question of whether a non-invertible symmetry is intrinsic, i.e. whether it can be related to an invertible symmetry by discrete gauging. We then describe the higher-dimensional origin of our results, and explain how the Anomaly and Symmetry TFTs, as well as N-ality defects, of class mathcal{S} theories can be obtained from compactification of a 7d Chern-Simons theory. Interestingly, we find that the Symmetry TFT for theories with intrinsically non-invertible symmetries can only be obtained by coupling the 7d Chern-Simons theory to topological gravity.

Complex saddles of three-dimensional de Sitter gravity via holography

We determine complex saddles of three-dimensional gravity with a positive cosmological constant by applying the recently proposed holography. It is sometimes useful to consider a complexified metric to study quantum gravity as in the case of the no-boundary proposal by Hartle and Hawking. However, there would be too many saddles for complexified gravity, and we should determine which saddles to take. We describe the gravity theory by three-dimensional SL$(2,\mathbb{C})$ Chern-Simons theory. At the leading order in the Newton constant, its holographic dual is given by Liouville theory with a large imaginary central charge. We examine geometry with a conical defect, called a de Sitter black hole, from a Liouville two-point function. We also consider geometry with two conical defects, whose saddles are determined by the monodromy matrix of Liouville four-point function. Utilizing Chern-Simons description, we extend the similar analysis to the case with higher-spin gravity.

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.

Resurgence, conformal blocks, and the sum over geometries in quantum gravity

In two dimensional conformal field theories the limit of large central charge plays the role of a semi-classical limit. Certain universal observables, such as conformal blocks involving the exchange of the identity operator, can be expanded around this classical limit in powers of the central charge c. This expansion is an asymptotic series, so — via the same resurgence analysis familiar from quantum mechanics — necessitates the existence of non-perturbative effects. In the case of identity conformal blocks, these new effects have a simple interpretation: the CFT must possess new primary operators with dimension of order the central charge. This constrains the data of CFTs with large central charge in a way that is similar to (but distinct from) the conformal bootstrap. We study this phenomenon in three ways: numerically, analytically using Zamolodchikov’s recursion relations, and by considering non-unitary minimal models with large (negative) central charge. In the holographic dual to a CFT2, the expansion in powers of c is the perturbative loop expansion in powers of ћ. So our results imply that the graviton loop expansion is an asymptotic series, whose cure requires the inclusion of new saddle points in the gravitational path integral. In certain cases these saddle points have a simple interpretation: they are conical excesses, particle-like states with negative mass which are not in the physical spectrum but nevertheless appear as non-manifold saddle points that control the asymptotic behaviour of the loop expansion. This phenomenon also has an interpretation in SL(2, ℝ) Chern-Simons theory, where the non-perturbative effects are associated with the non-Teichmüller component of the moduli space of flat connections.

How do axisymmetric black holes grow monopole and dipole hair?

We study the dynamical formation of scalar monopole and dipole hair in scalar Gauss-Bonnet theory and dynamical Chern-Simons theory. We prove that the spherically symmetric mode of the dipole hair is completely determined by the product of the mass of the spacetime and the value of the monopole hair. We then show that the dynamics of the $\ensuremath{\ell}=1$ mode of the dipole hair is intimately tied to the appearance of the event horizon during axisymmetric collapse, which results in the radiation of certain modes that could have been divergent in the future of the collapse. We confirm these analytical predictions by simulating the gravitational collapse of a rapidly rotating neutron star in the decoupling limit, both in scalar Gauss-Bonnet and dynamical Chern-Simons theory. Our results, combined with those in Hegade K. R. et al. [Phys. Rev. D 105, 064041 (2022)], provide a clear physical picture of the dynamics of scalar monopole and dipole radiation in axisymmetric and spherical gravitational collapse in these theories.

Quantum Riemann-Hilbert problems for the resolved conifold

We study the quantum Riemann-Hilbert problems determined by the refined Donaldson-Thomas theory on the resolved conifold. Using the solutions to classical Riemann-Hilbert problems in [6] we give explicit solutions in terms of multiple sine functions with unequal parameters. The new feature of the solutions is that the valid region of the quantum parameter q12=exp⁡(πiτ) varies on the space of stability conditions and BPS t-plane. Comparing the solutions with the partition function of refined Chern-Simons theory and invoking large N string duality, we find that the solution contains the non-perturbative completion of the refined topological string on the resolved conifold. Therefore solving the quantum Riemann-Hilbert problems provides a possible non-perturbative definition for the Donaldson-Thomas theory.

Rotational symmetry protected edge and corner states in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with non-trivial strong topological invariants (protected by internal symmetries) with additional "weak" topological indices. In this paper, we study the edge physics of systems with a non-trivial shift invariant, which is protected by either a continuous $\text{U}(1)_r$ or discrete $\text{C}_n$ rotation symmetry, along with internal $\text{U}(1)_c$ charge conservation. Specifically, we construct an interface between two systems which have the same Chern number but are distinguished by their Wen-Zee shift and, through analytic arguments supported by numerics, show that the interface hosts counter-propagating gapless edge modes which cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian sub-algebra framework for gapped boundaries while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and compute them in the case where the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in symmetry enriched topological phases.