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We apply Bayesian optimization and reinforcement learning to a problem in topology: the question of when a knot bounds a ribbon disk. This question is relevant in an approach to disproving the four-dimensional smooth Poincar\'e conjecture; using our programs, we rule out many potential counterexamples to the conjecture. We also show that the programs are successful in detecting many ribbon knots in the range of up to 70 crossings., Comment: 23 pages, 10 figures

Reducing a 6d fivebrane theory on a 3-manifold Y gives a q-series 3-manifold invariant ẑ(Y). We analyse the large-N behaviour of F_K = Ẑ(M_K), where M_K is the complement of a knot K in the 3-sphere, and explore the relationship between an a-deformed (a = qᴺ) version of F_K and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of F_K in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for a-deformed F_K for (2,2p+1)-torus knots. They suggest a further t-deformation based on superpolynomials, which can be used to obtain a t-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how F_K transforms under natural geometric operations on K indicates relations to quantum modularity in a new setting.

We initiate explicit computations of Vafa-Witten invariants of 3-manifolds, analogous to Floer groups in the context of Donaldson theory. In particular, we explicitly compute the Vafa-Witten invariants of 3-manifolds in a family of concrete examples relevant to various surgery operations (the Gluck twist, knot surgeries, log-transforms). We also describe the structural properties that are expected to hold for general 3-manifolds, including the modular group action, relation to Floer homology, infinite-dimensionality for an arbitrary 3-manifold, and the absence of instantons., 27 pages, 5 figures

By studying the properties of $q$-series $\widehat Z$-invariants, we develop a dictionary between 3-manifolds and vertex algebras. In particular, we generalize previously known entries in this dictionary to Lie groups of higher rank, to 3-manifolds with toral boundaries, and to BPS partition functions with line operators. This provides a new physical realization of logarithmic vertex algebras in the framework of the 3d-3d correspondence and opens new avenues for their future study. For example, we illustrate how invoking a knot-quiver correspondence for $\widehat{Z}$-invariants leads to many infinite families of new fermionic formulae for VOA characters., 85 pages, 3 figures, 6 tables

We study quantum fluctuations in the lightcone metric of the 4-d Einstein-Hilbert action via dimensional reduction to Jackiw-Teitelboim (JT) gravity. In particular, we show that, in Einstein gravity, the causal development of a region in flat Minkowski spacetime, near a horizon defined by light sheets, can be described by an effective two-dimensional dilaton theory. This enables us to make use of known solutions of the JT action, where the spacetime position of a horizon has quantum uncertainty due to metric fluctuations. This quantum uncertainty can be then directly related to the original 4-d light cone coordinates, allowing us to compute the uncertainty in the time of a photon to travel from tip-to-tip of a causal diamond in flat 4-d Minkowski space. We find that both Planck and infrared scales (with the latter set by the size of the causal diamond) enter the uncertainty in photon travel time, such that the quantum fluctuation in the arrival time may be observably large., 25 pages, 5 figures

We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus and SO(3) gauge group., Comment: 39 pages, 6 figures

75 pages, 4 figures; We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic $q$. Both types of invariants are labeled by extra data which plays an important role in the proposed relation. Bridging the two sides -- which until recently were developed independently, using very different methods -- opens many new avenues. In one direction, it allows to study (and perhaps even to formulate) $q$-series invariants labeled by spin$^c$ structures in terms of non-semisimple invariants. In the opposite direction, it offers new insights and perspectives on various elements of non-semisimple TQFT's, bringing the latter into one unifying framework with other invariants of knots and 3-manifolds that recently found realization in quantum field theory and in string theory.

Motivated by string dualities we propose topological gravity as the early phase of our universe. The topological nature of this phase naturally leads to the explanation of many of the puzzles of early universe cosmology. A concrete realization of this scenario using Witten's four dimensional topological gravity is considered. This model leads to the power spectrum of CMB fluctuations which is controlled by the conformal anomaly coefficients $a,c$. In particular the strength of the fluctuation is controlled by $1/a$ and its tilt by $c g^2$ where $g$ is the coupling constant of topological gravity. The positivity of $c$, a consequence of unitarity, leads automatically to an IR tilt for the power spectrum. In contrast with standard inflationary models, this scenario predicts $\mathcal{O}(1)$ non-Gaussianities for four- and higher-point correlators and the absence of tensor modes in the CMB fluctuations., 27 pages, 3 figures

We take a peek at a general program that associates vertex (or chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras.

We show how networks of Wilson lines realize quantum groups U_q(sl(m)), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface operators we reproduce known mathematical constructions of categorical representations and categorified quantum groups., Comment: 62 pages, 33 figures, minor changes, references added

We take a peek at a general program that associates vertex (or, chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras., Comment: 40 pp, a new figure, many improvements and references added

We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs., 119 pages, 10 figures and 20 tables

We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres-Douglas theories on $S^1 \times M_3$ with a non-trivial holonomy of a discrete global symmetry along the $S^1$. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern-Simons and Rozansky-Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the $(G_2)_1$ and $(F_4)_1$ Chern-Simons theories. Our construction is based on an intriguing relation between topologically twisted partition functions, wild Hitchin characters, and chiral algebras which, when combined together, relate Coulomb branch and Higgs branch data of the same 4d $\mathcal{N}=2$ theory. We test our proposal by applying localization techniques to the conjectural $\mathcal{N}=1$ UV Lagrangian descriptions of the $(A_1,A_2)$, $(A_1,A_3)$ and $(A_1,D_3)$ theories., Comment: 6 pages

We take a peek at a general program that associates vertex (or, chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras., 40 pp, a new figure, many improvements and references added

Searching for the simplest non-abelian 2d gauge theory with $\mathcal{N}=(0,2)$ supersymmetry and non-trivial IR physics, we propose a new duality for $SU(2)$ SQCD with $N_f = 4$ chiral flavors. The chiral algebra of this theory is found to be $\mathfrak{so}(8)_{-2}$, the same as in 4d $\mathcal{N}=2$ $SU(2)$ gauge theory with four hypermultiplets., Comment: 4 pages plus references; v2: typos corrected, references added

We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d $\mathcal{N}=(0,2)$ theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations., 67 pages, 11 figures

Searching for the simplest non-abelian 2d gauge theory with $\mathcal{N}=(0,2)$ supersymmetry and non-trivial IR physics, we propose a new duality for $SU(2)$ SQCD with $N_f = 4$ chiral flavors. The chiral algebra of this theory is found to be $\mathfrak{so}(8)_{-2}$, the same as in 4d $\mathcal{N}=2$ $SU(2)$ gauge theory with four hypermultiplets., 4 pages plus references; v2: typos corrected, references added

We study resurgence properties of partition function of SU(2) Chern-Simons theory (WRT invariant) on closed three-manifolds. We check explicitly that in various examples Borel transforms of asymptotic expansions posses expected analytic properties. In examples that we study we observe that contribution of irreducible flat connections to the path integral can be recovered from asymptotic expansions around abelian flat connections. We also discuss connection to Floer instanton moduli spaces, disk instantons in 2d sigma models, and length spectra of "complex geodesics" on the A-polynomial curve., 56 pages, 19 figures. v2: references added

We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previously by the second and third author. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. Finally, we prove that these dimensions form a one-parameter family of $1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra, which is a one-parameter family of Frobenius algebras. We construct this one-parameter family of Frobenius algebras as a deformation of the classical Verlinde algebra for $G$., Comment: 39 pages; references added, misprints corrected and exposition improved

Compactification of the 6d fivebrane theory on 3-manifolds is believed to result in a particular class of 3d N = 2 theories, whose physical properties contain rich information about geometry and topology of 3-manifolds. In this talk, I will survey the status of this subject, from basic ideas to what in financial markets are called “stress test” examples, which expose limitations, challenges, and open questions that require further development.

This is the seventh article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J.Teschner. It discusses an interesting class of observables localised on surfaces that attracts steadily growing attention. In the correspondence to conformal field theory some of these observables get related to a class of fields in two dimensions called degenerate fields. These fields satisfy differential equations that can be used to extract a lot of information on the correlation functions. Understanding the origin of these differential equations within gauge theory may help explaining the AGT-correspondence itself., 37 pages, see also overview article arXiv:1412.7145

This workshop focused on the interactions between string theory and geometry. Since its birth, the development of string theory has triggered major progress in mathematics. Experts from both communities from all over the world took part and created an environment which favored a fruitful scientific exchange. Among the key topics discussed during the workshop, we should mention derived algebraic geometry and topological field theory, localization formulas in supersymmetric quantum field theories, links between conformal field theory, susy and integrability, new trends in knot theory., Il Colle di Galileo, Vol 3, No 2 (2014)

Motivated by the connection between 4-manifolds and 2d N = (0, 2) theories, we study the dynamics of a fairly large class of 2d N = (0, 2) gauge theories. We see that physics of such theories is very rich, much as the physics of 4d N = 1 theories. We discover a new type of duality that is very reminiscent of the 4d Seiberg duality. Surprisingly, the new 2d duality is an operation of order three: it is IR equivalence of three different theories and, as such, is actually a triality. We also consider quiver theories and study their triality webs. Given a quiver graph, we find that supersymmetry is dynamically broken unless the ranks of the gauge groups and flavor groups satisfy stringent inequalities. In fact, for most of the graphs these inequalities have no solutions. This supports the folklore theorem that generic 2d N = (0, 2) theories break supersymmetry dynamically.

We propose a unified approach to a general class of codimension-2 defects in field theories with non-trivial duality symmetries and discuss various constructions of such "duality defects" in diverse dimensions. In particular, in d=4 we propose a new interpretation of the Seiberg-Witten u-plane by "embedding" it in the physical space-time: we argue that it describes a BPS configuration of two duality defects (at the monopole/dyon points) and propose its vast generalization based on Lefschetz fibrations of 4-manifolds., Comment: 6 pages, 4 figures. v2: references added

We revisit the role of loop and surface operators as order parameters for gapped phases of four-dimensional gauge theories. We show that in some cases surface operators are confined, and that this fact can be used to distinguish phases which are not distinguished by the Wilson-'t Hooft criterion. The long-distance behavior of loop and surface operators which are neither confined nor screened is controlled by a 4d TQFT. We construct these TQFTs for phases which are characterized by the presence of electrically and/or magnetically charged condensates. Interestingly, the TQFT describing a phase with a nonabelian monopole condensate is based on the theory of nonabelian gerbes. We also show that in phases with a dyonic condensate the low-energy theta-angle is quantized., 10 pages, latex

Motivated by the connection between 4-manifolds and 2d N=(0,2) theories, we study the dynamics of a fairly large class of 2d N=(0,2) gauge theories. We see that physics of such theories is very rich, much as the physics of 4d N=1 theories. We discover a new type of duality that is very reminiscent of the 4d Seiberg duality. Surprisingly, the new 2d duality is an operation of order three: it is IR equivalence of three different theories and, as such, is actually a triality. We also consider quiver theories and study their triality webs. Given a quiver graph, we find that supersymmetry is dynamically broken unless the ranks of the gauge groups and flavor groups satisfy stringent inequalities. In fact, for most of the graphs these inequalities have no solutions. This supports the folklore theorem that generic 2d N=(0,2) theories break supersymmetry dynamically., 31 pages, 12 figures

We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0,2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines / walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N=(0,2) theories and 3d N=2 theories, respectively, Comment: 81 pages, 18 figures. v2: misprints corrected, clarifications and references added. v3: additions and corrections about lens space theory, 4-manifold gluing, smooth structures

We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic.", 24 pages, 4 figures

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N=2 and N=4 twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant. This paper is based on a lecture delivered at the International Congress on Mathematical Physics 2006, Rio de Janeiro, and at the RTN Workshop 2006, Napoli., 37 pages. Based on a talk given at the ICMP 2006 and at the RTN Workshop 2006

The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for other Lie groups and representations. In particular, we introduce a new triply graded theory categorifying the Kauffman polynomial, test it, and predict the Kauffman homology for several simple knots., 45 pages, harvmac

We consider families of theories with large N=4 superconformal symmetry. We define an index generalizing the elliptic genus of theories with N=2 symmetry. In contrast to the N=2 case, the new index constrains part of the non-BPS spectrum. Motivated by aspects of the AdS/CFT correspondence we study the index in the examples of symmetric product theories. We give a physical interpretation of the Hecke operators which appear in the expressions for partition functions of such theories. Finally, we compute the index for a nontrivial example of a symmetric product theory., 62 pages, harvmac; references added

The problem of finding a holographic CFT dual to string theory on AdS(3)xS(3)xS(3)xS(1) is examined in depth. This background supports a large N=4 superconformal symmetry. While in some respects similar to the familiar small N=4 systems on AdS(3)xS(3)xK3 and AdS(3)xS(3)xT4, there are important qualitative differences. Using an analog of the elliptic genus for large N=4 theories we rule out all extant proposals -- in their simplest form -- for a holographic duality to supergravity at generic values of the background fluxes. Modifications of these extant proposals and other possible duals are discussed., 88 pages, 2 figures, harvmac; references added

The bulk partition function of pure Chern-Simons theory on a three-manifold is a state in the space of conformal blocks of the dual boundary RCFT, and therefore transforms non-trivially under the boundary modular group. In contrast the bulk partition function of AdS(3) string theory is the modular-invariant partition function of the dual CFT on the boundary. This is a puzzle because AdS(3) string theory formally reduces to pure Chern-Simons theory at long distances. We study this puzzle in the context of massive Chern-Simons theory. We show that the puzzle is resolved in this context by the appearance of a chiral "spectator boson" in the boundary CFT which restores modular invariance. It couples to the conformal metric but not to the gauge field on the boundary. Consequently, we find a generalization of the standard Chern-Simons/RCFT correspondence involving "nonholomorphic conformal blocks" and nonrational boundary CFTs. These generalizations appear in the long-distance limit of AdS(3) string theory, where the role of the spectator boson is played by other degrees of freedom in the theory., Comment: 43 pages, harvmac

We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as target space. Mirror symmetry and various string and M-theory dualities involving G_2 manifolds may be incorporated into this framework. To give some examples, we construct two non-compact manifolds with G_2 structures: one with a K3 fibration, and one with a torus fibration and a metric of G_2 holonomy. Kaluza-Klein reduction of the latter solution gives abelian BPS monopoles in 3+1 dimensions., Comment: 40 pages, 2 figures, LaTeX

Using D2-brane probes, we study various properties of M-theory on singular, non-compact manifolds of G_2 and Spin(7) holonomy. We derive mirror pairs of N=1 supersymmetric three-dimensional gauge theories, and apply this technique to realize exceptional holonomy manifolds as both Coulomb and Higgs branches of the D2-brane world-volume theory. We derive a ``G_2 quotient construction'' of non-compact manifolds which admit a metric of G_2 holonomy. We further discuss the moduli space of such manifolds, including the structure of geometrical transitions in each case. For completeness, we also include familiar examples of manifolds with SU(3) and Sp(2) holonomy, where some of the new ideas are clarified and tested., Comment: 79 pages, Latex

We find a one-parameter family of new G₂ holonomy metrics and demonstrate that it can be extended to a two-parameter family. These metrics play an important role as the supergravity dual of the large N limit of four-dimensional supersymmetric Yang–Mills. We show that these G₂ holonomy metrics describe the M theory lift of the supergravity solution describing a collection of D6-branes wrapping the supersymmetric three-cycle of the deformed conifold geometry for any value of the string coupling constant.

We discuss twisted states of AdS orbifolds which couple to N = 2 chiral primary operators not invariant under exchange of the gauge factors. Kaluza-Klein reduction on the fixed circle gives the correct conformal dimensions of operators in the superconformal theory and involves some aspects of monopole dynamics in the non-trivial background. As a byproduct we found evidence for decoupling of U(1) factors in the four-dimensional gauge theory.

We discuss twisted states of AdS orbifolds which couple to N=2 chiral primary operators not invariant under exchange of the gauge factors. Kaluza-Klein reduction on the fixed circle gives the correct conformal dimensions of operators in the superconformal theory and involves some aspects of monopole dynamics in the non-trivial background. As a byproduct we found evidence for decoupling of U(1) factors in the four-dimensional gauge theory., 10 pages, harvmac, new states considered, exposition improved

The low energy effective actions of the N = 2 SUSY SU(N_c) QCD are considered at the symmetric point on the moduli space. The classes of such theories have similar spectral curves. This fact allows us to show that all these models have the same structure of the coupling matrix and to show that the N_f = 2N_c spectral curve can not be presented as a double covering of the sphere. We calculate first instanton contributions to the coupling matrix and get nonperturbative β-functions in the SU(2) gauge theory with non-zero bare masses of the matter hypermultiplets.

A física teòrica, la Teoria Quàntica de Camps (TQC) és un marc teòric extremadament reeixit que combina la Relativitat Especial amb la Mecànica Quàntica, permetent el diseny de models físics de les partícules subatómiques i quasipartícules que descriuen els aspectes més fundamentals de la matèria amb una precisió increïblement alta. Entre aquestes teories, la TQC Chern-Simons és una especial, que no només descriu fenòmens topològics a la física com ara l’Efecte Hall Quàntic, sinó que encaixa amb la noció del que es coneix com una Teoria Topològica de Camps Quàntics (TTCQ). Va ser fent servir aquests axiomes de les TTCQs que Edward Witten va mostrar al 1989 com d’estretament relacionada està la teoria de Chern-Simons amb l’àmbit d’invariants polinòmics que apareixen a la Teoria de Nusos, com ara el ben conegut polinomi de Jones. En aquests darrers anys, investigacions en aquest camp han donat lloc a nous i més poderosos invariants d’enllaços i, a través de cirurgies de Dehn sobre ells, de 3-varietats també. Per exemple, la sèrie de Gukov-Manolescu recentment proposta el 2020 —denotada FK(x, q)— és un invariant conjectural de complements de nusos que, en cert sentit, continua analíticament els polinomis de Jones colorejats. Poc després, Sunghyuk Park va introduir l’enfoc de la Matriu R de Gran Color corresponent a sl(2,C) per estudiar FK per trenats positius i calcular FK per a diversos nusos i enllaços. Aquest procediment ha estat així mateix extès per Angus Gruen a totes les altres àlgebres de Lie sl(n+1) més enllà de sl(2). En aquest treball, després d’un extens repàs sobre els anteriorment esmentats conceptes, abordem la família so(2n) d’àlgebres de Lie semisimples sobre els complexos a la classificació de Cartan, centrant-nos principalment en el cas so(8) atrets per la simetria triple al seu diagrama de Dynkin D4. En física teórica, la Teoría Cuántica de Campos (TCC) es un marco teórico extremadamente exitoso que combina la Relatividad Especial con la Mecánica Cuántica, permitiendo el diseño de modelos físicos de las partículas subatómicas y cuasipartículas que describen los aspectos más fundamentales de la materia con una precisión increíblemente alta. Entre dichas teorías, la TCC Chern-Simons es una especial, que no sólo describe fenómenos topológicos en física tales como el Efecto Hall Cuántico, sino que encaja con la noción de lo que se conoce como una Teoría Topológica de Campos Cuánticos (TTCC). Fue utilizando estos axiomas de las TTCCs que Edward Witten mostró en 1989 cómo de estrechamente relacionada está la teoría de Chern-Simons con el ámbito de invariantes polinómicos que aparecen en la Teoría de Nudos, tales como el bien conocido polinomio de Jones. En estos últimos años, investigaciones en este campo han dado lugar a nuevos y más poderosos invariantes de enlaces y, a través de cirugías de Dehn sobre ellos, así mismo de 3-variedades. Por ejemplo, la serie de Gukov-Manolescu recientemente propuesta en 2020 —denotada FK(x, q)— es un invariante conjetural de complementos de nudos que, en cierto sentido, continúa analíticamente los polinomios de Jones coloreados. Poco después, Sunghyuk Park introdujo el enfoque de la Matriz R de Gran Color correspondiente a sl(2,C) para estudiar FK para trenzados positivos y calcular FK para varios nudos y enlaces. Este procedimiento ha sido a su vez extendido por Angus Gruen a todas las otras álgebras de Lie sl(n+1) más allá de sl(2). En la presente obra, tras un extenso repaso sobre los anteriormente mencionados conceptos, abordamos la família so(2n) de álgebras de Lie semisimples sobre los complejos en la clasificación de Cartan, centrándonos principalmente en el caso so(8) atraídos por la simetría triple en su diagrama de Dynkin D4. In theoretical physics, Quantum Field Theory (QFT) is an extremely successful theoretical framework combining both Special Relativity and Quantum Mechanics, enabling to design physical models of subatomic particles and quasiparticles describing the most fundamental aspects of matter with an incredibly high accuracy. Among these theories, the Chern-Simons QFT is a special one, not only describing topological phenomena in physics such as the Quantum Hall Effect, but also fitting the notion of what is known as a Topological Quantum Field Theory (TQFT). It was by using the axioms of TQFTs that Edward Witten showed back in 1989 how closely related the Chern-Simons theory is to the realm of polynomial invariants appearing in Knot Theory, such as the well-known Jones polynomial. In the past years, further research in this field has led to new and more powerful invariants of links and, by means of Dehn surgeries on them, of 3-manifolds as well. For instance, the Gukov-Manolescu series proposed recently in 2020 —denoted FK(x, q)— is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. Shortly after, Sunghyuk Park introduced the Large Color R-matrix approach for sl(2,C) to study FK for some simple links, giving a definition of FK for positive braid knots and computing FK for various knots and links. This procedure has in turn been extended by Angus Gruen to all other Lie algebras sl(n+1) beyond sl(2). In this work, after a broad review on the above mentioned background, we move on to the family so(2n) of complex semisimple Lie algebras in Cartan’s classification, mainly focusing on the so(8) case attracted by the three-fold symmetry in its Dynkin diagram D4. Outgoing

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the A-polynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the "color behavior" of the colored sl(2) knot homology, and eventually to a similar version for the colored HOMFLY homology. Furthermore, this deformation is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology., Comment: 48 pages, 9 figures. Comments are welcome