We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.
We briefly review the instanton-torus knot duality found in 5d SUSY gauge theories with one compact dimension. The fermion condensate turn out to be the generating function for the torus knot polynomials colored by the fundamental representation.
A number of results for the level-rank duality of $G(N)_K$ $\leftrightarrow$ $G(K)_N$ Chern-Simons theory are summarized, with emphasis on the applications to knot and link invariants. Explicit examples for $SU(2)_K$ $\leftrightarrow$ $SU(K)_2$ illustrate general results. A criterion to distinguish torus knots and links from hyperbolic knots and links, based on tables constructed by Kaul for one and two strand invariants, is presented. Possible symmetries of hyperbolic knot and link invariants are discussed. The level-rank duality of torus knot and link invariants of minimal models is examined
We formulate large $N$ duality of $\mathrm{U}(N)$ refined Chern-Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $Ω$-background. This form enables us to relate refined Chern-Simons invariants of a torus knot/link in $S^3$ to refined BPS invariants in the resolved conifold. Assuming that the extra $\mathrm{U}(1)$ global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2-M5 bou…
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in …
Knotted solutions to electromagnetism are investigated as an independent subsector of the theory. We write down a Lagrangian and a Hamiltonian formulation of Bateman's construction for the knotted electromagnetic solutions. We introduce a general definition of the null condition and generalize the construction of Maxwell's theory to massless free complex scalar, its dual two form field, and to a massless DBI scalar. We set up the framework for quantizing the theory both in a path integral approach, as well as the canonical Dirac method for a constrained system. We make several observations abo…
We establish the duality between the torus knot superpolynomials or the Poincaré polynomials of the Khovanov homology and particular condensates in $Ω$-deformed 5D supersymmetric QED compactified on a circle with 5d Chern-Simons(CS) term. It is explicitly shown that $n$-instanton contribution to the condensate of the massless flavor in the background of four-observable, exactly coincides with the superpolynomial of the $T(n,nk+1)$ torus knot where $k$ - is the level of CS term. In contrast to the previously known results, the particular torus knot corresponds not to the partition function of t…
We reconsider topological string realization of SU(N) Chern-Simons theory on S^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) in two variables x,p depending on the t'Hooft coupling parameter Q=e^{Ng_s}. Its vanishing locus is the quantum corrected moduli space of a special Lagrangian brane L_K, associated to K, probing the large N dual geometry, the resolved conifold. Using a generalized SYZ conjecture this leads to the statement that for every such Lagrangian brane L_K we get a distinct mirror of the resolved conifold given by uv=A_K(x,p;Q). Perturbative corrections…
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" and (quantum) A-polynomial structures. We pay a special attention to identity between the A-polynomial equations for knots and Baxter equations for quantum relativistic integrable systems, related through Seiberg-Witten theory to 5d super-Yang-Mills models and through the AGT rel…
The BCJ decomposition is a highly non-trivial property of gauge theories. In this paper we systematically construct an effective Lagrangian, whose Feynman rules automatically produce the BCJ numerators. The effective Lagrangian contains non-local terms. The difference between the standard Yang-Mills Lagrangian and the effective Lagrangian simplifies to zero.
We find general non-linear lagrangians of a U(1) field invariant under electric-magnetic duality. They are characterized by an arbitrary function and go to the Maxwell theory in the weak field limit. We give some explicit examples which are generalizations of the Born-Infeld theory.
We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry.
We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Marino. Comparing to the recent proposal by Aganagic and Vafa for knots on S^3, we demonstrate that th…
Using two different methods inspired by duality transformations we present the equivalence between effective Lagrangians for massive vector mesons using a vector field and an antisymmetric tensor field. This completes the list of explicit field transformations between the various effective Lagrangian methods to describe massive vector and axial vector mesons.
The formalism of spacetime dependent lagrangians developed in Ref.1 is applied to the Sine Gordon and massive Thirring models.It is shown that the well-known equivalence of these models (in the context of weak-strong duality) can be understood in this approach from the same considerations as described in [1] for electromagnetic duality. A further new result is that all these can be naturally linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of {\it noncommuting coodinates} residing on the boundaries.…
We study the action of the $SL(2;R)$ group on the noncommutative DBI Lagrangian. The symmetry conditions of this theory under the above group will be obtained. These conditions determine the extra U(1) gauge field. By introducing some consistent relations we observe that the noncommutative (or ordinary) DBI Lagrangian and its $SL(2;R)$ dual theory are dual of each other. Therefore, we find some $SL(2;R)$ invariant equations. In this case the noncommutativity parameter, its $T$-dual and its $SL(2;R)$ dual versions are expressed in terms of each other. Furthermore, we show that on the effective …
Starting from lagrangian field theory and the variational principle, we show that duality in equations of motion can also be obtained by introducing explicit spacetime dependence of the lagrangian. Poincare invariance is achieved precisely when the duality conditions are satisfied in a particular way. The same analysis and criteria are valid for both abelian and nonabelian dualities. We illustrate how (1)Dirac string solution (2)Dirac quantisation condition (3)t'Hooft-Polyakov monopole solutions and (4)a procedure emerges for obtaining {\it new} classical solutions of Yang-Mills (Y-M) theory. …
We explore the properties of polynomial Lagrangians for chiral $p$-forms previously proposed by the last named author, and in particular, provide a self-contained treatment of the symmetries and equations of motion that shows a great economy and simplicity of this formalism. We further use analogous techniques to construct polynomial democratic Lagrangians for general $p$-forms where electric and magnetic potentials appear on equal footing as explicit dynamical variables. Due to our reliance on the differential form notation, the construction is compact and universally valid for forms of all r…
We apply the spacetime dependent lagrangian formalism [1] to the action in general relativity. We obtain a Barriola-Vilenkin type monopole solution by exploiting theelectrogravity duality of the vacuum Einstein equations and using a modified definition of empty space. An {\it upper bound} is obtained on the monopole mass ${\tt M}$, ${\tt M}\leq e^{(1-α)/α}/(1-α)^{2}{\tt G}$ where $α= 2k $ is the global monopole charge. Keywords: global monopole, electrogravity duality, holographic principle. PACS: 11.15.-q, 11.27.+d, 14.80.Hv, 04.
We describe a Lagrangian defining the preon sector of the knot model. The preons are the elements of the fundamental representation of SLq(2), and unexpectedly agree with the preons conjectured by Harari and by Shupe. The leptons, neutrinos, up and down quarks, described as $j=3/2$ representations, and the electroweak vectors, described as $j=3$ representations, of SLq(2) also have the preon composition required by the schemes of Harari and of Shupe. The coupling constants and masses required by the preon Lagrangian introduce form factors that are simple functions of the knot model parameters.…
In these notes we describe how to formulate the Lagrangian insertion technique in a way that mimics generalized unitarity. We introduce a notion of cuts in real space and show that the cuts of the correlators in the super-correlators/super-amplitudes duality correspond to generalized unitarity cuts of the equivalent amplitudes. The cuts consist of correlation functions of operators in the chiral part of the stress-tensor multiplet as well as other half-BPS operators. We will also discuss the application of the method to other correlators as well as non-planar contributions.
We study the implications of target-space duality symmetries for low-energy effective actions of various four-dimensional string theories. In the heterotic case such symmetries can be incorporated in simple orbifold examples. At present a similar statement cannot be made about the simplest type IIB orientifolds due to an obstruction at the level of gravitational anomalies. This fact confirms previous doubts concerning a conjectured heterotic-type IIB orientifold duality and shows that target-space symmetries can be a powerful tool in studying relations between various string theories at the le…
We elaborate on a new representation of Lagrangians of 4D nonlinear electrodynamics including the Born-Infeld theory as a particular case. In this new formulation, in parallel with the standard Maxwell field strength $F_{αβ}, \bar{F}_{\dotα\dotβ}$, an auxiliary bispinor field $V_{αβ}, \bar{V}_{\dotα\dotβ}$ is introduced. The gauge field strength appears only in bilinear terms of the full Lagrangian, while the interaction Lagrangian $E$ depends on the auxiliary fields, $E = E(V^2, \bar V^2)$. The generic nonlinear Lagrangian depending on $F,\bar{F}$ emerges as a result of eliminating the auxili…
This is a rough transcript of talks given at the Workshop on Groups & Algebras in M Theory at Rutgers University, May 31--Jun 04, 2005. We review the basic motivation for a pre-geometric formulation of nonperturbative String/M theory, and for an underlying eleven-dimensional electric-magnetic duality, based on our current understanding of the String/M Duality Web. We explain the concept of an emerging spacetime geometry in the large N limit of a U(N) flavor matrix Lagrangian, distinguishing our proposal from generic proposals for quantum geometry, and explaining why it can incorporate curved s…
The global U-dualities of extended supergravity have played a central role in differentiating the distinct classes of extremal black hole solutions. When the U-duality group satisfies certain algebraic conditions, as is the case for a broad class of supergravities, the extremal black holes enjoy a further symmetry known as Freudenthal duality (F-duality), which although distinct from U-duality preserves the Bekenstein-Hawking entropy. Here it is shown that, by adopting the doubled Lagrangian formalism, F-duality, defined on the doubled field strengths, is not only a symmetry of the black hole …
It has been argued based on electric-magnetic duality that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four-dimension. And the Euler characteristic of Khovanov homology is the Jones polynomial which corresponds to the partition function of twisted $N=4$ super Yang-Mills theory. Moreover, Lee-Yang type phase transition is found in the topological twisted super Yang-Mills theory.
We study the issue of stability of static soliton-like solutions in some non-linear field theories which allow for knotted field configurations. Concretely, we investigate the AFZ model, based on a Lagrangian quartic in first derivatives with infinitely many conserved currents, for which infinitely many soliton solutions are known analytically. For this model we find that sectors with different (integer) topological charge (Hopf index) are not separated by an infinite energy barrier. Further, if variations which change the topological charge are allowed, then the static solutions are not even …
We give a proof of the equivalence of the electric-magnetic duality on one side and helicity conservation of the tree level amplitudes on the other side within general models of nonlinear electrodynamics. Using modified Feynman rules derived from generalized normal ordered Lagrangian we discuss the interrelation of the above two properties of the theory also at higher loops. As an illustration we present two explicit examples, namely we find the generalized normal ordered Lagrangian for the Born-Infeld theory and derive a semi-closed expression for the Lagrangian of the Bossard-Nicolai model (…
We construct a Lagrangian for general nonlinear electrodynamics that features electric and magnetic potentials on equal footing. In the language of this Lagrangian, discrete and continuous electric-magnetic duality symmetries can be straightforwardly imposed, leading to a simple formulation for theories with the $SO(2)$ duality invariance. When specialized to the conformally invariant case, our construction provides a manifestly duality-symmetric formulation of the recently discovered ModMax theory. We briefly comment on a natural generalization of this approach to $p$-forms in $2p+2$ dimensio…
We generalize previous results on target space duality to the case where there are background fields and the sigma model lagrangian has a potential function.
We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting structure involving representation/spin degeneracy of branes ending on branes which features in the large N dual description of Chern-Simons theory. This leads to a refinement of the integer invariants for links and knots. We illustrate our results with explicit computations on the Chern-Simons side.
We show how to formulate the phenomenon of gaugino condensation in a super-Yang-Mills theory with a field-dependent gauge coupling described with a linear multiplet. We prove the duality equivalence of this approach with the more familiar formulation using a chiral superfield. In so doing, we resolve a longstanding puzzle as to how a linear-multiplet formulation can be consistent with the dynamical breaking of the Peccei-Quinn symmetry which is thought to occur once the gauginos condense. In our approach, the composite gauge degrees of freedom are described by a real vector superfield, $V$, ra…
In this work, we revisit the duality between a self-dual non-gauge invariant theory and a topological massive theory in $3+1$ dimensions. The self-dual Lagrangian is composed by a vector field and an antisymmetric field tensor whereas the topological massive Lagrangian is build using a $B \wedge F$ term. Though the Lagrangians are quite different, they yield to equations of motion that are connected by a simple dual mapping among the fields. We discuss this duality by analyzing the degrees of freedom in both theories and comparing their propagating modes at the classical level. Moreover, we em…
A new representation of Lagrangians of 4D nonlinear electrodynamics is considered. In this new formulation, in parallel with the standard Maxwell field strength F, an auxiliary bispinor (tensor) field V is introduced. The gauge field strength appears only in bilinear terms of the full Lagrangian, while the interaction Lagrangian E depends on the auxiliary fields, E = E(V). Two types of self-duality inherent in the nonlinear electrodynamics models admit a simple characterization in terms of the function E. The continuous SO(2) duality symmetry between nonlinear equations of motion and Bianchi i…
We present a systematic study of the constraints coming from target-space duality and the associated duality anomaly cancellations on orbifold-like 4-D strings. A prominent role is played by the modular weights of the massless fields. We present a general classification of all possible modular weights of massless fields in Abelian orbifolds. We show that the cancellation of modular anomalies strongly constrains the massless fermion content of the theory, in close analogy with the standard ABJ anomalies. We emphasize the validity of this approach not only for (2,2) orbifolds but for (0,2) model…
A generalization of duality transformations for arbitrary Lorentz tensors is presented, and a systematic scheme for constructing the dual descriptions is developed. The method, a purely Lagrangian approach, is based on a first order parent Lagrangian, from which the dual partners are generated. In particular, a family of theories which are dual to the massive spin two Fierz-Pauli field, both free and coupled to an external source, is constructed in terms of a third rank tensor which is taken as the basic configuration variable.
The argument of Hodge duality symmetry is introduced starting from the electromagnetic field. Introducing bosonic string theory, O(d,d) duality symmetry can be implemented when there exist d-symmetries, which allows one to write Hodge-dual fields. A tree-level effective gravitational action of bosonic string theory coupled with the dilaton field is considered. This theory inherits the Busher's duality of its parent string theory. The dilaton field can be recast into the Weyl's mode of the metric tensor in the Jordan frame. This maps the effective one-loop bosonic string theory of gravity into …
We present the details of the novel framework for Lagrangian field theories that are Lorentz-invariant and lead to at most second order equations of motion. The use of antisymmetric structure is of crucial importance. The general ghost-free Lagrangians are constructed and then translated into the language of differential forms. The ghost-freeness has a geometric nature. A novel duality is proposed which generalizes the Hodge duality in Maxwell's theory. We discuss how the well-established theories are reformulated and propose many new theories.
The quantum mechanical transition between a free particle Lagrangian and the Klein Gordon field description of a free particle (particle wave duality) is conjectured to extend to an analogous construction of relativistically invariant wave equations associated with strings and branes, which we propose to call brane-wave duality. Electromagnetic interactions in the two systems are discussed. It is emphasised that all integrable free field theories, including those of Dirac-Born-Infeld type, are associated with Lagrangians equivalent to divergences on the space of solutions of the equations of m…
Topological defects attract much recent interest in high-energy and condensed matter physics because they encode (non-invertible) symmetries and dualities. We study codimension-1 topological defects from a hamiltonian point of view, with the defect location playing the role of `time'. We show that the Weinstein symplectic category governs topological defects and their fusion: each defect is a lagrangian correspondence, and defect fusion is their geometric composition. We illustrate the utility of these ideas by constructing S- and T-duality defects in string theory, including a novel topology-…
In this article we study the noncommutative description of the DBI Lagrangian and its T-dual counterpart. We restrict the freedoms of the noncommutativity parameters of these Lagrangians. Therefore the noncommutativity parameter, the effective metric, the effective coupling constant of the string and the extra modulus ${\tilde Φ}$ of the effective T-dual theory, can be expressed in terms of the closed string variables $g$, $B$, $g_s$ and the noncommutativity parameter of the effective theory of open string.
We present the string Lagrangian with manifest T-duality. Not only zero-modes but also all string modes are doubled. The gravitational field is an O(D,D) gauge field. We give a Lagrangian version of the section condition for the gauge invariance which compensates the O(D,D) transformation from the gravitational field and the GL(2D) coordinate transformation. We also show the gauge invariance of the line element of the manifest T-duality space and the O(D,D) condition on the background. Different sections describe dual spaces.
Bosonization of the gauged, massive Thirring model in 2+1-dimensions produces a Maxwell-Chern-Simons gauge theory, coupled to a dynamical, massive vector field. Exploiting the Master Lagrangian formalism, two dual theories are constructed, one of them being a gauge theory. The full two-point functions of both the interacting fields are computed in the path integral quantization scheme. Furthermore, some new dual models, derived from the original master lagrangian and valid for different regimes of coupling parameters, are constructed and analysed in details.
We continue the study of n-point correlation functions of half-BPS protected operators in N=4 super-Yang-Mills theory, in the limit where the positions of the adjacent operators become light-like separated. We compute the l-loop corrections by making l Lagrangian insertions. We argue that there exists a simple relation between the (n+l)-point tree-level correlator with l Lagrangian insertions and the integrand of the n-particle l-loop MHV scattering amplitude, as obtained by the recent momentum twistor construction of Arkani-Hamed et al. We present several examples of this new duality, at one …
We show that there exists a special compactification of QCD on $\mathbb{R}^3 \times S^1$ in which the theory has a domain where continuous chiral symmetry breaking is analytically calculable. We give a microscopic derivation of the chiral lagrangian, the chiral condensate, and the Gell-Mann-Oakes-Renner relation $m_π^2 f_π^2 = m_q \langle \bar{q} q \rangle$. Abelian duality, monopole operators, and flavor-twisted boundary conditions, or a background flavor holonomy, play the main roles. The flavor twisting leads to the new effect of fractional jumping of fermion zero modes among monopole-insta…
It is shown that the canonical quantization of the free massive Kalb-Ramond and Curtright-Freund Lagrangians leads to the same theory obtained from the canonical quantization of the free Proca and Klein-Gordon Lagrangians. The duality in the presence of interaction is explored in the context of the Feynman rules and beyond. It is pointed out that the equivalence between massive dual models without gauge symmetry is rooted in an ambiguity of coordinate choices.
We show that there is a function of one variable's worth of Lagrangians for a single Maxwell field coupled to gravity whose equations of motion admit electric-magnetic duality. Such Lagrangians are given by solutions of the Hamilton-Jacobi equation for timelike geodesics in Witten's two-dimensional black hole. Among them are the Born-Infeld Lagrangian which arises in open string theory. We investigate the effect of the axion and the dilaton in the open superstring case and we show that this theory loses its electric-magnetic duality invariance when one considers the higher order electromagneti…
