Classical boundary field theory of Jacobi sigma models by Poissonization
Ion V. Vancea
备注：
16 pages. References updated. Prepared for the Proceedings of 4th International Conference on Holography, String Theory and Discrete Approach, Hanoi, Vietnam, 2020
In this paper, we are going to construct the classical field theory on the boundary of the embedding of $\mathbb{R} \times S^{1}$ into the manifold $M$ by the Jacobi sigma model. By applying the poissonization procedure and by generalizing the known method for Poisson sigma models, we express the fields of the model as perturbative expansions in terms of the reduced phase space of the boundary. We calculate these fields up to the second order and illustrate the procedure for contact manifolds.
Using the concept of Jacobi-Lie group and Jacobi-Lie bialgebra, we generalize the definition of Poisson-Lie symmetry to Jacobi-Lie symmetry. In this regard, we generalize the concept of Poisson-Lie T-duality to Jacobi-Lie T-duality and present Jacobi-Lie T-dual sigma models on Lie groups, which have Jacobi-Lie symmetry. Using this symmetry, new cases of duality appear and some examples are given. This generalization may provide insights to understand the quantum features of Poisson-Lie T-duality, in a more satisfactory way.
We introduce a two-dimensional sigma model associated with a Jacobi manifold. The model is a generalisation of a Poisson sigma model providing a topological open string theory. In the Hamiltonian approach first class constraints are derived, which generate gauge invariance of the model under diffeomorphisms. The reduced phase space is finite-dimensional. By introducing a metric tensor on the target, a non-topological sigma model is obtained, yielding a Polyakov action with metric and B-field, whose target space is a Jacobi manifold.
The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories - recently introduced by the authors - which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic man…
We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In t…
The O(3) non-linear sigma model is investigated using multi Hamilton-Jacobi formalism. The integrability conditions are investigated and the results are in agreement with those obtained by Dirac's method. By choosing an adequate extension of phase space we describe the transformed system by a set of three Hamilton-Jacobi equations and calculate the corresponding action.
We first study a free particle on an $(n-1)$-sphere in an extended phase space, where the originally second-class Hamiltonian and constraints are now in strong involution. This allows for a Schrödinger representation and a Hamilton-Jacobi formulation of the model. We thereby obtain the free particle energy spectrum corresponding to that of a rigid rotator. We extend these considerations to a modified version of the field theoretical O(3) nonlinear sigma model, and obtain the corresponding energy spectrum as well as BRST Lagrangian.
We show that the Wess-Zumino-Novikov-Witten (WZW) model on the Heisenberg Lie group $H_{4}$ has Jacobi-Lie symmetry with four dual Lie groups. We construct Jacobi-Lie T-dual sigma models with one of their Jacobi-Lie bialgebra and show that the original model is equivalent to the $H_{4}$ WZW model. The conformality of the dual sigma model up to one-loop order is also investigated.
The construction of physical models with local time-reparametrization invariance is reviewed. Negative-energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton-Jacobi theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker type. After a discussion of empty universes, the FLRW theory is extended with homogeneous scalar fields generically described by a $\sg$-model …
We calculate the elliptic genus of two dimensional abelian gauged linear sigma models with (2,2) supersymmetry using supersymmetric localization. The matter sector contains charged chiral multiplets as well as Stueckelberg fields coupled to the vector multiplets. These models include theories that flow in the infrared to non-linear sigma models with target spaces that are non-compact Kahler manifolds with U(N) isometry and with an asymptotically linear dilaton direction. The elliptic genera are the modular completions of mock Jacobi forms that have been proposed recently using complementary ar…
We study the constraint structure of the O(3) nonlinear sigma model in the framework of the Lagrangian, symplectic, Hamilton-Jacobi as well as the Batalin-Fradkin-Tyutin embedding procedure.
Magnetized Baryonic layer and a novel BPS bound in the gauged-Non-Linear-Sigma-Model-Maxwell theory in (3+1)-dimensions through Hamilton-Jacobi equation
We derive the explicit form of the superpropagators in presence of general boundary conditions (coisotropic branes) for the Poisson Sigma Model. This generalizes the results presented in Cattaneo and Felder's previous works for the Kontsevich angle function used in the deformation quantization program of Poisson manifolds without branes or with at most two branes. The relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons P_n with particular "alternating" boundary conditions. In presence of more than three bra…
We describe a novel deformation of the 3-dimensional sigma model with hyperkähler target, which arises naturally from the compactification of a 4-dimensional $\mathcal{N}=2$ theory on a hyperkähler circle bundle (Gibbons-Hawking space). We derive the condition for which the deformed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and find that supersymmetry implies it is a holomorphic section of a certain holomorphic line bundle over the hyperkähler target. We study explicitly the case where the original 4-di…
We study a class of two-dimensional N=(2,2) sigma models called squashed toric sigma models, using their Gauged Linear Sigma Models (GLSM) description. These models are obtained by gauging the global U(1) symmetries of toric GLSMs and introducing a set of corresponding compensator superfields. The geometry of the resulting vacuum manifold is a deformation of the corresponding toric manifold in which the torus fibration maintains a constant size in the interior of the manifold, thus producing a neck-like region. We compute the elliptic genus of these models, using localization, in the case when…
The 1-loop anomalies of a d-dimensional quantum field theory can be computed by evaluating the trace of the regulated path integral jacobian matrix, as shown by Fujikawa. In 1983, Alvarez-Gaumé and Witten observed that one can simplify this evaluation by replacing the operators which appear in the regulator and in the jacobian by quantum mechanical operators with the same (anti)commutation relations. By rewriting this quantum mechanical trace as a path integral with periodic boundary conditions for a one-dimensional supersymmetric nonlinear sigma model, they obtained the chiral anomalies for s…
Squashed toric sigma models are a class of sigma models whose target space is a toric manifold in which the torus fibration is squashed away from the fixed points so as to produce a neck-like region. The elliptic genera of squashed toric-Calabi-Yau manifolds are known to obey the modular transformation property of holomorphic Jacobi forms, but have an explicit non-holomorphic dependence on the modular parameter. The elliptic genus of the simplest one-dimensional example is known to be a mixed mock Jacobi form, but the precise automorphic nature for the general case remained to be understood. W…
We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kahler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function. The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the ro…
These notes grew out of a lecture course on mathematical methods of classical physics for students of mathematics and mathematical physics at the master's level. Also, physicists with a strong interest in mathematics may find this text useful as a resource complementary to existing textbooks on classical physics. Topics include Lagrangian Mechanics, Hamiltonian Mechanics, Hamilton-Jacobi Theory, as well as Classical Field Theory formulated in the language of jet bundles. The latter topic also covers important examples of field theories such as sigma models, gauge theory, and Einstein's theory …
The holographic renormalization group (RG) is reviewed in a self-contained manner. The holographic RG is based on the idea that the radial coordinate of a space-time with asymptotically AdS geometry can be identified with the RG flow parameter of the boundary field theory. After briefly discussing basic aspects of the AdS/CFT correspondence, we explain how the notion of the holographic RG comes out in the AdS/CFT correspondence. We formulate the holographic RG based on the Hamilton-Jacobi equations for bulk systems of gravity and scalar fields, as was introduced by de Boer, Verlinde and Verlin…
