We investigate the Hamiltonian formulation of f (T ) gravity and find that there are five degrees of freedom. The six first class constraints corresponding to the local Lorentz transformation in Teleparallel gravity become second class constraints in f (T ) gravity, which leads to the appearance of three extra degrees of freedom and the violation of the local Lorentz invariance in f (T ) gravity.In general, there are D − 1 extra degrees of freedom for f (T ) gravity in D dimensions, and this implies that the extra degrees of freedom correspond to one massive vector field or one massless vector field with one scalar field. * Electronic address: mli@itp.ac.cn † Electronic address:

We prove that, in general, the first law of black hole thermodynamics, δQ = T δS, is violated in f (T ) gravity. As a result, it is possible that there exists entropy production, which implies that the black hole thermodynamics can be in non-equilibrium even in the static spacetime. This feature is very different from that of f (R) or that of other higher derivative gravity theories. We find that the violation of first law results from the lack of local Lorentz invariance in f (T ) gravity.By investigating two examples, we note that f ′′ (0) should be negative in order to avoid the naked singularities and superluminal motion of light. When f ′′ (T ) is small, the entropy of black holes in f (T ) gravity is approximatively equal to f ′ (T ) 4 A.

It is recently claimed by Nekrasov and Shatashvili that the N = 2 gauge theories in the Ω background with ǫ 1 = , ǫ 2 = 0 are related to the quantization of certain algebraic integrable systems. We study the special case of SU(2) pure gauge theory, the corresponding integrable model is the A 1 Toda model, which reduces to the sine-Gordon quantum mechanics problem. The quantum effects can be expressed as the WKB series written analytically in terms of hypergeometric functions. We obtain the magnetic and dyonic expansions of the Nekrasov theory by studying the property of hypergeometric functions in the magnetic and dyonic regions on the moduli space. We also discuss the relation between the electric-magnetic duality of gauge theory and the action-action duality of the integrable system.The nonperturbative properties of quantum field theories have been one of the most active research subjects during the past few decades, we have known a lot of information through various analytical or numerical methods. The four dimensional Yang-Mills theory stands as one of the few most attractive field models. One of the milestones of studying the supersymmetric gauge theories is the Seiberg-Witten solution of the four dimensional N = 2 gauge theory [1], which results in a fully analytic understanding of a large class of supersymmetric gauge theories. Their solution is based on typical features of supersymmetric gauge theories, i.e. the holomorphic structure of prepotential and the electric-magnetic duality of gauge theory. By analyzing the vacuum structure of the moduli space and the related monodromy problem, Seiberg and Witten discovered that the low energy physics of the gauge theory is encoded in a geometric object, an elliptic curve, the prepotential can be obtained through the periods of a holomorphic differential one form along the two conjugate homology cycles. The periods can be written as hypergeometric functions on the moduli space, they manifest the electric-magnetic duality in a very explicit way: the electric-magnetic duality group of the gauge theory is the same as the discontinuous reparametrization group of the elliptic curve. The solution is valid on the whole moduli space. In some region the theory is a weakly coupled electric theory, in some region the electric theory is strongly coupled, but it can be reformulated as a weakly coupled magnetic theory. By choosing suitable quantities as the fundamental degrees of freedom, we can either expand the effective action in terms of the electric fields or in terms of the magnetic (or dyonic) fields. Subsequent works have extended the solution to the N = 2 theory with more general gauge groups and with matters, it is also found that these solution can be interpreted in the context of string theory, see review [2,3].The original work of Seiberg and Witten is reinterpreted in [4] from a different viewpoint. The hard part of solving the N = 2 gauge theory is the sum of the instanton contributions, but the multi-instanton measure on moduli space grows very complicated as t...

We investigate the Hawking radiation cascade from the five-dimensional charged black hole with a scalar field coupled to higher-order Euler densities in a conformally invariant manner. We give the semi-analytic calculation of greybody factors for the Hawking radiation. Our analysis shows that the Hawking radiation cascade from this five-dimensional black hole is extremely sparse. The charge enhances the sparsity of the Hawking radiation, while the conformally coupled scalar field reduces this sparsity.

Considering the unexpected similarity between the thermodynamic features of charged AdS black holes and that of the van der Waals fluid system, we calculate the number densities of black hole micromolecules and derive the thermodynamic scalar curvature for the small and large black holes on the coexistence curve based on the so-called Ruppeiner thermodynamic geometry. We reveal that the microscopic feature of the small black hole perfectly matches that of the ideal anyon gas and that the microscopic feature of the large black hole matches that of the ideal Bose gas. More importantly, we investigate the issue of molecular potential among micromolecules of charged AdS black holes and point out explicitly that the well-known experiential Lennard-Jones potential is a feasible candidate to describe interactions among black hole micromolecules completely from a thermodynamic point of view. The behavior of the interaction force induced by the Lennard-Jones potential coincides with that of the thermodynamic scalar curvature. Both the Lennard-Jones potential and the thermodynamic scalar curvature offer a clear and reliable picture of microscopic structures for the small and large black holes on the coexistence curve for charged AdS black holes.