We calculate the black hole entropy in Loop Quantum Gravity as a function of the horizon area and provide the exact formula for the leading and sub-leading terms. By comparison with the BekensteinHawking formula we uniquely fix the value of the 'quantum of area' in the theory.The Bekenstein-Hawking formula gives the leading term in the entropy of the black hole in the formwhere l P is the Planck length and A is the black hole horizon area. According to the common belief, the entropy is always connected with the logarithm of the number of microscopic states realizing a given macroscopic state. The fact that the entropy in (1) is proportional to the area (and not as is usual to the volume) has led to the formulation of the so called holographic principle.It was however difficult to find the microscopic states that could account for such an entropy. The problem was attacked in two different approaches -in string theory [1]-[5] and in Loop Quantum Gravity [6]-[10] (see also the review [11]). The latter approach is based on a quantum theory of geometry. The basic geometric operators were introduced in [12,13] and a detailed description of the quantum horizon geometry was introduced using the U(1) Chern-Simons theory in [10] where one can find the detailed discussion of the states on the horizon of the black hole (see also [14]). However, the procedure introduced in [10] for state counting contained a spurious constraint on admissible sequences and the number of the relevant horizon states is underestimated. The correct method of counting was proposed in [14] but that analysis provides only lower and upper bounds on the number of states. The purpose of this paper is to rectify this situation. We will provide the correct value 1

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In this paper we explicitly prove the invariance of the time-dependent string gravity Lagrangian with up to four derivatives under the global O(d, d) symmetry.

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