Holography and Black Holes

We study the two-dimensional free symplectic fermion with anti-periodic boundary condition. This model has negative norm states with naive inner product. This negative norm problem can be cured by introducing a new inner product. We demonstrate that this new inner product follows from the connection between the path integral formalism and the operator formalism. This model has negative central charge, c = −2, and we clarify how CFT2 with negative central charge can have the non-negative norm. We introduce α-vacua in which the Hamiltonian is seemingly non-Hermitian. In spite of non-Hermiticity we find that the energy spectrum is real. We also compare a correlation function with respect to the α-vacua with that of the de Sitter space.