By Craps, B

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**Example text**

Hence, two physical states that differ by an exact state are physically equivalent. The physical Hilbert space is defined as a set of equivalence classes of BRST-closed modulo BRST-exact states. This is called the cohomology of QB . 61) and, in the R sector, plus the analogous conditions for the tilded modes in the case of closed strings. These conditions can be imposed on the Fock space states before applying the BRST procedure. Since one can check that {QB , bn } = Ln and [QB , βr ] = Gr , the conditions Eq.

Strings and branes with SX = 1 4πα′ √ d2 σ gg ab ∂a X µ ∂b X ν ηµν . 2) M In the previous formulae, λ is a constant, the meaning of which will become clear soon, χ is the Euler number of the world-sheet M and ηµν is the spacetime Minkowski metric of signature (− + + · · · +). Further, σ ≡ (σ 1 , σ 2 ) denotes the world-sheet coordinates. The fields X µ (σ) describe the embedding of the world-sheet in spacetime, and gab (σ) is a Euclidean world-sheet metric of signature (+, +). Note that this metric is non-dynamical; eliminating it from the action by using its classical equation of motion gives rise to the Nambu-Goto action, which is proportional to the area of the world-sheet.

It turns out that this closed string theory, called type I closed unoriented string theory is inconsistent (for instance, there are spacetime gravitational anomalies) unless one adds open strings to the theory in a very precise way. In order to add those open strings, we have to generalize the open strings we have discussed so far. The two endpoints of an oriented open string are special points, and distinct from each other. Therefore, it is possible to assume that the open string carries “charges” at its end points.