A First Course in String Theory

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Physical Meaning of Virasoro Operators

The Virasoro operators are fundamental in the context of two-dimensional conformal field theory (CFT) and string theory. They arise from the study of the symmetries of two-dimensional surfaces, particularly in how these surfaces can be mapped conformally (i.e., preserving angles) onto one another.

Conformal Symmetry

The Virasoro operators are associated with the Virasoro algebra, which is an infinite-dimensional Lie algebra that extends the algebra of diffeomorphisms on a circle. This algebra captures the symmetries of two-dimensional conformal transformations. In physical terms, these transformations are crucial for understanding how physical theories behave under changes of coordinates on the worldsheet of strings or in two-dimensional quantum field theories.

Role in String Theory

In string theory, the Virasoro operators are derived from the quantization of the string’s motion. They correspond to the modes of oscillation of the string and are denoted as ​. The operator ​ is particularly significant because it acts as the Hamiltonian for the system, determining the energy levels of the string states. The commutation relations among these operators encode important physical information, such as the constraints on physical states (the Virasoro constraints) that must be satisfied for a consistent theory.

Physical States and Constraints

The Virasoro constraints arise from the requirement that physical states must be invariant under the action of the Virasoro operators. This leads to the notion of physical states in the theory, which are those that satisfy the condition for . This condition ensures that the states are not affected by the higher modes of the Virasoro algebra, which correspond to unphysical degrees of freedom.

Applications in CFT

In the context of conformal field theory, the Virasoro operators play a crucial role in the conformal bootstrap approach, where they help in constructing correlation functions and understanding the structure of the theory. The Virasoro conformal blocks, which are special functions derived from these operators, are essential for calculating observables in CFTs.

Summary

In summary, the Virasoro operators are central to the mathematical formulation of two-dimensional conformal field theories and string theory. They encapsulate the symmetries of these theories, dictate the physical state conditions, and facilitate the computation of physical observables. Their significance extends beyond mathematics into the core of theoretical physics, influencing our understanding of fundamental interactions and the structure of spacetime.

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2024.09.08 Sunday ACHK