Virasoro Algebra

Definition

The Virasoro algebra is an infinite-dimensional Lie algebra that arises in the study of conformal field theory and string theory, characterized by its generators which satisfy specific commutation relations. It extends the Witt algebra by adding a central charge, making it crucial for understanding the symmetries of two-dimensional conformal field theories and their associated physical implications.

5 Must Know Facts For Your Next Test

1. The Virasoro algebra consists of generators $L_n$ for $n \in \mathbb{Z}$ and a central element $C$, with commutation relations given by $[L_m, L_n] = (m-n)L_{m+n} + \frac{C}{12}(m^3 - m)\delta_{m+n,0}$.
2. It is fundamental to string theory because it provides the algebraic structure necessary to describe the symmetries of two-dimensional surfaces in a consistent manner.
3. In representation theory, the Virasoro algebra has an infinite number of representations that correspond to different conformal blocks in conformal field theory.
4. The central charge $C$ plays a vital role in determining the modular invariance and physical properties of the theory, influencing aspects like critical dimensions.
5. The Virasoro algebra is intimately connected to various mathematical structures, including vertex operator algebras, which have implications in both mathematics and physics.

Review Questions

• How does the Virasoro algebra relate to the Witt algebra, and why is this relationship important in understanding conformal symmetries?
  • The Virasoro algebra can be seen as an extension of the Witt algebra by introducing a central charge. This relationship is important because while the Witt algebra captures basic symmetry transformations through derivations, the addition of the central charge allows for a richer structure that encompasses physical phenomena in conformal field theories. Understanding this connection helps in analyzing how these algebras apply to real-world physics, particularly in two-dimensional systems.
• Discuss the role of central charge in the Virasoro algebra and its impact on representation theory within conformal field theory.
  • The central charge in the Virasoro algebra affects how representations are constructed and classified within conformal field theory. It determines crucial aspects such as modular invariance and partition functions. Different values of the central charge correspond to different physical theories and influence properties like scaling dimensions of primary fields. Thus, it is essential for accurately describing physical systems and their symmetries.
• Evaluate how the properties of the Virasoro algebra influence both mathematical structures like vertex operator algebras and physical theories such as string theory.
  • The properties of the Virasoro algebra shape both mathematical constructs like vertex operator algebras and physical frameworks like string theory by providing an essential symmetry structure. In mathematics, it facilitates understanding various algebraic forms and representations that underpin more complex theories. In physics, these properties ensure consistency in string theory models by dictating how strings behave under conformal transformations. This dual relevance illustrates how abstract mathematical concepts can have profound implications across different scientific disciplines.

Related terms

Conformal Field Theory: A quantum field theory that is invariant under conformal transformations, playing a significant role in statistical mechanics and string theory.

Witt Algebra:

An infinite-dimensional Lie algebra generated by the derivations of formal power series, forming a basis for understanding the structure of the Virasoro algebra.

Central Charge: A parameter that appears in the Virasoro algebra's commutation relations, influencing the representation theory and physical characteristics of conformal field theories.