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Semi-infinite cohomology and string theory.
I B Frenkel1, H Garland, G J Zuckerman
1Department of Mathematics, Yale University, New Haven, CT 06520.
Summary
We introduce semi-infinite cohomology for graded Lie algebras, revealing its analogy to de Rham cohomology. This theory connects to string theory, identifying physical subspaces and proving the no-ghost theorem.
Area of Science:
- Mathematical Physics
- Algebraic Geometry
- String Theory
Background:
- Semi-infinite cohomology, introduced by Feigin, is a developing area in mathematical physics.
- Understanding the structure of Lie algebras is crucial for theoretical physics, particularly in string theory and quantum field theory.
Purpose of the Study:
- To develop the theory of semi-infinite cohomology for graded Lie algebras.
- To establish an analogy between semi-infinite cohomology and de Rham cohomology in Kähler geometry.
- To explore the applications of this theory in string theory, specifically concerning the Virasoro algebra and Fock modules.
Main Methods:
- Development of the theoretical framework for semi-infinite cohomology.
- Proof of a vanishing theorem for specific modules.
- Application of the theory to the Virasoro algebra and Fock module.
- Establishing connections to gauge-invariant string theories.
Main Results:
- The relative semi-infinite cohomology exhibits a structure analogous to de Rham cohomology.
- A vanishing theorem is proven for a specific class of modules.
- The zero cohomology of the Virasoro algebra and Fock module is identified as the physical subspace.
- The no-ghost theorem is derived as a consequence of the theory.
- Profound connections between semi-infinite cohomology and gauge-invariant string theories are revealed.
Conclusions:
- Semi-infinite cohomology provides a powerful framework for understanding graded Lie algebras and their relation to physics.
- The established analogy with de Rham cohomology offers new geometric insights.
- The theory has direct implications for string theory, unifying concepts like physical subspaces and the no-ghost theorem.
- Further exploration of connections to gauge-invariant interacting string theories and geometric realizations of infinite-dimensional Lie algebras is indicated.