Dimensions in string theory
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Dimensions in String Theory
Critical Dimension in String Theory
String theory posits that the universe is composed of one-dimensional "strings" rather than point particles. A critical aspect of string theory is the concept of the critical dimension, which is the specific number of dimensions in which a consistent string theory can be formulated. For instance, in the context of string theories with a background metric that is a product of Minkowski space and an $\mathrm{SU}(N)$ or $\mathrm{SO}(N)$ group manifold, the critical dimension is derived with the inclusion of a Wess-Zumino term associated with the compactified dimension. This implies that the compactified radius is quantized in units of the string tension .
Nonlocality and Geometric Shape of Strings
Nonlocality is a fundamental feature of string theory, distinguishing it from point-particle theories. At sufficiently large scales, strings appear structureless, akin to particles. This scale is unambiguously determined in free-string theory. The operational notions of worldsheet and target spacetime dimensions in string theory are clarified and found to be in mutual agreement, providing a coherent understanding of how strings transition from extended objects to point-like particles .
Entanglement Entropy in Two-Dimensional String Theory
Entanglement entropy serves as a diagnostic tool for understanding the emergence of locality in string theory. In two-dimensional string theory, the entanglement entropy is computed in the large-N matrix quantum mechanics dual to the theory. This computation reveals a logarithmically large but finite contribution corresponding to the short-distance entanglement of the tachyon field in the emergent spacetime. This entanglement is regulated by a nonperturbative "graininess" of space, highlighting the intricate relationship between spacetime and quantum entanglement .
Compactifications and Emergent Symmetries
Compactifications of string theory to lower dimensions, such as from six to four dimensions, reveal emergent symmetries and dualities. For example, compactifications of the 6d E-string theory on Riemann surfaces with punctures and non-abelian flat connections lead to the identification of N=1 field theories in four dimensions. These compactifications shed light on emergent symmetries in various 4d N=1 SCFTs and predict new exceptional dualities and symmetries .
Dynamics in Various Dimensions
The dynamics of string theories in dimensions greater than or equal to four are complex and interconnected through various dualities. For instance, eleven-dimensional supergravity can arise as a low-energy limit of the ten-dimensional Type IIA superstring. Additionally, a duality between the heterotic string and Type IIA superstrings controls the strong coupling dynamics of the heterotic string in five, six, and seven dimensions, implying S-duality for both heterotic and Type II strings .
Dimension-Changing Solutions
String theory also allows for dimension-changing solutions, where transitions between different dimensional string theories occur. These transitions involve the readjustment of the string-frame metric and dilaton gradient, maintaining the central charge of the worldsheet theory even as the number of dimensions changes. Such solutions connect supersymmetric and non-supersymmetric string theories, demonstrating the flexibility and richness of string theory in accommodating various dimensional frameworks .
Conclusion
String theory's exploration of dimensions reveals a complex and interconnected framework where critical dimensions, nonlocality, entanglement entropy, compactifications, and dynamic transitions between dimensions play crucial roles. These insights not only deepen our understanding of the fundamental nature of the universe but also highlight the intricate and multifaceted nature of string theory as a theoretical construct.
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