I understand AdS/CFT correspondence as follows:
AdS/CFT is a powerful tool that helps physicists solve complex problems in gravity and particle physics by relating them to each other.
What's the need of relating gravity and particle physics at all?
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2$\begingroup$ The quote doesn't say they "need" to be related, it says they turn out to be related so that solving one kind of problem helps you solve the other... If there is a need to relate them, it is first of all because gravity and particles both exist in the same reality, so a correct physical description has to accommodate them both... And then as it turns out, they may be related in all kinds of profound and unexpected ways, though the theoretical ideas about this, are still somewhat lacking in experimental confirmation. $\endgroup$– Mitchell PorterCommented 18 hours ago
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The connection between gravity and particle physics wasn't sought out just to satisfy a theoretical "wish" but rather it emerged naturally from understanding the universe more deeply.
In the framework of perturbative string theory, starting from a two-dimensional quantum field theory (the string world sheet theory) and demanding internal consistency, one finds that the theory must include a massless spin-2 excitation. By analyzing how this spin-2 mode interacts at low energies, one recovers an effective action that coincides with Einstein’s equations of General Relativity. This emergence of a spin-2 graviton is not imposed by hand; it arises from the requirement of anomaly cancellation and conformal invariance at the quantum level on the string world sheet.
In the context of AdS/CFT correspondence:
The AdS/CFT correspondence is a theoretical framework in string theory and quantum gravity that postulates a duality between two different types of physical theories. The path integrals, correlation functions, and spectra of one theory match exactly to those of the other.
The AdS/CFT correspondence states an equality between partition functions: $$Z_{\text{gravity in AdS}}[\phi_{\text{boundary}}] = Z_{\text{CFT}}[J]$$ $Z_{\text{gravity in AdS}}$ is the path integral of a gravitational theory (string theory or a supergravity limit) defined on an Anti-de Sitter (AdS) background, and $Z_{\text{CFT}}$ is the generating functional of correlation functions for a corresponding conformal field theory (CFT) living on the boundary of that AdS space. This equality specifies a one-to-one mapping between all observables (such as correlation functions, spectrum of states, and response to external sources) in the CFT and gravitational quantities in the bulk AdS theory.
The AdS/CFT correspondence rigorously shows that the symmetry groups and operator spectra of the two theories match in a precise way. The isometry group of $AdS_{\text{d+1}}$ space is isomorphic to the conformal group in d-dimensions. It means the underlying symmetry structure that defines a CFT in d-dimensions exactly matches the symmetry of a (d+1)-dimensional gravitational spacetime.
To summarize, It’s not that physicists arbitrarily decided to relate gravity and particle physics. Instead, the theories and mathematical structures designed to address fundamental questions about quantum fields, spacetime, and consistency conditions brought these areas together.
