In the context of the partition function, the term "partitions" typically refers to the way in which the states of a system are summed or integrated over to calculate the partition function. The partition function itself is a sum (or integral) over all possible states or configurations of a system, weighted by their respective Boltzmann factors (in statistical mechanics) or exponential of the action (in quantum field theory). ### Partition Function in General In statistical mechanics and quantum field theory, the **partition function** is a central mathematical object that encodes information about the thermodynamic properties of a system. It is a function of temperature (and other parameters) that sums over all possible states of the system, weighted by their Boltzmann factors (for statistical mechanics) or action (for quantum field theory). The partition function, $Z$, is defined as: $Z = \sum_i e^{-\beta E_i}$​ where: - $\beta = \frac{1}{k_B T}$​, with $k_B$​ being the Boltzmann constant and $T$ the temperature. - $E_i$​ are the energy levels of the system. In the path integral formulation of quantum field theory, the partition function is expressed as: $Z = \int \mathcal{D}\phi \, e^{-S[\phi]}$ where: - $\mathcal{D}\phi$ represents the functional integration over all field configurations. - $S[\phi]$ is the action of the field $\phi$. The partition function allows us to calculate important thermodynamic quantities such as free energy, entropy, and specific heat. It also plays a crucial role in determining the expectation values of observables and correlation functions. ### Partition Function in the Context of AdS/CFT Correspondence The **AdS/CFT correspondence** posits a duality between a gravitational theory in an $(n+1)$-dimensional [[Anti-de Sitter (AdS) space]] and a [[Conformal Field Theory]] (CFT) defined on the nnn-dimensional boundary of the [[AdS space]]. This duality is often summarized as: $\text{AdS}_{n+1} \leftrightarrow \text{CFT}_n$​ In this context, the partition functions on both sides of the correspondence are related in a profound way. Specifically, the partition function of the CFT on the boundary is equivalent to the partition function of the bulk gravitational theory with specified boundary conditions. Mathematically, this relationship is often expressed as: $Z_{\text{CFT}}[J] = Z_{\text{AdS}}[\phi_0]$ where: - $Z_{\text{CFT}}$ is the partition function of the [[CFT]] with source $J$. - $Z_{\text{AdS}}[\phi_0]$ is the partition function of the bulk AdS theory with boundary conditions $\phi_0$​ related to the source $J$. In more detail: - **Boundary Conditions**: In the AdS/CFT correspondence, the fields in the bulk AdS space are constrained by boundary conditions at the AdS boundary. These boundary conditions are determined by the sources $J$ in the boundary CFT. - **Holographic Principle**: The correspondence exemplifies the holographic principle, where the partition function of a higher-dimensional gravitational theory (AdS) is fully encoded by a lower-dimensional non-gravitational theory (CFT). ### Example: $AdS_5/CFT_4$​ Correspondence Consider the specific case of the $AdS_5​/CFT_4$​ correspondence: - The gravitational theory in the bulk AdS5_55​ space might be Type IIB string theory or supergravity. - The corresponding CFT on the 4-dimensional boundary is typically the $\mathcal{N}=4$ supersymmetric Yang-Mills theory. In this case, the partition function of the $\mathcal{N}=4$ SYM theory (a highly symmetric and conformally invariant field theory) on the boundary is dual to the partition function of the bulk gravitational theory in $AdS_5$​. Calculating one can provide insights into the properties of the other, especially in strongly coupled regimes where direct computation in the field theory is challenging. ### Importance 1. **Thermodynamic and Quantum Information**: The partition function in AdS/CFT correspondence allows for the computation of thermodynamic quantities and quantum information properties of strongly coupled systems using gravitational duals. 2. **Gauge/Gravity Duality**: It provides a concrete realization of the gauge/gravity duality, showing how a quantum field theory without gravity can describe a higher-dimensional theory with gravity. 3. **Black Hole Thermodynamics**: The partition function is crucial for studying the thermodynamics of black holes in AdS space, including phenomena like Hawking radiation and black hole entropy. ### Conclusion The partition function in the context of the [[AdS-CFT correspondence|AdS/CFT correspondence]] serves as a bridge between gravitational theories in higher-dimensional AdS spaces and conformal field theories on their lower-dimensional boundaries. This relationship encapsulates the core idea of the holographic principle and provides a powerful tool for studying strongly coupled quantum field theories and quantum gravity. Through this duality, complex problems in quantum field theory can be mapped onto more tractable problems in classical gravity, yielding insights into both fields. # References ```dataview Table title as Title, authors as Authors where contains(subject, "partition function") sort title, authors, modified, desc ```