Symmetry Theories, Wigner's Function, Compactification, and Holography

TL;DR

The paper addresses how global symmetry data of a D-dimensional QFT can be packaged in a $(D+1)$-dimensional SymTFT/SymTh bulk without fixing an absolute polarization, including mixed states of relative QFTs and boundary conditions. It develops and applies Wigner's quasi-probability function as a phase-space language to encode this data, including continuum, generalized, and discretized variants, and a wavefunction interpretation in the doubled Hilbert space. It then applies the framework to Abelian Chern-Simons theory, 6D SCFTs with 2-form symmetries, and holographic setups such as string compactifications and AdS/CFT, including thermofield double gluing and ER=EPR-like connections. The results provide a polarization-free, ensemble-capable description of symmetry data in gravity and string theory, with potential implications for entanglement structure and topological boundary conditions.

Abstract

The global symmetry data of a $D$-dimensional absolute quantum field theory can sometimes be packaged in terms of a $(D+1)$-dimensional bulk system obtained by extending along an interval, with a relative QFT$_D$ at one end and suitable gapped / free boundary conditions at the other end. The partition function of the QFT$_D$ can then be interpreted as a wavefunction depending on background fields. However, in some cases, it is not possible or simply cumbersome to fix an absolute form of the symmetry data. Additionally, it is also of interest to consider entangled and mixed states of relative QFTs as well as entangled and mixed states of gapped / free boundary conditions. We argue that Wigner's quasi-probabilistic function on phase space provides a physical interpretation of the symmetry data in all such situations. We illustrate these considerations in the case of string compactifications and holographic systems.

Figures (8)

• Figure 1: (i): Schematic depiction of the SymTFT formalism and its wavefunction interpretation. We decompress an absolute QFT to a relative QFT $|Z\rangle$ and a topological / free boundary $\langle B|$ with bulk a SymTFT / SymTh $\mathcal{S}$. The overlap is then $Z(B)$, the partition function with prescribed boundary conditions. (ii): Similarly, mixed states of relative QFTs $\rho_Z=\sum_{ij}z_{ij}|Z_i\rangle \langle Z_j|$ and topological boundary conditions are decompressed into a $|\rho_Z\rangle\!\rangle$ and $\langle \!\langle \rho_B|$ which are ket and bra to the SymTFT / SymTh $\mathcal{S}\otimes \overline{\mathcal{S}}$.
• Figure 2: Wigner's function for the pure state $\widehat{\rho}=|Z\rangle \langle Z|$ avoids any reference to topological / free boundary conditions and can be rephrased purely in reference to two sets of physical boundary conditions (red). The operators $\mathbb{U}_{\Phi}, \mathcal{C}, \mathbb{U}_{\Phi}^\dagger$ may be interpreted as $(-1)$-form symmetry operators which are codimension-one interfaces (green) in the SymTFT / SymTh slab. Subfigures (i) and (ii) show the same image, with the latter sketched in fewer dimensions.
• Figure 3: We sketch steps relating "unfolding" the doubled SymTFT sandwich to a sum of SymTFT sandwiches with two physical boundary conditions. (i): Starting point, i.e., the initial doubled configuration with partition function given by Wigner's function. Boundary conditions are determined by two pure states. (ii): We separate out two operators from the boundary condition, which are indicated by green dots and are supported in one of the two systems tensored to give the overall doubled system. (iii): The boundary condition trivializes to the identity operator $\text{Id}$ (with respect to a single copy of the SymTFT) upon separating out these two operators. We can therefore reconnect the two SymTFT supports to give a single connected system. (iv): Finally, we make $\rho_Z=\sum_{ij}z_{ij}|Z_i\rangle \langle Z_j|$ explicit and fuse the two green dots to $\mathbb{U}_{B}^{[s]}\:\!\mathcal{C} \:\! \mathbb{U}_{B}^{[r] \, \dagger}$.
• Figure 4: We sketch defect and symmetry operators in presence of a $\mathbb{U}_{\mathbb{B}}^{[s]}\:\! \mathcal{C} \:\!\mathbb{U}_{\mathbb{B}}^{[r],\dagger}$ insertion viewed as a codimension-one SymTFT operator / wall. (i): Defect operators piercing this wall may be dressed along their intersection with it (black dot). (ii) and (iii): Deforming a symmetry operator across the wall, it may remain attached thereto (dashed line). When genuine operators are mapped onto genuine operators, the dashed line is absent / trivial.
• Figure 5: Depiction of the SymTFT / SymTh sliver in a single (i) and multi-throat (ii) string background. In the multi-throat configuration, the tensor product of the physical boundary conditions produces a pure state relative QFT and the bulk geometry results in a mixed state for topological boundary conditions.