Path Integral Surgery in Quantum Field Theory

• Path Integral Surgery is a robust technique employing codimension-one defect operators to 'cut' and 'glue' path integrals, establishing a bridge between discrete topology and continuum quantum systems.
• It systematically uses gluing and cutting operators along with Heisenberg-group averaging to construct replica manifolds and calculate entanglement measures in various quantum systems.
• The method extends to gauge theories, quantum codes, and holographic frameworks, enabling concrete computations of knot invariants, topological phases, and entropic quantities.

The path integral surgery technique is a rigorous methodology for implementing "cutting" and "gluing" operations within the framework of quantum field theory and quantum statistical mechanics. It provides a powerful, general-purpose toolkit for the construction of replica manifolds, the calculation of entanglement measures, and the path-integral-based analysis of topological quantum phases, gauge theories, and topological invariants. This approach encompasses both non-perturbative constructions in Chern–Simons theory and the systematic averaging over operator insertions in more general quantum mechanical, field-theoretic, and gauge-theoretic contexts, thereby linking discrete topological and continuum quantum frameworks.

1. Codimension-One Defects and Path Integral Slicing

At the technical core of path integral surgery is the implementation of codimension-one defect operators to "slice" or "cut" the path integral at a specific surface or time $\tau_0$. For a Euclidean path integral over a manifold $\mathcal{M}$, such cuts are realized by the sequential insertion and functional integration over two classes of operators:

• Gluing operators: To enforce equality of field configurations before and after the slice, introduce

$$\int\!D J\,\exp\left(2\pi i\int d^d x\,J(x)[\phi(\tau_0^+,x)-\phi(\tau_0^-,x)]\right) = \delta[\phi(\tau_0^+)-\phi(\tau_0^-)]\,.$$

• Cutting operators: To fully disconnect adjacent slices, insert a projector onto zero conjugate momentum:

$$\int\!D K\,\exp\left(i\int d^d x\,K(x)\pi(\tau_0,x)\right) = 2\pi\,\delta[\pi(\tau_0,x)]\,.$$

In operator formalism, the defects are defined as $$\hat J[J] = \exp(2\pi i\int d^d x\,J(x)\hat\Phi(\tau_0,x))$$, and $$\hat K[K] = \exp(i\int d^d x\,K(x)\hat\Pi(\tau_0,x))$$. Functionally integrating over $J$ and $K$ enforces delta-function or projection constraints at the slice. This methodology generalizes to multiple slices for constructing replica geometries or to arbitrary locations in spacetime for imposing entanglement cuts and junctions (Radwan, 23 Oct 2025).

2. Heisenberg-Group Averaging and Replica Construction

Path integral surgery acquires a group-theoretic structure through the introduction of correlated averaging over a set of operators forming a unitary representation of the (discrete) Heisenberg group—the Weyl operators $e^{iJ\hat X}e^{iK\hat P}$. Group-averaging projects onto subspaces invariant under certain symmetries:

$\frac{1}{|G|}\sum_{g \in G} D(g) \rho D^{-1}(g)$

projects a density matrix $\rho$ onto its commutant with respect to $G$.

In the path integral, a cyclic product of such operator insertions,

$$\prod_{j=0}^{n-1} \int \frac{dK_j\,dJ_j}{2\pi} \langle e^{iJ_j X} e^{iK_j P} e^{-iJ_{j+1} X} \rangle_\beta,$$

computes ratios such as $Z(n\beta)/Z(\beta)^n$, yielding the Rényi entropies for quantum statistical systems (Radwan, 23 Oct 2025).

3. Explicit Stepwise Surgical Procedures and Entropic Quantities

The surgical method can be formulated as a precise sequence:

1. Start with the partition function on an $n$-fold cover (e.g., $Z(n\beta) = \mathrm{Tr}[e^{-n\beta H}]$).
2. Insert $n$ cutting operators $\hat K_j$ at "time" slices $\tau_j = j \beta$ to break the circle into $n$ arcs; insert gluing operators $\hat J_j$ to rejoin these segments.
3. Functional integration over $\{K_j, J_j\}$ restores the original manifold, effecting $Z(n\beta) \to Z(\beta)^n$.
4. Inverting this procedure yields

$$\frac{Z(n\beta)}{Z(\beta)^n} = \prod_{j=0}^{n-1} \int \frac{dK_j\,dJ_j}{2\pi} \langle e^{iJ_j X}e^{iK_j P}e^{-iJ_{j+1} X} \rangle_\beta\,,$$

which computes the Rényi entropy

$$S_n = \frac{1}{1-n} \ln \frac{Z(n\beta)}{Z(\beta)^n}\,.$$

This technique enables the rigorous computation of entanglement measures from first principles via path integral manipulation, without explicit modification of the underlying spacetime manifold (Radwan, 23 Oct 2025).

4. Generalization to Gauge Theories

Path integral surgery extends naturally to gauge theories, with distinct treatments required for Abelian and non-Abelian cases:

• Abelian Gauge Theories (e.g., Maxwell): Use temporal gauge $A_0 = 0$ to define the extended Hilbert space, project onto physical states via Gauss's law, and insert

$$\exp(2\pi i \int J \cdot \pi), \quad \exp(i \int K \cdot A)$$

as surgery operators. Gauge invariance enforces $\partial \cdot J = 0$ and restricts $K$ to gauge-invariant combinations.

• Non-Abelian Gauge Theories ($U(N)$ matrix models): The reduced density matrix from partial tracing over color indices lacks manifest gauge invariance. Introduce auxiliary bifundamental fields $Q^a_i$, $X^a_{~i}$, and Lagrange multipliers to "blow up" the partial trace into a gauge-invariant form. This leads to replica partition functions with a quiver gauge theory structure:

$$\mathrm{Tr}[\rho^n] = \int \Bigl[\prod_{\text{quiver nodes}} d\Omega dX dQ dV\Bigr] \exp(-S_{\text{core}}[Q,X;\rho] - S_{\text{aux}}[Q,X,\Omega,V])\,,$$

manifesting color entanglement as a physical observable and realizing the replica manifold as a quiver theory (Radwan, 23 Oct 2025).

5. Applications: Topological Invariants and Quantum Codes

Path integral surgery is foundational to the computation of topological invariants and the analysis of quantum codes:

• Chern–Simons Theory and Surgery Invariants: In both Abelian and non-Abelian Chern–Simons theories, path integral surgery underlies the computation of Reshetikhin–Turaev invariants. For $U(1)$ theory, the nonperturbative partition function and Wilson loop averages on a closed manifold $M$ are expressed as sums over torsion in $H_1(M,\mathbb{Z})$, with modular $S$- and $T$-group elements implementing topological surgeries (Guadagnini et al., 2014).
• Non-Abelian Path Integral Surgery: In Chern–Simons theory with compact $G$, the path-integral formalism rigorously implements surgery by gluing functional integrals over boundary holonomies, employing modular S-matrix weights. This procedure exactly recovers the Rosso–Jones formula for torus knot invariants via a sequence of discreteness, evaluation, surgical gluing, and transformation between the modular and knot-theoretic frameworks (Hahn, 2015).
• Topological Quantum Codes: Via the path integral approach, lattice surgery on fault-tolerant quantum circuits is realized as a topological surgery in spacetime. The merging and splitting of code regions, implemented as bridge geometries in the path integral, yield projective logical measurements, such as two-qubit $ZZ$ projections, and directly instantiate topological error correction thresholds via anyon worldline homology (Bauer, 2024).

6. Large-$N$ and Holographic Path Integral Surgery

The method generalizes to large-$N$ vector and gauge theories and has implications for holography:

• Large-$N$ Vector Models: By introducing Hubbard–Stratonovich fields, components can be traced out, yielding modified saddle-point equations that deform the large-$N$ effective action. The relative weighting of replica and non-replica components is controlled via the ratio $M/N$, directly influencing the saddle and, consequently, the entanglement structure (Radwan, 23 Oct 2025).
• Holography and Wormhole Connections: The GKPW dictionary suggests that the correlated path integral average over boundary sources $\{J_j, K_j\}$ maps to modified boundary conditions for the dual gravitational bulk. Known in simple models (e.g., SYK/JT gravity), such configurations generate Euclidean wormhole geometries, corresponding to the entanglement structure of replica manifolds. In non-Abelian gauge/gravity dual pairs, partial tracing over color directions may source new bulk topologies—implying the emergence of duals to generalized RT (Ryu–Takayanagi) surfaces in internal spaces (Radwan, 23 Oct 2025).

7. Table: Path Integral Surgery‐Related Contexts

Context	Key Surgery Operation	Reference
Abelian Chern–Simons	Fluctuation–torsion decomposition, $U(1)$ modular S	(Guadagnini et al., 2014)
Non-Abelian CS/Knots	Modular S-matrix gluing, Rosso–Jones formula	(Hahn, 2015)
Quantum Circuits/Codes	Lattice surgery via spacetime bridge/topological cut	(Bauer, 2024)
Entanglement Entropies	Replica manifold by Heisenberg-averaging	(Radwan, 23 Oct 2025)
Large-N/Holography	Saddle deformation, replica wormholes	(Radwan, 23 Oct 2025)

8. Significance and Outlook

The path integral surgery technique unifies the treatment of replica constructions, topological quantum invariants, and gauge-invariant entanglement measures via codimension-one operator slicing and surgical group averaging. It supports rigorous derivations of knot polynomials in Chern–Simons theory, enables explicit design of topological quantum operations in quantum codes, and extends to gauge–gravity duality contexts, suggesting new directions in bulk/boundary correspondence and the structure of entanglement in quantum gravity.

The method's generality and algorithmic flexibility make it applicable to a wide array of modern problems in mathematical physics, quantum information, and high-energy theory. Current research avenues include extension to more general operator averaging, its ramifications in non-Abelian gauge theories with quiver structures, and the precise mapping between path-integral surgical averages and higher-dimensional quantum error correction or holographic codes (Radwan, 23 Oct 2025, Hahn, 2015, Guadagnini et al., 2014, Bauer, 2024).