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openPhysics, High Energy Physics - TheoryPosted 29 Aug 2026 by qurore

A general mass formula for topological boundary conditions of non-Abelian 3d TQFTs

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Problem statement

Dymarsky and Shapere (arXiv:2602.00224) define the mass of the set of topological boundary conditions of a 3d TQFT: a total weighted count of those boundary conditions, equal to the TQFT partition function averaged over all closed 3d manifolds, which they interpret as the renormalized partition function of TQFT gravity. For Abelian TQFTs they evaluate this mass for any bosonic theory and show that it reduces to known mass formulas for particular families of classical codes. Outside the Abelian case, the state of the art is a single family of examples: n + n-bar copies of the Ising modular tensor category. The unsolved problem is the general non-Abelian mass formula. Concretely: for an arbitrary bosonic 3d TQFT whose modular tensor category admits topological boundary conditions (equivalently, is a Drinfeld center), express the weighted count of its topological boundary conditions in closed form in terms of intrinsic categorical data, and prove that the defining average over closed 3-manifolds converges, or give it a canonical regularization with a proof of well-definedness. A resolution is a big win for three reasons. First, it would extend Siegel-style mass formulas beyond their classical home. The mass formulas of number theory and coding theory count lattices and codes weighted by inverse automorphisms. A general non-Abelian mass formula would establish the same kind of statement for a genuinely new class of algebraic objects, the Lagrangian algebras of modular tensor categories, and would show how far the analogy between arithmetic averaging and gravitational path integrals actually reaches. Second, in the TQFT gravity program (arXiv:2405.20366), sums over 3d topologies are interpreted through ensembles of boundary theories. The mass is exactly such a sum, so a general formula would fix the quantitative side of that program beyond Abelian theories and would constrain the measure with which rational CFTs and gapped boundaries appear in the ensemble. Third, counting the gapped boundaries of a non-Abelian topological order is a hard combinatorial problem in its own right, relevant to condensed matter and quantum information. A mass formula gives a global constraint on that count which direct enumeration of Lagrangian algebras cannot reach with current methods.

Current state

The objects to be counted are well defined. Topological (gapped) boundary conditions of a 3d TQFT are classified by Lagrangian algebras in its modular tensor category: Kapustin and Saulina worked out the Abelian Chern-Simons case (arXiv:1008.0654), Kitaev and Kong built lattice models for gapped boundaries and domain walls (arXiv:1104.5047), and Davydov, Mueger, Nikshych, and Ostrik placed the existence question in the Witt group framework (arXiv:1009.2117). What is missing is the count. On the averaging side, Maloney and Witten (arXiv:2006.04855) and Afkhami-Jeddi, Cohn, Hartman, and Tajdini (arXiv:2006.04839) showed that the average of Narain CFT partition functions over moduli space is given by an Eisenstein series. Dymarsky and Shapere connected quantum codes, lattices, and CFTs (arXiv:2009.01244) and formulated TQFT gravity and ensemble holography (arXiv:2405.20366). Romaidis and Runkel studied mapping class group averages of CFT correlators (arXiv:2309.14000). Recent work develops the Abelian ensemble picture further (arXiv:2509.26052) and studies automorphism-weighted ensembles from TQFT gravity (arXiv:2511.04311). The linked paper (arXiv:2602.00224) evaluates the mass of topological boundary conditions for any Abelian bosonic TQFT, where it reduces to code mass formulas, computes one non-Abelian family, n + n-bar copies of the Ising modular tensor category, and generalizes the construction to 5d Abelian 2-form Chern-Simons theories. The general non-Abelian case stays open for two reasons. The code and lattice structure that makes the Abelian count tractable is absent for a general modular tensor category. And the sum over all closed 3-manifolds that defines the mass has not been given a convergence proof or a canonical regularization at that level of generality.

Resolution criteria

The Grand Challenge is solved when one paper, or a coherent series of papers, does all of the following: 1. States a mass formula for the topological boundary conditions of an arbitrary bosonic non-Abelian 3d TQFT whose modular tensor category admits topological boundary conditions (equivalently, is a Drinfeld center), or of an explicitly characterized class of such theories strictly larger than the union of Abelian TQFTs and n + n-bar copies of the Ising modular tensor category. 2. Makes the definition precise for that class: the weighting of the boundary conditions and the sum over closed 3-manifolds are both defined, and the sum is proven to converge or is given a canonical regularization whose well-definedness is proven rather than assumed. 3. Reproduces the known special cases: the Abelian mass formulas of arXiv:2602.00224, including their reduction to code mass formulas, and the n + n-bar Ising result. 4. Evaluates the formula in closed form for at least one non-Abelian example outside the Ising family, and checks it against a direct enumeration of Lagrangian algebras in at least one case where that enumeration is independently known.

Citations

  1. A. Dymarsky, A. Shapere, "Mass formula for topological boundary conditions from TQFT gravity", arXiv:2602.00224
  2. A. Dymarsky, A. Shapere, "TQFT gravity and ensemble holography", arXiv:2405.20366
  3. A. Kapustin, N. Saulina, "Topological boundary conditions in abelian Chern-Simons theory", arXiv:1008.0654
  4. A. Kitaev, L. Kong, "Models for gapped boundaries and domain walls", arXiv:1104.5047
  5. A. Davydov, M. Mueger, D. Nikshych, V. Ostrik, "The Witt group of non-degenerate braided fusion categories", arXiv:1009.2117
  6. A. Maloney, E. Witten, "Averaging Over Narain Moduli Space", arXiv:2006.04855
  7. N. Afkhami-Jeddi, H. Cohn, T. Hartman, A. Tajdini, "Free partition functions and an averaged holographic duality", arXiv:2006.04839
  8. A. Dymarsky, A. Shapere, "Quantum stabilizer codes, lattices, and CFTs", arXiv:2009.01244
  9. I. Romaidis, I. Runkel, "CFT correlators and mapping class group averages", arXiv:2309.14000
  10. N. Angelinos, "Abelian 3D TQFT gravity, ensemble holography and stabilizer states", arXiv:2509.26052
  11. A. Barbar, "Automorphism-weighted ensembles from TQFT gravity", arXiv:2511.04311

Linked papers