Baldridge-McCarty TQFT versus SU(2) Spin Networks: What’s the difference?

The Question I Always Get

When I give talks on the n-color homologies, I am often asked whether the underlying topological quantum field theory (TQFT) is based on $SU(2)$ , $SU(n)$ , or $SL(n,\mathbb{C})$ . It’s a fair question – the way we describe the theory can sound like these familiar frameworks. Experts hear terms like spin networks and see diagrams with “spins” or “colors” on edges, and naturally assume we’re working with a standard group-based TQFT. In fact, we are not using any of those classical groups. The Baldridge & McCarty (B&M) framework for the n-color homology TQFT is completely different in its algebraic formulation. In this post, I’ll explain the difference between the B&M framework and traditional spin network formulations, and clear up why people often confuse our TQFT with an $SU(2)$ , $SU(n)$ or $SL(n,\mathbb{C})$ theory. Along the way, we’ll see how the new approach is based on the cohomology ring of $\mathbb{C}P^{n-1}$ and the representation theory of $U(\mathfrak{sl}_n)$ , and why we sometimes still use the language of spin networks as a convenient analogy. Finally, I’ll touch on the origins of these ideas in graph coloring (going back to Penrose) and why this matters for mathematical physics today.

Spin Networks: A Quick Refresher

To set the stage, let’s recall what spin networks are in the classical sense. Spin networks were introduced by Roger Penrose in 1971 as diagrams encoding group representations and their invariant couplings, with the goal of tackling combinatorial problems like the Four Color Theorem [2]. Typically associated with $SU(2)$ , a spin network assigns representations (such as spin- $\tfrac{1}{2}$ or spin-1) to edges and couples them at vertices according to group-theoretic rules. When three edges meet, the requirement that they fuse to an invariant singlet imposes constraints — akin to angular momentum rules in quantum mechanics. These diagrams later became central to the study of quantum invariants (like the Jones polynomial) and formed the basis for loop quantum gravity models [4]. The essential idea is this: spin networks represent group data combinatorially, with edges corresponding to irreducible representations, vertices encoding invariant tensors, and the whole diagram evaluates to a number by contracting these tensors.

Now, because of this heritage, when someone sees an “n-color” theory with diagrams and mentions of “spin-1/2 lines” or “spin-1 lines,” they immediately think “Oh, this must be an $SU(n)$ or $SL(n,\mathbb{C})$ TQFT!” After all, $SU(n)$ would have an $n$ -dimensional fundamental representation, and $SL(n,\mathbb{C})$ is the complexified version often used in Chern–Simons theory, etc. And $SU(2)$ is the classic case $n=2$ . It’s a reasonable guess. However, in our case that guess is wrong.

The n-Color TQFT of Baldridge & McCarty

So, what is the B&M n-color TQFT based on, if not a familiar Lie group like $SU(n)$ ? The short answer: it’s based on a commutative algebra that comes from algebraic topology and combinatorics, not directly from a Lie group. In particular, the state-space in this theory is built from the cohomology ring of $\mathbb{C}P^{n-1}$ (complex projective space of dimension $n-1$ ). This cohomology ring is a well-known Frobenius algebra:

As an algebra (over $\mathbb{Z}$ or $\mathbb{Q}$ ), $H^*(\mathbb{C}P^{n-1}) \cong \mathbb{Z}[x]/(x^n)$ . In plain terms, it’s generated by a degree- $2$ element $x$ (think of the Kähler class/hyperplane class) with the relation $x^n=0$ . The vector space has dimension $n$ , with a basis ${1, x, x^2, \dots, x^{n-1}}$ . There is also a natural bilinear form given by integration (Poincaré duality) that makes it a Frobenius algebra.

This algebra is the foundation of our TQFT. Why $\mathbb{C}P^{n-1}$ ? Because it provides exactly n “colors” (basis elements) and a simple multiplication rule encoding how colors combine. It turns out to be the right tool to categorify the n-color polynomial — an invariant that counts proper colorings of graphs when evaluated at $q=1$ (cf. Section 3 of [1]). Essentially, the B&M TQFT assigns the algebra $H^*(\mathbb{C}P^{n-1})$ to a circle (the “state space”), and uses the algebra’s multiplication and comultiplication (coming from the Frobenius structure) to define operators for splitting or joining circles (cobordisms). In this way, a 2-dimensional CW complex built on a surface can be evaluated, and one obtains homology groups whose graded Euler characteristic evaluated at $q=1$ yields the Penrose polynomial of a graph evaluated at $n$ (cf. [1] or [2]). The homology theory produced is called the n-color homology. Our recent paper develops this thoroughly [1], and even shows how it could lead to a new approach to the Four Color Theorem (by constructing 4-colorings via spectral sequences).

A second structural ingredient in the B&M framework is the use of representation theory from the universal enveloping algebra $U(\mathfrak{sl}_n)$ , where $\mathfrak{sl}_n$ is the Lie algebra of traceless $n\times n$ matrices. At first glance, this might suggest a connection to $SL(n,\mathbb{C})$ gauge theory — and indeed, $\mathfrak{sl}_n$ is its Lie algebra. But in the B&M framework, this structure is not used to define a gauge group or a field theory in the usual geometric sense. Instead, $U(\mathfrak{sl}_n)$ functions purely as an algebraic toolkit. It provides the representation-theoretic machinery to encode how color states interact at vertices, how loops trace out combinatorial data, and how enhanced states evolve across differentials. Much like tensor categories underpin fusion rules in spin networks, here $U(\mathfrak{sl}_n)$ helps formalize how quantum wires in ribbon graphs can join, split, and close into Wilson loop–like objects.

However, this is not “purely abstract algebra” either — it has a meaningful link to physics. Specifically, the framework connects to the zero-temperature limit of the $Q = n$ antiferromagnetic Potts model, a statistical mechanics model where colorings correspond to spin configurations and the partition function counts proper colorings. The B&M TQFT provides a categorification of this count, enriching it into a homological invariant whose Euler characteristic also counts proper colorings. So while the use of $U(\mathfrak{sl}_n)$ is algebraic rather than field-theoretic, the overall structure is still physically motivated, capturing the combinatorics of the Potts model within a homological TQFT setting. In that sense, the algebra is doing physical work — just not in the way a Yang–Mills field or a gauge connection would.

In summary, the B&M n-color TQFT is built on a combinatorial-algebraic foundation: the ring $H^*(\mathbb{C}P^{n-1})$ (a commutative Frobenius algebra) provides the state space and basic operations, and $U(\mathfrak{sl}_n)$ representation theory underlies the coupling rules.

Colors vs. Spins: Same Diagrams, Different Algebra

If the B&M framework isn’t based on $SU(n)$ , why do we keep using terms like “spin- $\tfrac{1}{2}$ ” or draw two strands to represent something? While the language of spin networks certainly helps build intuition — especially for readers familiar with $SU(2)$ and quantum topology — this visual language actually has deeper geometric and physical meaning in our context. The strands and vertices in the B&M $n$ -color TQFT are not merely suggestive drawings; they encode how disks (2-cells) are glued into a ribbon graph to form a closed surface. For example, a “double strand” along an edge labeled by $x^2$ and $x^3$ , respectively, corresponds to parts of the boundary of two such disks — they are part of two loops that will each receive a color (an idempotent state in the spectral sequence limit) in the state-sum model. These closed loops represent physical degrees of freedom in the Potts model: they are the regions that hold the color states, and the rules for how those colors can interact (e.g., being distinct on adjacent regions) are imposed via the algebra’s structure.

In this light, what looks like “spin- $\tfrac{1}{2}$ ” or “spin-1” in the diagram isn’t just a mnemonic borrowed from $SU(2)$ representation theory. In the B&M framework, it reflects the topological structure of the ribbon graph’s blowup and the algebraic behavior of its associated TQFT. In the blowup, each original edge in the graph becomes a ribbon labeled “2” — interpreted as a spin-1 line — which splits into two quantum wires when the ribbon is resolved. These wires either connect the boundary of one disk directly to another (a straight ribbon) or introduce a twist (a half-twist ribbon), depending on the state. Meanwhile, the edges along the blown-up vertex cycles are labeled “1” — the spin- $\tfrac{1}{2}$ lines — and represent how color flux enters or exits each disk.

The cohomological states associated to these loops, such as $x^2$ or $x^3$ , describe how many quantum wires (i.e., units of color flux) enter the boundary of each disk. These are not multiple lines drawn in parallel but rather powers of the generator $x$ , indicating the algebraic “weight” assigned to the loop via the enhanced state. This structure determines how face colorings interact across the surface. In particular, only certain combinations of wires (e.g., color flows) can meet consistently at a vertex, and these rules are governed by the multiplication and comultiplication in the Frobenius algebra — not by the branching rules of an $SU(n)$ spin network, which are trivially satisfied by the fixed labels 2 and 1 on the original and cycle edges, respectively.

So while the diagrams resemble spin networks and borrow their visual grammar, they are doing something different. They encode the propagation of color information through the surface in a way that reflects the physical constraints of the Potts model (especially in the zero-temperature antiferromagnetic limit), but using new algebraic tools. This is not just an analogy — it’s a different topological model, rooted in graph colorings and homological algebra, but still referring to physical processes.

Concretely, in our $n$ -color state space, the generator $x$ (representing one unit of color) behaves in a way that loosely resembles a spin- $\tfrac{1}{2}$ line. For example, when $n = 2$ , the cohomology ring $H^*(\mathbb{C}P^1)$ is 2-dimensional with basis ${1, x}$ and the relation $x^2 = 0$ . This reflects the idea that two units of color on a merged face map to zero in homology via the differential, which includes the multiplication map — a situation analogous to two spin- $\tfrac{1}{2}$ representations fusing into a singlet. In the ribbon graph framework, each quantum wire along an edge can be labeled by $x$ , and these wires correspond to two distinct faces in a given enhanced state. When the edge is resolved as a 0-smoothing (with straight strands), these faces may be separate (let’s assume); but in the 1-smoothing (with a half-twist), the strands merge the two regions into a single face. In the chain complex, this corresponds to applying the multiplication map $m(x \otimes x) = x^2$ , which equals zero when $n = 2$ . So the enhanced state resulting from such a merge carries a weight of zero, a partial condition to be a nontrivial homology class (it’s differential is zero on this edge map). This is how the topological behavior of edge resolutions is tightly coupled with the algebraic structure of the Frobenius algebra, and why $x^2 = 0$ has geometric and combinatorial content beyond a formal relation.

For general $n$ , each strand in the blown-up ribbon graph can carry a cohomology class $x^i$ for $0 \le i \le n - 1$ , representing an assignment of $i$ units of “color” to that face. These labels come from the basis of the Frobenius algebra $H^*(\mathbb{C}P^{n-1})$ , where each monomial $x^k$ spans a one-dimensional space. In the chain complex, a 0-smoothing that corresponds to an edge where two faces remain distinct — each carrying their own label, say $x^i$ and $x^j$ . A 1-smoothing, by contrast, can merge these two faces into one, and the differential applies the multiplication map $m(x^i \otimes x^j) = x^{i+j}$ .

This multiplication is only nonzero if $i + j < n$ , due to the relation $x^n = 0$ . So if the merged label would be $x^n$ or higher, the resulting state maps to zero under the differential — and thus lies in the kernel for that edge map. That is, it is potentially a cycle in the complex, though not necessarily a nontrivial homology class. The structure of the differential, then, encodes constraints on how color weights propagate when two regions are joined. While visually we might draw $i$ strands to represent $x^i$ , these are not multiple tensor lines like in spin networks. Rather, it is better to think of each $x^i$ label is a single state on a loop — the boundary of a disk — and the ribbon structure ensures the correct topology for tracking how these states interact and cancel through the chain differential.

Rather than a crude cutoff, $\mathbb{C}[x]/(x^n)$ can be seen as a carefully selected $n$ -dimensional slice of an infinite-dimensional Hilbert space. In this framing, terms like $x^3 \otimes x$ vanish in $C[x]/(x^4)$ not because they’re meaningless, but because they’ve been systematically excluded — like discarding high-energy modes in a field theory. As $n \to \infty$ , we recover the full polynomial algebra, but at finite $n$ , we gain a tractable, algebraically structured model that captures the “effective” combinatorics of color interactions with manageable complexity. (Surprisingly, the solution to the cyclic double cover conjecture plays a crucial role in this setup, ensuring that proper colorings exist for large enough $n$ ! See Theorem 7.7 or Conjecture 8.12 in [1].)

The TQFT state-sum interprets each closed loop in the graph as carrying color information — captured algebraically by idempotents in a (different but related) Frobenius algebra. These loops represent the faces of the surface, and the coloring rule that “adjacent faces must differ” is woven into how the algebra assigns values to those loops and how it combines them across edges. That said, this enforcement isn’t always immediate or strict. Depending on which version of the algebra we use — and there are a few flavors — the rule might be enforced exactly, or only approximately at first, with improper colorings gradually canceled out in the homology in the spectral sequence limit.

In other words, the algebra can be tuned to behave more or less sharply with respect to coloring constraints. In its purest form, the initial algebra might still allow “illegal” colorings to show up in intermediate calculations, but ensures that by the end of the process — after all the homological spectral sequence bookkeeping is done — only proper colorings survive. So while the diagrams may look like spin networks and the fusion rules feel familiar, the real power of the B&M framework lies in how the algebra carefully manages color flow, sometimes softly, sometimes strictly, but always with an eye toward the final goal: face colorings that obey the rules.

Not an $SU(2)$ , $SU(n)$ , or $SL(n,\mathbb{C})$ Theory — But Something Close

Let’s address a common point of confusion head-on: the B&M $n$ -color TQFT is not built from any standard Lie group TQFT like $SU(2)$ , $SU(n)$ , or $SL(n,\mathbb{C})$ , even if it borrows some of the visual and algebraic language those frameworks made familiar. What’s actually happening is subtler and more foundational: the B&M framework is preparing the algebraic structure of a gauge theory — using ribbon graphs, Frobenius algebras, and spin-network-like diagrams to lay down the mathematical language one would need to describe connections, holonomies, and Wilson loop observables — but all within a purely combinatorial and topological context.

Not $SU(2)$ : Yes, the B&M construction takes inspiration from Khovanov homology [3], which categorifies the $SU(2)$ Jones polynomial using a cube-of-resolutions approach. But that’s where the similarity ends. Khovanov’s theory is rooted in 3D Chern–Simons theory and quantum groups like $U_q(\mathfrak{sl}_2)$ . In contrast, the B&M TQFT is entirely 2-dimensional. It is defined on ribbon graphs drawn on surfaces, where the edges behave like quantum wires: the original graph edges carry a “spin-1” label, and the cycle edges from blown-up vertices carry “spin- $\tfrac{1}{2}$ .” These are not group-theoretic labels, but algebraic proxies for how 2-cells (disks) are glued into the surface.

Not $SU(n)$ (for $n > 2$ ): It’s tempting to assume an $SU(n)$ structure — after all, $\mathfrak{sl}_n$ appears in the background, and $H^*(\mathbb{C}P^{n-1}) \cong \mathbb{Z}[x]/(x^n)$ has rank $n$ , matching the number of “colors.” But this isn’t a quantum group or Chern–Simons TQFT: there are no path integrals, fusion rings, or root-of-unity deformations (cf. [5]). The number $n$ reflects the combinatorics of proper graph colorings, not a global symmetry group.

Yet the structure is gauge-flavored. Each ribbon edge encodes how local color data — assigned to adjacent face(s) of the edge — is transported across the graph, with its resolution (straight or with a half-twist) determining how these face states combine. After resolving all edges, the resulting state graph consists of spin- $\tfrac{1}{2}$ quantum wires tracing loops around disks in the ribbon surface. These loops carry cohomological labels like $x^i$ , and in the graded theory, each closed loop contributes $[n]_q$ , which is a shifted factor of

$1 + q + \cdots + q^{n-1}$ ,

the graded dimension of $H^*(\mathbb{C}P^{n-1})$ . In this way, loops act as Wilson loop analogs, and the full state-sum captures their interactions much like quantum dimensions appear in lattice gauge theory.

This resemblance goes deeper in future work. As one moves between strata in the moduli space of metric ribbon graphs — via I–H moves, for example — these “Wilson loop” contributions begin to encode how vector potentials transition across strata. The current theory thus serves as a linearized precursor to a gauge-theoretic TQFT with connections and defects still to come. Interestingly, the complexity of the underlying graphs — the number of vertices, edges, and faces — can act as a combinatorial scaffold from which gauge-like behavior emerges. Our algebraic framework lays the groundwork for a full gauge-theoretic framework on the moduli space of metric ribbon graphs, where such structural complexity may govern the appearance of effective gauge symmetries and effective field-like couplings. (I am speculating a bit here, but not without evidence — see for example [9].)

But for now, B&M stay firmly within an algebraic framework. Instead of using representations via tensor products, the B&M framework encodes color interactions using the cup product in a Frobenius algebra. Instead of building a Hilbert space from irreducible representations, we assemble a chain complex whose homology filters out improper colorings. The resemblance to gauge theory is real, but what’s being constructed is the algebraic shadow of such a theory — an intermediate framework that captures its representation-theoretic and combinatorial bones, before geometry or dynamics are added.

Not $SL(n,\mathbb{C})$ : This confusion arises mainly from the use of $\mathfrak{sl}_n$ in constructing the Frobenius algebra, particularly through the symmetric algebra $\mathbb{C}[x]/(x^n)$ and its representation-theoretic analogies. But here, $\mathfrak{sl}_n$ serves purely as an algebraic toolkit — not as the Lie algebra of a gauge group. There are no complex gauge fields, holomorphic vector bundles, flat $SL(n,\mathbb{C})$ connections, or analytic continuations to a complex TQFT [6]. What the B&M framework offers instead is a combinatorial model that mirrors some of the algebraic footprint of such theories: cohomological data in place of representation rings, diagrammatic resolutions encoding local constraints, and loop contributions that track graded dimensions. While it may not yet carry geometric structure, it sets the stage for one — and may eventually interface with the moduli-theoretic landscape complex gauge theories inhabit.

So what is it? The B&M $n$ -color homology is a new type of 2D TQFT, one that blends the diagrammatics of spin networks, the algebra of cohomology rings, and the physics of the Potts model — all to recast a coloring problem as a topological state-sum [1]. It constructs algebraic data on edges (like vector bundles), defines local gluing rules (like gauge-invariant couplings), and interprets closed loops in the state-sum as Wilson loop analogs, each contributing a weight that counts or encodes colorings. It doesn’t begin with a gauge group, but it assembles much of the scaffolding that a gauge theory would require. In that sense, it’s a combinatorial precursor to a gauge-theoretic TQFT — laying the algebraic groundwork for connections, holonomies, and Wilson loops without appealing to a gauge group.

What it lacks — and what makes it fundamentally distinct from gauge-theoretic TQFTs like Chern–Simons theory — is a geometric or analytical origin. There’s no functional on a moduli space of connections, no critical point equations, and no differential arising from flow lines of a gauge-theoretic action. The B&M differential is constructed algebraically from face-merging and Frobenius multiplication, not from counting solutions to Yang–Mills equations. In that sense, it bypasses the analytical machinery of gauge theory while still mirroring much of its structural DNA.

This places B&M in a conceptual position very similar to where Khovanov homology stood in the early 2000s [3, 7]. At the time, Khovanov and Lee had built a categorified invariant of knots using purely algebraic and combinatorial tools — a 2D TQFT defined from a Frobenius algebra, diagrammatic resolutions, and chain complexes. There was no gauge theory in sight. Only later, through the work of Witten and others, did a gauge-theoretic interpretation begin to emerge, connecting the homology to critical points of a functional on flat connections and differentials arising from solutions to gauge-theoretic flow equations on $S^3 \times \mathbb{R}$ . The B&M theory may follow a similar trajectory: born in algebra, but with eyes towards geometry (I am a gauge theorists, after all!).

Origins in Graph Coloring and Penrose’s Ideas

It’s worth noting that the inspiration and raison d’être of the n-color homology theory is deeply tied to graph coloring problems, and this has a historical echo in Penrose’s original spin networks. Penrose introduced spin networks in part to tackle the Four Color Problem – he considered what happens if you formally allow “negative $\tfrac{1}{2}$ -dimensional” representations, trying to force a constraint that effectively encodes the four-colorability of a planar map [2]. While that approach didn’t directly solve the four-color theorem, it planted the seed that ideas from quantum/angular momentum (like spin networks) could inform graph coloring. Fast forward to today: the B&M framework can be seen as a modern, rigorous version of that philosophy. By using a TQFT and homological algebra, we categorify graph coloring polynomials and create tools to attack problems like the four color theorem. In fact, one of our results is that if you focus on the case $n=4$ (four colors) and look at the spectral sequence arising from our homology, it never collapses prematurely – suggesting a potential constructive proof of the Four Color Theorem via homological means. That’s a big deal: it connects a notoriously hard combinatorial problem to modern topology and category theory — and lays the groundwork for future links with gauge theory on moduli spaces.

From a mathematical physics perspective, this is exciting because it broadens the landscape of TQFTs. We usually think of TQFTs as emerging from physical theories (like Chern–Simons or BF theory) or from quantum groups. Here, we have a TQFT born from a purely combinatorial setting — graph colorings — but constructed using tools that echo those in physics: state-sum models, spin network-like evaluations, and algebraic structures with Frobenius trace pairings. The connection to the Potts model, whose partition function counts proper colorings in a statistical mechanics setting, suggests that such TQFTs can categorify not just polynomials, but physically meaningful quantities. Thus, the B&M framework shows, once again, that physics-inspired techniques can illuminate even the most classically combinatorial problems.

Conclusion: New Framework, Classic Flavor

At first glance, the B&M $n$ -color homology TQFT might resemble a classic spin network theory — but under the hood, it’s something quite different. Instead of relying on $SU(2)$ or $SU(n)$ gauge groups, it draws from the cohomology ring of $\mathbb{C}P^{n-1}$ and an array of algebraic techniques rooted in $U(\mathfrak{sl}_n)$ representation theory. While we borrow the language of “spins” (spin- $\tfrac{1}{2}$ and spin-1) to build intuition, these are different than angular momenta of quantum particles. The result is a framework that categorifies graph colorings, recovers the Penrose polynomial at a value as an Euler characteristic, and defines new homologies with implications for long-standing problems like the Four Color Theorem.

But there’s more to it than just combinatorics. The theory mirrors many structural features of gauge theory — vector bundles, Wilson loop analogues, local constraints at vertices. It even hints at deeper connections to statistical physics through its relationship with the Potts model. As the framework evolves, it lays the groundwork for future gauge-theoretic constructions on the moduli space of metric ribbon graphs, potentially with connections, stratified transitions, and topological defects. What begins as a “combinatorial TQFT” is starting to look like the algebraic skeleton of a much richer physical theory.

So the next time you hear “ $n$ -color TQFT,” don’t jump to “Ah, an $SU(n)$ theory.” Instead, think: “A graph-coloring TQFT that blends cohomology, category theory, and physics intuition…that may be pointing the way to something deeper.” It’s a different animal, albeit one that shares aspects with spin networks, lattice gauge theory, and topological quantum field theory. Hopefully this article gives a glimpse at the broader landscape opening up: a new kind of TQFT, emerging from combinatorics and homological algebra, but tied to modern geometry and physics.

References

[1] S. Baldridge and B. McCarty, A topological quantum field theory approach to graph coloring, arXiv:2303.12010 [math.GT] (2023). – Introduces the n-color TQFT and n-color homologies, defining a Frobenius algebra based on $H^*(\mathbb{C}P^{n-1})$ and developing homology theories that categorify graph colorings. (The foundational paper for the framework discussed here.)

[2] R. Penrose, Applications of negative dimensional tensors, in Combinatorial Mathematics and its Applications, D.J.A. Welsh (ed.), Academic Press, London, 1971, pp. 221–244. – The original paper introducing spin networks. Penrose uses diagrammatic tensors (later known as spin networks) and even speculates on using them for the Four Color Problem by considering “negative” dimensions. A classic piece of inspiration behind using physics-like methods in graph theory.

[3] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), no. 3, 359–426; arXiv:math.QA/9908171. – Khovanov’s landmark paper that categorified the $SU(2)$ Jones polynomial, creating what we now call Khovanov homology. This is an example of a homology theory arising from an $SU(2)$ TQFT (Chern–Simons theory), cited here for contrast with the n-color homology approach (which is not $SU(2)$ -based but was inspired by this categorification idea).

[4] C. Rovelli and L. Smolin, Spin networks and quantum gravity, Phys. Rev. D 52, 5743–5759 (1995); arXiv:gr-qc/9505006. – A modern application of spin networks in loop quantum gravity. This paper reinterprets Penrose’s diagrams in a fully quantum-geometric setting and provides the context for using graphs to encode physical data — conceptually adjacent to the B&M framework, though focused on quantum spacetime rather than coloring.

[5] V. G. Turaev and O. Y. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31 (1992), no. 4, 865–902. – A foundational example of a state-sum TQFT constructed from quantum groups. The Turaev–Viro model shows how combinatorial data on graphs and triangulations can produce topological invariants — a key point of comparison for understanding how the B&M framework differs by using cohomology rings instead of quantum groups.

[6] E. Witten, Analytic continuation of Chern–Simons theory, in Chern–Simons Gauge Theory: 20 Years After, eds. J. E. Andersen, H. U. Boden, A. Hahn, and B. Himpel, AMS/IP Studies in Advanced Mathematics, Vol. 50, American Mathematical Society, Providence, RI, 2011, pp. 347–446; arXiv:1001.2933 [hep-th]. – Discusses the complexification of Chern–Simons theory to gauge groups like $SL(n,\mathbb{C})$ , involving holomorphic structures and analytic continuation of path integrals. Included here to contrast with the combinatorial, non-analytic nature of the B&M TQFT.

[7] E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988), 353–386. – A seminal paper introducing TQFT as a field-theoretic framework to study manifold invariants. While rooted in physics, it laid the mathematical foundations of TQFT now used across geometry, topology, and category theory. Cited here for background on the general idea of TQFT and its interface with physics.

[8] C. Vafa, Topological Mirrors and Quantum Rings, in Essays on Mirror Manifolds, ed. S.-T. Yau, International Press, Hong Kong, 1992, pp. 96–119; arXiv:hep-th/9111017.
— Discusses the structure of Frobenius algebras, quantum cohomology, and moduli spaces in the context of topological string theory and mirror symmetry. Though the B&M framework is combinatorial, it shares conceptual features with this algebra-geometry duality, especially in its use of cohomology rings to encode physical or enumerative data.

[9] P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, Gapped Lineon and Fracton Models on Graphs, Phys. Rev. B 107, 125121 (2023); arXiv:2210.03727 [cond-mat.str-el].
— This paper explores how $\mathbb{Z}_N$ gauge theories with fracton-like excitations can emerge from graph-based lattice models. It demonstrates that graph-theoretic properties, such as the number of vertices and cycles, control topological degeneracies and constrain particle mobility, illustrating how emergent gauge structures arise from discrete combinatorial complexity.