Surfaceology: A Geometric Formulation of Gluons

• Surfaceology is a combinatorial-geometric formulation that encodes gluon dynamics through surfaces, arcs, and triangulations to capture kinematic, color, and propagation data.
• It replaces traditional field vertices with sums over non-crossing curve covers, leveraging binary geometry and arc complexes for gauge-invariant amplitude construction.
• The approach unifies perturbative and nonperturbative phenomena by linking amplitude construction with effective gluon mass generation via surface curvature and lattice boundary effects.

The surfaceology formulation of gluons provides a geometric and combinatorial approach to gauge theory amplitudes and gluon dynamics, systematically encoding kinematic, color, and propagation structure in terms of surfaces, curves, and associated cluster variables. Stemming from advances in both the scattering equations approach and the combinatorial/cluster-algebraic program, surfaceology unifies amplitude construction and non-perturbative gauge phenomena with a precise correspondence: gluon data and interactions are encoded as binary-geometry operations on arc complexes and higher surfaces. At its core, surfaceology replaces the field-theoretic description of cubic and quartic vertices with a sum (or integral) over covers of surfaces by non-crossing curves (arcs), with physical singularities, polarization dependence, and even mass generation arising from geometric/topological data of these surfaces and their triangulations.

1. Binary Geometry, Arc Complexes, and u-Variables

Surfaceology begins by considering a connected orientable surface $S$ (of genus $g$, $h$ boundaries, and $m$ marked points). The set $\mathcal{A}(S)$ of homotopy classes of non-self-intersecting arcs connecting boundary punctures encodes the graph-theoretic content of all planar/fatgraph representations of Feynman diagrams. Assigning a positive variable $u_\gamma$ to each arc $\gamma$, cluster-like algebraic (or "binary-geometry") relations are imposed: $$u_{\alpha\gamma}u_{\beta\delta} = u_{\alpha\beta}u_{\gamma\delta} + u_{\alpha\delta}u_{\beta\gamma}$$ for any quadrilateral of boundary segments $(\alpha, \beta, \gamma, \delta)$. The set of binary coordinates $\{u_\gamma\}$ parametrizes the space of all triangulations of $S$ and underlies the combinatorial structure of both scalar and gluon amplitudes (Arkani-Hamed et al., 2023).

For each arc $\gamma$, a kinematic invariant is associated,

$$X_\gamma = \sum_{1 \leq a < b \leq m} s_{ab}\; \theta_\gamma(a, b)$$

with $s_{ab}=2p_a\cdot p_b$ and $\theta_\gamma(a, b)=1$ if $a,b$ are separated by $\gamma$, $0$ otherwise. For the cubic scalar theory, the purely combinatorial amplitude at fixed topology is

$$m_S^{\phi^3}(s_{ab}) = \sum_{T \in \text{Triangulations of }S} \prod_{\gamma \in T} \frac{1}{X_\gamma}$$

which, at tree-level for $S$ a disk, reduces to the well-known trivalent expansion (Arkani-Hamed et al., 2023).

2. Surfaceology and the Scaffolded Gluon Construction

Gluon amplitudes are constructed from the scalar scaffolding prescription: one scatters $2n$ colored scalars in $n$ flavor pairs $(L_i,R_i)$, then fuses each pair to define a gluon with momentum $k_i = p_{L_i} + p_{R_i}$ and polarization $\epsilon_i$. The scaffolded kinematic shift for any $X_\gamma$ is given by

$$K_\gamma = X_\gamma + \sum_{i=1}^n u_\gamma(L_i, R_i)\, (2\epsilon_i \cdot k_i) + \sum_{i<j}\left[ u_\gamma(L_i, L_j) + u_\gamma(R_i, R_j) - u_\gamma(L_i, R_j) - u_\gamma(R_i, L_j) \right](2\epsilon_i \cdot \epsilon_j)$$

The resulting sum over triangulations,

$$M_S^{YM}(k_i, \epsilon_i) = \sum_{T \in \text{Triangulations}} \prod_{\gamma \in T} \frac{1}{K_\gamma}$$

is multilinear in the $\epsilon_i$, gauge invariant, and correctly factorizes on all physical poles, both at tree and loop level (Arkani-Hamed et al., 2023). This construction reproduces the field-theoretic and, at tree-level, bosonic open-string gluon amplitudes, up to normalization and kinematic shifts.

Surfaceology at tree-level coincides with the CHY (Cachazo–He–Yuan) scattering-equation formulation for gluons, in which the amplitude is an integral over $n$ punctures on the sphere (or disk) with localization by scattering equations. The integrand product is given by the Parke-Taylor color factor and a Pfaffian encoding kinematic and polarization data, which emerges as the multilinear part of the surfaceology scaffolded construction (Cachazo et al., 2013, Arkani-Hamed et al., 2023).

3. Curved Gauge Slices: Gluon Propagation, Mass, and Geometry

Beyond the purely algebraic and tree-level S-matrix role of surfaceology, the geometric formulation extends to the nonperturbative dynamics of gluons in confining gauge theories. Here, the gauge slice (the locus in field configuration space defined by a fixed gauge condition and, crucially, by blocks of constant dimension-2 gluon condensate, $\langle A^2(x)\rangle = \alpha^2$) is promoted to a curved manifold $\Sigma_\alpha$ embedded in space-time. The induced metric and connection on $\Sigma_\alpha$ render the gluon kinetic term

$L = -\frac{1}{4} (D_\mu A_\nu^a - D_\nu A_\mu^a)^2$

with $D_\mu$ the manifold covariant derivative (Kim et al., 2015, Lee et al., 2015). The resulting quadratic operator includes curvature couplings: $$L = \frac{1}{2}A^a_\mu\left(D^2g^{\mu\nu} - D^\mu D^\nu + R^{\mu\nu} \right)A^a_\nu$$ In the physically relevant case of maximally symmetric slices ($R^{\mu\nu} = m^2 g^{\mu\nu}$), the gluon propagator in momentum space becomes

$$G_{ab}^{\mu\nu}(k) = \delta_{ab} \frac{1}{k^2 - m^2}\left(g^{\mu\nu} - \frac{k^\mu k^\nu}{k^2}\right)$$

Thus the scalar curvature $R$ generates an effective, gauge-invariant mass $m^2=R/4$ for the gluon. This resolves the phenomenological tension between IR observations of massive gluonic modes and fundamental gauge invariance, attributing mass to surface curvature induced by vacuum condensates and gauge domain structure (Kim et al., 2015, Lee et al., 2015).

4. Leading Singularities, Graph Coverings, and All-Orders Structure

Surfaceology admits a powerful formulation of the leading singularities (maximal cuts) of gluon amplitudes at any genus or loop order. In field theory, these are calculated by gluing on-shell three-point vertices along internal lines and saturating with appropriate spin sums. Surfaceologically, the combinatorial problem is to enumerate all possible, non-overlapping coverings of the fatgraph by curves so that each edge is covered once. The gluon leading singularity $LS(\mathcal{G})$ for a graph $\mathcal{G}$ with $E$ internal lines is

$$LS(\mathcal{G}) = \sum_{\{\text{coverings}\}} \prod_{C} X_C \times (-1)^{\#\, \text{extensions}}$$

where $X_C$ are the kinematic invariants associated to each curve $C$ and the sign tracks the orientation/extension structure (Carrôlo et al., 18 Dec 2025).

At loop level, spin sum effects and BRST/ghost sector corrections alter the exponents associated with closed curves. Specifically, in the open-string surface integral

$$I_\Sigma(X) = \int_{y_P > 0} \prod_{P \in \mathcal{T}} \frac{dy_P}{y_P^2} \prod_C u_C^{X_C}$$

the exponent for a closed curve $\Delta_J$ is

$$\Delta_J = \begin{cases} 1-D & \text{if } \Delta_J \sim \text{(loop boundary)}\ -D & \text{otherwise} \end{cases}$$

ensuring correct unitarity and gauge structure (Carrôlo et al., 18 Dec 2025).

5. Surfaceology in Strings and $\alpha'$-Deformed Gauge Theories

In the string-theoretic context, surfaceology naturally encodes the open-string gluon amplitude as a sum over shifted Tr$(\phi^3)$ building blocks, with scaffolding and cycles on the surface determining both purely gluonic and mixed gluon-scalar configurations (Cao et al., 30 Apr 2025). The full superstring n-gluon amplitude is expressed as a sum over $(2n-3)!!$ shifted amplitudes structured by even-length cycles, perfect matchings, and prescribed kinematic shifts.

Explicitly, after performing the n scaffolding residues, the amplitude is written as

$$A_n^{\text{super}} = \sum_{\rho \subset \{1,\dots, n-1\}} T_\rho A_n^{\text{mixed bos.}}(\rho_s|\bar\rho_g)$$

with nested commutator prefactors $T_\rho$ corresponding to field strength traces and the combinatorics guaranteed to cancel unphysical poles such as tachyons and $F^3$ vertices. The curve integral formulation provides manifest gauge invariance and direct correspondence to combinatorial surface structures, streamlining the traditional RNS or pure-spinor approach (Cao et al., 30 Apr 2025).

6. Surface Terms, Lattice QCD, and Nonperturbative Gauge Boundary Effects

Surfaceology manifests physically through the explicit role of surface (or boundary) terms in the nonperturbative QCD Hamiltonian. On the lattice, the energy-momentum tensor's continuity equation, when integrated over a finite region (e.g., the hadron volume), yields non-vanishing flux terms on the spatial boundary. These surface integrals must be included in the Hamiltonian,

$$H_{\text{lat}}^{\text{corr}} = H_{\text{quarks}} + H_{\text{gluons}} + S_{\text{surf}}$$

with $$S_{\text{surf}} = a^3 \sum_{\vec{x} \in \partial V} \sum_{i=1}^3 T_{\text{lat}}^{i0}(\vec{x})$$ (Nayak, 2018). This refined Hamiltonian corrects the extraction of physical hadron energy from lattice correlators and directly reflects the necessity of accounting for gluonic field penetration and confinement at the surface—a feature geometrically natural in the surfaceology paradigm.

7. Phenomenological and Mathematical Implications

The surfaceology formulation provides a unified framework for encoding and extracting all essential physical properties of gluon amplitudes: multilinearity, gauge invariance, and factorization emerge from the binary-geometry and triangulation properties of surfaces (Arkani-Hamed et al., 2023). In addition, the interplay between vacuum condensates and geometric structure yields gauge-invariant mass generation, explaining IR gluon mass in hadrons and offering plausible candidates for dark-matter gluonics at cosmological scales (Lee et al., 2015). All-order matching of leading singularities and unitarity is dynamically realized by the combinatorial summation over arc coverings and curve classes, with singularities controlled by curve exponents and binary-geometric relations (Carrôlo et al., 18 Dec 2025). In string theory and $\alpha'$-expansion, surfaceology governs not only the leading gauge content but also organizes field-strength trace contributions, $\alpha'$ corrections, and the cancellation of unphysical states in superstring amplitudes (Cao et al., 30 Apr 2025).

Overall, the surfaceology paradigm furnishes a precise combinatorial-geometric dictionary for gluon interactions, encompassing both S-matrix and propagator-level phenomena, and enabling unification of field theory, string theory, and nonperturbative gauge dynamics.