Abelian Polyhedral Gauge Theories

• Abelian polyhedral gauge theories are defined by convex polyhedral constraints on U(1) charges arising from anomaly cancellation in 6D supergravity.
• They impose algebraic and geometric restrictions on gauge groups and charge lattices, ensuring a finite set of consistent gauge and matter structures for T < 9.
• String theory realizations, like those in F-theory, match charge lattices and anomaly coefficients to enforce positivity and geometric consistency.

Abelian polyhedral gauge theories are gauge-theoretic frameworks in which abelian gauge fields and their charge structures are constrained by algebraic and geometric conditions that carve out polyhedral (i.e., convex, typically finite) regions in the space of anomaly coefficients and charge assignments. These theories play a fundamental role in the structure of six-dimensional (6D) $\mathcal{N}=(1,0)$ supergravities with both abelian and non-abelian gauge sectors, and their mathematical analysis elucidates deep connections between anomaly cancellation, the allowed spectrum of matter, charge lattices, and the geometric realization of string vacua. The “polyhedral” characterization refers to the fact that the anomaly and positivity constraints define convex cones or polyhedra in charge space, restricting the gauge and matter content to lie within finite, well-defined regions determined by the gauge and gravitational anomaly coefficients.

1. Structure of Gauge Groups, Abelian Sectors, and Charge Lattices

In 6D $\mathcal{N}=(1,0)$ supergravity, the total gauge group is generically a product

$$\mathcal{G} = \left(\prod_{\kappa} G_\kappa\right) \times \left(\prod_{i=1}^{V_A} U(1)_i\right)$$

with nonabelian factors $G_\kappa$ and $V_A$ abelian $U(1)$ factors (Park et al., 2011). Hypermultiplet matter representations $I$ carry $U(1)$ charges $q_{I,i}$ under these abelian factors. The charge assignments, as well as the number and nature of the abelian factors, are strongly constrained by anomaly cancellation conditions.

The key anomaly polynomial must factorize in a Green–Schwarz-like fashion: $$I_8 = \frac{1}{2}\, \Omega_{\alpha\beta} X_4^\alpha X_4^\beta, \quad X_4^\alpha = \frac{1}{2} a^\alpha \operatorname{tr}R^2 + \sum_\kappa b^\alpha_{(\kappa)} \operatorname{tr}F^2_{(\kappa)} + \sum_{i,j} b^\alpha_{(ij)} F_i F_j$$ with $a$, $b_{(\kappa)}$, $b_{(ij)}$ (“anomaly coefficients”) valued in an $SO(1,T)$ space (T = number of tensor multiplets). The inner products $a\cdot b$, $b\cdot b$ enter the anomaly equations for the abelian sector: $$a \cdot b_{ij} = -\frac{1}{6} \sum_I M_I q_{I,i} q_{I,j}, \qquad b_{i} \cdot b_{j} = \sum_I M_I (q_{I,i} q_{I,j})^2$$ where $M_I$ is the multiplicity in representation $I$.

U(1) charge vectors $q_i$ are subject to a $GL(V_A)$ ambiguity, since redefinitions of the $U(1)$ basis (compactly, $SL(V_A,\mathbb{Z})$ in the compact case) can mix charges linearly. Physically inequivalent charge assignments are thus equivalence classes under these automorphisms.

2. Anomaly Cancellation, Positivity, and Polyhedral Constraints

Abelian anomaly cancellation requirements reduce to systems of polynomial equations in the $U(1)$ charges, determined by inner products among anomaly coefficients and weighted by the representations’ charge multiplicities. The analysis introduces the $SO(1,T)$-valued polynomial

$P(x) = \sum_i x^i b_i$

which must satisfy

$$a \cdot P(x) = -6 \sum_I M_I f_I(x)^2, \qquad P(x) \cdot P(x) = \sum_I M_I f_I(x)^4, \quad f_I(x) = \sum_i q_{I,i} x^i$$

These are “polyhedral” or convex constraints: For fixed non-abelian matter content, the space of consistent $U(1)$ charge assignments is cut out as a convex polyhedron in the space of charge vectors.

When $T < 9$, the gravitational anomaly coefficient $a$ is timelike (with $a^2 = 9-T > 0$). This restricts the possible kernel of the inner product: there is no nonzero vector $b$ orthogonal and null with respect to $a$, i.e., $b^2 = 0$, $a\cdot b = 0$. Consequently, abelian charge vectors must be linearly independent. Additionally, positivity of kinetic terms requires that for some unit vector $j$ in $SO(1,T)$,

$K_{ij} = j \cdot b_{ij}$

is positive definite. These polynomial and positivity bounds, together with the factorization property, sharply restrict the space of allowed $(\mathcal{G}, q_{I,i})$ data.

Quantitative bounds for the number of abelian factors $\mathsf{V}_A$ are derived: $V_A \leq (T+2)N + 2(T+2)$ with $N$ the number of distinct nonabelian matter representations (cf. equations (3.23) and (3.24) in (Park et al., 2011)).

3. Classification: Finiteness vs. Infinite Charge Families

For $T < 9$, the core result is: The number of possible gauge/matter structures—i.e. combinatorics of gauge group and matter representations, modulo equivalences of $U(1)$ charge lattices—is finite. The allowed $U(1)$ charge assignments for a fixed nonabelian “seed” are constrained to a finite region in charge space.

There are two types of infinite families:

• Trivial infinite families, where charges are related by $SL(V_A,\mathbb{Z})$ (or $GL(V_A)$) redefinitions, e.g., in $G_0\times U(1)^2$ by taking arbitrary coprime linear combinations $Q = r q_1 + s q_2$.
• Nontrivial infinite families, where even after modding by redefinitions, there are genuinely distinct, quantum-consistent charge assignments. For example, the $SU(13)\times U(1)$ theory with four two-index antisymmetrics, six fundamentals, and 23 singlets admits infinitely many inequivalent solutions (explicitly constructed by ansatz for charges parameterized by $a$, $f$ and satisfying number-theoretic constraints such as integrality and unimodularity, see equations (4.15)–(4.18) in (Park et al., 2011)).

For $T\geq 9$, anomaly cancellation allows infinite families by adding “lightlike” abelian factors: extra $U(1)$’s not coupled to any charged fields, or choices where $b^2=0=a\cdot b$. However, such cases typically fall outside the reach of known string constructions (e.g., in F-theory).

4. String Theoretic Realizations and Polyhedral Geometry

Explicit string compactifications (notably, F-theory and heterotic K3 models) often automatically satisfy the low-energy anomaly constraints, and further impose geometric and topological restrictions on the admissible gauge/matter structures:

• Kodaira constraint and the properties of the elliptically fibered Calabi–Yau geometry can bound the number of abelian factors and fix the allowed charge lattices.
• In known string models, the finite set of allowed nonabelian representations, and the corresponding finiteness in “gauge/matter structures,” agree with what is permitted by the field-theoretic anomaly/positivity constraints.
• Nevertheless, some infinite families allowed by low-energy anomalies (especially for $T\geq 9$) do not appear in string constructions, suggesting the necessity of as-yet-undiscovered quantum or geometric consistency conditions.

The term “polyhedral” is used to describe the anomaly bounds because the admissible set of charge vectors and anomaly coefficients is carved out as a convex polyhedron or cone in parameter space.

5. Physical Implications and Landscape Structure

For 6D ${\cal N}=(1,0)$ theories with $T<9$:

• The set of distinct allowed gauge/matter structures—including all nonabelian and abelian factors, up to charge-lattice equivalences—is strictly finite, even though there can be an infinite variety of $U(1)$ charge assignments for any fixed nonabelian seed. Most of these are related by lattice automorphisms, but “exceptional” families reflect deeper arithmetic or geometric structure.
• In string realizations, matching charge lattices, anomaly coefficients, and positivity domains often requires matching with limiting processes in the geometry (such as the structure of the Mordell–Weil group in F-theory).
• When $T\geq 9$, or in effective field theories not embeddable in string theory, infinite families point to gaps in our understanding of quantum consistency, motivating the formulation of new constraints possibly rooted in geometric topology or number theory.

The detailed structure of abelian polyhedral gauge theories thus underpins the finite “landscape” of consistent six-dimensional supergravity models—sharply contrasting with the naive expectation of an unbounded class of $U(1)$ extensions. The interplay of algebraic, geometric, and number-theoretic consistency ensures that only a limited array of such theories can be physically realized, while illuminating areas where further constraints are yet to be elucidated.

6. Mathematical Formulation and Key Equations

The mathematical underpinnings are expressed in the relations among anomaly coefficients, charge vectors, and matter multiplicities. Essential identities are: $$\begin{align*} I_8 &= \frac{1}{2}\, \Omega_{\alpha\beta}\, X_4^\alpha X_4^\beta \ a \cdot b_{ij} &= -\frac{1}{6} \sum_I M_I q_{I,i} q_{I,j} \ b_i \cdot b_j &= \sum_I M_I\, (q_{I,i} q_{I,j})^2 \ V_A &\leq (T+2)N + 2(T+2) \end{align*}$$ The polynomial $P(x)=\sum_i x^i b_i$ satisfies

$$a\cdot P(x) = -6 \sum_I M_I f_I(x)^2, \quad P(x)\cdot P(x) = \sum_I M_I f_I(x)^4,\quad f_I(x) = \sum_i q_{I,i} x^i$$

with all positivity and factorization constraints manifest.

7. Outlook and Future Directions

Abelian polyhedral gauge theories, as illuminated by anomaly constraints and positivity in 6D $\mathcal{N}=(1,0)$ supergravity, provide a blueprint for the systematic classification of low-energy consistent effective theories. They clarify why only a restricted set of possible gauge and matter structures can be realized in string theory and establish necessary (though not always sufficient) low-energy conditions.

Several future directions remain:

• Identifying and proving the quantum or geometric constraints that further rule out infinite families for $T\geq 9$ or those not realized in known string constructions.
• Formulating a complete “UV-swampland” criterion for $U(1)$ charges and charge lattice structures beyond anomaly cancellation.
• Relating the polyhedral geometry of anomaly constraints more closely to the discrete and arithmetic invariants arising in string compactifications via F-theory and heterotic geometry.
• Extending the analysis to incorporate higher-form abelian symmetries, generalized global symmetries, or more intricate polyhedral structures in the landscape of 6D (and higher-dimensional) theories.

These advances promise not only to complete the classification of consistent abelian gauge enhancements in string theory but also to clarify the landscape/swampland boundary for effective field theories with abelian gauge factors in any dimension.