Scalar, gauge and theory-family data for swampland conjectures

A swampland-claim datum specifies the dimension/frame/Planck normalization, quantum-gravity or EFT class, scalar target and metric, potential or gauge group/coupling/charge data, cutoff and tower spectrum, asymptotic/family limit, and whether a proposed inequality is a conjecture, a controlled conditional result or a model-specific example.

Here an effective field theory (EFT) means a low-energy description with a specified retained field content, cutoff, and domain of validity; a quantum-gravity class names the proposed ultraviolet completion family and its stated restrictions. A frame specifies the metric used to measure masses, distances, and energies. Throughout, “order-one” means a fixed positive dimensionless constant independent of the family limit unless the chosen conjectural version states otherwise.

More explicitly, in natural units $\mathrm{\hslash }=c=1$, choose an integer spacetime dimension $d\ge 3$, a spacetime signature and a frame. In an Einstein frame write the gravitational term as $M_{P,d}^{d-2}R\mathrm{/}2$, with $M_{P,d}^{d-2}:=(8\pi G_d{)}^{-1}$; thus $M_{P,d}$ has mass dimension one. In $d=4$ use the earlier four-dimensional Einstein-frame scalar-EFT datum: its smooth Lorentzian spacetime $\mathcal{X}$, smooth scalar field $\varphi :\mathcal{X}\to \mathcal{T}$, kinetic term $-G_{ij}(\varphi )g^{\mu \nu }{\mathrm{\partial }}_{\mu }{\varphi }^i{\mathrm{\partial }}_{\nu }{\varphi }^j\mathrm{/}2$, and constant reduced Planck mass $M_P>0$. A change of frame is additional supplied data, and bounds are compared only after translating all masses and couplings into the same frame.

The scalar data consist of a finite-dimensional smooth target manifold $\mathcal{T}$, a smooth positive-definite Riemannian metric $G$, and any smooth potential $V:\mathcal{T}\to \mathbb{R}$. When spacetime scalar configurations are used, supply a smooth Lorentzian $d$-manifold $(\mathcal{X},g)$ of the declared signature and a smooth map $\varphi :\mathcal{X}\to \mathcal{T}$. A scalar-target path is instead a continuous map $\gamma :J\to \mathcal{T}$ on a specified real interval $J\subseteq \mathbb{R}$; require piecewise $C^1$ regularity on each compact subinterval whenever its length is used. It need not be the restriction of $\varphi$ to a spacetime curve. Whenever a gradient or Hessian bound is proposed, $V$ is at least $C^2$, $\mathrm{\mid }\mathrm{\nabla }V\mathrm{\mid }$ means the norm induced by $G$, and Hessian eigenvalues are generalized eigenvalues relative to $G$ (equivalently, eigenvalues in a $G$-orthonormal frame); a least eigenvalue requires a nonzero-dimensional target. If a distance is used, it is the infimum of $G$-lengths of piecewise-$C^1$ paths between the specified points, with the infimum of an empty path set equal to $+\mathrm{\infty }$; distance to an asymptotic region is a separately specified limit of such distances, not an assumed global coordinate.

Gauge data name the compact internal gauge group $K$ and its global form, kinetic normalization and coupling, and allowed charge data. Here “compact” means compact as a topological group; the global form is the actual group, including any quotient by a discrete central subgroup, rather than only its Lie algebra. A one-dimensional unitary character is a continuous homomorphism $\chi :K\to U(1)$, where $U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ under multiplication. The character group $X^{\ast }(K):={\mathrm{Hom}}_{\mathrm{c}\mathrm{t}\mathrm{s}}(K,U(1))$ has pointwise multiplication. A lattice here means a free abelian group of finite rank, that is, a group isomorphic to ${\mathbb{Z}}^r$ for an integer $r\ge 0$; a general character group need not be a lattice. For example, for the discrete compact group $K=\mathbb{Z}\mathrm{/}2\mathbb{Z}$, a character sends its generator to $1$ or $-1$, so $X^{\ast }(K)\cong \mathbb{Z}\mathrm{/}2\mathbb{Z}$, which is not a lattice. For a torus $K=U(1{)}^r$ with declared circle coordinates, the characters are ${\chi }_q(z_1,\dots ,z_r)={\prod }_{j=1}^r{z}_j^{q_j}$ for $q\in {\mathbb{Z}}^r$; thus $X^{\ast }(K)\cong {\mathbb{Z}}^r$ is its character lattice. In particular, for $U(1)$ a declared unit-weight convention identifies this lattice with $\mathbb{Z}$, without asserting a unit-charge dynamical state. The allowed charge data are distinguished from the charges actually carried by dynamical states. A kinetic matrix, charge basis, and any duality frame are specified whenever needed; non-Abelian charge data are representations of the stated global group, not an unlabelled integer.

An EFT datum also gives a positive Wilsonian cutoff ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$, its field/spacetime regime, and a tower spectrum, when relevant: an indexed collection of states with masses $m_n(\lambda )$ in a stated family parameter $\lambda$. The meaning of the characteristic tower gap $m_{\mathrm{t}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}$ and its asymptotic scaling are declared for that tower. If a species scale is invoked, state the counting convention $N(E)$, the number of distinct species of mass below $E$ with any degeneracy convention declared, and use only the parametric relation ${\mathrm{\Lambda }}_{\mathrm{s}\mathrm{p}}^{d-2}\sim M_{P,d}^{d-2}\mathrm{/}N({\mathrm{\Lambda }}_{\mathrm{s}\mathrm{p}})$ within its range of validity; this estimate is not an exact identity. The omitted states, derivative range, and any control assumptions needed for the effective description are stated.

Finally specify the class of quantum-gravity theories or EFTs, the family and limiting procedure, the normalization of each proposed dimensionless inequality, and an epistemic label: conjecture, controlled conditional result with all extra hypotheses, or model-specific example. The label records status; it does not establish the inequality, derive a postulate, or convert an example into a universal claim.

Four-dimensional Einstein-frame multi-scalar cosmology EFT data