Negative Geometries in Mathematical Physics

• Negative geometries are mathematical constructs characterized by graph-dependent negativity conditions, negatively curved metrics, and negative interaction entropies.
• They are applied to analyze scattering amplitudes in N=4 SYM, study Finsler and Hilbert geometries, and model programmable negative curvature in active materials.
• Research on negative geometries provides insights into all-loop resummations, polylogarithmic structures, and holographic measures of complexity, influencing both theoretical and applied physics.

Searching arXiv for recent and foundational papers on “negative geometries” across the main usages represented in the literature. First, a direct search for the phrase and closely related amplitudes usage. Now a broader search covering hyperbolic / negatively curved geometry usages that are explicitly described as “negative geometries” in the supplied literature. “Negative geometries” denotes several distinct but structurally related constructions across contemporary mathematical physics. In the amplitudes program, it refers to Amplituhedron-related regions in which mutual positivity conditions between loop lines are replaced by graph-dependent negativity conditions, with canonical forms that organize the integrand of the logarithm of the four-point amplitude and the corresponding Wilson loop with Lagrangian insertion (2112.06956). In differential geometry and materials science, it denotes spaces or surfaces with negative Gaussian, flag, or weighted Ricci curvature, including Hilbert and Funk geometries, anti-cones, Beltrami pseudospheres, and hyperbolic lattices (Ohta, 2012, Modes et al., 2015, Taioli et al., 2015, Yu et al., 2020). In Casimir physics, the term is used for scattering geometries whose channel structure yields negative interaction entropy over an intermediate temperature range (Umrath et al., 2015). In holography, it has also been used for spacetimes in which the vacuum-subtracted maximal volume, interpreted as complexity of formation, becomes negative once compact directions are included (Engelhardt et al., 2021).

1. Sign-flipped canonical geometries in scattering amplitudes

In planar $\mathcal N=4$ super-Yang-Mills theory, the four-point loop Amplituhedron is a positive geometry in the space of loop lines $AB_i$ in momentum twistor space. For each loop line, the one-loop conditions are

$$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$

supplemented at higher loop order by mutual positivity $\langle AB_iAB_j\rangle>0$ for all $i\neq j$. Negative geometries keep the one-loop conditions but replace mutual positivity by graph-dependent negativity,

$$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$

with no condition otherwise (2112.06956).

The canonical forms of these negative geometries furnish the integrand of the logarithm of the four-point amplitude. Organizing the expansion by connected graphs $G$, one has

$$\widetilde\Omega_L=\sum_{\substack{G \text{ connected}\ \#\text{vertices}=L}}(-1)^{E(G)}\,\widetilde\Omega_G,$$

so connected negative geometries play for $\log M$ the role that the Amplituhedron plays for $M$. Freezing one loop line $AB_i$0 and integrating the remaining loops yields an IR-finite observable depending on a single cross ratio,

$AB_i$1

equivalently the normalized quadrangular Wilson loop with a single Lagrangian insertion (2112.06956).

A particularly simple subsector is formed by tree negative geometries. Their canonical forms factorize over vertices and edges, and their all-loop sum is governed by a nonlinear differential equation,

$AB_i$2

with $AB_i$3. This produces an all-coupling expression for the tree contribution to $AB_i$4 and to $AB_i$5, and the resulting $AB_i$6 shares the main qualitative characteristics of the known exact cusp anomalous dimension (2112.06956).

2. Multi-loop analytic structure, cycle expansions, and the ABJM variant

The negative-geometry expansion admits a further organization by the number of internal cycles in the graph. One-cycle graphs are next-to-leading in the expansion over cycles, and geometric Landau analysis shows that the integrated four-point one-cycle negative geometry has branch point singularities only at

$AB_i$7

to all loop orders. In this setting, the Landau analysis is geometric in the sense that candidate singularities are retained only when the corresponding cut solution lies on the boundary of the negative geometry; otherwise they are spurious (Paranjape et al., 24 Apr 2026).

At three loops, explicit integration of all four-loop negative geometries shows that the number of internal cycles is closely linked to the depth of polylogarithms. Tree geometries produce depth-1 structures, one-cycle geometries produce depth-2 structures, and higher-cycle geometries yield more intricate transcendental patterns. The same analysis shows that higher-cycle diagrams are suppressed if one considers separate odd and even zeta contributions to $AB_i$8, while certain convergent infinite series of one-cycle diagrams admit all-loop resummations (Dixon et al., 27 May 2026).

In ABJM theory, the four-point amplituhedron is obtained by projecting the 4D amplituhedron to a 3D symplectic locus. The logarithm of the amplitude again decomposes into negative geometries, but only connected bipartite graphs survive. Integrating $AB_i$9 loop variables produces IR-finite, dual-conformally invariant functions of a single cross ratio, and this structure provides a direct route to the ABJM cusp anomalous dimension (Henn et al., 2023). At four loops, explicit integration shows that the infrared divergence of a box-type negative geometry is weaker than that of the tree-type geometries, so only tree-type negative geometries contribute to the four-loop cusp anomalous dimension (Li, 2024). A complementary differential-equation analysis of the same four-loop ABJM sector finds an apparent simplicity in the leading singularities of the integrated results in the frame where the unintegrated loop variable goes to infinity, and it suggests an alternating sign pattern for the integrated negative geometries in the Euclidean region (Lagares et al., 2024).

3. Negatively curved metric-measure geometries

In Finsler geometry, Hilbert and Funk geometries on a bounded, strongly convex domain $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$0 are canonical examples of negatively curved non-Riemannian spaces. The Hilbert metric is the symmetric cross-ratio metric on $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$1, while the Funk metric is its non-symmetric forward version, with

$$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$2

When $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$3 is smooth and $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$4 is strongly convex, both arise from smooth Finsler norms (Ohta, 2012).

Ohta’s analysis computes the weighted Ricci curvature with respect to the Lebesgue measure $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$5. In the Funk case,

$$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$6

so the weighted Ricci curvature is constant and negative. In the Hilbert case, one obtains uniform bounds

$$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$7

which provide a negative lower bound independent of the concrete shape of $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$8 (Ohta, 2012).

These geometries are negative in several distinct senses. Their flag curvatures are constant and negative: $$\langle AB_i12\rangle,\langle AB_i23\rangle,\langle AB_i34\rangle,\langle AB_i14\rangle>0,\qquad \langle AB_i13\rangle,\langle AB_i24\rangle<0,$$9 for Funk and $\langle AB_iAB_j\rangle>0$0 for Hilbert. With Lebesgue measure, they satisfy $\langle AB_iAB_j\rangle>0$1 with $\langle AB_iAB_j\rangle>0$2, hence Brunn–Minkowski, Bishop–Gromov, Laplacian comparison, and Bochner–Weitzenböck inequalities follow from the general $\langle AB_iAB_j\rangle>0$3 theory. The paper also notes that, in Hilbert geometry, nonnegative weighted Ricci curvature for some finite $\langle AB_iAB_j\rangle>0$4 cannot occur for any measure, emphasizing that the negative Ricci character is intrinsic rather than an artifact of a particular density (Ohta, 2012).

4. Embedded, programmable, and lattice realizations of negative curvature

In thin active materials, negative geometry can be programmed through an induced metric. For an initially flat nematic sheet with azimuthal director field, activation by light or heat changes natural lengths by $\langle AB_iAB_j\rangle>0$5 along the director and by $\langle AB_iAB_j\rangle>0$6 transversely, producing the target metric

$\langle AB_iAB_j\rangle>0$7

When $\langle AB_iAB_j\rangle>0$8, circles have surplus perimeter relative to radius, and the sheet relieves this incompatibility by buckling into an anti-cone with negative Gaussian curvature concentrated at the apex. The integrated apex curvature is

$\langle AB_iAB_j\rangle>0$9

with $i\neq j$0, so $i\neq j$1 implies $i\neq j$2. The stretch-free state requires azimuthal displacements $i\neq j$3, and the resulting bend-minimizing shapes are smooth, aster-like, and can become re-entrant in the azimuthal coordinate for large deformations (Modes et al., 2015).

A complementary realization is the Beltrami pseudosphere, a surface of revolution with constant Gaussian curvature $i\neq j$4. In upper-half-plane coordinates the Lobachevsky metric is

$i\neq j$5

while on the Beltrami pseudosphere it takes the form

$i\neq j$6

This surface realizes a horocyclic sector of hyperbolic geometry in $i\neq j$7, but by Hilbert’s theorem it necessarily terminates at a singular boundary, the Hilbert horizon $i\neq j$8. In a trivalent carbon lattice the total curvature $i\neq j$9 implies six excess heptagons, and the defect pattern is governed by a non-Euclidean crystallographic group, specifically a loxodromic subgroup of $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$0 (Taioli et al., 2015).

Negative curvature also appears in hyperbolic lattices built from regular tessellations $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$1. For polygon area $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$2, the curvature is

$$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$3

At fixed $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$4, only discrete pairs $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$5 are allowed, so the curvature is “quantized” by the tessellation. On such lattices, Bloch’s theorem fails, but one can still define tight-binding Hamiltonians with pseudospin-dependent gauge phases and realize a non-Euclidean analogue of the quantum spin Hall effect. Hyperbolic edge dominance and quantized curvature lead to generalized Hofstadter butterflies and to topological immunity that is robust only in narrow spectral–magnetic windows in highly curved lattices (Yu et al., 2020).

5. Scattering geometry and negative Casimir entropy

In Casimir physics, “negative geometries” refers to geometrical configurations in which certain scattering channels contribute to the Casimir free energy so that the associated interaction entropy becomes negative over some temperature range. The Casimir free energy is written in scattering form as

$$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$6

Here $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$7 is an interaction entropy, not the total thermodynamic entropy, so there is no requirement that it be positive (Umrath et al., 2015).

The central result is that geometry and dissipation generate negative Casimir entropies through closely analogous mechanisms but in distinct scattering channels. In plane–sphere and sphere–sphere geometries, negative entropies occur even for perfectly reflecting objects because polarization-mixing channels vanish at zero Matsubara frequency. In Drude metals, the transverse electric channel loses its zero-frequency contribution and yields a dissipative negative entropy. In both cases the common structural feature is a scattering channel whose contribution to the free energy is nonzero at $$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$8 but vanishes in the high-temperature limit (Umrath et al., 2015).

The paper disentangles these mechanisms channel by channel in the sphere–sphere geometry. The TM contribution remains positive; the TE channel gives the dissipative negative dip for Drude metals; and the polarization-mixing channel gives the geometric negative dip even for perfect conductors. Beyond the Rayleigh limit, TE scattering becomes comparable to TM scattering, and negative Casimir entropies can then occur for Drude-type metals at large distances provided the dissipation strength is sufficiently small (Umrath et al., 2015).

6. Holographic negative volume geometries

In holography, the phrase has been used for spacetimes in which the vacuum-subtracted maximal spatial volume—the Complexity=Volume candidate for complexity of formation—becomes negative. The proposal is

$$\langle AB_iAB_j\rangle<0 \quad \text{if nodes } i,j \text{ are connected by an edge,}$$9

Earlier positivity results required asymptotically AdS$G$0 boundary conditions and a weak curvature condition. The paper shows that once compact directions are included, both of the natural volume prescriptions can have arbitrarily negative complexity of formation in asymptotically AdS$G$1 supergravity (Engelhardt et al., 2021).

The mechanism is geometrical. In the four-dimensional truncation one has an Einstein–scalar theory with a tachyonic scalar above the BF bound, while in eleven dimensions the uplift restores good energy conditions but destroys asymptotically AdS$G$2 behavior. The full spacetime is asymptotically AdS$G$3, not asymptotically AdS$G$4, so the hyperbolic comparison arguments behind $G$5 no longer apply. Explicit one-sided and two-sided solutions then exhibit vacuum-subtracted maximal volumes that are negative and, in some families, unbounded below (Engelhardt et al., 2021).

The same work also finds time-dependent examples in which complexity decreases at late times, including both single-sided geometries and two-sided wormholes. In particular, it constructs a cosmological wormhole with simultaneously negative and decreasing complexity of formation as computed by volume. The paper emphasizes a distinguished role for relevant primaries in these constructions and notes that identifying $G$6 with an absolute distance from a reference state would conflict with the basic properties expected of a complexity measure (Engelhardt et al., 2021).