An official website of the United States government
This content will become publicly available on May 5, 2026
SU(3) instanton homology for webs and foams
An instanton homology is constructed for webs and foams, using gauge theory with structure group SU (3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU (3) homology counts Tait colorings when theweb is planar. Unlike the SO(3) case, the SU (3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed.
more »
« less
- Award ID(s):
- 2105512
- PAR ID:
- 10644995
- Publisher / Repository:
- AxRiV
- Date Published:
- Format(s):
- Medium: X
- Institution:
- ArXiV
- Sponsoring Org:
- National Science Foundation
More Like this
-
Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q -series which satisfy some remarkable modularity properties and recover SU ( 2 ) {{\rm SU}(2)} quantum invariants of 3-manifolds at roots of unity.In particular, our work gives rise to a 2-variable refinement of the Z ^ {\widehat{Z}} -invariant.more » « less
-
Radio 2-colorings of graphs are a generalization of vertex colorings motivated by the problem of assigning frequency channels in radio networks. In a radio 2-coloring of a graph, vertices are assigned integer colors so that the color of two vertices u and v differ by at least 2 if u and v are neighbors, and by at least 1 if u and v have a common neighbor. Our work improves the best-known bounds for optimal radio 2-colorings of small hypercube graphs, a combinatorial problem that has received significant attention in the past. We do so by using automated reasoning techniques such as symmetry breaking and Cube and Conquer, obtaining that for n = 7 and n = 8, the coding-theory upper bounds of Whittlesey et al. (1995) are not tight. Moreover, we prove the answer for n = 7 to be either 12 or 13, thus making a substantial step towards answering an open problem by Knuth (2015). Finally, we include several combinatorial observations that might be useful for further progress, while also arguing that fully determining the answer for n = 7 will require new techniques.more » « less
-
Under flavor SU(3) symmetry (SU(3)_F), the final-state particles in B → PPP decays (P is a pseudoscalar meson) are treated as identical, and the PPP must be in a fully-symmetric(FS) state, a fully-antisymmetric (FA) state, or in one of four mixed states. In this paper, we present the formalism for the FA states. We write the amplitudes for the 22 B → PPP decays that can be in an FA state in terms of both the SU(3)_F reduced matrix elements and diagrams. This shows the equivalence of diagrams and SU(3)_F. We also give 15 relations among the amplitudes in the SU(3)_F limit as well as the additional four that appear when the diagrams E/A/PA are neglected. We present sets of B → PPP decays that can be used to extract γ using the FA amplitudes. The value(s) of γ found in this way can be compared with the value(s) found using the FS states.more » « less
-
null (Ed.)A bstract We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ 4 .more » « less
