Examples/classes:
Types
Related concepts:
A closed Jacobi diagram is a connected undirected graph with oriented trivalent vertices and with an embedded oriented circle, regarded modulo cyclic identifications, if any.
Here is a picture of a typical Jacobi diagram:
Here the internal lines need not form a tree; the following is also a Jacobi diagram:
graphics from Sati-Schreiber 19c
If all the vertices sit on the circle, Jacobi diagrams specialize to chord diagrams.
Half the number of vertices of a Jacobi diagram is called its order
For example, these are the two closed Jacobi diagrams with 2 vertices (order 1):
and these are are the 10 closed Jacobi diagrams with 4 vertices (order 2):
graphics grabbed from Chmutov-Duzhin-Mostovoy 11
In applications to Vassiliev invariants in knot theory, Jacobi diagrams play the role of connected Feynman diagrams for Chern-Simons theory in the presence of one Wilson line (the circle is then the knot/Wilson line).
Terminology. Jacobi diagrams have originally been called “chinese character diagrams” (Bar-Natan 95, Def. 1.8) or “Feynman diagrams for Chern-Simons theory” (Kricker-Spence-Aitchison 97) or “circle diagrams” (Kneissler 97) or “round diagrams” (Willerton 99). The term “Jacobi diagram” (Chmutov-Duzhin-Mostovoy 11, Chapter 5, Jackson-Moffat 19, Section 13) alludes to the Jacobi identity – here also called the STU relation – via the fact that these diagrams are naturally labelled by a Lie algebra equipped with a Lie algebra representation (which is really just the interaction-aspect of their interpretation as Chern-Simons theory Feynman diagrams…).
(…)
For $R \in$ CRing a commutative ring, let $R\langle \mathcal{D}^t \rangle$ denote the $R$-linear span of the set $\mathcal{D}^t$ of Jacobi diagrams.
Then one traditionally writes
for the quotient spaces of the linear span of Jacobi diagrams by the STU relations:
Hence:
– hence a plain $R$-valued function on the set $\mathcal{D}^t$ of Jacobi diagrams which is invariant under the STU relations –
is called a framed weight system. See there for more.
graphics from Sati-Schreiber 19c
chord diagrams modulo 4T are Jacobi diagrams modulo STU:
graphics from Sati-Schreiber 19c
On the vector space $\mathcal{A}$ of Jacobi diagrams modulo STU-relations (equivalently chord diagrams modulo 4T relations) there is a system of linear maps (for $q \in \mathbb{Z}$, $q \neq 0$)
which respect the coalgebra structure and satisfy
and as such are (dually) analogous to the Adams operations on topological K-theory.
(Bar-Natan 95, Def. 3.11 & Theorem 7)
In fact, when evaluated in Lie algebra weight systems $w_{\mathbf{N}}$ and under the identification (see here) of the representation ring of a compact Lie group $G$ with the $G$-equivariant K-theory of the point, these Adams operations on Jacobi diagrams correspond to the Adams operations on equivariant K-theory:
For more see at Adams operation on Jacobi diagrams.
chord diagrams | weight systems |
---|---|
linear chord diagrams, round chord diagrams Jacobi diagrams, Sullivan chord diagrams | Lie algebra weight systems, stringy weight system, Rozansky-Witten weight systems |
Original articles
Dror Bar-Natan, On the Vassiliev knot invariants, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, pdf)
A. Kricker, B. Spence and I. Aitchison, Cabling the Vassiliev invariants, J. Knot Theory Ramifications 6 (1997) 327–358
Jan Kneissler, The number of primitive Vassiliev invariants up to degree 12 (arXiv:q-alg/9706022)
Simon Willerton, The Kontsevich integral and algebraic structures on the space of diagrams in: Knots in Hellas ’98. Series on Knots and Everything, vol. 24, World Scientific, 2000, 530–546 (arXiv:math/9909151)
Lecture notes
Textbook accounts
Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 5 of: Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 (arxiv:1103.5628, doi:10.1017/CBO9781139107846)
David Jackson, Iain Moffat, Section 13 of: An Introduction to Quantum and Vassiliev Knot Invariants, Springer 2019 (doi:10.1007/978-3-030-05213-3)
The following is a list of references that involve (weight systems on) chord diagrams/Jacobi diagrams in physics:
In quantum many body models for for holographic brane/bulk correspondence:
For a unifying perspective (via Hypothesis H) and further pointers, see:
Hisham Sati, Urs Schreiber, Differential Cohomotopy implies intersecting brane observables, Adv. Theor. Math. Phys. 26 4 (2022) [doi:10.4310/ATMP.2022.v26.n4.a4arXiv:1912.10425]
David Corfield, Hisham Sati, Urs Schreiber: Fundamental weight systems are quantum states Lett. Math. Phys. 113 112 (2023) [arXiv:2105.02871, doi:10.1007/s11005-023-01725-4]
Carlo Collari, A note on weight systems which are quantum states, Can. Math. Bull. (2023) [doi:10.4153/S0008439523000206, arXiv:2210.05399]
Review:
Since weight systems are the associated graded of Vassiliev invariants, and since Vassiliev invariants are knot invariants arising as certain correlators/Feynman amplitudes of Chern-Simons theory in the presence of Wilson lines, there is a close relation between weight systems and quantum Chern-Simons theory.
Historically this is the original application of chord diagrams/Jacobi diagrams and their weight systems, see also at graph complex and Kontsevich integral.
Dror Bar-Natan, Perturbative aspects of the Chern-Simons topological quantum field theory, thesis 1991 (spire:323500, proquest:303979053, BarNatanPerturbativeCS91.pdf)
Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)
Daniel Altschuler, Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 (arxiv:q-alg/9603010)
Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949-1000 (arXiv:math/9910139)
Alberto Cattaneo, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (arXiv:math/0307218)
Reviewed in:
Applied to Gopakumar-Vafa duality:
See also
Marcos Mariño, Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants, Commun. Math. Phys. 253 (2004) 25-49 (arXiv:hep-th/0207096)
Stavros Garoufalidis, Marcos Mariño, On Chern-Simons matrix models (pdf, pdf)
Interpretation of Lie algebra weight systems on chord diagrams as certain single trace operators, in particular in application to black hole thermodynamics
Discussion of (Lie algebra-)weight systems on chord diagrams as SYK model single trace operators:
Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$, JHEP 04 (2018) 146 (arXiv:1801.02696)
Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large $N$ expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 (arXiv:1806.03271)
Micha Berkooz, Prithvi Narayan, Joan Simón, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (arxiv:1806.04380)
following:
which in turn follows
With emphasis on the holographic content:
Micha Berkooz, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: Towards a full solution of the large $N$ double-scaled SYK model, JHEP 03 (2019) 079 (arxiv:1811.02584)
Vladimir Narovlansky, Slide 23 (of 28) of: Towards a Solution of Large $N$ Double-Scaled SYK, 2019 (pdf)
Micha Berkooz, Mikhail Isachenkov, Prithvi Narayan, Vladimir Narovlansky, Quantum groups, non-commutative $AdS_2$, and chords in the double-scaled SYK model [arXiv:2212.13668]
Herman Verlinde, Double-scaled SYK, Chords and de Sitter Gravity [arXiv:2402.00635]
Micha Berkooz, Nadav Brukner, Yiyang Jia, Ohad Mamroud, A Path Integral for Chord Diagrams and Chaotic-Integrable Transitions in Double Scaled SYK [arXiv:2403.05980]
and specifically in relation, under AdS2/CFT1, to Jackiw-Teitelboim gravity:
Andreas Blommaert, Thomas Mertens, Henri Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (arXiv:1806.07765)
Andreas Blommaert, Thomas Mertens, Henri Verschelde, Fine Structure of Jackiw-Teitelboim Quantum Gravity, JHEP 1909 (2019) 066 (arXiv:1812.00918)
Henry W. Lin, The bulk Hilbert space of double scaled SYK, J. High Energ. Phys. 2022 60 (2022) $[$arXiv:2208.07032, doi:10.1007/JHEP11(2022)060$]$
Henry W. Lin, Douglas Stanford, A symmetry algebra in double-scaled SYK $[$arXiv:2307.15725$]$
Micha Berkooz, Ohad Mamroud: A Cordial Introduction to Double Scaled SYK [arXiv:2407.09396]
Discussion of weight systems on chord diagrams as single trace observables for the non-abelian DBI action on the fuzzy funnel/fuzzy sphere non-commutative geometry of Dp-D(p+2)-brane intersections (hence Yang-Mills monopoles):
Sanyaje Ramgoolam, Bill Spence, S. Thomas, Section 3.2 of: Resolving brane collapse with $1/N$ corrections in non-Abelian DBI, Nucl. Phys. B703 (2004) 236-276 (arxiv:hep-th/0405256)
Simon McNamara, Constantinos Papageorgakis, Sanyaje Ramgoolam, Bill Spence, Appendix A of: Finite $N$ effects on the collapse of fuzzy spheres, JHEP 0605:060, 2006 (arxiv:hep-th/0512145)
Simon McNamara, Section 4 of: Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels, 2006 (spire:1351861, pdf, pdf)
Constantinos Papageorgakis, p. 161-162 of: On matrix D-brane dynamics and fuzzy spheres, 2006 (pdf)
Chord diagrams encoding Majorana dimer codes and other quantum error correcting codes via tensor networks exhibiting holographic entanglement entropy:
Alexander Jahn, Marek Gluza, Fernando Pastawski, Jens Eisert, Majorana dimers and holographic quantum error-correcting code, Phys. Rev. Research 1, 033079 (2019) (arXiv:1905.03268)
Han Yan, Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models, Phys. Rev. B 102, 161119 (2020) (arXiv:1911.01007)
Discussion of round chord diagrams organizing Dyson-Schwinger equations:
Nicolas Marie, Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, Communications in Number Theory and Physics, 7(2):251291, 2013 (arXiv:1210.5457)
Markus Hihn, Karen Yeats, Generalized chord diagram expansions of Dyson-Schwinger equations, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 (arXiv:1602.02550)
Paul-Hermann Balduf, Amelia Cantwell, Kurusch Ebrahimi-Fard, Lukas Nabergall, Nicholas Olson-Harris, Karen Yeats, Tubings, chord diagrams, and Dyson-Schwinger equations [arXiv:2302.02019]
Review in:
Appearance of horizontal chord diagrams in discussion of neutrino interactions in supernovae:
Last revised on January 10, 2020 at 09:15:29. See the history of this page for a list of all contributions to it.