nLab Jacobi diagram

Idea

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

$$order(\Gamma) \;\coloneqq\; \#Vertices(\Gamma)/2$$

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…).

Definition

Jacobi diagrams

(…)

STU-relations and weight systems

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

(1)$$\mathcal{A}^t \;\coloneqq\; R\langle \mathcal{D}^c \rangle/STU$$

for the quotient spaces of the linear span of Jacobi diagrams by the STU relations:

Hence:

A linear function

$$w \;\colon\; \mathcal{A}^t \longrightarrow R \,,$$

– 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

Properties

Relation to chord diagrams

chord diagrams modulo 4T are Jacobi diagrams modulo STU:

graphics from Sati-Schreiber 19c

Relation to Adams operations

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 $q \in \mathbb{Z}$, $q \neq 0$)

$$\psi^q \;\colon\; \mathcal{A} \longrightarrow \mathcal{A}$$

which respect the coalgebra structure and satisfy

$$\psi^{q_2} \circ \psi^q_1 \;=\; \psi^{q_1 \cdot q_2}$$

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:

$$w_{\mathbf{N}}(\psi^q D) \;=\; w_{\psi^q \mathbf{N}}(D) \,.$$

(Bar-Natan 95, Exc. 6.24 )

For more see at Adams operation on Jacobi diagrams.

Related concepts

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

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal weight systems are cohomology of loop space of configuration space

References

General

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

• Dror Bar-Natan, Alexander Stoimenow, The Fundamental Theorem of Vassiliev Invariants (arXiv:q-alg/9702009)

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)

Chord diagrams and weight systems in Physics

The following is a list of references that involve (weight systems on) chord diagrams/Jacobi diagrams in physics:

1. In Chern-Simons theory

2. In Dp-D(p+2) brane intersections

3. In quantum many body models for for holographic brane/bulk correspondence:

   1. In AdS2/CFT1, JT-gravity/SYK-model

   2. As dimer/bit thread codes for holographic entanglement entropy

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. 66 4 (2023) [doi:10.4153/S0008439523000206, arXiv:2210.05399]

Review:

• Carlo Collari, Weight systems which are quantum states, talk at QFT and Cobordism, CQTS (Mar 2023) $[$web, pdf$]$

In Chern-Simons theory

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.

from Bar-Natan 1991

• 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:

• Ismar Volić, Section 4 of: Configuration space integrals and the topology of knot and link spaces, Morfismos, Vol 17, no 2, 2013 (arxiv:1310.7224)

Applied to Gopakumar-Vafa duality:

• Dave Auckly, Sergiy Koshkin, Introduction to the Gopakumar-Vafa Large $N$ Duality, Geom. Topol. Monogr. 8 (2006) 195-456 (arXiv:0701568)

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)

For single trace operators in AdS/CFT duality

Interpretation of Lie algebra weight systems on chord diagrams as certain single trace operators, in particular in application to black hole thermodynamics

• Micha Berkooz, Prithvi Narayan, Joan Simón, Section 2.1 of Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (arxiv:1806.04380)

In $AdS_2/CFT_1$, JT-gravity/SYK-model

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:

• László Erdős, Dominik Schröder, Phase Transition in the Density of States of Quantum Spin Glasses, D. Math Phys Anal Geom (2014) 17: 9164 (arXiv:1407.1552)

which in turn follows

• Philippe Flajolet, Marc Noy, Analytic Combinatorics of Chord Diagrams, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), Formal Power Series and Algebraic Combinatorics, Springer 2000 (doi:10.1007/978-3-662-04166-6_17)

from Narovlansky 2019

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]

• Adel A. Rahman, Leonard Susskind: $p$-Chords, Wee-Chords, and de Sitter Space [arXiv:2407.12988]

In D$p$/D$(p+2)$-brane intersections

from Sati and Schreiber 19c

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)

As codes for holographic entanglement entropy

From Yan 20

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)

From Jahn and Eisert 21

For Dyson-Schwinger equations

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:

• Ali Assem Mahmoud, Section 3 of: On the Enumerative Structures in Quantum Field Theory (arXiv:2008.11661)

Other

Appearance of horizontal chord diagrams in discussion of neutrino interactions in supernovae:

• Duff Neill, Hanqing Liu, Joshua Martin, Alessandro Roggero: Scattering Neutrinos, Spin Models, and Permutations [arXiv:2406.18677]