Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects

A.Marshakov; A.Mironov; A.Morozov; S.Kharchev

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Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects

S.Kharchev, A.Marshakov, A.Mironov, A.Morozov

High Energy Physics - Theory · Submitted 31 Dec 1993

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $\tau$-function.

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arXiv IDhep-th/9312210 (version 1)

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CitationA.Marshakov, A.Mironov, A.Morozov, S.Kharchev. "Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects." arXiv:hep-th/9312210 (1993). https://pith.science/paper/MMRMYF5U

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.55 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.72 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of $GL$ group. The integration over "matter fields" can be interpreted as going over the {\it model} (the space of all highest weight representations) of $GL$. In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice. The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $\tau$-function. (This is generalization of the classical statement that individual $GL$ characters are always singular KP $\tau$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ string models. However, under certain circumstances the formal sum over representations should be evaluated by steepest descent method and this procedure leads to some more complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the simple "character phase" deserves further investigation.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of $GL$ group.	result 0.68	0.36	0.18	asserted 0.98
s1	The integration over "matter fields" can be interpreted as going over the {\it model} (the space of all highest weight representations) of $GL$.	interpretation 0.95	0.47	0.12	asserted 0.63
s2	In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice.	result 0.37	0.33	0.08	asserted 0.86
s3	The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes.	result 0.79	0.35	0.21	asserted 1.00
s4	This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $\tau$-function.	result 0.96	0.63	0.24	asserted 0.80
s5	(This is generalization of the classical statement that individual $GL$ characters are always singular KP $\tau$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ string models.	interpretation 0.71	0.45	0.09	asserted 0.92
s6	However, under certain circumstances the formal sum over representations should be evaluated by steepest descent method and this procedure leads to some more complicated elements of Grassmannian.	result 0.56	0.43	0.11	asserted 0.51
s7	This "Kontsevich phase" as opposed to the simple "character phase" deserves further investigation.	future work 1.00	0.23	0.12	hedged possibility 0.42

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.72.

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Original abstract

The Kazakov-Migdal model, if considered as a functional of external fields, can be always represented as an expansion over characters of $GL$ group. The integration over "matter fields" can be interpreted as going over the {\it model} (the space of all highest weight representations) of $GL$. In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice. The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $\tau$-function. (This is generalization of the classical statement that individual $GL$ characters are always singular KP $\tau$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ string models. However, under certain circumstances the formal sum over representations should be evaluated by steepest descent method and this procedure leads to some more complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the simple "character phase" deserves further investigation.

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Cited by 10 Pith papers

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As pointed out therein (see also the work of Kharchev, Marshakov, Mironov and Morozov [ref] in a different context), $h$ is diagonal with respect to the basis ${|λ⟩}_λ∈$ of the charge-$0$ sector, and the diagonal matrix elements have the contents-product form [equation] where [equation] Since the work Okounkov [ref] , hypergeometric tau functions have played a significant role as generating functions of Hurwitz numbers and many other combinatorial notions [ref] (see also Harnad's comprehensive review [ref] ).

$q$-Difference Kac-Schwarz Operators in Topological String Theory · Relation to hypergeometric tau functions · 0.81

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