Holomorphic Anomaly in Gauge Theories and Matrix Models

Albrecht Klemm; Min-xin Huang

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Holomorphic Anomaly in Gauge Theories and Matrix Models

Min-xin Huang, Albrecht Klemm

High Energy Physics - Theory · Submitted 19 May 2006

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

The authors' main claim The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.75 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.42 · compares with prior results 0.25 · mentions code or data availability 0.02.

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We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold.	contribution 0.90	0.91	0.16	asserted 1.00
s1	In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity.	method 0.90	0.95	0.10	asserted 1.00
s2	In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2).	result 1.00	0.89	0.16	asserted 1.00
s3	In the matrix model we fix the holomorphic ambiguity up to genus two.	result 0.40	0.92	0.09	asserted 1.00
s4	The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.	result 0.52	0.91	0.66	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 1.00.

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Original abstract

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.

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How later papers used it

58 later papers cite it

Evidence: substantial. 41 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 155 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 1 used its method or tool and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 2 of 12 papers with this label · 1 by the same authors

Following this paper, introduced the quantities S and X, whose total u-derivatives can be computed using the equations.

A note on rank 5/2 Liouville irregular block, Painlev\'e 1 and the ${\cal H}_0$ Argyres-Douglas theory (2023) · 0.96 · paraphrased: the licence does not allow quoting

By the same authors

Used the gap condition as in the case of pure gauge theory in this paper to fix the holomorphic ambiguity and find the genus two amplitude and its S-dual.

Holomorphicity and Modularity in Seiberg-Witten Theories with Matter (2009) (same authors) · 0.96 · paraphrased: the licence does not allow quoting

Compares with it 1 of 7 papers with this label

Checked explicitly in the SU(2) case that the genus g free energies of this paper agree with those of the gauge theory up to genus g=4.

New results in $\mathcal{N}=2$ theories from non-perturbative string (2017) · 0.89 · paraphrased: the licence does not allow quoting

How the literature uses it

58 arXiv papers cite this in 155 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 102 sentences without a confident label

Builds on it 2 papers

Using the generalized holomorphic anomaly equation and boundary condition we extend the methods of [ref] .

Direct integration for general Omega backgrounds · Introduction · 1.00
In fact we are going to show that for a generic gauge group of the ADE series the $Ω$-deformed prepotential satisfies a generalized recursion relation, thus extending the analysis of [ref] - Grimm:2007tm,Huang:2009md,Huang:2010kf,Huang:2011qx,Galakhov:2012gw,Billo:2013fi,Billo:2013jba [ref] for the SU$(2)$ theory and of [ref] where the SU$(N)$ theories were considered.

S-duality and the prepotential in N=2* theories (I): the ADE algebras · The recursion relation in the $Ω$-background · 0.93

Uses its method or tool 12 papers

Using the holomorphic anomaly recursion relation [ref] we have found this irregular conformal block up to order $8$ in $ϵ_1,2$.

A note on rank $\frac{3}{2}$ Liouville irregular block · Introduction · 1.00
We shall follow the approach in [ref] and treat the period $a$ as a flat coordinate in the holomorphic limit.

Holomorphicity and Modularity in Seiberg-Witten Theories with Matter · Gravitational couplings · 1.00
These free energies can be obtained by using topological recursion [ref] or, more conveniently in our case, by solving the holomorphic anomaly equations (HAE) of [ref] on the curve, as first pointed out in [ref] .

Large N instantons, BPS states, and the replica limit · The topological string free energies · 1.00

Uses its data or values 1 paper

The three point couplings are rational functions of $z_1, z_2$, and have been computed in [ref] [equation] and the others are related by symmetry.

Dijkgraaf-Vafa conjecture and beta-deformed matrix models · Review of Dijkgraaf-Vafa geometry · 0.93

Compares with it 7 papers

Therefore the genus one amplitude is not simply $F^(1)=-log(η(τ))$ as in the $N_f=0$ case [ref] but rather [equation] In the following we use the notation in [ref] and define [equation] $h:=b+2d$ and $X:=b/1728c^2d^2$.

Holomorphicity and Modularity in Seiberg-Witten Theories with Matter · $N_f=2$ · 0.86
For instance in the $SU(2)$ case we have checked explicitly that the genus $g$ free energies $F^D_g$ of [ref] agree with those of the gauge theory (computed in [ref] ) up to genus $g=4$.

New results in $\mathcal{N}=2$ theories from non-perturbative string · The matrix model · 0.89
In this paper we have carried out a comparison between the non perturbative results obtained for a $N=2$ SCFT of the AD type with the recursion relations coming from a conformal anomaly [ref] and those obtained from the AGT duality for irregular states [ref] , finding an agreement also keeping into account gravitational corrections.

On irregular states and Argyres-Douglas theories · Conclusions · 0.91

Cites it as background 18 papers

Note that ([ref]) only determines the quantity $ℱ^(g)$ up to addition by an holomorphic quantity $f^(g)$ called the genus $g$ holomorphic ambiguity [equation] Boundary conditions are needed to fix the holomorphic ambiguity $f^(g)$, these includes the boundary condition at the large complex structure limit and the gap condition at the conifold loci on $ℳ$ [ref] .

Differential Rings from Special K\"ahler Geometry · Holomorphic anomaly equations · 0.86
To set up the HAE we need various ingredients (we refer to [ref] for the relevant background).

Large N instantons, BPS states, and the replica limit · The topological string free energies · 0.99
Huang and Klemm [ref] discovered, and Huang, Klemm, and Quackenbush [ref] developed systematically, a sharper prediction, the conifold gap condition: in the suitably normalized flat coordinate $t$ given by the period of the vanishing cycle, the genus-$g$ free energy satisfies [equation] with every intermediate negative power and every logarithm absent.

Conifold Gap Theorem for Topological Recursion · Background · 0.85

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