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Complex eigenvalue instantons and the Fredholm determinant expansion in the Gross-Witten-Wadia model
High Energy Physics - Theory · Submitted 11 Aug 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that, after taking the 't Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.
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We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact "ghost instantons", which are extrema of the one-eigenvalue effective potential on the "unphysical sheet" of the spectral curve and have been discussed in detail recently by Mari\~no, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the 't Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. | contribution 0.98 | 0.92 | 0.06 | asserted 0.80 |
| s1 | We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. | method 0.95 | 0.95 | 0.06 | asserted 1.00 |
| s2 | The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. | result 1.00 | 0.85 | 0.53 | asserted 1.00 |
| s3 | We show that these are in fact "ghost instantons", which are extrema of the one-eigenvalue effective potential on the "unphysical sheet" of the spectral curve and have been discussed in detail recently by Mari\~no, Schiappa, and Schwick. | interpretation 0.64 | 0.47 | 0.12 | asserted 1.00 |
| s4 | Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. | contribution 0.67 | 0.31 | 0.06 | asserted 0.53 |
| s5 | We find that, after taking the 't Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order. | result 1.00 | 0.95 | 0.64 | asserted 1.00 |
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The paper
Original abstract
We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact "ghost instantons", which are extrema of the one-eigenvalue effective potential on the "unphysical sheet" of the spectral curve and have been discussed in detail recently by Mari\~no, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the 't Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.
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References
How the literature uses it
11 arXiv papers cite this in 27 sentences
Papers citing it · 19 sentences without a confident label
Builds on it 1 paper
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We generalize the matrix computation of non-perturbative corrections to the partition function from the $k=1$ case [ref] to general $k$ in both phases, and find perfect agreement with the dual string theory results.
Normalization of ZZ instanton amplitudes in type 0B minimal superstring theory · Introduction · 0.98
Uses its method or tool 2 papers
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Following the procedure described in appendix C of [ref] , we can write this contribution as [equation] The only thing that has been done in this step is to incorporate the correct combinatorial factors for pulling out the set of eigenvalues as described above, and to separate the Vandermonde factor into various pieces.
Giant graviton expansion from eigenvalue instantons · Introduction · 0.99
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Also, a calculation similar to that in Ref. [ref] gives us that, in the large-$N$ limit, [equation] up to one-loop order.
Normalization of ZZ instanton amplitudes in type 0B minimal superstring theory · General multi-instanton contribution in the ungapped phase · 0.81
Cites it as background 4 papers
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These results are well known, but we refer the reader to appendix C of the recent paper [ref] for a recent pedagogical exposition.
Giant graviton expansion from eigenvalue instantons · Introduction · 0.98
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The extension to the type 0B case is straightforward, and we carry it out as well.[ Type 0A minimal superstring theories are dual to double-scaled complex matrix integrals, whereas type 0B theories are dual to double-scaled Hermitian or unitary matrix integrals [ref] .
Instanton-Induced Closed-String Amplitudes in Minimal Superstring Theory at Subleading Order · Introduction and summary · 0.88
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This expansion has also been related to eigenvalue tunnelling [ref] .
Localization and wall-crossing of giant graviton expansions in AdS$_5$ · Introduction and summary of results · 0.89
What it cites · 67 sentences without a confident label
Uses the method or tool of 6 papers
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It does not denote complex conjugation.] as: [equation] where we define the super-Vandermonde determinant [equation] and the $PV$ denotes a particular regularization prescription required to render the integral finite [ref] .
New Instantons for Matrix Models · Ghost-instanton contribution from tunneling anti-eigenvalues · 0.90
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So, in principle, we can use the results of [ref] expressing the one-loop instanton contribution in terms of the perturbative data around the leading one-cut saddle point.
The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation · Complex eigenvalue instantons in the strong-coupling phase · 0.91
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So, in principle, we can use the results of [ref] expressing the one-loop instanton contribution in terms of the perturbative data around the leading one-cut saddle point.
Multi-instantons in minimal string theory and in matrix integrals · Complex eigenvalue instantons in the strong-coupling phase · 0.91
Compares with 3 papers
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Furthermore, the present direct eigenvalue/ghost instanton approach to deriving nonperturbative corrections to the asymptotics of matrix integrals has the advantage of being easily generalized to unitary matrix integrals with more complicated potentials, whereas the differential equation approach of [ref] is specific to the GWW potential.
Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition · Introduction and setup · 0.81
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This result agrees with the known planar free energy of the GWW model [ref] .
Nonperturbative effects and nonperturbative definitions in matrix models and topological strings · The perturbative expansion in the strong-coupling phase · 0.97
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This result agrees with the known planar free energy of the GWW model [ref] .
A Study of U(N) Lattice Gauge Theory in 2-dimensions · The perturbative expansion in the strong-coupling phase · 0.97
Cites as background 19 papers
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In this short note, we study the Gross-Witten-Wadia (GWW) integral [ref] [equation] The integral is over $N ×N$ unitary matrices $U$.
A Study of U(N) Lattice Gauge Theory in 2-dimensions · Introduction and setup · 0.97
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In the past, the result ([ref]) was derived using techniques such as orthogonal polynomials and by studying differential equations satisfied by various observables [ref] .
Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition · Summary and Discussion · 0.85
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In the past, the result ([ref]) was derived using techniques such as orthogonal polynomials and by studying differential equations satisfied by various observables [ref] .
Nonperturbative effects and nonperturbative definitions in matrix models and topological strings · Summary and Discussion · 0.85
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