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Instantons in N=2 magnetized D-brane worlds

Marco Billo, Marialuisa Frau, Igor Pesando, Paolo Di Vecchia, Alberto Lerda, Raffaele Marotta

TL;DR

This work analyzes instanton effects in ${ m N}=2$ super Yang–Mills theories engineered with wrapped magnetized D9 branes and E5 branes on a toroidal orbifold of type IIB string theory. It shows that 1-loop mixed annulus diagrams with mixed boundaries encode the running of the gauge coupling and threshold corrections in a way that, when expressed in the four-dimensional supergravity basis, the non-holomorphic Kahler factors cancel against the annulus contributions to yield a holomorphic, Wilsonian instanton action on the Coulomb branch. By computing the charged and flavored instanton sectors and their annulus amplitudes, the authors derive an explicit holomorphic structure of the instanton-induced prepotential and effective action, including a holomorphic redefinition of the dynamical scale through η-functions of the complex structure moduli. The results confirm that the Kahler metrics of adjoint and fundamental fields precisely compensate non-holomorphic pieces, providing a robust string-theoretic realization of holomorphic non-perturbative effects and offering a framework to extract Kähler data in more general ${ m N}=1$ settings. This advances understanding of how stringy instantons contribute to the low-energy action and highlights the role of threshold corrections in shaping holomorphic non-perturbative physics.

Abstract

In a toroidal orbifold of type IIB string theory we study instanton effects in N=2 super Yang-Mills theories engineered with systems of wrapped magnetized D9 branes and Euclidean D5 branes. We analyze the various open string sectors in this brane system and study the 1-loop amplitudes described by annulus diagrams with mixed boundary conditions, explaining their role in the stringy instanton calculus. We show in particular that the non-holomorphic terms in these annulus amplitudes precisely reconstruct the appropriate Kahler metric factors that are needed to write the instanton correlators in terms of purely holomorphic variables. We also explicitly derive the correct holomorphic structure of the instanton induced low energy effective action in the Coulomb branch.

Instantons in N=2 magnetized D-brane worlds

TL;DR

This work analyzes instanton effects in super Yang–Mills theories engineered with wrapped magnetized D9 branes and E5 branes on a toroidal orbifold of type IIB string theory. It shows that 1-loop mixed annulus diagrams with mixed boundaries encode the running of the gauge coupling and threshold corrections in a way that, when expressed in the four-dimensional supergravity basis, the non-holomorphic Kahler factors cancel against the annulus contributions to yield a holomorphic, Wilsonian instanton action on the Coulomb branch. By computing the charged and flavored instanton sectors and their annulus amplitudes, the authors derive an explicit holomorphic structure of the instanton-induced prepotential and effective action, including a holomorphic redefinition of the dynamical scale through η-functions of the complex structure moduli. The results confirm that the Kahler metrics of adjoint and fundamental fields precisely compensate non-holomorphic pieces, providing a robust string-theoretic realization of holomorphic non-perturbative effects and offering a framework to extract Kähler data in more general settings. This advances understanding of how stringy instantons contribute to the low-energy action and highlights the role of threshold corrections in shaping holomorphic non-perturbative physics.

Abstract

In a toroidal orbifold of type IIB string theory we study instanton effects in N=2 super Yang-Mills theories engineered with systems of wrapped magnetized D9 branes and Euclidean D5 branes. We analyze the various open string sectors in this brane system and study the 1-loop amplitudes described by annulus diagrams with mixed boundary conditions, explaining their role in the stringy instanton calculus. We show in particular that the non-holomorphic terms in these annulus amplitudes precisely reconstruct the appropriate Kahler metric factors that are needed to write the instanton correlators in terms of purely holomorphic variables. We also explicitly derive the correct holomorphic structure of the instanton induced low energy effective action in the Coulomb branch.

Paper Structure

This paper contains 23 sections, 169 equations, 5 figures.

Table of Contents

  1. Introduction
  2. $\mathcal{N}=2$ models from magnetized branes
  3. The gauge sector
  4. The matter sector
  5. Generalizations
  6. $\mathcal{N}=2$ instanton calculus from the string perspective
  7. Instanton moduli
  8. The neutral instanton sector
  9. The charged instanton sector
  10. The flavored instanton sector
  11. Instanton partition function
  12. Instanton induced prepotential and effective action
  13. The rôle of annulus amplitudes
  14. The mixed annuli and the running gauge coupling constant
  15. The explicit form of the annulus amplitude $\mathcal{A}_{5_a}$
  16. ...and 8 more sections

Figures (5)

  • Figure 1: The mixed disk representing the coupling of the adjoint scalar field $\bar{\phi}$ to the instanton fermionic moduli $\bar{\mu}$ and $\mu$.
  • Figure 2: The amplitude $\mathcal{D}_{9_a}(f)$: a disk whose boundary lies on the D$9_a$ branes, with the insertion of the gauge field at the quadratic order.
  • Figure 3: The amplitude $\mathcal{D}_{5_a}$: a disk whose boundary lies on $k$ wrapped E$5_a$ branes.
  • Figure 4: The amplitudes $\mathcal{A}_{9_a;9_a}(f)$ or $\mathcal{A}_{9_a;9_b}(f)$ correspond to annuli where one boundary lies on the D$9_a$ branes and carries two insertions of $f$, while the other boundary lies, respectively, on a D$9_a$ or a D$9_b$ brane.
  • Figure 5: The amplitudes $\mathcal{A}_{5_a;9_a}$ or $\mathcal{A}_{5_a;9_b}$ correspond to annuli where one boundary lies on the E5$a$ branes, while the other boundary lies, respectively, on a D$9_a$ or a D$9_b$ brane.