Fermionic modes of D-instanton wormholes from broken local supersymmetry

Abstract: In low-energy supergravity treatment of type IIB superstring on general D-instanton wormhole profiles in the bulk, we obtain non-vanishing scalar two-point functions in addition to the vanishing $\langle τ^* τ^* \rangle$ that corresponds to the BPS amplitude detected by two D-instantons at their respective boundaries. This is exploited to show that the modes of broken local supersymmetry in the bulk deliver the fermionic (diagonal) modes on the boundaries through the deformation by the form of current-current two point functions propagating on the tree level cylinder geometry. Our treatment is generalizable to multi D-instanton cases and general Euclidean branes.

• The paper establishes that D-instanton wormholes yield fermionic zero modes as Goldstone modes of broken local supersymmetry.
• It computes tree-level effective quadratic Lagrangians for bosonic and fermionic sectors, revealing key supercurrent correlators and BPS cancellations.
• The methodology bridges supergravity deformations and matrix model representations, underpinning nonperturbative analyses in string theory.

Fermionic Modes on D-Instanton Wormholes from Broken Local Supersymmetry

Context and Motivation

The computation of D-instanton effects within type IIB superstring theory is central for non-perturbative phenomena, moduli stabilization, and the dynamics of supersymmetric vacua. While the closed/open string duality has led to deep insights in the structure of amplitudes and effective actions, the precise mechanism by which fermionic zero modes associated with broken local supersymmetry manifest at D-instanton boundaries remains structurally underdeveloped, especially in the context of matrix models and Euclidean brane constructions. The study under discussion systematically connects the bulk fermionic (Nambu-Goldstone) modes of spontaneously broken local supersymmetry to boundary modes supported on general D-instanton wormhole backgrounds, in the low-energy supergravity regime of type IIB theory.

Bosonic and Fermionic Sector: Supergravity Analysis

The authors begin by revisiting the well-known paradigm in which the open string annulus amplitude (cylinder) vanishes due to BPS cancellation for two D-instanton boundary states, corresponding to the absence of net force between the instantons in supersymmetry-preserving backgrounds. In the supergravity regime, D-instantons source the axion-dilaton field $\tau = a + i e^{-\phi}$ (NS-NS and R-R zero-forms), and the classical profiles are given by

$$e^\phi(\mathbf{x}) = \sum_{i=1}^n \frac{c_i}{|\mathbf{x} - \mathbf{X}_i|^8}$$

for multi-instanton locations $\mathbf{X}_i$ with positive residues $c_i$. This solution describes a multi-centered wormhole in ten-dimensional Euclidean space, corresponding to the back-reaction sourced by $n$ D-instantons.

The effective quadratic Lagrangian for the bosonic fluctuations around these backgrounds exhibits a critical structure: the two-point function $\langle \tilde{\tau}^* \tilde{\tau}^* \rangle$ vanishes by symmetry, whereas other combinations—particularly $\langle \tilde{\tau} \tilde{\tau} \rangle$ and $\langle \tilde{\tau}^* \tilde{\tau} \rangle$—are nonzero. This discriminant property aligns with the selection rules for BPS and non-BPS amplitudes in such backgrounds.

For the fermionic sector, the analysis is cast in terms of $32$-component Dirac spinors, breaking down to Majorana-Weyl (MW) spinors in the physical regime. The local supersymmetry is realized via covariant derivatives involving $SL(2, \mathbb{R})$ composite connections sourced by the axion-dilaton, and the field content includes the dilatino ($$e^\phi(\mathbf{x}) = \sum_{i=1}^n \frac{c_i}{|\mathbf{x} - \mathbf{X}_i|^8}$$0) and gravitino ($$e^\phi(\mathbf{x}) = \sum_{i=1}^n \frac{c_i}{|\mathbf{x} - \mathbf{X}_i|^8}$$1). The key result is the identification that, with respect to the instanton backgrounds, certain supersymmetry transformations remain unbroken, corresponding to the vanishing of specific variations, while others are spontaneously broken, yielding massless bulk fermionic modes.

Emergence and Propagation of Boundary Fermions

A central technical result is the explicit demonstration that the broken local supersymmetry in the bulk delivers the diagonal, Grassmann-valued fermionic modes to the D-instanton boundaries. The procedure involves:

1. Introducing an infinitesimal, localized Grassmann parameter $$e^\phi(\mathbf{x}) = \sum_{i=1}^n \frac{c_i}{|\mathbf{x} - \mathbf{X}_i|^8}$$2 at each instanton position and showing, via Ward-Takahashi-type deformation, that these are the correct fermionic coordinates to be identified with the diagonal elements of the instanton moduli space in matrix models.
2. Deforming the supergravity action at the linearized level by these broken supersymmetry modes, leading to an effective action for $$e^\phi(\mathbf{x}) = \sum_{i=1}^n \frac{c_i}{|\mathbf{x} - \mathbf{X}_i|^8}$$3, which at leading order is quadratic in the Grassmann variables and mediated via a two-point function of the bulk supercurrents.
3. Demonstrating that, on the tree-level (cylinder topology), the exponentiated effective action captures the correct leading order dynamics, with the supercurrent two-point function yielding the interaction kernel (see equation (X_sq1) in the original text).

This mechanism is fundamentally excluded for the unbroken supersymmetry sector: the relevant gauge-invariant current-current correlators cannot mediate tree-level cylinder amplitudes, in full analogy with the structure of gauge and global symmetry currents in other QFT settings.

Implications and Extensions

The identification of D-instanton boundary fermions as localized modes of the broken bulk local supersymmetry provides a direct link between the supergravity description and the zero-dimensional super Yang-Mills (matrix model) representations of D-instantons. The result establishes that these emergent boundary fermions are Nambu-Goldstone modes of spontaneously broken local SUSY, and their proper integration is required for any collective dynamics (the "instanton gas" picture). Moreover, the structure of the effective action—specifically, its dependence on local derivatives and the string coupling—illuminates the role of the gravitino, whose longitudinal component mediates the coupling of these Goldstone fermions.

Extension to higher-dimensional Euclidean branes (such as D1-branes) is natural: the same spontaneous breaking of local supersymmetry and resulting Grassmann boundary modes emerge when analyzing the classical profiles involving higher p-form RR backgrounds, subject to analogous technical caveats regarding the profiles and residual symmetries. The necessity of integrating over the fermionic collective coordinates to capture the full nonperturbative contribution persists and is directly analogous to standard treatments of bosonic zero modes in instanton calculus.

Conclusion

The paper establishes a systematic framework connecting the fermionic zero modes on D-instanton boundaries to the broken local supersymmetry in bulk type IIB supergravity. Through a careful deformation analysis and computation of supercurrent two-point functions in the presence of D-instanton wormhole backgrounds, the structure and propagation of these modes are elucidated at the quadratic (leading tree-level) order. This lays the groundwork for both quantitative calculations of D-instanton-induced nonperturbative effects and for formal connections to matrix model representations. Future theoretical work is motivated in the direction of higher order corrections (beyond linearized SUSY deformation), extensions to more general Euclidean Dp-brane configurations, and the incorporation of these structures in precise computations of string amplitudes and moduli stabilization scenarios.