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arXiv:2510.07155 · Journal of High Energy Physics

Observables of boundary RG flows from string field theory

Jaroslav Scheinpflug, Martin Schnabl, Jakub Vošmera

Published Apr 9, 2026• 2 citations
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Abstract

A bstract We present simple explicit formulae for the change of the g -function, boundary state, boundary spectrum and structure constants between the endpoints of short boundary RG flows at next-to-leading order. The formulae are derived using open string field theory and tested on integrable RG flows between conformal boundary states in unitary Virasoro minimal models.

DOI
10.1007/jhep04(2026)065

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A bstract We present simple explicit formulae for the change of the g -function, boundary state, boundary spectrum and structure constants between the endpoints of short boundary RG flows at next-to-leading order. The formulae are derived using open string field theory and tested on integrable RG flows between conformal boundary states in unitary Virasoro minimal models.
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Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of (p, 2) → (p − 1, 1)⊕(p + 1, 1) flows.
Conformal boundary conditions in two-dimensional conformal field theories are still mostly an uncharted territory. Even less is known about the relevant boundary deformations that connect them. A natural approach to the problem is via conformal perturbation theory, which, however, becomes quickly intractable. Using the formalism of open string field theory, we construct the corresponding classical solution, from which we can easily extract the boundary theory data. As a simple illustration we calculate the boundary degeneracy g to next-to-leading order for a generic theory.
This thesis explores renormalisation group flows in integrable quantum field theories with boundaries, as described by the g-function. The main focus is on the g-function in the staircase model, the renormalisation group flow of which passes close to the unitary minimal models. This g-function is used to identify flows between boundary conditions both within and between the minimal models. In certain limits the mathcalMAm(+)\\mathcal{M}A_m^{(+)}mathcalMAm(+)​ theories which interpolate between pairs of minimal models emerge from the staircase model, and exact expressions for the g-function in these models are extracted from the staircase g-function. Perturbative tests on the mathcalMA4(+)\\mathcal{M}A_4^{(+)}mathcalMA4(+)​ g-function are discussed, as is initial work on the g-function for the mathcalMA4(−)\\mathcal{M}A_4^{(-)}mathcalMA4(−)​ theory, which describes flows that emerge when the bulk coupling is taken to have the opposite sign to that in mathcalMA4(+)\\mathcal{M}A_4^{(+)}mathcalMA4(+)​. Expressions are also found for excited state versions of the mathcalMAm(+)\\mathcal{M}A_m^{(+)}mathcalMAm(+)​ g-function, and these allow the unique identification of certain boundary flows.