$\mathcal N=4$ SYM line defect Schur index and semiclassical string

TL;DR

This paper analyzes the large N limit of the line defect Schur index in N=4 SYM using semiclassical string theory, revealing that previously unexplained contributions are due to supersymmetric Casimir energy effects.

Contribution

It provides a direct semiclassical string analysis of the giant graviton expansion, clarifying the origin of extra contributions in the defect index at large N.

Findings

Extra contributions are identified as supersymmetric Casimir energy.

Semiclassical string partition function matches gauge theory data at large N.

Boundary conditions and fluctuation modes are systematically analyzed.

Abstract

The giant graviton expansion of the line defect Schur index in four dimensional $\mathcal N=4$ $U(N)$ SYM was recently proposed in arXiv:2403.11543 to be captured in the dual string theory by counting fluctuations states of two half-infinite fundamental strings in $AdS_{5}\times S^{5}$ ending on the line defect and D3 brane giant. However, agreement with the gauge theory data for the defect line index at finite $N$ required the inclusion of ad hoc extra contributions with unclear origin. We discuss the large $N$ leading order contribution of the giant graviton expansion of the defect line index by a direct analysis of semiclassical string partition function in a twisted background. We discuss supersymmetric boundary conditions in the presence of the D3 brane and evaluate the quadratic fluctuations effective action by introducing a suitable projection of fluctuation modes. We show that the extra contributions to the single giant graviton correction found in arXiv:2403.11543 correspond to a supersymmetric Casimir energy contribution.