We define supersymmetric zeta functions and supersymmetric determinants, which can reveal spectral properties complementary to those captured by the supersymmetric indices. They play a crucial role in analyzing the Cardy-like behaviors of the supersymmetric indices and the supersymmetric Casimir energies associated with the supersymmetric partition functions. We investigate a variety of examples of the supersymmetric zeta functions and determinants for two-, four-, and six-dimensional supersymmetric field theories.

Japan

Yu Nakayama

China

Tadashi Okazaki

Supersymmetric zeta functions and determinants

We study configurations of two $$ \mathcal{N}=4 $$ super Yang-Mills theories of unitary gauge groups connected by the BPS interfaces involving line operators. We find strong evidence of S-duality of the configurations as precise matching of the line defect half-indices which enumerate the BPS local operators at the junctions of the interfaces and line operators. The interface line defect half-indices are expressible as intriguing closed-form formulae involving the $\textit{q}$-binomial coefficients and the principal specializations of the Schur functions.

Yasuyuki Hatsuda

Interface line operators in $$ \mathcal{N}=4 $$ SYM theories and supersymmetric indices

We propose SL(2, ℤ) dualities of supersymmetric boundary conditions in the three-dimensional supersymmetric field theories describing a semi-infinite M2-brane terminating on M5-branes. Specifically, we present dualities of boundary conditions for Abelian (quiver) ADHM theories and circular quiver Chern-Simons matter theories including the ABJM model. For the circular quiver Chern-Simons theories we take boundary conditions breaking a U(1) × U(1) gauge group to its diagonal subgroup which is decoupled. This can be generalized to break U(1) × U(1) , leaving a ℤ gauge theory. We find matching of the ’t Hooft anomalies and supersymmetric half-indices for all the proposed dual boundary conditions.

United Kingdom

Douglas Smith

SL(2, ℤ) dualities of boundary conditions in Abelian M2-brane SCFTs

We find matching pairs of the line defect indices for 3d supersymmetric Abelian gauge theories as strong evidence of dualities of the BPS line operators. They lead to novel duality maps of the BPS line operators for $$ \mathcal{N}\ge 4 $$ supersymmetric circular and linear quiver gauge theories which can be realized as brane configurations in Type IIB string theory, including SQED, ADHM and ABJM theories.

Hirotaka Hayashi

Tomoki Nosaka

Abelian dualities and line defect indices for 3d gauge theories

We analyze the supersymmetric defect indices of $$ \mathcal{N} $$ = 4 super Yang Mills theories which are simultaneously decorated by the BPS line operators and the boundary conditions. We demonstrate that the two-point functions of the boundary ’t Hooft lines of magnetic charges associated with the minuscule representations in the presence of the regular Nahm pole boundary conditions can be obtained by applying the Higgsing prescription to the half-indices of the Dirichlet boundary conditions. Accordingly, we find precise matching of the indices for pairs of the S-dual configurations with the Wilson lines and Neumann boundary conditions and those with the ’t Hooft lines and the regular Nahm pole boundary conditions. Alternatively, we analytically compute the indices by means of the inner product of the Macdonald polynomials to find the exact closed-form expressions.

S-duality of boundary lines in $$ \mathcal{N} $$ = 4 SYM theories and supersymmetric indices

We analyze the line defect correlation functions decorating the supersymmetric indices of $$ \mathcal{N} $$ = 4 super Yang-Mills theories with orthogonal and symplectic gauge groups. We obtain exact closed-form expressions for the correlators by means of combinatorial methods associated with the group characters, including the generalized Murnaghan-Nakayama rules. The exact results for the large $\textit{N}$ correlators lead to the gravity indices for the gravity dual spectrum of the fluctuations of the fundamental string wrapping $\textit{AdS}$ and those of the fat string, the D5-brane wrapping $\textit{AdS}$ × ℝℙ .

Hai Lin

$$ \mathcal{N} $$ = 4 line defect correlators of type BCD

We study the half-indices of $$ \mathcal{N} $$ = 4 super Yang-Mills theories of orthogonal and symplectic gauge groups. We find precise matching pairs of the half-indices as strong evidence of dualities of the half-BPS boundary conditions and interfaces. The half-indices encode the spectra of excitations on the holographically dual orbifold ETW brane and giant gravitons. The exact form of the index for the orbifold ETW giant gravitons is obtained from the giant graviton expansion of the half-index.

Orbifold ETW brane and half-indices

We examine the giant graviton expansions of the Coulomb and Higgs indices for the M2-brane SCFTs to find the closed-form expressions for the indices that encode the spectra of the 1/4-BPS M5-brane giant gravitons and the 1/3-BPS orbifold M5-brane giant gravitons. Consequently, we get exact forms of the twisted indices for the 6d (2, 0) theories describing a stack of N M5-branes which generalize the unrefined indices. We confirm that they are also beautifully expanded with respect to the indices for the M2-brane giant gravitons which are obtained from the Coulomb and Higgs indices for the M2-brane SCFTs upon the change of variables.

M2-M5 giant graviton expansions

We study the giant gravitons in the AdS 4 bagpipe geometries involving end-of-the-world (ETW) brane constructed by a single 5-brane and either two stacks or one stack of D3-branes in Type IIB string theory. From the exact formulae and giant graviton expansions of the half-indices for the half-BPS boundary conditions and interfaces in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills theory, we obtain the BPS spectra of the fluctuation modes of the AdS 4 bagpipe geometries including the ETW brane region.

Giant graviton expansions and ETW brane

The Coulomb and Higgs indices of the 3d $$ \mathcal{N} $$ = 4 U( ) ADHM theories can be decorated by line defect operators as the line defect correlators. We obtain exact closed-form expressions and various non-trivial algebraic relations for the correlators of the Wilson lines in the fundamental and (anti)symmetric representations by means of the Hall-Littlewood expansion, the Fermi-gas method and the residue calculation. From the large limit of the correlators we obtain the single particle gravity indices which are expected to encode the spectra of fluctuation modes on the gravity duals of line operators in M2-brane SCFTs.