Shell formulas for instantons and gauge origami

TL;DR

The paper introduces the shell formula, a universal framework that expresses partition functions whose poles are classified by $d$-dimensional Young diagrams through a single $\mathcal J$-factor built from the shell of each diagram. This approach unifies instanton counting in 5d pure SYM for classical gauge groups and a broad class of gauge origami configurations (magnificent four, tetrahedron instantons, DT counting, spiked instantons), translating JK-residue computations into recursive shell operations and revealing sign-structure via the shell data. The method yields closed-form, unrefined-limit expressions, clarifies brane realizations, and suggests deep links to topological vertex and quantum algebras through $qq$-characters. It opens avenues to include matter, exceptional groups, orbifold/CY settings, and higher-dimensional DT physics, providing a versatile toolkit for exact BPS counting and protected operator generating functions.

Abstract

We introduce the shell formula -- a framework capable of providing a unified description for various partition functions whose pole structures are classified by Young diagrams of arbitrary dimension. This formalism encompasses a wide range of physical systems, including instanton partition functions of 5d pure super Yang-Mills theory with classical gauge groups, as well as gauge origami configurations such as the magnificent four, tetrahedron instantons, Donaldson-Thomas invariants, and spiked instantons.

Figures (13)

• Figure 1: ADHM quiver diagram of D0-D4 system in the infinite adjoint mass limit as $\mathcal{N}=(0,2)$ SUSY QM. The black solid lines represent chiral multiplets, and the red dashed lines represent fermi multiplets. The circular nodes denote gauge groups, while the square nodes denote flavor groups. In this quiver, we have a $\mathop{\mathrm{U}}\nolimits(k)$ gauge group, a $\mathop{\mathrm{U}}\nolimits(N)$ flavor group, a fundamental chiral $I$, an anti-fundamental chiral $J$, two adjoint chirals $B_{1,2}$ respectively, and an adjoint fermi $\Lambda_{1}$.
• Figure 2: The instanton configuration of 5d pure $\mathop{\mathrm{U}}\nolimits(N)$ SYM can be constructed from type IIA D0-D4 brane system after integrating out the adjoint hypermultiplets. The D4-branes extend along two complex directions $\mathbb{C}_1$, $\mathbb{C}_2$, and the time direction $x^0$. D4$_{\alpha}$ refers to the $\alpha$-th D4-brane, while D0$_{\alpha,\boldsymbol{x}}$ denotes the D0-brane within D4$_{\alpha}$ corresponding to the instanton at coordinate $\boldsymbol{x}$ in the Young diagram. The red wavy line represents strings connecting D0$_{\alpha,\boldsymbol{x}}$ and D4$_{\beta}$, and the contribution of these strings to the index is $\mathcal{J}(\mathcal{X}_\alpha|\lambda_\beta)/\operatorname{sh}(-\mathcal{X}_\alpha+\mathcal{X}_\beta)$.
• Figure 3: On the $x^5$-$x^6$ plane, the horizontally extending lines represent D5-branes, while the vertical or slanted lines represent NS5-branes. For the $\mathop{\mathrm{SO}}\nolimits$ gauge group, we need to use an O5$^-$ or $\widetilde{\text{O5}}^-$ orientifold at the bottom, where, when it crosses an NS5-brane, it becomes an O5$^+$. For the $\mathop{\mathrm{SO}}\nolimits(2n)$ gauge group, we require $n$ D5-branes and an O5$^-$. For $\mathop{\mathrm{SO}}\nolimits(2n+1)$, we need $n$ D5-branes and an $\widetilde{\text{O5}}^-$Zafrir:2015ftn. The red wavy lines represent the effective contribution of strings connecting D1$_{\alpha,\boldsymbol{x}}$ and D5$_\beta$, which is identical to that of the $\mathop{\mathrm{U}}\nolimits(N)$ gauge group given in \ref{['D0-D4 string']}. The blue wavy lines correspond to strings connecting D1$_{\alpha,\boldsymbol{x}}$ and D5$_\beta$ whose orientation is reversed by the O-plane; their effective contribution matches the original one, subject to the replacement $\mathcal{X}_\alpha \to -\mathcal{X}_\alpha$.
• Figure 4: (a) The brane construction for the Sp gauge group is similar to Fig. \ref{['fig:SON brane']}. For the $\mathop{\mathrm{Sp}}\nolimits(2N)$ group, we require $N$ D5-branes, an O5$^+$, and NS5-branes. Furthermore, we can transform this into the brane construction for $\mathop{\mathrm{SO}}\nolimits(2N+8)$. The transformation process is as follows: (b) We bring two D5-branes from infinity via Higgsing to the vicinity of the O-plane and freeze them. At this point, both the left and right sides at the bottom consist of O5$^-$ plus two D5-branes, while the central part at the bottom consists of O5$^+$ plus two D5-branes. (c) Using the equivalence O5$^- + 2$ D5 $\sim$ O5$^+$, we can replace both sides of the bottom with O5$^+$. (d) Through the equivalence O5$^+$$\sim$ O5$^- + 2$ D5, the central part can be replaced with O5$^- + 4$ D5. Therefore, $\mathop{\mathrm{Sp}}\nolimits(2N)$ can be viewed as $\mathop{\mathrm{SO}}\nolimits(2N+8)$ with 4 Coulomb branch parameters $v_{N+1},\ldots,v_{N+4}$ fixed as Tab. \ref{['tab:SP poles']}, corresponding to the frozen D5-branes.
• Figure 5: The brane construction of $\mathop{\mathrm{Sp}}\nolimits$ gauge theory. The blue wavy lines represent the effective contribution of strings connecting the D1$_{\alpha,\boldsymbol x}$ branes and the frozen D5$_{\beta}$ branes, where $\beta = N+1, \dots, N+4$. Unlike the string contributions connecting D1 and ordinary D5 branes mentioned above \ref{['D0-D4 string']}, the contribution of these strings is simply $\mathcal{J}(\mathcal{X_{\alpha}}|\lambda_{\beta'})$. For the contribution of strings that have been projected by the O-plane and connect D1 and the frozen D5 branes, we also need to replace $\mathcal{X}_{\beta'}$ with $-\mathcal{X}_{\beta'}$.

Theorems & Definitions (4)

• Definition 2.1: Young diagram
• Definition 2.2: Shell of a Young diagram
• Definition 2.3: Charge of shellbox
• Definition 2.4: $\mathcal{J}$-factor