本文提出了6d little string theories的blowup方程，通过引入辅助磁通量求和以消除混合反常，并结合模性质求解方程，从而确定了little strings的elliptic genera。
摘要：
我们提出了6d little string theories的blowup方程，推广了Nakajima-Yoshioka关于Omega背景上4d/5d instanton partition functions的blowup方程。我们发现，与标准SQFTs的blowup方程不同，我们需要对blown-up $\mathbb{P}^1$上的辅助磁通量求和，该磁通量属于一个非动力学2-form规范场，在消除规范对称性的混合反常中发挥作用。我们通过显式例子证明，blowup方程与模性质结合时可以被求解，从而确定little strings的elliptic genera。
英文摘要：
We propose blowup equations for 6d little string theories which generalize Nakajima-Yoshioka's blowup equations for the 4d/5d instanton partition functions on Omega background. We find that unlike the blowup equations for standard SQFTs, we need to sum over auxiliary magnetic fluxes on the blown-up $ \mathbb{P}^1$ for a non-dynamical 2-form gauge field which plays a role in canceling the mixed anomalies of the gauge symmetries. We demonstrate with explicit examples that the blowup equations, when combined with the modular properties, can be solved in order to determine the elliptic genera of little strings.
Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional $\mathcal{N}=1$ supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on $SU(N)$ geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the $\bf r$ fields introduced in our previous paper…
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We propose novel functional equations for the BPS partition functions of 6d (1,0) SCFTs, which can be regarded as an elliptic version of Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic Calabi-Yau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in det…
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At high energy densities, fivebranes are populated by a Hagedorn phase of so-called "little strings", whose statistical mechanics underlies black fivebrane thermodynamics. A particular limit of this phase yields BTZ black holes in $AdS_3$, leading us to the idea that in this context fuzzballs and highly excited little strings are one and the same. We explore these ideas through an analysis of D-brane probes of fivebrane supertube backgrounds. String theory dynamics on these backgrounds is described by an exactly solvable null-gauged WZW model. We develop the formalism of null gauging on worlds…
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We study the elliptic genera of 6d strings based on their modular properties. They are weak Jacobi forms of weight 0, whose indices are determined from the 2d chiral anomalies. We propose the ansatz for the elliptic genera which reflects the analytic structure of instanton partition functions. Given a finite amount of initial BPS data, we completely determine the elliptic genera of 6d strings in various 6d SCFTs. We also apply our ansatz to study $\mathcal{N}=(2,0)$ and $(1,1)$ little strings as well as $\mathcal{N}=(1,0)$ heterotic little strings, for which T-duality of little string theories…
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We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the $Ω$-background with gauge group $U(N)$ and matter in the adjoint representation. We show that the instanton partition function of these theories can be written in terms of Kronecker-Eisenstein series, which in a particular limit of the deformation parameters of the $Ω$-background organise themselves into Greens functions of free scalar fields on a torus. We provide a concrete identification between (differences of) such propagators and Nekrasov subfunctions. The latter are…
As was shown recently, non-Abelian vortex strings supported in four-dimensional ${\cal N}=2$ supersymmetric QCD with the U(2) gauge group and $N_f=4$ quark multiplets (flavors) become critical superstrings. In addition to the translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional target space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space and a Calabi-Yau non-compact threefold, namely, the conifold. We study closed string sta…
