These notes are based on lectures given at the Erwin-Schrodinger Insitut in Vienna in 2006/07 and at the 2007 School on Attractor Mechanism in Frascati. Lecture I: special geometry from the superconformal point of view. Lecture II: black hole attractor mechanism, its underlying variational principle, and black hole partition functions. Lecture III: large and small BPS black holes in N=4 supergravity. Lecture IV: state counting for N=4 string compactifications. Appendix A: special geometry from the mathematical point of view. Appendix B: review of modular forms. Contains four problems which allow the readers to develop some of the key concepts by themselves.
We study the new case of the application of the JKLMR conjecture on the connection between the exact partition functions of $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on $S^2$ and special Kähler geometry on the moduli spaces of Calabi-Yau manifold $Y$. The last ones arise as manifolds of the supersymmetric vacua of the GLSM. We establish this correspondence using the Mirror symmetry in Batyrev's approach. Namely, starting from the two-moduli non-Fermat Calabi-Yau manifold $X$ we construct the dual GLSM with the supersymmetric vacua $Y$, which is the mirror for $X$. Kn…
We compute the exact partition function of the isotropic 6-vertex model on a cylinder geometry with free boundary conditions, for lattices of intermediate size, using Bethe ansatz and algebraic geometry. We perform the computations in both the open and closed channels. We also consider the partial thermodynamic limits, whereby in the open (closed) channel, the open (closed) direction is kept small while the other direction becomes large. We compute the zeros of the partition function in the two partial thermodynamic limits, and compare with the condensation curves.
Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected pecul…
We initiate the study of M-strings in the thermodynamic limit. In this limit the BPS partition function of M5 branes localizes on configurations with a large number of strings which leads to a reformulation of the partition function in terms of a matrix model. We solve this matrix model and obtain its spectral curve which can be interpreted as the Seiberg-Witten curve associated to the compactified M5 brane theory.
We continue to develop our method for effectively computating the special Kähler geometry on the moduli space of Calabi-Yau manifolds. We generalize it to all polynomial deformations of Fermat hypersurfaces.
The Geometry of Supersymmetric Partition Functions
考虑紧致流形M上的超对称场论，得到了其配分函数Z_M对参数依赖性的约束。我们的主要焦点是Z_M对M几何的依赖性，以及耦合到连续味对称的背景规范场。对于N=1的四维U（1）R对称理论，M必是一个具有Hermite度量的复流形.我们发现Z_M与度量无关，全纯地依赖于复结构模。背景规范场定义了M上的全纯向量丛，Z_M是相应丛模的全纯函数。我们还对具有U（1）R对称性的三维N=2理论进行了平行分析，其中M上的必要几何结构是具有横向Hermite度量的横向全纯叶理（THF）.我们再次发现Z_M与度规无关，全纯地依赖于THF的模。我们讨论了几个应用，包括与超对称指数相关的半纯于S^3 x S^1或S^2 x S^1的流形，以及半纯于S^3（压扁球）的流形。在Z_M已明确计算的例子中，我们的结果解释了它的许多观察到的性质。
Using the boundary string field theory (BSFT) techniques we study the boundary state and partition function for a dynamical (rotating-moving) D$p$-brane coupled to the electromagnetic and tachyonic background fields in superstring theory. By making use of the created partition function, the super BSFT action with a tachyonic field and Dirac-Born-Infeld type action will be constructed. By analyzing the obtained action interesting features will be revealed.
It is shown that the generating function of $\mathscr{N}=2$ topological strings, in the heterotic weak coupling limit, is identified with the partition function of a six-dimensional Melvin background. This background, which corresponds to an exact CFT, realises in string theory the six-dimensional $\varOmega$-background of Nekrasov, in the case of opposite deformation parameters $ε_1=-ε_2$, thus providing the known perturbative part of the Nekrasov partition function in the field theory limit. The analysis is performed on both heterotic and type I strings and for the cases of ordinary $\mathsc…
We consider string theory on $\text{AdS}_3 \times \text{S}^3 \times \mathbb{T}^4$ in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal $\text{AdS}_3$, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on ther…
We clarify the recently proposed method to compute a Special Kähler metric on a Calabi-Yau complex structures moduli space that uses the fact that the moduli space is a subspace of specific Frobenius manifold. We apply this method to computing the Special Kähler metric in a two-moduli non-Fermat model which has been unknown until now.
Orientifolds, Brane Coordinates and Special Geometry
本文报道了在D3-D 7-膜和通量的存在下，K3 x T2/Z2定向褶皱上IIB型真空的规范超重力分析。我们讨论了Minkowski真空对应的超对称临界点以及相关的模的固定，发现与以前的分析一致。一个重要的作用是通过选择的辛全纯部分的特殊几何进入计算的标量势。明确给出了相关的周期矩阵N。阐明了特殊几何与膜模的Born-Infeld作用量之间的关系。
A new method for explicit computation of the CY moduli space metric was proposed by the authors recently. The method makes use of the connection of the moduli space with a certain Frobenius algebra. Here we clarify this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point.
We consider the $Ω$-deformed $\mathcal{N}=2$ $SU(2)$ gauge theory in four dimensions with $N_{f}=4$ massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters $\varepsilon_{1}, \varepsilon_{2}$, the scalar field expectation value $a$, and the hypermultiplet masses ${\bf m}=(m_{1}, m_{2}, m_{3}, m_{4})$. Motivated by recent findings in the $\mathcal{N}=2^{*}$ theory, we explore the theories that are characterized by special fixed ratios $\varepsilon_{2}/\varepsilon_{1}$ and ${\bf m}/\varepsilon_{1}$ and propose a simple condition on the structure …
We develop an efficient method to compute the torus partition function of the six-vertex model exactly for finite lattice size. The method is based on the algebro-geometric approach to the resolution of Bethe ansatz equations initiated in a previous work, and on further ingredients introduced in the present paper. The latter include rational $Q$-system, primary decomposition, algebraic extension and Galois theory. Using this approach, we probe new structures in the solution space of the Bethe ansatz equations which enable us to boost the efficiency of the computation. As an application, we stu…
We apply differential geometry methods to the computation of the anomaly-induced hydrodynamic equilibrium partition function. Implementing the imaginary-time prescription on the Chern-Simons effective action on a stationary background, we obtain general closed expressions for both the invariant and anomalous part of the partition function. This is applied to the Wess-Zumino-Witten action for Goldstone modes, giving the equilibrium partition function of superfluids. In all cases, we also study the anomaly-induced gauge currents and energy-momentum tensor, providing explicit expressions for them…
We consider general charged Kasner-like solutions for the theory of five-dimensional supergravity coupled to Abelian vector multiplets in arbitrary space-time signature. These solutions, depending on the choice of coordinates, can be thought of as generalisations of Melvin/Rosen cosmologies, flux-branes and domain walls.
We study Bethe/gauge correspondence at the special locus of Coulomb moduli where the integrable system exhibits the splitting of degenerate levels. For this investigation, we consider the four-dimensional pure $\mathcal{N}=2$ supersymmetric $U(N)$ gauge theory, with a half-BPS surface defect constructed with the help of an orbifold or a degenerate gauge vertex. We show that the non-perturbative Dyson-Schwinger equations imply the Schrödinger-type and the Baxter-type differential equations satisfied by the respective surface defect partition functions. At the special locus of Coulomb moduli the…
The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometries of four- and five-dimensional vector multiplets. We formulate ${\cal N}= 2$ theories in terms of Hessian structures and give various concrete applications of Hessian geometry, ranging from static BPS black holes in four and five space-time dimensions to topological string theor…
本文研究了在余维为2的缺陷存在下，R^4 × S^1上具有极大超对称性的五维U（N）规范理论的Nekrasov配分函数。余维2的缺陷既可以用单值缺陷来描述，也可以通过耦合到R^2 x S^1上的某类三维双规范理论来描述。我们解释这些计算是如何与经典和量子可积系统。作为瞬子数的展开，我们检验了上述配分函数是一个椭圆可积多体系统的本征函数，该系统量子化了五维规范理论的塞伯格-威滕几何。
