16 Supersymmetries - a half-way meeting in the City

这笔赠款将用于组织一次研讨会，讨论N=2超对称四维规范理论及其在规范/弦对应下的弦对偶的最新发展。我们的目标是将该领域的领先国际研究人员聚集在一起，展示他们的最新成果，分析该领域的最新进展，并激发合作，以确定和解决该研究领域的突出问题。现代弦理论最重要的发展之一是发现了规范理论和弦理论之间的对偶。这些显著的二元性假设了非阿贝尔规范理论和量子引力理论的等价性，正如弯曲时空上的弦理论所描述的那样，在非微扰水平上揭开了这两种事先非常不同的理论之间的未知联系。这给了我们第一次通过弯曲时空上的弦理论来定量处理强耦合规范理论的方法，这解决了现代理论物理中的关键问题之一。这种对偶性研究最多的例子可能是最大超对称(32个超对称)N=4SU(N)超杨-米尔斯(SYM)理论和ADS_5x S^5上的对偶IIB型超弦理论。近年来，这种对偶偶变得相对容易理解，部分是因为保留了大量的超对称性，部分是因为对偶两边都发现了可积结构。N=2对偶(16个超对称)的例子在文献中是已知的，但通常了解得较少。同时，具有N=2超对称性的规范理论比N=4规范理论具有更丰富的结构；例如，N=2理论可以具有不同表示规范群的场，标准模型就是这样。继Seiberg和Witten的著名工作之后，N=2规范理论一直被视为折中之家：它们都具有许多现实特征，但由于超对称的存在，其中一个保留了很大的控制权。因此，对一般的N=2理论及其弦理论对偶的详细理解将有助于我们理解非微扰规范理论的动力学。最近，文献中出现了一些与16个超对称理论相关的新结果。仅举三个例子：首先，Gaiotto发现了非常一般的N=2超对称规范理论的一种优雅结构，并与Maldacena一起，找到了研究它们的引力对偶的形式。其次，一些作者(包括PI)一直在将研究最大超对称规范/弦对偶的可积性方法推广到较不超对称的情况(包括16个超对称的情况)，这表明这种方法可能也适用于N=2超对称四维规范理论的规范/弦对偶。最后，在Nekrasov以前的一些结果的基础上，Alday、Gaiotto和Tchikawa以及后来的Wyllard猜测了N=2超对称规范理论的某些子部门与Liouville和Toda理论之间的关系。总而言之，我们认为，在过去一年左右的时间里，出现了越来越多的文献，这些文献为16种超对称规范理论提供了新的线索。这表明，关于N=2超对称四维规范理论及其弦对偶的研讨会将特别及时，并可能刺激进一步的重大突破。

This grant will cover the costs of organising a workshop on recent developments in N=2 supersymmetric four-dimensional gauge theories and their string duals under the gauge/string correspondence. We aim to bring together the leading international researchers in the field to present their latest results, analyse the state-of-the-art of the field, and inspire collaborations to identify and tackle the outstanding problems in this research area.One of the most important developments in modern string theory has been the discovery of dualities between gauge theories and string theories. These remarkable dualities postulate the equivalence of non-abelian gauge theories and quantum theories of gravity, as described by string theories on curved spacetimes, unravelling an unexplored connection between these two, a priori very different, types of theories at a non-perturbative level. This has given us for the first time a quantitative handle on a strongly coupled gauge theory via string theory on a curved spacetime, which tackles one of the key questions in modern theoretical physics. Perhaps the most studied example of this duality is the maximally supersymmetric (32 supersymmetries) N=4 SU(n) super Yang-Mills (SYM) theory and the dual Type IIB superstring theory on AdS_5 x S^5. This dual pair has become relatively well understood in recent years, in part because of the large amount of supersymmetry preserved and in part because of integrable structures identified on both sides of the duality. Examples of N=2 dual pairs (16 supersymmetries) are known in the literature, but are in general less well understood. At the same time, gauge theories with N=2 supersymmetry have a much richer structure than N=4 gauge theories; for example N=2 theories can have fields in different representations of the gauge group, as is the case with the Standard Model. Following the celebrated work of Seiberg and Witten, N=2 gauge theories have been viewed as a half-way house: they have both many realistic features yet one retains a good deal of control due to the supersymmtry present. As a result, a detailed understanding of general N=2 theories and their string theory duals is lightly to shed light on our understanding of non-perturbative gauge theory dynamics.Recently, a number of novel results relating to theories with 16 supersymmetries have appeared in the literature. To name just three: Firstly, Gaiotto has identified an elegant construction of very general N=2 supersymmetric gauge theories, and, together with Maldacena, found a formalism for studying their gravity duals. Secondly, a number of authors (including the PI) have been extending the integrability approach used to study the maximally supersymmetric gauge/string dual pair to less supersymmetric cases (including cases with 16 supersymmetries), suggesting that such techniques may also be relevant for gauge/string dual pairs of N=2 supersymmetric four dimensional gauge theories. Finally, building on some previous results of Nekrasov, Alday, Gaiotto and Tachikawa and later Wyllard have conjectured a relation between certain sub-sectors of N=2 supersymmetric gauge theories and Liouville and Toda theories. To summarize, we believe that over the past year or so a growing body of literature has appeared which sheds new light on gauge theories with 16 supersymmetries. This suggests that a workshop on N=2 supersymmetric four-dimensional gauge theories and their string duals would be particularly timely, and may stimulate further significant breakthroughs.