Lattice Gauge Theories beyond QCD: large N, Supersymmetry and Orientifold planar equivalence

在基本相互作用（强，电弱和引力）。强相互作用可能是最难以捉摸的。虽然它们并不直接在宏观世界中表现出来，但强相互作用解释了物质的稳定性，因为它们负责将中子和质子束缚在原子核内的力。中子和质子是一个更广泛的粒子家族--强子的成员，它们之间都有强烈的相互作用。强子不是基本粒子：它们是由更基本的粒子，夸克和胶子组成的。在电磁学中，粒子通过它们的电荷相互作用;在强相互作用中，粒子通过一种电荷相互作用，这一次我们把这种电荷称为图像上的颜色。在自然界中有三种不同的色荷;我们说红、蓝和绿色夸克，用颜色来指粒子携带的特定电荷（这与我们在日常经验中看到的颜色无关）。描述夸克、胶子及其相互作用的理论是QCD。目前，QCD（其细节将在LHC进行测试）尚未完全理解：虽然我们对高能区（能量为1 GeV或更高）有明确的分析预测，但我们目前知道的分析计算技术不能用于低能区。不幸的是，这是最有趣的区域，因为例如核子内的夸克具有低能量。在没有分析预测的情况下，已经开发出一种成功的数值方法：基于Monte Carlo的晶格离散理论模拟。在这种方法中，理论是在一个离散的有限时空上制定的，空间为a，线性维数为L。这种简化的理论是用有限个积分来表示的，可以在（超级）计算机上执行。连续理论通过外推到极限L趋于无穷大和a趋于零而恢复。为了充分利用计算的预测能力，仍然需要分析理解。通过把QCD看作更一般的一类理论的特例，在这方面已经取得了很大的进展。在这个意义上，一种可能性是对耦合到夸克的SU（N）规范理论取大N极限。当N=3时，QCD恢复。大N理论比原来的N=3理论简单，由于N的有限值对它的修正可以表示为1/N的幂级数。最近的晶格已被证明是一个有用的工具，用于确定未知系数的幂级数在夸克的情况下。我们计划在理论中包括Nf夸克时也扩展这项工作。另一个有趣的方向是Orientifold平面等价。这是一个不同的大N极限，它将理论与广义夸克联系起来：SU（N）规范理论一边是反对称夸克，另一边是伴随夸克。这个框架的预测能力是由于这样一个事实，即后者与N=1的超级杨-米尔斯有关，而前者在N=3时简化为一个夸克的QCD。悬而未决的问题仍然存在，比如用正确数量的味道恢复QCD或从无限N的情况到N=3的修正的大小。我们项目的中心点之一就是回答这些问题。这将需要一个非平凡的推广目前使用的格QCD算法和技术。一旦我们开发了在SU（N）规范群的一般表示中模拟一般数量的费米子的技术，我们就有了解决其他有趣问题的工具，比如晶格上的N=1超级杨米尔斯和电弱对称性破缺机制的替代方案。这两个问题都具有高度的现象学相关性，并将在即将到来的LHC实验中进行研究。

Among the fundamental interactions (strong, electroweak and gravitational). the strong interactions are probably the most elusive. Although they do not manifest themselves directly in the macroscopic world, strong interactions account for the stability of matter, since they are responsible for forces that bind neutrons and protons inside nuclei. Neutrons and protons are members of a wider family of particles, the hadrons, all of which interact strongly. The hadrons are not elementary particles: they are made of more fundamental particles, quarks and gluons. In electromagnetism particles interact through their electric charge; in strong interactions particles interact through a charge which this time is called pictorially colour. In nature there are three different colour charges; we speak then of the red, blue and green quarks, referring by the colour to the particular charge carried by a particle (which has nothing to do with the colours we see in our everyday experience). The theory that describes quark, gluons and their interaction is QCD. Currently, QCD (whose fine details are going to be tested at LHC) is not fully understood: while we have clear analytical predictions for the high energy regime (energies of 1 GeV or more), analytical computation techniques we currently know can not be used in the low energy regime. Unfortunately this is the most interesting regime, since e.g. quarks inside nucleons have low energies. In the absence of analytical predictions, a successful numerical approach has been developed: Monte Carlo based simulations of the theory discretised on a lattice. In this approach, the theory is formulated on a discrete and finite spacetime of spacing a and linear dimension L. This simplified theory is formulated in terms of a finite number of integrals, which can be performed on a (super)computer. The continuous theory is recovered by extrapolating to the limits L going to infinity and a going to zero. In order to fully exploit the predictive power of the calculations, an analytical understanding is still needed. Much progress has been achieved on this side by considering QCD as a special case of a more general class of theories. One possibility in this sense is to take the large N limit for a SU(N) gauge theory coupled to quarks. QCD is recovered when N=3. The large N theory is simpler than the original N=3 one, and corrections to it due to a finite value of N can be expressed as a power series in 1/N. Recently the lattice has proved to be an useful tool for determining the unknown coefficients of that power series in the absence of quarks. We plan to extend that work also when Nf quarks are included in the theory. Another interesting direction is the Orientifold planar equivalence. This is a different large N limit that relates theories with generalised quarks: a SU(N) gauge theory with an antisymmetric quark on one side and a SU(N) gauge theory with an adjoint quark on the other. The predictive power of this framework is due to the fact that the latter is related to N=1 Super Yang-Mills, for which much is know analytically, while the former reduces to QCD with one quark when N=3. Open questions remain, like the size of the corrections to recover QCD with the correct number of flavours or to go from the infinite N case to N=3. One of the central points of our project is to answer those questions. This will require a non-trivial generalisation of currently used lattice QCD algorithms and techniques. Once we have developed the techniques to simulate a generic number of fermions in a generic representation of a SU(N) gauge group, we would have the tools to attack other interesting problems, like N=1 Super Yang-Mills on the lattice and alternative scenarios for the electroweak symmetry breaking mechanism. Both those problems have a high phenomenological relevance, and will be studied at the upcoming LHC experiments.