Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
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迈向路径积分、复杂鞍座和复兴的皮卡德-莱夫谢茨理论
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发表时间：
2015
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影响因子：
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通讯作者：
M. Unsal
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文献类型：
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作者：
A. Behtash;G. Dunne;T. Schaefer;T. Sulejmanpasic;M. Unsal
我们证明了一般欧氏路径积分的半经典分析必然要求作用量和测度的复化，并考虑复鞍解，证明了复鞍点在Picard-Lefschetz理论中有一个自然的解释.部分原因是在$mathbb{R}^[3]上带有伴随物质的QCD的半经典展开 当S ^1 $时，我们研究具有谐振简并最小值的玻色子和费米子（格拉斯曼）自由度的量子力学系统，以及具有谐振非简并最小值的（相关）纯玻色子系统。我们找到了全纯牛顿方程的精确有限作用非BPS反弹和bion解。我们不仅找到了真实的解，而且还找到了具有非平凡单值性的复解，最后找到了复多值解和奇异解。复杂的生物子是必要的，以获得正确的非微扰结构的这些模型。在超对称极限下，复解支配着基态性质，它们对半经典展开的贡献是与超对称代数保持一致所必需的。动作的多值性要么与隐藏的拓扑角度有关，要么与模糊性的重新消除有关。我们还表明，在近似的多瞬子描述的复杂的准零模顶针的集成产生的精确解的最显着的特点。虽然在量子场论中构造精确的复鞍更困难，但与近似顶针构造的关系表明，此类解可能是量子场论中近似生物子鞍的一些显着特征的基础。
We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions.We demonstrate that complex saddle points have a natural interpretation in terms of the Picard–Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on $mathbb{R}^3 imes S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic nondegenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct nonperturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.
关于 Lefschetz 顶针和复杂 Langevin 动力学的一些评论
影响因子：
5.4
作者：
G. Aarts;L. Bongiovanni;E. Seiler;D. Sexty
通讯作者：
D. Sexty