Some remarks on Lefschetz thimbles and complex Langevin dynamics

Aarts, Gert; Bongiovanni, Lorenzo; Seiler, Erhard; Sexty, Dénes

doi:10.1007/jhep10(2014)159

Cited by 67 publications

(69 citation statements)

References 75 publications

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“…To find the resolution of the problem we found in the present study, first we have to pin down what the real source of the failure is. For this, consideration on the basis of another approach to the functional integral with complex measure, the Lefschetz thimble [21][22][23][24][25], might provide useful insight. See also Ref.…”

Section: Resultsmentioning

confidence: 99%

Complex Langevin method applied to the 2DSU(2)Yang-Mills theory

2015

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Section: Resultsmentioning

confidence: 99%

Complex Langevin method applied to the 2DSU(2)Yang-Mills theory

2015

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“…We anticipate that HTAs will have crucial impact on the semiclassical analysis of many interesting quantum field theories and quantum mechanical systems. Our examples also show that in an attempt to perform lattice simulations using Lefschetz thimbles, e.g., [42][43][44][45], all thimbles whose multipliers are non-zero must be carefully summed over to correctly capture the dynamics of the theory.…”

Section: Discussionmentioning

confidence: 96%

Hidden Topological Angles in Path Integrals

et al. 2015

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We demonstrate the existence of hidden topological angles (HTAs) in a large class of quantum field theories and quantum mechanical systems. HTAs are distinct from theta-parameters in the lagrangian. They arise as invariant angle associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles). Physical effects of HTAs become most transparent upon analytic continuation in n f to non-integer number of flavors, reducing in the integer n f limit to a Z 2 valued phase difference between dominant saddles. In N = 1 super Yang-Mills theory we demonstrate the microscopic mechanism for the vanishing of the gluon condensate. The same effect leads to an anomalously small condensate in a QCD-like SU(N) gauge theory with fermions in the two-index representation. The basic phenomenon is that, contrary to folklore, the gluon condensate can receive both positive and negative contributions in a semi-classical expansion. In quantum mechanics, a HTA leads to a difference in semi-classical expansion of integer and half-integer spin particles.Introduction. Providing a non-perturbative continuum definition of the path integral in quantum field theory is a challenging but important problem [1]. There is growing evidence that, if an ordinary integral or a path integral admits a Lefschetz-thimble decomposition [2,3] or resurgent transseries expansion [4][5][6][7][8][9][10][11][12] then either of these methods gives this long-sought non-perturbative definition. If this is indeed the case, then we expect that these new methods will provide new and deep insight into quantum field theory and quantum mechanics formulated in terms of path integrals. In this article we introduce a new phenomenon of this kind, the appearance of hidden topological angles (HTAs).The main prescription associated with the Lefschetzthimble decomposition or the resurgent expansion is the following: Even if an ordinary integral or a path integral is formulated over real fields, the natural space that the critical points (saddles) ρ σ live in is the complexification of the original space of fields. However, the dimension of the crit-

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“…Both the complex saddle and the HTA are crucial for the argument. This example also demonstrates that in using Lefschetz thimbles, for example, either in Euclidean semi-classics or real time semi-classics (with sign problems) [110,111] or in lattice simulations [101][102][103][104], all thimbles whose Stokes multipliers are non-zero must be summed over. Numerical evidence for the correctness of this perspective is also given in [105][106][107][108][109].…”

Section: Hidden Topological Angles and Complex Saddlesmentioning

confidence: 88%

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

Dunne

Ünsal

2016

Annu. Rev. Nucl. Part. Sci.

112

170

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We present a broad conceptual introduction to some new ideas in non-perturbative QFT. The large-N orbifold-orientifold equivalence connects a natural large-N limit of QCD to QCD with adjoint fermions. QCD(adj) with periodic boundary conditions and double-trace deformation of Yang-Mills theory satisfy large-N volume independence, a type of orbifold equivalence. Certain QFTs that satisfy volume independence at N = ∞ exhibit adiabatic continuity at finite-N , and also become semi-classically calculable on small R 3 × S 1 . We discuss the role of monopole-instantons, and magnetic and neutral bion saddles in connection to mass gap, and center and chiral symmetry realizations. Neutral bions also provide a weak coupling semiclassical realization of infrared-renormalons. These considerations help motivate the necessity of complexification of path integrals (Picard-Lefschetz theory) in semi-classical analysis, and highlights the importance of hidden topological angles. Finally, we briefly review the resurgence program, which potentially provides a novel non-perturbative continuum definition of QFT. All these ideas are continuously connected.Keywords: large-N orbifold and orientifold equivalence, volume independence, double-trace deformations, adiabatic continuity, semi-classical calculability, magnetic and neutral bions, Picard-Lefschetz theory of path integration, and resurgence arXiv:1601.03414v1 [hep-th]

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Some remarks on Lefschetz thimbles and complex Langevin dynamics

Cited by 67 publications

References 75 publications

Complex Langevin method applied to the 2DSU(2)Yang-Mills theory

Complex Langevin method applied to the 2DSU(2)Yang-Mills theory

Hidden Topological Angles in Path Integrals

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

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