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We consider a hybrid Monte Carlo algorithm which is applicable to lattice theories defined on Lefschetz thimbles. In the algorithm, any point (field configuration) on a thimble is parametrized uniquely by the flow-direction and the flow-time defined at a certain asymptotic region close to the critical point, and it is generated by solving the gradient flow equation downward. The associated complete set of tangent vectors is also generated in the same manner. Molecular dynamics is then formulated as a constrained dynamical system, where the equations of motion with Lagrange multipliers are solved by the second-order constraint-preserving symmetric integrator. The algorithm is tested in the λφ 4 model at finite density, by choosing the thimbles associated with the classical vacua for subcritical and supercritical values of chemical potential. For the lattice size L = 4, we find that the residual sign factors average to not less than 0.99 and are safely included by reweighting and that the results of the number density are consistent with those obtained by the complex Langevin simulations. Summary and Discussion 21A Tangent vectors at the critical point of the thimble 2-(b) 222 See [5, 27] for reviews on this approach. 3 This model has also been simulated successfully by the dual variable method in [32][33][34]. 4 It has also been shown by the authors of [11] that in such models the symmetry property is preserved and the perturbation theory reproduces the same result as the original one. 5 Strictly speaking, the field configurations obtained in this simulation do not belong to the Lefschetz thimble associated with the Gaussian critical point (or the classical vacuum). Rather, they are obtained by projecting on to the tangent space at the critical point, although the tangent space is not necessarily in the same homology class as the thimble. Accordingly, the imaginary part of the action does not stay constant, and its exponent is included by reweighting. It has been claimed that this approximation is good enough to reproduce the silver blaze behavior in the λφ 4 model. 6 This Metropolis sampling method is based on a mapping between a thimble and its asymptotic "Gaussian" region close to the critical point.

A single-parameter family of covariant gauge fixing conditions in bosonic string field theory is proposed. It is a natural string field counterpart of the covariant gauge in the conventional gauge theory, which includes the Landau gauge as well as the Feynman (Siegel) gauge as special cases. The action in the Landau gauge is largely simplified in such a way that numerous component fields have no derivatives in their kinetic terms and appear in at most quadratic in the vertex.Comment: 24 page

We study a product rule and a difference operator equipped with Leibniz rule in a general framework of lattice field theory. It is shown that the difference operator can be determined by the product rule and some initial data through the Leibniz rule. This observation leads to a no-go theorem that it is impossible to construct any difference operator and product rule on a lattice with the properties of (i) translation invariance, (ii) locality and (iii) Leibniz rule. We present a formalism to overcome the difficulty by an infinite flavor extension or a matrix expression of a lattice field theory.

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