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Register a new userGluing II: Boundary Localization and Gluing Formulas
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Authors
Mykola Dedushenko
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We describe applications of the gluing formalism discussed in the companion paper. When a $d$-dimensional local theory $text{QFT}_d$ is supersymmetric, and if we can find a supersymmetric polarization for $text{QFT}_d$ quantized on a $(d-1)$-manifold $W$, gluing along $W$ is described by a non-local $text{QFT}_{d-1}$ that has an induced supersymmetry. Applying supersymmetric localization to $text{QFT}_{d-1}$, which we refer to as the boundary localization, allows in some cases to represent gluing by finite-dimensional integrals over appropriate spaces of supersymmetric boundary conditions. We follow this strategy to derive a number of `gluing formulas in various dimensions, some of which are new and some of which have been previously conjectured. First we show how gluing in supersymmetric quantum mechanics can reduce to a sum over a finite set of boundary conditions. Then we derive two gluing formulas for 3D $mathcal{N}=4$ theories on spheres: one providing the Coulomb branch representation of gluing, and another providing the Higgs branch representation. This allows to study various properties of their $(2,2)$-preserving boundary conditions in relation to Mirror Symmetry. After that we derive a gluing formula in 4D $mathcal{N}=2$ theories on spheres, both squashed and round. First we apply it to predict the hemisphere partition function, then we apply it to the study of boundary conditions and domain walls in these theories. Finally, we mention how to glue half-indices of 4D $mathcal{N}=2$ theories.
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We review some aspects of the cutting and gluing law in local quantum field theory. In particular, we emphasize the description of gluing by a path integral over a space of polarized boundary conditions, which are given by leaves of some Lagrangian foliation in the phase space. We think of this path integral as a non-local $(d-1)$-dimensional gluing theory associated to the parent local $d$-dimensional theory. We describe various properties of this procedure and spell out conditions under which symmetries of the parent theory lead to symmetries of the gluing theory. The purpose of this paper is to set up a playground for the companion paper where these techniques are applied to obtain new results in supersymmetric theories.
We construct a gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition at a rigid, regular, codimension one configuration. This construction plays an important role in establishing the relation between the gauged linear sigma model and the nonlinear sigma model in the presence of Lagrangian branes.
We provide a unifying entropy functional and an extremization principle for black holes and black strings in AdS$_4times S^7$ and AdS$_5times S^5$ with arbitrary rotation and generic electric and magnetic charges. This is done by gluing gravitational blocks, basic building blocks that are directly inspired by the holomorphic blocks appearing in the factorization of supersymmetric partition functions in three and four dimensions. We also provide an explicit realization of the attractor mechanism by identifying the values of the scalar fields at the horizon with the critical points of the entropy functional. We give examples based on dyonic rotating black holes with a twist in AdS$_4times S^7$, rotating black strings in AdS$_5times S^5$, dyonic Kerr-Newman black holes in AdS$_4times S^7$ and Kerr-Newman black holes in AdS$_5times S^5$. In particular, our entropy functional extends existing results by adding rotation to the twisted black holes in AdS$_4$ and by adding flavor magnetic charges for the Kerr-Newman black holes in AdS$_4$. We also discuss generalizations to higher-dimensional black objects.
Given two semigroups $langle Arangle$ and $langle Brangle$ in ${mathbb N}^n$, we wonder when they can be glued, i.e., when there exists a semigroup $langle Crangle$ in ${mathbb N}^n$ such that the defining ideals of the corresponding semigroup rings satisfy that $I_C=I_A+I_B+langlerhorangle$ for some binomial $rho$. If $ngeq 2$ and $k[A]$ and $k[B]$ are Cohen-Macaulay, we prove that in order to glue them, one of the two semigroups must be degenerate. Then we study the two most degenerate cases: when one of the semigroups is generated by one single element (simple split) and the case where it is generated by at least two elements and all the elements of the semigroup lie on a line. In both cases we characterize the semigroups that can be glued and say how to glue them. Further, in these cases, we conclude that the glued $langle Crangle$ is Cohen-Macaulay if and only if both $langle Arangle$ and $langle Brangle$ are also Cohen-Macaulay. As an application, we characterize precisely the Cohen-Macaulay semigroups that can be glued when $n=2$.
Farhi and others have introduced the notion of solving NP problems using adiabatic quantum com- puters. We discuss an application of this idea to the problem of integer factorization, together with a technique we call gluing which can be used to build adiabatic models of interesting problems. Although adiabatic quantum computers already exist, they are likely to be too small to directly tackle problems of interesting practical sizes for the foreseeable future. Therefore, we discuss techniques for decomposition of large problems, which permits us to fully exploit such hardware as may be available. Numerical re- sults suggest that even simple decomposition techniques may yield acceptable results with subexponential overhead, independent of the performance of the underlying device.
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