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The infinity-category of stabilized Liouville sectors [article]

Oleg Lazarev, Zachary Sylvan, Hiro Lee Tanaka
2021 arXiv   pre-print
We prove the surprising fact that the infinity-category of stabilized Liouville sectors is a localization of an ordinary category of stabilized Liouville sectors and strict sectorial embeddings.  ...  Moreover, we characterize the symmetric monoidal structure using a universal property, again producing a simple-as-possible criterion for verifying whether invariants are both continuously and multiplicatively  ...  The categorical results were not at all obligated to exist based on the geometry known to us when we first began this project.  ... 
arXiv:2110.11754v1 fatcat:zp7yvbstkvewtno6trqwodpgza

Geometry and rigidity of mapping class groups [article]

Jason Behrstock, Bruce Kleiner, Yair Minsky, Lee Mosher
2010 arXiv   pre-print
We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely  ...  As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S); a characterization of the image of the curve-complex projection  ...  Local homology via hulls. In Section 6 we use Σ-hulls in order to study the local homology properties of the asymptotic cone.  ... 
arXiv:0801.2006v4 fatcat:vemxgsr6i5bgzjyr6easym3j44

A DG guide to Voevodsky's motives [article]

A. Beilinson, V. Vologodsky
2008 arXiv   pre-print
We give a concise exposition of Voevodsky's theory of motives.  ...  Re- placing A by A pretr does not change A − → . (ii) The embedding A ֒→ A − → does not commute with infinite direct sums.  ...  First, we define a natural quasi-isomorphism compatible with the base change. It suffices to define (6.3.8) locally with respect to Z (in a way compatible with the base change).  ... 
arXiv:math/0604004v5 fatcat:qkqlxsjfanh3fjrjfbhytstoie

Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period [article]

Hansol Hong, Yu-Shen Lin, Jingyu Zhao
2019 arXiv   pre-print
We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic correspondence theorem for holomorphic discs.  ...  As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces which relates the log descendant Gromov-Witten invariants with the oscillatory integrals  ...  with suitable Maslov index will give 1-dimensional skeleton in the SYZ-base for X.  ... 
arXiv:1812.08845v2 fatcat:oyt35fmrvvcwljwqsf4fxgcvfa

Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory [article]

Pavel Mnev
2017 arXiv   pre-print
Lecture notes for the course "Batalin-Vilkovisky formalism and applications in topological quantum field theory" given at the University of Notre Dame in the Fall 2016 for a mathematical audience.  ...  In these lectures we give a slow introduction to the perturbative path integral for gauge theories in Batalin-Vilkovisky formalism and the associated mathematical concepts.  ...  Field ψ has local components ψ i α pxq with i the index of spanning the basis of the representation space R and α the spinor index; B { A " ř µ,α,β,i,j pγ µ q αβ pδ ij B µp T a q ij A a µ pxqq is the Dirac  ... 
arXiv:1707.08096v1 fatcat:7wamdgzltfdohlr7if7xantuc4

Generating Functionals for Spin Foam Amplitudes [article]

Jeff Hnybida
2014 arXiv   pre-print
We call this new basis the discrete-coherent basis. We focus our study on the 4-valent basis, which is the first non-trivial dimension, and is also the case of interest for Quantum Gravity.  ...  Finally we discuss the process of coarse graining moves at the level of the generating functionals and give a general prescription for arbitrary graphs.  ...  Furthermore, the Grassmannian can be embedded into complex projective space via the well known Plücker embedding Gr(2, C N ) → P(C N ∧ C N ) ∼ = CP ( N 2 )−1 (8.2) which maps 2d subspaces of C N with coordinates  ... 
arXiv:1411.2049v1 fatcat:ubvhijduyrdarhp56urhcxjidm

Geometrically Interpreting Higher Cup Products, and Application to Combinatorial Pin Structures [article]

Sri Tata
2020 arXiv   pre-print
In particular, we find that the 'quadratic refinement' property of Gaiotto-Kapustin can be derived geometrically using our vector fields and interpretation of ∪_i, together with a certain trivalent resolution  ...  We construct from a simplex and a branching structure a special frame of vector fields inside each simplex that allow us to interpret cochain-level formulas for the ∪_i as a generalized intersection product  ...  Acknowledgements We thank Maissam Barkeshli and Danny Bulmash for related discussions. And we acknowledge TA appointments and the Condensed Matter Theory Center at UMD for support.  ... 
arXiv:2008.10170v1 fatcat:46val7zi2zb3lignsbqfbclopm

Topological non-linear σ-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems [article]

Chenchang Zhu, Tian Lan, Xiao-Gang Wen
2018 arXiv   pre-print
A discrete non-linear σ-model is obtained by triangulate both the space-time M^d+1 and the target space K.  ...  Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear σ-models with π_1(K)= finite groups, π_2(K)=Z_2, and π_n>2(K)=0.  ...  The rank-25 tensorΩ 4 , as well as the weight tensors w 0 , w 1 , and w 2 , must satisfy certain conditions in order for the above path integral to be re-triangulation invariant.  ... 
arXiv:1808.09394v1 fatcat:bogorb5ktfhmlej2yhrrilywgi

A Simple Introduction to Particle Physics Part II [article]

Matthew B. Robinson, Tibra Ali, Gerald B. Cleaver
2009 arXiv   pre-print
In this paper (and the paper to follow), we continue our emphasis on gauge theories, but we do so with a more geometrical approach.  ...  We will conclude this paper with a brief discussion of general relativity, and save more advanced topics (including fibre bundles, characteristic classes, etc.) for the next paper in the series.  ...  Simply put, a tensor with n upper indices and m lower indices is called a "tensor of rank (n, m)". A tensor of rank (n, m) is an object that maps a tensor of rank (m, n) to R.  ... 
arXiv:0908.1395v1 fatcat:hq7msxxbzzhvxdbnqtkurirs4q