We introduce a variant of stable logarithmic maps, which we call punctured logarithmic maps. They allow an extension of logarithmic Gromov?Witten theory in which marked points have a negative order of tangency with boundary divisors.
As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with tropical geometry providing the underlying combinatorics.
Punctured Gromov?Witten invariants also play a pivotal role in the intrinsic construction of mirror partners by the last two authors, conjecturally relating to symplectic cohomology, and in the logarithmic gauged linear sigma model in work of Qile Chen, Felix Janda and Yongbin Ruan.
Frontmatter pp. i?iv
Abstract p. v
Contents pp. vii?viii 1
Introduction
pp. 1-9 2 Punctured maps
pp. 11-59 3 The stack of punctured maps
pp. 61-88 4 The perfect obstruction theory
pp. 89-1015 Splitting and gluing
pp. 103-132 A Contact orders
pp. 133-139 B Charts for morphisms of log stacks
pp. 141-144 C Functorial tropicalization and the category of points
pp. 145-152 References
pp. 153-156
We study the construction of the Φ 33
?
-measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer [J. Statist. Phys. 50 (1988), no. 3-4, 657?687]. This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the Φ 33
?
-measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the Φ 33
?
-measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors, we establish non-normalizability of the Φ 33
?
-measure. Due to the singularity of the Φ 33
?
-measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, we first construct a σ-finite version of the Φ 33
?
-measure and show that this measure is not normalizable. Furthermore, we prove that the truncated Φ 33
?
-measures have no weak limit in a natural space, even up to a subsequence.
We also study the dynamical problem for the canonical stochastic quantization of the Φ 33
?
-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic Φ 33
?
-model). By adapting the paracontrolled approach, in particular from the works by Gubinelli, Koch, and the first author [J. Eur. Math. Soc. 26 (2024), no. 3, 817?874] and by the authors [Mem. Amer. Math. Soc. 304 (2024), no. 1529], we prove almost sure global well-posedness of the hyperbolic Φ 33
?
-model and invariance of the Gibbs measure in the weakly nonlinear regime. In the globalization part, we introduce a new, conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the Boue?Dupuis variational formula and ideas from theory of optimal transport.
*
The Seiberg-Witten Floer spectrum is a stable homotopy refinement of the
monopole Floer homology of Kronheimer and Mrowka. The Seiberg?Witten Floer
spectrum was defined by Manolescu for closed, spin
c
3-manifolds with b
1
?
=0 in an S
1
-equivariant stable homotopy category and has been producing interesting topological applications. Lidman and Manolescu showed that the S
1
-equivariant homology of the spectrum is isomorphic to the monopole Floer homology.
For closed spin
c
3-manifolds Y with b
1
?
(Y)>0, there are analytic and homotopy-theoretic difficulties in defining the Seiberg?Witten Floer spectrum. In this memoir, we address the difficulties and construct the Seiberg?Witten Floer spectrum for Y, provided that the first Chern class of the spin
c
structure is torsion and that the triple-cup product on H
1
(Y;Z) vanishes. We conjecture that its S
1
-equivariant homology is isomorphic to the monopole Floer homology.
For a 4-dimensional spin
c
cobordism X between Y
0
?
and Y
1
?
, we define the Bauer?Furuta map on these new spectra of Y
0
?
and Y
1
?
, which is conjecturally a refinement of the relative Seiberg?Witten invariant of X. As an application, for a compact spin 4-manifold X with boundary Y, we prove a
8
10
?
-type inequality for X which is written in terms of the intersection form of X and an invariant κ(Y) of Y.
In addition, we compute the Seiberg?Witten Floer spectrum for some 3-manifolds.
Paperback
This book originated from the lecture notes of the online course “String Theory, Field Theory, and Holographic Theory” held at Southeast University in China in 2021. These lectures, given by influential young scholars, cover topics such as scattering amplitudes, supergravity, AdS/CFT and integrability, holographic theories beyond AdS/CFT, as well as the intersection of cosmology and particle physics in frontier research areas.
Preface
Preview
Lectures on AdS/CFT Correspondence and Integrability
Ryo Suzuki
pp. 1-62
Lectures on Non-perturbative Entanglement Properties in Quantum Field Theory
Yang Zhou
pp. 63-81
From Feynman Diagrams to On-shell Methods: A Modern Perspective of Quantum Field Theory
Gang Yang
pp. 82-118
An Introduction to Supergravity
Yi Pang
pp. 119-155
Topics on Particle Physics in the Early Universe
Zhongzhi Xianyu
pp. 156-189
Pages 198
Publish Date ISBN-13 Medium Binding Size Publish Status
2024-11-11 9781571464460 Print Paperback 7” x 10” In Print