71 research outputs found
Invertible QFTs and differential Anderson duals
This is the proceeding of a talk given at Stringmath 2022. We introduce a
Cheeger-Simons type model for the differential extension of Anderson dual to
generalized homology theory with physical interpretations. This construction
generalizes the construction of the differential Anderson dual to bordism
homology theories, given in a previous work of Yonekura and the author.Comment: 18 page
ALGEBRAIC TOPOLOGY AND PHYSICS (Women in Mathematics)
Recently, there has been a growing interest in the relations between algebraic topology and physics. Algebraic topology is used to classify physical systems, and it can be a very powerful tool to analyze physical problems in purely mathematical ways. In this talk, I explain this idea and some of my related works
Topological modular forms and the absence of all heterotic global anomalies
We reformulate the question of the absence of global anomalies of heterotic
string theory mathematically in terms of a certain natural transformation
,
from topological modular forms to the Anderson dual of string bordism groups,
using the Segal-Stolz-Teichner conjecture. We will show that this natural
transformation vanishes, implying that heterotic global anomalies are always
absent. The fact that plays an important
role in the process. Along the way, we also discuss how the twists of
can be described under the Segal-Stolz-Teichner conjecture, by
using the result of Freed and Hopkins concerning anomalies of quantum field
theories.
The paper contains separate introductions for mathematicians and for string
theorists, in the hope of making the content more accessible to a larger
audience. The sections are also demarcated cleanly into mathematically rigorous
parts and those which are not.Comment: 36 pages; v2: incorporates many suggestions by a helpful anonymous
refere
Remarks on mod-2 elliptic genus
For physicists: For supersymmetric quantum mechanics, there are cases when a
mod-2 Witten index can be defined, even when a more ordinary
-valued Witten index vanishes. Similarly, for 2d supersymmetric
quantum field theories, there are cases when a mod-2 elliptic genus can be
defined, even when a more ordinary elliptic genus vanishes. We study such mod-2
elliptic genera in the context of supersymmetry, and show
that they are characterized by mod-2 reductions of integral modular forms,
under some assumptions.
For mathematicians: We study the image of the standard homomorphism for or
, by relating them to the mod-2 reductions of integral modular forms.Comment: 31 page
Spectral convergence in geometric quantization --- the case of non-singular Langrangian fibrations
We develop a new approach to geometric quantization using the theory of
convergence of metric measure spaces. Given a family of K\"ahler polarizations
converging to a non-singular real polarization on a prequantized symplectic
manifold, we show the spectral convergence result of
-Laplacians, as well as the convergence result of quantum
Hilbert spaces. We also consider the case of almost K\"ahler quantization for
compatible almost complex structures, and show the analogous convergence
results
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