Arc geometry and algebra: foliations, moduli spaces, string topology and field theory

2014 ◽  
pp. 255-325 ◽  
Author(s):  
Ralph Kaufmann
Keyword(s):  
Field Theory ◽  
Moduli Spaces ◽  
String Topology ◽  
And Algebra

scholarly journals Toward AGT for Parabolic Sheaves

2020 ◽  
Author(s):  
Andrei Neguţ
Keyword(s):  
Field Theory ◽  
Gauge Theory ◽  
Conformal Field Theory ◽  
Moduli Spaces ◽  
Type A ◽  
Conformal Field ◽  
Toroidal Algebra ◽  
Agt Correspondence ◽  
K Theory ◽  
The Ideal

Abstract We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$ and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the Alday-Gaiotto-Tachikawa (AGT) correspondence between gauge theory with surface operators and conformal field theory.

scholarly journals Instanton Moduli Spaces and Bases in Coset Conformal Field Theory

2012 ◽  
Vol 319 (1) ◽  
pp. 269-301 ◽  
Author(s):  
A. A. Belavin ◽  
M. A. Bershtein ◽  
B. L. Feigin ◽  
A. V. Litvinov ◽  
G. M. Tarnopolsky
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Moduli Spaces ◽  
Conformal Field

scholarly journals The Batalin–Vilkovisky Algebra in the String Topology of Classifying Spaces

2019 ◽  
Vol 71 (4) ◽  
pp. 843-889
Author(s):  
Katsuhiko Kuribayashi ◽  
Luc Menichi
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Group ◽  
Hochschild Cohomology ◽  
General Theorem ◽  
String Topology ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Conformal Field ◽  
Connected Lie Group

AbstractFor almost any compact connected Lie group$G$and any field$\mathbb{F}_{p}$, we compute the Batalin–Vilkovisky algebra$H^{\star +\text{dim}\,G}(\text{LBG};\mathbb{F}_{p})$on the loop cohomology of the classifying space introduced by Chataur and the second author. In particular, if$p$is odd or$p=0$, this Batalin–Vilkovisky algebra is isomorphic to the Hochschild cohomology$HH^{\star }(H_{\star }(G),H_{\star }(G))$. Over$\mathbb{F}_{2}$, such an isomorphism of Batalin–Vilkovisky algebras does not hold when$G=\text{SO}(3)$or$G=G_{2}$. Our elaborate considerations on the signs in string topology of the classifying spaces give rise to a general theorem on graded homological conformal field theory.

scholarly journals Topological Field Theory Interpretation of String Topology

2003 ◽  
Vol 240 (3) ◽  
pp. 397-421 ◽  
Author(s):  
Alberto S. Cattaneo ◽  
Jürg Fröhlich ◽  
Bill Pedrini
Keyword(s):  
Field Theory ◽  
Topological Field Theory ◽  
String Topology ◽  
Topological Field ◽  
Theory Interpretation

scholarly journals Fundamental Theorems of Pure Mathematics & Absolute Geometry

2021 ◽  
Vol 4 (2) ◽  
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Fundamental Theorem ◽  
Absolute Geometry ◽  
Pure Mathematics ◽  
And Algebra ◽  
Fundamental Theorems

The superunified field theory consists of a row of discoveries in the realm of pure mathematics. It is two centuries ago that Karl Gauss unified higher arithmetic (number theory), algebra and geometry into what is called pure mathematics. The latter, however, still remains without its fundamental theorem despite that arithmetic and algebra, or even analysis, have their own.

scholarly journals A polyfold proof of the Arnold conjecture

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Benjamin Filippenko ◽  
Katrin Wehrheim
Keyword(s):  
Field Theory ◽  
Moduli Spaces ◽  
Symplectic Manifolds ◽  
Joint Work ◽  
Pseudoholomorphic Curves ◽  
Countable Collection ◽  
Arnold Conjecture ◽  
Detailed Proof ◽  
Symplectic Field Theory

AbstractWe give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from $${{\mathbb {C}}{\mathbb {P}}}^1\times M$$ C P 1 × M to $${{\mathbb {C}}}^+ \times M$$ C + × M and $${{\mathbb {C}}}^-\times M$$ C - × M , as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

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