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

2014 ◽  
pp. 255-325 ◽  
Author(s):  
Ralph Kaufmann
Author(s):  
Andrei Neguţ

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.


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

2019 ◽  
Vol 71 (4) ◽  
pp. 843-889
Author(s):  
Katsuhiko Kuribayashi ◽  
Luc Menichi

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.


2003 ◽  
Vol 240 (3) ◽  
pp. 397-421 ◽  
Author(s):  
Alberto S. Cattaneo ◽  
Jürg Fröhlich ◽  
Bill Pedrini

2021 ◽  
Vol 4 (2) ◽  

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.


2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Benjamin Filippenko ◽  
Katrin Wehrheim

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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