scholarly journals DUBROVIN’S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE

2017 ◽  
Vol 18 (3) ◽  
pp. 449-497 ◽  
Author(s):  
P. Dunin-Barkowski ◽  
P. Norbury ◽  
N. Orantin ◽  
A. Popolitov ◽  
S. Shadrin
Keyword(s):  
Field Theory ◽  
Spectral Curve ◽  
Frobenius Manifold ◽  
Simple Point ◽  
Topological Recursion ◽  
Cohomological Field Theory

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

scholarly journals Noncommutative cohomological field theory and GMS soliton

2002 ◽  
Vol 43 (2) ◽  
pp. 872-896 ◽  
Author(s):  
Akifumi Sako ◽  
Shin-Ichiro Kuroki ◽  
Tomomi Ishikawa
Keyword(s):  
Field Theory ◽  
Cohomological Field Theory

Random matrices and loop equations

2018 ◽  
Author(s):  
Bertrand Eynard
Keyword(s):  
Random Matrix ◽  
Matrix Theory ◽  
Spectral Curve ◽  
Coulomb Gas ◽  
Algebraic Equations ◽  
Saddle Point Approximation ◽  
Large N ◽  
Topological Recursion ◽  
Random Matrix Ensembles ◽  
Coulomb Gas Method

This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.

scholarly journals Towards formalization of the soliton counting technique for the Khovanov–Rozansky invariants in the deformed ℛ-matrix approach

2018 ◽  
Vol 33 (36) ◽  
pp. 1850221
Author(s):  
A. Anokhina
Keyword(s):  
Field Theory ◽  
Intermediate Stage ◽  
Diagram Technique ◽  
Mathematical Treatment ◽  
Matrix Approach ◽  
Detailed Review ◽  
Counting Technique ◽  
Cohomological Field Theory

We consider recently developed Cohomological Field Theory (CohFT) soliton counting diagram technique for Khovanov (Kh) and Khovanov–Rozansky (KhR) invariants.[Formula: see text] Although, the expectation to obtain a new way for computing the invariants has not yet come true, we demonstrate that soliton counting technique can be totally formalized at an intermediate stage, at least in particular cases. We present the corresponding algorithm, based on the approach involving deformed [Formula: see text]-matrix and minimal positive division, developed previously in Ref. 3. We start from a detailed review of the minimal positive division approach, comparing it with other methods, including the rigorous mathematical treatment.4 Pieces of data obtained within our approach are presented in the Appendices.

scholarly journals Matrix factorizations and cohomological field theories

2016 ◽  
Vol 2016 (714) ◽  
pp. 1-122 ◽  
Author(s):  
Alexander Polishchuk ◽  
Arkady Vaintrob
Keyword(s):  
Field Theory ◽  
Field Theories ◽  
Matrix Factorizations ◽  
Hypersurface Singularity ◽  
Algebraic Construction ◽  
Cohomological Field Theories ◽  
Cohomological Field Theory

AbstractWe give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity

scholarly journals Invariants of four-manifolds with flows via cohomological field theory

2014 ◽  
Author(s):  
Hugo Garcia-Compeân ◽  
Roberto Santos-Silva ◽  
Alberto Verjovsky
Keyword(s):  
Field Theory ◽  
Probability Measure ◽  
Vector Fields ◽  
Invariant Probability Measure ◽  
Yang Mills ◽  
Higher Dimensional ◽  
Coupling Dynamics ◽  
Four Manifolds ◽  
Cohomological Field Theory ◽  
Asymptotic Cycles

This chapter argues that the Jones–Witten invariants can be generalized for smooth, nonsingular vector fields with invariant probability measure on three-manifolds, thus giving rise to new invariants of dynamical systems. After a short survey of cohomological field theory for Yang–Mills fields, Donaldson–Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. The chapter studies the case of Kähler manifolds by using the Witten's consideration of the strong coupling dynamics of N = 1 supersymmetric Yang–Mills theories. The whole construction is performed by implementing the notion of higher-dimensional asymptotic cycles. In the process Seiberg–Witten invariants are also described within this context. Finally, the chapter gives an interpretation of the asymptotic observables of four-manifolds in the context of string theory with flows.

scholarly journals Gromov–Witten theory and cycle-valued modular forms

2018 ◽  
Vol 2018 (735) ◽  
pp. 287-315 ◽  
Author(s):  
Todor Milanov ◽  
Yongbin Ruan ◽  
Yefeng Shen
Keyword(s):  
Field Theory ◽  
Modular Forms ◽  
Generating Functions ◽  
Mirror Symmetry ◽  
Complex Structure ◽  
Field Theories ◽  
Witten Theory ◽  
Cohomological Field Theories ◽  
Cohomological Field Theory ◽  
Projective Lines

AbstractIn this paper, we review Teleman’s work on lifting Givental’s quantization of{\mathcal{L}^{(2)}_{+}{\rm GL}(H)}action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight(3,3,3),(2,4,4),(2,3,6). Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

scholarly journals COHOMOLOGICAL FIELD THEORY APPROACH TO MATRIX STRINGS

1999 ◽  
Vol 14 (25) ◽  
pp. 3979-4002 ◽  
Author(s):  
FUMIHIKO SUGINO
Keyword(s):  
Field Theory ◽  
String Theory ◽  
Partition Function ◽  
Three Dimensional ◽  
Exact Result ◽  
Theory Approach ◽  
Two Dimensional ◽  
Yang Mills ◽  
Infrared Limit ◽  
Cohomological Field Theory

In this paper we consider IIA and IIB matrix string theories which are defined by two-dimensional and three-dimensional super Yang–Mills theory with the maximal supersymmetry, respectively. We exactly compute the partition function of both of the theories by mapping to a cohomological field theory. Our result for the IIA matrix string theory coincides with the result obtained in the infrared limit by Kostov and Vanhove, and thus gives a proof of the exact quasiclassics conjectured by them. Further, our result for the IIB matrix string theory coincides with the exact result of IKKT model by Moore, Nekrasov and Shatashvili. It may be an evidence of the equivalence between the two distinct IIB matrix models arising from different roots.

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