The Eynard–Orantin recursion for the total ancestor potential

In this paper, we review Teleman’s work on lifting Givental’s quantization of ℒ + ( 2 ) ⁢ GL ⁢ ( H ) {\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 ) (3,3,3) , ( 2 , 4 , 4 ) (2,4,4) , ( 2 , 3 , 6 ) (2,3,6) . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Gu\'er\'e and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case $n=1$ and arbitrary $g$ using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hern\'andez Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$.Comment: v3: final journal version, 44 pages

We return in this chapter to the classical singular, Alexander-Spanier, de Rham, and Čech cohomology theories. It is shown that under suitable restrictions, these theories are equivalent to sheaf-theoretic cohomology. Homomorphisms induced by maps, cup products, and relative cohomology are also discussed at some length. In Section 3 the direct natural transformation between singular theory and de Rham theory, which is important in the applications, is considered.KeywordsExact SequenceCommutative DiagramOpen CoveringCohomology GroupNatural TransformationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the axiomatic systems of singular cohomology theory

G-Frobenius manifolds

We obtain a simple recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus $0$ and $1$ in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, restricted, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann, Manin, and Zagier.

The quantization of singular field theories, in particular, gauge theories, is one of the key problems in quantum field theory. This book - which addresses the reader acquainted with the foundations of quantum field theory - provides a comprehensive analysis of this problem and techniques for its solution. The main topics are canonical quantization and the path integral method. A description of the Lagrangian BRST quantization is included as are original results obtained by the authors: presentation of the physical spectrum of arbitrary gauge theories, quantization of singular theories with higher derivatives and with time-dependent constraints, and the treatment of relativistic point-particles.

On the structure of the topological phase of two-dimensional gravity

Quantum [formula omitted] Drinfeld–Sokolov hierarchy and quantum singularity theory

Mean field theory, topological field theory, and multi-matrix models

We have seen that the equivariant localization formalism is an excellent, conceptual geometric arena for studies of supersymmetric and topological field theories, and more generally of (quantum) integrability. Given that the Hamiltonians in an integrable hierarchy are functionals of action variables alone [106], the equivariant localization formalism might yield a geometric characterization of quantum integrability, and perhaps some deeper connection between quantum-integrable bosonic theories and supersymmetric quantum field theories. This is particularly interesting from the point of view of examining corrections to the localization formulas, which in the last Chapter we have seen reflect global properties of the theory. This would be of particular interest to analyse more closely, as it could then lead to a unified description of localization in the symplectic loop space, the supersymmetric loop space and in topological quantum field theory.KeywordsLoop SpaceEquivariant CohomologySupersymmetric Quantum MechanicLocalization FormulaEquivariant LocalizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we address the problem of classification of quasi‐homogeneous formal power series providing solutions of the oriented associativity equations. Such a classification is performed by introducing a system of monodromy local moduli on the space of formal germs of homogeneous semisimple flat ‐manifolds. This system of local moduli is well defined on the complement of the strictly doubly resonant locus, namely, a locus of formal germs of flat ‐manifolds manifesting both coalescences of canonical coordinates at the origin, and resonances of their conformal dimensions. It is shown how the solutions of the oriented associativity equations can be reconstructed from the knowledge of the monodromy local moduli via a Riemann–Hilbert–Birkhoff boundary value problem. Furthermore, standing on results of B. Malgrange and C. Sabbah, it is proved that any formal homogeneous semisimple flat ‐manifold, which is not strictly doubly resonant, is actually convergent. Our semisimplicity criterion for convergence is also reformulated in terms of solutions of Losev–Manin commutativity equations, growth estimates of correlators of ‐cohomological field theories, and solutions of open Witten–Dijkgraaf–Verlinde–Verlinde equations.

The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical [Formula: see text] filed theory whose completeness has been proved in [V. Karimipour and M. H. Zarei, Phys. Rev. A 85 (2012) 032316]. We give a new and general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical [Formula: see text] model. To this end, we start from various classical field theories containing models on arbitrary lattices and also [Formula: see text] lattice gauge theories. Then we convert them to a new classical field model on a nonplanar bipartite graph with imaginary kinetic terms. After that, we show that any polynomial function of the field in the corresponding Hamiltonian can approximately be converted to a [Formula: see text] term by adding enough numbers of vertices to the bipartite graph. In the next step, we give a few graphical transformations to convert the final nonplanar graph to a 2D rectangular lattice. We also show that the number of vertices which should be added grows polynomially with the number of vertices in the original model.

The structure of the UV divergencies in higher dimensional nonrenormalizable theories is analysed. Based on renormalization operation and renormalization group theory it is shown that even in this case the leading divergencies (asymptotics) are governed by the one-loop diagrams the number of which, however, is infinite. Explicit expression for the one-loop counter term in an arbitrary D-dimensional quantum field theory without derivatives is suggested. This allows one to sum up the leading asymptotics which are independent of the arbitrariness in subtraction of higher order operators. Diagrammatic calculations in a number of scalar models in higher loops are performed to be in agreement with the above statements. These results do not support the idea of the na\"ive power-law running of couplings in nonrenormalizable theories and fail (with one exception) to reveal any simple closed formula for the leading terms.

The construction of gauge invariant actions for arbitrary spin and bosonic string field theories