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We examine the color-kinematics duality within the BV formalism, highlighting its emergence as a feature of specific gauge-fixed actions. Our goal is to establish a general framework for studying the duality while investigating straightforward examples of off-shell color-kinematics duality. In this context, we revisit Chern-Simons theory as well as introduce new examples, including BF theory and 2D Yang-Mills theory, which are shown to exhibit the duality off-shell. We emphasize that the geometric structures responsible for flat-space color-kinematics duality appear for general curved spaces as well.

This paper introduces a concrete relation between genus zero closed Gromov-Witten invariants of Calabi-Yau threefolds and genus zero open Gromov-Witten invariants of a Lagrangian A-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold (X,ω) with a Hamiltonian U(1) action into two pieces glued along an invariant divisor. In this paper we study a quantum uplift of the cut construction defined in terms of equivariant gauged linear sigma models. The nexus between closed and open Gromov-Witten invariants is a quantum Lebesgue measure associated to a choice of cut, that we introduce and study. Integration of this measure recovers the equivariant quantum volume of the whole CY3, thereby encoding closed Gromov-Witten invariants. Conversely, the monodromies of the quantum measure around cycles in Kähler moduli space encode open Gromov-Witten invariants of a Lagrangian A-brane associated to the cut. Both in the closed and the open string sector we find a remarkable interplay between worldsheet instantons and semiclassical volumes regularized by equivariance. This leads to equivariant generating functions of GW invariants that extend smoothly across the entire moduli space, and which provide a unifying description of standard GW potentials. The latter are recovered in the non-equivariant limit in each of the different phases of the geometry.

In this work, we introduce an equivariant deformation of the B model on the sphere with a U(1)-action. We present the deformed supersymmetry transformations and corresponding Lagrangians and study observables in the supercharge cohomology. The inclusion of equivariance allows for the introduction of novel, position-dependent observables on the sphere, which have no counterparts in the conventional B model. Two specific cases we explore in detail are position-dependent superpotentials and complex structure deformations. In both instances, the theory exhibits notable differences from the standard B model, revealing intriguing new features.

We propose formulas for the large N expansion of the generating function of connected correlators of the β-deformed Gaussian and Wishart–Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of β with 1 / β and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit β = 1, these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic β, we define the higher genus Catalan polynomials C_g,v(β) whose coefficients are integer numbers. 

We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d sudefine these objects and study their dependence on equivariant parameters for non-compact toric Ka<spacing diaeresis>hler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov-Witten invariants of the target.

We study four dimensional gauge theories in the context of an equivariant extension of the Batalin-Vilkovisky (BV) formalism. We discuss the embedding of BV Yang-Mills (YM) theory into a larger BV theory and their relation. Partial integration in the equivariant BV setting (BV push-forward map) is performed explicitly for the abelian case. As result, we obtain a non-local homological generalization of the Cartan calculus and a non-local extension of the abelian YM BV action which satisfies the equivariant master equation.

We present a systematic study of N = (2, 2) supersymmetric non-linear sigma models on S-2 with the target being a Kahler manifold. We discuss their reformulation in terms of cohomological field theory. In the cohomological formulation we use a novel version of 2D self-duality which involves a U(1) action on S-2. In addition to the generic model we discuss the theory with target space equivariance corresponding to a supersymmetric sigmamodel coupled to a non-dynamical supersymmetric background gauge multiplet. We discuss the localization locus and perform a one-loop calculation around the constant maps. We argue that the theory can be reduced to some exotic model over the moduli space of holomorphic disks.

We solve the problem of determining the fundamental degrees of freedom underlying a generalized Kahler structure of symplectic type. For a usual Kahler structure, it is well-known that the geometry is determined by a complex structure, a Kahler class, and the choice of a positive (1, 1)-form in this class, which depends locally on only a single real-valued function: the Kahler potential. Such a description for generalized Kahler geometry has been sought since it was discovered in 1984. We show that a generalized Kahler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence between them, and the choice of a positive Lagrangian brane bisection, which depends locally on only a single real-valued function, which we call the generalized Kahler potential. Our solution draws upon, and specializes to, the many results in the physics literature which solve the problem under the assumption (which we do not make) that the Poisson structures involved have constant rank. To solve the problem we make use of, and generalize, two main tools: the first is the notion of symplectic Morita equivalence, developed by Weinstein and Xu to study Poisson manifolds; the second is Donaldson's interpretation of a Kahler metric as a real Lagrangian submanifold in a deformation of the holomorphic cotangent bundle.

We consider the matrix model of U(N) refined Chern-Simons theory on S-3 for the unknot. We derive a q-difference operator whose insertion in the matrix integral reproduces an infinite set of Ward identities which we interpret as q-Virasoro constraints. The constraints are rewritten as difference equations for the generating function of Wilson loop expectation values which we solve as a recursion for the correlators of the model. The solution is repackaged in the form of superintegrability formulas for Macdonald polynomials. Additionally, we derive an equivalent q-difference operator for a similar refinement of ABJ theory and show that the corresponding q-Virasoro constraints are equal to those of refined Chern-Simons for a gauge super-group U(N vertical bar M). Our equations and solutions are manifestly symmetric under Langlands duality q <-> t(-1) which correctly reproduces 3d Seiberg duality when q is a specific root of unity.

Motivated by M-theory, we study rank n K-theoretic Donaldson-Thomas theory on a toric threefold X. In the presence of compact four-cycles, we discuss how to include the contribution of D4-branes wrapping them. Combining this with a simple assumption on the (in)dependence on Coulomb moduli in the 7d theory, we show that the partition function factorizes and, when X is Calabi-Yau and it admits an ADE ruling, it reproduces the 5d master formula for the geometrically engineered theory on A(n-1) ALE space, thus extending the usual geometric engineering dictionary to n > 1. We finally speculate about implications for instanton counting on Taub-NUT.