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In Gérard et al. (Ann Sci Ecole Norm Sup 56:127–196, 2023), the Unruh state for massless fermions on a Kerr spacetime was constructed and the authors showed its Hadmard property in the case of very slowly rotating black holes a ≪ M . In this note, we extend this result to the full non extreme case a < M . [ABSTRACT FROM AUTHOR]

We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2. [ABSTRACT FROM AUTHOR]

We describe a framework for random pairwise comparisons matrices, inspired by selected constructions related to the so called inconsistency reduction of pairwise comparisons (PC) matrices. In order to build up structures on random pairwise comparisons matrices, the set up for (deterministic) PC matrices for non-reciprocal PC matrices is completed. Basic concepts such as inconsistency indices and geometric mean method are extended to random pairwise comparisons matrices and completed by new notions which seem useful to us. Two procedures for (random) inconsistency reduction are sketched, based on well-known existing objects, and a fiber bundle-like decomposition of random pairwise comparisons is proposed. [ABSTRACT FROM AUTHOR]

We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space S is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to S and thus define a quotient space of "gauge orbits" O . We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on O (up to a potential finite time blow-up) associated to the stochastic YMH flow. [ABSTRACT FROM AUTHOR]

We identify Whittaker vectors for W k (g) -modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G-bundles over P 2 for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure N = 2 four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods. [ABSTRACT FROM AUTHOR]

This paper constructs in the framework of algebraic quantum field theory (AQFT) the linear Chern–Simons/Wess–Zumino–Witten system on a class of 3-manifolds M whose boundary ∂ M is endowed with a Lorentzian metric. It is proven that this AQFT is equivalent to a dimensionally reduced AQFT on a 2-dimensional manifold B, whose restriction to the 1-dimensional boundary ∂ B is weakly equivalent to a chiral free boson. [ABSTRACT FROM AUTHOR]

We introduce the topologically twisted index for four-dimensional N = 1 gauge theories quantized on AdS 2 × S 1 . We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on AdS 2 × T 2 , with and without a boundary: in both instances we classify normalizability and boundary conditions for gauge, matter and ghost fields. The index is twisted as the dynamical fields are coupled to a R-symmetry background 1-form with non-trivial exterior derivative and proportional to the spin connection. After regularization, the index is written in terms of elliptic gamma functions, reminiscent of four-dimensional holomorphic blocks, and crucially depends on the R-charge. [ABSTRACT FROM AUTHOR]

We describe the topological A and B twists of 3d N = 4 theories of hypermultiplets gauged by N = 4 vector multiplets as certain deformations of the holomorphic–topological (HT) twist of those theories, utilizing the twisted superfields of Aganagic–Costello–Vafa–McNamara describing HT-twisted 3d N = 2 theories. We rederive many known results from this perspective, including state spaces on Riemann surfaces, deformations induced by flavor symmetries, the boundary VOAs of Costello–Gaiotto, and the category of line operators as proposed by Costello–Dimofte–Gaiotto–Hilburn–Yoo. Along the way, we show how the secondary product of local operators in the holomorphic–topological twist is related to the secondary product in the fully topological twist. [ABSTRACT FROM AUTHOR]

We consider sequences of solutions (ψ n , A n) n = 1 ∞ to Taubes's modified Seiberg–Witten equations, associated with a fixed volume-preserving vector field X on a 3-manifold and corresponding to arbitrarily large values of the strength parameter r n → ∞ . In Taubes's work, the asymptotic behavior of these solutions is related to the dynamics of X. We consider the rather unexplored case of sequences of solutions whose energy is not uniformly bounded as n → ∞ . Our first main result shows that when the energy grows more slowly than r n 1 / 2 , the limiting nodal set of the solutions converges to an invariant set of the vector field X. The main tool we use is a novel maximum principle for the solutions with the key property that it remains valid in the unbounded energy case. As a byproduct, in the usual case of sequences of solutions with bounded energy, we obtain a new, more straightforward proof of Taubes's result on the existence of periodic orbits that does not involve a local analysis or the vortex equations. Our second main result proves that, contrary to what happens in the bounded energy case, when the energy is unbounded there are no local restrictions to the limiting measures that may arise in the modified Seiberg–Witten equations. Furthermore, we obtain a connection between the dimension of the support of the limiting measure (as expressed through a d-Frostman property) and the energy growth of the sequence of local solutions we construct. [ABSTRACT FROM AUTHOR]

The existence of interacting higher-spin theories is tightly constrained by many no-go theorems. In this paper, we construct a chiral, higher-spin generalization of Yang–Mills theory in flat space which avoids these no-go theorems and has non-trivial tree-level scattering amplitudes with some higher-spin external legs. The fields and action are complex, so the theory is non-unitary and parity-violating, yet we find surprisingly compact formulae for all-multiplicity tree-level scattering amplitudes in the maximal helicity violating (MHV) sector, where the two negative helicity particles have identical but arbitrary spin. This is possible because the theory admits a perturbative expansion around its self-dual sector. Using twistor theory, we prove the classical integrability of this self-dual sector and show that it can be described on spacetime by an infinite tower of interacting massless scalar fields. We also give a twistor construction of the full theory and use it to derive the formula for the MHV amplitude. [ABSTRACT FROM AUTHOR]

We construct local solutions to the Yang–Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato–Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic arguments. In a companion work, we use the main results of this paper to propose a way toward the construction of 3D Yang–Mills measures. [ABSTRACT FROM AUTHOR]

In this paper we investigate a novel connection between the effective theory of M2-branes on (C 2 / Z 2 × C 2 / Z 2) / Z k and the q -deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver N = 4 Chern–Simons matter theory solves the q -Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local dP 5 and, via geometric engineering, five dimensional N f = 4 SU (2) N = 1 gauge theory on a circle. The results we find extend the known relation between ABJM theory, q -Painlevé III 3 , and topological strings on local P 1 × P 1 . From the mathematical viewpoint the quiver Chern–Simons theory provides a conjectural Fredholm determinant realisation of the q -Painlevé VI τ -function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to N f = 0 , corresponding to q -Painlevé III 3 . [ABSTRACT FROM AUTHOR]

In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$ -bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgs k-functional with Higgs self-interaction, we show that, provided $\dim (M) , for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local $L^2$ -derivative estimates, energy estimates and blow-up analysis. [ABSTRACT FROM AUTHOR]

We give a complete classification of twists of supersymmetric Yang–Mills theories in dimensions 2 ≤ n ≤ 10 . We formulate supersymmetric Yang–Mills theory classically using the BV formalism, and then we construct an action of the supersymmetry algebra using the language of L ∞ algebras. For each orbit in the space of square-zero supercharges in the supersymmetry algebra, under the action of the spin group and the group of R-symmetries, we give a description of the corresponding twisted theory. These twists can be described in terms of mixed holomorphic-topological versions of Chern–Simons and BF theory. [ABSTRACT FROM AUTHOR]

We study linearization of lattice gauge theory. Linearized theory approximates lattice gauge theory in the same manner as the loop O(n)-model approximates the spin O(n)-model. Under mild assumptions, we show that the expectation of an observable in linearized Abelian gauge theory coincides with the expectation in the Ising model with random edge-weights. We find a similar relation between Yang-Mills theory and 4-state Potts model. For the latter, we introduce a new observable. [ABSTRACT FROM AUTHOR]

We propose an effective framework for computing the prepotential of the topological B-model on a class of local Calabi–Yau geometries related to the circle compactification of five-dimensional N = 1 super Yang–Mills theory with simple gauge group. In the simply laced case, we construct Picard–Fuchs operators from the Dubrovin connection on the Frobenius manifolds associated with the extended affine Weyl groups of type ADE . In general, we propose a purely algebraic construction of Picard–Fuchs ideals from a canonical subring of the space of regular functions on the ramification locus of the Seiberg–Witten curve, encompassing non-simply laced cases as well. We offer several precision tests of our proposal for simply laced cases by comparing with the gauge theory prepotentials obtained from the K-theoretic blow-up equations, finding perfect agreement. Whenever there is more than one candidate Seiberg-Witten curve for non-simply laced gauge groups in the literature, we employ our framework to verify which one agrees with the K-theoretic blow-up equations. [ABSTRACT FROM AUTHOR]

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras. [ABSTRACT FROM AUTHOR]

We study the regular surface defect in the Ω -deformed four-dimensional supersymmetric gauge theory with gauge group SU(N) with 2N hypermultiplets in fundamental representation. We prove its vacuum expectation value obeys the Knizhnik–Zamolodchikov equation for the 4-point conformal block of the sl ^ N -current algebra, originally introduced in the context of two-dimensional conformal field theory. The level and the vertex operators are determined by the parameters of the Ω -background and the masses of the hypermultiplets; the cross-ratio of the 4 points is determined by the complexified gauge coupling. We clarify that in a somewhat subtle way the branching rule is parametrized by the Coulomb moduli. This is an example of the BPS/CFT relation. [ABSTRACT FROM AUTHOR]

We show a mathematical relation between the mod-two Atiyah–Patodi–Singer (APS) index of a massless Dirac operator and massive domain-wall fermion determinant. The domain-wall fermion is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher-dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow. [ABSTRACT FROM AUTHOR]

We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$. Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving 'almost flat' Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B400(1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993), Vol. 328 of NATO Advanced Science Institutes Series B: Physics, Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B303(1) 95–98). [ABSTRACT FROM AUTHOR]

We apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of ‘braided L ∞ -algebras’. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern–Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus. [ABSTRACT FROM AUTHOR]

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n - 3 -forms on K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n - 3 -planes in K n introduced by ABHY. [ABSTRACT FROM AUTHOR]

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds. [ABSTRACT FROM AUTHOR]

The five integrable vortex equations, recently studied by Manton, are generalized to include magnetic impurities of the Tong–Wong type. Under certain conditions, these generalizations remain integrable. We further set up a gauge theory with a product gauge group, two complex scalar fields and a general charge matrix. The second species of vortices, when frozen, are interpreted as the magnetic impurity for all five vortex equations. We then give a geometric compatibility condition, which enables us to remove the constant term in all the equations. This is similar to the reduction from the Taubes equation to the Liouville equation. We further find a family of charge matrices that turn the five vortex equations into either the Toda equation or the Toda equation with the opposite sign. We find exact analytic solutions in all cases and the solution with the opposite sign appears to be new. [ABSTRACT FROM AUTHOR]

We observe, utilize dualities in differential equations and differential inequalities (see Theorem 2.1), dualities between comparison theorems in differential equations (see Theorems E and 2.2), and obtain dualities in 'swapping' comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II (see Theorems 2.3 and 2.4) and lead to comparison theorems in Riemannian geometry (see Theorems 2.5 and 2.8) with analytic, geometric, PDE's and physical applications. In particular, we prove Hessian comparison theorems (see Theorems 3.1–3.5) and Laplacian comparison theorems (see Theorems 2.6, 2.7 and 3.1–3.5) under varied radial Ricci curvature, radial curvature, Ricci curvature and sectional curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei (2014) and Wei (2016). We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship (see Theorem 5.4). These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving the duality theorem and the unity theorem, generalizing the work of Andreotti and Vesentini (1965) and Wei (2020). We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds, the embedding theorem for weighted Sobolev spaces of functions on manifolds, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles, such as F-Yang-Mills fields (when F is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for F - harmonic maps (when F (t) = 1 p (2 t) p 2 , p > 1 , they become p-harmonic maps or harmonic maps if p = 2), and Dirichlet problems on starlike domains for vector bundle valued differential 1-forms and {tF}-harmonic maps (see Theorems 4.1, 7.3–7.7, 8.1, 9.1–9.3, 10.1, 11.2, 12.1 and 12.2), generalizing the work of Caffarelli et al. (1984) and Costa (2008), in which M = ℝn and its radial curvature K(r) = 0, the work of Wei and Li (2009), Chen et al. (2011, 2014), Dong and Wei (2011), Wei (2020) and Karcher and Wood (1984), etc. The boundary value problem for bundle-valued differential 1-forms is in contrast to the Dirichlet problem for p-harmonic maps to which the solution is due to Hamilton (1975) for the case p = 2 and RiemN ⩽ 0, and Wei (1998) for 1 < p < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of nontopological solutions to the self-dual Einstein-Maxwell-Higgs equations on a compact surface when the parameter ε > 0 is small enough and the total string number N is bigger than two. This improves the previously known results by removing any extra conditions on N. Based on degree theory and blowup analysis, we introduce a new method to obtain nontopological solutions. [ABSTRACT FROM AUTHOR]