We develop a contour integral formalism for computing the K-theoretic equivariant 3-vertex. Within the Jeffrey-Kirwan (JK) residue framework, we show that, by an appropriate choice of the reference vector, both the equivariant Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) 3-vertices can be extracted from the same integrand. We analyze three distinct limits of the PT 3-vertex, recovering the unrefined topological vertex, the refined topological vertex, and the Macdonald refined topological vertex. Higher-rank extensions of PT counting and the DT/PT correspondence are also explored. From a quantum algebraic perspective, we construct an operator version of the equivariant PT 3-vertex and term it the Pandharipande-Thomas $\textit{qq}$-character. We then discuss its connection with the quantum toroidal $${\mathfrak{g}\mathfrak{l}}_{1}$$ .

France

Taro Kimura

Japan

Go Noshita

Gauge origami and quiver W-algebras. Part IV. Pandharipande-Thomas $\textit{qq}$-characters

We further develop the BPS/CFT correspondence between quiver W-algebras/ -characters and partition functions of gauge origami. We introduce -characters associated with multi-dimensional partitions with nontrivial boundary conditions which we call Donaldson-Thomas (DT) -characters. They are operator versions of the equivariant DT vertices of toric Calabi-Yau three and four-folds. Moreover, we revisit the construction of the D8 -characters with no boundary conditions and give a quantum algebraic derivation of the sign rules of the magnificent four partition function. We also show that under the proper sign rules, the D6 and D8 -characters with no boundary conditions all commute with each other and discuss its physical interpretation.

Gauge origami and quiver W-algebras. Part III. Donaldson-Thomas

We analyze the angular momentum balance for a particle undergoing Thomas precession. The relationships among relativistic torque, the center of mass, and the center of inertia for a spinning particle are clarified. We show that spin precession is accompanied by orbital angular momentum precession, and present examples of the resulting out-of-plane motion.

Canada

Andrzej Czarnecki

Andrei Zelnikov

Thomas precession, relativistic torque, and non-planar orbits

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas et al. to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a “Rosetta Stone” between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant.

United States

Tyler Grover

Kory Stiffler

Patrick Vecera

Covariant and manifestly projective invariant formulation of Thomas-Whitehead gravity

We consider the Seiberg-Witten solution of pure   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauge theory in four dimensions, with gauge group SU(N). A simple exact series expansion for the dependence of the 2(N − 1) Seiberg-Witten periods a  I (u), a  DI (u) on the N − 1 Coulomb-branch moduli u  n  is obtained around the ℤ2N -symmetric point of the Coulomb branch, where all u  n  vanish. This generalizes earlier results for N = 2 in terms of hypergeometric functions, and for N = 3 in terms of Appell functions. Using these and other analytical results, combined with numerical computations, we explore the global structure of the Kähler potential K =   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mo>∑</mo> <mi>I</mi> </msub> </math>  $$ \frac{1}{2}{\sum}_I $$  Im(  <math> <mover> <mi>a</mi> <mo>¯</mo> </mover> </math>  $$ \overline{a} $$   I  a  DI ), which is single valued on the Coulomb branch. Evidence is presented that K is a convex function, with a unique minimum at the ℤ2N -symmetric point. Finally, we explore candidate walls of marginal stability in the vicinity of this point, and their relation to the surface of vanishing Kähler potential.

Eric D’Hoker

Thomas Dumitrescu

Emily Nardoni

Exploring the strong-coupling region of SU(N) Seiberg-Witten theory

Thomas-Whitehead (TW) gravity was recently introduced as a projective gauge theory of gravity over a d-dimensional manifold that embeds reparametrization invariance into the action functional for gravitation through the use of the Thomas-Whitehead connection. The projective invariance in this d-dimensional theory enjoys an intimate relationship with the Virasoro coadjoint elements found in string theory as one of the components of the connection, <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow></math>, is directly related to the coadjoint elements of the Virasoro algebra. TW gravity exploits projective Gauss-Bonnet terms in the action functional which allows the theory to collapse to Einstein’s theory of general relativity in the limit that <math><msub><mi>D</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math> vanishes. In this paper we develop the graded extension of TW gravity, super-TW gravity, in the framework of a DeWitt supermanifold. We construct the Lagrangian for super-TW gravity, give a detailed derivation of the classical field equations, and discuss the graded extension of the projective connection as a prelude to a future understanding of TW-supergravity (which has manifest supersymmetry) and its relationship to the super-Virasoro algebra.

Calvin Mera Sánchez

Vincent G. J. Rodgers

Graded extension of Thomas-Whitehead gravity

Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure SU(N)   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauge theory in four dimensions. At this point N − 1 mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by D’Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to SU(2) and SU(3) gauge theories, the large-N results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite N , finding exact agreement with our first calculation.

Efrat Gerchkovitz

Revisiting the multi-monopole point of SU(N)   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauge theory in four dimensions

We establish the existence of an infinite-dimensional fermionic symmetry in four-dimensional supersymmetric gauge theories by analyzing semiclassical photino dynamics in abelian   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 theories with charged matter. The symmetry is parametrized by a spinor-valued function on an asymptotic S 2 at null infinity. It is not manifest at the level of the Lagrangian, but acts non-trivially on physical states, and its Ward identity is the soft photino theorem. The infinite-dimensional fermionic symmetry resides in the same   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supermultiplet as the physically non-trivial large gauge symmetries associated with the soft photon theorem.

Temple He

Prahar Mitra

Andrew Strominger

Infinite-dimensional fermionic symmetry in supersymmetric gauge theories

We examine six-dimensional quantum field theories through the lens of higher-form global symmetries. Every Yang-Mills gauge theory in six dimensions, with field strength f (2), naturally gives rise to a continuous 1-form global symmetry associated with the 2-form instanton current J (2) ∼ ∗Tr (f (2) ∧ f (2)). We show that suitable mixed anomalies involving the gauge field f (2) and ordinary 0-form global symmetries, such as flavor or Poincaré symmetries, lead to continuous 2-group global symmetries, which allow two flavor currents or two stress tensors to fuse into the 2-form current J (2). We discuss several features of 2-group symmetry in six dimensions, many of which parallel the four-dimensional case. The majority of six-dimensional supersymmetric conformal field theories (SCFTs) and little string theories have infrared phases with non-abelian gauge fields. We show that the mixed anomalies leading to 2-group symmetries can be present in little string theories, but that they are necessarily absent in SCFTs. This allows us to establish a previously conjectured algorithm for computing the ’t Hooft anomalies of most SCFTs from the spectrum of weakly-coupled massless particles on the tensor branch of these theories. We then apply this understanding to prove that the a-type Weyl anomaly of all SCFTs with a tensor branch must be positive, a &gt; 0.

Clay Córdova

Kenneth Intriligator

2-Group global symmetries and anomalies in six-dimensional quantum field theories

Thomas-Whitehead (TW) gravity is a projectively invariant model of gravity over a d-dimensional manifold that is intimately related to string theory through reparametrization invariance. Unparametrized geodesics are the ubiquitous structure that ties together string theory and higher dimensional gravitation. This is realized through the projective geometry of Tracy Thomas. The projective connection, due to Thomas and later Whitehead, admits a component that in one dimension is in one-to-one correspondence with the coadjoint elements of the Virasoro algebra. This component is called the diffeomorphism field <math><msub><mi>D</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math> in the literature. It also has been shown that in four dimensions, the TW action collapses to the Einstein-Hilbert action with cosmological constant when <math><msub><mi>D</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math> is proportional to the Einstein metric. These previous results have been restricted to either particular metrics, such as the Polyakov 2D metric, or were restricted to coordinates that were volume preserving. In this paper, we review TW gravity and derive the gauge invariant TW action that is explicitly projectively invariant and general coordinate invariant. We derive the covariant field equations for the TW action and show how fermionic fields couple to the gauge invariant theory. The independent fields are the metric tensor <math><msub><mi>g</mi><mrow><mi>a</mi><mi>b</mi></mrow></msub></math>, the fundamental projective invariant <math><msub><msup><mi>Π</mi><mi>a</mi></msup><mrow><mi>b</mi><mi>c</mi></mrow></msub></math>, and the diffeomorphism field <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow></math>.