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 identify a minisuperspace of complex deformations of ABJM theory for which the partition function specifies the amplitude of an eternally inflating universe. The boundary theory predicts that the bosonic bulk is effectively in the Hartle-Hawking no-boundary state, with a subleading ‘tunneling’ contribution. This holographic model of inflation also reveals a close connection between the swampland distance and cobordism conjectures, and the condition for the asymptotic wave function to predict classical behavior in geometry and fields.

Belgium

United Kingdom

Thomas Hertog

Switzerland

Jef Pauwels

Victoria Venken

Holographic (eternal) inflation

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

<p>We show that the Kontsevich-Segal-Witten (KSW) criterion applied to the no-boundary state constrains anisotropic deformations of de Sitter space. We consider squashed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup><mo>×</mo><msup><mi>S</mi><mn>2</mn></msup></math> boundaries and find that, in both models, the KSW criterion excludes a significant range of homogeneous but anisotropic configurations. For squashed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>3</mn></msup></math> boundaries, the excluded range includes all surface geometries with negative scalar curvature, in line with dS/CFT reasoning. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup><mo>×</mo><msup><mi>S</mi><mn>2</mn></msup></math> boundaries, we find that KSW selects the low-temperature regime of configuration space where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> is sufficiently large compared to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>2</mn></msup></math>. In both models, the KSW criterion renders the semiclassical wave function normalizable, up to one-loop effects.</p>

Oliver Janssen

Joel Karlsson

Kontsevich-Segal criterion in the no-boundary state constrains anisotropy

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 construct wormholes supported by axion flux in the presence of a positive cosmological constant. The solutions describe compact, one-handle bodies colloquially known as kettlebell geometries. The wormholes are perturbatively stable, but regularity of the Euclidean geometry implies an upper bound on the axion flux. Viewed as no-boundary saddle points, wormholes are suppressed relative to the round sphere. The symmetric kettlebell with maximal axion density has vanishing Euclidean action. Continuing into the Lorentzian across the equator, the solutions describe two expanding branches of de Sitter space filled with an axion field that rapidly dilutes and which are connected by a quantum bounce across which the arrow of time reverses.

Sergio Aguilar-Gutierrez

Rob Tielemans

Netherlands

Jan Schaar

Thomas Riet

Axion-de Sitter wormholes

We show that the Kontsevich-Segal (KS) criterion, applied to the complex saddles that specify the semiclassical no-boundary wave function, acts as a selection mechanism on inflationary scalar field potentials. Completing the observable phase of slow-roll inflation with a no-boundary origin, the KS criterion effectively bounds the tensor-to-scalar ratio of cosmic microwave background fluctuations to be less than 0.08, in line with current observations. We trace the failure of complex saddles to meet the KS criterion to the development of a tachyon in their spectrum of perturbations.

Italy

Kontsevich-Segal Criterion in the No-Boundary State Constrains Inflation

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.