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 present a lattice calculation of the low energy constants of QCD with <math><msub><mi>N</mi><mi>c</mi></msub><mo>=</mo><mn>3</mn></math>, 4 and 5 colors and <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn></math> flavors of degenerate mass fermions. We fit data for the pseudoscalar meson mass, the pseudoscalar decay constant, and the axial Ward identity fermion mass to formulas from next-to-next-to leading order chiral perturbation theory. We extract the next to leading order low energy constants and study their behavior as a function of <math><msub><mi>N</mi><mi>c</mi></msub></math>. Preexisting analyses of <math><msub><mi>N</mi><mi>c</mi></msub><mo>=</mo><mn>3</mn></math> inform our fitting strategies.

Thomas DeGrand

Evan Wickenden

Lattice study of the chiral properties of large-<math><msub><mi>N</mi><mi>c</mi></msub></math> QCD

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

I make some simple observations about the calculation of weighted averages over energy of Minkowski space spectral densities from weighted averages over time of Euclidean space correlation functions, measured in lattice simulations. The correlator of two vector currents is used as an example, where it appears that a determination of a weighted average of the spectral function near the rho pole at the five percent level is possible from lattice simulations.

Remarks about weighted energy integrals over Minkowski spectral functions from Euclidean lattice data

I present a numerical study of the crossover between the low temperature chirally broken phase and the high temperature chirally restored phase in <math><mi>S</mi><mi>U</mi><mo>(</mo><msub><mi>N</mi><mi>c</mi></msub><mo>)</mo></math> gauge theory with <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mn>3</mn><mi>–</mi><mn>5</mn></mrow></math> colors and <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn></math> degenerate fermion flavors. Fermion masses span a range of intermediate values corresponding to pseudoscalar to vector meson masses <math><mo>(</mo><msub><mi>m</mi><mrow><mi>P</mi><mi>S</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mi>V</mi></msub><msup><mo>)</mo><mn>2</mn></msup><mo>∼</mo><mn>0.25</mn><mi>–</mi><mn>0.63</mn></math>. Observables include the temperature-dependent chiral condensate and screening masses. At each fermion mass these quantities show nearly identical temperature dependence across <math><msub><mi>N</mi><mi>c</mi></msub></math>.

Finite temperature properties of QCD with two flavors and three, four, and five colors

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>.

Samuel Brensinger

Kenneth Heitritter

General structure of Thomas-Whitehead gravity

Motivated by the calculation of observables in the decays <math><msub><mi>Λ</mi><mi>b</mi></msub><mo>→</mo><msub><mi>Λ</mi><mi>c</mi></msub><mrow><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>±</mo></msup><mo>)</mo></mrow><mo>ℓ</mo><mover><mi>ν</mi><mo>¯</mo></mover></math>, we present a calculation of form factors in the quark model. Our scheme combines a spectroscopic model with the internal wave functions and the Bakamjian-Thomas (BT) relativistic formalism to get the wave functions in motion. In the heavy quark limit, the current matrix elements provide the Isgur-Wise (IW) function. This limit is covariant, satisfies a large set of sum rules, and has been successfully applied to mesons. On the other hand, for baryons, we meet difficulties using standard spectroscopic models. This leads us to propose a phenomenological model, a Q-pointlike-diquark model, nonrelativistic, with harmonic oscillator forces, giving both a reasonable low-lying spectrum and the expected slope of the IW function. To begin, we extract this slope from lattice QCD data and find it to be around <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup><mo>∼</mo><mn>2</mn></math>, that we use as a guideline. We are not able to reproduce the right <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup></math> using certain typical standard <math><mrow><mtext>linear</mtext><mo>+</mo><mtext>Coulomb</mtext></mrow></math> potential models, both with three quarks <math><mi>Q</mi><mi>q</mi><mi>q</mi></math> or in a Q-pointlike-diquark picture. These difficulties seem to derive from the high sensitivity of <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup></math> to the structure of the light quark subsystem in a relativistic scheme. Finally, we present our model, and fixing its parameters to yield the correct spectrum and <math><msubsup><mi>ρ</mi><mi>Λ</mi><mn>2</mn></msubsup><mo>∼</mo><mn>2</mn></math>, we apply it to the calculation of observables. By studying Bjorken sum rule, we show that the inelastic IW function is large, and therefore, the transitions <math><msub><mi>Λ</mi><mi>b</mi></msub><mo>→</mo><msub><mi>Λ</mi><mi>c</mi></msub><mrow><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>,</mo><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>)</mo></mrow><mo>ℓ</mo><mover><mi>ν</mi><mo>¯</mo></mover></math> could be studied at LHCb. Interestingly, some observables in the <math><mi>τ</mi></math> case present zeroes for specific values of <math><msup><mi>q</mi><mn>2</mn></msup></math> that could be tests of the Standard Model.

D. Bečirević

A. Le Yaouanc

V. Morénas

L. Oliver

Heavy baryon wave functions, Bakamjian-Thomas approach to form factors, and observables in <math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>→</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><msup><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mo>±</mo></mrow></msup><mo>)</mo><mo>ℓ</mo><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> transitions