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.

France

Taro Kimura

Japan

Go Noshita

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

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

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

While the prompt $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  cross section and polarization have been measured with good precision as a function of transverse momentum, $$p_\mathrm{T}$$ <math><msub><mi>p</mi><mi>T</mi></msub></math> , those of the directly produced $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  are practically unknown, given that the cross sections and polarizations of the $$\chi _{{c}1} $$ <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math>  and $$\chi _{{c}2} $$ <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></math>  mesons, large indirect contributors to $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  production, are only known with rather poor accuracy. The lack of precise measurements of the $$\chi _{{c}J} $$ <math><msub><mi>χ</mi><mrow><mi>cJ</mi></mrow></msub></math>  polarizations induces large uncertainties in the level of their feed-down contributions to the prompt $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  yield, because of the polarization-dependent acceptance corrections. The experimental panorama of charmonium production can be significantly improved through a consistent and model-independent global analysis of existing measurements of $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math> , $$\psi \mathrm{(2S)}$$ <math><mrow><mi>ψ</mi><mrow><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></mrow></mrow></math>  and $$\chi _{c}$$ <math><msub><mi>χ</mi><mi>c</mi></msub></math>  cross sections and polarizations, faithfully respecting all the correlations and uncertainties. In particular, it is seen that the $$\chi _{{c}J} $$ <math><msub><mi>χ</mi><mrow><mi>cJ</mi></mrow></msub></math>  polarizations and feed-down fractions to $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  production have a negligible dependence on the $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  $$p_\mathrm{T}$$ <math><msub><mi>p</mi><mi>T</mi></msub></math> , with average values $$\lambda _\vartheta ^{\chi _{{c}1}} = 0.55 \pm 0.23$$ <math><mrow><msubsup><mi>λ</mi><mi>ϑ</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></msubsup><mo>=</mo><mn>0.55</mn><mo>±</mo><mn>0.23</mn></mrow></math> , $$\lambda _\vartheta ^{\chi _{{c}2}} = -0.39 \pm 0.22$$ <math><mrow><msubsup><mi>λ</mi><mi>ϑ</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></msubsup><mo>=</mo><mo>-</mo><mn>0.39</mn><mo>±</mo><mn>0.22</mn></mrow></math> , $$R^{\chi _{{c}1}} = (18.8 \pm 1.4)\%$$ <math><mrow><msup><mi>R</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></msup><mo>=</mo><mrow><mo>(</mo><mn>18.8</mn><mo>±</mo><mn>1.4</mn><mo>)</mo></mrow><mo>%</mo></mrow></math>  and $$R^{\chi _{{c}2}} = (6.5 \pm 0.5)\%$$ <math><mrow><msup><mi>R</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></msup><mo>=</mo><mrow><mo>(</mo><mn>6.5</mn><mo>±</mo><mn>0.5</mn><mo>)</mo></mrow><mo>%</mo></mrow></math> . The analysis also shows that $$(67.2 \pm 1.9)$$ <math><mrow><mo>(</mo><mn>67.2</mn><mo>±</mo><mn>1.9</mn><mo>)</mo></mrow></math> % of the prompt $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  yield is due to directly-produced mesons, of polarization constrained to remarkably small values, $$\lambda _\vartheta ^{\mathrm{J}/\psi } = 0.04 \pm 0.06$$ <math><mrow><msubsup><mi>λ</mi><mi>ϑ</mi><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></msubsup><mo>=</mo><mn>0.04</mn><mo>±</mo><mn>0.06</mn></mrow></math> .

Portugal

Pietro Faccioli

CERN

Carlos Lourenço

Germany

Thomas Madlener

From prompt to direct $$\mathrm{J}/\psi $$ <math><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></math>  production: new insights on the $$\chi _{{c}1} $$ <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math>  and $$\chi _{{c}2} $$ <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub></math>  polarizations and feed-down contributions from a global-fit analysis of mid-rapidity LHC data

We compute the elliptic genus of the D1/D7 brane system in flat space, finding a non-trivial dependence on the number of D7 branes, and provide an F-theory interpretation of the result. We show that the JK-residues contributing to the elliptic genus are in one-to-one correspondence with coloured plane partitions and that the elliptic genus can be written as a chiral correlator of vertex operators on the torus. We also study the quantum mechanical system describing D0/D6 bound states on a circle, which leads to a plethystic exponential formula that can be connected to the M-theory graviton index on a multi-Taub-NUT background. The formula is a conjectural expression for higher-rank equivariant K-theoretic Donaldson-Thomas invariants on ℂ 3.

Italy

Francesco Benini

Giulio Bonelli

Matteo Poggi

Alessandro Tanzini

Elliptic non-Abelian Donaldson-Thomas invariants of ℂ 3

In this paper, we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory—we find explicit formulas for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to <math><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow></math> torus knots and show that their classical generating functions, in the extremal limit and framing <math><mi>r</mi><mi>s</mi></math>, are generating functions of lattice paths under the line of the slope <math><mi>r</mi><mo>/</mo><mi>s</mi></math>. Generating functions of such paths satisfy extremal A-polynomial equations, which immediately follows after representing them in terms of the Duchon grammar. Moreover, these extremal A-polynomial equations encode Donaldson-Thomas invariants, which provides an interesting example of algebraicity of generating functions of these invariants. We also find a quantum generalization of these statements, i.e. a relation between motivic quiver generating functions, quantum extremal knot invariants, and <math><mi>q</mi></math>-weighted path counting. Finally, in the case of the unknot, we generalize this correspondence to the full HOMFLY-PT invariants and counting of Schröder paths.

Poland

Miłosz Panfil

Serbia

Marko Stošić

Piotr Sułkowski

Donaldson-Thomas invariants, torus knots, and lattice paths

Kinetic theory of Dirac fermions is studied within the matrix valued differential forms method. It is based on the symplectic form derived by employing the semiclassical wave packet build of the positive energy solutions of the Dirac equation. A satisfactory definition of the distribution matrix elements imposes to work in the basis where the helicity is diagonal which is also needed to attain the massless limit. We show that the kinematic Thomas precession correction can be studied straightforwardly within this approach. It contributes on an equal footing with the Berry gauge fields. In fact in equations of motion it eliminates the terms arising from the Berry gauge fields.

Türkiye

Ömer F. Dayi

Eda Kilinçarslan

A semiclassical kinetic theory of Dirac particles and Thomas precession