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

We examine dualities of two dimensional conformal field theories by applying the methods developed in previous works. We first derive the duality between <math><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>|</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>/</mo><mo>(</mo><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>⊗</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math> coset and Witten's cigar model or sine-Liouville theory. The latter two models are Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation of the three models is named FZZ-triality. These results are used to study correlator correspondences between large <math><mi>N</mi><mo>=</mo><mn>4</mn></math> super Liouville theory and a coset of the form <math><mi>Y</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math>, where <math><mi>Y</mi><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> consists of two <math><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>|</mo><mn>1</mn><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> and free bosons or equivalently two <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> cosets of <math><mi>D</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>;</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>)</mo></math> at level one. These correspondences are a main result of this paper. The FZZ-triality acts as a seed of the correspondence, which in particular implies a hidden <math><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math> in <math><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>|</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub></math> or <math><mi>D</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>;</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>1</mn></mrow></msub></math>. The relation of levels is <math><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>1</mn><mo>/</mo><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math>. We also construct boundary actions in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we generalize the FZZ-triality to the case with <math><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mi>n</mi><mo>|</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>/</mo><mo>(</mo><mi>S</mi><mi>L</mi><msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>⊗</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math> for arbitrary <math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math>.

Thomas Creutzig

Yasuaki Hikida

FZZ-triality and large super Liouville theory

We derive correspondences of correlation functions among dual conformal field theories in two dimensions by developing a “first order formulation” of coset models. We examine several examples, and the most fundamental one may be a conjectural equivalence between a coset (SL(n) k  ⊗SL(n) −1)/SL(n) k−1 and   <math> <mi>sl</mi> <mfenced> <mi>n</mi> </mfenced> </math>  $$ \mathfrak{sl}(n) $$  Toda field theory with generic level k. Among others, we also complete the derivation of higher rank FZZ-duality involving a coset SL(n + 1) k  /(SL(n) k  ⊗ U(1)), which could be done only for n = 2, 3 in our previous paper. One obstacle in the previous work was our poor understanding of a first order formulation of coset models. In this paper, we establish such a formulation using the BRST formalism. With our better understanding, we successfully derive correlator correspondences of dual models including the examples mentioned above. The dualities may be regarded as conformal field theory realizations of some of the Gaiotto-Rapčák dualities of corner vertex operator algebras.

Correlator correspondences for Gaiotto-Rapčák dualities and first order formulation of coset models

We examine a strong/weak duality between a Heisenberg coset of a theory with   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$   n  subregular   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$ -algebra symmetry and a theory with a   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$   n|1-structure. In a previous work, two of the current authors provided a path integral derivation of correlator correspondences for a series of generalized Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality. In this paper, we derive correlator correspondences in a similar way but for a different series of generalized duality. This work is a part of the project to realize the duality of corner vertex operator algebras proposed by Gaiotto and Rapčák and partly proven by Linshaw and one of us in terms of two dimensional conformal field theory. We also examine another type of duality involving an additional pair of fermions, which is a natural generalization of the fermionic FZZ-duality. The generalization should be important since a principal   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$ -superalgebra appears as its symmetry and the properties of the superalgebra are less understood than bosonic counterparts.

Devon Stockal

Correlator correspondences for subregular   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$ -algebras and principal   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$ -superalgebras

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

We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the   <math> <mi>sl</mi> <mfenced> <mn>2</mn> </mfenced> <mo>/</mo> <mi>u</mi> <mfenced> <mn>1</mn> </mfenced> </math>  $$ \mathfrak{sl}(2)/\mathfrak{u}(1) $$  coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from   <math> <mi>sl</mi> <mfenced> <mn>2</mn> </mfenced> </math>  $$ \mathfrak{sl}(2) $$  Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type   <math> <mi>sl</mi> <mfenced> <mrow> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mo>/</mo> <mfenced> <mrow> <mi>sl</mi> <mfenced> <mi>N</mi> </mfenced> <mo>×</mo> <mi>u</mi> <mfenced> <mn>1</mn> </mfenced> </mrow> </mfenced> </math>  $$ \mathfrak{sl}\left(N+1\right)/\left(\mathfrak{sl}(N)\times \mathfrak{u}(1)\right) $$  and investigate the equivalence to a theory with an   <math> <mi>sl</mi> <mfenced> <mrow> <mi>N</mi> <mo>+</mo> <mfenced> <mn>1</mn> </mfenced> <mi>N</mi> </mrow> </mfenced> </math>  $$ \mathfrak{sl}\left(N+\left.1\right|N\right) $$  structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for   <math> <mi>sl</mi> <mfenced> <mi>N</mi> </mfenced> </math>  $$ \mathfrak{sl}(N) $$  and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y 0,N,N+1[ψ] and Y  N,0,N+1[ψ  −1].

Higher rank FZZ-dualities

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