Duals of Feynman integrals. Part I. Differential equations

We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.

Genre poetics of R. Wagner’s “Parsifal” in the context of Gesamtkunstwerk idea

Background. Over the course of the last decades, musicology was marked by a revival of interest in the phenomenon of artistic synthesis. This paper considers it in view of “general unity” as one of techniques of understanding the 19th century culture. Such a mindset can be defined by various terms, including Gesamtkunstwerk. The scholars today do not narrow this term down to a mere synthesis of arts, but add immersion into cultural history to it. This allows us to view Gesamtkunstwerk as a way of cultural memory existence, and genre poetics of R. Wagner’s musical drama as a bearer of this memory. In this context, special place belongs to “Parsifal”, which in this article is presented as an absolute embodiment of the idea of Gesamtkunstwerk. The aim of the article is to reveal compositional devices, through which references of “Parsifal” poetics and other genres of different arts create “general unity” under the sign of “Gesamtkunstwerk”. A genre method, a systematic method based on interdisciplinary approach, and a comparative one are chosen as methods of research. Research materials used in the paper relate to different branches of Humanitarian knowledge, including theatre arts (S. Mokulsky), literature (M. Bakhtin), medievistics (A. Gurevich), musicology (M. Veremiova, N. Vieru, E. Makhrova, K. Richter, A. Philippov etc.). It is noted that the definition of “Bühnenweihfestspiel” indicated in the score speaks of the relatedness of this work to the genre of medieval mystery. We have detected some common features of “Parsifal” like slow-paced events, their rather low concentration in time and meticulous development of verbally-acoustic matter. In the musical drama under consideration both a word and a system of leitmotifs function as an instrument for the above-mentioned scrupulosity; thus, poetics of a mystery is transformed into poetics of musical drama. The references of “Parsifal” to medieval epics are noted. Its plot contains only essential for the main storyline events. Cumulative method is chosen as the basic one which defines the algorithm of compositional movement, making it somewhat discrete. At the same time, the continuous flow of leitmotif development contributes to endless disclosure of the sense and its symbolization. Duality of time contributes to appearance of spatiality in genre poetics of “Parsifal”, which also corresponds to the epic poetics. There are some references between genre poetics of “Parsifal” and such types of novel as “Erziehungsroman” and “adventure novel”. The features of “Erziehungsroman”, based on a choice of constantly evolving protagonist as a plot-creating device are embodied to the fullest extent in a presentation of the main hero of “Parsifal”. In the work he passes the way from a naïve youth to the saviour of The Holy Grail, doing the deed of all-embracing love, thus the portrayal of his image is done according to the principle of development. At the same time, there have been revealed differences between the last work of the composer and “Lohengrin”. Unlike Elsa, with her gradual transformation, Parsifal undergoes this process almost instantly. But R. Wagner, using leitmotiv system, reveals subconscious psychological development, thus using it as a nonverbal explanation of one of the essential plot events in the work. The links between “adventure novel” and “Parsifal” lie in the type of intrigue, localization of space-time, due to which it is differentiated into adventurous, legendary, psychological etc. The hero, shifting into another spacetime dimension, transforms his image. This paper shows it on example of Kundry. This heroine in her cyclic “death – resurrection” transformations completely belongs to poetics of the myth, although the composer here does not use it as an example, as he fills musical characteristics of Kundry with all the traits of opera character. The article explores references of “Parsifal” to Passions by J. S. Bach. It is pointed out that in this case R. Wagner does not follow their specific traits, as he chooses the most essential features of poetics of this genre. A сonclusion is made that genre poetics of “Parsifal”, reflecting the features of poetics of another genres, incorporates their essential traits, which are then intertwined into general texture of musical drama, creating inseparable fusion, according to the Gesamtkunstwerk idea.

A basis of analytic functionals for CFTs in general dimension

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition at infinity. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

On the conformal symmetry of exceptional scalar theories

The DBI and special galileon theories exhibit a conformal symmetry at unphysical values of the spacetime dimension. We find the Lagrangian form of this symmetry. The special conformal transformations are non-linearly realized on the fields, even though conformal symmetry is unbroken. Commuting the conformal transformations with the extended shift symmetries, we find new symmetries, which when taken together with the conformal and shift symmetries close into a larger algebra. For DBI this larger algebra is the conformal algebra of the higher dimensional bulk in the brane embedding view of DBI. For the special galileon it is a real form of the special linear algebra. We also find the Weyl transformations corresponding to the conformal symmetries, as well as the necessary improvement terms to make the theories Weyl invariant, to second order in the coupling in the DBI case and to lowest order in the coupling in the special galileon case.

The unitary representations of the Poincar\'e group in any spacetime dimension

An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D\geqslant 3D≥3 is presented. An exhaustive treatment is performed of the two most important classes of unitary irreducible representations of the Poincar'e group, corresponding to massive and massless fundamental particles. Covariant field equations are given for each unitary irreducible representation of the Poincar'e group with non-negative mass-squared.

High-dimensional Schwarzschild black holes in scalar–tensor–vector gravity theory

We obtain a high-dimensional Schwarzschild black hole solution in the scalar–tensor–vector gravity (STVG), and then analyze the influence of parameter $$\alpha $$ α associated with a deviation of the STVG theory from General Relativity on event horizons and Hawking temperature. We calculate the quasinormal mode frequencies of massless scalar field perturbations for the high-dimensional Schwarzschild STVG black hole by using the sixth-order WKB approximation method and the unstable null geodesic method in the eikonal limit. The results show that the increase of parameter $$\alpha $$ α makes the scalar waves decay slowly, while the increase of the spacetime dimension makes the scalar waves decay fast. In addition, we study the influence of parameter $$\alpha $$ α on the shadow radius of this high-dimensional Schwarzschild STVG black hole and find that the increase of parameter $$\alpha $$ α makes the black hole shadow radius increase, but the increase of the spacetime dimension makes the black hole shadow radius decrease. Finally, we investigate the energy emission rate of the high-dimensional Schwarzschild STVG black hole, and find that the increase of parameter $$\alpha $$ α makes the evaporation process slow, while the increase of the spacetime dimension makes the process fast.

Cosmological α′-corrections from the functional renormalization group

We employ the techniques of the Functional Renormalization Group in string theory, in order to derive an effective mini-superspace action for cosmological backgrounds to all orders in the string scale α′. To this end, T-duality plays a crucial role, classifying all perturbative curvature corrections in terms of a single function of the Hubble parameter. The resulting renormalization group equations admit an exact, albeit non-analytic, solution in any spacetime dimension D, which is however incompatible with Einstein gravity at low energies. Within an E-expansion about D = 2, we also find an analytic solution which exhibits a non-Gaussian ultraviolet fixed point with positive Newton coupling, as well as an acceptable low-energy limit. Yet, within polynomial truncations of the full theory space, we find no evidence for an analog of this solution in D = 4. Finally, we comment on potential cosmological implications of our findings.

Superconformal boundaries in 4 − ϵ dimensions

Boundaries in three-dimensional $$ \mathcal{N} $$ N = 2 superconformal theories may preserve one half of the original bulk supersymmetry. There are two possibilities which are characterized by the chirality of the leftover supercharges. Depending on the choice, the remaining 2d boundary algebra exhibits $$ \mathcal{N} $$ N = (0, 2) or $$ \mathcal{N} $$ N = (1) supersymmetry. In this work we focus on correlation functions of chiral fields for both types of supersymmetric boundaries. We study a host of correlators using superspace techniques and calculate superconformal blocks for two- and three-point functions. For $$ \mathcal{N} $$ N = (1) supersymmetry, some of our results can be analytically continued in the spacetime dimension while keeping the codimension fixed. This opens the door for a bootstrap analysis of the ϵ-expansion in supersymmetric BCFTs. Armed with our analytically-continued superblocks, we prove that in the free theory limit two-point functions of chiral (and antichiral) fields are unique. The first order correction, which already describes interactions, is universal up to two free parameters. As a check of our analysis, we study the Wess-Zumino model with a super-symmetric boundary using Feynman diagrams, and find perfect agreement between the perturbative and bootstrap results.

Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

Higher spins from exotic dualisations

At the free level, a given massless field can be described by an infinite number of different potentials related to each other by dualities. In terms of Young tableaux, dualities replace any number of columns of height hi by columns of height D − 2 − hi, where D is the spacetime dimension: in particular, applying this operation to empty columns gives rise to potentials containing an arbitrary number of groups of D − 2 extra antisymmetric indices. Using the method of parent actions, action principles including these potentials, but also extra fields, can be derived from the usual ones. In this paper, we revisit this off-shell duality and clarify the counting of degrees of freedom and the role of the extra fields. Among others, we consider the examples of the double dual graviton in D = 5 and two cases, one topological and one dynamical, of exotic dualities leading to spin three fields in D = 3.