arXiv subjects
Publications and source records attributed to Pietro Antonio Grassi.
We present a supersymmetric extension of the Hull-Lambert-Sen chiral boson model. The two-metric formulation developed by Hull provides a powerful and tractable framework for the quantization of chiral bosons, and we demonstrate that it admits a natural supersymmetric generalization. We construct the theory explicitly on the two-torus and analyze its coupling to external sources.
This work provides a complete analytical characterization of the transfer- matrix structure associated with the finite Kronig-Penney model consisting of one-dimensional arrays of Dirac delta potentials recently introduced by Figueroa et al. (2025). Although their study identified the emergence of specific transfer-matrix entries and related combinatorial coefficients, a rig- orous derivation of the corresponding closed-form expressions has not yet been established. By expressing the N th power of the unit-cell transfer ma- trix in terms of Chebyshev polynomials of the second kind, we obtain explicit closed-form representations for the global transmission and reflection ampli- tudes. The proposed formulation reveals a previously unrecognized structural correspondence between multiple quantum scattering processes, discrete con- volutional patterns, and hypercomplex combinatorial structures.
We review the construction of higher-form symmetries for supersymmetric theories using a supergeometry framework. This reveals an enlarged set of topological conserved supercurrents, including Chern-Weil symmetries and new geometric Chern-Weil symmetries built from invariant supermanifold forms. In N=1 super-Maxwell theory in three dimensions, we construct the corresponding operators and charged defects, with charges determined by a super-linking number between their supporting hypersurfaces. At the end we provide as an original unpublished contribution some hints on how to construct super-symTFT for Chern-Weil and geometric Chern-Weil symmetries directly from supergravity.
We propose a manifestly supersymmetric formulation of the Symmetry Topological Field Theory (SuSymTFT) for theories with supersymmetry. The SymTFT is a framework that helps organizing symmetries and anomalies of a QFT. Albeit a lot of activity in the field has been devoted to the construction of the SymTFT for the bosonic symmetry structure, the fermionic case has not been analyzed in detail. Here, we consider the most prominent example of theories exhibiting fermionic symmetries, that is supersymmetric models. These are most naturally formulated on supermanifolds, where the supergeometry approach allows for a manifest organization of the symmetries according to their fermionic grading. We provide the general construction of the SuSymTFT as a super-BF theory living in a (n|m)-dimensional supermanifold and check our proposal in two particular examples, the compact and the chiral super-bosons in two dimensions.
We study the $T\bar T$ deformation using its formulation as a CFT coupled to two-dimensional dynamical gravity. Working within the BRST formalism, we apply the intertwiner construction of arXiv:2411.08865 to obtain a unitary "dressing" map between undeformed CFT operators and elements of the BRST cohomology of the deformed theory. We identify the resulting "dressed" operators corresponding to CFT primaries as the physical observables of the deformed theory and show that they arise from a field-dependent change of coordinates, in agreement with what is expected for the $T\bar T$ deformation. We then give a non-perturbative definition of deformed correlation functions as BRST-invariant expectation values of dressed operators in the gauge theory. Finally, we verify that our construction reproduces known structural and perturbative results.
We compute the partition function for the $N=1$ spinning particle, including pictures and the large Hilbert space, and show that it counts the dimension of the BRST cohomology in two- and four-dimensional target space. We also construct a quadratic action in the target space. Furthermore, we find a consistent interaction as a derived bracket based on the associative product of world line fields, leading to an interacting theory of multiforms in space-time. Finally, we comment on the equivalence of the multiform theory with a Dirac fermion. We also identify the chiral anomaly of the latter with a Hodge anomaly for the multiform theory, which manifests itself as a deformation of the gauge fixing.
Powerful techniques have been developed in quantum field theory that employ algebras of local operators, yet local operators cannot create physical charged states in gauge theory or physical nonzero-energy states in perturbative quantum gravity. A common method to obtain physical operators out of local ones is to dress the latter using appropriate Wilson lines. This procedure destroys locality, it must be done case by case for each charged operator in the algebra, and it rapidly becomes cumbersome, particularly in perturbative quantum gravity. In this paper we present an alternative approach to the definition of physical charged operators: we define an automorphism that maps an algebra of local charged operators into a (non-local) algebra of physical charged operators. The automorphism is described by a formally unitary intertwiner mapping the exact BRS operator associated to the gauge symmetry into its quadratic part. The existence of an automorphism between local operators and the physical ones, describing charged states, allows to retain many of the results derived in local operator algebras and extend them to the physical-but-nonlocal algebra of charged operators as we discuss in some simple applications of our construction. We also discuss a formal construction of physical states and possible obstructions to it.
The BV formalism is a well-established method for analyzing symmetries and quantization of field theories. In this paper we use the BV formalism to derive partition functions of gauge invariant operators up to equations of motions and their redundancies of selected theories. We discuss various interpretations of the results, some dualities and relation to first quantized models.
In several elementary particle scenarios, self-dual fields emerge as fundamental degrees of freedom. Some examples are the $D = 2$ chiral boson, $D = 10$ Type IIB supergravity, and $D = 6$ chiral tensor multiplet theory. For those models, a novel variational principle has been proposed in the work of Ashoke Sen. The coupling to supergravity of self-dual models in that new framework is rather peculiar to guarantee the decoupling of unphysical degrees of freedom. We generalize this technique to the self-dual super Maxwell gauge theory in $D = 4$ Euclidean spacetime both in the component formalism and the superspace. We use the geometric tools of rheonomy and integral forms since they are very powerful geometrical techniques for the extension to supergravity. We show the equivalence between the two formulations by choosing a different integral form defined using a Picture Changing Operator. That leads to a meaningful action functional for the variational equations. In addition, we couple the model to a non-dynamical gravitino to extend the analysis slightly beyond the free case. A full-fledged self-dual supergravity analysis will be presented elsewhere.
We describe a family of twisted partition functions for the relativistic spinning particle models. For suitable choices of fugacities this computes a refined Euler characteristics that counts the dimension of the physical states for arbitrary picture and, furthermore, encodes the complete BV-spectrum of the effective space-time gauge theory originating from this model upon second quantization. The relation between twisted world-line partition functions and the spectrum of the space-time theory is most easily seen on-shell but we will give an off-shell description as well. Finally we discuss the construction of a space-time action in terms of the world-line fields in analogy to string field theory.
The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.
The Hodge dual operator, recently introduced for supermanifolds, is used to reformulate super Yang-Mills and supergravity in $D=4$. We first recall the definition of the Hodge dual operator for flat and curved supermanifolds. Then we show how to recover the usual super-Yang-Mills equations of motion for $N=1,2$ supersymmetry, and the obstacles (as seen from Hodge dual point of view) in the case $N \geq 3$. We reconsider several ingredients of supergeometry, relevant for a superspace formulation of supergravity, in terms of the Hodge dual operator. Finally we discuss how $D=4$ and $N=1$ supergravity is obtained in this framework.
We develop the theory of Free Integro-Differential Algebras (FIDA) extending the powerful technique of Free Differential Algebras constructed by D. Sullivan. We extend the analysis beyond the superforms to integral- and pseudo-forms used in supergeometry. It is shown that there are novel structures that might open the road to a deeper understanding of the geometry of supergravity. We apply the technique to some models as an illustration and we provide a complete analysis for D=11 supergravity. There, it is shown how the Hodge star operator for supermanifolds can be used to analyze the set of cocycles and to build the corresponding FIDA. A new integral form emerges which plays the role of the truly dual to 4-form $F^{(4)}$ and we propose a new variational principle on supermanifolds.
We make some considerations and remarks on D=11 supergravity and its integral form. We start from the geometrical formulation of supergravity and by means of the integral form technique we provide a superspace action that reproduces (at the quadratic level) the recent formulation of supergravity in pure spinor framework. We also make some remarks on Chevalley-Eilenberg cocycles and their Hodge duals.
We generalize the geometrical formulation of Wilson loops recently introduced in arXiv:2003.01729v2 to the description of Wilson Surfaces. For N=(2,0) theory in six dimensions, we provide an explicit derivation of BPS Wilson Surfaces with non-trivial coupling to scalars, together with their manifestly supersymmetric version. We derive explicit conditions which allow to classify these operators in terms of the number of preserved supercharges. We also discuss kappa-symmetry and prove that BPS conditions in six dimensions arise from kappa-symmetry invariance in eleven dimensions. Finally, we discuss super-Wilson Surfaces - and higher dimensional operators - as objects charged under global $p$-form (super)symmetries generated by tensorial supercurrents. To this end, the construction of conserved supercurrents in supermanifolds and of the corresponding conserved charges is developed in details.
As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold $\mathbb{C}^3/\mathbb{Z}_4$, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface $\mathbb F_2$. We determine the explicit Kähler geometry of the smooth resolved manifold $Y$, which is the total space of the canonical bundle of $\mathbb F_2$. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet-Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space $\mathbb P[1,1,2]$, while the other is the product of the ALE space $A_1$ by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.
Inspired by the analogy between different types of differential forms on supermanifolds and string fields in superstring theory, we construct new multilinear non-associative products of forms which yield an $A_\infty$-algebra.
We use the techniques of integral forms to analyse the easiest example of two dimensional sigma models on a supermanifold. We write the action as an integral of a top integral form over a D=2 supermanifold and we show how to interpolate between different superspace actions. Then, we consider curved supermanifolds and we show that the definitions used for flat supermanifold can also be used for curved supermanifolds. We prove it by first considering the case of a curved rigid supermanifold and then the case of a generic curved supermanifold described by a single superfield $E$.