Collective coordinates are frequently employed in path integrals to manage divergences caused by fluctuations around saddle points that align with classical symmetries. These coordinates parameterize a manifold of zero modes and more broadly provide judicious coordinates on the space of fields. However, changing from local coordinates around a saddle point to more global collective coordinates is remarkably subtle. The main complication is that the mapping from local coordinates to collective coordinates is generically multi-valued. Consequently one is forced to either restrict the domain of path integral in a delicate way, or otherwise correct for the multi-valuedness by dividing the path integral by certain intersection numbers. We provide a careful treatment of how to fix collective coordinates while accounting for these intersection numbers, and then demonstrate the importance of the fix for free theories. We also provide a detailed study of the fix for interacting theories and show that the contributions of higher intersections to the path integral can be non-perturbatively suppressed. Using a variety of examples ranging from single-particle quantum mechanics to quantum field theory, we explain and resolve various pitfalls in the implementation of collective coordinates.

Any non-trivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness -- that all "in" and "out" states lie in the standard (zero memory) Fock space -- and therefore leads to infrared divergences in the standard $S$-matrix amplitudes. We define an infrared finite scattering theory valid for general quantum field theories and quantum gravity. The (infrared finite) "superscattering" map $\$$ is defined as a map between "in" and "out" states which does not require any a priori choice of a preferred Hilbert space. We define a "generalized asymptotic completeness" which accommodates states with memory in the space of asymptotic states. We define a complete basis of improper states on any memory Fock space (called "BMS particle" states) which are eigenstates of the energy-momentum -- or, more generally, the BMS supermomentum -- that generalize the usual $n$-particle momentum basis to account for states with memory. We then obtain infrared finite $\$$-amplitudes defined as matrix elements of $\$$ in the BMS particle basis. This formulation of the scattering theory is a key step towards analyzing fine-grained details of the infrared finite scattering theory. In quantum gravity, invariance of $\$$ under BMS supertranslations implies factorization of $\$$-amplitudes as the frequency of one of the BMS particles vanishes. This proves an infrared finite analog of the soft graviton theorem. Similarly, an infrared finite soft photon theorem in QED follows from invariance of $\$$ under large gauge transformations. We comment on how one must generalize this framework to consider $\$$-amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories).

Geometric engineering is a collection of tools developed to establish dictionaries between local singularities in string theory and (supersymmetric) quantum fields. Extended operators and defects, as well as their higher quantum numbers captured by topological symmetries, can be encoded within geometric engineering dictionaries. In this paper we revisit and clarify aspects of these techniques, with special emphasis on 't Hooft anomalies, interpreted from the SymTFT perspective as obstructions to the existence of Neumann boundary conditions. These obstructions to gauging higher symmetries are captured via higher link correlators for the SymTFT on spheres. In this work, we give the geometric engineering counterpart of this construction in terms of higher links of topological membranes. We provide a consistency check in the context of 5D SCFTs with anomalous 1-form symmetries, where we give two independent derivations of the anomaly in terms of higher links, one purely field theoretical and the other purely geometrical. Along the way, we also recover the construction of non-invertible duality defects in 4D $\mathcal N=4$ SYM from a geometric engineering perspective.

It has been shown in recent works that JT gravity with matter with two boundaries has a type II$_\infty$ algebra on each side. As the bulk spacetime between the two boundaries fluctuates in quantum nature, we can only define the entanglement wedge for each side in a pure algebraic sense. As we take the semiclassical limit, we will have a fixed long wormhole spacetime for a generic partially entangled thermal state (PETS), which is prepared by inserting heavy operators on the Euclidean path integral. Under this limit, with appropriate assumptions of the matter theory, geometric notions of the causal wedge and entanglement wedge emerge in this background. In particular, the causal wedge is manifestly nested in the entanglement wedge. Different PETS are orthogonal to each other, and thus the Hilbert space has a direct sum structure over sub-Hilbert spaces labeled by different Euclidean geometries. The full algebra for both sides is decomposed accordingly. From the algebra viewpoint, the causal wedge is dual to an emergent type III$_1$ subalgebra, which is generated by boundary light operators. To reconstruct the entanglement wedge, we consider the modular flow in a generic PETS for each boundary. We show that the modular flow acts locally and is the boost transformation around the global RT surface in the semiclassical limit. It follows that we can extend the causal wedge algebra to a larger type III$_1$ algebra corresponding to the entanglement wedge. Within each sub-Hilbert space, the original type II$_\infty$ reduces to type III$_1$.

The state-of-the-art theoretical formalism for a covariant description of non-Gaussian fluctuation dynamics in relativistic fluids is discussed.

We start by defining two-particle operators that appear in celestial CFT. We then show how to compute their OPE coefficients with the known single-particle operators at tree level from multiparticle factorization channels, focusing on the leading contribution involving the two-particle states. These factorization channels only give us single-particle exchanges. To extract the multiparticle exchanges, we look at the $\overline{\rm MHV}$ gluon amplitudes and show how non-factorization channels contribute to two-particle terms in the single-helicity sector. This is a first step towards systematically computing the full celestial OPE.

We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their associated geometry, are completely fixed by their Yangian and permutation symmetries. As a new feature for the hexagonal fishnets, the star-triangle identity introduces an ambiguity in the graph representation of a given Feynman integral. This translates into a map between different geometric interpretations attached to a graph. We demonstrate explicitly how these fishnet integrals can be understood as Calabi-Yau varieties, whose Picard-Fuchs ideals are generated by the Yangian over the conformal algebra. In analogy to elliptic curves, which represent the simplest examples of fishnet integrals with four-point vertices, we find that the simplest examples of three-point fishnets correspond to Picard curves with natural generalisations at higher loop orders.

We fill some of existed gaps in the correspondence between Supersymmetric Quantum Mechanics and the Inverse Scattering Transform by extending the consideration to the case of paired stationary and non-stationary Hamiltonians. We formulate the corresponding to the case Goursat problem and explicitly construct the kernel of the non-local Inverse Scattering Transform, which solves it. As a result, we find the way of constructing non-hermitian Hamiltonians from the initially hermitian ones, that leads, in the case of real-valued spectra of both potentials, to pairing of ${\cal CPT/PT}$-invariant Hamiltonians. The relevance of our proposal to Quantum Optics and optical waveguides technology, as well as to non-linear dynamics and Black Hole Physics is briefly discussed.

The Shifman--Vainshtein--Zakharov (SVZ) sum rules provide a method to obtain trans-series expansions in many quantum field theories, in which exponentially small corrections are calculated by combining the operator product expansion with the assumption of vacuum condensates. In some solvable models, exact expressions for trans-series can be obtained from non-perturbative results, and this makes it possible to test the SVZ method by comparing its predictions to these exact trans-series. In this paper we perform such a precision test in the example of the fermion self-energy in the Gross--Neveu model. Its exact trans-series expansion can be extracted from the large $N$ solution, at the first non-trivial order in $1/N$. It is given by an infinite series of exponentially small corrections involving factorially divergent power series in the 't Hooft parameter. We show that the first two corrections are associated to two-quark and four-quark condensates, and we reproduce the corresponding power series exactly, and at all loops, by using the SVZ method. In addition, the numerical values of the condensates can be extracted from the exact result, up to order $1/N$.

We give an elementary proof of the following property of unitary, interacting four-dimensional $\mathcal{N}=2$ superconformal field theories: at large central charge $c$, there exist at least $\sqrt{c}$ single-trace, scalar superconformal primary operators with dimensions $\Delta \lesssim \sqrt{c}$ (suppressing multiplicative logarithmic corrections). This follows from a stronger, more refined bound on the spectral density in terms of the asymptotic growth rate of the central charge. The proof employs known results on the structure of Coulomb branch operators. Interpreted holographically, this bounds the possible degree of scale separation in semiclassical AdS$_5$ half-maximal supergravity. In particular, the bulk must contain an infinite tower of charged scalar states of energies parametrically below the large black hole threshold, $E_{\rm BH} \sim c$. We address the extreme case of AdS$_5$ pure supergravity, ruling it out as the asymptotic limit of certain sequences in theory space, though the general question remains open.

We review different notions of cuts appearing throughout the literature on scattering amplitudes. Despite similar names, such as unitarity cuts or generalized cuts, they often represent distinct computations and distinct physics. We consolidate this knowledge, summarize how cuts are used in various computational strategies, and explain their relations to other quantities including imaginary parts, discontinuities, and monodromies. Differences and nuances are illustrated on explicit examples.

We describe a method to extract an effective Lagrangian description for open bosonic strings, at zero transcendentality. The method relies on a particular formulation of its scattering amplitudes derived from color-kinematics duality. More precisely, starting from a $(DF)^2 + \text{YM}$ quantum field theory, we integrate out all the massive degrees of freedom to generate an expansion in the inverse string tension $\alpha^\prime$. We explicitly compute the Lagrangian terms through $\mathcal{O}(\alpha^{\prime 4})$, and target the sector of operators proportional to $F^4$ to all orders in $\alpha^\prime$.

The superconformal index of half-BPS states in ${\cal N}=4$ supersymmetric Yang-Mills with gauge group $U(N)$ admits an expansion in terms of giant gravitons, ${\cal I}_N(q)={\cal I}_\infty(q) \sum\limits_{m=0}^\infty q^{mN}\hat{\mathcal I}_m(q)$, where $m$ is the number of giant gravitons. We derive this expansion directly in supergravity from the class of half-BPS solutions due to Lin, Lunin, and Maldacena in type IIB supergravity. The moduli space of these configurations can be quantized using covariant quantization methods. We review how this quantization leads to the graviton index, ${\cal I}_\infty(q)$, and present a modification that leads to the precise expression for the expansion in terms of giant gravitons. Our proposal provides a derivation of the giant graviton expansion directly in terms of supergravity degrees of freedom. We also comment on how to derive the expansion in terms of the effective Fermi droplet picture.

Higher-group symmetries are combinations of higher-form symmetries which appear in various field theories. In this paper, we explain how higher-group symmetries arise in 10d and 11d supergravities when the latter are coupled to brane sources. Motivated by this observation, we study field theories at zero and finite temperature invariant under a class of continuous Abelian higher-group symmetries. We restrict the analysis to the low-energy regime where the dynamical field content exclusively consists of Goldstone fields arising from the spontaneous breaking of higher-group and spacetime symmetries. Invariant quantities are constructed and the phases of matter are classified according to the pattern of spontaneous symmetry breaking. With respect to supergravity, we highlight how such Goldstone effective theories provide a symmetry-based interpretation for the theories living on D/M-branes. As an explicit example we construct a 6-group invariant action for the bosonic M5 brane, consistent with the self-duality of the 3-form field strength on the brane. While the self-duality condition in the bosonic case needs to be imposed externally as a constraint at zero temperature, we find an equilibrium effective action for the bosonic M5 brane at finite temperature that inherently implements self-duality.

In the context of ghost-free infinite derivative gravity we consider the single graviton exchange either between two spinless particles or between a spinless particle and a photon. To this end, we compute the gravitational potential for both cases and derive the quantum correction that arises at the linearized level. In the local theory it is well-known that such a correction is in the form of a Dirac delta function. Here we show that, for the nonlocal theory and in contrast to the local theory, the quantum correction is smeared out and takes on non-zero values for a non-zero separation between the two particles.

We explore the possibility of avoiding cosmological singularity with a bounce solution in the early Universe. The main finding is that simple and well-known semiclassical correction, which describes the mixing of radiation and gravity in the effective action, may provide an analytic solution with a bounce. The solution requires a positive beta function for the total radiation term and the contraction of the Universe at the initial instant. The numerical estimate shows that the bounce may occur in an acceptable range of energies, but only under strong assumptions about the particle physics beyond the Standard Model.

Topological arguments are currently being used as a novel scheme to discern the properties of black holes while ignoring their detailed structure and specific field equations. Among various avenues of black hole physics, where this novel approach is being utilized, the phase transition in black hole thermodynamics lies at the forefront. There are several types of phase transition in black holes; such as the van der Waals type phase transition, Davies-type phase transition, extremal phase transition, and Hawking-Page (HP) transition. So far, the topological interpretation, where the critical point has been identified with the non-zero topological charge, has been obtained only for the van der Waals type phase transition and HP transition in different spacetimes. To complete the picture, here we provide the same interpretation for two other phase transitions: Davies-type phase transition and extremal phase transition. The entire analysis is general and is valid for any spacetime where these types of phase transitions are observed. More importantly, our analysis suggests that amid the apparent differences in these phase transitions, they share the same topological characteristics, \textit{i.e.} non-zero topological charge arising from different thermodynamic potentials in different types of phase transition.

Using the cosmological date sets, the cosmological parameters are constrained in this paper, with some well known form of Hubble parameter. To understand the dynamics of the Weyl type $f(Q,T)$, functional form $f(Q,T)$ has been introduced, where $Q$ and $T$ respectively represents the nonmetricity scalar and trace of energy-momentum tensor. Using the constrained values of the parameters, the other geometrical parameters are analysed and the accelerating behaviour has been shown. Further to get the complete evolutionary behaviour of the Universe, the dynamical system analysis has been performed.

We thoroughly study the induced gravitational wave interpretation of the possible gravitational wave background reported by PTA collaborations, considering the unknown equation of state $w$ of the early universe. We perform a Bayesian analysis of the NANOGrav data using the publicly available \textsc{PTArcade} code together with \textsc{SIGWfast} for the numerical integration of the induced gravitational wave spectrum. We focus on two cases: a monochromatic and a log-normal primordial spectrum of fluctuations. For the log-normal spectrum, we show that, while the results are not very sensitive to $w$ when the GW peak is close to the PTA window, radiation domination is out of the $2\sigma$ contours when only the infra-red power-law tail contributes. For the monochromatic spectrum, the $2\sigma$ bounds yield $0.1\lesssim w\lesssim0.9$ so that radiation domination is close to the central value. We also investigate the primordial black hole (PBH) counterpart using the peak formalism. We show that, in general terms, a larger width and stiffer equation of state alleviates the overproduction of PBHs. No PBH overproduction requires $w\gtrsim0.42$ up to 2-$\sigma$ level for the monochromatic spectrum. Furthermore, including bounds from the cosmic microwave background, we find in general that the mass range of the PBH counterpart is bounded by $10^{-5} M_\odot\lesssim M_{\rm PBH}\lesssim10^{-1} M_\odot$. Lastly, we find that the PTA signal can explain the microlensing events reported by OGLE for $0.42\lesssim w\lesssim 0.50$. Our work showcases a complete treatment of induced gravitational waves and primordial black holes for general $w$ for future data analysis.

Having spectral correlations that, over small enough energy scales, are described by random matrix theory is regarded as the most general defining feature of quantum chaotic systems as it applies in the many-body setting and away from any semiclassical limit. Although this property is extremely difficult to prove analytically for generic many-body systems, a rigorous proof has been achieved for dual-unitary circuits -- a special class of local quantum circuits that remain unitary upon swapping space and time. Here we consider the fate of this property when moving from dual-unitary to generic quantum circuits focussing on the \emph{spectral form factor}, i.e., the Fourier transform of the two-point correlation. We begin with a numerical survey that, in agreement with previous studies, suggests that there exists a finite region in parameter space where dual-unitary physics is stable and spectral correlations are still described by random matrix theory, although up to a maximal quasienergy scale. To explain these findings, we develop a perturbative expansion: it recovers the random matrix theory predictions, provided the terms occurring in perturbation theory obey a relatively simple set of assumptions. We then provide numerical evidence and a heuristic analytical argument supporting these assumptions.

Roy-equation analyses on lattice data of $\pi\pi$ scattering phase shifts at $m_\pi=391$MeV reveals that the lowest $f_0$ meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry (PRD 108, 034009). We use the $N/D$ method to partially recover crossing symmetry of the $O(N)$ $\sigma$ model amplitude at leading order of $1/N$ expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The $\sigma$ pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying $m_\pi$. As the temperature increases, the complex $\sigma$ poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidences that the lowest $f_0$ state extracted from experiments and lattice data plays the role of $\sigma$ meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when $m_\pi$ and temperature get large.

We review the condensation completion of a modular tensor category, which yields a fusion 2-category of codimension-1 and higher defects in a $2+1$D topological order. We apply the condensation completion to $2+1$D toric code model and a $\mathbbm Z_4$ chiral topological order. In both cases, we explicitly enumerate the $1$d and $0$d defects present in these topological orders, along with their fusion rules. We also talk about other applications of condensation completion: alternative interpretations of condensation completion of a braided fusion category; condensation completion of the category of symmetry charges and its correspondence to gapped phases with symmetry; for a topological order $\cC$, one can also find all gapped boundaries of the stacking of $\cC$ with its time-reversal conjugate through computing the condensation completion of $\cC$.

We construct a kinetic model for matter-radiation interactions where the hydrodynamic gradient expansion can be computed analytically up to infinite order in derivatives, in the fully non-linear regime, and for arbitrary flows. The frequency dependence of the opacity of matter is chosen to mimic the relaxation time of a self-interacting scalar field. In this way, the transient sector simulates that of a realistic quantum field theory. As expected, the gradient series is divergent for most flows. We identify the mechanism at the origin of the divergence, and we provide a successful regularization scheme. Additionally, we propose a universal qualitative framework for predicting the breakdown of the gradient expansion of an arbitrary microscopic system undergoing a given flow. This framework correctly predicts the factorial divergence of the gradient expansion in most non-linear flows and its breakdown due to stochastic fluctuations. It also predicts that jets may induce an ultraviolet divergence in the gradient expansion of quark matter hydrodynamics.

Light primordial black holes (PBHs) with masses $M_\mathrm{PBH}<10^9\mathrm{g}$ can interestingly dominate the Universe's energy budget and give rise to early matter-dominated (eMD) eras before Big Bang Nucleosyntesis (BBN). During this eMD era, one is met with an abundant production of induced gravitational waves (GWs) serving as a portal to constrain the underlying theory of gravity. In this work, we study this type of induced GWs within the context of string-inspired running-vaccuum models (StRVMs), which, when expanded around de Sitter backgrounds, include logarithmic corrections of the space-time curvature. In particular, we discuss in detail the effects of StRVMs on the source as well as on the propagation of these PBH-induced GWs. Remarkably, under the assumption that the logarithmic terms represent quantum gravity corrections in the PBH era, we show that GW overproduction can be avoided if one assumes a coefficient of these logarithmic corrections that is much larger than the square of the reduced Planck mass. The latter cannot characterise quantum gravity corrections, though, prompting the need for revision of the quantisation of StRVMs in different than de Sitter backgrounds, such as those characterising PBH-driven eMD eras. This non trivial result suggests the importance of light PBHs as probes of new physics.

We perform a comprehensive analysis of state-of-the-art waveform models, focusing on their predictions concerning kick velocity and inferred gravitational wave memory. In our investigation we assess the accuracy of waveform models using energy-momentum balance laws, which were derived in the framework of full, non-linear General Relativity. The numerical accuracy assessment is performed for precessing as well as non-precessing scenarios for models belonging to the \textit{EOB}, \textit{Phenom}, and \textit{Surrogate} families. We analyze the deviations of these models from each other and from Numerical Relativity waveforms. Our analysis reveals statistically significant deviations, which we trace back to inaccuracies in modelling subdominant modes and inherent systematic errors in the chosen models. We corroborate our findings through analytical considerations regarding the mixing of harmonic modes in the computed kick velocities and inferred memories.

We examine neutrino zeromode solutions on the electroweak $Z$-string and their effect on the stability of the string in the standard model and its extensions. We propose using topological invariants constructed from the momentum (and real) space topology of Green's functions, often used for investigating edge modes in condensed matter physics. We analyze the standard model and then examine type-I and type-II extensions of the neutrino sector as well as their hybrid. Based on this analysis, we also comment on proposals in the literature to stabilize the $Z$-string.

We consider a proposed alternative to quantum gravity, in which the spacetime metric is treated as classical, even while matter fields remain quantum. Consistency of the theory necessarily requires that the metric evolve stochastically. Here, we show that this stochastic behaviour leads to a modification of general relativity at low accelerations. In the low acceleration regime, the variance in the acceleration produced by the gravitational field is high in comparison to that produced by the Newtonian potential, and acts as an entropic force, causing a deviation from Einstein's theory of general relativity. We show that in this "diffusion regime", the entropic force acts from a gravitational point of view, as if it were a contribution to the matter distribution. We compute how this modifies the expectation value of the metric via the path integral formalism, and find that an entropic force driven by a stochastic cosmological constant can explain galactic rotation curves without needing to evoke dark matter. We caution that a greater understanding of this effect is needed before conclusions can be drawn, most likely through numerical simulations, and provide a template for computing the deviation from general relativity which serves as an experimental signature of the Brownian motion of spacetime.