### Bounds on Species Scale and the Distance Conjecture

The species scale $\Lambda_s\leq M_{pl}$ serves as a UV cutoff in the gravitational sector of an EFT and can depend on the moduli of the theory as the spectrum of the theory varies. We argue that the dependence of the species scale $\Lambda_s (\phi)$ on massless (or light) modes $\phi^i$ satisfies $M_{pl}^{d-2} |\Lambda_s'/\Lambda_s|^2< \mathcal{O}(1)$. This bound is true at all points in moduli space including also its interior. The argument is based on the idea that the short distance contribution of massless modes to gravitational terms in the EFT cannot dramatically affect the black hole entropy. Based on string theory arguments we expect the $\mathcal{O}(1)$ constant in this bound to be equal to ${1\over {d-2}}$ as we approach the boundary of the moduli space. However, we find that the slope of the species scale can approach its asymptotic value from above as we go from interior points to the boundaries, thereby implying that the constant in the bound must be larger than ${1\over {d-2}}$. The bound on the variation of the species scale also implies that the mass of towers of light modes cannot go to zero faster than exponential in field distance in accordance with the Distance Conjecture.

### Non-BPS path to the string lamppost

We provide further motivation for the string lamppost principle in 9d supergravities. Using a blend of ideas which includes Swampland conjectures, finiteness of black hole entropy, and classification of SCFTs, we show that infinite distance limits that keep BPS states heavy must decompactify to type IIA supergravity on an interval. Without relying on string theory, we provide bottom-up explanations for various UV features of the theory, such as the physics near the orientifold branes and the worldvolume theories of different stacks of non-perturbative 8-branes. We also provide a Swampland argument for the countability of the number of inequivalent string limits up to dualities which is a strong result with applications beyond this work.

### Tree-Level Color-Kinematics Duality from Pure Spinor Actions

Pure spinor actions of various gauge theories are of Chern-Simons form, and they come with natural $\text{BV}^{\square}$-algebra structures that manifest tree-level color-kinematics duality. This has been observed before in arXiv:2112.11452 for the currents of 10D supersymmetric Yang-Mills theory. Here, we show that this observation can be extended to the scattering amplitudes as well as to theories with matter. In particular, we show that the BLG, ABJM and ABJ models possess tree-level color-kinematics duality without having to resort to explicit computations. The $\text{BV}^{\square}$-algebra structure explicitly contains the kinematic Lie algebra in the form of a derived bracket. However, regularization issues obstruct the seemingly implied lift to the loop level, and we give reasons to believe that this is a general feature of manifestly CK-dual actions close to Yang-Mills theory. We also explain the link of our ordinary gauge-matter CK-duality to the 3-Lie algebra form of CK-duality previously discussed in the literature.

### Dualities of 3D $\mathcal{N}=1$ SQCD from Branes and non-SUSY deformations

We study the dynamics of an 'electric' $\mathcal{N}=1$ 3D $U(N_c)_{k,k+\frac{N_c}{2}}$ SQCD theory. By embedding the theory in string theory, we propose that the theory admits a 'magnetic' dual and analyse the low energy dynamics of the theory using its dual. When $\frac{N_f}{2} \ge\frac{N_c}{2}-k$ the IR dynamics is described by either a TQFT for large quark masses, or a Grassmanian and a Wess-Zumino (WZ) term for small masses. We also consider non-supersymmetric mass deformations and RG flows in the vicinity of the SUSY point and find agreement between the IR of the electric and its magnetic dual. When $\frac{N_f}{2} < \frac{N_c}{2}-k$ supersymmetry is broken and the IR dynamics is a described by a TQFT accompanied by a Goldstino. We also discuss SQCD theories based on $SO$/$USp$ gauge groups.

### Aspects of Holographic Entanglement Entropy in Cubic Curvature Gravity

In this thesis we explore general aspects of the entanglement entropy (EE) for Conformal Field Theories (CFTs) dual to Cubic Curvature Gravity. We derived a covariant expression for the EE by using a scheme inherited from the bulk renormalization method through extrinsic counterterms. We evaluate this functional in different entangling regions to calculate CFT data. In particular, we compute the $t_4$ coefficient of the 3-point function of the stress-tensor correlator by considering a deformed entangling region. We observe that there is a discrepancy between the outcomes attained through the employment of the EE functional for minimal and non-minimal splittings. We find that only the $t_4$ obtained from the non-minimal functional agrees with previous results in the literature that were computed by splitting-independent CFT methods for specific theories such as the massless graviton case.

### Dynamical Compactification with Matter

In this work, we study cosmological solutions of the 8-dimensional Einstein Yang-Mills theory coupled to a perfect-fluid matter. A Yang-Mills instanton of extra dimensions causes a 4-dimensional expanding universe with dynamic compactification of the extra dimensions. To construct physically reliable situations, we impose the null energy condition on the matter. This energy condition is affected by the extra dimensions. Then, we consider cosmological constant to grasp the structure of the solution space. Even in this simple case, we find several interesting solutions, such as bouncing universes and oscillatory solutions, eventually arriving at a de Sitter universe with stabilized compact dimensions. In addition, we consider a class of matters whose energy density depends on the volume of the extra dimensions. This case shows another set of bouncing universes. Also, a real scalar with potential is taken into account. The scalar field model admits de Sitter solutions due to the choice of potential, and we demonstrate how potentials can be constructed using flow equations.

### Energy from Ellwood invariant for solutions involving $X^0$ variables

For some classical solutions $\Psi_\mathrm{sol}$ in Witten's bosonic string field theory, it was proven that energy of the solution is proportional to the Ellwood invariant $\mathrm{Tr}(\mathcal{V}\Psi_\mathrm{sol})$ with $\mathcal{V}=c\bar{c}\partial X^0\bar{\partial}X^0$. We examine the relation for solutions involving $X^0$ variables. As a result, we obtain that the relation may not hold for such solutions. Namely, there is a possibility that the energy is not proportional to the Ellwood invariant.

### 't Hooft lines of ADE-type and Topological Quivers

We investigate 4D Chern-Simons theory with ADE gauge symmetries in the presence of interacting Wilson and 't Hooft line defects. We analyse the intrinsic properties of these lines' coupling and explicate the building of oscillator-type Lax matrices verifying the RLL integrability equation. We propose gauge quiver diagrams Q$_{G}^{\mu }$ encoding the topological data carried by the Lax operators and give several examples where Darboux coordinates are interpreted in terms of topological bi-fundamental matter. We exploit this graphical description $\left( i\right)$ to give new results regarding solutions in representations beyond the fundamentals of $sl_{N}$, $% so_{2N}$ and $e_{6,7}$, and $\left( ii\right)$ to classify the Lax operators for simply laced symmetries in a unified E$_{7}$ CS theory. For quick access, a summary list of the leading topological quivers Q$% _{ADE}^{\mu }$ is given in the conclusion section [Figures 29.(a-e), 30.(a-d) and 31.(a-d)].

### One-loop algebras and fixed flow trajectories in adjoint multi-scalar gauge theory

We study the one loop renormalisation of 4d $SU(N)$ Yang-Mills theory with $M$ adjoint representation scalar multiplets related by $O(M)$ symmetry. General $M$ are of field theoretic interest, and the 4d one loop beta function of the gauge coupling $g^2$ vanishes for the case $M=22$, which is intriguing for string theory. This case is related to D3 branes of critical bosonic string theory in $D=22+4=26$. An RG fixed point could have provided a definition for a purely bosonic AdS/CFT, but we show that scalar self-couplings $\lambda$ ruin one-loop conformal invariance in the large $N$ limit. There are real fixed flows (fixed points of $\lambda/g^2$) only for $M\ge 406$, rendering one-loop fixed points of the gauge coupling and scalar couplings incompatible. We develop and check an algebraic approach to the one-loop renormalisation group which we find to be characterised by a non-associative algebra of marginal couplings. In the large $N$ limit, the resulting RG flows typically suffer from strong coupling in both the ultraviolet and the infrared. Only for $M\ge 406$ fine-tuned solutions exist which are weakly coupled in the infrared.

### Mutual information of subsystems and the Page curve for Schwarzschild de-Sitter black hole

In this work, we show that the two proposals associated to the mutual information of matter fields can be given for an eternal Schwarzschild black hole in de-Sitter spacetime. These proposals also depicts the status of associated entanglement wedges and their roleplay in obtaining the correct Page curve of radiation. The first proposal has been give for the before Page time scenario, which shows that the mutual information $I(R_{H}^{+}:R_{H}^{-})$ vanishes at a certain value of the observer's time $t_{b_{H}}=t_{H}$ (where $t_{H}\ll \beta_{H}$). We claim that this is the Hartman-Maldacena time at which the entanglement wedge associated to $R_{H}^{+}\cup R_{H}^{-}$ gets disconnected and the fine-grained radiation entropy has the form $S(R_{H})\sim \log(\beta_{H})$. The second proposal depicts the fact that just after the Page time, when the replica wormholes are the dominating saddle-points, the mutual information $I(B_{H}^{+}:B_{H}^{-})$ vanishes as soon as the time difference $t_{a_{H}}-t_{b_{H}}$ equals the scrambling time. Holographically, this reflects that the entanglement wedge associated to $B_{H}^{+}\cup B_{H}^{-}$ jumps to the disconnected phase at this particular time-scale. Furthermore, these two proposals lead us to the correct time-evolution of the fine-grained entropy of radiation as portrayed by the Page curve. We have also shown that similar observations can be obtained for the radiation associated to the cosmological horizon.

### On Perturbative Quantum Gravity with a Cosmological Constant

We discuss how the incorporation of a cosmological constant affects the perturbative quantization of (effective) Quantum General Relativity. To this end, we derive the gravitational Slavnov--Taylor identities and appropriate renormalization conditions for the cosmological constant. Additionally, we calculate the corresponding Feynman rules for any vertex valence and with general gauge parameter. Furthermore, we provide the BRST setup and generate the Faddeev--Popov ghost and the symmetric ghost via a gauge fixing fermion and a gauge fixing boson, respectively. Finally, we study the transversality of the graviton propagator and the graviton three-valent vertex.

### Transformations of currents in sigma-models with target space supersymmetry

We develop a framework for systematic study of symmetry transformations of sigma-model currents in a special situation, when symmetries have a well-defined projection onto the target space. We then apply this formalism to pure spinor sigma-models, and describe the resulting geometric structures in the target space (which in our approach includes the pure spinor ghosts). We perform a detailed study of the transformation properties of currents, using the formalism of equivariant cohomology. We clarify the general structure of the dilaton zero mode, study the contact terms in the OPE of BRST currents, and derive some relations between currents and vertex operators which perhaps have not been previously acknowledged. We also clarify the geometrical meaning of the " minimalistic " BV action for pure spinors in AdS.

### Coupled vector Gauss-Bonnet theories and hairy black holes

We study vector-tensor theories in which a 4-dimensional vector field $A_{\mu}$ is coupled to a vector quantity ${\cal J}^{\mu}$, which is expressed in terms of $A_{\mu}$ and a metric tensor $g_{\mu \nu}$. The divergence of ${\cal J}^{\mu}$ is equivalent to a Gauss-Bonnet (GB) term. We show that an interacting Lagrangian of the form $f(X)A_{\mu}{\cal J}^{\mu}$, where $f$ is an arbitrary function of $X=-(1/2)A_{\mu}A^{\mu}$, belongs to a scheme of beyond generalized Proca theories. For $f(X)=\alpha={\rm constant}$, this interacting Lagrangian reduces to a particular class of generalized Proca theories. We apply the latter coupling to a static and spherically symmetric vacuum configuration by incorporating the Einstein-Hilbert term, Maxwell scalar, and vector mass term $\eta X$ ($\eta$ is a constant). Under an expansion of the small coupling constant $\alpha$ with $\eta \neq 0$, we derive hairy black hole solutions endowed with nonvanishing temporal and radial vector field profiles. The asymptotic properties of solutions around the horizon and at spatial infinity are different from those of hairy black holes present in scalar-GB theories. We also show that black hole solutions without the vector mass term, i.e., $\eta=0$, are prone to ghost instability of odd-parity perturbations.

### Apparent dark matter as a non-local manifestation of emergent gravity

We disclose a close correspondence between Verlinde's Emergent Gravity (VEG) theory and the non-local gravity theories. Such non-local effects can play crucial role at small distances as well as in large scale structures. In particular, we argue that the emergent gravity effectively is a manifestation of the entanglement entropy and can modify Newton's law of gravity as well as address the flat rotation curves of spiral galaxies. In the cosmological setup, we have considered three different models for the apparent dark matter density. In the first model, we have found that Friedmann equations get modified due to the presence of the apparent dark matter (DM) in such a way that Newton's constant of gravity shifts as $G\rightarrow G_N=G\left(1+\zeta\right)$, where $\zeta$ is a dimensionless small parameter. This modification basically coincides with the modified gravity (MOG) theory. Using the flat rotating curves we estimate $\zeta \sim 10^{-7}$. Interestingly enough, for such a model, we find out that by rescaling the radial coordinate, $r$, the curvature space constant, $k$, and the scale factor of the universe, $a$, the effect of apparent DM can change the geometry of the universe and can shift the curvature space constant as $k_{\star}=k (1-\zeta)$. Finally, we study a more realistic model applied to the whole universe with evolving densities and we address the Hubble tension problem in the context of Verlindes emergent gravity using the look-back time quantity.

### Shannon entropy in quasiparticle states of quantum chains

In this paper, we investigate the Shannon entropy of the total system and its subsystems, as well as the subsystem Shannon mutual information, in quasiparticle excited states of free bosonic and fermionic chains and the ferromagnetic phase of the spin-1/2 XXX chain. Our focus is on single-particle and double-particle states, and we derive various analytical formulas for free bosonic and fermionic chains in the scaling limit. These formulas are also applicable to magnon excited states in the XXX chain under certain conditions. We discover that, unlike entanglement entropy, Shannon entropy does not separate when two quasiparticles have a large momentum difference. Moreover, in the large momentum difference limit, we obtain universal results for quantum spin chains that cannot be explained by a semiclassical picture of quasiparticles.

### Some generalizations of Mirzakhani's recursion and Masur-Veech volumes via topological recursions

The Chekhov-Eynard-Orantin's topological recursion gives a Laplace dual representation of the Mirzakhani's recursion for the moduli space of bordered hyperbolic Riemann surfaces. Via the Andersen-Borot-Orantin's geometric recursion, a twist of the topological recursion is proposed, and a recursion for the Masur-Veech polynomials is uncovered. The purpose of this article is to explore the generalization of the Mirzakhani's recursion based on physical two dimensional gravity models related to the Jackiw-Teitelboim gravity. For generalized Mirzakhani's recursions involving a Masur-Veech type twist, we derive Virasoro constraints and cut-and-join equations, and also show some computations of generalized volumes for the physical two dimensional gravity models.

### Non-Gaussianity in rapid-turn multi-field inflation

We show that theories of inflation with multiple, rapidly turning fields can generate large amounts of non-Gaussianity. We consider a general theory with two fields, an arbitrary field-space metric, and a potential that supports sustained, rapidly turning field trajectories. Our analysis accounts for non-zero field cross-correlation and does not fix the power spectra of curvature and isocurvature perturbations to be equal at horizon crossing. Using the $\delta N$ formalism, we derive a novel, analytical formula for bispectrum generated from multi-field mixing on super-horizon scales. Rapid-turn inflation can produce a bispectrum with several potentially large contributions that are not necessarily of the local shape. We exemplify the applicability of our formula with a fully explicit model and show that the new contributions indeed can generate a large amplitude of local non-Gaussianity, $f_{\rm NL}^{\rm loc}\sim {\cal O}(1)$. These results will be important when interpreting the outcomes of future observations.