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We consider the von Neumann entropy of a thermal mixed state in quantum systems derived from mirror curves, where the kinetic terms are exponential functions of the momentum operators. Using the mathematical results on the asymptotics of the energy eigenvalues, we compute the asymptotic entropy in high temperature limit and compare with that of the conventional models. We discuss the connections with some folklores in quantum gravity, particularly on the finiteness of entropy., 10 pages, no figure

Recently, a novel measure for the complexity of operator growth is proposed based on Lanczos algorithm and Krylov recursion method. We study this Krylov complexity in quantum mechanical systems derived from some well-known local toric Calabi-Yau geometries, as well as some non-relativistic models. We find that for the Calabi-Yau models, the Lanczos coefficients grow slower than linearly for small $n$'s, consistent with the behavior of integrable models. On the other hand, for the non-relativistic models, the Lanczos coefficients initially grow linearly for small $n$'s, then reach a plateau. Although this looks like the behavior of a chaotic system, it is mostly likely due to saddle-dominated scrambling effects instead, as argued in the literature. In our cases, the slopes of linearly growing Lanczos coefficients almost saturate a bound by the temperature. During our study, we also provide an alternative general derivation of the bound for the slope., 26 pages, 26 figures. v2: some significant improvements on numerical calculations, conclusions modified, references added

We propose a novel modular anomaly equation for the unflavored Schur index in the $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory. The vanishing conditions overdetermine the modular ambiguity ansatz from the equation, thus together they are sufficient to recursively compute the exact Schur indices for all $SU(N)$ gauge groups. Using the representations as MacMahon's generalized sum-of-divisors functions and Jacobi forms, we then prove our proposal as well as elucidate a general formula conjectured by Pan and Peelaers., Comment: 14 pages. v2: journal version, some improvements, references added

Recently, a novel bootstrap method for numerical calculations in matrix models and quantum mechanical systems is proposed. We apply the method to certain quantum mechanical systems derived from some well-known local toric Calabi-Yau geometries, where the exact quantization conditions have been conjecturally related to topological string theory. We find that the bootstrap method provides a promising alternative for the precision numerical calculations of the energy eigenvalues. An improvement in our approach is to use a larger set of two-dimensional operators instead of one-dimensional ones. We also apply our improved bootstrap methods to some non-relativistic models in the recent literature and demonstrate better numerical accuracies., 21 pages, 21 figures

We study the instanton partition functions of two well-known superconformal field theories with mass deformations. Two types of anomaly equations, namely, the modular anomaly and holomorphic anomaly, have been discovered in the literature. We provide a clean solution to the long standing puzzle about their precise relation, and obtain some universal formulas. We show that the partition function is invariant under the SL(2,Z) duality which exchanges theories at strong coupling with those of weak coupling., Comment: 5 pages. v2: journal version

We extend the direct integration method of the holomorphic anomaly equations to general Omega backgrounds for pure SU(2) N=2 Super-Yang-Mills theory and topological string theory on non-compact Calabi-Yau threefolds. We find that an extension of the holomorphic anomaly equation, modularity and boundary conditions provided by the perturbative terms as well as by the gap condition at the conifold are sufficient to solve the generalized theory in the above cases. In particular we use the method to solve the topological string for the general Omega backgrounds on non-compact toric Calabi-Yau spaces. The conifold boundary condition follows from that the N=2 Schwinger-Loop calculation with BPS states coupled to a self-dual and an anti-self-dual field strength. We calculate such BPS states also for the decompactification limit of Calabi-Yau spaces with regular K3 fibrations and half K3s embedded in Calabi-Yau backgrounds., 37 pages, 2 pstex figures

We consider the transmission of classical information over a quantum channel by many senders, which is a generalization of the two-sender case. The channel capacity region is shown to be a convex hull bound by the von Neumann entropy and the conditional von Neumann entropies. The result allows a reasonable distribution of channel capacity over the senders.

We analyze the trajectories for multifield DBI inflation, which can arise in brane inflation models, and show that the trajectories are the same as in typical slow roll inflation. We calculate the power spectrum and find that the higher derivative terms of the DBI action lead to a suppression of the contribution from the isocurvature perturbations. We also calculate the bispectrum generated by the isocurvature perturbation, and find that it leads to distinctive features., 7 pages, 4 figures; v2. references added; v3. typos corrected, matches published version

We compute the leading order contribution to the four-point function of the primordial curvature perturbation in a class of single field models where the inflationary Lagrangian is a general function of the inflaton and its first derivative. This class of models includes string motivated inflationary models such as DBI inflation. We find that the trispectrum for some range of parameters could potentially be observed in future experiments. Moreover, the trispectrum can distinguish DBI inflation from other inflation models with large non-Gaussianities which typically have a similar bispectrum. We also derive a set of consistency conditions for n-point functions of the primordial curvature perturbation in single field inflation, generalizing Maldacena's result for 3-point functions., v2: 5 pages, journal version. v3, v4: formulae for the momentum dependence of trispectrum corrected, main conclusions remain unchanged, this version supersedes the journal version. v5: minor revision with an added reference

The topological string partition function Z=exp(lambda^{2g-2} F_g) is calculated on a compact Calabi-Yau M. The F_g fulfill the holomorphic anomaly equations, which imply that Z transforms as a wave function on the symplectic space H^3(M,Z). This defines it everywhere in the moduli space of M along with preferred local coordinates. Modular properties of the sections F_g as well as local constraints from the 4d effective action allow us to fix Z to a large extend. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovos theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic., Comment: LaTeX, 57 pages, 2 eps-figures

This thesis consists of two parts. In the first part we study some topics in $\mathcal{N}=1$ supersymmeric gauge theory and the relation to matrix models. We review the relevant non-perturbative techniques for computing effective superpotential, such as Seiberg-Witten curve. Then we review the proposal of Dijkgraaf and Vafa that relates the glueball superpotentials to the computation in matrix models. We then consider a case of multi-trace superpotential. We perform the perturbative computation of glueball superpotential in this case and explain the subtlety in identifying the glueball superfield. We also use these techniques to study phases of $\mathcal{N}=1$ gauge theory with flavors. In the second part we study topics in AdS/CFT correspondence and its plane wave limit. We review the plane wave geometry and BMN operators that corresponding to string modes. Then we study string interactions in the case of a highly curved plane wave background, and demonstrate the agreements between calculations of string interaction amplitudes in the two dual theories. Finally we study D3-brane giant gravitons and open string attached to them. Giant gravitons are non-perturbative objects that have very large R-charge., Comment: Ph.D. Thesis. 115 pages. v2: references added

We consider non-planar contributions to the correlation functions of BMN operators in free N = 4 super Yang Mills theory. We recalculate these non-planar contributions from a different kind of diagram and find some exact agreements. The vertices of these diagrams are represented by free planar three point functions, thus our calculations provide some interesting identities for correlation functions of BMN operators in N = 4 super Yang Mills theory. These diagrams look very much like loop diagrams in a second quantized string field theory, thus these identities could possibly be interpreted as natural consequences of the pp-wave/CFT correspondence., 21 pages, 10 figures. V3: journal version, slight changes in abstract and introduction. V4: minor typos and correction

Exploiting insights on strings moving in pp-wave backgrounds, we show how open strings emerge from N = 4 SU(M) Yang-Mills theory as fluctuations around certain states carrying R-charge of order M. These states are dual to spherical D3-branes of AdS_5 x S^5 and we reproduce the spectrum of small fluctuations of these states from Yang Mills theory. When G such D3-branes coincide, the expected G^2 light degrees of freedom emerge. The open strings running between the branes can be quantized easily in a Penrose limit of the spacetime. Taking the corresponding large charge limit of the Yang-Mills theory, we reproduce the open string worldsheets and their spectra from field theory degrees of freedom., V4: Typos and anomalous dimension calculation corrected, 20 pages

In this paper we calculate three-string interaction from light cone string field theory in pp-wave. We find exact agreements with the free planar three point functions of non-chiral BMN operators of ${\cal N} = 4$ super Yang Mills. The three string interaction vertex involving the Neumann matrices was derived in a recent paper hep-th/0204146. We explicitly calculate the bosonic Neumann matrices in the limit of large $\mu p^{+} \alpha^{'}$ . Using the Neumann matrices we are able to compute the cubic interactions of three string modes in a pp-wave background., Comment: 9 pages, no figure. V2: typos corrected, some changes in introduction, a mistake in Section 2 and Appendix corrected. V3: more typos corrected and minor changes