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During cosmic timescales, supermassive binary black holes (SMBBHs) form by galaxy merg- ers, where each galaxy hosts a supermassive black hole (SMBH) at its center. By studying SMBBHs, we can gain insights into galaxy evolution and black hole growth. However, the typical separation between black holes in SMBBHs is usually below 1 pc, making them dif- ficult to resolve using direct imaging or photometry. To be able to distinguish binary black holes (BBHs) from typical AGN powered by single black holes (SBHs), we conducted this research project to develop a new diagnostic method to identify a unique feature of SMBBHs, which can be used to distinguish AGN powered by BBHs from those powered by SBHs. The basic idea of this method is that BBHs have different configurations compared with those of SBHs, such as the circumbinary disk enveloping the whole binary system, the mini- disks around each black hole, and the streams between the circumbinary disk and minidisks. It is these different configurations of accretion disks of black holes that lead to the difference in the spectral energy distributions (SEDs), which in turn will differently photoionize the broad line region (BLR) and produce different strengths of broad emission lines. Thus, it is these differences in strengths of broad emission lines that we expect to see between two different configurations of black holes, either binary or single black holes. By identifying these differences in line strengths, we can distinguish between binary and single black holes. In order to achieve this goal of distinguishing BBHs from SBHs, we organized our project in the steps below: (1) obtained BBH SEDs from Gutiérrez et al., 2022, (2) generated SBH SEDs using XSPEC modeling code OPTXAGNF, (3) produced single-cloud models, which represent the BLR, input BBH and SBH SEDs into the models, and simulated the single0cloud response with a photoionization code Cloudy, (4) built cloud-ensemble models, which represent a more realistic BLR, input BBH and SBH SEDs into the models, and simulated the emission-line response within those clouds using a broad emission line mapping code BELMAC. In step (3), we have explored the differences in line ratios between BBH and SBH for a representative single-cloud photoionization model. The emission lines we used here are: Si IV λ1400Å, C III] λ1909Å, C IV λ1549Å, Mg II λ2798Å, and Lyα λ1216Å. It turned i out that differences do exist, but they are too small to be identified in observational data. Furthermore, we have investigated the line equivalent widths predicted by SBH and BBH models respectively. By doing so, we found some apparent differences between BBHs and SBHs in some specific emission lines: Lyα, CIV, and Hα. However, these differences vanish at the highest mass of black holes (109M⊙). In step (4), we continued the investigation of the equivalent width between SBH and BBH BLR cloud-ensemble models and found that some emission lines show the difference between BBHs and SBHs, such as CIV in the case of BH mass 107M⊙, U ∝ r−2, and log n/1 cm−3 = 10.5, 11.0. For the highest mass in the case of U ∝ r−2, the results are similar to the one in single-cloud models: no difference is shown between BBHs and SBHs across all emission lines. Most importantly, in this step, we found that for CIV λ1549 Å in the case of a black hole mass of 107M⊙, s = −2, and log n/1 cm−3 = 10.5, only the SBH EW falls inside the range of the observed range in SDSS DR7 Quasar Catalog while the BBH EW falls outside the range and becomes an outlier. This is what we want to find to distinguish BBHs from SBHs in the observational data. 

A bstract We study monodromy defects in O ( N ) symmetric scalar field theories in d dimensions. After a Weyl transformation, a monodromy defect may be described by placing the theory on S 1 × H d− 1 , where H d− 1 is the hyperbolic space, and imposing on the fundamental fields a twisted periodicity condition along S 1 . In this description, the codimension two defect lies at the boundary of H d− 1 . We first study the general monodromy defect in the free field theory, and then develop the large N expansion of the defect in the interacting theory, focusing for simplicity on the case of N complex fields with a one-parameter monodromy condition. We also use the ϵ -expansion in d = 4 − ϵ , providing a check on the large N approach. When the defect has spherical geometry, its expectation value is a meaningful quantity, and it may be obtained by computing the free energy of the twisted theory on S 1 × H d− 1 . It was conjectured that the logarithm of the defect expectation value, suitably multiplied by a dimension dependent sine factor, should decrease under a defect RG flow. We check this conjecture in our examples, both in the free and interacting case, by considering a defect RG flow that corresponds to imposing alternate boundary conditions on one of the low-lying Kaluza-Klein modes on H d− 1 . We also show that, adapting standard techniques from the AdS/CFT literature, the S 1 × H d− 1 setup is well suited to the calculation of the defect CFT data, and we discuss various examples, including one-point functions of bulk operators, scaling dimensions of defect operators, and four-point functions of operator insertions on the defect.