Blowup Equations for 6d SCFTs. I

We propose novel functional equations for the BPS partition functions of 6d (1,0) SCFTs, which can be regarded as an elliptic version of Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic Calabi-Yau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In companion papers, we will study the elliptic blowup equations for all other non-Higgsable clusters.Comment: 52 pages, 3 figure

Compact Visibility Representation of Plane Graphs

The visibility representation (VR for short) is a classical representation of plane graphs. It has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. It is known that there exists a plane graph $G$ with $n$ vertices where any VR of $G$ requires a grid of size at least (2/3)n x((4/3)n-3) (width x height). For upper bounds, it is known that every plane graph has a VR with grid size at most (2/3)n x (2n-5), and a VR with grid size at most (n-1) x (4/3)n. It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely with size at most c_h n x c_w n with c_h < 1 and c_w < 2$). In this paper, we provide the first VR construction with this property. We prove that every plane graph of n vertices has a VR with height <= max{23/24 n + 2 Ceil(sqrt(n))+4, 11/12 n + 13} and width <= 23/12 n. The area (height x width) of our VR is larger than the area of some of previous results. However, bounding one dimension of the VR only requires finding a good st-orientation or a good dual s^*t^*-orientation of G. On the other hand, to bound both dimensions of VR simultaneously, one must find a good$st$-orientation and a good dual s^*t^*-orientation at the same time, and thus is far more challenging. Since st-orientation is a very useful concept in other applications, this result may be of independent interests

Revisiting the distance, environment and supernova properties of SNR G57.2+0.8 that hosts SGR 1935+2154

We have performed a multi-wavelength study of supernova remnant (SNR) G57.2+0.8 and its environment. The SNR hosts the magnetar SGR 1935+2154, which emitted an extremely bright ms-duration radio burst on 2020 Apr 28 (The Chime/Frb Collaboration et al. 2020; Bochenek et al. 2020). We used the 12CO and 13CO J=1-0 data from the Milky Way Image Scroll Painting (MWISP) CO line survey to search for molecular gas associated with G57.2+0.8, in order to constrain the physical parameters (e.g., the distance) of the SNR and its magnetar. We report that SNR G57.2+0.8 is likely impacting the molecular clouds (MCs) at the local standard of rest (LSR) velocity V_{LSR} ~ 30 km/s and excites a weak 1720 MHz OH maser with a peak flux density of 47 mJy/beam. The chance coincidence of a random OH spot falling in the SNR is <12%, and the OH-CO correspondence chance is 7% at the maser spot. This combines to give < 1% false probability of the OH maser, suggesting a real maser detection. The LSR velocity of the MCs places the SNR and magnetar at a kinematic distance of d=6.6 +/- 0.7 kpc. The nondetection of thermal X-ray emission from the SNR and the relatively dense environment suggests G57.2+0.8 be an evolved SNR with an age $t>1.6 \times 10^4$ (d/6.6 kpc) yr. The explosion energy of G57.2+0.8 is lower than $2 \times 10^{51}(n_0/10 cm^{-3})^{1.16} (d/~6.6 kpc)^{3.16}$ erg, which is not very energetic even assuming a high ambient density $n_0$ = 10 cm$^{-3}$. This reinforces the opinion that magnetars do not necessarily result from very energetic supernova explosions.Comment: 9 pages, 5 figures, accepted for publication in the Astrophysical Journa

The linearized second law for any higher curvature gravity with the scalar and the electromagnetic fields

The first law of black hole thermodynamics is suitable for any diffeomorphism invariant gravity, and the entropy in the first law is the Wald entropy which is highly dependent on the non-minimal coupling interactions in the theory of gravity. However, whether the Wald entropy still satisfies the second law needs to be investigated. The entropy of black holes obeying the linearized second law in arbitrary high-order curvature gravity is given, which can be written as the Wald entropy with correction terms. It indicates that the Wald entropy is not commonly obeying the linearized second law for any high-order curvature gravity. When the interactions of gravity with matter fields are included in the theory of gravity, the entropy of black holes obeying the linearized second law has not been obtained in this case. Considering any high-order curvature gravity with the scalar and the electromagnetic fields, from the Raychaudhuri equation, the entropy obeying the linearized second law is generally obtained, which can be expressed as the Wald entropy with correction terms as well. The entropy does not include the contribution from the electromagnetic fields, and the correction terms contain the contribution from the minimal coupling interaction between gravity and the scalar fields. Since the entropy satisfying the linearized second law depends only on the non-minimal coupling interaction of gravity in previous research, this result upends our understanding of the entropy of black holes obeying the linearized second law in any gravitational theory with matter fields.Comment: 17 page