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Register a new userSymmetries of the refined D1/D5 BPS spectrum
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We examine the large $N$ 1/4-BPS spectrum of the symmetric orbifold CFT Sym$^N(M)$ deformed to the supergravity point in moduli space for $M= K3$ and $T^4$. We consider refinement under both left- and right-moving $SU(2)_R$ symmetries of the superconformal algebra, and decompose the spectrum into characters of the algebra. We find that at large $N$ the character decomposition satisfies an unusual property, in which the degeneracy only depends on a certain linear combination of left- and right-moving quantum numbers, suggesting deeper symmetry structure. Furthermore, we consider the action of discrete symmetry groups on these degeneracies, where certain subgroups of the Conway group are known to play a role. We also comment on the potential for larger discrete symmetry groups to appear in the large $N$ limit.
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We consider states of the D1-D5 CFT where only the left-moving sector is excited. As we deform away from the orbifold point, some of these states will remain BPS while others can `lift. We compute this lifting for a particular family of D1-D5-P states, at second order in the deformation off the orbifold point. We note that the maximally twisted sector of the CFT is special: the covering surface appearing in the correlator can only be genus one while for other sectors there is always a genus zero contribution. We use the results to argue that fuzzball configurations should be studied for the full class including both extremal and near-extremal states; many extremal configurations may be best seen as special limits of near extremal configurations.
We briefly review the microscopic modeling of black holes as bound states of branes in the context of the soluble D1-D5 system. We present a discussion of the low energy brane dynamics and account for black hole thermodynamics and Hawking radiation rates. These considerations are valid in the regime of supergravity due to the non-renormalization of the low energy dynamics in this model. Using Maldacena duality and standard statistical mechanics methods one can account for black hole thermodynamics and calculate the absorption cross section and the Hawking radiation rates. Hence, at least in the case of this model black hole, since we can account for black hole properties within a unitary theory, there is no information paradox.
In the context of the fuzzball programme, we investigate deforming the microscopic string description of the D1-D5 system on T^4xS^1 away from the orbifold point. Using conformal perturbation theory and a generalization of Lunin-Mathur symmetric orbifold technology for computing twist-nontwist correlators developed in a companion work, we initiate a program to compute the anomalous dimensions of low-lying string states in the D1-D5 superconformal field theory. Our method entails finding four-point functions involving a string operator O of interest and the deformation operator, taking coincidence limits to identify which other operators mix with O, subtracting the identified conformal family to isolate other contributions to the four-point function, finding the mixing coefficients, and iterating. For the lowest-lying string modes, this procedure should truncate in a finite number of steps. We check our method by showing how the operator dual to the dilaton does not participate in mixing that would change its conformal dimension, as expected. Next we complete the first stage of the iteration procedure for a low-lying string state of the form partial X partial X barpartial X barpartial X and find its mixing coefficient. Our main qualitative result is evidence of operator mixing at first order in the deformation parameter, which means that the string state acquires an anomalous dimension. After diagonalization this will mean that anomalous dimensions of some string states in the D1-D5 SCFT must decrease away from the orbifold point while others increase.
Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $mathbb{P}^2$, and make them explicit for $U(4)$.
F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function $Z_{top}$ of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing $Z_{top}$ in terms of contributions $Z_beta$ at base degree $beta$ of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kaehler class of the elliptic fiber and elliptic parameters the couplings and mass parameters. The indices with regard to the multiple elliptic parameters are fixed by the refined holomorphic anomaly equations, which we show to be completely determined from knowledge of the chiral anomaly of the corresponding SCFT. We express $Z_beta$ as a quotient of weak Jacobi forms, with a universal denominator inspired by its pole structure as suggested by the form of $Z_{top}$ in terms of 5d BPS numbers. The numerator is determined by modularity up to a finite number of coefficients, which we prove to be fixed uniquely by imposing vanishing conditions on 5d BPS numbers as boundary conditions. We demonstrate the feasibility of our approach with many examples, in particular solving the E-string and M-string theories including mass deformations, as well as theories constructed as chains of these. We make contact with previous work by showing that spurious singularities are cancelled when the partition function is written in the form advocated here. Finally, we use the BPS invariants of the E-string thus obtained to test a generalization of the Goettsche-Nakajima-Yoshioka $K$-theoretic blowup equation, as inspired by the Grassi-Hatsuda-Marino conjecture, to generic local Calabi-Yau threefolds.
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