We study a rich set of four-dimensional superconformal field theories with both central charges i... more We study a rich set of four-dimensional superconformal field theories with both central charges identical: a ¼ c. These are constructed via the diagonal N ¼ 2 or N ¼ 1 gauging of the flavor symmetry G of a collection of N ¼ 2 Argyres-Douglas theories of type D p ðGÞ, with or without adjoint chiral multiplets, in 2106.12579 and 2111.12092. We compute superconformal indices of some theories where the rank of G is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in detail. We find that most of these theories flow to interacting superconformal field theories with a ¼ c in the infrared.
Communications in Mathematical Physics, Jan 24, 2023
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum erro... more We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to e... more We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to elliptically fibered fourfolds Y dual in F-theory to a gauge theory with gauge group G. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. It follows that Chern and Pontryagin numbers are independent on a choice of a crepant resolution. We present the results for the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, theÂ-genus, and the curvature invariant X 8 that appears in M-theory. We also show that certain characteristic classes are independent on the choice of the Kodaria fiber characterizing the group G. That is the case of ∫ Y c 2 1 c 2 , the arithmetic genus, and theÂ-genus. Thus, it is enough to know ∫ Y c 2 2 and the Euler characteristic χ(Y) to determine all the Chern numbers of an elliptically fibered fourfold. We consider the cases of G = SU(n) for (n = 2, 3, 4, 5, 6, 7), USp(4), Spin(7), Spin(8)
We provide an overview of the string landscape and the Swampland program. Our review of the strin... more We provide an overview of the string landscape and the Swampland program. Our review of the string landscape covers the worldsheet and spacetime perspectives, including vacua and string dualities. We then review and motivate the Swampland program from the lessons learned from the string landscape. These lecture notes are aimed to be self-contained and thus can serve as a starting point for researchers interested in exploring these ideas. These notes are an expanded version of two courses The String Landscape and the Swampland taught by C. Vafa at Harvard University in 2018 with a focus on the landscape, written by M. J. Kang with additional material from N. B. Agmon, and in 2022 with a focus on the Swampland, written by A. Bedroya.
We study a rich set of four-dimensional superconformal field theories (SCFTs) with both central c... more We study a rich set of four-dimensional superconformal field theories (SCFTs) with both central charges identical: a = c. These are constructed via the diagonal N = 2 or N = 1 gauging of the flavor symmetry G of a collection of N = 2 Argyres-Douglas theories of type D p (G), with or without adjoint chiral multiplets, in 2106.12579 and 2111.12092. We compute superconformal indices of some theories where the rank of G is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in detail. We find that most of these theories flow to interacting SCFTs with a = c in the infrared.
The Higgs branch of 4d N = 2 SCFTs can be analyzed via the Hilbert series of the Higgs branch or,... more The Higgs branch of 4d N = 2 SCFTs can be analyzed via the Hilbert series of the Higgs branch or, in special cases, by computing the Hall-Littlewood index. For any class S theory corresponding to a genus-zero Riemann surface, they are conjectured to be identical. We present several families of counterexamples. We find that for any class S theory with four or more Z 2-twisted punctures, they do not match. We construct 3d mirrors for such theories and analyze their Coulomb branch Hilbert series to compute the Higgs branch Hilbert series of the 4d theory. We further construct a = c theories in class S using the twisted punctures, and these theories, which includes the D 4 (SU (2n + 1)) theories, have Hall-Littlewood index different from the Hilbert series of the Higgs branch. We conjecture that this is the case for all a = c theories with non-empty Higgs branch, including N ≥ 3 SCFTs.
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-tri... more We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E 9−d with d = 1, 2, 3, 4 whose generic fibers are normal elliptic curves of degree d. We then analyze the characteristic numbers of the Q 7-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E 5 , E 6 , and E 7-models. Finally, we examine the characteristic numbers of G-models with G = SO(n) with n = 3, 4, 5, 6 and G = PSU(3) whose Mordell-Weil groups are respectively Z 2Z and Z 3Z. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.
We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb ... more We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefold with an I 2 +I 4 collision of singularities. The resulting gauge theory has a semi-simple Lie algebra su(2) ⊕ sp(4) or su(2) ⊕ su(4). We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representation by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is Z 2 but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplift. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.
We consider all 4D N ¼ 2 theories of class S arising from the compactification of exceptional 6D ... more We consider all 4D N ¼ 2 theories of class S arising from the compactification of exceptional 6D (2, 0) SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4D theories has another origin as a 6D (1, 0) SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6D theory is identified with a complex structure deformation of ðe n ; e n Þ minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class S construction. We explain how the plurality of origins makes manifest some aspects of 4D SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4D theories from different origins via mirror symmetry.
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum erro... more We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
We consider all 4d N = 2 theories of class S arising from the compactification of exceptional 6d ... more We consider all 4d N = 2 theories of class S arising from the compactification of exceptional 6d (2, 0) SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d (1, 0) SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of (e n , e n) minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class S construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.
We consider a d-dimensional unitary conformal field theory with a compact Lie group global symmet... more We consider a d-dimensional unitary conformal field theory with a compact Lie group global symmetry G and show that, at high temperature T and on a compact Cauchy surface, the probability of a randomly chosen state being in an irreducible unitary representation R of G is proportional to (dim R) 2 exp[−c 2 (R)/(b T d−1)]. We use the spurion analysis to derive this formula and relate the constant b to a domain wall tension. We also verify it for free field theories and holographic conformal field theories and compute b in these cases. This generalizes the result in 2109.03838 that the probability is proportional to (dim R) 2 when G is a finite group. As a by-product of this analysis, we clarify thermodynamical properties of black holes with non-abelian hair in anti-de Sitter space.
We study a rich set of four-dimensional superconformal field theories with both central charges i... more We study a rich set of four-dimensional superconformal field theories with both central charges identical: a ¼ c. These are constructed via the diagonal N ¼ 2 or N ¼ 1 gauging of the flavor symmetry G of a collection of N ¼ 2 Argyres-Douglas theories of type D p ðGÞ, with or without adjoint chiral multiplets, in 2106.12579 and 2111.12092. We compute superconformal indices of some theories where the rank of G is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in detail. We find that most of these theories flow to interacting superconformal field theories with a ¼ c in the infrared.
Communications in Mathematical Physics, Jan 24, 2023
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum erro... more We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to e... more We compute characteristic numbers of crepant resolutions of Weierstrass models corresponding to elliptically fibered fourfolds Y dual in F-theory to a gauge theory with gauge group G. In contrast to the case of fivefolds, Chern and Pontryagin numbers of fourfolds are invariant under crepant birational maps. It follows that Chern and Pontryagin numbers are independent on a choice of a crepant resolution. We present the results for the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, theÂ-genus, and the curvature invariant X 8 that appears in M-theory. We also show that certain characteristic classes are independent on the choice of the Kodaria fiber characterizing the group G. That is the case of ∫ Y c 2 1 c 2 , the arithmetic genus, and theÂ-genus. Thus, it is enough to know ∫ Y c 2 2 and the Euler characteristic χ(Y) to determine all the Chern numbers of an elliptically fibered fourfold. We consider the cases of G = SU(n) for (n = 2, 3, 4, 5, 6, 7), USp(4), Spin(7), Spin(8)
We provide an overview of the string landscape and the Swampland program. Our review of the strin... more We provide an overview of the string landscape and the Swampland program. Our review of the string landscape covers the worldsheet and spacetime perspectives, including vacua and string dualities. We then review and motivate the Swampland program from the lessons learned from the string landscape. These lecture notes are aimed to be self-contained and thus can serve as a starting point for researchers interested in exploring these ideas. These notes are an expanded version of two courses The String Landscape and the Swampland taught by C. Vafa at Harvard University in 2018 with a focus on the landscape, written by M. J. Kang with additional material from N. B. Agmon, and in 2022 with a focus on the Swampland, written by A. Bedroya.
We study a rich set of four-dimensional superconformal field theories (SCFTs) with both central c... more We study a rich set of four-dimensional superconformal field theories (SCFTs) with both central charges identical: a = c. These are constructed via the diagonal N = 2 or N = 1 gauging of the flavor symmetry G of a collection of N = 2 Argyres-Douglas theories of type D p (G), with or without adjoint chiral multiplets, in 2106.12579 and 2111.12092. We compute superconformal indices of some theories where the rank of G is low, performing a refined test for unitarity, and further determine the relevant and marginal operator content in detail. We find that most of these theories flow to interacting SCFTs with a = c in the infrared.
The Higgs branch of 4d N = 2 SCFTs can be analyzed via the Hilbert series of the Higgs branch or,... more The Higgs branch of 4d N = 2 SCFTs can be analyzed via the Hilbert series of the Higgs branch or, in special cases, by computing the Hall-Littlewood index. For any class S theory corresponding to a genus-zero Riemann surface, they are conjectured to be identical. We present several families of counterexamples. We find that for any class S theory with four or more Z 2-twisted punctures, they do not match. We construct 3d mirrors for such theories and analyze their Coulomb branch Hilbert series to compute the Higgs branch Hilbert series of the 4d theory. We further construct a = c theories in class S using the twisted punctures, and these theories, which includes the D 4 (SU (2n + 1)) theories, have Hall-Littlewood index different from the Hilbert series of the Higgs branch. We conjecture that this is the case for all a = c theories with non-empty Higgs branch, including N ≥ 3 SCFTs.
We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-tri... more We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E 9−d with d = 1, 2, 3, 4 whose generic fibers are normal elliptic curves of degree d. We then analyze the characteristic numbers of the Q 7-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E 5 , E 6 , and E 7-models. Finally, we examine the characteristic numbers of G-models with G = SO(n) with n = 3, 4, 5, 6 and G = PSU(3) whose Mordell-Weil groups are respectively Z 2Z and Z 3Z. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.
We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb ... more We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefold with an I 2 +I 4 collision of singularities. The resulting gauge theory has a semi-simple Lie algebra su(2) ⊕ sp(4) or su(2) ⊕ su(4). We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representation by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is Z 2 but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplift. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.
We consider all 4D N ¼ 2 theories of class S arising from the compactification of exceptional 6D ... more We consider all 4D N ¼ 2 theories of class S arising from the compactification of exceptional 6D (2, 0) SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4D theories has another origin as a 6D (1, 0) SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6D theory is identified with a complex structure deformation of ðe n ; e n Þ minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class S construction. We explain how the plurality of origins makes manifest some aspects of 4D SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4D theories from different origins via mirror symmetry.
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum erro... more We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
We consider all 4d N = 2 theories of class S arising from the compactification of exceptional 6d ... more We consider all 4d N = 2 theories of class S arising from the compactification of exceptional 6d (2, 0) SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d (1, 0) SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of (e n , e n) minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class S construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.
We consider a d-dimensional unitary conformal field theory with a compact Lie group global symmet... more We consider a d-dimensional unitary conformal field theory with a compact Lie group global symmetry G and show that, at high temperature T and on a compact Cauchy surface, the probability of a randomly chosen state being in an irreducible unitary representation R of G is proportional to (dim R) 2 exp[−c 2 (R)/(b T d−1)]. We use the spurion analysis to derive this formula and relate the constant b to a domain wall tension. We also verify it for free field theories and holographic conformal field theories and compute b in these cases. This generalizes the result in 2109.03838 that the probability is proportional to (dim R) 2 when G is a finite group. As a by-product of this analysis, we clarify thermodynamical properties of black holes with non-abelian hair in anti-de Sitter space.
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Papers by Monica Kang