# Classifying all possible BPS configurations in string theory

Is there a classification of all possible BPS configurations in string theory? It has to include Calabi-Yau orbifolds, intersecting D-branes, coincident D-branes, etc. . To simplify matters, take the limit of zero string coupling.

Thanks.

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This question is hopeless. There are just too many of these. Is all you are looking for a literature review? You can embed practically any vacuum space of any N=2 theory in a BPS configuration, so such a review would have to encompass at least one active subfield with hundreds of papers. –  Ron Maimon Sep 3 '11 at 4:47

The short answer is no. There isn't even a complete classification of calabi-yau manifolds themselves, much less of all the interesting things that can be put inside them. Then there is the related problem of classifying all 2d conformal field theories, also unsolved. However, if one looks at a fixed vacuum, eg, flat 10d space, then a classification is possible. In more complicated situations the classification, (of D-branes anyway) is achieved through K theory (http://arxiv.org/pdf/hep-th/9810188v2.pdf). By the way, taking the string coupling to be zero doesn't simplify the problem at all. BPS objects will stay BPS as we vary the coupling because they are protected by the SUSY algebra.

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This presentation is based on a reading of "Four-qubit entanglement from string theory" L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, W. Rubens.

There exist mathematical correspondences between black holes and entanglement theory. 3-Q-bit theory is related to BPS black holes and 4-Q-bit entanglements are related to extremal black holes. And this leads to STU theory of coset QFTs with entanglement and entropy correspondences. The Stochastic Local Operations and Classical Communication (SLOCC) is a relationship between states in entanglement and teleportation. Two states are SLOCC related by a teleporation if they can be inter-converted to each other in a reversible manner with some probability of success. This uses group theory, where the group $G_{SLOCC}$ for this process is an $N$-partite system of q-bits with some group $GL(2,~C)$. The states further transform as a $(2~2,~...,~2)$. $$G_{SLOCC}~=~SL(2,~C)_1\otimes SL(2,~C)_2\otimes\dots\otimes SL(2,~C)_N,$$ which is the group that acts on the moduli space of black holes and is the U duality group. A composite state is then, $$|\psi_{12...N}\rangle~=~SL(2,~C)_1\times SL(2,~C)_2\times\dots\otimes SL(2,~C)_N|\phi_{12...N}\rangle$$ So this is an $N$-partite quantum information system where the entanglements are determined by the group element $G_{SLOCC}$ and polynomials of this group. This is the moduli space for black holes composed of Q-bits and the U-duality group.

For a $2$ Q-bit system this construction is apparent. You have a state of the form $\sum_{ij}a_{ij}|i,j\rangle$ for $i$ and $j$ running form $0$ to $1$. The elements $a_{ij}$ transform as $(2,~2)$ of the $G_{SLOCC}$. The invariant element is the determinant of these matrices so $det(a_{ij})$ transformed under the $G_{SLOCC}$ into $$det(a_{ij})~\rightarrow~det(a'_{ij})~=~det(U_{i'i}a_{ij}U'^\dagger_{j'j})~=~det(a_{ij})$$ with the obvious result on the determinant of a product that the transformation elements have unit determinant. The entanglement entropy is given by this measure so $S_{ij}~=~4|det(a_{ij})|^2$. For multipartite systems the same rule generally applies, but the matrix interpretation is different. For an $N$-partite system the entanglement entropy is given by a $2\times 2\times\dots\times 2$ (N times) set of elements. This leads to the entangled states $|00\rangle~+~|11\rangle$ and $|01\rangle~+~|10\rangle$ (without normalization) for singlet and triplet entangled states.

For a $3$ q-bit system the amplitudes are elements $a_{ijk}$ --- a $2\times 2\times 2$ element that is not a matrix, which has no diagonalization procedure. We then have to focus on invariants. The determinant is replaced by a hyperdeterminant that transforms as a $(2,~2,~2)$, and there are elements $\sigma_i,~\sigma_j$, and $\sigma_k$, co-invariants, that transform as $(3,~1,~1),~ (1,~3,~1)$ and $(1,~1,~3)$ of the $G_{SLOCC}$. So these four then construct entanglement measures $S_{ijk},~ S_{ij}$, $S_{ik}$ and $S_{jk}$, for the GHZ state and the bipartite elements which for a W-state.

For a $3$ Q-bit system we focus on invariants. The determinant is replaced by a hyperdeterminant that transforms as a $({\bf 2},~{\bf 2},~{\bf 2})$, and there are elements $\sigma_i,~ \sigma_j$ and $\sigma_k$, co-invariants, that transform as $({\bf 3},~{\bf 1},~{\bf 1})$ $({\bf 1},~{\bf 3},~{\bf 1})$ and $({\bf 1},~{\bf 1},~{\bf 3})$ of the $G_{SLOCC}$. These four construct entanglement measures $S_{ijk}$, $S_{ij}$, $S_{ik}$ and $S_{jk}$. $S_{ABC}$ is a tri-partite entanglement entropy with $$S_{ABC}~=~S_{A(BC)}~-~S_{AB}~-~S{AC}$$

The bipartite elements are pair entanglements and the tripartite involves a triplet which is entanglement. So $B~=~Bob$ is maximally entangled with $A~=~Alice$ and $C~=~Carl$ in the tripartite state. Where if I trace out $A$ $B$ or $C$ there is a complete mixed state, classical information due to tracing, and no correlation, while with the three bipartite entanglements Bob's entanglement is with $A$ and $C$, and a tracing out of his quantum state entanglement continues to have the $A-C$ entanglement. The tripartite entanglement corresponds to a large black hole, while the set of bipartite entangled states a small black hole. The tripartite state is also called the GHZ state for Greenberger, Horne, Zeilinger formulation of a three particle entanglement.

The N-partite entanglement is different from the standard bipartite entanglement. These correspond to three separable state, bipartite states plus one separated, nonseparable bipartite states and a tripartite GHZ state. These states in black hole logic correspond to small black holes $1/2$ supersymmetric, small black holes with two BPS charges and $1/4$ supersymmetric, two black holes with $3$ charges and $1/4$ supersymmetric and, the GHZ state is $4$ charges and $1/8$ supersymmetric.

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@Lawrence that is all fine, but the question asks about BPS configurations. What is the relationship between BPS states, black holes and multipartite entanglement? –  user346 Mar 22 '11 at 4:53
ps: the down-vote is not mine. I'm hoping for a clarification from you before voting either way, or not voting, as the case may be. –  user346 Mar 22 '11 at 10:20
I am not sure what much else there is. Classifying BPS black holes is a mathematical problem, and that is what I outlined. I am not sure what else is needed. This is an ongoing research area, and what I wrote is an outline of this work. It is an area of work which I do follow pretty closely as it is related to something I am doing. –  Lawrence B. Crowell Mar 22 '11 at 14:11
I'm just asking for a clarification. Are all black holes in string theory given by BPS states? (I'm a string theory novice so you'll forgive me if that sounds like an ignorant question. I understand they are certain monopole solutions of a gauge field which saturate the BPS bound - which gives you the heaviest possible "black" p-brane solutions.) If so then this is an incredibly beautiful correspondence. But sometimes such things can be too good to be true! That is why I ask. –  user346 Mar 22 '11 at 14:32
I should have mentioned this works with IIB strings --- oops! There is a lot of group theory involved. The upshot is that different 3-qubit configurations define BPS black holes. –  Lawrence B. Crowell Mar 24 '11 at 3:08