superalgebra

and

supergeometry

# Contents

## Idea

### General

In supersymmetric quantum field theory with extended supersymmetry, certain extremal supermultiplets have some of the supersymmetries retained (have 0-eigenvalue under some of the supersymmetry generators). These are called Bogomol’nyi–Prasad–Sommerfield saturated solutions.

They exist also in some models of soliton theory (English Wikipedia: Bogomol’nyi–Prasad–Sommerfield bound). The fact that a certain fraction (typically one half or fourth of supersymmetry generators) of supersymmetry is retained implies the saturation of the BPS-bound, which does make sense a bit more generally. The retained generators generate a nontrivial subalgebra of the full supersymmetry algebra and carry conserved charges; the mass is exactly determined in terms of these charges.

BPS states play a central role in the investigation of moduli spaces of classical vacua as they form part of the moduli problem which is often the most tractable.

Several mathematical theories in geometry are interpreted as counting BPS-states in the sense of integration on appropriate compactification of the moduli space of BPS-states in a related physical model attached to the underlying geometry: most notably the Gromov-Witten invariants, Donaldson-Thomas invariants and the Thomas-Pandharipande invariants; all the three seem to be deeply interrelated though they are defined in rather very different terms. The compactification of the moduli space involves various stability conditions.

### In supergravity

In the context of supergravity BPS states correspond to super spacetimes admitting Killing vectors. These notably include extremal black brane solutions.

### In superstring theory

Specifically in superstring theory BPS states in target space correspond to string states on the worldsheet which are annihilated by the left-moving (say) half of the Dirac-Ramond operator. These are counted by the Witten genus, see at Witten genus – Relation to BPS states.

The degeneracy of BPS states in string theory has been used to provide a microscopic interpretation of Bekenstein-Hawking entropy of black holes, see at black holes in string theory.

## Formalization in higher differential geometry

The following are some observations on the formalization of BPS states from the nPOV, in higher differential geometry. More details are in classicalinhigher, section 3.3 and dcct, section 1.2.15.3.3. See also current algebra – As a homotopy Lie algebra.

Let $\mathbb{R}^{d-1,1|N}$ be a super-Minkowski spacetime, let $(d,N,p)$ be in the brane scan and write

$\phi \coloneqq \bar \psi \wedge E^{\wedge p} \wedge \psi \in \Omega^{p+2}(\mathbb{R}^{d-1,1|N})$

for the correspoding super Lie algebra cocycle, as discussed at Green-Schwarz action functional, see (FSS 13) for the perspective invoked here.

Consider then $X$ a super-spacetime locally modeled on $\mathbb{R}^{d-1,1|N}$ as a Cartan geometry, solving the relevant supergravity equations of motion (e.g. 11-dimensional supergravity for $d= 11$, heterotic supergravity for $d = 10$ and $N = (1,0)$, type IIA supergravity for $d = 10$ and $N= (1,1)$ or type IIB supergravity for $d = 10$ $N= (2,0)$).

This means in particular that $X$ carries a super differential form

$\omega \in \Omega^{p+2}(X)$

which is definite on $\phi$. This is the curvature of the WZW-term which defines the relevant super p-brane sigma-model with target space $X$.

By (AGIT 89) $X$ is a BPS state to the extent that it carries Killing spinors which form a central Lie algebra extension of a sub-algebra of the supersymmetry algebra (i.e. of the super translation Lie algebra) by $H^p_{dR}(X)$ which is classified by the cocycle given by

$(\epsilon_1, \epsilon_2) \mapsto \omega(\epsilon_1,\epsilon_2) \in \Omega^p(X)_{cl}/im(\mathbf{d}_{dR}) \,.$

Now we observe that by (hgpII, theorem 3.3.1) this is precisely the 0-truncation of the super-Poisson bracket Lie n-algebra $\mathfrak{Pois}(X,\omega)$ induced by regarding $(X,\omega)$ as an pre-n-plectic supermanifold and restricting along the inclusion of the Killing vectors/Killing spinors into all the Hamiltonian vector fields.

$H^p_{dR}(X) \to \tau_0 \mathfrak{Pois}(X,\omega) \to Vect_{Ham}(X)$

(Here we are using that if an n-type is an extension of a 0-type, then its 0-truncation is still an extension by the 0-truncation of the original homotopy fiber.)

The elements in $H^p_{dR}(X)$ here are precisely the $p$-brane charges, as discussed in (AGIT 89, p. 8).

Hence $X$ is the more BPS the more odd-graded elements there are in $\tau_0 \mathfrak{Pois}(X,\omega)$ (or its restriction to super-isometries). Hence $X$ is a 1/2 BPS state of supergravity if the odd dimension of this is half that of $\mathbb{R}^{d-1,d|N}$, it is 1/4 BPS if the odd dimension is one fourth of that of $\mathbb{R}^{d-1,d|N}$, etc.

Notice that if

$\array{ && \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} \\ & {}^{\mathllap{\mathbf{L}_{WZW}}}\nearrow & \downarrow^{\mathrlap{F_{(-)}}} \\ X &\stackrel{\omega}{\longrightarrow}& \mathbf{\Omega}^{p+2}_{cl} }$

is a prequantization of $\omega$, i.e. an actual WZW term with curvature $\omega$, then $\mathfrak{Pois}(X,\omega)$ is supposed to be the Lie differentiation of the stabilizer group of $\mathbf{L}_{WZW}$, which is the quantomorphism n-group $QuantMorph(\mathbf{L}_{WZW})$. (This Lie differentiation statement is strictly shown only for $p = 0$ and $p = 1$ in dcct but clearly should hold generally.)

Hence we may regard $\mathbf{QuantMorph}(\mathbf{L}_{WZW})$ (or its restriction to super-isometries) as the Lie integration of the brane-charge extended supersymmetry algebra. By the discussion at conserved current – In higher differential geometry this is indeed the n-group of conserved currents of $\mathbf{L}_{WZW}$ regraded as a local Lagrangian, and so this conceptually connects back to the considerations in (AGIT 89).

## Examples

### In 11d Supergravity

In 11-dimensional supergravity (M-theory) there are four kinds of BPS states (e.g. Stelle 98, section 3 EHKNT 07):

## References

### General

The BPS bound derives its name from

• (E. B. Bogomolnyj) Е. Б. Богомольний, Устойчивость классических решений, Яд. Физ. 24 (1976) 449–454

• M. K. Prasad, Charles Sommerfield, Exact classical solution for ‘t Hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett. 35 (1975) 760–762.

The original article identifying the role of BPS states in supersymmetric field theory is

Further developments are in

### Introductions, surveys and lectures

An introduction that starts at the beginning and then covers much of the ground in some detail is

• Greg Moore, PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories (pdf)

A survey of progress on the most general picture is in

• Katzutoshi Ohta, BPS state counting and related physics (2005) (pdf)

### In supergravity

Discussion of extremal/BPS black branes in supergravity (especially in 11-dimensional supergravity and 10d type II supergravity) includes

Discussion of more general classification of solutions to supergravity preserving some supersymmetry, i.e. admitting some Killing spinors includes

The conceptual identification of the relevant brane-charge extension of the supersymmetry algebra as that of the conserved currents of the Green-Schwarz super p-brane sigma models is due to

The structure found in that article matches the general extension structure discussed in

in view of

Discussion of the relation of that to the traditional discussion of current algebras is in

### Spectral networks

Revised on February 23, 2015 22:18:30 by Urs Schreiber (82.113.98.114)