You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We consider the bosonic sectors of supergravity theories in ten and eleven dimensions which correspond to the low energy limits of string theories and M-theory. The solutions of supergravity field equations are known as supergravity backgrounds and the number of preserved supersymmetries in those backgrounds are determined by Killing spinors. We provide some examples of supergravity backgrounds which preserve different fractions of supersymmetry. An important invariant for the characterization of supergravity backgrounds is their Killing superalgebras which are constructed out of Killing vectors and Killing spinors of the background. After constructing Killing superalgebras of some special supergravity backgrounds, we discuss about the possibilities of the extensions of these superalgebras to include the higher degree hidden symmetries of the background.'
address: 'School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom\'
author:
- Ümit Ertem
title: Supergravity backgrounds and symmetry superalgebras
---

Introduction
============

The unification of fundamental forces of nature is one of the biggest aims in modern theoretical physics. Most promising approaches for that aim includes the ten dimensional supersymmetric string theories and their eleven dimensional unification called M-theory. There are five different string theories in ten dimensions which are type I, type IIA and IIB and heterotic $E_8\times E_8$ and $SO(32)$ theories. However, some dualities called T-duality, S-duality and U-duality between strong coupling and weak coupling limits of these theories can be defined and these dualities can give rise to one unified M-theory in eleven dimensions [@Green; @Schwarz; @Witten; @Townsend1; @Duff; @Becker; @Becker; @Schwarz]. The main common property of these ten and eleven dimensional theories is that their low energy limits correspond to the supergravity theories in those dimensions. This is the main reason for supergravity theories to attract increasing attention in recent research literature [@Van; @Proeyen; @Freedman; @Van; @Proeyen]. Understanding supergravity theories is important for knowing the dynamics of massless fields in string theories and finding the backgrounds that strings can propagate.

Supergravity theories are extensions of General Relativity to obtain an action that is invariant under supersymmetry transformations. For the consistency and invariance of the theory one has to define higher spin fermionic fields and extra bosonic fields with appropriate supersymmetry transformations. There are different consistent supergravity theories in different dimensions. The low energy limits of string and M-theories are described by the bosonic sectors of ten and eleven dimensional supergravity theories. Bosonic sectors of supergravity theories correspond to taking fermionic fields and their variations to be zero in the full theory. The solutions of bosonic supergravity field equations are called supergravity backgrounds. An important invariant that is used in the classification of supergravity backgrounds is the number of preserved supersymmetries in those backgrounds. The number of supersymmetries are described by Killing spinors of the background which correspond to the spinors that are solutions of the differential equation results from the variation of the gravitino field [@Townsend2]. This differential equation also defines the spinor covariant derivative in terms of the extra bosonic fields of the theory.

The amount of supersymmetry in a supergravity background is the main tool for the classification of supersymmetric supergravity backgrounds. Constructing the Killing superalgebras of a background is an important step to reach this classification. A Killing superalgebra is a Lie superalgebra that consists of the isometries of the background which correspond to the Killing vectors and the number of preserved supersymmetries that constitutes the Killing spinors of the background. These constituents of the superalgebra are even and odd parts of it respectively. In the definition of Killing superalgebras, the spinorial Lie derivative and Dirac currents that are constructed from two Killing spinors and correspond to Killing vectors are used [@O; @Farrill; @Meessen; @Philip; @O; @Farrill; @Jones; @Moutsopoulos]. To have a well defined Lie superalgebra structure, the Jacobi identities of odd and even parts of the superalgebra must be satisfied. The dimension of the Killing superalgebra is related to the homogeneity structure of the background [@O; @Farrill; @Hustler1; @O; @Farrill; @Hustler2]. By constructing the Killing superalgebras of different supergravity backgrounds, one can obtain the geometric and supersymmetric properties of those backgrounds. On the other hand, classification of supergravity backgrounds can be considered in different contexts. Besides the construction of Killing superalgebras, there are also methods that use $G$-structures and spinorial geometry to attack on the classification problem [@Gauntlett; @Pakis; @Gauntlett; @Gutowski; @Pakis; @Gauntlett; @Martelli; @Pakis; @Waldram; @Gran; @Papadopoulos; @Roest; @Gran; @Gutowski; @Papadopoulos; @Roest]. However, we will focus on the method of constructing Killing superalgebras in the rest of the paper.

In this review paper, we consider the current status of the subject and discuss about the possibble extensions of the structures, which can give hints about how to achive some developments regarding the problem. The paper is organized as follows. In section 2, we summarize the bosonic sectors of different supergravity theories in ten and eleven dimensions which are the low energy limits of string theories and M-theory. In section 3, we provide some examples of solutions of those supergravity theories that are called supersymmetric supergravity backgrounds. Section 4 contains the construction and properties of Killing superalgebras in general supergravity backgrounds and some possible extensions of them to higher order forms. In section 5, we provide the summary and discussion before concluding the paper.

Bosonic sectors of supergravity theories
========================================

The low energy limits of string theories and M-theory are ten and eleven dimensional supergravity theories. To obtain the solutions that strings can propagate consistently, we consider the bosonic sectors of supergravities. These correspond to the backgrounds of the theories by taking the fermionic fields to be zero and supersymmetry variations of fermionic fields give rise to Killing spinor equations whose solutions give the number of supersymmetries preserved by the background. In general, a supergravity background consists of a Lorentzian spin manifold $M$, a metric $g$ and some other bosonic fields defined on $M$ depending on the type and dimension of the supergravity theory. The metric and bosonic fields satisfy some field equations generalizing the Einstein-Maxwell equations. A supergravity background is called supersymmetric if it admits nonzero Killing spinors that are solutions of the Killing spinor equation. In this section, we will introduce the bosonic sectors of supergravity theories in ten and eleven dimensions.

Eleven-dimensional supergravity
-------------------------------

The maximum dimension that a supergravity theory can be consistently constructed is eleven and eleven dimensional supergravity is the low energy limit of M-theory [@Cremmer; @Julia; @Scherk]. Bosonic sector of eleven-dimensional supergravity consists of a Lorentzian metric $g$ and a closed 4-form $F$ which is the field strength of a 3-form $A$, namely $F=dA$. This theory can be considered as a generalization of Einstein-Maxwell theory to eleven dimensions with a generalized field strength $F$. The action of the theory is written as follows $$S=\frac{1}{12\kappa_{11}^2}\int\left(R_{ab}\wedge*e^{ab}-\frac{1}{2}F\wedge*F-\frac{1}{6}A\wedge F\wedge F\right)$$ where $\kappa_{11}$ is the eleven dimensional gravitational coupling constant, $R_{ab}$ are curvature 2-forms, $e^a$ are coframe basis constructed from the metric $g$ with $e^{ab}=e^a\wedge e^b$ and $*$ is the Hodge star operation. Note that the second term in the action is in the form of the Maxwell action and the last term is the topological Chern-Simons term. Varying with respect to $e^a$ and $A$ gives rise to the following field equations; $$\begin{aligned}
Ric(X,Y)*1&=&\frac{1}{2}i_XF\wedge*i_YF-\frac{1}{6}g(X,Y)F\wedge*F\\
d*F&=&\frac{1}{2}F\wedge F\end{aligned}$$ where $X,Y$ are vector fields, $Ric$ is the Ricci tensor, $*1$ is the volume form and $i_X$ is the interior derivative or contraction with respect to $X$. On the other hand, the spinor covariant derivative is modified by the existence of the extra bosonic fields in supergravity theories. Variation of the gravitino field gives a spinor equation whose solutions are Killing spinors $${\cal{D}}_X\epsilon=0$$ with $\epsilon$ is a spinor and the modified spinor covariant derivative is defined in terms of ordinary spinor covariant derivative $\nabla_X$ as $${\cal{D}}_X=\nabla_X+\frac{1}{6}i_XF-\frac{1}{12}\tilde{X}\wedge F$$ and $\tilde{X}$ is the 1-form that corresponds to the metric dual of $X$. The solutions of the Killing spinor equation determine the number of preserved supersymmetries in a supergravity background.

Type IIA and IIB supergravities
-------------------------------

By compactifying eleven-dimensional supergravity on $S^1$ with the Kaluza-Klein reduction method, one obtains the ten-dimensional type IIA supergravity which is the low energy limit of type IIA string theory [@Duff; @Nilsson; @Pope]. Besides the fermionic fields gravitino and dilatino, type IIA supergravity includes several bosonic fields that are the graviton described by the metric $g$, a real scalar field called dilaton $\phi$, a 2-form gauge potential $B_2$ with field strength $H_3=dB_2$ and 1-form and 3-form Ramond-Ramond gauge potentials $C_1$ and $C_3$ with field strengths $F_2=dC_1$ and $F_4=dC_3$. The bosonic sector of IIA supergravity correspond to taking the fermionic fields to be zero.

By defining a new field strength $$\tilde{F}_4=F_4-C_1\wedge H_3$$ with the property $$d\tilde{F}_4=-F_2\wedge H_3$$ the bosonic action of IIA supergravity is written as follows $$\begin{aligned}
S&=&\frac{1}{2\kappa_{10}^2}\int e^{-2\phi}\left(R_{ab}\wedge*e^{ab}+4d\phi\wedge*d\phi\right)\nonumber\\
&&-\frac{1}{4\kappa_{10}^2}\int\left(e^{-2\phi}H_3\wedge*H_3+F_2\wedge*F_2+\tilde{F}_4\wedge*\tilde{F}_4\right)\nonumber\\
&&-\frac{1}{4\kappa_{10}^2}\int B_2\wedge F_4\wedge F_4.\end{aligned}$$

The Killing spinors in type IIA supergravity satisfy two equations since there are two fermionic fields in the theory. The differential condition comes from the variation of the gravitino and the equation corresponds to the condition of being parallel with respect to the following modified spinor covariant derivative $${\cal{D}}_X=\nabla_X-\frac{1}{4}(i_XH_3)z-\frac{1}{8}e^{\phi}(i_XF_2)z+\frac{1}{8}e^{\phi}\tilde{X}\wedge F_4$$ where $z$ is the ten dimensional volume form. The algebraic condition comes from the variation of the dilatino and reads as follows $$\left(-\frac{1}{3}(d\phi)z+\frac{1}{6}H_3-\frac{1}{4}e^{\phi}F_2+\frac{1}{12}e^{\phi}F_4z\right)\epsilon=0.$$

There is another ten dimensional supergravity theory which is called type IIB supergravity and can not be obtained from the eleven dimensional theory [@DallAgata; @Lechner; @Tonin]. Fermionic content of IIB theory is same as in IIA case, however the bosonic sector differs from it. Besides the metric $g$, dilaton $\phi$ and the 2-form $B_2$ with field strength $H_3=dB_2$, it also contains 0-form, 2-form and 4-form Ramond-Ramond gauge potentials $C_0$, $C_2$ and $C_4$ with field strengths $F_1=dC_0$, $F_3=dC_2$ and self-dual $F_5=dC_4=*F_5$. By defining the new combinations of field strengths $$\begin{aligned}
\tilde{F_3}&=&F_3-C_0\wedge H_3\nonumber\\
\tilde{F_5}&=&F_5-\frac{1}{2}C_2\wedge H_3+\frac{1}{2}B_2\wedge F_3,\end{aligned}$$ the action of the bosonic sector of the type IIB supergravity can be written as $$\begin{aligned}
S&=&\frac{1}{2\kappa_{10}^2}\int e^{-2\phi}\left(R_{ab}\wedge*e^{ab}+4d\phi\wedge*d\phi\right)\nonumber\\
&&-\frac{1}{4\kappa_{10}^2}\int\left(e^{-2\phi}H_3\wedge*H_3+F_1\wedge*F_1+\tilde{F}_3\wedge*\tilde{F}_3+\tilde{F}_5\wedge*\tilde{F}_5\right)\nonumber\\
&&-\frac{1}{4\kappa_{10}^2}\int C_4\wedge H_3\wedge F_3.\end{aligned}$$ In fact, it is only a pseudo-action since only after externally imposing the self-duality condition $F_5=*F_5$, the field equations of the theory can be obtained from it.

The differential Killing spinor equation in type IIB supergravity is written from the following modified spinor covariant derivative [@Jones; @Smith] $${\cal{D}}_X=\nabla_X+\frac{1}{4}i_XH_3\otimes\sigma_3-\frac{1}{16}e^{\phi}\tilde{X}\wedge\left(F_1\otimes i\sigma_2-\tilde{F_3}\otimes\sigma_1+\tilde{F_5}\otimes i\sigma_2\right)$$ where $\sigma_i$ are Pauli matrices and the algebraic Killing spinor equation is $$\left(\frac{1}{2}(d\phi-ie^{\phi}dC_0)+\frac{1}{4}(ie^{\phi}\tilde{F_3}-H_3)\right)\epsilon=0.$$

Type I and heterotic supergravities
-----------------------------------

The low energy limit of type I superstring theory is a ten-dimensional supergravity theory coupled with a super Yang-Mills theory with gauge group $SO(32)$ in ten dimensions [@Chapline; @Manton]. This low energy limit is called as type I supergravity theory and it contains the metric $g$, the dilaton $\phi$, a Ramond-Ramond 2-form $C_2$ and the non-abelian Yang-Mills $SO(32)$ gauge connection $A_1$ as bosonic fields. The field strengths are defined as $$\begin{aligned}
F_2&=&dA_1+A_1\wedge A_1\nonumber\\
\tilde{F_3}&=&dC_2+\frac{1}{4}\omega_3\nonumber\end{aligned}$$ where $\omega_3=A_1\wedge dA_1+\frac{2}{3}A_1\wedge A_1\wedge A_1$ is the Chern-Simons form of the gauge connection. The bosonic action of type I supergravity is defined in terms of these field strengths as follows $$S=\int\left[e^{-2\phi}\left(R_{ab}\wedge*e^{ab}+4d\phi\wedge*d\phi\right)-\frac{1}{2}\tilde{F_3}\wedge*\tilde{F_3}-e^{-\phi}F_2\wedge*F_2\right].$$ The variation of the gravitino gives rise to the following spinor covariant derivative which is used in the definition of the Killing spinors; $${\cal{D}}_X=\nabla_X-\frac{1}{8}e^{\phi}\tilde{F_3}\tilde{X}.$$ Besides the gravitino, there are two more fermionic fields which are dilatino and gaugino and hence we have the following two algebraic Killing spinor equations $$\begin{aligned}
\left(d\phi+\frac{1}{2}e^{\phi}\tilde{F_3}\right)\epsilon&=&0\nonumber\\
F_2\epsilon&=&0.\end{aligned}$$

The heterotic supergravity differs from type I supergravity in its fermionic sector. Since we consider only the bosonic sectors of supergravities, we can write the bosonic action of the heteroric supergravity in terms of the metric $g$, the dilaton $\phi$, a 2-form $B_2$ with field strength $H_3=dB_2$ and 1-form non-abelian $SO(32)$ or $E_8\times E_8$ gauge potential $A_1$ with the field strength $F_2$ as in the type I case; $$S=\int e^{-2\phi}\left(R_{ab}\wedge*e^{ab}+4d\phi\wedge*d\phi-\frac{1}{2}H_3\wedge*H_3-\frac{1}{2}F_2\wedge*F_2\right).$$ The differential Killing spinor equation is determined by the following spin connection $${\cal{D}}_X=\nabla_X-\frac{1}{4}i_XH_3.$$ Although the fermionic sector of heterotic and type I supergravities are not same, the algebraic Killing spinor equations of heterotic supergravity has a similar form as in the type I case; $$\begin{aligned}
\left(d\phi+\frac{1}{2}H_3\right)\epsilon&=&0\nonumber\\
F_2\epsilon&=&0.\end{aligned}$$

Supergravity backgrounds
========================

There are various special solutions for eleven-dimensional and ten-dimensional supergravity theories. One of the methods that can give hints about obtaining all solutions is to find a way of classification for them. However, the complete classification of all supergravity backgrounds has not been achieved yet. On the other hand, for some special cases such as for backgrounds that have maximal supersymmetry and for symmetric space backgrounds, the classification problem can be accomplished. In this section, we consider some special solutions of the supergravity theories and summarize the classification results for symmetric and maximally supersymmetric backgrounds.

In eleven dimensions, spinor space is a 32-dimensional real space and hence the space of the solutions of Killing spinor equation can be at most 32-dimensional. An important invariant for a supersymmetric supergravity background is the fraction of preserved supersymmetries in that background which is denoted as $\nu=\frac{1}{32}n$ and $n$ is the dimension of the Killing spinor space. There are various numbers of known supergravity backgrounds which have different fractions of preserved supersymmetries. Indeed, the known solutions correspond to $n=0,1,2,3,4,5,6,...,8,...,12,...,16,...,18,...,20,...,22,...,24,...,26,...,32$ [@O; @Farrill; @Meessen; @Philip]. The numbers that does not appear in the list correspond to the cases of the fraction of preserved supersymmetries whose exact forms are not known. Moreover, for $n=30$ and $31$, it is known that there is no supersymmetric supergravity backgrounds [@Gran; @Gutowski; @Papadopoulos; @Bandos; @Azcarraga; @Varela; @Gran; @Gutowski; @Papadopoulos; @Roest2; @O; @Farrill; @Gadhia]. It is also proved that for the cases of $\nu>\frac{1}{2}$, those supergravity backgrounds are locally homogeneous [@O; @Farrill; @Hustler1; @O; @Farrill; @Hustler2]. For $\nu=1$, the solutions are called maximally supersymmetric or BPS solutions and examples for this case includes eleven-dimensional flat Minkowski spacetime and Freund-Rubin backgrounds such as $AdS_7\times S^4$ and $AdS_4\times S^7$. There are also half-BPS, namely $\nu=\frac{1}{2}$, solutions of the eleven-dimensional supergravity. Examples for this case are M-wave, Kaluza-Klein monopole and M2- and M5-brane solutions [@Acharya; @O; @Farrill; @Hull; @Spence].

Half-BPS solutions
------------------

A gravitational $pp$-wave is defined as a spacetime with a parallel null vector. A supersymmetric gravitational $pp$-wave which is called the M-wave is a solution of eleven dimensional supergravity and its metric is given as follows [@Hull] $$ds^2=2dx^+dx^-+a(dx^+)^2+(dx^9)^2+g_{ij}dx^idx^j,$$ where $i,j=1$ to $8$, $x^+,x^-$ are light cone coordinates, $a$ is an arbitrary function with $\partial_-a=0$ and $g_{ij}$ is a family of metrics dependent on $x^+$. The holonomy group of the manifold on which $g_{ij}$ is defined is contained in $Spin(7)$ and the following property is satisfied $$\partial_+\Omega=\lambda\Omega+\Psi$$ with $\Omega$ is the self-dual $Spin(7)$-invariant Cayley 4-form, $\lambda$ a smooth function of $(x^+,x^-)$ and $\Psi$ is a anti self-dual 4-form. The closed 4-form $F$ of eleven-dimensional supergravity vanishes in this solution. The metric (21) is a supersymmetric solution of eleven-dimensional supergravity if and only if it is Ricci-flat. By dropping the $x^9$ coordinate, one can also obtain a solution of ten-dimensional supergravity which is written in the following form $$ds^2=2dx^+dx^-+a(dx^+)^2+g_{ij}dx^idx^j.$$

There is also a brane solution of eleven-dimensional supergravity with the following metric which describes a number of parallel M2-branes [@Duff; @Stelle]; $$g=H^{-2/3}g_{2+1}+H^{1/3}g_8$$ where $g_{2+1}$ is the metric on the three-dimensional Minkowskian woldvolume $E^{2,1}$ of the branes and $g_8$ is the metric on the eight-dimensional Euclidean space $E^8$ transverse to the branes. $H$ is a harmonic function on $E^8$ and can be chosen as $$H(r)=1+\frac{a^6}{r^6}$$ with $r$ is the radial coordinate and $a^6=2^5\pi^2N{l_p}^6$. Here $N$ is the number of coincident membranes at $r=0$ and $l_p$ is the eleven-dimensional Planck length. The harmonic function has the property $\lim_{r\rightarrow\infty}H(r)=1$. The closed 4-form field in this background corresponds to $$F=\pm z_{2+1}\wedge dH^{-1}$$ where $z_{2+1}$ is the volume form of the brane worldvolume. The M2-brane solution preserves half of the supersymmetries namely $\nu=\frac{1}{2}$. However, it interpolates between two maximally supersymmetric solutions; near the brane horizon $r\ll a$, it corresponds to the maximally supersymmetric Freund-Rubin background $AdS_4\times S^7$ and for infinitely far away from the brane $r\rightarrow\infty$, it corresponds to the flat Minkowski space $E^{10,1}$.

There is a similar brane solution describing a number of parallel M5-branes with metric and 4-form are given as follows [@Guven] $$\begin{aligned}
g&=&H^{-1/3}g_{5+1}+H^{2/3}g_5\\
F&=&\pm3*_5H\end{aligned}$$ where $g_{5+1}$ is the metric on the six-dimensional Minkowskian worldvolume $E^{5,1}$ of the branes, $g_5$ and $*_5$ is the metric and Hodge dual on the five-dimensional Euclidean space $E^5$ transverse to the branes. The harmonic function $H$ is defined as $$H(r)=1+\frac{a^3}{r^3}$$ where $a^3=\pi N{l_p}^3$, $N$ is the number of coincident fivebranes at $r=0$ and $\lim_{r\rightarrow\infty}H(r)=1$. M5-brane preserves half of the supersymmetries and it interpolates between the maximally supersymmetric Freund-Rubin background $AdS_7\times S^4$ and flat Minkowski space $E^{10,1}$.

The ten-dimensional type IIB supergravity also has a brane solution which is called D3-brane with the metric given by [@Horowitz; @Strominger] $$g=H^{-1/2}g_{3+1}+H^{1/2}g_6$$ where $g_{3+1}$ is the metric on the worldvolume $E^{3,1}$ of the brane and $g_6$ is the Euclidean metric on the transverse space $E^6$ to the brane. The self-dual 5-form $F_5$ is defined on $S^5\subset E^6$ and has quantized flux and the dilaton $\phi$ is constant. The harmonic function $H$ is defined as $$H(r)=1+\frac{a^4}{r^4}$$ with $a^4=4\pi gN{l_s}^4$ and $g$ is the string coupling constant and $l_s$ is the string length. $N$ is the number of parallel D3-branes at $r=0$. This solution interpolates between flat Minkowski space at infinity and $AdS_5\times S^5$ at the near horizon limit.

Maximally supersymmetric backgrounds
------------------------------------

The solutions of supergravity theories which have maximum number of Killing vector fields and maximum number of Killing spinors are called maximally supersymmetric supergravity backgrounds. In this case the dimension of the space of Killing spinors is 32 and we have $\nu=1$.

In eleven dimensions, there are four classes of solutions that have maximal supersymmetry. First one is the flat Minkowski spacetime $M^{1,10}$ with $F=0$. Two of the other non-trivial classes include the Freund-Rubin backgrounds of which case we have a four-dimensional and seven-dimensional split of spacetime $M=M^4\times M^7$ and the total metric is written as the sum of the metrics of the split spacetimes; $$ds^2={ds_4}^2+{ds_7}^2.$$ One of the split spacetimes has positive and the other one has negative constant curvatures. First case of this type of solutions corresponds to the following background; $$\begin{aligned}
AdS_7(-7R)\times S^4(8R)\nonumber\\
F=\sqrt{6R}z_{S^4}\end{aligned}$$ where the numbers in paranthesis correspond to the constant scalar curvatures of the relevant backgrounds with $R>0$ and $z_{S^4}$ is the volume form of $S^4$. The second case is the following spacetime; $$\begin{aligned}
AdS_4(8R)\times S^7(-7R)\nonumber\\
F=\sqrt{-6R}z_{AdS^4}\end{aligned}$$ with $R<0$ and $z_{AdS^4}$ is the volume form of $AdS_4$. The fourth class of maximally supersymmetric solutions is a one parameter family of symmetric plane waves with the metric and the 4-form $$\begin{aligned}
g&=&2dx^+dx^--\frac{1}{36}\mu^2\left(4\sum_{i=1}^3(x^i)^2+\sum_{i=4}^9(x^i)^2\right)(dx^-)^2+\sum_{i=1}^9(dx^i)^2\nonumber\\
F&=&\mu dx^-\wedge dx^1\wedge dx^2\wedge dx^3\end{aligned}$$ where $\mu$ is a real number.

In ten dimensions, the only maximally supersymmetric solution of type I, heterotic and IIA supergravities is the flat Minkowski spacetime. However, in type IIB case we have two non-trivial classes. The first one is the Freund-Rubin background $$\begin{aligned}
AdS_5(-R)\times S^5(R)\nonumber\\
F_5=\sqrt{\frac{4R}{5}}\left(z_{AdS_5}-z_{S^5}\right)\end{aligned}$$ where $F_5$ is the self-dual 5-form and $R>0$. The second one is a family of symmetric plane waves with the following metric and self-dual 5-form $F_5$ $$\begin{aligned}
g&=&2dx^+dx^--\frac{1}{4}\mu^2\sum_{i=1}^8(x^i)^2(dx^-)^2+\sum_{i=1}^8(dx^i)^2\nonumber\\
F_5&=&\frac{1}{2}\mu dx^-\wedge\left(dx^1\wedge dx^2\wedge dx^3\wedge dx^4+dx^5\wedge dx^6\wedge dx^7\wedge dx^8\right).\end{aligned}$$

The classification of all supergravity backgrounds is one of the main goals of the study of supergravity theories. The homogeneity theorem for supergravity backgrounds says that a supergravity background which preserve more then half of the supersymmetries must be homogeneous [@O; @Farrill; @Hustler1; @O; @Farrill; @Hustler2]. A homogeneous spacetime can be characterized as a spacetime of which the tangent space at any point is spanned by Killing vectors that are constructed by squaring the Killing spinors of the background. If a homogeneous background also corresponds to a symmetric space, then we call it a symmetric supergravity background. The classification of symmetric supergravity backgrounds in eleven-dimensional and ten-dimensional type IIB cases are achieved in [@O; @Farrill; @Papadopoulos]. However, the full classification of all supergravity backgrounds is still an open problem.

Killing superalgebras
=====================

An important invariant that characterizes the supersymmetric supergravity backgrounds is the Killing superalgebra of that background [@O; @Farrill; @Meessen; @Philip; @O; @Farrill]. A Killing superalgebra $\mathfrak{g}$ has a Lie superalgebra structure which consists of a $\mathbb{Z}_2$-graded algebra that is a direct sum of two components $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$. The first component $\mathfrak{g}_0$ of the Lie superalgebra is called the even part and has a Lie algebra structure, the second one $\mathfrak{g}_1$ is called the odd part and corresponds to a module of $\mathfrak{g}_0$. A Lie bracket $[ , ]$ on a Lie superalgebra is defined as a bilinear multiplication $$[ , ]:\mathfrak{g}_i\times\mathfrak{g}_j\longrightarrow\mathfrak{g}_{i+j}$$ where $i, j=0, 1$ and satisfies the following skew-supersymmetry and super-Jacobi identities for $a, b, c$ are elements of $\mathfrak{g}$ and $|a|$ denotes the degree of $a$ which corresponds to 0 or 1 depending on that $a$ is in $\mathfrak{g}_0$ or $\mathfrak{g}_1$ respectively $$\begin{aligned}
[a,b]&=&-(-1)^{|a||b|}[b,a]\nonumber\\
\left[a,[b,c]\right]&=&[[a, b], c]+(-1)^{|a||b|}[b, [a, c]].\end{aligned}$$ For a Killing superalgebra, the even and odd parts of it are defined as bosonic and fermionic parts of the superalgebra. The bosonic part of the Killing superalgebras of supergravity backgrounds corresponds to the Lie algebra of Killing vector fields which are generated by Killing spinors. A Killing vector field $K$ is an isometry of the background which means that the Lie derivative of the metric $g$ with respect to $K$ is zero $${\cal{L}}_Kg=0.$$ Killing vector fields in the Killing superalgebra also preserve the other bosonic fields in the corresponding supergravity background. The fermionic part of the Killing superalgebra consists of the Killing spinors of the background. The supersymmetric properties of a supergravity background can be found from the Killing superalgebra which consists of the supersymmetry generators and isometries of the background.

Three operations corresponding to the Lie brackets in the superalgebra can be defined for the even (bosonic) and odd (fermionic) parts of a Killing superalgebra. Since the even part corresponds to the algebra of Killing vector fields, the Lie bracket defined on it is the ordinary Lie bracket for vector fields $$[\,,\,]:\mathfrak{g}_0\times\mathfrak{g}_0\longrightarrow\mathfrak{g}_0.$$ The action of the even part to the odd part is defined as the Lie derivative of spinor fields with respect to a Killing vector; $${\cal{L}}:\mathfrak{g}_0\times\mathfrak{g}_1\longrightarrow\mathfrak{g}_1.$$ The Lie derivative of a Killing spinor will be again a Killing spinor. Spinor Lie derivative is defined only with respect to Killing vectors and it is induced from the Lie derivative on differential forms as sections of the Clifford bundle of the manifold [@Lichnerowicz; @Kosmann]. It is written in terms of the spinor covariant derivative and a 2-form constructed from a Killing 1-form as follows $${\cal{L}}_K\psi=\nabla_K\psi+\frac{1}{4}(d\tilde{K})\psi$$ where $\tilde{K}$ is the Killing 1-form that is the metric dual of the Killing vector $K$. The third operation which takes two odd elements to obtain an even element is described by the squaring map of spinors $$\mathfrak{g}_1\times\mathfrak{g}_1\longrightarrow\mathfrak{g}_0.$$ This map is defined in terms of the spin invariant inner product on spinors [@Benn; @Tucker]. The Clifford product of a spinor $\psi$ with a dual spinor $\bar{\phi}$ can be written as a sum of differential forms by using the spinor inner product $( , )$. This decomposition is known as the Fierz identity; $$\psi\bar{\phi}=(\psi,\phi)+(\psi,e_a\phi)e^a+(\psi,e_{a_2a_1}\phi)e^{a_1a_2}+...+(-1)^{\lfloor\frac{n}{2}\rfloor}(\psi,z\phi)z$$ where $\lfloor\rfloor$ is the floor function and $z$ is the volume form. The squaring map corresponds to the projecting the Fierz identity onto the 1-form component. If $\psi$ and $\phi$ are Killing spinors, then the dual of the resulting 1-form is a Killing vector and for $\psi=\phi$, it is called the Dirac current $V_{\psi}$ of $\psi$.

In this way, all the needed brackets are defined for the Killing superalgebra. However, to obtain a Lie superalgebra structure, the Jacobi identities of the algebra must be satisfied. There are four Jacobi identities correspond to the $[\mathfrak{g}_0,\mathfrak{g}_0,\mathfrak{g}_0]$, $[\mathfrak{g}_0,\mathfrak{g}_0,\mathfrak{g}_1]$, $[\mathfrak{g}_0,\mathfrak{g}_1,\mathfrak{g}_1]$ and $[\mathfrak{g}_1,\mathfrak{g}_1,\mathfrak{g}_1]$ components. The first one is the ordinary Jacobi identity for the Lie algebra of Killing vector fields. The second and third ones are equivalent to the following properties of the spinor Lie derivative defined on spinors $$[{\cal{L}}_{K_1},{\cal{L}}_{K_2}]\psi={\cal{L}}_{[K_1,K_2]}\psi$$ $${\cal{L}}_K(\psi\bar{\phi})=({\cal{L}}_K\psi)\bar{\phi}+\psi\overline{{\cal{L}}_K\phi}.$$ The fourth Jacobi identity is the vanishing of the Lie derivative of a spinor with respect to the Dirac current of itself; $${\cal{L}}_{V_{\psi}}\psi=0.$$ This is not a trivial result and it has been proved for different cases in [@O; @Farrill; @Meessen; @Philip; @O; @Farrill; @Jones; @Moutsopoulos].

To construct the Killing superalgebra of a supergravity background, one needs to know the isometry algebra of the background which consists of the Killing vectors on that background and the algebra consisting of the space of Killing spinors. However, by using the cone construction [@Bar], which states that there is a one to one correspondence between the Killing spinors of a background and the parallel spinors of a background corresponding to the metric cone over the first one, one can also construct the Killing superalgebra without knowing the exact form of Killing spinors. For some special examples such as Freund-Rubin backgrounds in eleven dimensions, this kind of a construction can be achieved. For the $AdS_4\times S^7$ spacetime, the isometry algebra consists of the direct product of isometry algebras of the component spaces. Isometry algebra of $AdS_4$ is $\mathfrak{so}(3,2)$ and of $S^7$ is $\mathfrak{so}(8)$, so the isometry algebra of $AdS_4\times S^7$ is $\mathfrak{so}(3,2)\times\mathfrak{so}(8)$ and by using the cone construction procedures the Killing spinor space can also be obtained [@O; @Farrill]. As a result, the Killing superalgebra of this background is the following orthosymplectic Lie superalgebra; $$\mathfrak{osp}(8|4).$$ As another example, the isometry algebra of $AdS_7\times S^4$ is $\mathfrak{so}(6,2)\times\mathfrak{so}(5)$ and the Killing superalgebra is $$\mathfrak{osp}(6,2|4).$$ In eleven dimensional Minkowski background, the isometries generated by Killing spinors are translational Killing vector fields and the Killing superalgebra correspond to the supertranslation ideal of the Poincare superalgebra. However, one can also consider all isometries of the background that also contain rotational Killing vector fields and extend the Killing superalgebra to a symmetry superalgebra corresponding to the Poincare superalgebra. The Killing superalgebras of the fourth class of eleven dimensional maximally supersymmetric backgrounds denoted in (35) can be obtained by taking contractions of the Killing superalgebras of Freund-Rubin backgrounds.

Killing superalgebras play an important role in the classification of supergravity backgrounds. Maximally supersymmetric backgrounds which have $\nu=1$ and the minimally superymmetric ones can be classified completely. However, the classification of less than maximal and more than minimal supersymmetric bakgrounds is related to the Killing superalgebras of them. In eleven dimensions, the backgrounds preserving $\nu>\frac{1}{2}$ fraction of supersymmetry are locally homogeneous and this result is achieved by first constructing the Killing superalgebras and then obtaining the dimension of the traslational component of the squaring map from Killing spinors to Killing vectors. In ten dimensions, local homogeneity is guaranteed for $\frac{1}{2}\leq\nu\leq\frac{3}{4}$ and this can be proved from the construction of the Killing superalgebras [@O; @Farrill; @Jones; @Moutsopoulos].

Extension to higher order superalgebras
---------------------------------------

Isometries of a background which are generated by Killing spinors constitute the even part of the Killing superalgebras. If one considers all isometries of the background (not necessarily generated by Killing spinors), then one can define more general symmetry superalgebras of the background. Besides this fact, the squaring map of spinors can be extended to define higher order spinor bilinears. Although there are some attempts to include these higher order objects in the symmetry superalgebras, the construction of the so-called maximal superalgebras is still an open problem [@O; @Farrill; @Jones; @Moutsopoulos; @Simon; @Medeiros; @Hollands]. It is known that the Killing vector fields have higher order antisymmetric generalizations to hidden symmetries and these hidden symmetries are called Killing-Yano (KY) forms. A KY $p$-form $\omega$ is defined as the solution of the following equation $$\nabla_X\omega=\frac{1}{p+1}i_Xd\omega.$$ Moreover, one can also define the generalizations of Dirac currents to higher-degree forms. From Fierz identity (45), one can define projection operators $\wp_p$ that projects onto the $p$-form component of the Clifford product of a spinor with its dual. The generalized currents are called $p$-form Dirac currents and defined as follows $$\wp_p(\psi\bar{\psi})=(\psi,e_{a_p...a_2a_1}\psi)e^{a_1a_2...a_p}.$$ It is shown in [@Acik; @Ertem] that these $p$-form Dirac currents of twistor spinors correspond to conformal Killing-Yano forms and for the Killing spinors case they correspond to the KY forms. It is also known that KY forms satisfy a graded Lie algebra structure in constant curvature space-times [@Kastor; @Ray; @Traschen]. The following Schouten-Nijenhuis bracket is defined for a $p$-form $\alpha$ and a $q$-form $\beta$ $$\left[\alpha, \beta\right]_{SN}=i_{X_a}\alpha\wedge\nabla_{X^a}\beta+(-1)^{pq}i_{X_a}\beta\wedge\nabla_{X^a}\alpha$$ and corresponds to a Lie bracket for KY forms in constant curvature space-times. This means that the Killing superalgebras can be extended to include KY forms in some constant curvature supergravity backgrounds. On the other hand, a generalized Lie derivative on spinor fields with respect to KY forms have to be defined and the Jacobi identites also have to be satisfied. The spinorial Lie derivatives with respect to Killing vector fields also correspond to the symmetry operators of the Dirac equation in curved backgrounds. Symmetry operators of the Dirac equation that contain higher degree forms are also constructed in curved backgrounds by using KY forms [@Benn; @Charlton; @Benn; @Kress; @Acik; @Ertem; @Onder; @Vercin; @Cariglia; @Krtous; @Kubiznak]. Hence, these operators are natural candidates for the generalized Lie derivatives of extended superalgebras. The construction of these extended Killing superalgebras can give rise to new hints about the classification problem of supergravity backgrounds. As a side remark, KY forms are also used in relation to $G$-structures in the supergravity context [@Papadopoulos].

Discussion
==========

Ten and eleven dimensional supergravity backgrounds correspond to the spacetimes that strings can propagate in a well defined manner. So, the complete classification of supergravity backgrounds can give hints about the possible string backgrounds and the unification of string theories. There are many known solutions of supergravity theories which correspond to the backgrounds that have different fraction of preserved supersymmetries. Finding the common geometrical properties of these backgrounds and the complete classification of them according to the preserved supersymmetries is one of the main problems in supergravity and string theory. In some special cases, the classification problem is generally understood, however, in most cases it is not completely achieved yet. To achieve this aim, finding the invariants of these backgrounds is an important step. One of the main invariants of supersymetric supergravity backgrounds is their Killing superalgebras and they are constructed out of the isometries and Killing spinors of the background.

To construct a Killing superalgebra in a special supergravity background, one needs to know the Lie algebras of Killing vectors and Killing spinors. However, without knowing the Killing spinors of the background, one can also find the odd part of the superalgebra by using the cone construction and the knowledge about the parallel spinors. In this construction, the Lie derivatives of spinor fields and the Dirac currents of spinors are used and satisfying the Jacobi identities of the superalgebra completes the construction procedure. By this way, the Killing superalgebras of some supersymmetric backgrounds are obtained in the literature. Structure of these Killing superalgebras in different backgrounds and relations between them give a way to classify these supersymmetric backgrounds. For example, by using the properties of Killing superalgebras, the local homogeneity of a supergravity background that preserves more than half of supersymetries is proved [@O; @Farrill; @Hustler1; @O; @Farrill; @Hustler2].

Obtaining more hints about the classification problem can be possible by extending the Killing superalgebras to higher order geometric objects. Killing vector fields have natural antisymmetric generalizations to higher-degree forms which correspond to KY forms. Dirac currents also have generalizations to higher-degree components that are called $p$-form Dirac currents. The correspondence between the $p$-form Dirac currents and KY forms is proved in [@Acik; @Ertem]. So, extending the Killing superalgebras to include KY forms and $p$-form Dirac currents with a new definition of the generalized spinor Lie derivative can be possible. The properties of these extended superalgebras may give new insights to the classification of supergravity backgrounds.

The author thanks Özgür Aç[i]{}k, Jose M. Figueroa-O‘Farrill and Andrea Santi for useful discussions on the subject. This work is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) grant BİDEB 2219.

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