M2-brane

<h2 id="description">Description</h2>
<p>The M2-brane solution can be found<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> by requiring 
  
    
      
        (
        P
        o
        i
        n
        c
        a
        r
        e
        
          )
          
            3
          
        
        ×
        S
        O
        (
        8
        )
      
    
    {\displaystyle (Poincare)_{3}\times SO(8)}
  
 symmetry of the solution and solving the <a href="/facts/Supergravity/jeNk3aAB">supergravity</a> equations of motion with the <a href="/facts/P-brane/fPHTVchc">p-brane</a> ansatz. The solution is given by a <a href="/facts/Metric_tensor_(general_relativity)/lJoCMdve">metric</a> and three-<a href="/facts/Differential_form/T9gyHtMv">form</a> gauge field which, in isotropic coordinates, can be written as 
</p>

d
                
                  s
                  
                    M
                    2
                  
                  
                    2
                  
                
              
              
                
                =
                
                  
                    (
                    
                      1
                      +
                      
                        
                          q
                          
                            r
                            
                              6
                            
                          
                        
                      
                    
                    )
                  
                  
                    −
                    
                      
                        2
                        3
                      
                    
                  
                
                d
                
                  x
                  
                    μ
                  
                
                d
                
                  x
                  
                    ν
                  
                
                
                  η
                  
                    μ
                    ν
                  
                
                +
                
                  
                    (
                    
                      1
                      +
                      
                        
                          q
                          
                            r
                            
                              6
                            
                          
                        
                      
                    
                    )
                  
                  
                    
                      1
                      3
                    
                  
                
                d
                
                  x
                  
                    i
                  
                
                d
                
                  x
                  
                    j
                  
                
                
                  δ
                  
                    i
                    j
                  
                
              
            
            
              
                
                  F
                  
                    i
                    
                      μ
                      
                        1
                      
                    
                    
                      μ
                      
                        2
                      
                    
                    
                      μ
                      
                        3
                      
                    
                  
                
              
              
                
                =
                
                  ϵ
                  
                    
                      μ
                      
                        1
                      
                    
                    
                      μ
                      
                        2
                      
                    
                    
                      μ
                      
                        3
                      
                    
                  
                
                
                  ∂
                  
                    i
                  
                
                
                  
                    (
                    
                      1
                      +
                      
                        
                          q
                          
                            r
                            
                              6
                            
                          
                        
                      
                    
                    )
                  
                  
                    −
                    1
                  
                
                ,
                
                μ
                =
                1
                ,
                …
                ,
                3
                
                i
                =
                4
                ,
                …
                ,
                11
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}ds_{M2}^{2}&=\left(1+{\frac {q}{r^{6}}}\right)^{-{\frac {2}{3}}}dx^{\mu }dx^{\nu }\eta _{\mu \nu }+\left(1+{\frac {q}{r^{6}}}\right)^{\frac {1}{3}}dx^{i}dx^{j}\delta _{ij}\\F_{i\mu _{1}\mu _{2}\mu _{3}}&=\epsilon _{\mu _{1}\mu _{2}\mu _{3}}\partial _{i}\left(1+{\frac {q}{r^{6}}}\right)^{-1},\quad \mu =1,\ldots ,3\quad i=4,\ldots ,11,\end{aligned}}}

<p>where 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
 is the <a href="/facts/Minkowski_space/vDkfkQJn">flat-space</a> metric and the distinction has been made between world volume 
  
    
      
        
          x
          
            μ
          
        
      
    
    {\displaystyle x^{\mu }}
  
 and transverse 
  
    
      
        
          x
          
            i
          
        
      
    
    {\displaystyle x^{i}}
  
 coordinates. The constant 
  
    
      
        q
      
    
    {\displaystyle q}
  
 is proportional to the <a href="/facts/Noether%2527s_theorem/VFjqrEvz">charge</a> of the brane which is given by the integral of 
  
    
      
        F
      
    
    {\displaystyle F}
  
 over the boundary of the transverse space of the brane.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/String_theory/HUJFIPYH">String theory</a></li>
<li><a href="/facts/Membrane_(M-theory)/fPHTVchc">Membrane (M-theory)</a></li>
<li><a href="/facts/M-theory/5KJgrr9P">M-theory</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>K. Stelle, "BPS Branes in Supergravity" <a href="https://arxiv.org/abs/hep-th/9803116" target="_blank">https://arxiv.org/abs/hep-th/9803116</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>A. Miemiec, I. Schnakenburg "Basics of M-theory" <a href="https://arxiv.org/abs/hep-th/0509137" target="_blank">https://arxiv.org/abs/hep-th/0509137</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

M2-brane open-in-new

M2-brane