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R.G.J
R.G.J
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A neat way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components (complex) which are reduced to 16 after imposing the Majorana-Weyl condition consecutively in that order(which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then one should start from $\mathcal{N}=1$ SYM in four dimensions and not with $\mathcal{N}=4$ SYM.

16 supercharges:
D=10, $\mathcal{N}$ = 1 $~\to~$ D=6, $\mathcal{N}$ = 2 $~\to~$ D=4, $\mathcal{N}$ = 4 $~\to~$ D=3, $\mathcal{N}$ = 8 $~\to~$ D=2, $\mathcal{N} = (8,8)$

8 supercharges:
$\hspace{33mm}$ D=6, $\mathcal{N}$ = 1 $~\to~$ D=4, $\mathcal{N}$ = 2 $~\to~$ D=3, $\mathcal{N}$ = 4

4 supercharges:
$\hspace{64mm}$ D=4, $\mathcal{N}$ = 1 $~\to~$ D=3, $\mathcal{N}$ = 2

A neat way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components which are reduced to 16 after imposing the Majorana-Weyl condition (which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then one should start from $\mathcal{N}=1$ SYM in four dimensions and not with $\mathcal{N}=4$ SYM.

A neat way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components (complex) which are reduced to 16 after imposing the Majorana-Weyl condition consecutively in that order(which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then one should start from $\mathcal{N}=1$ SYM in four dimensions and not with $\mathcal{N}=4$ SYM.

16 supercharges:
D=10, $\mathcal{N}$ = 1 $~\to~$ D=6, $\mathcal{N}$ = 2 $~\to~$ D=4, $\mathcal{N}$ = 4 $~\to~$ D=3, $\mathcal{N}$ = 8 $~\to~$ D=2, $\mathcal{N} = (8,8)$

8 supercharges:
$\hspace{33mm}$ D=6, $\mathcal{N}$ = 1 $~\to~$ D=4, $\mathcal{N}$ = 2 $~\to~$ D=3, $\mathcal{N}$ = 4

4 supercharges:
$\hspace{64mm}$ D=4, $\mathcal{N}$ = 1 $~\to~$ D=3, $\mathcal{N}$ = 2

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Source Link
R.G.J
R.G.J
  • 795
  • 6
  • 14

A neaterneat way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components which are reduced to 16 after imposing the Majorana-Weyl condition (which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then I willone should start from $\mathcal{N}=1$ SYM in four dimensions because it will match the counting and not with $\mathcal{N}=4$ SYM.

A neater way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components which are reduced to 16 after imposing the Majorana-Weyl condition (which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then I will start from $\mathcal{N}=1$ SYM in four dimensions because it will match the counting and not with $\mathcal{N}=4$ SYM.

A neat way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components which are reduced to 16 after imposing the Majorana-Weyl condition (which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then one should start from $\mathcal{N}=1$ SYM in four dimensions and not with $\mathcal{N}=4$ SYM.

Source Link
R.G.J
R.G.J
  • 795
  • 6
  • 14

A neater way of deciding which higher-dimensional theory to start from is to count the number of spinor components. In ten dimensions, a spinor has $2^{[d/2]} = 32$ components which are reduced to 16 after imposing the Majorana-Weyl condition (which can only be done where d mod 8 = 2). Therefore, $\mathcal{N}=1$ SYM in ten dimensions has 16 components (real !)

A Majorana spinor in four dimensions has 4 components. So, suppose you want $\mathcal{N}=4$ SYM in four dimensions (which has 16 [four copies of four components) you want to start with $\mathcal{N}=1$ SYM in ten dimensions.

$\mathcal{N}=1$ SYM in four dimensions similarly has 4 components. So, if I want to construct a supersymmetric theory in two dimensions with four supercharges i.e $\mathcal{N}$ = (2,2) SYM (which has single component after imposing Majorana-Weyl), then I will start from $\mathcal{N}=1$ SYM in four dimensions because it will match the counting and not with $\mathcal{N}=4$ SYM.