Questions tagged [superspace-formalism]
The Green-Schwarz formalism, or the superspace-formalism, are formalisms for supersymmetry with explicit spacetime supersymmetry.
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questions
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Expanding superfields: inconsistency of notation?
If I have a wavefunction of a fermion field $\Psi[\psi]$ I can expand it like so about some vacuum:
$$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...)$$
Now all ...
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0answers
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Determination of Torsion constraints in ${\cal N} = 1$, D = 10 Superspace
For the on-shell theory, containing the graviton $e_m^{\ \ \ a}$, gravitino $\psi_m^{\ \ \ \alpha}$, dilaton $\phi$, dilatino $\lambda$ and 3-form $H_{n m p}$, one has to demand that the SUSY algebra ...
2
votes
1answer
45 views
Why is an action built from superfields guaranteed to be supersymmetric?
Given a superfield (in 0+1 spacetime + 2 superspace coordinates)
$$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$
and given the standard supercharges ...
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0answers
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Supersymmetry Generator Definition for ${\cal N }= 1$
I am studying SYM $\mathcal{N}$ = 1 in D = 10, and using the bimodular representations for the 32x32 gamma matrices $\Gamma^a$. This means that I work with the off-diagonal 16x16 matrices, which I ...
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votes
1answer
40 views
Anti-Commutator of derivatives of Grassmann variables
How do I evaluate the anti-commutator of $\frac{\partial}{\partial\chi}$ and $\frac{\partial}{\partial\eta}$ when both $\chi$ and $\eta$ are Grassmann variables?
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vote
1answer
68 views
What are Grassmann numbers in field theory?
I've been struggling with the use of Grassmann numbers in QFT e.g. Peskin and Schroeder. They are introduced as "numbers" whose product is antisymmetric, and associative (this isn't said, but used in ...
2
votes
1answer
258 views
${\cal N} = 1$ SUSY Non-renormalization theorem
In Ref. 1, on Page 53, the ${\cal N} = 1$ SUSY non-renormalization theorem is derived. One first specifies the symmetries of the general ${\cal N} = 1$ SUSY action in the superspace formalism, and ...
3
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1answer
82 views
Auxiliary Grassmann variables in supergeometry
I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory.
In texts like [1] or [2] the authors introduced auxiliary Grassmann odd variables to ...
0
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1answer
37 views
Grassmann-odd extra dimensions and gravity
Take a world with $D=3+n$ space-time dimensions, where $n$ are extra space-like dimensions.
With extra-dimensional newton gravity
$$F=G_N(D)\dfrac{Mm}{r^{2+n}}$$
Can $n$ affect IF the extra ...
2
votes
1answer
54 views
Could you get real space from Grassmann numbers?
You can get a vector field from a pair of spinor fields with $A_\mu(x)=\psi(x) \gamma_\mu \overline{\psi}(x)$. Using this fact could you define a space-time vector in terms of Grasman numbers?
Say ...
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0answers
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Super-gauge transformation in two dimensional $\mathcal{N}= (0,2)$ superspace
I'm trying to couple matter to $\mathcal{N}=(0,2)$ SYM in 2d using superfield formalism. There are some paper (this on Sec. 6, or this on Sec. 3 [whose notation will be used here]) that construct what ...
7
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1answer
135 views
2D ${\cal N}=(2,2)$ Super Yang-Mills with Superspace
I'm reading this famous paper by Witten. There is the expression of field strength for the abelian vector multiplet (eq. (2.16)):
$$\Sigma = \frac{1}{\sqrt{2}}\bar{D}_+D_- V\;.\tag{2.16}$$
I'm ...
3
votes
1answer
152 views
Are there SUSY Lagrangian terms that are not D-term nor F-term?
I've read that a way to construct supersymmetric invariant lagrangian could be either to integrate a superfield in the whole superspace, i.e. in all anticommuting coordinates (D-term), or in half of ...
2
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1answer
95 views
Supergravity action as a total integral, over 4 spacetime and 4 Grassmann coordinates
Wess and Bagger, in their Supersymmetry and Supergravity, give the action for a global SUSY, ${\cal N}=1$, $D=4$, Yang-Mills gauge model as an integral over the 4 spacetime coordinates and 4 Grassmann ...
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1answer
70 views
Super Field Strength Identity
I work on a introduction into Super-symmetry. In the course we define
\begin{equation}
D_{\alpha} = \frac{\partial}{\partial \theta^{\alpha}} - i \sigma^{\mu}_{\alpha \dot{\alpha}} \bar{\theta}^{\dot{\...
2
votes
1answer
45 views
Ambiguity of Time Derivative of Superfunctions
I think that there is an ambiguity for defining the time derivative of a superfunction on the phase space of pseudo-classical mechanics of Grassmann numbers.
Let $\xi$ be a Grassmann odd number. Its ...
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1answer
68 views
Questions about the Ferrara-Zumino Multiplet
These questions arose while reading the paper ``Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity" by Komargodski and Seiberg (arXiv:1002.2228)
The Ferrara-Zumino ...
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3answers
253 views
What is the definition of $\delta^{m|n}$ and $\delta^{m}$?
I am reading the paper. What is the definition of $\delta^{m|m}$ and $\delta^{m+k}$ in (1.1) and (1.3) on pages 2,3? Are they some kind of delta function?
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vote
3answers
195 views
Grassmann numbers & supermanifolds
I'm asking this question because I'm currently trying to learn about Super Symmetry but I'm having trouble understanding the concept of super-space and super-manifold.
I read that in super-spaces you ...
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votes
1answer
405 views
Should the complex conjugate of a derivative of a Grassmann number include a sign?
Take a real Grassmann variable, by which I mean $\theta=\theta^*$. We have
$$\int d\theta~ \theta =1,\qquad \frac{\partial}{\partial\theta}\theta=1$$
If I define the conjugation of Grassmann ...
3
votes
1answer
262 views
Fayet-Iliopoulos terms
It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in ...
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votes
1answer
106 views
State counting in the d = 1+2, $\cal{N} = 2$ vector multiplet
The question is from Box 8.2, page 282 of the book "Gauge Gravity Duality" by Ammon and Erdmenger. The link to the specific page from Google Books is here.
According to the authors, a $\mathcal{N} = ...
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0answers
155 views
Projective superspace: why extra bosonic coordinates
I'm studying the projective superspace formalism for N = 4 supersymmetric $\sigma$-models in two dimensions. My question is: why do we need the extra bosonic coordinates for the manifest action?
I ...
3
votes
2answers
242 views
Do the Grassmann coordinates in the superfield formalism have any physical meaning?
In the superfield formalism we consider fields in a space who has four so called bosonic coordinates $x^{\nu}$ and four so called fermionic coordinates $\theta_1$,$\theta_2$,$\bar{\theta_1}$,$\bar{\...
3
votes
0answers
177 views
What mathematical structure describes superspace and superfields?
In every book related to supersymmetry I have encountered at some point the idea of superspace is introduced. Superspace is presented as a space spanned by 4 "normal" directions and 4 Grassmannian ...
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1answer
84 views
Why is the mass dimension of anticommutingcoordinates $[Mass]−1/2$
I am reading a review about supersymmetry and in page 29 I have read that the mass dimension of the Grassmann anticommuting coordinates is $-1/2$. Why this? why don't they have the same mass ...
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votes
1answer
444 views
Basic question about superspace, Grassmann numbers and world sheet supersymmetry
So, I'm trying to read the section on superspace from the book on string theory by Becker, Becker and Schwarz, and I realized that I've been stuck on something simple for a while. Some relevant ...
3
votes
1answer
320 views
Supersymmetry transformations as coordinate transformations
Usually, a supersymmetry transformation is carried out on bosonic and fermionic fields which are functions of the coordinates (or on a superfield which is a function of real and fermionic coordinates)....
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votes
1answer
205 views
${\cal N}=4$ SYM in terms of ${\cal N}=1$: The $SO(6)$ in the Yukawa term
I'm trying to write ${\cal N}=4$ SYM in terms of ${\cal N}=1$ superfields. I have the Lagrangian
$$\mathcal{L}=\frac{1}{16 k} \int d^2 \sigma \text{Tr} \big[W^a W_a\big]+c.c+\int d^4\theta \text{Tr}\...
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votes
0answers
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Information about fields and superfields [closed]
I want some explanation of fields and superfields (types and components), and what the relationship between them and representation of a group.
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vote
1answer
169 views
Pre-gauge-fixed superspace action of the RNS superstring
When writing down the the action of the RNS superstring in superspace, all of the sources I have checked (BBS, GSW, Polchinski) seem to just write down the action in conformal gauge, that is
$$
S_{\...
2
votes
1answer
222 views
Relation between $\mathcal{N}=2$ d=4 SUSY and $\mathcal{N}=4$ d=3 SUSY
What is meant with the fact that supersymmetry with $\mathcal{N}=4$ in three (2+1) dimensions is equivalent to supersymmetry with $\mathcal{N}=2$ in ordinary four (3+1) dimensions?
Which way does one ...
3
votes
1answer
727 views
Are there two types of D-term and two types of F-term in SUSY?
I've noticed that one can obtain D-terms either by integrating a vector superfield (the vector multiplet) over superspace or by integrating a Kahler potential over superspace. In both cases we get ...
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0answers
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Do commutation relations exist between superfields?
To quantize a theory, Klein gordon field for example, commutation relations are stablished. Or anticommuting ones in the fermionic case.
If I have the Wess.Zumino model or the free model:
$$S~=~\int\...
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votes
2answers
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What is kappa symmetry?
On page 180 David McMohan explains that to obtain a (spacetime) supersymmetric action for a GS superstring one has to add to the bosonic part
$$
S_B = -\frac{1}{2\pi}\int d^2 \sigma \sqrt{h}h^{\alpha\...
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0answers
224 views
Can mass dimension of a field be viewed as another 'quantum number'?
While studying SUSY in 4D, I noticed the dynamical chiral superfield has dimension [GeV], whereas the dynamical vector superfield (for gauge theories) is unitless. Because I was introduced to the ...
18
votes
4answers
3k views
Grassmann paradox weirdness
I'm running into an annoying problem I am unable to resolve, although a friend has given me some guidance as to how the resolution might come about. Hopefully someone on here knows the answer.
It is ...
18
votes
2answers
2k views
Kähler potential vs full effective potential
In evaluating the vacuum structure of quantum field theories you need to find the minima of the effective potential including perturbative and nonperturbative corrections where possible.
In ...
