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Commuting/anticommuting properties of fermionic ghost fields in BRST Quantization

I was reading the paper "Batalin-Vilkovisky analysis of supersymmetric systems" (by Laurent Baulieu and others). I am struggling to understand how commutation/anticommutation relations of ...
Aravind Madhavan's user avatar
3 votes
2 answers
33 views

Graded cyclic properties in tensor calculus formalism of supergravity

I am trying to understand the chapter 4 of https://arxiv.org/abs/hep-th/0204035. I want to obtain equation 4.19 in this article. First let me summarized some equations we need Denoting the gauge ...
phy_math's user avatar
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2 votes
1 answer
98 views

Proof of Batalin-Fradkin-Vilkovisky (BFV) theorem using BRST operator and graded Poisson bracket algebra

In the proof of Batalin-Fradkin-Vilkovisky (BFV) theorem one has to determine how the path integral measure changes. The set of canonical variables $$\varphi = (Q^A;P_{A}) =(q,\eta;\pi,\mathscr{P})\...
Faber Bosch's user avatar
-1 votes
1 answer
100 views

Confusion about whether a fermion field and its conjugate as an Grassmann number

I'm confused about what "a Grassmann-odd number" really means and how does it apply to fermions. In some text, it says that "if $\varepsilon \eta+\eta \varepsilon =0 $, then $\...
Errorbar's user avatar
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0 answers
25 views

Looking for a source to explain the Process of Topological twisting

as the title suggests I am looking for papers or other material that explains the notion of Topological twisting as it appears in the context of certain SUSY algebras. Concretely I am interested in ...
1 vote
0 answers
55 views

Literature on Representation Theory of Graded Lie Algebras

I am currently studying 'advanced' representation theory, including topics like super-Lie algebras. I've come across various gradings (excluding the $\mathbb{Z}_2$ grading), such as how to select odd ...
2 votes
1 answer
74 views

Berezin Integration, confirming an measure is invariant

I am working through the Mirror Symmetry book, available here. I already had a question about an earlier part of the same Exercise 9.2.1 on page 157: We are given the following action with one boson ...
Gleeson's user avatar
  • 213
2 votes
1 answer
132 views

Confirming an action is invariant under a supersymmetric transformation

I am studying chapter 9 of the book Mirror Symmetry, available here. My question is relating to page 156/157 where Supersymmetry is being introduced for the first time in QFT in 0-dimensions. We are ...
Gleeson's user avatar
  • 213
1 vote
1 answer
51 views

Difference between $\mathcal{N}=2$ and $\mathcal{N}=(1,1)$ SUSY

In supersymmetry algebra, $\mathcal{N}$ refers to $I=1,2,.. \mathcal{N} $ in $Q^{I}_{\alpha}$. My question is what does it mean to write $\mathcal{N}=(1,1)$ superalgebra?
htr's user avatar
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1 vote
1 answer
130 views

SUSY Algebra Anti-commutation Relation

I am trying to understand one of the anti-commutation relations of SUSY algebra. The lecture notes "Supersymmetry and Extra Dimensions" (PDF) taken by Flip Tanedo, says on p.29, due to the ...
Alex's user avatar
  • 85
6 votes
2 answers
317 views

Grassmann parameter in supersymmetry

Let's consider a free Wess-Zumino Lagrangian given by $$\mathcal{L} = \partial^{\mu}\overline{\phi}\partial_{\mu}\phi + i\psi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi\tag{1}$$ Whose ...
user avatar
2 votes
1 answer
152 views

Bosonic representation of delta function for Grassmann-even quantity

Suppose I have 2 Grassmann scalars $\theta$ and $\bar{\theta}$ and form the bosonic quantity $X = \bar{\theta}\theta$. Is there a purely bosonic representation of the delta function $\delta(X - \...
eric's user avatar
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1 vote
0 answers
67 views

Why must Lie superalgebras always contain both $Q$ and $\bar{Q}$?

In four dimensions Lie superalgebras naturally arise by relaxing the presupposition in the Coleman-Mandula theorem that the symmetry is not classified by a Lie algebra. It is then typically stated ...
Wintermute's user avatar
2 votes
1 answer
68 views

How to construct irreducible representations of supergroup $SU(N|N)$?

In physics, it is quite common to use Young tableaux to denote the irreducible representations of Lie groups, such as $SU(N)$. There are many excellent textbooks explaining how to construct them. ...
123123's user avatar
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0 answers
49 views

Generalized Stokes theorem in superspace

Do the generalized Stokes theorem apply in superspace? Any issues or uncommon behaviour of the gradient, divergence and rotational in superspace?
riemannium's user avatar
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0 votes
0 answers
22 views

Symmetries and equations of superspheres and other superspaces

What are the symmetries and the most studied/most standard examples of superspaces? I include exceptional superalgebras and infinite-dimensional spaces. Bonus: Do quantum groups apply in the above ...
riemannium's user avatar
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1 vote
0 answers
77 views

Matrix representation of Grassmann variables and Berezin Integrals

In this question, the problem of finding matrix representations for a set of Grassmann variables is discussed. How can this representation be used in Berezin integrals or Grassmann derivatives? Can ...
Lucas Baldo's user avatar
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2 votes
1 answer
110 views

Functions on Superspace

I find the definition of a function $\phi$ on a superspace $(z,\theta)$ confusing for the following reason. $\phi(z,\theta)$ can be expanded as $$\phi(z,\theta) = \phi_0(z) + \theta \psi(z)\tag{1}.$$ ...
Bronsteinx's user avatar
5 votes
1 answer
287 views

Contradictory results for Berezin integral

Say $$u = (u_1, \dots, u_{2n}) = (\xi_1, \eta_1, \dots \xi_n, \eta_n)\tag{1}$$ is a vector of Grassmann variables. For an antisymmetric bosonic matrix $A$ we know that $$ \int e^{\frac{1}{2} \sum_{a,...
almostsurely's user avatar
3 votes
1 answer
366 views

What does it exactly mean by right and left functional derivatives?

In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions. I do not quite see what it means by that. For example, let us ...
Keith's user avatar
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2 votes
1 answer
642 views

Complex valued Grassmann variables $(\theta \eta)^* $, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$

Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom. We start by accepting the ...
Марина Marina S's user avatar
3 votes
0 answers
108 views

Anti-de Sitter superalgebra from preserved symmetry

Following Freedman & Van Proeyen's "Supergravity", the $\mathcal{N}=1,\, D=4$ AdS supergravity action is $$ \dfrac{1}{2\kappa^2}\int d^4x \; e \left[R(\omega) - \bar{\psi}_\mu \gamma^{\...
phenolphthalein's user avatar
3 votes
1 answer
261 views

Interaction with odd number of fermionic fields?

Can there exist an interacting part of the Hamiltonian with odd number of fermionic operators? In other words, can we have a vertex which couples an odd number of fermions (there can also be 1, 2, or ...
RedGiant's user avatar
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1 vote
1 answer
108 views

Jacobi identity of the anti-bracket

I'm currently reading a volume 2 of Weinberg's QFT, and am puzzled by the Jacobi identities of the anti-bracket.  The anti-bracket is defined using the anti-field $\chi^n$ and $\chi_n^{‡}$ as follows $...
Siam's user avatar
  • 1,363
4 votes
1 answer
305 views

Berezin integral of a Grassmann field

Consider a time dependent Grassmann field i.e. $\theta(t)$. Now, consider the following Berezin integral $$\int [\mathcal{D}\theta] ~\prod_{t}\dot{\theta}\tag{1}$$ where $\dot{\theta}$ time derivative ...
Dr. user44690's user avatar
1 vote
1 answer
71 views

Conventions for graded wedge product in supergeometry

There are two conventions for the graded exterior product on superspace (see https://ncatlab.org/nlab/show/signs+in+supergeometry): $$\alpha \wedge \beta = (-1)^{pq+|\alpha||\beta|}\beta\wedge\alpha \;...
Gabriel Caro Mendoza's user avatar
1 vote
1 answer
125 views

Odd or Even symplectic structure in BV formalism

I am studying Batalin-Vilkovisky formalism. I am a little bit confused on what an odd (or even) symplectic structure is (i know what the degree of the underlying 2-form is). I can not find a clear ...
Far's user avatar
  • 13
1 vote
2 answers
167 views

Logarithm of Grassmann numbers

What is $\log \theta$ where $\theta$ is a Grassmann number such that $\theta^2 = 0$. How does one then look at its logarithm? Does it even make sense?
Dr. user44690's user avatar
3 votes
2 answers
360 views

Integration with complex Grassmann numbers

I have a question about a convention from Peskin & Schroeder, namely that $$\int d\theta^{*}\, d\theta \, (\theta \theta^*) = 1,$$ where $\theta$ and $\theta^*$ are independent Grassmann numbers. ...
Matteo Macchini's user avatar
1 vote
1 answer
37 views

Superbundles and spin in Dirac equation

The Dirac operator $D$ of a Clifford bundle $S$ is a first order differential operator on $C^{\infty}(S)$. The Clifford bundles considered are $\mathbb{Z} / 2 \text { -graded }$ (superbundles). This ...
David's user avatar
  • 394
1 vote
1 answer
168 views

How do I prove that the product of chiral superfields is itself a chiral superfield?

I am currently learning about $\mathcal{N}=(2,2)$ supersymmetry and have come up against what is probably a really silly question. The $\mathcal{}N=(2,2)$ superspace consists of bosonic coordinates $x^...
CoffeeCrow's user avatar
3 votes
0 answers
93 views

Derivation of anti-commutation relations of massive supermultiplet generators [closed]

In almost all intro to supersymmetry notes the commutation relations are given between the generators and their conjugates however, I can not find any proofs of them anywhere and am struggling to ...
Barnsandmaths's user avatar
2 votes
1 answer
585 views

Right derivative of Grassmann number and associated anti-commutation relation

I am reading chapter 3 of Anomalies in quantum field theory by Reinhold Bertlmann and I found one statement that I don't know how to prove. First of all he defined the right derivative on the ...
ocf001497's user avatar
  • 766
1 vote
1 answer
39 views

Question for ${\cal N}=1$ supersymmetry representations

Please see this lecture note: https://arxiv.org/abs/1011.1491. In section "2.2.5 Massless supermultiplet" the author defines a Casimir and says it is zero. How can we confirm it? We take the ...
KoKo_physmath's user avatar
2 votes
1 answer
144 views

Precise zero energy bound for supersymmetry

Usually we can shift the energy $E$ by any amount $\delta$ to redefine the lowest energy as $$ E + \delta. $$ However, in supersymmetry, there is a precise $E=0$ must be true, so that the supercharge $...
ann marie cœur's user avatar
1 vote
1 answer
145 views

Supersymmetry infinitesimal variation

In Wess & Bagger, chapter 3, the infinitesimal supersymmetric transformation is defined as: $$ \delta_\xi \psi = i\sqrt{2} \ \sigma^m \bar{\xi}\partial_m A + \sqrt{2} \ \xi F$$ and $$\delta_\xi A =...
BVquantization's user avatar
2 votes
2 answers
324 views

How do fermionic operators transform?

In quantum mechanics, if we have an operator $\Omega$, then under the transformation $T$, with infinitesimal generator $G$ (i.e. $T(\epsilon)=1-i\epsilon G + \ldots$), then operator transforms as $$\...
awsomeguy's user avatar
  • 857
1 vote
1 answer
116 views

Ward identity of QED - whether the fields are all $c$-number fields

I am following Sidney Coleman's lectures of Quantum Field Theory. At the end of ch.32, he derived the Ward identity for the 1PI generating functional $\Gamma[\psi,\bar{\psi},A_{\mu}]$ for QED: \begin{...
ocf001497's user avatar
  • 766
2 votes
1 answer
431 views

Right vs Left Derivatives

Let $\theta$ be a fermionic quantity and $f(\theta)=f(0)+\theta\frac{\partial f}{\partial\theta}=f(0)+\frac{\partial_r f}{\partial\theta}\theta$. Under a variation $\theta\mapsto\theta+\delta\theta$ ...
Ivan Burbano's user avatar
  • 3,915
1 vote
0 answers
47 views

Jacobi Identity in de Sitter Superalgebra

In the book "Supergravity" (by D. Freedman and A. Van Proeyen) they talk about why $\mathcal{N}=1$ de-sitter superalgebra is impossible to construct (Section 12.6.1). Basically de-Sitter algebra is a ...
Ari's user avatar
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3 votes
2 answers
205 views

SUSY $\mathcal{N}=1$ algebra

Given the definitions $$ P_\mu= -i\partial_\mu $$ $$ Q_\alpha=-i(\partial_\alpha-(\sigma^\mu\bar{\theta})_\alpha\partial_\mu) $$ $$ \bar{Q_\dot{\alpha}}=+i(\bar{\partial}_\dot{\alpha}-({\theta}\sigma^...
twisted manifold's user avatar
3 votes
1 answer
103 views

$\mathcal{N} \ge 2$ Supersymmetry massive supermultiplets

In Bertolinis SUSY notes [https://people.sissa.it/~bertmat/susycourse.pdf] we have defined: $$ \{Q^I_\alpha,\bar{Q}_\dot{\beta}^J\}=2m\delta_{\alpha\dot{\beta}}\delta^{IJ}\tag{3.24} $$ $$ \{Q^I_\alpha,...
twisted manifold's user avatar
0 votes
1 answer
2k views

What is the meaning of a Grassmann variable?

I don't seem to understand the concept of a Grassmann-variable. When studying superspaces and superfields I am told that the coordinates being used are $$(x^\mu, \theta_\alpha, \bar{\theta}_\alpha)$$ ...
user7077252's user avatar
1 vote
1 answer
99 views

Are there cases where the use of the Grassmann variables simplifies computations in the usual bosonic analysis?

When one introduces complex numbers and complex analysis one can then use the new machinery to solve some real-analysis problems. A lamppost example is computing integrals via residues. I think I've ...
Weather Report's user avatar
2 votes
1 answer
354 views

Is the expectation value of a Fermi field operator a Grassmann number?

It's often noted that Bosonic fields result from quantizing classical field theories defined on a regular numbers, whereas Fermionic fields arise when quantizing a classical field theory defined on ...
Abe Levitan's user avatar
2 votes
1 answer
64 views

Unitarity/Hermiticity condition for $osp(m,n|\mathbb{C})$ superalgebra

According to Dictionary on Lie Superalgebras (page 82), the compact form of $OSP(m,n|\mathbb C)$ Lie superalgebra must satisfy $M^{\text st}H\,M=1$ and $M^{\ddagger}M=1$ (is this the unitarity ...
Patrick El Pollo's user avatar
7 votes
1 answer
2k views

What exactly are "Grassmann-valued fields"?

Peskin & Schroeder define a Grassmann field $\psi(x)$ as a function whose values are anticommuting numbers, that can be written as : [p.301 eq. 9.71] $$\psi(x) = \sum\psi_i \phi_i(x),\tag{9.71}$$ ...
user341440's user avatar
2 votes
1 answer
67 views

Expand superspace function into component form

In 2D (1,1) superconformal field theory, the invariant "distance" between two points $Z_1=(z_1,\theta_1)$ and $Z_1=(z_1,\theta_1)$ in superspace is $$Z_{12}=z_1-z_2-\theta_1\theta_2.$$ My question ...
phys_student's user avatar
2 votes
2 answers
564 views

One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?

In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent: $$Q_{B}^{2}=0.\tag{74.32}$$ I don't know why.
hehehehehehe's user avatar
8 votes
1 answer
339 views

What's the space of eigenvalues/field configurations for a fermion?

In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\...
alexchandel's user avatar