# Questions tagged [supersymmetry]

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### Mathematical motivation for supergeometry

Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) ...
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### Link between Extended Supergauge Group and Twistor Theory

In a 1974 paper on supergauge transformations in four dimensions, Wess and Zumino considered an extended supergauge group which contains the conformal group as a subgroup. An interesting thing about ...
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### Superspace derivation of supersymmetric non-linear sigma model in Supersolutions by Deligne and Freed

I am having a little trouble understanding passage from the linear to the non-linear sigma model in Section 4.1 of Supersolutions by Deligne and Freed. Most of my confusion comes down to the ... 160 views

### Supersymmetric SYK Model in 3D?

In a 2017 article More on supersymmetric and 2d analogs of the SYK model by Murugan, Stanford and Witten, the authors take a model called the SYK model (named after Sachdev, Ye and Kitaev) and study ...
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### What is a BPS state and why is it the cohomology of a moduli space?

The notion of a BPS state has existed in physics for a long time: I do not understand it completely, but I get the impression they are the supersymmetric analogue of ground states, and then physicists ...
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### Is there Z_n graded supersymmetry?

I have tried searching for something similar to what is described below, but to no avail. It would be great if somebody could show some right references, where this has been done, or explain why such ...
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### Chain rule for the superderivative

A one dimensional complex supermanifold $X$ is locally described by an ordinary complex coordinate $z$ and an anticommuting coordinate $\theta$, $\theta^2 = 0$. The superderivative is the square root ...
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### Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$: $$(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0$$ which defines the anti-...
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### Notation on supergeometry — parity

I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...
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### Spectral Flow Invariance for Calabi-Yau Sigma Models

I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows. If $X$ is a Calabi-Yau $d$-... 194 views

### References for superhomology

This question concerns topological string theory. It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau ...
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### Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Suppose we have a Lie superalgebra with triangular decomposition: \begin{equation} \mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-} \end{equation} I've seen it stated ...
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### Variation Formula of APS $\eta$-Invariant and Chern-Simons Theory

In Perturbative Expansion of Chern-Simons Theory with Noncompact Gauge Group, the author proved the variation formula of the APS $\eta$-invariant. I have a few questions about their proof. I will ...
Are there any mathematical explanations for the following surprising facts? $$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$ and $$\int_{Gr(2,6)} c_{\text{top}}... 6 votes 1 answer 294 views ### Branching from E(6) to SO(10) \times U(1) In E(6) inspired models of supersymmetry, the inclusion of Lie subgroups$$ SO(10) \times U(1) \hookrightarrow E_6 $$is important object of interest. See here for my motivating example. In ... 7 votes 1 answer 517 views ### Implications of gauge symmetry breaking on the spectral side of geometric Langlands? Let G be a complex reductive algebraic group and X be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted N=4 SUSY Yang--Mills theory is \mathfrak{h}^*/W (... 6 votes 1 answer 1k views ### Question on Witten’s paper “Supersymmetry and Morse theory” EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)). This well known article applies some tools developed by physicists (e.g. path integrals) to ... 2 votes 0 answers 135 views ### Two definitions of super-Virasoro algebra Let A=\mathbb C[x,\epsilon] where x is an even variable and \epsilon is an odd variable (thus A is a commutative super-algebra). Let \mathfrak g denote the Lie super-algebra of vector fields ... 8 votes 1 answer 147 views ### Are there Type III codes with small but nonzero "index"? Recall that a Type III code of rank r is a linear subspace C \subset \mathbb F_3^r which is self-dual for the standard inner product. (These occur only when r is divisible by 4.) Elements of ... 10 votes 1 answer 223 views ### Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24? For the purposes of this question, a rank-r (integral) lattice is a full-rank discrete subgroup L \subset \mathbb R^r such that \langle \ell, \ell' \rangle \in \mathbb Z for all \ell \in L. It ... 5 votes 0 answers 243 views ### Localization principle in integration over supermanifolds This post is closely related to the post Localization principle in supersymmetry and can be considered as a continuation of it, although independent. In § 9.3 of the book "Mirror symmetry" (K. Hori ... 11 votes 1 answer 565 views ### supersymmetry and the de Rham complex In Alvarez-Gaume's paper "Supersymmetry and the index theorem" there is given a certain supersymmetric Lagrangian whose quantization, apparently, leads to the de Rham Laplacian on the exterior ... 10 votes 4 answers 1k views ### Geometric or conceptual way to understand supersymmetry algebra Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ... 6 votes 0 answers 211 views ### Physical effects in supersymmetric theories of the underlying supermanifold being split or non-split? Given an m-dimensional C^{\infty} manifold M with an n-dimensional vector bundle E over M, one can use the transition functions of the manifold and the bundle to construct an (m,n)-... 18 votes 7 answers 2k views ### Supermanifolds — elementary introduction? I am looking for an elementary but mathematically precise introductory text on supermanifolds in a modern differential geometric setting. Elementary in the sense that there is plenty of motivation for ... 2 votes 0 answers 227 views ### What exactly is the role of the mysterious manifold underlying the definition of a superspace? In the intro to chapter 12.3 of this book about the applications of coherent states, it says that classical spaces for bosons are real or complex vector spaces or manifolds, whereas classical spaces ... 4 votes 1 answer 714 views ### Definition of orthosymplectic supergroups I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard. The first version of the ... 3 votes 1 answer 153 views ### Reference request: coordinate ring of OSP(2p|n) In the paper, the orthosymplectic supergroup OSP(2p|n) is defined as follows. Let A = A_0 \oplus A_1 be a supercommutative superalgebra, where elements in A_0 are even and elements in A_1 are ... 1 vote 1 answer 166 views ### Construct super Poisson brackets on the coordinate rings of Lie super groups On line 7 of page 61 of the book a guide to quantum groups, a Poisson bracket is defined on \mathbb{C}[GL_n] for every classical r-matrix as follows. Let V be a vector space with a basis v_1, \... 2 votes 1 answer 84 views ### Trace of the chiral matrix of a subspace Let (V,Q) be a pair consisting of a \mathbb{C}-vector space V together with a nondegenerate bilinear form Q and let V_0\subseteq V be a linear subspace such that Q\vert_{V_0} is ... 8 votes 1 answer 376 views ### How do you get the spectral curve from a Calabi-Yau? In N=2 Quantum Field Theories and Their BPS Quivers by Alim, Cecotti, Córdova, Espahbodi, Rastogi and Vafa the authors give a recipe which constructs from a pair (C,\phi) consisting of a Riemann ... 2 votes 0 answers 72 views ### \mathbb{Z}_2 graded analog of row operations for supermatrices I'm working on some research involving supermatrices, and I was wondering if there was a \mathbb{Z}_2 graded analog of row operations for supermatrices. It seems to me that it makes sense to have ... 10 votes 2 answers 345 views ### What is the "quaternionic" super Brauer group? In addition to the two reasonably well-known categories \mathrm{SuperVect}_{\mathbb R} and \mathrm{SuperVect}_{\mathbb C} of real and complex super vector spaces, each of which is monoidally ... 10 votes 1 answer 1k views ### Mathematics of Chiral Rings Let A be a graded vector space, and suppose that two commuting differentials d_1 and d_2 of degree +1 act on A, such that A equipped with either is a chain complex. We now construct C(A), ... 3 votes 1 answer 123 views ### An intrinsic supergeometric description of the Green–Schwarz supersymmetric action The Green-Schwarz action is a natural supersymmetric extension of the Polyakov action (with a B-field which I will omit in what follows since it is not relevant to the question). For a morphism X:... 29 votes 1 answer 929 views ### Is there a symmetric monoidal 2-category "SuperDuperVect"? Recall that the category \mathrm{SuperVect}, as a category, consists of pairs of vector spaces, thought of as formal direct sums V \oplus W\,\Pi, where \Pi is the "odd line". (Called &... 3 votes 1 answer 227 views ### projective representation of supergroup In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group SO(... 5 votes 1 answer 605 views ### Localization principle in supersymmetry In \S 9.3 of the book "Mirror symmetry" (Vafa, Zaslow eds.) the authors formulate the following general localization principle for computation of integrals with respect to both even and odd ... 1 vote 2 answers 339 views ### Witten index non-trivial in the context of Quantum Mechanics? Let H be a self-adjoint Hamiltonian and H admits a decomposition into closed operators D,D^*, such that we have H = D^*D. I will now consider the one-dimensional case on a compact set: So ... 3 votes 0 answers 605 views ### Orthosymplectic group, matrix representations We have the orthosymplectic osp(n,m|2k). The bosonic part is so(n,m)\times sp(2k). The lie algebra generators are given in eg http://cds.cern.ch/record/524737/files/0110257.pdf where the group ... 1 vote 0 answers 137 views ### Double vector bundle vs vector bundle over supermanifolds Double vector bundle is roughly a vector bundle over (horizontal) vector bundle. Vector bundle is a supermanifold in a nature way (non-nature on the other way around). My question: is a double ... 3 votes 1 answer 553 views ### A good reference for learning about super-differentiation & super-integration? I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ... 3 votes 0 answers 103 views ### Analytic stuctures on \mathbb R^n and the nilpotent ideal of supermanifolds I have two questions which are somewhat related: (a) It is a well known result (of Freedman?) in differential topology that \mathbb R^4 has exotic smooth structures. Apparently, it is known that ... 1 vote 0 answers 203 views ### Classification of (almost) contact structures on S^3 Question: Is there a classification of almost contact or contact structures on S^3? What is it and references? The motivation of this question is as follows: (1) There is one paper showing that a ... 7 votes 1 answer 798 views ### wrapping M5-branes on a Riemann surface AdS/CFT gives us a way to use geometry to study field theory! I am trying to wrap M5-branes on a Riemann surface \Sigma_{g}. In my problem, for a Riemann surface in 11d, the normal bundle is max SO(... 11 votes 1 answer 474 views ### Is there a version of supersymmetry for homogeneous spaces? The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime \mathbb R^n and signature; I will write \mathrm{SO}(n) for the corresponding group of orthogonal ... 8 votes 1 answer 586 views ### Matrix-tree theorem via supersymmetry (i.e. Grassman algebras) The matrix-tree theorem states the number of spanning trees of a graph G is equal to a modified determinant of the adjacency matrix or "graph Laplacian", \Delta_G:$$\#\{ \text{spanning ...
For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the ...