Abstract
We present a few types of integral transforms and integral representations that are very useful for extending to supergeometry many familiar concepts of differential geometry. Among them we discuss the construction of the super Hodge dual, the integral representation of picture changing operators of string theories and the construction of the super-Liouville form of a symplectic supermanifold.
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Notes
We consider metrics of arbitrary signature; The sign of g fixes the sign of the overall coefficient \(\frac{\sqrt{\left| g\right| }}{g}=\pm \frac{1}{\sqrt{\left| g\right| }}.\)
Here and in the following, we adopt the convention that d is an odd operator (so \(\mathrm{d}x\) is a odd form but \(\frac{\partial }{\partial x}\) is an even vector). A change of parity is necessary because we want \(\eta \) to be an odd variable.
We must integrate generically monomials of the type \(\left( \xi \right) ^{p+q-k}\left( \mathrm{d}x\right) ^{k}\) and the Berezin integration selects \(k=p+q-n.\)
As pointed out in the previous section, to make contact with the standard physical literature we adopt the conventions that d is an odd operator and \(\mathrm{d}x\) (an odd form) is dual to the even vector \(\frac{\partial }{\partial x}\). The same holds for the odd variables \(\theta .\) As clearly explained, for example, in the appendix of the paper [25], if one introduces also the natural concept of even differential (to make contact with the standard definition of cotangent bundle of a manifold) our cotangent bundle (that we consider as the bundle of one-forms) should, more appropriately, be denoted by \(\Pi T^{*}.\)
The normalization coefficients chosen in the definitions of the duals of \(\rho _{\left( r,l\right) }\) and \(\rho _{\left( r|j\right) }\) lead to the usual duality on \(\Omega ^{\left( p|0\right) }\):
$$\begin{aligned} \star \star \rho _{\left( r,p-r\right) }=(-1)^{p(p-n)}\rho _{\left( r,p-r\right) } \end{aligned}$$In the literature, see [13] and also [18], one finds pseudodifferential forms of distributional type which belong to the spaces \(\Omega ^{(p|q)}\) where p denotes the form degree and q the picture number with \(0 \le q \le m\) (for picture number we intend the number of Dirac delta functions assuming that a given pseudodifferential form can be decomposed in terms of them). Those with \(q=m\) denote the Bernstein-Leites integral forms.
For example, if \(\omega =b\mathrm{d}b\) then \(\mathcal {T}(\omega )=i\delta ^{\prime }(\mathrm{d}\theta )\) and if \(\omega =\delta ^{\prime }(b)\mathrm{d}b\) then \(\mathcal {T}(\omega )=-i\mathrm{d}\theta .\) The imaginary factors could be eliminated introducing a normalization factor in the definition of the integral transformation.
There is also the possibility to increase the picture to a number between zero and the maximum value. In that case, we have pseudodifferential forms (i.e., forms with picture \(<m\)); however, since we do not use them in the present work, we leave aside such a possibility.
For a function f(x), we have \(\left[ \mathcal {F} \frac{\mathrm{d}f}{\mathrm{d}x}\right] (p)=ip\mathcal {F}(f)(p)\) and this is usually written as \(\mathcal {F}\left( \frac{\mathrm{d}}{\mathrm{d}x}\right) =ip\).
References
Castellani, L., Catenacci, R., Grassi, P.A.: The geometry of supermanifolds and new supersymmetric actions. Nucl. Phys. B 899, 112 (2015). arXiv:1503.07886 [hep-th]
Castellani, L., Catenacci, R., Grassi, P.A.: Supergravity actions with integral forms. Nucl. Phys. B 889, 419 (2014). arXiv:1409.0192 [hep-th]
Castellani, L., Catenacci, R., Grassi, P.A.: Hodge dualities on supermanifolds. Nucl. Phys. B 899, 570 (2015). arXiv:1507.01421 [hep-th]
Manin, Yu.I.: Gauge fields and complex geometry, 2nd edn. Springer (1977)
Deligne, P., Morgan, J.: Notes on supersymmetry (following Joseph Bernstein). In: Quantum fields and strings: a course for mathematicians, vols. 1, 2, Princeton (1996–1997)
Rogers, A.: Supermanifolds, Theory and applications. World Scientific Publ. (2007)
Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D.H.: The geometry of supermanifolds. Kluwer Academic Publishers, Dordrecht (1991)
Witten, E.: Notes on supermanifolds and integration. arXiv:1209.2199 [hep-th]
Voronov, T., Zorich, A.: Integration on vector bundles. Funct. Anal. Appl. 22, 94–103 (1988)
Kalkman, J.: BRST model for equivariant cohomology and representatives for the equivariant Thom class. Commun. Math. Phys. 153, 447 (1993)
Majid, S.: Hodge star as braided Fourier transform. arXiv:1511.00190 [math.QA]
Majid, S.: The self-representing Universe. In: Eckstein, M., Heller, M., Szybka, S. (eds.) Mathematical Structures of the Universe, Copernicus Center. CRC Press (2014, We thank the author for making us aware of this reference)
Bernstein, J., Leites, D.: Integral forms and the Stokes formula on supermanifolds. Funct. Anal. Appl. 11(1), 55–56 (1977)
Bernstein, J., Leites, D.: How to integrate differential forms on supermanifolds. Funct. Anal. Appl. 11, 70–71 (1977)
Voronov, F.F.: Quantization on supermanifolds and an analytic proof of the Atiyah-Singer index theorem. J. Soviet Math. (Translated in) 64(4), 9931069 (1993) [Original: Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., vol. 38, pp. 3–118. VINITI, Moscow (1990)]
Voronov, T.: Geometric integration theory on supermanifolds. Soviet Scientific Review, Section C: Mathematical Physics, 9, Part 1. Harwood Academic Publisher, Chur (1991) (Second edition 2014)
Voronov, T.: Quantization of forms on the cotangent bundle. Commun. Math. Phys. 205(2), 315–336 (1999). arXiv:math/9809130 [math.DG]
Berezin, F.A.: Differential forms on supermanifolds. Sov. J. Nucl. Phys. 30, 605–609 (1979)
Friedan, D., Martinec, E.J., Shenker, S.H.: Conformal invariance, supersymmetry and string theory. Nucl. Phys. B 271, 93 (1986)
Belopolsky, A.: New geometrical approach to superstrings. arXiv:hep-th/9703183
Belopolsky, A.: Picture changing operators in supergeometry and superstring theory. arXiv:hep-th/9706033
Belopolsky, A.: De Rham cohomology of the supermanifolds and superstring BRST cohomology. Phys. Lett. B 403, 47–50 (1997)
Castellani, L., Catenacci, R., Grassi, P.A.: The integral form of supergravity. arXiv:1607.05193 [hep-th]
Catenacci, R., Debernardi, M., Grassi, P.A., Matessi, D.: Čech and de Rham cohomology of integral forms. J. Geom. Phys. 62, 890 (2012). arXiv:1003.2506 [math-ph]
Voronov, T.: On volumes of classical supermanifolds. arXiv:1503.06542v1 [math.DG]
Witten, E.: Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys. 252, 189 (2004). arXiv:hep-th/0312171
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We thank Paolo Aschieri for valuable discussions.
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Castellani, L., Catenacci, R. & Grassi, P.A. Integral representations on supermanifolds: super Hodge duals, PCOs and Liouville forms. Lett Math Phys 107, 167–185 (2017). https://doi.org/10.1007/s11005-016-0895-x
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DOI: https://doi.org/10.1007/s11005-016-0895-x
Keywords
- Geometric integration theory
- Supermanifolds and graded manifolds
- Analysis on supermanifolds or graded manifolds