nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

An Overview of B-branes in Gauged Linear Sigma Models

verfasst von : Nafiz Ishtiaque

Erschienen in: Superschool on Derived Categories and D-branes

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We review the BPS D-branes in gauged linear sigma models corresponding to toric Calabi–Yau (CY) varieties preserving \({\mathscr {N}}=2_B\) supersymmetry, and their relation to stable low energy branes. The chiral sectors of these low energy branes are described mathematically by various derived categories in various parts of the CY Kähler moduli space \({\mathscr {M}}_K\). For a fixed \({\mathscr {M}}_K\), all these descriptions should in fact be equivalent in a categorical sense and we review some aspects of this equivalence from a physical perspective. This is a short summary of the results of the comprehensive work by Herbst, Hori and Page [3] with some elementary commentary.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Introduction to Topological String Theories

Nur mit Berechtigung zugänglich

Of particular relevance to the topic of this review is the mirror symmetry, where D-branes underlie both the homological approach of Kontsevich [1] and the T-duality perspective of Strominger, Yau, and Zaslow [2].

Elements of the chiral and the twisted chiral rings that can be placed at a boundary.

In a categorical sense, which we will make a bit clearer over the course of the review.

Which implies that the supersymmetry variation of this Lagrangian is a total derivative, and in the absence of boundaries the action is invariant.

An R-symmetry is a symmetry that acts nontrivially on the fermionic coordinates of the superspace and is not part of the space-time isometry.

It may seem a little redundant to introduce them in the first place. They serve the purpose of closing the supersymmetry algebra (on the fields) off shell (without using equations of motion) which is of paramount importance in some cases, such as in discussing supersymmetry in curved backgrounds, but they play no significant role for us at the moment.

The equations \(\mathrm {d}W = 0\) are called the F-term equations. Also note that, as defined, the classical space of vacua \(\mathrm {Vac}^\mathrm {cl}_{ r}\) has a metric induced from the Fubiny-Study type metric on the toric variety \(X_{ r}\). If we could add all quantum corrections corresponding to integrating out all the massive modes, we would find that this metric (which appears in the kinetic term for the \(x^i\)’s) gets modified to a Ricci-Flat (Calabi–Yau) metric [10].

More generally, fields that transform under faithful representations of the gauge group.

Unlike fields from vector multiplets which transform under the adjoint representation and therefore can break the gauge group only upto the maximal torus.

More specifically, the Kähler geometry.

Since, if x is a solution of (14) for some r then \(\sqrt{\xi } x\) is a solution of (14) for \(\xi r\).

The singular nature arises from integrating out the \(\sigma _a\)’s (which become massless on these loci) and trying to keep only the chiral multiplet fields as dynamical, which only works well away from the singularity.

Classically we can only see the effects of the real FI parameters but after including quantum corrections, the true parameters of the low energy theories will be the complexified FI parameters (4).

Compared to the scale of theory set by the gauge coupling e, which has mass dimension 1.

The one loop contribution to the effective potential is exact due to a non-renormalization theorem for the twisted superpotential.

For \(r>0\), the base \({\mathbb {C}}\mathbb {P}^2\) is the quotient \(\{{\mathbb {C}}^3\backslash \varDelta _+ \} / {\mathbb {C}}^\times \) by the complexified gauge group, where the \({\mathbb {C}}^3\) is spanned by \((x^1, x^2, x^3)\), and \(x^4\) becomes the coordinate on the fiber. For \(r<0\), \(\{{\mathbb {C}}\backslash \varDelta _-\}/{\mathbb {C}}^\times \) is a point which is invariant under a discrete gauge group \(\mathbb {Z}_3 \subset U(1)\) and \((x^1, x^2, x^3)\) spans the \({\mathbb {C}}^3\) carrying a nontrivial representation of \(\mathbb {Z}_3\).

The symbol \({\mathscr {P}}\) means path ordered which is a prescription to make sense of the exponential of the integral of a matrix valued connections in the cases where the matrices from different points are non-commuting.

A representation \(U(1) \ni \lambda \mapsto \lambda ^{\frac{a}{b}}\) for some \(a,b \in \mathbb {Z}\) with \(\mathrm {gcd}(a,b)=1\), is an |a|-fold representation of the |b|-fold cover of U(1).

Note that when we mention string we simply mean that our space is one dimensional. In the honest string theory one needs to further couple the world-sheet SCFT to a ghost system to gauge world-sheet diffeomorphism.

The rest of the minimal boundary action to cancel the variation of the world-sheet integral of the rest of the terms in (3) is not going to play any part in our discussion, so we are not going to talk about them. They can be found in [3, 12, 13].

The actual \(2_B\) supersymmetry is generated by constant \(\varepsilon \), so in a variation generated by time dependent \(\varepsilon \), any term that behaves as \({\mathscr {O}}(\dot{\varepsilon })\) can be ignored as far as symmetry preservation is concerned, but such terms help to recognize various contributions to the Noether charge associated to the symmetry.

We use the same notation to denote the gradings of \({\mathscr {V}}_W\) as we did for \({\mathscr {V}}\), just by replacing \({\mathscr {V}}\) with \({\mathscr {V}}_W\).

A time slice of the worldsheet.

The chiral sector is invariant under any continuous world-sheet metric deformation.

The chiral ring is also an aglebra over \({\mathbb {C}}\), but it is more commonly referred to as a ring.

In the language of Sect. 1.B, each brane \({\mathscr {B}}_i\) comes with an index set \({\mathscr {I}}_i\) of the Chan–Paton indices, so that for each index \(k \in {\mathscr {I}}_i\) we have a basic boundary condition \({\mathscr {B}}_i(k)\) associated to the brane \({\mathscr {B}}_i\), and the Hilbert space for the open string between \({\mathscr {B}}_1\) and \({\mathscr {B}}_2\) has the following decomposition: \({\mathscr {H}}({\mathscr {B}}_1, {\mathscr {B}}_2) = \bigoplus _{\begin{array}{c} i \in {\mathscr {I}}_1 \\ j \in {\mathscr {I}}_2 \end{array}} {\mathscr {H}}({\mathscr {B}}_1(i), {\mathscr {B}}_2(j))\).

Which are automatically chiral due to the nilpotence of \(\varvec{Q}_B\).

Which simply says that the expectation value of the supersymmetry variation of an operator in a supersymmetric vacuum is zero.

This is a nontrivial result, for example, it is not true for the twisted chiral ring (\(\varvec{Q}_A\)-cohomology) which receives world-sheet instanton corrections.

Let us say \(\psi _+^i {\mathscr {O}}\) is chiral for some operator \({\mathscr {O}}\). Then \(0 = [\varvec{Q}_\mathrm {bulk}, \psi _+^i {\mathscr {O}}] = -2(\partial _+ x^i) {\mathscr {O}}- \psi _+^i [\varvec{Q}_\mathrm {bulk}, {\mathscr {O}}]\) where \(2\partial _+ = \partial _0 + \partial _1\). If \({\mathscr {O}}\) is chiral then \(2(\partial _+ x^i) {\mathscr {O}}= 0\) implies \({\mathscr {O}}\) is zero and so is \(\psi _+^i {\mathscr {O}}\). If \({\mathscr {O}}\) is not chiral then \(2(\partial _+ x^i) {\mathscr {O}}+ \psi _+^i [\varvec{Q}_\mathrm {bulk}, {\mathscr {O}}] = 0\) implies \({\mathscr {O}}\) must be of the form \(\psi _+^i {\mathscr {O}}'\) for some \({\mathscr {O}}'\) in which case \(\psi _+^i {\mathscr {O}}= \psi _+^i \psi _+^i {\mathscr {O}}' = 0\) since \(\psi _+^i\) is anti-commuting.

The action of \({\mathscr {N}}=(2,2)\) supersymmetry on the fields of GLSM can be found in [3, 7].

Which is a sum of the form degree (as \(\overline{\psi }_+^i + \overline{\psi }_-^i\) has R-charge 1) and the degree of the \({\mathrm {Hom}}({\mathscr {V}}_1, {\mathscr {V}}_1)\) part (which is due to the \(x^i\)’s and \(\overline{x}^i\)’s).

In addition to the (differential) \(\mathbb {Z}\)-grading by \(U(1)_V\).

The condition of chirality \(\varvec{Q}_B({\mathscr {O}}) = [\varvec{Q}_B, {\mathscr {O}}] = 0\) for \({\mathrm {Hom}}({\mathscr {V}}_1, {\mathscr {V}}_2)\)-valued holomorphic functions is equivalent to the condition for \({\mathscr {O}}: {\mathscr {C}}({\mathscr {B}}_1) \rightarrow {\mathscr {C}}({\mathscr {B}}_2)\) to be a cochain map (see (59)).

A morphism between two \(\mathbb {Z}^k\)-graded \({\mathscr {R}}_W\)-modules is a morphism that preserves the grading, i.e., the morphisms themselves have \(\mathbb {Z}^k\)-degree zero, in our context this means that the morphisms are gauge invariant.

A quasi-isomorphism is a cochain map that induces an isomorphism of cohomology. Recall that in physical terms, cochain maps are chiral ring elements.

Perhaps we should say “if \(\pi _r({\mathscr {C}}({\mathscr {B}}_1))\) and \(\pi _r({\mathscr {C}}({\mathscr {B}}_2))\) are quasi-isomorphic,” but we will not put much effort into distinguishing a brane from its complex.

In passing from a homotopy category to the derived category one introduces formal inverses for the quasi-isomorphisms (and all compositions of morphisms involving these inverses) so that they can be treated as genuine isomorphisms. Also note that, this conclusion is consistent with the, by now standard, result that the low energy branes in a geometric phase are objects in the derived category of coherent sheaves supported on the CY target space of the IR sigma model [1, 17, 18].

If we have non-trivial superpotential we need to mention that as well in the notation for the GLSM B-brane category.

The following complex represents the cochain complex (48), which in the absence of a superpotential is the same as (45). We are using the notation \({\mathscr {W}}\) introduced in the context of Wilson line branes (see “Wilson line branes” in Sect. 3.1) instead of using the Chan–Paton spaces to highlight the gauge charges, which plays a crucial role in the computations of the next section. The superscript on \({\mathscr {W}}\) denotes the R-charge.

We use an underline to point out the R-charge (differential degree) zero part of a complex.

A boundary condition can involve literally boundary conditions for the fields at the boundary along with boundary actions supported on the boundary that help preserve some of the symmetries of the bulk theory.

We are using the term “rotation” loosely, the actual symmetry depends on the details of the theory, it can be U(N), SO(N), Sp(N), etc.

The dual distinguishes the lower index from the upper in terms of representations of groups acting on \({\mathscr {V}}\). This is important for oriented strings, for unoriented strings the relevant representations must be self-dual.

Maxim Kontsevich. Homological Algebra of Mirror Symmetry. 1994.

Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T duality. Nucl. Phys., B479:243–259, 1996.

Manfred Herbst, Kentaro Hori, and David Page. Phases Of N \(=\) 2 Theories In 1+1 Dimensions With Boundary. 2008.

T. Bridgeland, A. King, and M. Reid. Mukai implies McKay: the McKay correspondence as an equivalence of derived categories. ArXiv Mathematics e-prints, August 1999.

Y. Kawamata. Log Crepant Birational Maps and Derived Categories. ArXiv Mathematics e-prints, November 2003.

D. Orlov. Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities. ArXiv Mathematics e-prints, March 2005.

Edward Witten. Phases of N=2 theories in two-dimensions. Nucl. Phys., B403:159–222, 1993. [AMS/IP Stud. Adv. Math.1,143(1996)].

Paul S. Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Anton Kapustin, Gregory W. Moore, Mark Gross, Graeme Segal, Balázs Szendröi, and P. M. H. Wilson. Dirichlet branes and mirror symmetry, volume 4 of Clay Mathematics Monographs. AMS, Providence, RI, 2009.

A. Schwimmer and N. Seiberg. Comments on the N \(=\) 2, N \(=\) 3, N \(=\) 4 Superconformal Algebras in Two-Dimensions. Phys. Lett., B184:191–196, 1987.

10.

Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow. Mirror symmetry, volume 1 of Clay mathematics monographs. AMS, Providence, RI, 2003. With a preface by Vafa.

11.

Lance J. Dixon, Jeffrey A. Harvey, C. Vafa, and Edward Witten. Strings on Orbifolds. Nucl. Phys., B261:678–686, 1985.

12.

Simeon Hellerman, Shamit Kachru, Albion E. Lawrence, and John McGreevy. Linear sigma models for open strings. JHEP, 07:002, 2002.

13.

Kentaro Hori. Linear models of supersymmetric D-branes. In Symplectic geometry and mirror symmetry. Proceedings, 4th KIAS Annual International Conference, Seoul, South Korea, August 14-18, 2000, pages 111–186, 2000.

14.

Wolfgang Lerche, Cumrun Vafa, and Nicholas P. Warner. Chiral Rings in N \(=\) 2 Superconformal Theories. Nucl. Phys., B324:427–474, 1989.

15.

Edward Witten. Mirror manifolds and topological field theory. 1991. [AMS/IP Stud. Adv. Math.9,121(1998)].

16.

Ashoke Sen. Tachyon condensation on the brane anti-brane system. JHEP, 08:012, 1998.

17.

Michael R. Douglas. D-branes, categories and N=1 supersymmetry. J. Math. Phys., 42:2818–2843, 2001.

18.

Eric R. Sharpe. D-branes, derived categories, and Grothendieck groups. Nucl. Phys., B561:433–450, 1999.

Titel: An Overview of B-branes in Gauged Linear Sigma Models
verfasst von: Nafiz Ishtiaque
Verlag: Springer International Publishing
Buch: Superschool on Derived Categories and D-branes
Print ISBN: 978-3-319-91625-5

Electronic ISBN: 978-3-319-91626-2

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-91626-2_16

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"