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D-Brane Central Charge and Landau–Ginzburg Orbifolds

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Abstract

We propose a formula for the exact central charge of a B-type D-brane that is expected to hold in all regions of the Kähler moduli space of a Calabi–Yau. For Landau–Ginzburg orbifolds we propose explicit expressions for the mathematical objects that enter into the central charge formula. We show that our results are consistent with results in FJRW theory and the hemisphere partition function of the gauged linear sigma model.

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Notes

  1. Our convention is that the A-twist corresponds to twisting by the axial R-charge \(U(1)_A=U(1)_{R}-U(1)_{L}\) and the B-twist to the twist by the vector R-charge \(U(1)_V=U(1)_{L}+U(1)_{R}\).

  2. In a nutshell, the boundary state describing the boundary CFT with A/B-boundary conditions will be a state in the (cc)/(ac) Hilbert space, respectively. Hence A/B-boundary conditions will naturally couple to (cc)/(ac) local operators.

  3. This bound is a consequence of the \({\mathcal {N}}=2\) SCFT algebra.

  4. We remark that the dimension of \({\mathcal {H}}_{\mathrm {Mar}}\) (and \({\mathcal {H}}_{\mathrm {def}}\)) is not necessarily the same for each topological ring. Usually it is not.

  5. Here we are being vague. \({\mathcal {M}}_{\mathrm {BPS}}\) can stand for the moduli of stable maps if we are working in the A-twisted sigma model, and hence referring Gromov–Witten (GW) theory, or for the moduli of maps satisfying the Witten equation (as in FJRW theory) and so on.

  6. Here \(K_{0}({\mathcal {D}})\) denotes the Grothendieck group of the triangulated category \({\mathcal {D}}\) and can be defined in general (see [41] for a review).

  7. The involution for a general compact complex manifold was studied in [46].

  8. There can be situations where \(\mathrm {det}({\overline{\rho }}_{m}(J))=-1\) but after addition of extra massive fields, one can construct an equivalent orbifold theory where \(\mathrm {det}({\overline{\rho }}_{m}(J))=1\) [52].

  9. See also [37] for a first principle derivation from orbifold defects.

  10. Here we view \({\tilde{v}}^{T}M^{-T}\), \({\tilde{v}}\in {\mathbb {Z}}^{N}\) as an element of \({{\,\mathrm{Aut}\,}}(W^{T})\), acting on \(y_{\alpha }\) as \(y_{\alpha }\mapsto \exp (2\pi i ({\tilde{v}}^{T}M^{-T})_{\alpha })y_{\alpha }\). Then, we can define an element \(\varphi ({\tilde{v}}^{T}M^{-T})\in G^{*}=\mathrm {Hom}(G,{\mathbb {C}}^{*})\) through the embedding of G into \({{\,\mathrm{Aut}\,}}(W)\) by mapping \(g \in G\) to \(v^{T}M^{-1}\) (for some \(v\in {\mathbb {Z}}\)) and then to \(\exp (2\pi i v^{T}M^{-1}{\tilde{v}})\in {\mathbb {C}}^{*}\).

  11. See Footnote \(^{18}\).

  12. See Sects. 2 and 6 of [68] and Section 2.4 of [69] for a short review.

  13. For the sake of exposition we ignore many further technical details in the description below.

  14. We do not multiply \(I^s\) with a factor of \(u^J\) as in [14, 19] since we require the \(e_J\) component \(I_{T,J}\) of \(I_T\) to be the form \(I_{T,J} = 1 + O(u)\).

  15. Due to shortage of fonts, we denote GLSM B-branes with the same letter as B-branes in the CFT in Sect. 2.

  16. In [63] the more general case of \(\mathrm {rk}{\mathsf {C}}<h\) is considered.

  17. This is shown in [63] and \(\mu \) denotes the moment map associated to the action of \({\mathsf {G}}\) on \({\mathsf {V}}\).

  18. It would be interesting to study the GLSMs one gets from \(q^{\mathrm {LG}}\) and \(q^{\mathrm {ext}}\). For a closely related discussion, see [90, 91].

  19. For other quintic Calabi–Yaus and different quotients by subgroups of \(\mathrm {Aut}(W)\) in the context of the Berglund-Hübsch-Krawitz mirror construction, see [93].

  20. A further interesting example is the “canonical” matrix factorization \({\overline{Q}}=\sum _{i=1}^5\phi _i\eta _i+\phi _i^4{\bar{\eta }}_i\) that has also been used in [19].

  21. Note that the labeling \({\widehat{\Gamma }}_{\ell _1,\ell _2}\) does not coincide with the labeling of (3.65) which has been defined using the labeling of the (ac)-ring.

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Acknowledgements

We thank Huai-Liang Chang, David Erkinger, Huijun Fan, Kentaro Hori, Hans Jockers, Maximilian Kreuzer, Wolfgang Lerche, Greg Moore, Daniel Pomerleano, Thorsten Schimannek, Yefeng Shen, Dmytro Shklyarov, and Harald Skarke for discussions and comments. JK thanks BICMR, CERN, and YMSC for hospitality. MR thanks Fields Institute, Caltech and Universidad Andres Bello for hospitality. ES thanks CERN and the University of Melbourne for hospitality. JK was partially supported by the Austrian Science Fund (FWF): [P30904-N27]. ES acknowledges support from NSFC grant No. 11431001. MR acknowledges support from the National Key Research and Development Program of China, grant No. 2020YFA0713000, and the Research Fund for International Young Scientists, NSFC grant No. 11950410500.

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Appendices

Alternative Derivation of \(q^{\mathrm {ext}}\)

In this appendix we recall the mirror map for Landau–Ginzburg orbifolds following [58, 60] which suggests why our definition of \(q^{\mathrm {ext}}\) could have a direct interpretation in terms of the (ac)-ring elements.

Consider an element in \({\mathcal {H}}^{(a,c)}\) of \((W,G,{\bar{\rho }}_{m},{\mathbb {C}}_L^{*})\), which we can represent as

$$\begin{aligned} \prod _{j\in I^{\gamma J^{-1}}} \phi _{j}^{l_{j}}|0\rangle ^{(a,c)}_{\gamma }, \qquad l_j \in {\mathbb {Z}}_{\ge 0}, j=1,\dots N\, . \end{aligned}$$
(A.1)

We choose a set of generators \({\bar{g}}_1,\dots ,{\bar{g}}_N\) of \({{\,\mathrm{Aut}\,}}(W^T)\) (as in (3.36) with M replaced by \(M^T\)) and define the element \({\bar{\gamma }} \in {{\,\mathrm{Aut}\,}}(W^{T})\) by

$$\begin{aligned} {\bar{\gamma }}:=\prod _{j\in I^{\gamma J^{-1}}}{\bar{g}}_{j}^{l_{j}+1}. \end{aligned}$$
(A.2)

By the isomorphism \({{\,\mathrm{Aut}\,}}(W^T) \cong {{\,\mathrm{Aut}\,}}(W)\), this element gets mapped to \(\gamma J^{-1} \in {{\,\mathrm{Aut}\,}}(W)\). By (3.36) this defines exponents \(v_1,\dots , v_N\) via

$$\begin{aligned} \gamma J^{-1}=\prod _{\alpha \in I^{{\bar{\gamma }}}}g_{\alpha }^{v_{\alpha }+1}. \end{aligned}$$
(A.3)

Then the mirror map is

$$\begin{aligned} \prod _{i\in I^{\gamma J^{-1}}}\phi _{j}^{l_{j}}|0\rangle ^{(a,c)}_{\gamma }\longleftrightarrow \prod _{\alpha \in I^{{\bar{\gamma }}}} y_{\alpha }^{v_\alpha }|0\rangle ^{(c,c)}_{{\bar{\gamma }}}. \end{aligned}$$
(A.4)

A few remarks are in order. This mirror map is shown to be one-to-one in the case of the so called atomic invertible polynomials [54, 58] of chain and Fermat type and there is some ambiguity for the case of loop type. Once this ambiguity is fixed, the mirror map gives an isomorphism between \({\mathcal {H}}^{(a,c)}\) of (WG) and \({\mathcal {H}}^{(c,c)}\) of \((W^{T},G^{T})\), where \(G^{T} \equiv G^{\vee }\) in (3.41). Then, the map is extended to an isomorphism when W is a sum of atomic invertible polynomials, just by taking the tensor product of the individual maps [58]. More important for us is that this map is an isomorphism even without projecting to gauge invariant states [58, 60], namely it provides an isomorphism between the spaces \(H^{(a,c)}\) of (WG) and \(H^{(c,c)}\) of \((W^{T},G^{T})\).

Now we come to the matrix q of Sect. 3.3. When a state belongs to a narrow sector of \(H^{(a,c)}\) it is clear that (A.4) maps it to a state in the untwisted sector \(H^{(c,c)}_{e}\) of the form \(\prod _{\alpha =1}^{N}y_{\alpha }^{v_{\alpha }}|0\rangle _{e}^{(c,c)}\). In particular, the marginal deformations \({\mathcal {O}}_{\gamma _a}\in {\mathcal {H}}^{(a,c)}_{(-1,1),\gamma _a}\) (cf. (3.32) and (3.33)) are mapped to states in \({\mathcal {H}}^{(c,c)}_{e,(1,1)}\) given by a vector \(v_a\in {\mathbb {Z}}^N\). The condition that the R-charges (with respect to \(J^{\vee }\in G^{\vee }\)) are 1 is equivalent to the condition \(1=J^{\vee }\cdot v_a\), i.e. we can identify the state with a monomial deformation \(\prod _{\alpha =1}^{N}y_{\alpha }^{v_{\alpha }} \) of \(W^{T}\). Furthermore, since also \(g^{\vee }\cdot v=0\) for all \(g^{\vee }\in G^{\vee }\), we find that \(v_a \in {\mathcal {A}}_{{{\,\mathrm{ext}\,}}}\). In fact, one can argue that \(v_a \in {\mathcal {A}}_{\mathrm {LG}}\) given in (3.45). We can repeat the procedure to obtain the matrix q in (3.43) by replacing the set \({\mathcal {A}}_{\mathrm {ext}}\) by \({\mathcal {A}}_{\mathrm {LG}}\). The resulting \(h\times (h+N)\) matrix then agrees with the matrix \(q^{\mathrm {LG}}\) defined in (3.33).

Since we will also make a connection to the gauged linear sigma model and geometry, we have found that it is useful to define an extended matrix \(q^{{{\,\mathrm{ext}\,}}}\) that also captures deformations which may seem unnatural from the Landau–Ginzburg point of view. A working hypothesis is that one should at least include unprojected sectors \(H^{(a,c)}_\gamma \) satisfying

$$\begin{aligned} F_{R}({\mathcal {O}}_{\gamma ,\mu })-F_{L}({\mathcal {O}}_{\gamma ,\mu }) \in \{1,2\}. \end{aligned}$$
(A.5)

This includes the marginal deformations, both narrow and broad, but can also include further sectors that may be empty after projection. These correspond to trivial monomial deformations of \(W^T\). The broad marginal deformations cannot be identified with monomial deformations of \(W^{T}\). Ignoring the latter, we can define an extended matrix \(q^{\mathrm {ext}}\in \mathrm {Mat}_{{\hat{h}}\times ({\hat{h}}+N)}({\mathbb {Q}})\) by

$$\begin{aligned} q^{{{\,\mathrm{ext}\,}}}_{{\hat{a}},{\hat{b}}}=\delta _{{\hat{a}},{\hat{b}}}, \qquad q^{{{\,\mathrm{ext}\,}}}_{{\hat{a}},{\hat{h}}+j}=-\theta ^{\gamma ^{-1}_{{\hat{a}}}}_{j}\qquad \text { \ for \ } \left\{ \begin{array}{l}{\hat{a}},{\hat{b}}=1,\ldots ,{\hat{h}} \ge h\\ j=1,\ldots ,N. \end{array} \right. \end{aligned}$$
(A.6)

We claim that the matrix \(q^{{{\,\mathrm{ext}\,}}}\) defined in (3.46) agrees with the matrix defined in (3.43).

Details on Examples

1.1 Quintic

The Landau–Ginzburg periods of the quintic can be found in [4]. There are two bases of periods. These can be obtained, for instance, by solving the Picard-Fuchs equation of the mirror quintic characterized by

$$\begin{aligned} \phi _1^5+\phi _2^5+\phi _3^5+\phi _4^5+\phi _5^5-5\psi \phi _1\phi _2\phi _3\phi _4\phi _5. \end{aligned}$$
(B.1)

The Gepner point is at \(\psi =0\). This is related to the large complex structure coordinate z via \(z=-(5\psi )^{-5}\). Comparing with the I-function we have \(u=-5\psi \).

One basis of periods is given by

$$\begin{aligned} \varpi _j=-\frac{1}{5}\sum _{m=1}^{\infty }\omega ^{2m}\frac{\Gamma \left( \frac{m}{5}\right) (5\psi )^m}{\Gamma (m)\Gamma \left( 1-\frac{m}{5}\right) ^4}\omega ^{jm}\qquad j=0,\ldots 4, \end{aligned}$$
(B.2)

with \(\omega \equiv J=e^{\frac{2\pi i}{5}}\). Under monodromy around the Gepner point at \(\psi =0\) the periods transform as \(\varpi _j\rightarrow \varpi _{j+1}\), modulo the relation \(\sum _{j=0}^{4}\varpi _j=0\).

There is a second basis given by

$$\begin{aligned} {\hat{\varpi }}_k=\sum _{n=0}^{\infty }\frac{\Gamma \left( n+\frac{k}{5}\right) ^5}{\Gamma (5n+k)}(5\psi )^{5n+k}\qquad k=1,\ldots ,4. \end{aligned}$$
(B.3)

The two bases are related via

$$\begin{aligned} \varpi _j=-\frac{1}{5}\frac{1}{(2\pi i)^4}\sum _{k=1}^4\omega ^{jk}(-1+\omega ^k)^4{\hat{\varpi }}_k. \end{aligned}$$
(B.4)

1.2 Two-parameter example 1

Two bases of LG periods of the two-parameter degree 8 example have been discussed in [102]. The periods can be obtained by solving the Picard-Fuchs equation of the mirror hypersurface characterized by the equation

$$\begin{aligned} \phi _1^8+\phi _2^8+\phi _3^4+\phi _4^4+\phi _5^4-8\psi \phi _1\phi _2\phi _3\phi _4\phi _5-2\phi \phi _1^4\phi _2^4. \end{aligned}$$
(B.5)

The Gepner point is at \((\psi ,\phi )=0\). The relation to coordinates \((z_1,z_2)\) at the large complex structure point is given by

$$\begin{aligned} z_1z_2^{\frac{1}{2}}=-(8\psi )^{-4}\qquad z_2=(2\phi )^{-2}. \end{aligned}$$
(B.6)

The Picard-Fuchs operators at the Landau–Ginzburg point are

$$\begin{aligned} {\mathcal {L}}_1= & {} 32\psi ^2\theta _{\psi }^2\theta _{\phi }-\phi (\theta _{\psi }-1)(\theta _{\psi }-2)(\theta _{\psi }-3)\nonumber \\ {\mathcal {L}}_2= & {} 16\theta _{\phi }(\theta _{\phi }-1)-\phi ^2(4\theta _{\phi }+\theta _{\psi })^2. \end{aligned}$$
(B.7)

One basis of periods is given by

$$\begin{aligned} \varpi _j(\psi ,\phi )=-\frac{1}{4}\sum _{m=1}^{\infty }\frac{(-1)^m\alpha ^{mj}\Gamma \left( \frac{m}{4}\right) }{\Gamma (m)\Gamma \left( 1-\frac{m}{4}\right) ^3}(2^{12}\psi ^4)^{\frac{m}{4}}u_{-\frac{m}{4}}((-1)^j\phi ), \end{aligned}$$
(B.8)

with \(\alpha =e^{\frac{2\pi i}{8}}\) and

$$\begin{aligned} u_{\nu }(\phi )=(2\phi )^{\nu } {}_2F_1\left( -\frac{\nu }{2},-\frac{\nu }{2}+\frac{1}{2};1;\frac{1}{\phi ^2} \right) . \end{aligned}$$
(B.9)

Since the Landau–Ginzburg point is at \(\phi =0\) we have to analytically continue \({}_2F_1\) to \(\phi =0\). This gives a sum of two terms

$$\begin{aligned}&u_{\nu }(\pm \phi )=\frac{1}{4\pi i}\frac{1-e^{i\pi \nu }}{\Gamma (-\nu )}\sum _{m=0}^{\infty }\frac{\Gamma \left( -\frac{\nu }{2}+m\right) ^2}{\Gamma (2m+1)}(2\phi )^{2m}\nonumber \\&\quad \mp \frac{1}{4\pi i}\frac{1+e^{i\pi \nu }}{\Gamma (-\nu )}\sum _{m=0}^{\infty }\frac{\Gamma \left( -\frac{\nu }{2}+\frac{1}{2}+m\right) ^2}{\Gamma (2m+2)}(2\phi )^{2m+1}. \end{aligned}$$
(B.10)

There is a second basis given by

$$\begin{aligned} \xi _r(\psi ,\phi )= & {} \sum _{n=0}^{\infty }\frac{\Gamma \left( n+\frac{r}{4}\right) ^4}{\Gamma (4n+r)}(2^{12}\psi ^4)^{n+\frac{r}{4}}(-1)^nu_{-\left( n+\frac{r}{4}\right) }(\phi )\nonumber \\ \eta _r(\psi ,\phi )= & {} \sum _{n=0}^{\infty }\frac{\Gamma \left( n+\frac{r}{4}\right) ^4}{\Gamma (4n+r)}(2^{12}\psi ^4)^{n+\frac{r}{4}}u_{-\left( n+\frac{r}{4}\right) }(-\phi ). \end{aligned}$$
(B.11)

Upon evaluating the I-function one gets four contributions, depending on whether some combinations of the summation variables are even or odd. Let is give some intermediate steps of the calculation. The expression we want to rewrite is

$$\begin{aligned} I_{LG}(u)= & {} - \sum _{m,n,r,s}\frac{(-1)^{G(k,q)}u_1^{4n+r-1}u_2^{2m+s}}{\Gamma (1+2m+s)\Gamma (4n+r)}\nonumber \\&\frac{\Gamma \left( \left\langle -n-\frac{r}{4}\right\rangle \right) ^3\Gamma \left( \left\langle -\frac{n}{2}-\frac{r}{8}-m-\frac{s}{2}\right\rangle \right) ^2}{\Gamma \left( 1-\left( n+\frac{r}{4}\right) \right) ^3\Gamma \left( 1-\left( \frac{n}{2}+\frac{r}{8}+m+\frac{s}{2}\right) \right) ^2}\phi _{4n+r,2m+s} \end{aligned}$$
(B.12)

Applying the reflection formula to the denominator of the second quotient produces a term

$$\begin{aligned} \frac{(-1)^n}{\pi ^5}\sin ^3\frac{r}{4}\sin ^2\left( \frac{r}{8}+\frac{n+s}{2}\right) \Gamma \left( n+\frac{r}{4}\right) ^3\Gamma \left( m+\frac{r}{8}+\frac{n+s}{2}\right) ^2 \end{aligned}$$
(B.13)

Next, we apply the reflection formula to the \(\Gamma (\langle \cdot \rangle )\) which simplifies to \(\Gamma \left( \left\langle -\frac{r}{4}\right\rangle \right) ^3\Gamma \left( \left\langle -\frac{r}{8}-\frac{n+s}{2} \right\rangle \right) ^2\). Here we have to distinguish between even and odd \(n+s\). With \(a\in {\mathbb {Z}}\) and using \(\Gamma \left( \langle -\frac{k}{d}\rangle \right) = 1- \frac{k}{d}\) if \(d\in {\mathbb {Z}}_{>0}\) and \(k\in \{1,\dots ,d-1\}\), we get

  • \(n+s=2a\)

    $$\begin{aligned} \frac{(-1)^n}{\pi ^5}\sin ^3\left( \frac{r}{4}\right) \sin ^2\left( \frac{r}{8}+a\right) \Gamma \left( \left\langle -\frac{r}{4}\right\rangle \right) ^3\Gamma \left( \left\langle -\frac{r}{8}-a \right\rangle \right) ^2=\frac{(-1)^s}{\Gamma \left( \frac{r}{4}\right) ^3 \Gamma \left( \frac{r}{8}\right) ^2}\nonumber \\ \end{aligned}$$
    (B.14)
  • \(n+s=1+2a\)

    $$\begin{aligned} \frac{(-1)^n}{\pi ^5}\sin ^3\left( \frac{r}{4}\right) \sin ^2\left( \frac{r}{8}+\frac{2a+1}{2}\right) \Gamma \left( \left\langle -\frac{r}{4}\right\rangle \right) ^3\Gamma \left( \left\langle -\frac{r}{8}-a-\frac{1}{2}\right\rangle \right) ^2=-\frac{(-1)^s}{\Gamma \left( \frac{r}{4}\right) ^3\Gamma \left( \frac{r}{8}+\frac{1}{2}\right) ^2}\nonumber \\ \end{aligned}$$
    (B.15)

Combining all the expressions, we arrive at the result in the main text.

1.3 Four-parameter example

1.3.1 Differential operators

The GKZ differential operators at the Landau–Ginzburg point are

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_1&= 9\, {u_{{1}}}^{3} \theta _{{2}}\theta _{{3}}\theta _{{4}} \left( \theta _{{3}}-1 \right) \left( \theta _{{2}}-1 \right) \\&\quad + {u_{{2}}}^{2 }{u_{{3}}}^{2}u_{{4}}\left( \theta _{{1}}-1 \right) \left( \theta _{{1}}-2 \right) \left( \theta _{{1}}-3 \right) \left( \theta _{{2}}+\theta _{{1}}+2\,\theta _{{3}} \right) \left( 2\,\theta _{{2}}+\theta _{{1}}+\theta _{{3}} \right) \\ {\mathcal {L}}_2&= 3\, u_{{3}}\theta _{{2}} \left( \theta _{{2}}-1 \right) +{u_{{2}}}^{2}\theta _{{3}} \left( 2\,\theta _{{2}}+\theta _{{1}}+\theta _{{3}} \right) \\ {\mathcal {L}}_3&= 3\, u_{{2}}\theta _{{3}} \left( \theta _{{3}}-1 \right) + {u_{{3}}}^{2}\theta _{{2}} \left( \theta _{{2}}+\theta _{{1}}+2\,\theta _{{3}} \right) \\ {\mathcal {L}}_4&= 729\,\theta _{{4}} \left( \theta _{{4}}-1 \right) \left( \theta _{{4}}- 2 \right) + {u_{{4}}}^{3} \left( \theta _{{1}}+3\,\theta _{{4}} \right) ^{3} \end{aligned} \end{aligned}$$
(B.16)

where \(\theta _i=u_i\frac{\partial }{\partial u_i}\) and the \(u_i\) being the coordinates at the Landau–Ginzburg point. It can be shown that the I-function satisfy the GKZ equations.

1.3.2 FJRW invariants

The twisted sectors \({\mathcal {H}}_{\gamma }\) corresponding to the marginal deformations are given by \(\gamma \in G^{(2)}\) as in (6.79). Hence, there are two types of invariants

$$\begin{aligned} \mathrm {FJRW}_{n_1,n_2,n_3,n_4}&:=\left\langle \left( e_{J^2}\right) ^{n_1}\left( e_{J^3g}\right) ^{n_2}\left( e_{J^3g^2}\right) ^{n_3}\left( e_{J^4}\right) ^{n_4} \right\rangle _{0,n_1+n_2+n_3+n_4}\nonumber \\ \mathrm {FJRW}^0_{n_1,n_2,n_3,n_4}&:=\left\langle \left( e_{J^2}\right) ^{n_1}\left( e_{J^3g}\right) ^{n_2}\left( e_{J^3g^2}\right) ^{n_3}\left( e_{J^4}\right) ^{n_4} \tau _1\left( e_J\right) \right\rangle _{0,n_1+n_2+n_3+n_4+1}. \end{aligned}$$
(B.17)

In Table 5 we give the first few non-zero invariants organized in terms of \(|n|=\sum _in_i\).

Table 5 The invariants \(\mathrm {FJRW}_{n_1,n_2,n_3,n_4}\) and \(\mathrm {FJRW}^0_{n_1,n_2,n_3,n_4}\)

PALP and Landau–Ginzburg Orbifolds

The program poly.x of software package PALP [59] is capable of analyzing Calabi–Yau Landau–Ginzburg orbifolds. It implements the results of [10, 11]. The option poly.x -L provides the information on how the twisted sectors contribute to the Hodge numbers. Since this option has not been discussed in detail in the PALP manual [106], we provide a detailed explanation here.

In general, there is no need to specify W only the group G needs to be entered. If \(G=\langle J \rangle \) we simply enter the numbers \(d, w_1, \dots , w_N\) where \(q_j=\frac{w_j}{d}\), \(j=1,\dots ,N\). For illustration, consider the example from Sect. 6.4: \(W = x_1^9 + x_2^9 + x_3^9 + x_4^3 + x_5^3\) with \(d=9\) and \(w=(1,1,1,3,3)\) and \(G=\langle J\rangle \).

./poly.x -L

type degree and weights [d w1 w2 ...]: 9 1 1 1 3 3

sec[0] th= 0 0 0 0 0 QL= 0/9 dQ= 0 q00+=1 q11+=112 q22+=112 q33+=1

sec[1] th= 1 1 1 3 3 QL= 0/9 dQ= 3 q03+=1

sec[2] th= 2 2 2 6 6 QL= 9/9 dQ= 1 q12+=1

sec[3] th= 3 3 3 0 0 QL= 6/9 dQ= 1 q12+=2

sec[4] th= 4 4 4 3 3 QL= 9/9 dQ= 1 q12+=1

sec[5] th= 5 5 5 6 6 QL=18/9 dQ=-1 q21+=1

sec[6] th= 6 6 6 0 0 QL=15/9 dQ=-1 q21+=2

sec[7] th= 7 7 7 3 3 QL=18/9 dQ=-1 q21+=1

sec[8] th= 8 8 8 6 6 QL=27/9 dQ=-3 q30+=1

WittenIndex=-216, Trace=236

9 1 1 1 3 3 M:145 5 N:7 5 V:4,112 [-216]

Here sec[i] corresponds to \({\mathcal {H}}^{(c,c)}_\gamma \) with \(\gamma = J^i\), th= i1 i2 ... iN corresponds to \(\theta ^\gamma = ( \frac{i_1}{d} , \dots , \frac{i_N}{d})\).

The value of QL corresponds to \(q_+\), the value of dQ corresponds to \(d_\gamma - 2{{\,\mathrm{age}\,}}(\gamma )\) with the notation as in Sect. 4.3. Finally, the pair (ij) in qij corresponds to \((i,{{\widehat{c}}}-j)\) in the sector \({\mathcal {H}}^{(c,c)\, i,j}_\gamma \cong {\mathcal {H}}_{\mathrm {FJRW},\gamma }^{i,{{\hat{c}}}-j}\), and the value of qij+= corresponds to \(\dim {\mathcal {H}}_\gamma ^{(c,c)\,i,j}\). Only the sectors with qij> 0 are displayed.

Note that it is easy to spot the broad sectors by looking for 0’s among \(\mathtt{th}\). In this example, there are three broad sectors, sec[0], sec[3], sec[6]. The untwisted sector has an odd number of zero phases, hence it is odd. While the \(J^3\)– and \(J^6\)–twisted sectors have an even number of zero phases and therefore contribute to the even part of \({\mathcal {H}}_{\mathrm {FJRW}}\).

The penultimate line gives the Witten index and the sum of all Hodge numbers, as computed by the Poincaré polynomial of the chiral ring. For comparison, the last line lists the numbers of points and vertices of the corresponding M– and N–lattice polytopes. This is explained in great detail in [106].

For a bigger group, one needs to add to d w1 w2 ... wN the further generators g in the form /Zn: k1 k2 ... kN. Here, n is the order of the generator, and k1 k2 ... kN are related to the phases of g by \(\theta _j^g = \frac{k_j}{n}\). In the example above, we consider now the group \(G = \langle J, g \rangle \) where the generator g acts on \({\mathbb {C}}^5\) by \(\text {diag}(1,1,1,\zeta _3,\zeta _3^2)\), \(\zeta _3^3 = 1\).

./poly.x -L

type degree and weights [d w1 w2 ...]: 9 1 1 1 3 3 /Z3: 0 0 0 1 2

sec[0:0] th= 0 0 0 0 0 QL= 0/9 dQ= 0 q00+=1 q11+=56 q22+=56 q33+=1

sec[0:1] th= 0 0 0 3 6 QL= 3/9 dQ= 0 q11+=28 q22+=28

sec[0:2] th= 0 0 0 6 3 QL= 3/9 dQ= 0 q11+=28 q22+=28

sec[1:0] th= 1 1 1 3 3 QL= 0/9 dQ= 3 q03+=1

sec[2:0] th= 2 2 2 6 6 QL= 9/9 dQ= 1 q12+=1

sec[3:1] th= 3 3 3 3 6 QL= 9/9 dQ= 1 q12+=1

sec[3:2] th= 3 3 3 6 3 QL= 9/9 dQ= 1 q12+=1

sec[4:0] th= 4 4 4 3 3 QL= 9/9 dQ= 1 q12+=1

sec[5:0] th= 5 5 5 6 6 QL=18/9 dQ=-1 q21+=1

sec[6:1] th= 6 6 6 3 6 QL=18/9 dQ=-1 q21+=1

sec[6:2] th= 6 6 6 6 3 QL=18/9 dQ=-1 q21+=1

sec[7:0] th= 7 7 7 3 3 QL=18/9 dQ=-1 q21+=1

sec[8:0] th= 8 8 8 6 6 QL=27/9 dQ=-3 q30+=1

WittenIndex=-216, Trace=236

9 1 1 1 3 3 /Z3: 0 0 0 1 2 M:67 5 N:13 5 V:4,112 [-216]

In this case, sec[i;j] corresponds to \({\mathcal {H}}^{(c,c)}_\gamma \) with \(\gamma = J^ig^j\). The remaining quantities have the same meaning as above.

Note that we can also determine \(\mathrm {S}\mathrm {L}(N,{\mathbb {C}}) \cap {{\,\mathrm{Aut}\,}}^{\text {diag}}(W)\) as follows:

./poly.x -fv | ./cws.x -N

Degrees and weights ‘d1 w11 w12 ... d2 w21 w22 ...’ or

‘#lines #columns’ (= ‘PolyDim #Points’ or ‘#Points PolyDim’):

9 1 1 1 3 3

Type the 20 coordinates as dim=4 lines with #pts=5 columns:

9 1 1 1 3 3 /Z9: 4 8 0 0 6 /Z3: 2 0 0 0 1 /Z3: 1 0 0 2 0

Hence, we read off that \(\mathrm {S}\mathrm {L}(N,{\mathbb {C}}) \cap {{\,\mathrm{Aut}\,}}^{\text {diag}}(W) \cong \mu _9 \times \mu _9 \times \mu _3 \times \mu _3\). This works, however, only for Fermat polynomials W. More generally, we can enter the exponent matrix M explicitly as follows: We remove from M the column (or row) corresponding to the highest weight. Then we shift the entries by \(-1\). Consider the example \(W=x_1^7 + x_2^7 + x_3^7 + x_2x_4^3+ x_3x_5^3\).

./poly.x -fv | ./cws.x -N

Degrees and weights ‘d1 w11 w12 ... d2 w21 w22 ...’ or

‘#lines #columns’ (= ‘PolyDim #Points’ or ‘#Points PolyDim’):

4 5

6 -1 -1 -1 -1

-1 6 -1 -1 -1

-1 -1 6 -1 -1

-1 0 -1 2 -1

Type the 20 coordinates as dim=4 lines with #pts=5 columns:

7 1 1 1 2 2 /Z21: 9 9 0 4 20 /Z7: 5 3 0 6 0

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Knapp, J., Romo, M. & Scheidegger, E. D-Brane Central Charge and Landau–Ginzburg Orbifolds. Commun. Math. Phys. 384, 609–697 (2021). https://doi.org/10.1007/s00220-021-04042-w

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