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D-Branes And Mirror Symmetry

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Kentaro Hori, Amer Iqbal, Cumrun Vafa

High Energy Physics - Theory · Submitted 25 May 2000 · 63 references

Pith's verdict: A foundational paper whose boundary-condition classification and brane-creation mechanism hold up; the helix 'explanation' is a charge-lattice correspondence, not a worldsheet derivation, and the paper mostly says so.

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CitationAmer Iqbal, Cumrun Vafa, Kentaro Hori. "D-Branes And Mirror Symmetry." arXiv:hep-th/0005247 (2000). https://pith.science/paper/IKOOH5WC
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The paper

The paper claims

The paper identifies A-type D-branes in Landau-Ginzburg models with the wavefront trajectories from critical points of the superpotential, and uses mirror symmetry to map holomorphic D-branes to these Lagrangian cycles.

The reading

The authors set out to determine which boundary conditions preserve half of the bulk $(2,2)$ supersymmetry on a two-dimensional worldsheet, in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. They find two basic families: A-type boundary conditions require a middle-dimensional Lagrangian cycle (a cycle on which the Kähler form vanishes) whose image in the $W$-plane is a straight line, while B-type conditions require a complex cycle on which $W$ is constant. The central payoff is a dictionary between D-branes and the geometry of the superpotential: in a Landau-Ginzburg model the wavefront swept out by a critical point is itself a D-brane, and when its image crosses another critical value, Picard-Lefshetz monodromy creates new branes with computable multiplicity. Mirror symmetry then identifies these Lagrangian cycles with D-branes wrapped on holomorphic cycles and carrying holomorphic bundles on the Kähler side. If the dictionary is right, open-string spectra, soliton numbers, and fusion rules on Fano varieties and minimal models can be read off from intersections of classical cycles rather than solved directly.

What carries the argument

The load-bearing object is the wavefront cycle $gamma_a$: the set of all points reached by BPS soliton trajectories from a critical point $a$, whose image $W(gamma_a)$ is a straight half-line in the $W$-plane. It is Lagrangian because the Kähler form is invariant under the flow generated by the gradient of $W$, and its intersection with a slightly tilted copy of $gamma_b$ computes the Witten index $I(a,b)=Tr(-1)^F$ between the corresponding boundary conditions. The Picard-Lefshetz formula $gamma'_a=gamma_a pm (Delta_a circ Delta_b)gamma_b$ is the brane-creation rule. In minimal models the cycles are wedges between rays in the $X$-plane, so their intersection numbers are SU(2) fusion coefficients; period integrals $int_gamma e^{-beta W/2}Omega$ reproduce boundary-state overlaps and the modular $S$-matrix.

Extended reading notes

Core claim

The paper's central claim is that the BPS D-branes of a $(2,2)$ Landau-Ginzburg model are not extra data beyond the superpotential: they are the middle-dimensional cycles swept out by the gradient-flow wavefronts emanating from the critical points, with $W$ mapping each such cycle to a straight ray. This makes the D-brane charge lattice the relative homology of $W$ and makes brane creation under deformation a Picard-Lefshetz monodromy: when the ray from one critical point passes through another critical value, the cycle is shifted by the intersection number times the crossed cycle, so new branes appear. Under mirror symmetry, a D-brane wrapped on a holomorphic cycle of a Kähler manifold is mirror to one of these Lagrangian cycles; shifting the worldsheet theta angle by $2n$ corresponds to tensoring by $O_X(-n)$. The same intersection calculus reproduces the soliton-number/R-charge relation of massive theories, the SU(2) level-$k$ Verlinde fusion coefficients as intersection numbers of LG D-branes, and the modular $S$-matrix as a period integral. In the Fano case it explains the helix structure of exceptional bundles: mutations of bundles are mirror to Picard-Lefshetz moves on the LG cycles.

Load-bearing premise

The load-bearing premise is that the overlap between an A-type boundary state and a chiral field equals the period integral $int_gamma e^{-beta W/2}Omega$, which the paper derives under assumptions—$X$ Stein, $c_1(X)=0$, and vanishing periods of compact two-cycles—that it asserts but does not verify for the minimal-model and Fano cases where the result is later used.

Editorial extensions

If this is right

  • Open-string spectra between holomorphic bundles on a Fano variety are computable from intersections of straight-line cycles in the mirror LG model, with Ext groups matching the ground states.
  • Soliton numbers of a massive $(2,2)$ theory are entries of the monodromy matrix built from vanishing-cycle intersections, giving the spectrum of $exp(2pi iR)$ at the UV fixed point.
  • For SU(2) level $k$, the Verlinde fusion coefficients and the modular $S$-matrix have a geometric realization as intersection numbers and periods of LG D-branes.
  • Mutations of exceptional bundles and the helix property on Fano varieties are mirror to Picard-Lefshetz monodromy and brane creation.
  • For non-compact Calabi-Yau threefolds, the local mirror LG periods reduce BPS state counting to minimal cycles and string junctions in an elliptic fibration, connecting to probe BPS states in F-theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dictionary suggests a practical algorithm: for any LG mirror, draw the rays from critical values, count intersections, and read off Ext groups and fusion rules without solving the sigma model; only crossings of rays under deformation should change the open-string spectrum.
  • If the period-integral identification can be extended beyond the Stein, $c_1(X)=0$, and vanishing-period assumptions, modular matrices for more general LG orbifolds could be computed from classical periods, giving a purely algebraic route to boundary states.
  • The SU(2) result points to a higher-rank generalization: LG orbifolds with several fields might realize Verlinde algebras of higher-rank WZW models as intersection rings of their D-branes, though the paper does not attempt it.
  • The F-theory section hints that wall-crossing of BPS states in engineered four-dimensional theories is governed by the same Picard-Lefshetz moves; making that precise would connect the brane-creation rule to BPS spectrum computations.
Original abstract

We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K\"ahler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,687 input tokens, $0.0002 reported.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K\"ahler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. background 0.53 0.82 0.03 asserted 0.91
s1 We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. contribution 0.81 0.95 0.26 asserted 1.00
s2 We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. result 0.68 0.96 0.16 asserted 1.00
s3 Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K\"ahler manifolds. result 0.80 0.92 0.16 asserted 1.00
s4 In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. result 0.52 0.84 0.14 asserted 0.99
s5 Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. result 0.88 0.95 0.13 asserted 1.00
s6 This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra. interpretation 0.52 0.90 0.12 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. BPS soliton map to straight line in the W-plane. Soliton solutions exist for each intersection point of vanishing cycles. Lines in the W-plane which are homotopic to the straight line (doted lines) can also be used to calculate soliton numbers. Let us fix a point w in the W-plane. For each critical point a of W, we choose an arbitrary path in the W-plane emanating from W(a) and ending on w, but not passing through other critical values. This yields N cycles ∆a over W−1 (w) and it is known [14] that these form a complete basis for the middle-dimensional homology cycles of W−1 (w).…
Figure 2. As the positions of critical values change in the W-plane, the choice of the vanishing cycles relevant for computing the soliton numbers change. In particular the homology element corresponding to vanishing cycle a changes ∆a → ∆′ a and we need to find out how it changes. Picard-Lefshetz theory gives a simple formula for this change. In particular it states that ∆ ′ a = ∆a ± (∆a ◦ ∆b)∆b . (2.10) The sign in the above formula is determined once the orientation of the cycles are fixed and will depend on the handedness of the crossing geometry (see [3]). This is perhaps most familiar
Figure 3. The cycles emanating from the critical points. The images in the W-plane are the straight lines emanating from the critical values and extending to the infinity in the real positive direction. Let B be the region of Cn where ReW is larger than a fixed value which is chosen sufficiently large. The non-compact cycles γa can be viewed as elements of the homology 8
Figure 4. The images in the W-plane of γa and γb (left); and γa and γ ′ b (right). The second will give rise to“intersection number”. As we will see in the next section, this contains a certain information on D-branes in the LG model. former case the “intersection number” is of course zero. In the latter case as shown in
Figure 5. The two solitons of the IP1 model. such that their trajectories in the x-plane lie on two half-circles as shown in
Figure 6. Intersecting lines in the W-plane and the corresponding intersecting cycles in the x-space. intersect each other and the intersection number in this case is two. We now turn to the study of solitons of the IPN−1 sigma model. The LG theory mirror to the non-linear sigma model with IPN−1 target space has superpotential [2] W(X) = N X−1 k=1 Xk + λ X1 · · · XN−1 . (2.19) This superpotential has N critical points given by X (a) i = e 2πia N i = 1, · · · , N − 1 ; a = 0, · · · , N − 1 , (2.20) with the critical values wa ≡ W(X~ (a) ) = Ne 2πia N . (2.21) Here unlike the pr
Figure 7. The strip R × I Let us consider a supersymmetric sigma model on a K¨ahler manifold X of dimension n with a superpotential W. We denote the K¨ahler metric with respect to local complex coordinates z i as gi¯. We formulate the theory on the strip Σ = R × I where I is an 14
Figure 8. The image of D-branes in the W-plane and a path between them boundary condition. We note that there is no such configuration if ImW(a) < ImW(b). In such a case I(a, b) = 0. Now let us compute the index. We are considering the situation as depicted in
Figure 9. A Graph of ∆x1 as a function of w1 = ReA = ReB. Corresponding to the situation in
Figure 10. Open and closed string channels function from the closed string view point. On the other hand, one can interpret it from the point view of open strings. Let Hab be the space of states on the interval of length T with a and b as the left and the right boundary conditions and let H(T) be the Hamiltonian generating the evolution in the circle direction. If the theory has spin half 31
Figure 11. The semi-infinite cigar leading to Πa i = RRha|φii semi-infinite cigar where the boundary condition a is imposed at the outgoing boundary and the operator φi is inserted at the tip (see
Figure 12. Separation into two regions the region Σ1. We recall that (Q+φac)(x) = H x G+φac(x) etc, where the contour integral is along a small circle around the point x. We can deform the contour of the G− integral from the small circle around x to two boundary circles of Σ1; S 1 and S 1 mid. The one on S 1 mid can be considered as the supercharge acting on the state at the boundary of Σ2. Since Σ2 is infinitely long, the state that appears at the boundary is a ground state. Thus, the contour integral along S 1 mid vanishes. The one on S 1 turns into i H S1 G+ by the boundary conditi
Figure 13. As the image of D-brane in the W-plane passes through a critical value, new D-branes are created whose image start from the crossed critical value. lation functions of the 2-dimensional theory. For example if we consider the 2-dimensional theory on a cylinder with one boundary ending on the γ1 (t), then the continuity of the correlation function with this boundary condition as a function of t demands that as we change t from t0 − ǫ to t0 + ǫ we would have to also add to the correlation function the correlator involving boundaries on the γ2 brane with multiplicity factor +∆1 ◦ ∆2.…
Figure 14. A convex arrangement of critical values in the W-plane is the most convenient one for deriving the monodromy action by 2π rotation. convexity compatible with the ordering of Im(W) as shown in the figure. This can be done, by deforming the coefficients of W if necessary. As we increase α from 0 to π we rotate the image of branes in the W-plane counter clockwise. As discussed in the previous section, during this process we create new branes. In particular the action of brane creation in the basis of branes emanating from the critical points γi is rather simple: The rotation of branes…
Figure 15. To get the index in the open string sector stretched between γn1n2 and γr1r2 we have to rotate the image in the W-plane of the γr1r2 brane in the positive direction and compute the corresponding intersection number. The fact that we have obtained (k + 2)(k + 1) of such branes at the conformal point is very encouraging as that is exactly the same as the predicted number of Cardy states, as already discussed. Moreover, if we consider the range of parameters where 0 ≤ n1 < n2 ≤ k + 1 we see that |n1 − n2| ∈ {1, · · · , k + 1} and n1 + n2 ∈ {0, · · · 2k + 2}. The range of these…
Figure 16. and relates to the statement that if we change n1 → n2 and n2 → n1 + k + 2 we get the same brane back up to a flip in the orientation (reflected in the shift in s). k=2: < < < < ^ ^ (0,1,1) (2,-3,-1) k=4: (0,1,1) (4,-5,-1)
Figure 17. The intersection number of the branes is 1 if the corresponding asymptotes alternate. Otherwise it is zero. where m = m2 − m1 (to show this and write all inequalities in terms of inequalities with equal signs we used the fact that m2 − m1 and l2 + l1 are equal mod 2). This is precisely the condition for the SU(2) level k algebra and we have thus derived (5.9) from a purely LG point of view. 5.4 Period Integrals and Boundary States As was discussed in the context of LG models [2] and also in section 3 of this paper there is a natural pairing between the A-model boundary states and…
Figure 18. The image in the Wf-plane of the cycle γθ . critical point p0 and the straight line in the Wf-plane emanates outward from the critical value we0. Thus, γθ is the wavefront trajectory emanating from the critical point p0. Since the image in the Wf-plane has the slope θ/b1 and the boundary condition of the fermions is rotated as (6.60), the boundary condition indeed preserves the supersymmetry Q and Q† in (6.49). When θ is an integer multiple of 2π, as we have noted before, in the non-linear sigma model limit the corresponding B-field is integral and the coupling of the worldsheet to
Figure 19. a) {O(0),θ = 0} = O(0) , b) {O(0),θ = 2π} = O(−1), c) {O(0),θ = −2π} = O(1) or O(−2), O(−3),..., it is impossible for rotating the cycle so that the images in the W-plane are parallel to the real axis without passing through other critical values. One can avoid this cross-over by bending the branes although it results in the breaking of the supersymmetry. Bending in the clockwise direction, we obtain the collection of bundles {O(0), · · · , O(n + 1)} which are exceptional collections as we have seen above. By partially changing the direction of bending, we can obtain other…
Figure 20. Three exceptional collections which are related to each other by mutations. B-type. 72
Figure 21. The Picard-Lefshetz Monodromy, leading to brane creation, can be interpreted in terms of left mutations on the mirror side that we can identify C ′′ 2 with the mirror of L0L1(O(2)). In other words the above process 73
Figure 22. The Branes depicted here are mirrors of L1(O(2))
Figure 23. The mirror of the tangent bundle is the D-brane shown in (b). action on γ0. 2 Now we come back to the question of counting the soliton numbers. The soliton num￾ber between w0 and w−k can be computed by first left mutating the O(k) brane through O(k−1), ..., O(1) and then taking its inner product with O(0). Since ch(L1 · · ·Lk−1O(k)) = Pk−1 i=0 (−1)i  n+1 i  ch(O(k − i)), hence soliton numbers between two different vacua w0 and w−k is equal to µi,i−k = µ0,−k = χ(O(0), L1 · · ·Lk−1O(k)) (7.24) = k X−1 i=0 (−1)i
Figure 24. Toric diagram for IP2 , its three blow ups and IP1 × IP1 . The superpotential of the mirror LG theory can be written from the toric data given in the figure. Let {v (a) = (v (a) 1 , v (a) 2 ) | a = 1, · · · , N} be the set of vertices. Then [2] W(X) = X N a=1 CaX v (a) 1 1 X v (a) 2 2 , (7.25) where Ca are complex numbers. Only a subset of Ca are actually physical since some of them can be absorbed by rescaling Xi . 7.4.1 IP2 From the discussion in the previous section we know that the lines shown in the
Figure 25. Two exceptional collections related by a left mutation. 7.4.2 B1 The superpotential is given by W(X1, X2) = X1 + X2 + e −t X1X2 + e −tE1 X1 , (7.27) where tE1 = A(E) − i θ(E) is the complexified K¨ahler parameter of the exceptional cycle, E1. After rescaling the coordinates we can express the superpotential in the following form which will be more useful for the later discussion, W(X1, X2) = e − t 3 (X1 + X2 + 1 X1X2 + e −tE1+ 2 3 t X1 ). (7.28) We define µ1 = e −tE1+ 2 3 t , then the critical values of the superpotential given above are w = e − t 3 (3 y∗ + 2 µ1 y 2 ∗ ), where µ1y…
Figure 26. The effect of transformation µ1 7→ µ1e iθ on the critical values. V V V1 2 3
Figure 27. The pull back from IP2 to B1 of {O(−1), O(0), O(1)}. with the critical value wb1 we take µ1 to be very small and positive so that wb1 is far from other critical values. In this case the critical value is on the negative real axis as shown in the
Figure 28. Brane creation as the critical value wb1 undergoes a full rotation for µ1 7→ µ1e 2πi . ch( bV1) = e c1(O(0)−[E1] − ch(O(0)) = (1, [E]1, − 1 2 ) − (1, 0, 0) = (0, −[E1], − 1 2 ). Thus bV1 is actually a sheaf with support on the exceptional divisor E1. The collection of sheaves {V1, V2, V3, bV1} is an exceptional collection since χ( bV1, bV1) = 1 and χ( bV1, Vi) = 0 for i = 1, 2, 3. V V V V1 1 2 3
Figure 29. The mirror of an exceptional collection of bundles on B1. It is interesting to consider the effect of transformation e −t 7→ e −t+2πi. Since µ1 = e −tE1+ 2 3 t we see that under the above transformation µ1 7→ µ1e 4πi 3 . From the previous discussion it would seem that the critical point wb1 undergoes a rotation by an angle 4π 3 . This, however, is not the case. From eq. (7.28) it is clear that because of the overall factor e − t 3 a phase transformation e −t− 7→ e −t+2πi rotates all the critical values by an angle 2π 3 . 79
Figure 30. The effect of the transformation e −t 7→ e −t+2πi on the exceptional collection. and bV ′ 1 are V1, V2, V3 and bV1 bundles in the background where the B-field through the cycle l has been turned on, where l along with E1 forms a basis of H2(B1, ZZ) such that the self-intersection number of l is plus one, l 2 = 1. Denoting the cohomology class dual to l by [l] we get ch(V ′ k ) = e c1(Vk)−[l] = e (k−3)[l] = (1,(k − 3)[l], 1 2 (k − 3)2 ) (7.34) Thus we see that V ′ k is the pull back of O(k −3) bundle from IP2 to B1 and can be written as {V ′ 1 , V ′ 2 , V ′ 3
Figure 31. Left mutations of the exceptional collection to obtain an exceptional collection for soliton counting. Sij = χ(Ei , Ej ) =   1 −1 −3 +3 0 1 +1 −2 0 0 1 −3 0 0 0 1   (7.37) In other words there is one soliton between vacua 2, 1 and 2, 3, three solitons between 1, 3 and 1, 4 and 3, 4 and two solitons between vacua 2, 4. Of course, as we change the K¨ahler parameters of the manifold the position of the vacua change and the number of solitons change, as reviewed in section 2. 7.4.3 B2 The superpotential of the LG theory mirror to non-linear sigma model with B2 is giv
Figure 32. Brane creation as the critical value wb2 undergoes a rotation for µ2 7→ µ2e 2πi . an exceptional collections for which the corresponding semi-infinite lines are such that any two of them do not enclose a critical point. To obtain such an exceptional collection we consider the right mutations shown in
Figure 33. Left mutations of the exceptional collection on B2 to obtain an exceptional collection for soliton counting. {V1, V2, V3, bV1, bV2} 7→ {V1, V2, LV3 ( bV1), LV3 ( bV2), V3} (7.42) 7→ {V1, LV2LV3 ( bV1), LV2LV3 ( bV2), V2, V3} 7→ {V1, LV2LV3 ( bV1), LV2LV2 ( bV2), LV2 (V3), V2} =: {E1, E2, E3, E4, E5} . The soliton numbers between different vacua are now given by χ(Ei , Ej ) where the vacua are labeled in the counter clockwise direction as shown in
Figure 34. Brane creation as the critical value wb3 undergoes a rotation for µ3 7→ µ3e 2πi . infinite lines corresponding to this collection are shown in
Figure 36. Left mutations of the exceptional collection (shown in
Figure 37. Critical values when e − t1 2 and e − t2 2 are real. semi-infinite line starting at w1) is mirror to the trivial bundle O(0, 0) 1 . Note that the transformation e −ti 7→ e −tie 2πi has the following effect on the critical values, e −t1 7→ e −t1 e 2πi =⇒ w1 ↔ w3 , w2 ↔ w4 , e −t2 7→ e −t2 e 2πi =⇒ w1 ↔ w2 , w3 ↔ w4 . (7.51) 1A bundle O(a, b) on F0 is a rank one bundle with first Chern class c1 = a l1 + b l2, where l1 and l2 are the generators of H2(F0) such that l1 ◦ l1 = l2 ◦ l2 = 0 and l1 ◦ l2 = 1. 87
Figure 38. The effect of transformation e −t2 7→ e −t2 e 2πi on the critical values. ch(V2) = ch(O(0, 0)) + B-field = e c1(O(0,0))−l2 = 1 − l2 = (1, −l2, 0). (7.52) Thus we can identify V2 with O(0, −1). The effect of the first transformation of eq. (7.51) on the cycle C1 is shown in
Figure 39. The effect of transformation e −t1 7→ e −t1 e 2πi on the critical values. mirror to C3 is such that ch(V3) = ch(O(0, 0)) + B-field = e c1(O(0,0))−l1 = 1 − l1 = (1, −l1, 0). (7.53) Thus we can identify V3 with O(−1, 0). Now consider the transformation (e −t1 , e−t2 ) 7→ (e iθe −t1 , eiθe −t2 ), θ ∈ [0, 2π] . (7.54) The effect of this transformation on the critical values and the cycle C1 is shown in
Figure 40. The effect of the combined transformation (e −t1 ,e−t2 ) 7→ e 2πi(e −t1 ,e−t2 ). Thus we can identify V4 with O(−1, −1). The set of bundles {O(−1, −1), O(−1, 0), O(0, −1), O(0, 0)} is an exceptional collec￾tion and using it we can calculate the soliton counting matrix. But first since we do not want any three vacua to be collinear we deform the configuration using eq. (7.54) for |θ| << 1. And also we need to transform this collection of exceptional bundles into the one shown in
Figure 41. A right mutation of the exceptional collection to obtain another exceptional collection for soliton counting. {O(−1, −1), O(−1, 0), O(0, −1), O(0, 0)} 7→ {O(−1, 0), RO(−1,0)(O(−1, −1)), O(0, −1), O(0, 0)} =: {E1 , E2 , E3 , E4} . The soliton counting matrix Sij is then given by Sij = χ(Ei , Ej ) =   1 −2 0 2 0 1 2 0 0 0 1 2 0 0 0 1   . (7.56) 89
Figure 42. The effect of the transformation µ1 7→ µ1e 2πi on the critical values wbi for IP5 . Consider the case of
Figure 43. Brane creation as wb1 passes through the mirror of the trivial line bundle. 91
Figure 44. Bundles which are mirror of the lines starting at wbi . 8 D-Brane in String Theory and Mirror Symmetry It is natural to ask how the map between D-branes corresponding to sheaves and the Lagrangian submanifolds in the Landau-Ginzburg model works in the case of conformal theories. In the context of Gepner models, the structure of Cardy states and their sigma model interpretation have been studied [49]. our constructions of Cardy states in terms of Lagrangian submanifolds of LG models lead to a deeper geometric insight in this regard. Since we have considered the case of minimal…
Figure 45. The relation with the probe theory and its BPS states is now clear. The degenerating 2The existence of the 2nd type of closed 2-cycle is precisely the reason this fibration can be used to construct 5-brane web description of 5D E0 field theory which can also be obtained via geometric engineering from the CY threefold M [63]. 98
Figure 46. Positions of the degenerate fibers of WB1 One can check that this matrix produces correct Ramond charges for the chiral fields. B2: In this case there are five degenerate fibers as shown in
Figure 47. Positions of the degenerate fibers of WB2 . γi ◦ γj =   0 −1 −1 −3 3 0 0 0 −1 2 0 0 0 −1 2 0 0 0 0 −3 0 0 0 0 0   . (8.22) B3: The six degenerate fibers in this case are shown in
Figure 48. Positions of the degenerate fibers of WB3 . γi ◦ γj =   0 −1 −1 −1 −3 3 0 0 0 0 −1 2 0 0 0 0 −1 2 0 0 0 0 −1 2 0 0 0 0 0 −3 0 0 0 0 0 0   . (8.23) F0: The four degenerate fibers in this case are shown in
Figure 49. Positions of the degenerate fibers of WF0 . is given by γi ◦ γj =   0 −2 0 2 0 0 2 0 0 0 0 2 0 0 0 0   . (8.24) One can check that these matrices give the correct Ramond charge for the chiral fields. Acknowledgement We would like to thank M. Aganagic, C. Bachas, A. Chari, S. Coleman, M. Gutperle, R. Myers, A. Polishchuk, R. Thomas, S.T. Yau and E. Zaslow for valuable discussions. 101

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References
How later papers used it

142 later papers cite it

Evidence: substantial. 128 of the citing papers share no author with it, and 21 of those carry a confident label from Pith's citation classifier.

Pith read 26 of the 278 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

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142 arXiv papers cite this in 279 sentences

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Papers citing it · 177 sentences without a confident label

Builds on it 2 papers
  • We shall now investigate quantum effects for the alternative formulation of equivariant B-branes given in Section 4.3,[We shall not study the quantum effects for the first formulation, since the main quantum correction is the running of the FI parameters, and the FI parameters do not enter the boundary conditions in that formalism.] since we shall use this formulation for the proof of mirror symmetry in the following section.[The following is a generalization of the analysis given in Section 6 of [ref] to the case of multiple $U(1)$ gauge groups. ]

    Open Gauged Sigma Models, Equivariant Branes, and Equivariant Homological Mirror Symmetry · Quantum Corrections · 0.98

  • [ref] from sheaves on $X$ to sheaves on the loop space $LX$.

    A 3d Gauge Theory/Quantum K-Theory Correspondence · Defect entropies and vortex counting · 0.86

Uses its method or tool 10 papers
  • To find Picard-Fuchs equations for periods of mirrors, in this section, we study the open string theory with A-boundary in the Landau-Ginzburg model [ref] , and these boundary configurations are called A-branes or Lagrangian submanifolds in the context of LG models denoted by $γ_i$.

    Nonabelian mirrors and Gromov-Witten invariants · Picard-Fuchs equations for periods · 0.90

  • Following [ref] , we can rewrite the elliptic fibration in an almost Weierstrass form as follows.

    The Physical Mirror Equivalence for the Local P^2 · The Double Fibration · 0.99

  • The integral of such a one-form over a wedge is then given by the contour integral [ref] [equation] with $k=0,1,2$.

    Symmetries and M-theory-like Vacua in Four Dimensions · The x46 LG orientifold model · 0.86

Uses its data or values 1 paper
  • For instance, in [ref] the Witten index was computed for all such boundary conditions and it was found that it is $±1$ for two neighboring smallest wedges $(b,x_c, b+1)$ and $(b+1, x_c, b+2)$.

    Topological defects and SUSY RG flow · Branes and flows · 0.82

Compares with it 8 papers
  • Notice that the quasimap SYZ mirror family [ref] is entirely written in terms of symplectic-geometric information such as complexified Kähler parameters and quasidisk invariants of $X$, and it coincides with the mirror [ref] predicted by physical arguments [ref] .

    Quasimap SYZ for toric Calabi-Yau manifolds · Wall-crossing for quasidisk invariants and the SYZ mirror · 0.97

  • Notice that the SYZ mirror family [ref] is entirely written in terms of complexified Kähler parameters and disk counting invariants of $X$, and it agrees with the prediction [ref] using physical arguments [ref] .

    SYZ mirror symmetry for toric varieties · The SYZ construction · 0.94

  • In the presence of mass and FI parameters, the boundary conditions in 3d $=4$ theories share many properties with boundary conditions in A-twisted Landau-Ginzburg models [ref] , which generate Fukaya-Seidel categories [ref] .

    Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory · Two Dimensions · 0.92

Cites it as support 3 papers
  • As proven in [ref] and reviewed in the appendix [ref] the correlation function of the 2d theory on a half-cylinder of radius $R$, with such a boundary at one end and a choice of Ramond vacuum at the other end, is a flat section of the spectral connection $∇$ with the same $R, ζ$!

    Surface Operators in N=2 4d Gauge Theories · Hitchin systems from surface operators · 0.97

  • [ref] it was shown that the Lagrangian $A$–branes ${γ_i}$ span the relative homology group $H_1(,)$ (where $⊂$ is the region near the punctures where $Re(e^iθ W)≫1$), so all the axioms for an ideal triangulation are satisfied.

    Classification of complete N=2 supersymmetric theories in 4 dimensions · Lagrangian $A$–branes as ideal triangulations · 0.96

  • Note that this definition of morphisms is in agreement with the fact, found in [ref] that the open string index between A-branes in a LG model satisfying $(W(p_k))< (W(p_k'))$ always vanishes.

    Exponential BPS graphs and D-brane counting on toric Calabi-Yau threefolds: Part II · A view from the Fukaya category · 0.98

Cites it as background 65 papers
  • For open string, Aganagic and Vafa [ref] conjectured a relationship between genus $0$ open string invariants and the equation for the mirror curve given by the Hori-Iqbal-Vafa construction [ref] .

    Open String Invariants and Mirror Curve of the Resolved Conifold · Introduction · 0.87

  • B-branes on anomalous models have been studied in the context of mirror symmetry of massive theories [ref] and the interpretation in terms of flows in the moduli space in [ref] for Fano and general type hypersurfaces in $^N$.[In mathematics similar cases have been analyzed in [ref] , but mostly limited to Fano varieties or general type hypersurfaces in $^N$.] We want to unify [ref] with the categorical approach of [ref] .

    B-brane transport in anomalous (2,2) models and localization · Introduction · 0.88

  • The mirror of a toric Calabi-Yau manifold $X$ is predicted to be a family of affine hypersurfaces in $^2 ×(^×)^n-1$ [ref] explicitly written as [equation] where the coefficients $Č_i ∈$ are constants (without loss of generality, we will set $Č_m = 1$) subject to the constraints [equation] here $t = (t_1, …, t_r)$ are coordinates on the mirror complex moduli $ℳ̌_B := 𝕂^∨⊗_ ^×≅(^×)^r$. $X̌_t$ is Calabi-Yau since [equation] is a nowhere vanishing holomorphic volume form on $X̌_t$.

    Quasimap SYZ for toric Calabi-Yau manifolds · Toric Calabi-Yau manifolds and their physical mirrors · 0.89

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